INDIAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS Vol. 10, No. 1 (Special Issue), Jan-June 2019 EDITORIAL BOARD Patron of ISIAM: Prof. K.R. Sreenivasan EDITOR-IN-CHIEF Prof. Abul Hasan Siddiqi Sharda University, President, Indian Society of Industrial and Applied Mathematics, (ISIAM), India, [email protected] MANAGING EDITORS 1. 2. 3. Prof. Pammy Manchanda, Department of Mathematics, Guru Nanak Dev University, Amritsar, India, [email protected] Prof. Rashmi Bhardwaj, USBAS, Nonlinear Dynamics Research Lab, Guru Gobind Singh Indraprastha University, New Delhi, India, [email protected] Prof. A. A. Khan, Rochester Institute of Technology, Rochester, U.S.A., [email protected] EDITORIAL BOARD International 1. Prof. Zafer Aslan, Istanbul Ayadin University, Turkey, [email protected] 2. Prof. Martin Brokate, Technical University Munich, Germany, [email protected] 3. Prof. O. Christensen, Technical University of Denmark, [email protected] 4. Prof. Iain S. Duff, Rutherford Appleton Laboratory, U.K., [email protected] 5. Prof. Pavel Exner, Department of Theoretical Physics, Nuclear Physics Institute, Academy of Sciences, Near Prague, Czech Republic, [email protected] 6. Prof. Graeme Fairweather, American Mathematical Society 416 Fourth Street, Ann Arbor, MI, U.S.A., [email protected] 7. Prof. Yuri Farkov, Russian Presidential Academy, Moscow, Russia, [email protected] 8. Prof. H.G. Feichtinger, University of Vienna, Austria, [email protected] 9. Prof. Sandor Fridli, Budapest, Hungary, [email protected] 10. Prof. A. Gupta, University of British Columbia, Canada, [email protected] 11. Prof. Rene Lozi, University of Nice Sophia Antipolis, France, [email protected] 12. Prof. M. Lawati, Sultan Qaboos University, Oman (Ex. Chairman), [email protected] 13. Prof. Guenter Leugering, Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany, [email protected] 14. Prof. P. Maass, Universität Bremen, FB 3, Mathematik und Informatik Zentrum für Technomathematik, Germany, [email protected] 15. Prof. M. Moonis, Department of Neurology Worcester, MA 01655, U.S.A., [email protected] 16. Prof. M. Zuhair Nashed, University of Central Florida, U.S.A., [email protected] 17. Prof. Fahima Nekka, Biotechnology Focus, Montreal University, Canada, [email protected] 18. Prof. M. Yu. Rasulova, Institute of Nuclear Physics, Uzbekistan, [email protected] 19. Prof. M. Skopina, Euler Institute of Mathematical Sciences, Saint Petersburg State University, Russia, [email protected] 20. Prof. D. Walnut, George Mason University, U.S.A., [email protected] 21. Prof. A.I. Zayed, DePaul University, U.S.A., [email protected] Indian 22. Prof. R.C. Singh, Department of Physics, School of Basic Sciences and Research, Sharda University, Greater Noida, UP, India, [email protected] 23. Prof. G.D.V. Gowda, TIFR Centre for Applicable Mathematics Sharada Nagar, Chikkabommasandra, India, [email protected] 24. Prof. U.B. Desai, IIT Hyderabad, India, [email protected] 25. Prof. Narendra K. Gupta, Department of Applied Mechanics Indian Institute of Technology, Delhi, India, [email protected] 26. Prof. S. Kesavan, Institute of Mathematical Sciences, Chennai, India, [email protected] 27. Prof. A.K. Pani, IIT Bombay, India, [email protected] 28. Prof. G. Rangarajan, IISc, Bangaluru, India, [email protected] 29. Prof. U.P. Singh, (Ex. VC, Purvanchal University) Rajendra Nagar, East Gorakhpur, India, [email protected] Indexed/Abstracted with: NAAS Rating 2017-3.98; EBSCO Discovery; INFOBASE INDEX (IB Factor 2017-3.1); CNKI Scholar; Summon (ProQuest); Google Scholar; Cite Factor; ISRA-JIF; J-Gate; IIJIF; DRJI; Indian Citation Index (ICI) Preface An International Conference on Analysis and its Applications (ICAA-2017) was held at Department of Mathematics, Aligarh Muslim University, Aligarh, India during November 20-22, 2017, under the DRS (SAP-II) programme of the University Grants Commission, Government of India. This conference focused on wide range of topics related to analysis and its applications, namely, Nonlinear Analysis, Operator Theory, Fixed Point Theory, Sequence Spaces and Matrix Transformations, Modern Methods in Summability and Approximation Theory, Set-valued Analysis, Variational Analysis including Variational Inequalities, Convex Analysis, Smooth and Non-smooth Analysis, Wavelet Analysis, Fourier Analysis, etc.. This conference was an intense academic affair where approximately 208 participants were actively engaged which include 30 foreign delegates from 20 countries of the world, 90 out-station participants beside 88 internal ones. We had excellent speakers from India as well as abroad. Following such very successful conference, we decided to publish a special issue dedicated to this event. We have received more than 40 research articles for this conference proceeding. After peer review process, only 18 research articles have been accepted for the publication in this special issue. Sincere thanks to the following sponsors of the conference for their continuous support: Aligarh Muslim University (AMU), University Grants Commission (UGC), Science and Engineering Research Board-Department of Science and Technology (DST-SERB), National Board of Higher Mathematics (NBHM), Council of Scientific and Industrial Research (CSIR) and the Indian National Science Academy (INSA). Finally, we appreciate Indian Society of Industrial and Applied Mathematics for doing excellent job in bringing out this special issue. Editors Rais Ahmad Mijanur Rahaman Ordering Information Indian Journal of Industrial and Applied Mathematics provides an unique opportunity to disseminate contemporary work on applications of mathematics to emerging areas. Editorial board comprising distinguished experts from different regions of the world will endeavor to prove the point that mathematics is the mother of all technologies. Peer reviewed research papers, and popular review articles introducing new fields of research and innovative ideas may enhance image of mathematics among general public and also attract future generation towards mathematics. The journal provides a forum for sharing thoughts and information interdisciplinary in nature through its hard and soft copies. New Mathematical concepts, tools and techniques having potential of solving problems of emerging technologies and financial markets will form the basis of this co-operation and exchange of ideas. The Indian Society of Industrial and Applied Mathematics (ISIAM) will make all out efforts that the journal achieves its target. Subscription Rates for the year: 2019 Editorial Subscription Indian (Rs.) Foreign (US$) Print** 2200 200 Online* 1770 131 Frequency 2 *Print subscription includes complimentary online limited access **Online subscription includes limited access (5 concurrent users) of current subscription Site Licensing Price: INR 8,850.00 & USD 655.00 For Consortia Pricing, Please Contact at [email protected] IndianJournals.com A product of Diva Enterprises Pvt. Ltd. Indian Journal of Industrial and Applied Mathematics Vol. 10, No. 1 (Special Issue), Jan-June 2019 Contents 1. On the Zeros of a Polynomial with Restricted Coefficients Bilal Dar 1-10 2. Projection Method for Bounded and Unbounded Solution of Hypersingular Integral Equations of the First Kind Z.K. Eshkuvatov and Anvar Narzullaev 11-37 3. Discrete Legendre Projection Methods for the Fredholm Integral Equations of the Second Kind Jitendra Kumar Malik and Bijaya Laxmi Panigrahi 38-50 4. Some Curvature Conditions on Nearly Cosymplectic Manifolds G¨ulhan Ayar, Pelin Tekin and Nesip Aktan 51-58 5. Second-order Characterization of Invex Functions and Its Applications in Optimization Problems M.T. Nadi and J. Zafarani 59-70 6. Approximation of Periodic Functions via Statistical β-summability and Its Applications to Approximation Theorems B.B. Jena, S.K. Paikray and U.K. Misra 71-86 7. On Semi-infinite Optimization Problem Under the Generalized Convexity Pushkar Jaisawal, Vivek Laha and S.K. Mishra 87-108 8. A Strict Constraint Qualification in Vector Optimization Muskan Kapoor and C.S. Lalitha 109-120 9. Fixed Point Theorems for Generalized [a, b, c] p-nonexpansive Mappings in Banach Spaces 121-144 D.R. Sahu and Satyendra Kumar 10. On Generalized Operator Mixed Vector Quasi-equilibrium Problem Rais Ahmad, Haider Abbas Rizvi, Mijanur Rahaman and Javid Iqbal 145-151 11. On Approximation of Signals in the Generalized Zygmund Class Using (E, 1) (N, pn)-summability Means of Fourier Series Tejaswini Pradhan, Susanta Kumar Paikray and Umakanta Misra 152-164 12. n-tupled Coincidence Point Results for Nonlinear Contraction in Partially Ordered Complete Metric Spaces Mohammad Imdad, Aftab Alam, Anupam Sharma and Mohammad Arif 165-181 13. On Starlikeness and Convexity Properties of the Generalized Bessel-Hypergeometric Function Nusrat Raza and Eman S.A. AbuJarad 182-201 14. A New Iterative Class for Finding Common Fixed Points of a Finite Family of Generalized Total Asymptotically Nonexpansive and Multivalued Mappings in Hyperbolic Spaces Shamshad Husain and Nisha Singh 202-219 15. On a Class of Generalised ( p, q) Bernstein Operators Lakshmi Narayan Mishra, Shikha Pandey and Vishnu Narayan Mishra 220-233 16. Dhage Iteration Method for Approximating Solutions of IVPs of Nonlinear Second Order Hybrid Neutral Functional Differential Equations Bapurao C. Dhage 234-246 17. A New Version of KKM Theorem with Geodesic Convex Hull with an Application on Hadamard Manifold J.C. Yao 247-254 18. The Relaxed Resolvent Operator For Solving Fuzzy Variational Inclusion Problem Involving Ordered RME Set-Valued Mapping Iqbal Ahmad, Abul H. Siddiqi and Rais Ahmad 255-268 Printed & Published by: Diva Enterprises Pvt. Ltd. on behalf of Indian Society of Industrial and Applied Mathematics, Printed at: Spectrum, 208 A/14A, Savitri Nagar, New Delhi-110017, Published at: Diva Enterprises Pvt.Ltd. B-9, A-Block, L.S.C., Naraina Vihar, New Delhi-110028, India, Editor in Chief: Prof. A.H. Siddiqi INDIAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019, pp. 1–10 DOI: On the Zeros of a Polynomial with Restricted Coefficients Bilal Dar Department of Mathematics Govt. Degree College Boys, Pulwama-192301 Email: [email protected], [email protected] Abstract: This paper focuses on the problem concerning the location and the number of zeros of polynomials in a specific region when their coefficients are restricted with special conditions. We obtain extensions of some classical results concerning the number of zeros of a polynomial in a prescribed regions by subjecting the real and imaginary parts of its coefficients to certain restrictions. Mathematics Subject Classifications (2010): 30A99, 30E10, 41A10. Keywords: Polynomial, Zeros, Eneström-Kakeya theorem. 1. INTRODUCTION If P(z) = nj=0 a j z j is a polynomial of degree n such that an ≥ an−1 ≥ ... ≥ a1 ≥ a0 > 0, then P(z) has all its zeros in |z| ≤ 1. This famous result is known as Eneström-Kakeya theorem, for reference see (section 8.3 of 11). In the literature, for example see [1–12], there exist various extensions and generalizations of Eneström-Kakeya theorem. Taking account of the restrictions on the coefficients of a polynomial allows for establishing improved bounds and here, in this paper, we impose some restrictions on the coefficients of polynomials in order to count the number of zeros in a certain region. The following result concerning the number of zeros of a polynomial in a closed disk can be found in Titchmarsh’s classic ”The Theory of Functions”, see [13, page 171, 2nd edition]. Theorem A. Let F(z) be analytic in |z| ≤ R. Let |F(z)| ≤ M in |z| ≤ R and suppose F(0) = 0. Then for 0 < δ < 1, the number of zeros of F(z) in the disk |z| ≤ Rδ does not exceed M 1 . log 1 |F(0)| log δ I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 2 Bilal Dar Regarding the number of zeros in |z| ≤ 12 and by putting a restriction on the coefficients of a polynomial similar to that of the Eneström-Kakeya theorem, Mohammad [9] used a special case of Theorem A to prove the following result Theorem B. If P(z) = nj=0 a j z j is a polynomial of degree n such that 0 < a0 ≤ a1 ≤ ... ≤ an , then the number of zeros of P(z) in |z| ≤ 12 does not exceed 1+ an 1 . log log2 a0 The above result of Mohammad [9] was generalized in different ways for example see [1–3,5,6,11]. Recently Ahmad, Rasool and Liman [1] proved the following result under less restrictive conditions on the coefficients of a polynomial. Theorem C. If P(z) = nj=0 a j z j is a polynomial of degree n with complex coefficients such that for some λ ≥ 1 and 0 ≤ k ≤ n, |an | ≤ |an−1 | ≤ ... ≤ |ak+1 | ≤ λ|ak | ≥ |ak−1 | ≥ ... ≥ |a0 | and |arg a j − β| ≤ α ≤ π2 , 1 ≤ j ≤ n, for some real α and β. then the number of zeros of P(z) in |z| ≤ 12 does not exceed M 1 log , log2 |a0 | where M =2λ|ak |cosα + 2(λ − 1)|ak |sinα + |an |(sinα − cosα + 1) + 2sinα n−1 |a j | + 2(λ − 1)|ak |. j=0 In this paper, we further weaken the hypotheses of the above results by considering a larger class of polynomials and obtain extensions of some classical results concerning the number of zeros of a polynomial in a prescribed region. Theorem 1. If P(z) = nj=0 a j z j is a polynomial of degree n such that for some t > 0, λ ≥ 1 and 0 ≤ k ≤ n, 0 < |a0 | ≤ t|a1 | ≤ ... ≤ t k−1 |ak−1 | ≤ λt k |ak | ≥ t k+1 |ak+1 | ≥ ... ≥ t n |an | and |arg a j − β| ≤ α ≤ π2 , 1 ≤ j ≤ n for some real α and β. Then for 0 < δ < 1, the number of zeros of P(z) in the disk |z| ≤ δt does not exceed M 1 , log t|a0 | log 1δ I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 On the Zeros of a Polynomial with Restricted Coefficients 3 where M =t|a0 |(1 − cosα − sinα) + 2sinα n−1 |a j |t j+1 + t n+1 |an |(1 + sinα − cosα) j=0 + 2t k+1 |ak | λcosα + (λ − 1)(1 + sinα) . In particular taking t = 1 in Theorem 1, we get the following: Corollary 1. If P(z) = nj=0 a j z j is a polynomial of degree n with complex coefficients such that for some λ ≥ 1 and 0 ≤ k ≤ n, 0 < |a0 | ≤ |a1 | ≤ ... ≤ |ak−1 | ≤ λ|ak | ≥ |ak+1 | ≥ ... ≥ |an | and |arg a j − β| ≤ α ≤ π2 , 1 ≤ j ≤ n, for some real α and β. Then for 0 < δ < 1, the number of zeros of P(z) in the disk |z| ≤ δ does not exceed 1 M log |a0 | log 1δ where M =|a0 |(1 − cosα − sinα) + 2sinα n−1 |a j | + |an |(1 + sinα − cosα) j=0 + 2|ak | λcosα + (λ − 1)(1 + sinα) . Taking δ = exceed 1 2 in the Corollary 1, we get the number of zeros of P(z) in the disk |z| ≤ 1 2 does not 1 M , log log2 |a0 | where the value of M is same as in Corollary 1. Since 0 ≤ α ≤ π2 , we have 1 − cosα − sinα ≤ 0. So the value of M given in the Corollary 1 is less than or equal to 2sinα n−1 j=0 |a j | + |an |(1 + sinα − cosα) + 2|ak | λcosα + (λ − 1)(1 + sinα) and hence Corollary 1 implies Theorem C. With k = n and λ = 1 in Corollary 1, the hypothesis becomes 0 < |a0 | ≤ |a1 | ≤ ... ≤ |an | and the value of M becomes |a0 |(1 − cosα − sinα) + 2sinα n−1 j=0 |a j | + |an |(1 + sinα + cosα) and which by the same reasoning as above is less or I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 4 Bilal Dar equal to |an |(1 + sinα + cosα) + 2sinα n−1 |a j |. j=0 This shows Corollary 1 implies a result of Pukhta [10, Theorem 1]. Remark 1. For λ = 1, Theorem 1 reduces to a result of Gardnar and Shields [5, Theorem 1]. Theorem 2. If P(z) = nj=0 a j z j is a polynomial of degree n with complex coefficients. If Re(a j ) = α j , I m(a j ) = β j , 0 ≤ j ≤ n and for some t > 0, λ ≥ 1, we have 0 = α0 ≤ tα1 ≤ ... ≤ t k−1 αk−1 ≤ λt k αk ≥ t k+1 αk+1 ≥ ... ≥ t n αn , 0 ≤ k ≤ 1, then for 0 < δ < 1, the number of zeros of P(z) in the disk |z| ≤ δt does not exceed M 1 , log t|a0 | log 1δ where M =(|αn | − αn )t n+1 + (|α0 | − α0 )t + 2λαk t k+1 + 2(λ − 1)|αk |t k+1 + 2 n |β j |t j+1 . j=0 Taking t = 1 in Theorem 2, we get the following: Corollary 2. If P(z) = nj=0 a j z j is a polynomial of degree n with complex coefficients. If Re(a j ) = α j , I m(a j ) = β j , 0 ≤ j ≤ n and for some λ ≥ 1, we have 0 = α0 ≤ α1 ≤ ... ≤ αk−1 ≤ λαk ≥ αk+1 ≥ ... ≥ αn , 0 ≤ k ≤ 1, then for 0 < δ < 1, the number of zeros of P(z) in the disk |z| ≤ δ does not exceed 1 M , log |a0 | log 1δ where M =(|αn | − αn ) + (|α0 | − α0 ) + 2λαk + 2(λ − 1)|αk | + 2 n |β j |. j=0 Corollary 2, the hypothesis becomes 0 < α0 ≤ α1 ≤ ... ≤ αn and With k = n, λ = 1 and α0 > 0 in the value of M becomes 2(αn + nj=0 |β j |), therefore, Corollary 2 reduces to a result of Pukhta [10]. For β j = 0, 1 ≤ j ≤ n and δ = 12 , Corollary 2 reduces to a result of Dewan and Bidkham [3]. Remark 2. For δ = Theorem 2.1]. 1 2 and t = 1, Theorem 2 reduces to a result of Rasool, Ahmad and Liman [12, I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 On the Zeros of a Polynomial with Restricted Coefficients 5 Theorem 3. If P(z) = nj=0 a j z j is a polynomial of degree n with complex coefficients. If Re(a j ) = α j , I m(a j ) = β j , 0 ≤ j ≤ n. Suppose that for some t > 0, λ ≥ 1 and 0 ≤ k ≤ n, we have 0 = α0 ≤ tα1 ≤ ... ≤ t k−1 αk−1 ≤ λt k αk ≥ t k+1 αk+1 ≥ ... ≥ t n αn and for some μ ≥ 1, 0 ≤ l ≤ n, we have β0 ≤ tβ1 ≤ ... ≤ t l−1 βl−1 ≤ μt l βl ≥ t l+1 βl+1 ≥ ... ≥ t n βn . Then for 0 < δ < 1, the number of zeros of P(z) in the disk |z| ≤ δt does not exceed 1 M , log t|a0 | log 1δ where M =(|αn | − αn )t n+1 + (|α0 | − α0 )t + (|βn | − βn )t n+1 + (|β0 | − β0 )t + 2λαk t k+1 + 2μβl t l+1 + 2(λ − 1)|αk |t k+1 + 2(μ − 1)|βl |t l+1 . We have some consequences of Theorem 3. If we take t = λ = μ = 1 in Theorem 3, we get the following result. Corollary 3. If P(z) = nj=0 a j z j is a polynomial of degree n with complex coefficients. If Re(a j ) = α j , I m(a j ) = β j , 0 ≤ j ≤ n. Suppose that for 0 ≤ k ≤ n, we have 0 = α0 ≤ α1 ≤ ... ≤ αk−1 ≤ αk ≥ αk+1 ≥ ... ≥ αn and for 0 ≤ l ≤ n, we have β0 ≤ β1 ≤ ... ≤ βl−1 ≤ βl ≥ βl+1 ≥ ... ≥ βn . Then for some 0 < δ < 1, the number of zeros of P(z) in the disk |z| ≤ δ does not exceed 1 M , log |a0 | log 1δ where M =(|αn | − αn ) + (|α0 | − α0 ) + (|βn | − βn ) + (|β0 | − β0 ) + 2αk + 2βl . With k = l = n and α0 > 0 and β0 > 0 in Corollary 3, the hypothesis becomes 0 < α0 ≤ α1 ≤ ... ≤ αn and 0 < β0 ≤ β1 ≤ ... ≤ βn and the value of M becomes 2(αn + βn ) and hence the number of zeros of P(z) in the disk |z| ≤ δ, 0 < δ < 1, does not exceed 1 2(αn + βn ) . log |a0 | log 1δ With t = λ = μ = 1 and k = l = 0, Theorem 3 gives the following result. Corollary 4. If P(z) = nj=0 a j z j is a polynomial of degree n with complex coefficients. If Re(a j ) = α j , I m(a j ) = β j , 0 ≤ j ≤ n. Suppose 0 = α0 ≥ α1 ≥ ... ≥ αn and β0 ≥ β1 ≥ ... ≥ βn . Then for 0 < δ < 1, the number of zeros of P(z) in the disk |z| ≤ δ does not exceed 1 M , log |a0 | log 1δ I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 6 Bilal Dar where M = (|αn | − αn ) + (|α0 | − α0 ) + (|βn | − βn ) + (|β0 | − β0 ). Besides obtaining bounds for the zeros of polynomials with restriction on their coefficients, it is equally important to locate the zeros of polynomials with arbitrary coefficients. Here, we have been able to prove the following result. m j=0 |a j | 1p , p > 1, then all the zeros of the polynomial P(z) = z n + 1 m m−1 am z + am−1 z + ... + a1 z + a0 are contained in the disk |z| ≤ k, where k ≥ max 1, |am | n−m Theorem 4. Let β = p is the largest positive root of the equation n−m 1 1 q + = 1, z − |am | z q − 1 − β q = 0, p q where 0 ≤ m ≤ n − 1. Remark 3. The special case of the above theorem for m = n − 1 was proved by Joyal, Labelle and Rahman [7, Theorem 2]. 2. PROOFS OF THEOREMS We need the following lemma for the proofs of theorems. Lemma 1. For any two complex numbers b0 and b1 such that |b0 | ≥ |b1 | and |arg b j − β| ≤ α ≤ π , j = 0, 1 for some real β, then 2 |b0 − b1 | ≤ (|b0 | − |b1 |)cosα + (|b0 | + |b1 |)sinα. The above lemma is due to Govil and Rahman [6]. Proof of Theorem 1. Consider the polynomial F(z) = (t − z)P(z) = (t − z) n ajz j j=0 = n ta j z − j j=0 = ta0 + a j z j+1 j=0 n j=1 = ta0 + n n ta j z j − n a j−1 z j − an z n+1 j=1 (ta j − a j−1 )z j − an z n+1 . j=1 I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 On the Zeros of a Polynomial with Restricted Coefficients 7 For |z| = t, we have |F(z)| ≤ t|a0 | + n |ta j − a j−1 |t j + |an |t n+1 j=1 = t|a0 | + k−1 |ta j − a j−1 |t j + |tak − ak−1 |t k j=1 + |tak+1 − ak |t k+1 + n |a j−1 − ta j |t j + |an |t n+1 j=k+2 = t|a0 | + k−1 |ta j − a j−1 |t j + t k |tak + λtak − λtak − ak−1 | j=1 +t k+1 |tak+1 + λak − λak − ak | + n |a j−1 − ta j |t j + |an |t n+1 j=k+2 ≤ t|a0 | + k−1 |ta j − a j−1 |t j + t k |λtak − ak−1 | + t k+1 |λak − tak+1 | j=1 + 2(λ − 1)|ak |t k+1 + n |a j−1 − ta j |t j + |an |t n+1 j=k+2 k−1 (t|a j | − |a j−1 |)cosα + (|a j−1 | + t|a j |)sinα t j ≤ t|a0 | + j=1 + (λt|ak | − |ak−1 |)cosα + (λt|ak | + |ak−1 |)sinα t k + (λ|ak | − t|ak+1 |)cosα + (λ|ak | + t|ak+1 |)sinα t k+1 + 2(λ − 1)|ak |t k+1 + n (|a j−1 | − t|a j |)cosα + (|a j−1 | + t|a j |)sinα t j j=k+2 + |an |t n+1 (by using Lemma 1) = t|a0 | − (t|a0 | + t n+1 |an |)cosα + 2λt k+1 |ak |cosα + 2(λ − 1)t k+1 |ak | − (t|a0 | + 2|ak |t k+1 )sinα + 2sinα n−1 |a j |t j+1 + t n+1 |an |sinα + 2λt k+1 |ak |sinα + |an |t n+1 j=0 = t|a0 |(1 − cosα − sinα) + 2sinα n−1 |a j |t j+1 + t n+1 |an |(1 + sinα − cosα) j=0 + 2t k+1 |ak | λcosα + (λ − 1)(1 + sinα) = M. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 8 Bilal Dar Now F(z) is analytic in |z| ≤ t and |F(z)| ≤ M for |z| = t and F(0) = 0. So by Theorem A and the Maximum Modulus Theorem, the number of zeros of F (and hence of P) in |z| ≤ δt does not exceed M 1 . log t|a0 | log 1δ Hence the Theorem 1 follows. Proof of Theorem 2. Consider the polynomial F(z) = (t − z)P(z) = (t − z) n ajz j j=0 n = ta0 + (ta j − a j−1 )z j − an z n+1 j=1 = (α0 + iβ0 )t + n (α j + iβ j )t − (α j−1 + iβ j−1 ) z j − (αn + iβn )z n+1 j=1 = (α0 + iβ0 )t + n n (α j t − α j−1 )z + i (β j t + β j−1 )z j − (αn + iβn )z n+1 . j j=1 j=1 For |z| = t, we have |F(z)| ≤ (|α0 | + |β0 |)t + n |α j t − α j−1 |t j j=1 + n (|β j |t + |β j−1 |)z j + (|αn | + |βn |)t n+1 j=1 = (|α0 | + |β0 |)t + k−1 n (α j t − α j−1 )t j + |tαk − αk−1 |t k + (α j−1 − tα j )t j j=1 + |tαk+1 − αk |t k+1 + j=k+2 n−1 |β j |t j+1 + |βn |t n+1 j=1 + |β0 |t + n−1 |β j |t j+1 + (|αn | + |βn |)t n+1 j=1 I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 On the Zeros of a Polynomial with Restricted Coefficients = t|α0 | + k−1 (tα j − α j−1 )t j + t k |tαk + λtαk − λtαk − αk−1 | + j=1 n 9 (α j−1 − tα j )t j j=k+2 + t k+1 |tαk+1 + λαk − λαk − αk | + 2 n |β j |t j+1 + |αn |t n+1 j=0 ≤ t|α0 | + k−1 (tα j − α j−1 )t j + t k (λtαk − αk−1 ) + t k+1 (λαk − tαk+1 ) j=1 + 2(λ − 1)|αk |t k+1 + n (α j−1 − tα j )t j + 2 j=k+2 n |β j |t j+1 + |αn |t n+1 j=0 = (|α0 | − α0 )t + (|αn | − αn )t n+1 + 2λt k+1 αk + 2(λ − 1)t k+1 |αk | + 2 n |β j |t j+1 j=0 = M. The result follows as in the proof of theorem 1. Proof of Theorem 3. Proceeding on the same lines of proof of Theorem 2, the proof of Theorem 3 follows. Proof of Theorem 4. We have P(z) = z n + am z m + am−1 z m−1 + ... + a1 z + a0 , so that |P(z)| ≥ |z|n − |am ||z|m − |am−1 z m−1 + ... + a1 z + a0 |. (2.1) By using the well-known Holder’s inequality, we have |am−1 z m−1 + ... + a1 z + a0 | 1p m−1 q1 m−1 p jq ≤ |a j | |z| j=0 |z| − 1 |z|q − 1 β|z|m < 1 . (|z|q − 1) q =β mq q1 I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS j=0 Vol. 10, No. 1 (Special Issue), Jan–June 2019 10 Bilal Dar From (2.1), we get |P(z)| > |z|n − |am ||z|m − = |z|m (|z|q − 1) 1 q β|z|m 1 (|z|q − 1) q n−m 1 − |am |)(|z|q − 1) q − β . (|z| Hence |P(z)| > 0 if 1 (|z|n−m − |am |)(|z|q − 1) q − β ≥ 0 i.e, if (|z|n−m − |am |)q (|z|q − 1) − β q ≥ 0. Define F(z) = (z n−m − |am |)q (z q − 1) − β q . (2.2) 1 We observe that F |am | n−m < 0 and F(1) < 0. Also limz→+∞ F(z) = +∞, we see from (2.2) 1 that the largest positive zero k of F(z) is greater than or equal to max 1, |am | n−m . This implies |P(z)| > 0 if F(|z|) > 0 and hence |z| > k. This proves the result. REFERENCES [1] I. Ahmad, T. Rasool and Abdul Liman, Zeros of certain polynomials and analytic functions with restricted coefficients, J. Class. Anal., 4(2014), 149–157. [2] K. K. Dewan, Extremal properties and coefficient estimates for polynomials with restricted zeros and on location of zeros of polynomials, Ph.D. Thesis, Indian Institute of Technology, Delhi, 1980. [3] K. K. Dewan and M. Bidkham, On the Eneström-Kakeya theorem, J. Math. Anal. Appl., 180(1993), 29—36. [4] R. B. Gardner and N. K. Govil, On the location of the zeros of a polynomial, J. Approx. Theory, 78(1994), 286–292. [5] R. Gardner and Brett Shields, The number of zeros of a polynomial in a disk, J. Class. Anal., 3(2013), 167–176. [6] N. K. Govil and Q. I. Rahman, On the Eneström-Kakeya theorem, Tôhoku Math. Jour., 20(1968), 126–136. [7] A. Joyal, G. Labelle and Q. I. Rahman, On the location of zeros of polynomials, Canad. Math. Bull., 10(1967), 53–63. [8] M. Marden, Geometry of Polynomials, Math. Surveys, No.3, Amer, Math. Soc., Providence, R.I., 1966. [9] Q. G. Mohammad, On the zeros of the polynomials, Amer. Math. Monthly, 72(1965), 631–633. [10] M. S. Pukhta, On the zeros of a polynomial, Appl. Math, 2(2011), 1356–1358. [11] Q. I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, Oxford University Press, 2002. [12] T. Rasool, I. Ahmad and Abdul Liman, On zeros of polynomials with restricted coefficients, Kyungpook Math. J., 55(2015), 807–816. [13] E. C. Titchmarsh, The Theory of Functions, 2nd Edition, Oxford University Press, London, 1939. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 INDIAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019, pp. 11–37 DOI: Projection Method for Bounded and Unbounded Solution of Hypersingular Integral Equations of the First Kind Z.K. Eshkuvatov1,2∗ and Anvar Narzullaev1 1 Faculty of Science and Technology, Universiti Sains Islam Malaysia (USIM), Nigeri Sembilan, Malaysia 2 Institute for Mathematical Research, Universiti Putra Malaysia, Selangor, Malaysia ∗ Corresponding author Email: [email protected] Abstract: In this note, we consider a hypersingular integral equations (HSIEs) of the first kind on the interval [−1, 1] with the assumption that kernel of the hypersingular integral is constant on the diagonal of the domain D = [1, −1] × [−1, 1]. Projection method together with Chebyshev polynomials of the first and second kinds are used to find bounded and unbounded solutions of HSIEs respectively. Exact calculations of hypersingular and singular integrals for Chebyshev polynomials allows us to obtain high accurate approximate solution. Gauss-Chebyshev quadrature with Gauss-Lobotto nodes are presented as the high accurate computation of regular kernel integrals. Six examples are provided to verify the validity and accuracy of the proposed method. Comparisons with other methods are also given. Numerical examples reveals that approximate solutions are exact if solution of HSIEs is of the polynomial forms with corresponding weights. It is worth to note that proposed method works well for large value of node points n and errors are drastically decreases. Comparisons of SPU times are also shown to demonstrate effectiveness of the method and less complexity computations. Mathematics Subject Classification: 65R20, 45E05. Keywords: Integral equations, Hypersingular integral equations, Chebyshev polynomials, Approximation, Convergence. 1. INTRODUCTION It is known that a closed-form solution of singular and hypersingular integral equations (HSIEs) of the form a(t)ϕ(t) + 1 b(t) 1 ϕ(t) = dt + L(x, t)ϕ(t)dt = f (x), p = {1, 2}, −1 < x < 1, π −1 (t − x) p −1 I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS (1.1) Vol. 10, No. 1 (Special Issue), Jan–June 2019 12 Z.K. Eshkuvatov and Anvar Narzullaev is only possible in exceptional cases. Comprehensive presentations and extensive literature survey associated with all methods of solutions of singular integral equations (SIEs) (the case p = 1 in Equation (1.1)) of the first and second kinds are found in [1–17] and literatures cited there in. The methods of solution of HSIEs (the case p = 2 in Equation (1.1)) are much less elaborated. In [18–31] the quadrature, projection, semi-analytical solution, Galerkin and collocation methods, homotopy perturbation method, reproducing kernel method, for solving HSIEs of the first and second kinds (1.1) were developed and justified by imposing some conditions on the kernels and righthand sides of the equations. In several physical problems such as aerodynamics, hydrodynamics, and elasticity, one encounters the integral equations of the form (see [22]) K (x, t) 1 1 + L (x, t) dt = f (x) , −1 < x < 1. (1.2) = ϕ(t) 1 π −1 (t − x)2 Hadamard ([20], 1952) was first to introduce the HSIEs in solving Cauchy’s problem for hyperbolic differential equations. He defined the HSIEs as the finite part of a divergent integral whose singularities are arranged at the endpoints of the integration interval for one-dimensional integrals or on the boundary of the domain for multidimensional integrals. These integrals came to be known as Hadamard’s finite part integrals or simply Hadamard’s integrals. For HSIEs: Chan et al. ([21], 2003) investigate HSIEs of the form Iα (Tn , m, r ) = Iα (Un , m, r ) = 1 −1 1 −1 m−1/2 Tn (s) 1 − s 2 ds, |r | < 1, m ∈ N, (s − r )α m−1/2 Un (s) 1 − s 2 ds, |r | < 1, (s − r )α where Tn (x) and Un (x) are the Chebyshev polynomials of the first and second kinds, respectively, and the exact solution are derived for the cases α = {1, 2, 3, 4} and m = {0, 1, 2, 3}. Mandal and Bera [22] considered Equation (1.2) with K (x, t) = 1 and solved it by polynomial expansion method and proved the exactness of the method for linear density function. Mandal and Bhattacharya [23] proposed approximate numerical solutions of some classes of integral equations by using Bernstein polynomials as basis. The method was explained with illustrative examples. As well as the convergence of the method is established for each class of integral equations. Martin [24] obtain the analytic solution to the simplest one-dimensional hypersingular integral equation i.e. the case of K (x, t) = 1 and L(x, t) = 0 in Equation (1.2). Abdulkawi et al. [26], considered the finite part integral equation (1.2) with K (x, t) = 1 and showed the exactness of the proposed method for the linear density function and illustrated it with examples. Nik and Eshkuvatov [27] have used the complex variable function method to formulate the multiple curved crack problems into hypersingular integral equations of the first kind in more general case and these hypersingular integral equations are solved numerically for the unknown function, which are later used to find the stress intensity factor (SIF). For the SIEs: Capobianco et al. [3, 4] studied collocation and quadrature methods for singular integro-differential equations of Prandtl’s type in weighted Sobolev spaces by using Langrange polynomials as basis. Elliot [7,8] proposed classical collocation method for the approximate I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Projection Method and Error Bound for HSIEs 13 solution of complete singular integral equations with Cauchy kernel taken over the arc (-1, 1) and proved that under reasonable conditions, the approximate solutions converge to the solution of the original equation as well as Petrov-Galerkin methods is applied to solve SIEs of second kind and proved convergence of the proposed method, respectively. Ioakimidis [11, 12] proposed a classical collocation and Galerkin methods for the numerical solution of SIEs of the first kind (1.2) involving a finite part integrals with double pole singularity in the case of K (x, t) = 1. As an application of the method to the problem of a straight crack under an exponential normal loading distribution is also made and showed the rapid convergence of the obtained numerical results for the stress intensity factors at the crack tips. Eshkuvatov et al. [13, 14] consider SIEs (1.1) in the case p = 1, of the first and second kind and proposed approximate methods based on Chebyshev polynomials for all type of solutions of the first kind and bounded solution of the second kind respectively. Exactness of the method is also shown. Boykov et al. [28, 29] proposed method asymptotically optimal and optimal in order algorithms for numerical evaluation of one-dimensional hypersingular integrals with fixed and variable singularities and described a spline-collocation method and its justification for the solution of one-dimensional hypersingular integral equations, polyhypersingular integral equations, and multi-dimensional hypersingular integral equations, respectively. Golberg [19] consider Equation (1.2) with the kernel K (x, t) = 1 and proposed projection method with the truncated series of Chebyshev polynomials of the second kind together with Galerkin and collacation methods. The author established the uniform convergence and obtained convergence rates for their algorithms for solving a class of Hadamard singular integral equations. By extending the results of Golberg [19] for the Equation (1.2), where the kernel K (x, t) is a constant on the diagonal of the domain D = [−1, 1] × [−1, 1], we outline collocations and Galerkin methods together with Gauss-Chebyshev quadrature. For the unique solution of the bounded and unbounded cases the following conditions ϕ(−1) = ϕ(1) = 0, 1 1 ϕ(x)d x = C. π −1 (1.3) are imposed respectively. The structure of this note is arranged as follows: In Section 2.1, the detail derivations of the projection method is presented followed by Gauss-Chebyshev quadrature in Section 2.2 with adaptation weighted kernel integrals. Section 3 deals with six examples to verify the validity and accuracy of the proposed method and outlines computation of complexity and economization of CPU time by comparing with other methods. Finally, the conclusion and acknowledgment are presented in Section 4. 2. DESCRIPTION AND DISCRETIZATION OF THE METHOD 2.1. Description of the method Since kernel in Equation (1.2) is constant on the diagonal we can write it as follows K (x, t) = c0 + (t − x)Q(x) + (t − x)2 Q 1 (x, t), c0 = 0. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS (2.1) Vol. 10, No. 1 (Special Issue), Jan–June 2019 14 Z.K. Eshkuvatov and Anvar Narzullaev where Q(x) is smooth function and Q 1 (x, t) is square integrable kernel. Taking into account Equation (2.1) we are able to write Equation (1.2) in the form c0 1 ϕ(t) 1 1 Q(x) 1 ϕ(t) = − dt + dt + L(x, t)ϕ(t)dt = f (x), −1 < x < 1, (2.2) π −1 (t − x)2 π π −1 −1 t − x where L(x, t) = Q 1 (x, t) + L 1 (x, t), and the first integral in Equation (2.2) is being understood as the Hadamard finite-part. Main aim is to find bounded and unbounded solution of Equation (2.2) satisfying condition (1.3). Hence, we search solution in the form ϕ(x) = ωi (x)u(x), i = {1, 2}, (2.3) √ 1 − x 2 , ω2 (x) = (2.4) where ω1 (x) = √ 1 , 1−x 2 Substituting (2.3) into (2.2) yields c0 1 ωi (t) Q(x) 1 ωi (t) u(t) dt u(t) dt + π −1 (t − x)2 π −1 t − x 1 1 + ωi (x)L(x, t) u(t) dt = f (x), π −1 −1 < x < 1, (2.5) Introducing notations c0 1 ωi (x) = Hi u = u(t)dt, π −1 (t − x)2 Ci u = Q(x) 1 ωi (x) − u(t)dt, π −1 t − x Li u = 1 π (2.6) 1 −1 L(x, t)ωi (x)u(t)dt, leads to the operator equation Hi u + Ci u + L i u = f, i = {1, 2}, f ∈ L 2ρ , u ∈ L 1ρ , (2.7) where the spaces L 2ρ and L 1ρ are defined as weighted Hilbert space and subspace of Hilbert respectively. It is known that the hypersingular operator Hg can be considered as differential Cauchy operator i.e., 1 1 ω(t) d d Cg u = − u(t)dt . (2.8) Hg u = dx d x π −1 t − x I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Projection Method and Error Bound for HSIEs 15 Therefore, Equation (2.7) may be viewed as an integro-differential Prandtl’s type equation. On the other hand we know that (Mason [32]) √ 1 1 1 − t 2 Un (t) C g Un (x) = − dt = −Tn+1 (x), n = 1, 2, ..., π −1 (t − x) 1 1 − C g Tn (x) = π −1 (2.9) Tn (t) 1 − t 2 (t − x) dt = Un−1 (x), n = 1, 2, ..., For n = 0 we have C g U0 (x) = −T1 (x), C g T0 (x) = 0, where Tn (x) and Un (x) are the Chebyshev polynomials of the first and second kinds respectively, defined by Tn (x) = cos(nθ ), Un (x) = sin(n + 1)θ , x = cos θ, 0 ≤ θ ≤ π. sin θ By differentiating Equation (2.9) (see Chan [21]), we obtain √ 1 1 1 − t 2 Un (t) Hg Un (x) = dt = −(n + 1)Un (x), n = 1, 2, ..., − π −1 (t − x)2 1 1 Hg Tn (x) = − π −1 Tn (t) 1 dt = 2 2 1 − x2 1 − t (t − x) n+1 n−1 Un−2 (x) − Un (x) , n = 1, 2, ..., 2 2 (2.10) for n = 0 Hg U0 (x) = −U0 (x), Hg T0 (x) = 0, Moreover, Tn+1 (x) = 1 [Un+1 (x) − Un−1 (x)] , n = 0, 1, 2, ..., 2 xUn (x) = 1 [Un+1 (x) + Un−1 (x)] , n = 0, 1.2, ... 2 Un (x) = (2.11) 1 [Tn (x) − Tn+2 (x)], n = 0, 1, 2, ..., 2(1 − x 2 ) where U−1 (x) = 0. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 16 Z.K. Eshkuvatov and Anvar Narzullaev It can easily be shown from (2.6), (2.9), (2.10) and (2.11) that H1 φn,1 (x) = −c0 (n + 1) φn,1 (x), n = 1, 2, . . . n+1 c0 n−1 φn−2,1 (x) − φn,1 (x) , n = 1, 2, . . . H2 φn,2 (x) = 1 − x2 2 2 Q(x) φn+1,1 (x) − φn−1,1 (x) , n = 1, 2, . . . , C1 φn,1 (x) = − 2 Q(x) C2 φn,2 (x) = − φn−1,1 (x), n = 1, 2, . . . 2 (2.12) where φ−1,1 (x) = 0 and orthonormal polynomials are defined as φi,1 (x) = φi,2 (x) = ⎧ ⎪ ⎪ ⎪ ⎨ 2 Ui (t), i = 0, 1, ..., n π ⎪ ⎪ ⎪ ⎩ 2 Ti (x), i = 1, 2, ..., n π 1 T0 (x), i = 0, π (2.13) Equations (2.12) -(2.13) are crucial to the rest of our analysis. To find an approximate solution of Equation (2.5), u (t) is approximated by u(t) ∼ = u n,i (t) = n b(n) j,i φ j,i (t), i = {1, 2}. (2.14) j=0 Therefore approximate solution of Equation (2.2) can be written as ϕ(x) ≈ ϕn,i (x) = ωi (x) n b(n) j,i φ j,i (x), i = {1, 2}, (2.15) j=0 where unknown coefficients b(n) j,i need to be defined. To do these end let us consider two cases: r Case 1, (i = 1). This is the case where the solution of Equation (2.7) is bounded at the end of the interval [−1, 1]. Substituting (2.14) into (2.5) and using Equation (2.12) - (2.13) yields n Q(x) (n) (n) (n) −c0 ( j + 1)b(n) b U U (x) + − b (x) + b ψ (x) j j,1 j j+1,1 j−1,1 j,1 j,1 2 j=0 + Q(x) (n) b Un+1 (x) = 2 n,1 π f (x), 2 I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS (2.16) Vol. 10, No. 1 (Special Issue), Jan–June 2019 Projection Method and Error Bound for HSIEs 17 n n where U−1 (x) = 0, b−1,1 = bn+1,1 = 0 and 1 ψ j,1 (x) = π 1 L(x, t) 1 − t 2 U j (t)dt. (2.17) 1 n such as roots (i) For the collocation method we choose the suitable collocation points {xi }i=1 2 of Un+1 (x) or Tn+1 (x) or (1 − x )Un−1 (x). Then Equation (2.16) leads to a system of linear equation n Q(xk ) (n) (n) (n) (n) b j+1,1 − b j−1,1 U j (xk ) + b j,1 ψ j,1 (xk ) −c0 ( j + 1)b j,1 U j (xk ) + 2 j=0 + Q(x) (n) π bn,1 Un+1 (xk ) = f (xk ), k = 0, 1, ..., n, 2 2 (2.18) (n) (n) provided that b−1,1 = bn+1,1 = 0. Solving the system of Equation (2.18) for the unknown coefficients b(n) j,1 , j = 0,. . . , n and (n) substituting the values of b j,1 into Equation (2.15) yields the numerical solution of Equation (2.2). (ii) For the Galerkin’s method by taking into account (2.12) and (2.14), we consider the following residual √ √ c0 1 1 − t 2 u n,1 (t) Q(x) 1 1 − t 2 = − u n,1 (t)dt dt + Rn,1 (x) = π −1 (t − x)2 π −1 t − x + 1 π 1 L(x, t) 1 − t 2 u n,1 (t)dt − f (x) −1 n 2 Q(x) (n) (n) (n) (n) −c0 ( j + 1)b j,1 U j (x) + (b j+1,1 − b j−1 )U j (x) + b j,1 ψ j,1 (x) π j=0 2 = + 2 Q(x) (n) b Un+1 (x) − f (x), π 2 n,1 (2.19) in which satisfies the conditions 1 −1 1 − τ 2 Rn,1 (τ )φk,1 (τ )dτ = 0, k = 0, ..., n, I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS (2.20) Vol. 10, No. 1 (Special Issue), Jan–June 2019 18 Z.K. Eshkuvatov and Anvar Narzullaev It is known (Mason [32]) that 1 2 π 1 − x 2 φm,1 (t)φn,1 (t)dt = −1 1 −1 1 1 − x 2 Um (t)Un (t)dt = −1 ⎧ 1 1 ⎪ ⎪ ⎨ φm,2 (t)φn,2 (t) π −1 dt = √ 2 ⎪ 2 1 1−x ⎪ ⎩ π −1 1, 0, T0 (t)T0 (t) dt, √ 1, 1 − x2 = Tm (t)Tn (t) 0, dt. √ 1 − x2 m = n, m = n (2.21) m = n, m = n. (2.22) Due to orthogonality conditions (2.21), residual in Equation (2.20) reduces to the system of algebraic equation for finding the unknown coefficients {bnj,1 }nj=0 . 1 (n) 1 (n) (n) Q(x) bk+1,1 + − bk−1,1 (b − b(n) j−1,1 ) Qφ j,1 , φk,1 2 2 j=0 j+1,1 n (n) − c0 (k + 1)bk,1 + + n b(n) j,1 ψ j,1 , φk,1 ρ j=0 1 (n) + bn,1 Qφn+1,1 , φk,1 2 (n) (n) with b−1,1 = bn+1,1 = 0 and f, g ρ ρ Qφ j,1 , φk,1 ρ = = f, φk,1 ρ , k = 0, ...n, (2.23) denotes the inner product with weight function i.e., f, φk,1 ρ ρ = 1 −1 1 1 − x 2 f (x)φk,1 (x)d x. −1 1 − x 2 (Q(t) − Q(x))φ j,1 (x)φk,1 (x)d x. r Case 2 (i = 2). In this case, solution of Equation (2.7) will be unbounded at the end of the interval [−1, 1]. Substitute (2.14) into (2.5) and use Equation (2.12) to get n j=1 b(n) j,2 c0 · 1 − x2 j +1 j −1 U j−2 (x) − U j (x) + Q(x)U j−1 (x) + ψ j,2 (x) 2 2 (n) ψ0,2 (x) = + b0,2 π f (x), 2 (2.24) T j (t) L(x, t) √ dt. 1 − t2 (2.25) where U−1 (x) = 0 and ψ j,2 (x) = 1 π 1 1 I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Projection Method and Error Bound for HSIEs 19 To have unique solution of Equation (2.24) we impose condition (1.3) to yield C= 1 π 1 −1 ϕ(t)dt = n 1 (n) 2 1 T j (t)dt b j,2 √ π j=0 π −1 1 − t 2 2 (n) 2 b j,2 + π π j=1 n (n) = b0,2 1 −1 T j (t)T0 (t) (n) dt = b0,2 √ 2 1−t 2 . π From this it follows that (n) =C b0,2 π . 2 (2.26) Substituting (2.26) into (2.24) yields n b(n) j,2 j=1 c0 · 1 − x2 j +1 j −1 U j−2 (x) − U j (x) + Q(x)U j−1 (x) + ψ j,2 (x) = f 1 (x), 2 2 (2.27) π f (x) + ψ0,2 (x) · C 2 (i) To solve Equation (2.27) for unknown parameters b(n) j,2 using collocation method we n such as roots of Un (x) or Tn (x) or (1 − x 2 )Un−2 (x). choose the suitable node points {xi }i=1 Then Equation (2.27) leads to a system of linear equation where f 1 (x) = n j=1 b(n) j,2 c0 · 1 − xk2 j +1 j −1 U j−2 (xk ) − U j (xk ) + Q(x)U j−1,1 (xk ) + ψ j,2 (xk ) 2 2 = f 1 (xk ), k = 1, 2, ..., n. (2.28) Solving the system of Equation (2.28) for the unknown coefficients b(n) j,2 , j = 1, . . . , n and (n) substituting the values of b j,2 , j = 0, 1, ..., n into Equation (2.15) yields the numerical solution of Equation (2.2). (ii) In the case of unbounded solution for the Galerkin’s method we consider the following residual u n,2 (t) u n,2 (t) c0 1 Q(x) 1 = √ − √ dt dt + Rn,2 (x) = 2 2 π −1 1 − t (t − x) π 1 − t 2 (t − x) −1 + 1 π 1 u n,2 (t) L(x, t) √ dt − f (x). 1 − t2 −1 I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS (2.29) Vol. 10, No. 1 (Special Issue), Jan–June 2019 20 Z.K. Eshkuvatov and Anvar Narzullaev From (2.12), (2.14), (2.26) and (2.29) it follows that n 2 (n) c0 j +1 j −1 U U b j,2 · (x) − (x) j−2 j π j=1 1 − x2 2 2 Rn,2 (x) = 2 (n) b ψ0,2 (x) · C − f (x) +Q(x)U j−1,1 (x) + ψ j,2 (x) + π 0,2 ⎧ ⎡ ⎤ n−1 3 j + 3 2 ⎨ c0 n − 1 j − 1 (n) (n) (n) (n) bn,2 Un (x) + b j+2,2 − b j,2 U j (x)⎦ · ⎣ b2,2 U0 (x) − = π ⎩ 1 − x2 2 2 2 2 j=1 ⎫ n−1 ⎬ (n) + b j,2 Q(x)U j−1 (x) + ψ j,2 (x) + C · ψ0,2 (x) − f (x). (2.30) ⎭ j=1 Since rational function seats on the denominator, residual in (2.30) satisfies the conditions 1 (1 − τ 2 )3/2 Rn,2 (τ )φk,1 (τ )dτ = 0, k = 1, ..., n. (2.31) −1 Equation (2.31) reduces to the system of algebraic equation for finding the unknown coefficients {bnj,2 }nj=1 , due to (2.12) and (2.21), c0 n k + 3 (n) k − 1 (n) bk+2,2 − bk,2 + b(n) j,2 Qφ j−1,1 , φk,1 2 2 j=1 + C ψ0,2 , φk,1 ρ = f, φk,1 , ρ ρ + ψ j,2 , φk,1 k = 1, ...n − 1, ρ (2.32) where b−1,2 = bn+1,2 = 0. For k = n we have n − 1 (n) (n) bn,2 + b j,2 Qφ j−1,1 , φn,1 2 j=1 n − c0 ρ + ψ j,2 , φn,1 ρ + C ψ0,2 , φn,1 ρ = f, φn,1 , ρ (2.33) where f, g ρ denotes the inner product with weight function i.e., 1 f, φk,1 ρ = (1 − x 2 )3/2 f (x)φk,1 (x)d x, −1 1 (1 − x 2 )3/2 Q(x)φ j,1 (x)φk,1 (x)d x, Qφ j , φk,1 ρ = −1 1 (1 − x 2 )3/2 ψ j,2 (x)φk,1 (x)d x. ψ j,2 , φk,1 ρ = −1 Solving the system of Equation (2.33) for the unknown coefficients b(n) j,2 , j = 1, . . . , n and into Equation (2.15) yields the numerical solution of Equation substituting the values of b(n) j,2 (2.2). I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Projection Method and Error Bound for HSIEs 21 2.2. Discretization of the method In the Section 2.1, we have described two methods which deals with many form of weighted kernel integrals. In this section, we develop Gauss-Chebyshev quadrature formula with Gauss-Lobotto nodes for weighted kernel integrals. It is known that many weighted kernel integrals have not exact solution. So that we need suitable quadrature for numerical computation of weighted kernel integrals In Kythe [35], states that the Gauss quadrature formula of the form b w(x) f (x)d x = a n Ai f (xi ), (2.34) i=0 is exact for all f ∈ P2n+1 if the weights Ai and the nodes xi can be found for different orthogonal polynomials approximation of f (x) on the interval [a, b]. In particular, if [a, b] = [−1, 1] and ω1 (x), ω2 (x) are defined by (2.4), the orthogonal polynomials are the Chebyshev polynomials Tn (t), Un (t) of the first and second kind respectively then resulting formulas of Equation (2.34) are known as Gauss-Chebyshev rule. Let us define the nodes ξi,1 and ξi,2 as the zeros of Tn+1 (x) and Un+1 (x) respectively, ξi,1 = cos (2i − 1)π 2(n + 1) ξi,2 = cos , i = 1, 2, ..., n + 1 (2.35) iπ , i = 1, 2, ..., n + 1. n+2 (2.36) Lemma 1. Open Gauss-Chebyshev rule (in some literature it is called Meller quadrature) is given as 1 −1 where Ak,1 = 1 f (t)dt = Ak,1 f (tk ) + Rn.1 ( f ). √ 1 − t2 k=1 n+1 (2.37) π and tk is defined by (2.35). Similarly n+1 1 −1 1 − t 2 f (t)dt = n+1 Ak,2 f (tk ) + Rn.2 ( f ), (2.38) k=1 π (1 − tk2 ) and tk is defined by (2.36). The word “open” is used for not including n+2 endpoints. We usually omit “open” since all Gaussian rules with positive weight function are of the open type. where Ak,2 = Proper Derivation. Let us show proper derivation of Gauss-Chebyshev quadrature formulas (QF) (2.37)-(2.38) which is different than Kythe [35] and Israilov [36]. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 22 Z.K. Eshkuvatov and Anvar Narzullaev It is known that (see Israilov [36, pp. 338]) coefficients of the interpolation quadrature formula with weight function b ρ(t) f (t)dt = a n Ak f (xk ), (2.39) Pn (t) dt, (t − tk ) (2.40) k=1 are defined as 1 Ak = Pn (tk ) b a ρ(t) where tk are the zeros of interpolation polynomial Pn (t). In view of (2.39)-(2.40) and (2.9), the coefficients of the QF (2.37) is Ak,1 = 1 Tn+1 (tk ) 1 −1 1 1 Tn+1 (t) dt = πUn (tk ), √ 2 Tn+1 (tk ) 1 − t (t − tk ) (2.41) It is easy to check that Tn+1 (x) = (n + 1)Un (x), therefore Ak,1 = π . n+1 To derive QF (2.38) we take tk , k = 1, 2, ..., n + 1 as the zeros of Chebyshev polynomials of the second kind Un+1 (t) = 0 which is defined by (2.36). In a similar way as (2.41), by applying (2.9), we obtain 1 √ 1 1 − t 2 Un (t) 1 dt = (−π Tn+1 (tk )). (2.42) Ak,2 = Un (tk ) −1 (t − tk ) Un (tk ) Since tk , k = 1, 2, ..., n + 1 are the zeros of Un+1 (x) and 1 (Un+1 (x) − Un−1 (x)), 2 d n+2 1 n Un (x) = U U (x) − (x) , n−1 n+1 dx 1 − x2 2 2 Tn+1 (x) = (2.43) (2.44) π (1 − tk2 ) . n+2 The proof of main theorem based on the following theorems given in Kythe [35]. we have Ak,2 = Theorem 1 (Johnson and Riess 1977). Gaussian QF has precision 2n + 1 only if the points xi , i = 0, 1, ..., n are the zeros of φn+1 (x), where φn+1 (x) are orthogonal polynomials. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Projection Method and Error Bound for HSIEs Theorem 2. If f ∈ C 2n+2 [a, b], then the error of Gaussian QF is given by f 2n+2 (ξ ) b 2 Rn ( f ) = Iab ( f ) − In ( f ) = ρ(x)Wn+1 (x)d x, ξ ∈ [a, b] (2n + 2)! a 23 (2.45) where Wn+1 (x) is the polynomials of degree n + 1 with n + 1 distinct zeros anf ρ(x) is a weight function. Main theorem is formulated as follows Theorem 3. If f ∈ C 2n+2 [−1, 1], then the error of Gauss-Chebyshev QFs (2.37) and (2.38) are given by π Rn,1 ( f ) = f 2n+2 (ξ1 ) , ξ1 ∈ [−1, 1], (2n + 2)! 22n+1 π Rn,2 ( f ) = (2.46) f 2n+2 (ξ2 ) , ξ2 ∈ [−1, 1]. (2n + 2)! 22n+3 Proof. In Israilov [36] shown the relationship between polynomial Wn+1 in (2.45) and Chbyshev polynomials of the first and second kind as fallows Tn+1 (x) Wn+1 (x) = , 2n (2.47) Un+1 (x) Wn+1 (x) = . 2n+1 On the other hand it is known that Tn+1 , Tn+1 = 1 −1 Un+1 , Un+1 = T 2 (t) π √n+1 dt = , 2 2 1−t (2.48) 1 −1 2 1 − t 2 Un+1 (t)dt = π . 2 From (2.47) - (2.48) it follows that Rn,1 ( f ) = f 2n+2 (ξ1 ) (2n + 2)! f 2n+2 (ξ2 ) Rn,2 ( f ) = (2n + 2)! 1 −1 √ 1 2 π f 2n+2 (ξ1 ) , Tn+1 (t)dt = 2n+1 2n 2 (2n + 2)! 1 − t2 2 1 (2.49) 1 −1 1 − t2 1 22n+2 2 Un+1 (t)dt = π 22n+3 f 2n+2 (ξ2 ) . (2n + 2)! Now we extend Gauss-Chebyshev QF (2.37) - (2.38) for the weight kernel integrals (2.6) which is given in Section 2.1. In many problems of HSIEs regular kernel L(x, t) will be given as convolution type m L(x, t) = ci (x)di (t). (2.50) i=1 I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 24 Z.K. Eshkuvatov and Anvar Narzullaev In the case of convolution type kernel (2.50), for the regular kernel in (2.6), the Gauss-Chebyshev QF in Lemma 1 has the form 1 L 1 (u) = π −1 −1 L(x, t) 1 − t 2 u(t)dt m n 2 1 1 ci (x)b(n) j,1 π i=1 j=0 π −1 = m n n+1 1 − t 2 di (t)U j (t)dt (2.51) ci (x)b(n) j,1 Ak,1 f 1i, j (tk ). i=1 j=0 k=1 Similarly L 2 (u) = 1 π −1 L(x, t) √ −1 1 1 − t2 u(t)dt m n 2 1 1 1 ci (x)b(n) di (t)T j (t)dt √ j,2 π i=1 j=0 π −1 1 − t 2 = m n n+1 (2.52) ci (x)b(n) j,2 Ak,2 f 2i, j (tk ). i=1 j=0 k=1 where Ak,1 = 1 − tk2 , f 1i, j (tk ) = di (tk )U j (tk ), n+2 Ak,2 = 1 , f 2i, j (tk ) = di (tk )T j (tk ). n+1 For non convolution regular kernel L(x, t) case, we have the following Gauss-Chebyshev QF L 1 (u) = 1 π = −1 −1 L(x, t) 1 − t 2 u(t)dt n 2 (n) 1 1 b j,1 π j=0 π −1 n n+1 1 − t 2 L(x, t)U j (t)dt (2.53) b(n) j,1 Ak,1 f 1(x, tk ). j=0 k=1 I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Projection Method and Error Bound for HSIEs 25 In the same way we obtain 1 L 2 (u) = π = −1 −1 1 L(x, t) √ u(t)dt 1 − t2 n 2 (n) 1 1 1 b j,2 L(x, t)T j (t)dt √ π j=0 π −1 1 − t 2 n n+1 (2.54) b(n) j,1 Ak,2 f 2(x, tk ). j=0 k=1 where Ak,1 = 1 − tk2 , f 1 (x, tk ) = L(x, tk )U j (tk ), n+2 Ak,2 = 1 , f 2 (x, tk ) = L(x, tk )T j (tk ). n+1 3. NUMERICAL RESULTS We very often need to use polynomial values of the Chebyshev polynomials for the numerical computation. 3.1. Case 1, i = 1. Bounded solution Example 1. Consider the following HSIEs 1 π 1 −1 ϕ(t) 1 dt + (t − x)2 π 1 −1 16x 3 t 3 ϕ(t)dt = 16x − 31x 3 , (3.1) with the conditions ϕ(±1) = 0. It is not hard to verify that the exact solution of Equation (3.1) is ϕ(x) = 1 − x 2 (8x 3 − 4x). (3.2) Solution: We search bounded solution of HSIEs in the form ϕ(x) ∼ = ϕn (x) = 2 π 1 − x2 I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS n b(n) j,1 U j (x). (3.3) j=0 Vol. 10, No. 1 (Special Issue), Jan–June 2019 26 Z.K. Eshkuvatov and Anvar Narzullaev Table 1. Chebyshev polynomials of the first and second kind n Tn (x) Un (x) 0 1 1 1 x 2x 2 2x 2 − 1 4x 2 − 1 3 4x 3 8x 4 4 − 3x − 8x 2 8x 3 − 4x +1 16x 4 16x 5 − 20x 3 + 5x 5 64x 7 7 32x 5 − 32x 3 + 6x − 48x 4 + 18x 2 −1 64x 6 − 80x 4 + 24x 2 − 1 − 112x 5 + 56x 3 − 7x 128x 7 − 192x 5 + 80x 3 − 8x 32x 6 6 − 12x 2 + 1 Since c0 = 1, Q(x) = 0 and L(x, t) = 16x 3 t 3 , from the Equation (2.16) and (2.17) with n = 3 it follows that 3 π (16x − 31x 3 ), 2 b(n) j,1 −( j + 1)U j (x) + ψ j,1 (x) = j=0 where ψ j,1 (x) is defined as ψ j,1 (x) = 1 π 1 −1 (3.4) (16x 3 t 3 ) 1 − t 2 U j (t)dt. Since t3 = 1 [U3 (t) + 2U1 (t)] , 8 (3.5) we obtain ψ0,1 (x) = 0, ψ1,1 (x) = 2x 3 , ψ2,1 (x) = 0, ψ3,1 (x) = x 3 , ψ j,1 (x) = 0, j > 3. (3.6) Substituting (3.6) into (3.4) and taking into account Table 1, yields (n) (n) (n) (n) − b0,1 + 2(x 3 − 2x)b1,1 − 3(4x 2 − 1)b2,1 + (16x − 31x 3 )b3,1 = π (16x − 31x 3 ). 2 (3.7) Equating the same power of x yields (n) (n) (n) (n) b0,1 = b1,1 = b2,1 = 0, b3,1 = π . 2 Substituting all values of b(n) j,1 , j = {0, 1, 2, 3} into (3.3), gives ϕ(x) = 1 − x 2 U3 (x) = 1 − x 2 (8x 3 − 4x) , which is identical to the exact solution (3.2). I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Projection Method and Error Bound for HSIEs 27 Let us apply Galerkin method for Example 1. Since c0 = 1, Q(x) = 0, and φ j,1 (x) are defined by (2.13), residual in Equation (2.19) is Rn,1 (x) = n 2 (−( j + 1))b(n) j,1 U( x) + π j=0 2 3 (n) (n) 2x b1,1 + x 3 b3,1 − 16x − 31x 3 . π In view of (3.5) and 2x = U1 (x) with n = 3, it can be written as 2 21 (n) (n) (U3 (x) + 2U1 (x)) 2b1,1 (−( j + 1))b(n) + b3,1 j,1 U( x) + π j=0 π8 31 − 8U1 (x) − (U3 (x) + 2U1 (x)) . 8 3 Rn,1 (x) = (3.8) Multiplying (3.8) by φk,1 (τ ), and integrating over [−1, 1] yields 1 −1 1 − τ 2 R3,1 (τ )φk,1 (τ )τ = 0, k = {0, 1, 2, 3}. (3.9) From (3.9) it follows that (3) (3) b0,1 = b2,1 = 0, (3.10) and ⎧ 1 1 ⎪ (n) (n) (n) ⎪ ⎪ −2b1,1 2b1,1 − + + b3,1 ⎪ ⎨ 4 4 π = 0, 2 (3.11) ⎪ ⎪ 31 1 ⎪ (n) (n) (n) ⎪ ⎩ −4b3,1 2b1,1 + + + b3,1 8 8 π = 0. 2 Solving Equation (3.11), gives π . 2 (n) (n) = 0, b3,1 = b1,1 (n) Exact solution can be obtained by substituting the values of bi,1 , (3.3). (3.12) j = {0, 1, 2, 3} into Equation Example 2. Let us investigate the following HSIEs 1 π 1 −1 (1 + 2(t − x)) 1 ϕ(t)dt + (t − x)2 π 1 −1 I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS 1 2x 3 e t ϕ(t)dt = f (x), 2 (3.13) Vol. 10, No. 1 (Special Issue), Jan–June 2019 28 Z.K. Eshkuvatov and Anvar Narzullaev where f (x) = −16x 4 − 40x 3 + 4x 2 + 22x + 1 + π 2x e . 32 The exact solution of Equation (3.13) is ϕ(x) = 1 − x 2 (8x 3 + 4x 2 − 4x − 1). (3.14) Solution: Comparing (3.13) with (2.2) we get c0 = 1, Q(x) = 2, L(x, t) = 1 2x 3 e t . 2 From (2.16) and (2.17) we obtain n (n) (n) (n) (n) −( j + 1)b(n) j,1 + b j+1,1 − b j−1,1 U j (x) + b j,1 ψ j,1 (x) + bn,1 Un+1 (x) = j=0 π f (x), (3.15) 2 n n where U−1 (x) = 0, b−1,1 = bn+1,1 = 0 and 1 ψ j,1 (x) = π 1 1 1 2x 3 e t 2 1 − t 2 U j (t)dt. (3.16) From (3.16) and orthogonality conditions (2.21) it follows that ψ0,1 (x) = 0, ψ1,1 (x) = 1 2x 1 2x e , ψ2,1 (x) = 0, ψ3,1 (x) = e , ψ j,1 (x) = 0, j > 3. 16 32 (3.17) Substituting (3.17) into (3.15) and choosing collocation points xi as the root of Tn+1 (x) which is given in Equation (2.35), we have the following system of algebraic equations n (n) (n) −( j + 1)b(n) + b − b j,1 j+1,1 j−1,1 U j (x i ) + bn,1 Un+1 (x i ) j=0 + 1 (n) (n) 2b1,1 e2xi = + b3,1 32 π f (xi ), i = 0, ..., n, 2 (3.18) Solving Equation (3.18) at the collocation points (2.35) for the different value of n, we obtain the numerical solution of Equation (3.13). The errors of numerical solution of Equation (3.13) are summarized in Table 2. Table 2 reveals that method proposed here is very accurate and stable. The large value of n = {50, 500} shows that proposed method is exact for HSIEs (3.13). Actually, it can be shown that proposed method is exact for n = 3 only. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Projection Method and Error Bound for HSIEs 29 Table 2. Numerical solution of Example 2 Exact value of HSIEs (3.13) Errors Rn , n = 5 Errors Rn , n = 50 Errors Rn , n = 500 −0.01412481476 5.0480 × 10−016 4.7878 × 10−016 4.8121 × 10−016 −0.00011175527 1.0973 × 10−015 4.9398 × 10−015 2.4761 × 10−015 -0.725 0.65698032427 1.5543 × 10−015 3.5527 × 10−015 2.5313 × 10−015 -0.436 0.75715136458 1.2212 × 10−015 1.0123 × 10−015 1.0214 × 10−015 −1.4433 × 10−015 −1.8874 × 10−015 x -0.9999 -0.901 -0.015 0.015 0.436 × 10−016 −0.93902134227 2.2204e −1.05895384758 4.4409 × 10−016 1.0014 × 10−016 −1.5543 × 10−015 −1.18843460478 −1.1102 × 10−015 −2.6645 × 10−015 −3.5527 × 10−015 0.725 0.86171092460 1.0014 × 10−016 −2.8866e × 10−015 −3.6637 × 10−015 0.901 1.94987172916 1.7764 × 10−015 −2.4425 × 10−015 −3.5527 × 10−015 0.09895288143 9.7145 × 10−16 1.0024 × 10−016 −8.3267 × 10−016 0.9999 Example 3. Solve HSIEs of the form 1 π 1 −1 K (x, t) + L(x, t) ϕ(t)dt = f (x), (t − x)2 (3.19) where K (x, t) = 2 + t x(t − x), L 1 (x, t) = 1 1 + , t +2 x +2 and √ √ √ √ 20 3 10x 2 10 √ 10(2 − 3) f (x) = − (2 − 3 + x) + 10(2 − 3)x + (2 3 − 3) + . − (2 + x)2 x +2 3 x +2 The exact solution of Equation (3.19) is ϕ(x) = 1 − x2 10 . x +2 (3.20) Remark. In this example, main kernel K (x, t) is given as convolution form but on the diagonal K (x, x) = const. On the other hand regular kernel L(x, t) is not convolution type. Here we present experimentally that the proposed method can work well even the regular kernel L(x, t) in Equation (3.19) is not in the convolution type and solution is not of the polynomial form. Solution. Suitable changes in Equation (3.19) leads to 2 π 1 −1 ϕ(t)dt x2 + (t − x)2 π 1 −1 1 ϕ(t)dt + t−x π 1 −1 x+ 1 1 + ϕ(t)dt = f (x), x +2 t +2 I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS (3.21) Vol. 10, No. 1 (Special Issue), Jan–June 2019 30 Z.K. Eshkuvatov and Anvar Narzullaev Table 3. Numerical solution of Example 3 x Exact, (3.20) Errors Rn , n = 6 Errors [25], n, m = 6, Errors Rn , n = 26, Errors [25], n = m = 26, 0.1414037 3.5663 × 10−4 1.0852 × 10−4 5.2180 × 10−015 1.4040 × 10−10 3.9473984 1.2235 × 10−4 3.5502 × 10−4 −1.2879 × 10−015 1.8203 × 10−9 -0.725 5.4019519 2.4011 × 10−3 1.88998 × 10−4 1.7764 × 10−015 0.6194 × 10−9 -0.436 5.7541347 2.2201 × 10−3 3.0648 × 10−4 −1.5987 × 10−014 0.2622 × 10−9 5.0372166 −1.6201 × 10−3 0.8678 × 10−4 −1.4211 × 10−014 0.1739 × 10−9 4.9622208 −1.601 × 10−3 1.9992 × 10−4 −1.1546 × 10−014 1.8552 × 10−9 0.436 3.6943623 1.3001 × 10−3 3.3917 × 10−4 −9.7700 × 10−015 0.1970 × 10−9 0.725 2.5275188 −0.9487 × 10−3 0.6333 × 10−4 −2.6645 × 10−015 0.2347 × 10−9 0.901 1.4954122 2.0388 × 10−4 1.2746 × 10−4 −4.8850 × 10−015 0.1340 × 10−9 0.0471408 1.0831 × 10−4 0.3333 × 10−4 1.4225 × 10−015 3.5400 × 10−10 -0.9999 -0.901 -0.015 0.015 0.9999 So that c0 = 2, Q(x) = x 2 and L(x, t) = x + 1 1 + . From Equations (2.15) - (2.17) and x +2 t +2 (3.21) we obtain 1 1 x2 x2 (n) (n) x+ + g0,1 b0,1 + −2 + b1,1 + Un+1 (x)bn,1 2 x +2 2 2 n x 2 (n) (n) (n) b −2( j + 1)b(n) U = + + − b (x) + b g j j,1 j,1 j+1,1 j−1,1 j,1 2 j=1 where b−1,1 = bn+1,1 = 0 and g j,1 1 = π π f (x), 2 (3.22) 1 √ −1 1 − t2 U j (t)dt. t +2 We choose the collocation points xi as in Equation (2.35). Solving Equation (3.22) for the unknown coefficients b(n) j,1 for different values of n and substituting it into (2.15), we obtain the numerical solution of Equation (3.19). Errors of numerical solution of Equation (3.19) and comparisons with the method presented in Eshkuvatov [25] are given in Table 3. where n is a number of collocation points and m denotes the number of selection functions. Table 3 shows that when x comes close to the end points of the interval (−1, 1) or in the middle of the interval, errors decreases drastically and when n = 26 the errors reached to almost zero. It reveals that proposed method is suitable for HSIEs when solution is bounded. On the other hand proposed method is dominated over the method proposed in Eshkuvatov at al. [25]. In [25] n stands for number of nodes and m is for number of selection function. Example 4 (Mandal and Bera [22]). Let us consider the following HSIEs 1 π 1 −1 ϕ(t) 1 dt + (t − x)2 π 1 −1 (t + x) ϕ(t)dt = 1 + 2x, I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS (3.23) Vol. 10, No. 1 (Special Issue), Jan–June 2019 Projection Method and Error Bound for HSIEs 31 The exact solution of Equation (3.23) is 4 1 − x 2 (9 + 10x). 31 ϕ(x) = − (3.24) Solution: Comparing (3.23) with (2.2) we get c0 = 1, Q(x) = 0 L(x, t) = x + t. Equations (2.16) and (2.17) yields n # " b(n) j,1 −( j + 1)U j (x) + ψ j,1 (x) = j=0 π (1 + 2x), 2 (3.25) where 1 ψ j,1 (x) = π 1 (t + x) 1 − t 2 U j (t)dt. 1 Due to orthogonality condition (2.21), we obtain ψ0,1 (x) = x 1 , ψ1,1 (x) = ψ j,1 (x) = 0, j ≥ 2. 2 4 (3.26) Substituting (3.26) into (3.25) and choosing collocation points xi as given in Equation (??), the system of algebraic equations (3.25) has the form n xi 1 (n) b0,1 + b1,1 (−( j + 1))b(n) = j,1 U j (x i ) + 2 4 j=0 π (1 + 2xi ), i = 0, ..., n, 2 (3.27) Solving Equation (3.27) at the collocation points (2.35) for the different value of n, we obtain the numerical solution of Equation (3.23). The comparison errors of Equation (3.23) are summarized in Table 4. Table 4, reveals that proposed method (2.18) and Mandal’s method [22] are very accurate to this example for small value of n but Table 5 shows that CPU time of proposed method is much more less than Mandal’s method [22]. On the other hand for large value of n computational complexity of Mandal’s method is much more higher than the proposed method. On the other hand proposed method can be used for any value of n. In Table 4, we are able to compute for “n = {3, 7}” only. It can be shown that the method proposed here is exact for Example 4 with only n = 2. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 32 Z.K. Eshkuvatov and Anvar Narzullaev Table 4. Comparison results of Example 4 x Exact solution (3.24) Error of proposed method (2.15) Errors of Mandal and Bera [22] 0.0079350 1.73 × 10−018 5.55 × 10−017 -0.688 −0.19851699 1.11 × 10−016 −1.94 × 10−016 -0.118 −1.00198275 2.22 × 10−016 0.00 × 10+00 −1.30437141 2.22 × 10−016 0.00 × 10+00 −1.48700461 2.22 × 10−016 4.44 × 10−016 −0.15481294 2.27 × 10−017 8.33 × 10−017 0.0079350 6.94 × 10−018 3.09 × 10−016 -0.688 −0.19851699 5.55 × 10−017 3.61 × 10−016 -0.118 −1.00198275 2.22 × 10−016 0.00 × 10+00 −1.30437141 2.22 × 10−016 0.00 × 10+00 −1.48700461 0.00 × 10+00 −8.88 × 10−016 −0.15481294 2.27 × 10−017 −6.11 × 10−016 n=3 -0.998 0.118 0.688 0.998 n=7 -0.998 0.118 0.688 0.998 Table 5. CPU time (in seconds). Comparisons for Example 4 Number of points n CPU Prop. method (3.23) Mandal and Bera [22] 3 0.3121 0.6022 7 0.9409 1.5328 10 1.6526 20 5.204 30 11.3804 50 32.4334 3.2. Case 2 i = 2. Unbounded solution Example 5. Consider the following HSIEs with corresponding condition 1 π 1 π 1 −1 ϕ(t) 1 dt + (t − x)2 π 1 −1 16x 3 t 3 ϕ(t)dt = 1, (3.28) 1 3 ϕ(t)dt = . 2 −1 It is not hard to verify that the exact solution of Equation (3.28) is ϕ(x) = √ 1 1− x2 (1 + x 2 ). I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Projection Method and Error Bound for HSIEs 33 Solution: Approximate solution is searched as n 1 ϕ(x) = ϕn (x) = √ b(n) j,2 φ j,2 (x). 1 − x 2 j=0 (3.29) Since c0 = 1, Q(x) = 0 and L(x, t) = 16x 3 t 3 , from the Equation (2.27) it follows that n b(n) j,2 j=1 1 · 1 − xk2 j +1 j −1 (n) ψ0,2 (x) = U j−2 (xk ) − U j (xk ) + ψ j,2 (x) + b0,2 2 2 π , (3.30) 2 where U−1 (x) = 0 and 1 ψ j,2 (x) = π 1 −1 T j (t) (16x 3 t 3 ) √ dt, j = 0, 1, ... 1 − t2 It is easy to check that t3 = 1 (T3 (t) + 3T1 (x). 4 (3.31) Applying (3.31) and using orthogonality condition (2.22), we obtain ψ0,2 (x) = 0, ψ1,2 (x) = 6x 3 , ψ2,2 (x) = 0, ψ3,2 (x) = 2x 3 , ψ j,2 (x) = 0, j ≥ 4. With these values of ψ j,2 (x), Equation (3.30) with n = 3 has the form 3 j=1 b(n) j,2 1 · 1 − xk2 j +1 j −1 U j−2 (xk ) − U j (xk ) 2 2 + 6b1,2 x 3 + 2b3,2 x 3 = π , 2 (3.32) and taking into account Table 1, it follows that 6b1,2 x 3 + 2b2,2 + b3,2 (8x + 2x 3 ) = π . 2 (3.33) By equating the same powers of x in Equation (3.33) and take into account (2.26), we arrive at b0,2 = π3 , b1,2 = 0, b2,2 = 22 π1 , b3,2 = 0. 22 (3.34) Exact solution is achieved when we substitute Equation (3.34) into (3.29) i.e. 1 ϕn (x) = √ 1 − x2 $ 3 2 1 2 + π 2 2 T2 (x) π % I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS 1 π =√ (1 + x 2 ). 2 1 − x2 Vol. 10, No. 1 (Special Issue), Jan–June 2019 34 Z.K. Eshkuvatov and Anvar Narzullaev Example 6. Let HSIEs with corresponding condition be given by 1 π 1 π 1 −1 1 −1 (1 + 2(t − x)) 1 ϕ(t)dt + 2 (t − x) π 1 −1 1 2x 3 e t ϕ(t)dt = 4 + 8x, 2 (3.35) ϕ(t)dt = 1. The exact solution of Equation (3.35) is 1 (4x 2 − 1). ϕ(x) = √ 1 − x2 (3.36) Solution: Comparing Equation (3.35) with (2.2) we obtain c0 = 1, Q(x) = 2 and L(x, t) = Let approximate solution be searched as (3.29), then Equation (2.28) becomes n b(n) j,2 j=1 1 · 1 − x2 e2x t 3 . 2 j +1 j −1 U j−2 (x) − U j (x) + 2U j−1,1 (x) + ψ j,2 (x) 2 2 (3.37) +b0,2 ψ0,2 (x) = π (4 + 8x), 2 where ψ j,2 (x) = 1 π −1 1 e2x t 3 T j (t) dt, j = 0, 1, ... √ 2 1 − t2 Taking into account (3.31) and orthogonality conditions (2.22), we get ψ0,2 (x) = 0, ψ1,2 (x) = 3 2x 1 2x e , ψ2,2 (x) = 0, ψ3,2 (x) = e , ψ j,2 (x) = 0, j ≥ 4. 16 16 With these values of ψ j,2 (x), Equation (3.37) takes the form n j=1 b(n) j,2 1 · 1 − x2 j +1 j −1 U j−2 (x) − U j (x) + 2U j−1 (x) 2 2 3e2x e2x + b3,2 = +b1,2 16 16 π (4 + 8x), 2 (3.38) For n = 3 and applying Table 1, we arrive at (3) (3) (3) (3) 2b1,2 + b2,2 (2 + 4x) + b3,2 (8x + 2(4x 2 − 1)) + b1,2 I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS 2x 3e2x (3) e + b3,2 = 16 16 π (4 + 8x). (3.39) 2 Vol. 10, No. 1 (Special Issue), Jan–June 2019 Projection Method and Error Bound for HSIEs 35 Table 6. Numerical solution of Example 6 Exact of HSIEs (3.35) Errors Rn , n = 3 Errors Rn , n = 12 Errors Rn , n = 30 212.0807707 1.13687 × 10−13 2.2991 × 10−11 4.99057 × 10−11 1.600728589 8.88178 × 10−16 1.49089 × 10−14 6.02045 × 10−15 -0.436 −0.266255779 1.1657 × 10−15 1.42511 × 10−13 6.66389 × 10−13 -0.015 −0.999212418 3.33067 × 10−16 1.17653 × 10−13 7.672987 × 10−13 −0.999212418 5.55112 × 10−16 1.14435 × 10−13 8.02681 × 10−13 −0.266255779 7.77156 × 10−16 1.145391 × 10−13 2.04732 × 10−13 0.725 1.600728589 4.44089 × 10−16 1.08045 × 10−13 1.64476 × 10−13 0.9999 212.0807707 1.13687 × 10−13 5.60651 × 10−12 2.77061 × 10−11 x -0.999 -0.725 0.015 0.436 Equating like powers of x from both sides of Equation (3.39), we get π . 2 (3) (3) (3) b1,2 = b3,2 = 0, b2,2 =2 (3.40) (3) To find b0,2 we impose second condition of (3.35) to yield (3) = b0,2 π . 2 (3.41) Substitute Equations (3.40) - (3.41) into (3.29) ϕ(x) = √ 1 1− x2 (T0 (x) + 2T2 (x)) = √ 1 1 − x2 (4x 2 − 1). (3.42) which is identical with the exact solution (3.36). Let us solve the problem 6 numerically. To do end this we solve Equation (3.38) at the collocation points (2.36) and substitute it into (3.29). Results are summarized in Table 6. 4. CONCLUSIONS In this note, we have developed projection method for solving HSIEs of the first kind, where the main kernel K (x, t) is const on the diagonal of domain D = [−1, 1] × [−1, 1]. Collocation and Galerkin method are used to obtain a system of algebraic equations for the unknown coefficients. Stable computational scheme and high accurate Gauss-Chebyshev QF leads to high accurate and stable method. Developed method is suitable for both bounded and unbounded solutions of HSIEs. Examples 1 − 4 verify that the developed method is very accurate and stable for almost any kernel K (x, t) of HSIEs of the first kind. Especially, Example 3 shows that method converges very fast even if exact solution is in the rational form. Examples 5-6, show that developed method is exact whenever solution of HSIEs is the product of wight function and polynomials. Table 6 shows that approximate solution is stable for large value of n. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 36 Z.K. Eshkuvatov and Anvar Narzullaev ACKNOWLEDGEMENT This work was supported by Universiti Sains Islam Malaysia (USIM) under Research Management Center (RMC), Project code: PPP/USG-0216/FST/30/15316. Authors are grateful for sponsor and financial support of the RMC of USIM. REFERENCES [1] D. Berthold, W. Hoppe, B. Silbermann: A fast algorithm for solving the generalized airfoil equation. Volume 43, Issues 12, 25 November 1992, pp. 185–219. [2] A. Chakrabarti, G. Vanden Berge, Approximate Solution of Singular Integral Equations, Appl. Math. Lett., Vol. 17, 2004, pp. 553–559. [3] Maria Rosaria Capobianco, Giuliana Criscuolo, Peter Junghanns. 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(Uzbek) I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 INDIAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019, pp. 38–50 DOI: Discrete Legendre Projection Methods for the Fredholm Integral Equations of the Second Kind Jitendra Kumar Malik∗ and Bijaya Laxmi Panigrahi1 Department of Mathematics, Sambalpur University, Odisha-768019, India (∗ Corresponding author) Email: ∗ [email protected], 1 [email protected] Abstract: In this paper, we consider the discrete Galerkin and discrete collocation methods to solve the Fredholm integral equations of the second kind using Legendre polynomial bases. Using sufficiently accurate numerical quadrature rule, we obtain the error bounds for discrete Legendre solutions and iterated discrete Legendre solution for L 2 norm in both Legendre Galerkin and Legendre collocation method. We also obtain the superconvergence results for iterated Legendre Galerkin solution over Legendre Galerkin solution in L 2 -norm. Numerical example is presented to illustrate the theoretical results. Keywords: Discrete projection methods, Fredholm integral equations, Legendre polynomial bases. 2010 Mathematical Subject Classifications: 45B05, 65R20, 47G10. 1. INTRODUCTION Consider the following integral operator K defined on X = L 2 ([−1, 1]) or C([−1, 1]) by 1 Ku(s) = k(s, t)u(t) dt, s ∈ [−1, 1]. (1.1) −1 We are interested to solve the Fredholm integral equations of the second kind (I − K)u = f. (1.2) It is required to obtain approximate solutions because the above problem can not be solved exactly. There are so many different methods have been developed to find the approximate solutions of equations (1.2), see for example [6, 9, 10]. Spectral methods have developed rapidly in the past two decades and this method applied successfully in many fields. The Galerkin, collocation and their discretized versions are the commonly used spectral projection methods for finding numerical I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Discrete Legendre Projection Methods for the Fredholm Integral Equations of the Second Kind 39 solutions of various integral equations, see for example [3,4,7,12] and references therein. Legendre spectral approximation method for eigenvalue problem of a compact integral operator is developed in [13]. In this paper, we use discrete Legendre spectral projection methods to solve the Fredholm integral equations of the second kind and evaluate the error bounds for approximate solutions and the iterated solutions with the exact solutions. We show that the iterated solution converges faster than approximate solution in discrete Legendre Galerkin method. We organize this paper as follows. In Section 2, we discuss the discrete Legendre Galerkin and discrete Legendre collocation methods for Fredholm integral equations. In Section 3, we discuss the convergence rates for the projection (Galerkin and collocation) methods for Fredholm integral equations. In Section 4, we present numerical examples. 2. DISCRETE LEGENDRE PROJECTION METHODS Let L 2 be the space of complex valued square integrable functions on [−1, 1] with the inner product f, g = 1 −1 f (t) g(t)dt, f, g ∈ L 2 , 1 and norm f L 2 = f, f 2 . Let X = C[−1, 1] ⊂ L 2 [−1, 1] be the space of complex valued continuous functions on [−1, 1]. Consider the Fredholm integral equations of the second kind u(s) − 1 −1 k(s, t)u(t)dt = f (s), s ∈ [−1, 1], (2.1) where the kernel k(s, t) ∈ C([−1, 1] × [−1, 1]) and f are known functions and u is the unknown function in X to be determined. Let 1 Ku(s) = k(s, t)u(t) dt, s ∈ [−1, 1]. (2.2) −1 Then K is a compact linear integral operator on C[−1, 1] and L 2 [−1, 1]. Then the equation (2.1) can be written in an operator form as (I − K)u = f, (2.3) where K is a compact linear integral operator, f is a given function, u is an unknown to be determined and I be the identity operator defined on X. To describe the discrete Legendre projection methods for the solution of Fredholm integral equations of the second kind (2.3), we let Xn = span{φ0 , φ1 , · · · , φn } be the sequence of Legendre polynomial subspaces of X of degree ≤ n, where {φ0 , φ1 , · · · , φn } forms an orthonormal basis for Xn . The φi ’s are given by 2i + 1 L i (s), i = 0, 1, . . . , n, φi (s) = 2 I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 40 Jitendra Kumar Malik and Bijaya Laxmi Panigrahi where L i ’s are the Legendre polynomials of degree ≤ i. The Legendre polynomials can be generated by the following recurrence relation L 0 (s) = 1, L 1 (s) = s, s ∈ [−1, 1], and for i = 1, 2, · · · , n − 1, (i + 1)L i+1 (s) = (2i + 1)s L i (s) − i L i−1 (s), s ∈ [−1, 1]. Since φi and φ j ’s are polynomials, note that φi , φ j = 1 −1 φi (t)φ j (t)dt = 1 −1 φi (t)φ j (t)dt = δi, j , (2.4) for i, j = 0, 1, . . . , n. Now we will discuss on discrete methods. We choose a numerical integration scheme 1 f (t)dt −1 M(n) w p f (t p ), (2.5) p=1 where M(n) is a constant depend upon n, and (i) w p are the weights such that w p > 0, p = 1, 2, · · · , M(n). (2.6) (ii) the above quadrature rule has degree of precision d, which is atleast 2n, that is 1 −1 f (t)dt = M(n) w p f (t p ) (2.7) p=1 for all polynomial of degree ≤ 2n ≤ d. For the notational convenience, from now on we set M(n) = M. Using the above quadrature rule (2.5), we define the discrete inner product f, g M = M w p f (t p )g(t p ), f, g ∈ C[−1, 1]. (2.8) p=1 For the approximation of K, using the rule (2.5), the Nyström operator Kn is defined by (Kn u)(s) = M w p k(s, t p )u(t p ). (2.9) p=1 I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Discrete Legendre Projection Methods for the Fredholm Integral Equations of the Second Kind 41 Now, we set the following notations. Let C r [−1, 1] denote the space of r -times continuously differentiable complex valued function on [−1, 1]. For u ∈ C r [−1, 1], let ur,∞ = max{u (i) ∞ : 1 ≤ i ≤ r }, where u (i) denote the i-th derivative of u. Assume k(., .) ∈ C d [−1, 1] × [−1, 1], where d is the degree of precision of the numerical quadrature rule and d ≥ 2n > n ≥ r > 1. For fixed s ∈ [−1, 1], we denote ks (t) = k(s, t). Since w j > 0 and 2= 1 −1 ds = M wi , (2.10) i=1 it follows that for j = 0, 1, . . . , d, M ∂j (Kn u)( j) ∞ = sup |(Kn u)( j) (s)| = sup w p j ks (t p )u(t p ) ≤ 2u∞ k j,∞ , ∂s s∈[−1,1] s∈[−1,1] p=1 where k j,∞ = ∂ i+l i l ks (t). Then ∂s ∂t s,t∈[−1,1],0≤i,l≤ j sup Kn ud,∞ = max{(Kn u)( j) ∞ : 0 ≤ j ≤ d} ≤ ckd,∞ u∞ , j (2.11) where c is a constant independent of n. Also, for j = 0, 1, . . . , d, we have (Ku)( j) ∞ = sup |(Ku)( j) (s)| = sup s∈[−1,1] s∈[−1,1] 1 −1 ∂j k (t)u(t)dt ≤ 2u∞ k j,∞ . s ∂s j Thus, Kud,∞ ≤ cu∞ kd,∞ , (2.12) where c is a constant independent of n. We quote the following theorem, which gives the error bound of Nyström operator (2.9) with the operator K (2.2). Theorem 2.1. ( [7]) Let k(., .) ∈ C d [−1, 1] × [−1, 1], then for any u ∈ C d [−1, 1], we have (Kn − K)u∞ ≤ cn −d kd,∞ ud,∞ , (2.13) where c is a constant independent of n. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 42 Jitendra Kumar Malik and Bijaya Laxmi Panigrahi Lemma 2.2. Let Kn be the Nyström operator defined by (2.9). Assume that k(., .) ∈ C d ([−1, 1] × [−1, 1]) and d ≥ 2n > n ≥ r > 1, then (K − Kn )K∞ = O(n −d ). Proof. By replacing u with Ku in equation (2.13) and using the estimate (2.12), we obtain (K − Kn )Ku∞ ≤ cn −d kd,∞ Kud,∞ ≤ cn −d k2d,∞ u∞ , (2.14) where c is a constant independent of n. This completes the proof. Lemma 2.3. ([2]) Let X be a Banach space and S ⊂ X is a relatively compact set. Assume that T and Tn are bounded linear operators from X into X satisfying Tn ≤ c, for all n ∈ N, and for each x ∈ S, Tn − T → 0 as n → ∞, where c is a constant independent of n. Then Tn − T → 0 uniformly for all x ∈ S. Discrete Legendre orthogonal projection operator: To discuss on the discrete Legendre Galerkin methods, we need to introduce the discrete orthogonal projection operator. Discrete orthogonal projection namely hyper interpolation operator QnG : X → Xn (Sloan [14]) is defined by QnG u = n u, φ j M φ j , u ∈ X, (2.15) j=0 for j = 0, 1, . . . , n and QnG satisfy QnG u, φ M = u, φ M , for all φ ∈ Xn . Now we quote some properties of QnG from ( [7, 14]). Lemma 2.4. Let QnG : X → Xn , be the hyperinterpolation operator defined as above. Then the following results hold (i) For any u ∈ X, √ QnG u L 2 ≤ 2u∞ , (2.16) and √ QnG u − u L 2 ≤ 2 2 inf u − u n ∞ → 0, as n → ∞. u n ∈Xn (ii) In particular, for u ∈ C ([−1, 1]), r QnG u − u L 2 ≤ cn −r u (r ) ∞ , (2.17) where c is a constant independent of n and n ≥ r . I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Discrete Legendre Projection Methods for the Fredholm Integral Equations of the Second Kind 43 Note that for any u ∈ C r [−1, 1], using Jackson’s theorem ( [?]) and the estimate (2.10), we get 1 1 2 2 u − QnG u, u − QnG u M = min u − χ , u − χ M χ∈Xn = = min M χ∈Xn M 12 i=1 wi 1 2 i=1 ≤ wi (u − χ )2 (ti ) inf u − χ ∞ χ∈Xn √ 2cn −r ur,∞ , (2.18) where c is a constant independent of n and n ≥ r . Using orthogonal projection QnG , the discrete Legendre Galerkin method for the Fredholm integral equations of the second kind is u nG − QnG Kn u nG = QnG f, (2.19) where u nG ∈ Xn is the approximation of u. We define the iterated solution as ũ nG = Kn u nG + f . Discrete Legendre interpolatory projection operator: Let {τ0 , τ1 , . . . , τn } be the zeros of the Legendre polynomial of degree n + 1 and define the interpolatory projection QCn : X → Xn by QCn u ∈ Xn , QCn u(τi ) = u(τi ), i = 0, 1, . . . , n, u ∈ X. (2.20) Lemma 2.5. ( [4, 8]) Let QCn : X → Xn be the interpolatory projection operator defined by (2.20). Then the following conditions hold: (i) QCn u L 2 ≤ cu∞ , u ∈ C[−1, 1], where c is a constant independent of n. (ii) There exists a constant c > 0 such that for any u ∈ X, u − QCn u L 2 ≤ c inf u − φ∞ → 0, as n → ∞. φ∈Xn (iii) For any u ∈ C r ([−1, 1]), there exists a constant c independent of n such that u − QCn u L 2 ≤ cn −r u (r ) ∞ . (2.21) Using interpolatory projection QCn , the discrete Legendre collocation method for the Fredholm integral equations of the second kind is u Cn − QCn Kn u Cn = QCn f, (2.22) where u Cn ∈ Xn is the approximation of u. We define the iterated solution as ũ Cn = Kn u Cn + f . Remark 2.6. If M = n + 1 and the quadrature points used in the discrete inner product (2.8) and the collocation nodes in (2.20) are the same, the discrete orthogonal projection operator QnG reduces to the interpolatory projection operator, i.e., in such case QnG and QCn are the same. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 44 Jitendra Kumar Malik and Bijaya Laxmi Panigrahi 3. CONVERGENCE RATES In this section, we will discuss on the convergence rates of the approximate and iterated approximate solutions with the exact solution for the Fredholm integral equations of the second kind using discrete Legendre Galerkin and discrete Legendre collocation methods. At first, we will evaluate the convergence rates by using discrete Legendre Galerkin methods. 3.1. Discrete Legendre Galerkin Method: The discrete Legendre Galerkin method for the Fredholm integral equations of the second kind is u nG − QnG Kn u nG = QnG f. First, we will prove that the operator QnG Kn is ν-convergent to K in L 2 -norm, then we will analyze the existence and error bounds of the approximate and iterated approximate solutions by using discrete Legendre Galerkin methods. Theorem 3.1. QnG Kn is ν-convergent to K in L 2 -norm. Proof. Consider QnG Kn L 2 ≤ p1 Kn ∞ , where p1 is a constant independent of n. From Theorem-2.1, we see that {Kn } converges to K pointwise. Hence, {Kn } is pointwise bounded and since X is Banach space, by Uniform Boundedness principle we have {Kn } is uniformly bounded, i.e., Kn ∞ ≤ p2 , where p2 is a constant independent of n. This shows that QnG Kn L 2 is uniformly bounded. By using the estimate (2.17), we obtain (QnG − I)Ku L 2 ≤ cn −r (Ku)(r ) ∞ ≤ cn −r kr,∞ u∞ , (3.1) where c is a constant independent of n. Next consider |(QnG Kn − K)u(t)| = |(QnG Kn − QnG K + QnG K − K)u(t)| ≤ |QnG (Kn − K)u(t)| + |(QnG − I)Ku(t)|. Now using the estimates (2.16), (3.1) and (2.13), we see (QnG Kn − K)u L 2 ≤ QnG (Kn − K)u L 2 + (QnG − I)Ku L 2 ≤ c(Kn − K)u∞ + (QnG − I)Ku L 2 ≤ cn −r kr,∞ ur,∞ + cn −r kr,∞ u∞ → 0, (3.2) as n → ∞ and c is a constant independent of n. This gives that QnG Kn is pointwise converges to K. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Discrete Legendre Projection Methods for the Fredholm Integral Equations of the Second Kind 45 Let B = {u ∈ X : u ≤ 1} be a closed unit ball in C([−1, 1]). Since K is compact operator, the set S = {Ku : u ∈ B} is a relatively compact set in C([−1, 1]). Then by Lemma-2.3, we have (QnG Kn − K)K L 2 = sup{(QnG Kn − K)Ku L 2 : u ∈ B} = sup{(QnG Kn − K)u L 2 : u ∈ S} → 0 as n → ∞. Since QnG is uniformly bounded in L 2 norm over Kn u, u ∈ B and Kn is compact, S = {QnG Kn u : u ∈ B} is a relatively compact set. Thus (QnG Kn − K)QnG Kn L 2 = sup{(QnG Kn − K)QnG Kn u L 2 : u ∈ B} = sup{(QnG Kn − K)u L 2 : u ∈ S} → 0 as n → ∞. Combining all these results leads to the first result that QnG Kn is ν-convergent to K in L 2 -norm. This completes the proof. Since QnG Kn is ν-convergent to K, for all small n, the spectrum of QnG Kn inside consists of G G G , λn,2 , · · · , λn,m counted accordingly to their algebraic multiplicities inside m eigenvalues, say λn,1 . Let λnG = G G G λn,1 + λn,2 + · · · + λn,m m denote their arithmetic mean and we approximate λ by λnG . Let P S and PnS be the spectral projections of K and QnG Kn , respectively, associated with their corresponding spectral inside . Lemma 3.2. Assume (I − K)−1 exists on X. If the approximate operator QnG Kn is ν-convergent to K in L 2 -norm, then there is a positive integer N such that for all n ≥ N , the inverse (I − QnG Kn )−1 exists as linear operator defined on X and there exist positive constants c1 independent of n such that for all n ≥ N , (I − QnG Kn )−1 L 2 ≤ c1 . Proof. The proof is very obvious from [1] with Theorem-3.1. Theorem 3.3. Let k(., .) ∈ C d [−1, 1], d ≥ 2n > n ≥ r > 1 and u ∈ X be the exact solution of equation (2.3). Let u nG be the discrete Legendre Galerkin approximation of u defined by (2.19). Then the following holds u − u nG L 2 = O(n −r ). Proof. From the equation (2.3) and (2.19), we obtain u nG − u = (I − QnG Kn )−1 Qn f − (I − K)−1 f = (I − QnG Kn )−1 [QnG − QnG K − I + QnG Kn ]u. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS (3.3) Vol. 10, No. 1 (Special Issue), Jan–June 2019 46 Jitendra Kumar Malik and Bijaya Laxmi Panigrahi Then by taking the norm on both sides and using the last Theorem with the estimates (2.17), (2.16) and (2.13) we obtain u nG − u L 2 ≤ ≤ c1 (QnG − I)u L 2 + QnG (Kn − K)u L 2 ) √ cn −r u (r ) ∞ + c1 2(Kn − K)u∞ ≤ cn −r u (r ) ∞ + cn −d kd,∞ ud,∞ , where c and c1 are the constants independent of n. Thus, u nG − u L 2 = O(n − min{r,d} ) = O(n −r ). This completes the proof. Theorem 3.4. Let k(., .) ∈ C d [−1, 1], d ≥ 2n > n ≥ r > 1 and ũ nG = Ku nG + f be the iterated discrete Legendre Galerkin approximation of u. Then the following holds u − ũ nG L 2 = O(n −2r ). Proof. We have ũ nG = Kn u nG + f . It follows that QnG ũ nG = u nG . Now by using the estimates (2.3) and (2.13), we get u − ũ nG L 2 = = Ku + f − ( f + Kn QnG ũ nG ) L 2 Ku − Kn QnG ũ nG L 2 ≤ (K − Kn )u L 2 + Kn u − Kn QnG ũ nG L 2 ≤ cn −d kd,∞ ud,∞ + Kn (u − u nG ) L 2 , (3.4) where c is a constant independent of n. By using the estimate (3.8) and the Lemma-3.2, we obtain Kn (u − u nG ) L 2 = Kn (I − QnG Kn )−1 (QnG − QnG K − I + QnG Kn )u L 2 ≤ (I − QnG Kn )−1 L 2 Kn (QnG − QnG K − I + QnG Kn )u L 2 ≤ c1 Kn (QnG − I)u L 2 + c1 Kn (QnG Kn − QnG K)u L 2 , (3.5) where c1 is a constant independent of n. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Discrete Legendre Projection Methods for the Fredholm Integral Equations of the Second Kind 47 Now the first term in the right hand side of the above equation, we use the orthogonality of QnG |Kn (QnG − I)u(s)| = M w p k(s, t p )(QnG − I)u(t p ) p=1 = (QnG − I)u, ks M = (QnG − I)u, (QnG − I)ks M M = w p (QnG − I)u(t p )(QnG − I)ks (t p ) p=1 Set ks = ls . Now by using Cauchy Schwarz inequality, we obtain |Kn (QnG − I)u(s)| ≤ M w p |(QnG − I)ls (t p )|2 M 1/2 p=1 ≤ ≤ w p |(QnG − I)u(t p )|2 1/2 p=1 1/2 (QnG − I)ls , (QnG − I)ls M (QnG cn −2r ls(r ) ∞ u (r ) ∞ , 1/2 − I)u, (QnG − I)u M where c is a constant independent of n. Thus, Kn (QnG − I)u L 2 ≤ cn −2r kr,∞ u (r ) ∞ . (3.6) Now the second term of the right hand side of (3.5), |Kn (QnG Kn − QnG K)u(s)| = M w p k(s, t p )QnG (Kn − K)u(t p ) p=1 ≤ QnG (Kn − K)u L 2 M w 2p |k(s, t p )|2 1/2 p=1 ≤ c(Kn − K)u∞ ks ∞ ≤ cn −d ks ∞ kd,∞ ud,∞ , where c is a constant independent of n. Thus, Kn (QnG Kn − QnG K)u L 2 ≤ cn −d kr,∞ kd,∞ ud,∞ . (3.7) Now by using equation (3.6), (3.7) and (3.5) in (3.4), we obtain u − ũ nG L 2 = O(n −min{d,2r } ) = O(n −2r ). This completes the proof. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 48 Jitendra Kumar Malik and Bijaya Laxmi Panigrahi 3.2. Discrete Legendre collocation methods: The discrete Legendre collocation methods for the Fredholm integral equations of the second kind is u Cn − QCn Kn u Cn = QCn f. First, we will prove that the operator QCn Kn is ν-convergent to K in L 2 -norm, then we will analyze the existence and error bounds of the approximate and iterated approximate solutions by using discrete Legendre collocation methods. Theorem 3.5. QCn Kn is ν-convergent to K in L 2 -norm. Proof. The proof is similar to the proof of Theorem-3.1. So, we omit it. Since QCn Kn is ν-convergent to K, for all small n, in L 2 -norm, the spectrum of QCn Kn inside consists of m eigenvalues say λCn,1 , λCn,2 , . . . , λCn,m counted accordingly to their algebraic multiplicities inside ( Chatelin [5], Osborn [11]). Let λ̂Cn = λCn,1 + λCn,2 + · · · + λCn,m m , denote their arithmetic mean and we approximate λ by λ̂Cn . Let PnC be the spectral projections of QCn Kn , associated with their corresponding spectral inside . Lemma 3.6. Assume (I − K)−1 exists on X. If the approximate operator QCn Kn is ν-convergent to K, then there is a positive integer N such that for all n ≥ N , the inverse (I − QCn Kn )−1 exists as linear operator defined on X and there exist positive constants c2 independent of n such that for all n ≥ N , (I − QCn Kn )−1 L 2 ≤ c2 . Proof. The proof is very obvious from [1] with Theorem-3.5. Theorem 3.7. Let k(., .) ∈ C d [−1, 1] and u be exact solution of equation (2.3). Let u Cn be the discrete Legendre collocation approximation of u defined by (2.19). Then the following holds u − u Cn L 2 = O(n −r ). Proof. From the equation (2.3) and (2.19), we obtain u Cn − u = (I − QCn Kn )−1 Qn f − (I − K)−1 f = (I − QCn Kn )−1 [QCn − QCn K − I + QCn Kn ]u. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS (3.8) Vol. 10, No. 1 (Special Issue), Jan–June 2019 Discrete Legendre Projection Methods for the Fredholm Integral Equations of the Second Kind 49 Table 1. n u − unG L 2 u − ũ nG L 2 u − unC L 2 2 3 4 3.201944e-03 1.230043e-05 4.512378e-06 4.716913e-05 2.145267e-07 1.237801e-09 3.918328e-03 8.281007e-05 2.178939e-07 5 6 9.120781e-07 1.456890e-08 2.267891e-10 8.776551e-12 3.456712e-08 2.345689e-09 Then by taking the norm on both sides and using the last Theorem with the estimates (2.17), (2.16) and (2.13) we obtain u Cn − u L 2 ≤ c1 (QCn − I)u L 2 + QCn (Kn − K)u L 2 ) √ ≤ cn −r u (r ) ∞ + c1 2(Kn − K)u∞ ≤ cn −r u (r ) ∞ + cn −d kd,∞ ud,∞ where c1 and c are the constants independent of n. Thus, u Cn − u L 2 = O(n − min{r,d} ) = O(n −r ). This completes the proof. Remark 3.8. By using a sufficiently accurate numerical quadrature rule, we obtained that the discrete Legendre Galerkin and iterated discrete Legendre Galerkin solution converges with the orders O(n −r ) and O(n −2r ), respectively in L 2 norm, where r being the smoothness of the solution and d ≥ 2n be the degree of the precision of the numerical quadrature. Also, we obtained that the discrete Legendre collocation solution converges with the order O(n −r ) in L 2 norm. 4. NUMERICAL RESULT In this section, we present a numerical example. Choose the approximating subspace Xn to be the Legendre polynomial subspaces of degree less than equal to n. In Table 1, we present the errors for discrete Legendre Galerkin, iterated discrete Legendre Galerkin and discrete Legendre collocation solutions with exact solution for Fredholm integral equations of the second kind (2.3) in L 2 -norm. We denote u nG , ũ nG and u Cn be the discrete Legendre Galerkin, iterated discrete Legendre Galerkin and discrete Legendre collocation solution, respectively. For different values of n, we compute u nG , ũ nG , u Cn . The computed errors in L 2 -norm are presented in the following Table. Example 4.1. We consider the following integral equation 1 k(s, t)u(t)dt = f (s), s ∈ [−1, 1], u(s) − −1 I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 50 Jitendra Kumar Malik and Bijaya Laxmi Panigrahi with the kernel k(s, t) = st and the function f (s) = s, where the exact solution is given by u(s) = 3s. ACKNOWLEDGEMENT The second author is supported by a research grant from Science and Engineering Research Board, DST, India-SR/FTP/MS-010/2012. REFERENCES [1] M. Ahues, A. Largillier and B. V. Limaye, Spectral computations for bounded operators, Chapman and Hall/CRC, New York, 2001. [2] K. E. Atkinson, The Numerical solution of Integral Equations of the Second Kind. Cambridge University Press,,Cambridge, 1997. [3] Guo Ben-yu, Spectral methods and their applications, World Scientific, 1998. [4] C. Canuto and M. Y. Hussaini and A. Quarteroni and T.A. Zang, Spectral methods, Springer-Verlag Berlin Heidelberg, 2006. [5] F. Chatelin, Spectral approximation of linear operators, Academic Press, New York, 1983. [6] Z. Chen, G. Long and G. Nelakanti, The discrete multi-projection method for Fredholm integral equations of the second kind, J. Integral Equations and Appl., 19 (2007), 143–162. [7] P. Das, G. Nelakanti and G. Long. Discrete Legendre spectral projection methods for Fredholm-Hammerstein integral equations. J. Comp. Appl. Math., 278 (2015) 293–305. [8] M. A. Golberg, C.S. Chen, Discrete projection methods for integral equations, Computational Mechanics Publications, Southampton (1997). [9] R.P. Kulkarni, G. Nelakanti, Spectral approximation using iterated discrete Galerkin method, Numer. Funct. Anal. Optim., 23 (2002) 91–104. [10] G. Long, G. Nelakanti, B.L. Panigrahi and M.M. Sahani, Discrete multi-projection methods for eigen-problems of compact integral operators, Appl. Math. Comput., 217 (2010), 3974–3984. [11] J. E. Osborn, Spectral approximation for compact operators, Math. Comp., 29 (1975), 712–725. [12] B. L. Panigrahi and G. Nelakanti, Legendre multi-projection methods for solving eigenvalue problems for a compact integral operator, J. Comput. Appl. Math., 239 (2013), 135–151. [13] B. L. Panigrahi and G. Nelakanti, Superconvergence of Legendre projection methods for the eigenvalue problem of a compact integral operator, J. Comput. Appl. Math., 235 (2011), 2380–2391. [14] I. H. Sloan, Polynomial interpolation and hyperinterpolation over general regions, Journal of Approximation Theory, 83 (2) (1995), 238–254. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 INDIAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019, pp. 51–58 DOI: Some Curvature Conditions on Nearly Cosymplectic Manifolds Gülhan Ayar1∗ , Pelin Tekin2 and Nesip Aktan3 1 Karamanoğlu Mehmetbey University,Kamil Ö zdağ Science Faculty, Department of Mathematics Trakya University, Faculty of Science, Department of Mathematics, Edirne/TURKEY 3 Necmettin Erbakan University, Faculty of Sciences, Department of Mathematics-Computer Sciences, Konya/TURKEY (∗ Corresponding author) Email: ∗ [email protected], 2 [email protected], 3 [email protected] 2 Abstract: The aim of this study is to show that η−Einstein nearly cosymplectic manifolds are constant curvature manifolds. Also in this work, we present a complete connected nearly cosymplectic manifold is M−projectively flat if and only if it is either isometric to the sphere S n or the real projective space. 2000 Mathematics Subject Classiffication. 53C15, 53D15, 53C25. Keywords: Contact Manifolds, Cosymplectic Manifolds, Nearly Cosymplectic manifolds 1. INTRODUCTION Cosymplectic manifold is an odd dimensional counterpart of a Kähler manifold which is defined by Lipperman 1959 [12] and Blair 1967 [5]. An almost contact metric structure (ϕ, ξ, η, g) is said to be cosymplectic if it is normal and both and η are closed [2]. In 1971, Blair [1] defined an almost contact manifolds with Killing structure tensors as a nearly cosymplectic manifold. The other study of Blair with Showers in 1972, was on the topological aspect of this manifolds. Nearly Kähler manifolds were defined by Gray [9] as almost Hermitian manifolds (M, J, g) such that the covariant derivative of the almost complex structure with respect to the Levi-Civita connection is skew-symmetric, that is (∇ X J )X = 0, I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 52 Gülhan Ayar, Pelin Tekin and Nesip Aktan for every vector field X on M. Notice that in the defining condition of a nearly Kähler manifold, only the symmetric part of ∇ J vanishes, in contrast to the Kähler case where ∇ J = 0. Nearly cosymplectic manifolds were defined in the same spirit starting from cosymplectic or sometimes called coKähler manifolds. A smooth manifold M endowed with an almost contact metric structure (ϕ, ξ, η, g) is said to be nearly cosymplectic if (∇ X ϕ)X = 0, (1.1) for every vector field X on M [7]. Recently, nearly Sasakian and nearly cosymplectic manifolds was studied by CappellettiMontano, B., Dileo, G. in [6]. In that paper, it is proved for five-dimensional nearly cosymplectic manifolds that any such manifold is Einstein with positive scalar curvature. It is also worth remarking that (1-parameter families of) examples of both nearly Sasakian and nearly cosymplectic structures are provided by every 5-dimensional manifold endowed with a Sasaki-Einstein SU (2)structure. In the light of literature studies, in this paper, after given some basic concepts we show that an η−Einstein nearly cosymplectic manifolds are constant curvature manifolds and a nearly cosymplectic manifold M n is M−projectively flat if and only if it is either locally isometric to the elliptic space or has constant scalar curvature. Also we show that a complete connected nearly cosymplectic manifold is M−projectively flat if and only if it is either isometric to the sphere S n or the real projective space. 2. PRELIMINARIES Let (M, ϕ, ξ, η, g) be an n = (2m + 1)- dimensional almost contact Riemannian manifold, where ϕ is a (1, 1)-tensor field, ξ is the structure vector field, η is a 1-form and g is the Riemannian metric. It is well known that the (ϕ, ξ, η, g)-structure satisfies the conditions [4] ϕξ = 0, η(ϕ X ) = 0, ϕ 2 X = −X + η(X )ξ, η(ξ ) = 1, (2.1) η(X ) = g(X, ξ ), (2.2) g(ϕ X, ϕY ) = g(X, Y ) − η(X )η(Y ), (2.3) for any vector fields X and Y on M. From the above definition, ϕ is skew-symmetric with respect to g, so that the bilinear form := g(., ϕ.) defines a 2-form on M, called fundamental 2-form. An almost contact metric manifold such that dη = 2 is called a contact metric manifold. In this case η is a contact form, i.e., η ∧ (dη)n = 0 everywhere on M [7]. A nearly cosymplectic manifold is an almost contact metric manifold (M, ϕ, ξ, η, g) such that (∇ X ϕ)Y + (∇Y ϕ)X = 0, (2.4) for all vector fields X, Y on M [7]. Clearly, this condition is equivalent to (∇ X ϕ)X = 0. It is known that in a nearly cosymplectic manifold the Reeb vector field ξ is Killing and satisfies ∇ξ ξ = 0 and I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Some Curvature Conditions on Nearly Cosymplectic Manifolds 53 ∇ξ η = 0. The tensor field h of type (1, 1) defined by ∇X ξ = h X (2.5) is skew symmetric and anti-commutes with ϕ. It satisfies hξ = 0, η ◦ h = 0 and ∇ξ ϕ = ϕh = 1 Lξ ϕ. 3 Also the following formulas hold [7, 8]: g((∇ X ϕ)Y, h Z ) = η(Y )g(h 2 X, ϕ Z ) − η(X )g(h 2 Y, ϕ Z ), (2.6) (∇ X h)Y = g(h 2 X, Y )ξ − η(Y )h 2 X, (2.7) tr (h 2 ) = constant, (2.8) R(Y, Z )ξ = η(Y )h 2 Z − η(Z )h 2 Y, (2.9) S(ξ, Z ) = −η(Z )tr (h 2 ), (2.10) S(ϕY, Z ) = S(Y, ϕ Z ), ϕ Q = Qϕ, (2.11) S(ϕY, ϕ Z ) = S(Y, Z ) + η(Y )η(Z )tr (h 2 ), (2.12) g((∇ X ϕ)ϕY, Z ) = g(∇ X ϕ)Y, ϕ Z ) + η(Z )g(h X, Y ) + η(Y )g(h X, Z ), (2.13) g(∇ϕ X ϕ)Y, Z ) = g(∇ X ϕ)Y, ϕ Z ) + η(Z )g(h X, Y ) + η(X )g(h Z , Y ), (2.14) g(∇ϕ X ϕ)ϕY, Z ) = −g(∇ X ϕ)Y, Z ) + η(X )g(h Z , ϕY ) + η(Y )g(h X, ϕ Z ). (2.15) 3. η−EINSTEIN NEARLY COSYMPLECTIC MANIFOLDS Definition 1. Let M be a nearly cosymplectic manifold, for every X, Y ∈ χ (M). If M satisfies the condition S(X, Y ) = ag(X, Y ) + bη(X )η(Y ) (3.1) then M is an η−Einstein manifold where a, b : M −→ R is a function. Proposition 1. Let M be an n dimensional nearly cosymplectic manifold. If M is an η−Einstein manifold then a + b = −tr (h 2 ). Proof. From (3.1) and (2.4), S(ξ, ξ ) = a + b = −tr (h 2 ) = constant can be obtained easily. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 54 Gülhan Ayar, Pelin Tekin and Nesip Aktan Proposition 2. Let M be a nearly cosymplectic manifold of dimension n. If M is η−Einstein manifold, then a and b are constant. Proof. M is a n dimensional η−Einstein nearly cosymplectic manifold and let {ei }, i = 1, 2, . . . , n be orthonormal basis on every point of the tangent space. If we write X = Y = ei in (3.1) for 1 ≤ i ≤ n, we get n S(ei , ei ) = a i=1 n g(ei , ei ) + b i=1 n η(ei )η(ei ) i=1 and scalar curvature r as follows, r a+b = (2m + 1)a + b = −tr (h 2 ) r = 2ma − tr (h 2 ). (3.2) If we take the derivative of r, ∇Y r = 2 = 2 = 2 = 2 g((∇ei Q)Y, ei ) g(∇ei QY − Q∇ei Y, ei ) g(∇ei (aY + bη(Y )ξ ) − a∇ei Y − bη(∇ei Y )ξ, ei ) g(ei (a)Y + a∇ei Y + ei (b)η(Y )ξ + bη(∇ei Y )ξ + bg(Y, ∇ei ξ )ξ = +bη(Y )∇ei ξ − a∇ei Y − bη(∇ei Y )ξ, ei ) 2 g(ei (a)Y + ei (b)η(Y )ξ + bg(Y, hei )ξ + bη(Y )hei , ei ) 2 ei (a)g(Y, ei ) + ei (b)η(Y )η(ei ) + bg(hei , Y )η(ei ) + bη(Y )g(hei , ei ) 2 g(∇a, ei )g(Y, ei ) + ξ (b)η(Y ) − bg(hY, ξ ) + bη(Y )tr (h 2 ) = 2Y (a) + 2ξ (b)η(Y ). = = Finally writing this equation as 2mY (a) = 2Y (a) + 2ξ (b)η(Y ), let us take Y = ξ then, mξ (a) = (m − 1)ξ (a) = =⇒ ξ (a) + ξ (b) ξ (b) ξ (a) = 0, ξ (b) = 0. If we consider a + b = −tr (h 2 ), a= r + tr (h 2 ) , n−1 b= −tr (h 2 )n + r n−1 (3.3) so we have a and b are constant. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Some Curvature Conditions on Nearly Cosymplectic Manifolds 55 Corollary 1. Let M be a n dimensional η−Einstein nearly cosymplectic manifold. The condition η−Einstein for nearly cosymplectic manifolds are characterized by the equality r + tr (h 2 ) −n tr (h 2 ) + r S(X, Y ) = g(X, Y ) + η(X )η(Y ). n−1 n−1 (3.4) Proof. The proof is clear from (3.1) and (3.3) Theorem 1. Let M be a n dimensional η−Einstein nearly cosymplectic manifold. The Ricci tensor of M is η−parallel if and only if M is a constant curvature manifold. Proof. By using (3.4) and (2.3), we obtain (∇U S)(Y, Z ) = dr (U ) −n tr (h 2 ) + r [g(ϕY, ϕ Z )] + [g(Y, hU )η(Z ) + η(Y )g(z, hU )] n−1 n−1 and then (∇U S)(ϕY, ϕ Z ) = dr (U ) [g(ϕY, ϕ Z )] . n−1 Hence we have desired result. 4. THE M-PROJECTIVE CURVATURE TENSOR In 1971, Pokhariyal and Mishra [13] defined a tensor field W * on a m-dimensional Riemannian manifold as W * (X, Y )Z = R(X, Y )Z − 1 [S(Y, Z )X − S(X, Z )Y + g(Y, Z )Q X − g(X, Z )QY ] 2(m − 1) (4.1) so that W * (X, Y, Z , U )de f = g(W * (X, Y )Z , U ) = W * (Z , U, X, Y ) (4.2) Wi*jkl wi j w kl = Wi jkl wi j w kl (4.3) and where Wi*jkl and Wi jkl are components of W * and W respectively and wkl is a skew-symmetric tensor [14, 18]. Such a tensor field W * is known as M−projective curvature tensor. The properties of M−projective curvature tensor in Sasakian and Kähler manifolds are defined and studied in [14,15]. He has also shown that it bridges the gap between conformal curvature tensor, conharmonic curvature tensor and concircular curvature tensor on one side and H −projective curvature tensor on the other. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 56 Gülhan Ayar, Pelin Tekin and Nesip Aktan The Weyl projective curvature tensor W , concircular curvature tensor C and conformal curvature tensor V are given by [16] 1 {S(Y, Z )X − S(X, Z )Y }, m−1 (4.4) r {g(Y, Z )X − g(X, Z )Y } m(m − 1) (4.5) W (X, Y )Z = R(X, Y )Z − on the M-projective curvature tensor C(X, Y )Z = R(X, Y )Z − and 1 V (X, Y )Z = R(X, Y )Z − (m−2) [S(Y, Z )X − S(X, Z )Y + g(Y, Z )Q X − g(X, Z )QY ] r + (m−1)(m−2) {g(Y, Z )X − g(X, Z )Y }. (4.6) Following theorems and corollary are given in [20]. Theorem 2. The M−projective and Weyl projective curvature tensors of the Riemannian manifold M are linearly dependent if and only if M is an Einstein manifold. Theorem 3. The necessary and sufficient condition for a Riemannian manifold to be an Einstein manifold is that the M−projective curvature tensor W * and concircular curvature tensor C are linearly dependent. Theorem 4. A Riemannian manifold becomes an Einstein manifold if and only if conformal and M−projective curvature tensors of the manifold are linearly dependent. Corollary 2. In a Riemannian manifold M, the followings are equivalent i) M is an Einstein manifold ii) M-projective and Weyl projective curvature tensors are linearly dependent iii) M-projective and concircular curvature tensors are linearly dependent iv) M-projective curvature and conformal curvature tensors are linearly dependent. 5. M−PROJECTIVELY FLAT NEARLY COSYMPLECTIC MANIFOLDS In view of W * = 0, (4.1) becomes R(X, Y )Z = 1 [S(Y, Z )X − S(X, Z )Y + g(Y, Z )Q X − g(X, Z )QY ]. 2 (n − 1) (5.1) Replacing Z by ξ in (5.1) and then using some curvature relations of nearly cosymplectic manifolds, we obtain 1 2(n−1) η(X )h 2 Y − η(Y )h 2 X ) = 2 −η(Y )X tr (h ) − η(X )Y + tr (h 2 ) + η(Y )Q X − η(X )QY . I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS (5.2) Vol. 10, No. 1 (Special Issue), Jan–June 2019 Some Curvature Conditions on Nearly Cosymplectic Manifolds 57 Again putting Y = ξ in above relation and using some curvature relations of nearly cosymplectic manifolds, we have Q X = tr (h 2 )X − 2 (n − 1) (h 2 )X ⇔ S(X, Y ) = tr (h 2 )g(X, Y ) − 2 (n − 1) g(h 2 X, Y ) (5.3) and r = (n − 2)tr (h 2 ). (5.4) We know that the eigenvalues of the symmetric operator h 2 are constant and since h is skewsymmetric, the nonvanishing eigenvalues of h 2 are negative [7]. So, if we write h 2 (X ) = −λ2 X , then we obtain tr (h 2 ) = −nλ2 . Putting this equation in the above relations; we obtain Q X = (n − 2) (λ2 )X (5.5) r = n(n − 2)λ (5.6) and In consequence of (5.5), (5.1) becomes R(X, Y )Z = (n − 2) (λ2 ) [g(Y, Z )X − g(X, Z )Y ] (n − 1) (5.7) 2 ) We can easily see that (n−2)(λ > 0. (n−1) A space form is said to be elliptic if and only if the sectional curvature tensor is positive [24]. Thus, we can state: Theorem 5. A n−dimensional nearly cosymplectic manifold M is M−projectively flat if and only if it is either locally isometric to the elliptic space E n (C) or M has constant scalar curvature (n − 2 2)tr (h 2 ) where C = (n−2)λ . (n−1) Corollary 3. A n−dimensional complete connected nearly cosymplectic manifold M is (n−1)λ2 n M−projectively flat if and only if it is either isometric to the sphere S of radius or the n−1 real projective space S n / {±I } . ACKNOWLEDGEMENT This work is supported by Necmettin Erbakan University Scientific Research Projects Coordination Unit. REFERENCES [1] Blair, D. E., Almost Contact Manifolds with Killing Structure, Tensors I. Pac. J. Math., 39(1971), 285–292. [2] Blair, D. E., The theory of quasi-Sasakian structures, J. Diff. Geom., 1(1967), 331–345. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 58 Gülhan Ayar, Pelin Tekin and Nesip Aktan [3] Blair, D. E., Showers, D.K., Almost Contact Manifolds with Killing Structures Tensors II. J. Differ. Geom., 9(1974), 577–582. [4] Blair, D. E., Contact manifolds in Riemannian geometry, Lecture Notes in Mathematics, 509(1976), Springer-Verlag, Berlin. [5] Blair, D. E., Goldberg S. I.: Topology of almost contact manifolds, J. Differential Geometry, 1 (1967), 347–354. [6] Cappelletti-Montano, B., Dileo, G., Nearly Sasakian Geometry and SU(2)-structures, Ann. Mat. Pura Appl., (IV) 195(2016), 897–922. [7] De Nicola, A., Dileo, G. & Yudin, I., On Nearly Sasakian and Nearly Cosymplectic Manifolds, Annali di Matematica (2017). https://doi.org/10.1007/s10231-017-0671-2 [8] Endo, H., On the Curvature Tensor of Nearly Cosymplectic Manifolds of Constant φ-sectional curvature. An. Stiit. Univ. “Al. I. Cuza” Iasi. Mat. (N.S.), (2005), 439–454. [9] Gray, A., Nearly Kahler Manifolds, J. Differential Geom., 4 (1970), 283–309. [10] Jun J.B., De U.C., Pathak G. . On Kenmotsu manifolds, J. Korean Math. Soc., 42(2005): 435–445. [11] Kenmotsu, K., A Class of Almost Contact Riemannian Manifolds, Tohoku Math. J., 24(1972), 93–103. [12] Libermann, P., Sur les automorphismes infinit´esimaux des structures symplectiques et de atructures de contact, 1959, Coll. G´eom. Diff. Globale, pp. 37–59. [13] Yano, K. and Kon, M., Structures on Manifolds, Series in Pure Math, Vol 3,(1984) World Sci. G.P.l. Pokhariya and R.S. Mishra, Curvature tensor and their relativistic significance II, Yokohama Mathematical Journal, 19 (1971), 97–103. [14] R.H. Ojha, A note on the M-projective curvature tensor, Indian J. Pure Appl. Math., 8, 12 (1975), 1531–1534. R. [15] H. Ojha, M-projectively flat Sasakian manifolds, Indian J. Pure Appl. Math., 17, 4 (1986), 481–484. [16] R.S. Mishra, Structures on a Differentiable manifold and their applications, Chandrama Prakashan, 50-A Bairampur House Allahabad 1984. [17] B.Y. Chen, Geometry of Submanifolds, Marcel Dekker, Inc. New York 1973. [18] S. Tanno, Curvature tensors and non-existence of killing vectors, Tensor N.S. 22 (1971), 387–394. [19] A. Ghosh, T. Koufogiorgos and R. Sharma, Conformally flat contact metric manifolds, J. Geom., 70(2001), 66–76. [20] S.K. Chaubey and R.H. Ojha, On the m-projective curvature tensor of a Kenmotsu manifold, Differential Geometry Dynamical Systems, Geometry Balkan Press, 12, (2010), 2–60. [21] M. Boothby and R.C. Wong, On contact manifolds, Ann. Math., 68 (1958), 421–450. [22] S. Sasaki and Y. Hatakeyama, On differentiable manofolds with certain structures which are closely related to almost contact structure, Tohoku Math. J., 13 (1961), 281–294. [23] F. Ö. Zengin, M-projectively flat Spacetimes, Math. Reports, 14(64), 4(2012), 363–370. [24] S. Kobayashi and K. Nomizu , Foundations of Differential Geometry, Vol 1, Interscience Publish., Newyork-London, 1963. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 INDIAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019, pp. 59–70 DOI: Second-order Characterization of Invex Functions and Its Applications in Optimization Problems M.T. Nadi1 and J. Zafarani2∗ 1 University of Isfahan, Isfahan, 81745-163, Iran University of Isfahan and Sheikhbahaee University, Isfahan, Iran ∗ Corresponding author Email: [email protected] 2 Abstract: We give a second-order characterization for invex functions by using the Clarke subdifferentials, analogous to the positive semidefinite property for convex functions. Furthermore, some applications of these characterizations in optimization problems are obtained. Keywords: Invexity; Second-order characterization; Second-order optimality conditions; Clarke subdifferential. 2010 Mathematics Subject Classification. 47J05; 90C33. 1. INTRODUCTION AND PRELIMINARIES Convexity, monotonicity and their generalizations are very useful and applicable in optimization, economy, engineering and many other sciences. The notion of invexity is a very important generalization of convexity which was introduced by Hanson as a condition that guarantees Kuhn-Tuker conditions to be sufficient for optimality in non-linear programming [12]. This concept, named afterwards by Carven [8], which is an abbreviation of invariant convex. Some extensions of invexity and their applications presented also by others. For example, Kaul and Kaur [21] introduced quasiinvex and pseudoinvex functions. The concept of preinvex functions and its applications have studied by Jeyakumar and Weir [16]. See, also [24, 30] and the references therein for other generalizations of invexity and their properties. Also, the notion of invexity have been utilized frequently as a useful tool in optimization; see, e.g., [27–29]. Invariant monotonicity, some of its extension and applications have been studied by many authors; see, e.g., [14,15,19,20,22]. The relationships between various kinds of generalized invexity and generalized monotonicity of corresponding subdifferentials are very impressive; see, e.g., [13]. It seems that the second-order characterization (characterization of a kind of generalized convex function by its corresponding second-order subdifferential) is more useful. It is well known in the classic mathematical analysis that the second-order differential of a function f : Rn −→ R at its I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 60 M.T. Nadi and J. Zafarani optimal point is positive semi-definite and conversely, positive definiteness of second-order differential (Hessian matrix) at a critical point (where ∇ f vanishes), ensures the optimality of the reference point. So, the second order differential of a function can be used as a strong tool in optimization and characterization of a convex function. The concept of Hessian, for non-differentiable case, generalized by Mordukhovich [25], by using the coderivative of set-valued mappings. Afterwards, this generalization used for characterization of maximal monotone mappings by Poliquin and Rockafellar [31]. Also, the second order characterization of a convex function and some of its generalizations, have been studied by many authors. Crouziex and Ferland [9], characterized the quasiconvex and pseudoconvex functions, by their second order differential. Luc and Schaible, investigated this characterization for class of C 1,1 functions [23]. For, more study in this field, see [3–5]. As a useful study in second order characterization, we can mention [4], which gives a characterization for a continuously differentiable convex function on the Hilbert space, by its second order Fréchet and Limiting subdifferentials. Also recently, we obtained some characterizations of the convexity in terms of second order subdifferentials [26]. The second order generalized directional derivative, also have been studied by many authors, for obtaining some necessary and sufficient conditions of optimality; see, e.g., [2, 7, 10, 11, 17, 18]. Here, our work is based on generalized second order Clarke subdifferential, and generalized Hessian which introduced by Luc and Schaible [23]. Indeed, we obtain an invariant case which is a generalization of Hessian. Throughout this paper, we suppose X is an arbitrary Hilbert space. Definition 1.1. Let f : X → R be a locally Lipschitz function at x ∈ X and v any vector in X . The Clarke generalized directional derivative of f at x in direction v, denoted by f ◦ (x, v), is defined by f ◦ (x, v) = lim sup y→x,t↓0 f (y + tv) − f (y) . t Definition 1.2. Let f : X → R be a locally Lipschitz function at x ∈ X and v any vector in X . The Clarke generalized subdifferential of f at x, denoted by ∂ f (x), and defined by ∂ f (x) = {ξ ∈ X : f ◦ (x, v) ≥ ξ, v , for any v ∈ X }. For more details and properties of the Clarke subdifferential, we refer the reader to [6]. Let K be a nonempty subset of X , η : X × X → X be a vector valued function and F : X ⇒ X be a set-valued mapping. Definition 1.3. A set K is said to be invex with respect to η : X × X → X , when for any x, y ∈ K and 0 ≤ λ ≤ 1, y + λη(x, y) ∈ K . I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Characterization of Invex Functions 61 Definition 1.4. A vector valued function η : X × X → X is said to be skew, if η(x, y) + η(y, x) = 0, for any x, y ∈ X. The following assumptions are frequently used in the literature: ASSUMPTION A: Let K be an invex set with respect to η, and f : K → R. Then f (y + η(x, y)) ≤ f (x) for any x, y ∈ K . ASSUMPTION C: Let η : X × X → X . Then, for any x, y ∈ X and for any λ ∈ [0, 1], η(y, y + λη(x, y)) = −λη(x, y), η(x, y + λη(x, y)) = (1 − λ)η(x, y). Remark 1.1. [33] It is easy to see that Assumption C implies η(y + λη(x, y), y) = λη(x, y). Definition 1.5. Let K be an invex subset of X with respect to η : X × X → X . The function f : K → R is said to be preinvex with respect to η, if for any x, y ∈ K and λ ∈ [0, 1], f (y + λη(x, y)) ≤ λ f (x) + (1 − λ) f (y). Definition 1.6. A differentiable function f : X → R is said to be invex with respect to η, if for any x, y ∈ K , one has ∇ f (y), η(x, y) ≤ f (x) − f (y). (1.1) Definition 1.7. A locally Lipschitz function f : K ⊆ X → R is said to be invex with respect to η, if for any x, y ∈ K and any ξ ∈ ∂ f (y), one has ξ, η(x, y) ≤ f (x) − f (y). (1.2) Remark 1.2. A function f is said to be invex in a neighborhood of x, if either (5) or (6) holds for all y such that y − x is sufficiently small. Remark 1.3. Note that, in the above definitions by letting η(x, y) = x − y, we reduce to the convex case. Indeed, both preinvex and invex functions reduce to convex function, and invex set, to convex set. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 62 M.T. Nadi and J. Zafarani Definition 1.8. [13] A set valued mapping F : X ⇒ X ∗ is said to be invariant monotone on K with respect to η, if for any x, y ∈ K and any u ∈ F(x), v ∈ F(y), one has v, η(x, y) + u, η(y, x) ≤ 0. In Section 2, we give some second order necessary and sufficient conditions for nondifferentiable invex functions, and in Section 3, we present some applications of our results in optimization. 2. MAIN RESULTS We begin with a necessary condition for a twice differentiable function which guarantees the invexity and afterwards we obtain a similar result for C 1,1 functions (i.e. functions with locally Lipschitz differential). In the following, we denote the differential of η(., y) (differential of η, relative to the first argument) by ηx (., y). Proposition 2.1. Let f : Rn → R be an invex function with respect to η : Rn × Rn → Rn , which is skew, f be twice differentiable at x ∈ Rn and η(., x) be differentiable at x. Then ηx (x, x)u, D 2 f (x)u ≥ 0, for any u ∈ Rn . Proof. The differential mapping F = f is η − monotone, since f is η − invex. Suppose on the contrary that, ηx (x, x)u, D f (x)u < 0 for some u ∈ Rn . By definitions of D F(x)u and ηx (x, x)u, we have η(x + tu, x) − η(x, x) F(x + tu) − F(x) and D F(x)u = lim . t→0 t→0 t t ηx (x, x)u = lim Thus, for sufficiently small t > 0, we conclude that η(x + tu, x) − η(x, x) F(x + tu) − F(x) , < 0, t t which implies that η(x + tu, x) − η(x, x), F(x + tu) − F(x) < 0. By letting y = x + tu, we have η(y, x), F(y) − F(x) < 0, which contradicts η − monotonicit y of F. Therefore, when D F(x) exists, for any u ∈ Rn we conclude that ηx (x, x)u, D f (x)u ≥ 0. Now, we present an analogous second-order necessary condition for C 1,1 functions. We denote the Clarke subdifferential of the mapping F : Rn → Rn at x, by ∂ F(x). Theorem 2.1. Suppose that f : Rn → R is C 1,1 , invex function with respect to an skew η : Rn × Rn → Rn , where η is differentiable in the first argument at x and continuous. Then ηx (x, x)u, x ∗ u ≥ 0, for any u ∈ Rn and x ∗ ∈ ∂ f (x). I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Characterization of Invex Functions 63 Proof. Since f = F : Rn → Rn is a locally Lipschitz mapping, by Rademacher Theorem [6], we have: ∂ F(x) = conv{lim D F(xi ) : xi → x and F is differentiable at xi }. Thus, for any x ∗ ∈ ∂ F(x), we can find x1∗ , x2∗ ∈ Rn and two sequences xi1 → x and xi2 → x such that D F(xi1 ) → x1∗ , D F(xi2 ) → x2∗ and x ∗ is a convex combination of x1∗ and x2∗ , which means x ∗ = λx1∗ + (1 − λ)x2∗ , for some λ ∈ [0, 1]. First, we show that ηx (x, x)u, D F(xi1 )u ≥ 0 for sufficiently large i. Suppose, on the contrary that, there exists a subsequence (xi1k ) of (xi1 ) such that ηx (x, x)u, D F(xi1k )u < 0. Therefore, by definition of differential, we have F(xi1k + tu) − F(xi1k ) η(x + tu, x) − η(x, x) and D F(xi1k )u = lim . t→0 t→0 t t ηx (x, x)u = lim Thus, for each xi1k we can find sufficiently small tk > 0 such that η(x + tk u, x) − η(x, x), F(xi1k + tk u) − F(xi1k ) < 0. By continuity of η at both arguments, for sufficiently large k, we conclude that η(xi1k + tk u, xi1k ) − η(xi1k , xi1k ), F(xi1k + tk u) − F(xi1k ) < 0, which contradicts η-monotonicity of F, since we know that f is invex with respect to η. We can use the foregoing argument for x2∗ and (xi2 ), and we conclude for sufficiently large i, that ηx (x, x)u, D f (xi2 )u ≥ 0. Therefore, we have ηx (x, x)u, x1∗ u ≥ 0 and ηx (x, x)u, x2∗ u ≥ 0, which implies that ηx (x, x)u, x ∗ u = ηx (x, x)u, (λx1∗ + (1 − λ)x2∗ )u ≥ 0. The following example, shows that some times to characterize the invexity of a function by second order condition is easier than checking by the first order condition. Example 2.1. Consider the following C 1,1 function f : R → R, −x 2 + x, x ≤ 0 f (x) = x > 0. x 2 + x, Let η(x, y) = x 3 − y 3 . An easy calculation implies that ⎧ x <0 ⎨ −2, [−2, 2], x = 0 ∂ f (x) = ⎩ 2, x > 0, I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 64 M.T. Nadi and J. Zafarani which means that ηx (x, x)u, x ∗ u = 3x 2 u 2 x ∗ < 0, by letting x = −1 and any arbitrary u ∈ R. The foregoing function is not also invex with respect to η(x, y) = x y 2 − yx 2 , by using a similar calculation for x = 1. It is more difficult to check the invexity of this function with respect to each of the above η, by using the first order characterization conditions. Note that we can find some η, such that f is invex with respect to η, since f has not any stationary f (y) . When η is not point ( where f vanishes). Indeed, f is invex with respect to η(x, y) = f (x)− f (y) differentiable relative to the first argument, but f is twice differentiable, we can also, present an analogous of the above Theorem, by a similar argument, as bellow. We denote the Clarke subdifferential of η, relative to the first argument by ∂x η. Proposition 2.2. Let f : Rn → R be an invex function with respect to η : Rn × Rn → Rn , which is skew, f be twice differentiable at x ∈ Rn and η(., x) be Lipschitz in the first argument. Then ξ u, D 2 f (x)u ≥ 0, for any u ∈ Rn and ξ ∈ ∂x η(x, x). The following example illustrate that we can use the above result for denying the invexity of a twice differentiable function, when η is Lipschitz with respect to the first argument. Example 2.2. Consider the twice differentiable function f : R → R as f (x) = x 3 + x 2, −x 3 + x 2 , x ≤0 x > 0, and η : R × R → R as bellow: ⎧ −x + y, ⎪ ⎪ ⎨ x − y, η(x, y) = |x| − |y| = x + y, ⎪ ⎪ ⎩ −x − y, x x x x ≤ 0, y ≥ 0, y > 0, y < 0, y ≤0 ≥0 <0 > 0. An easy calculation implies that ∂x η(0, 0) = [−1, 1]. Since we can find some ξ ∈ ∂x η(0, 0) such that ξ u, D 2 f (0)u = 2ξ u 2 < 0 for any arbitrary u ∈ R, therefore the second order necessary condition in Proposition 2.2 dose not hold. Hence, f is not invex with respect to η. Although we can find some η such that f is invex with respect to it. Remark 2.1. The above results are the natural extensions of convex case. In fact by replacing η(x, y) with x − y, we have the classical form of Hessian. In the following, we give some second-order sufficient conditions for invex functions. We begin with a sufficient condition for twice differentiable case in the next theorem. First, we need the following lemma, which is an extension of monotone case. We follow a similar technique as in Lemma 3.4 of [5], to establish our lemma. Definition 2.1. We say that a set valued mapping F : Rn ⇒ Rn is semi-locally invariant monotone at x̄ ∈ dom F with respect to η, if there exists a neighborhood U of x̄ such that for any x, y ∈ U I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Characterization of Invex Functions 65 and any u ∈ F(x), v ∈ F(y), one has v, η(x, y) + u, η(y, x) ≤ 0. Also, we say that F is semi-locally monotone on an invex set K ⊆ Rn , if it is semi-locally invariant monotone at any point of K . We need the following useful concept to prove the following lemma. Definition 2.2. [1] Let K ⊆ Rn be a nonempty invex set with respect to η, and x and u, two arbitrary points of K . A set Puv is said to be a closed η − path joining the points u and v = u + η(x, u) (contained in K ) if Puv := {y = u + λη(x, u) : λ ∈ [0, 1]}. Analogously, an open η − path joining the points u and v = u + η(x, u) (contained in K) is the following set ◦ := {y = u + λη(x, u) : λ ∈ (0, 1)}. Puv Lemma 2.1. Let F : Rn ⇒ Rn be a semi-locally η − monotone mapping on Rn and η be skew satisfying Assumption C and η(., y) be onto for any y ∈ Rn . Then F is η − monotone on Rn . Proof. Suppose that u 1 and u 2 be arbitrary, v1 ∈ F(u 1 ) and v2 ∈ F(u 2 ). We will show that v2 − v1 , η(u 2 , u 1 ) ≥ 0. Since η(., u 1 ) is onto, there exists u ∈ X such that u 2 = u 1 + η(u, u 1 ). First, we show that v2 − v1 , η(u, u 1 ) ≥ 0. Consider η − path, Pu 1 u 2 := {u 1 + λη(u, u 1 ) : 0 ≤ λ ≤ 1}. By semi-local η-monotonicity of F, for any x ∈ Pu 1 u 2 , there exists γx > 0 such that y2 − y1 , η(x2 , x1 ) ≥ 0 for any x1 , x2 ∈ Bγx (x) and y1 ∈ F(x1 ), y2 ∈ F(x2 ). Now, n the compactness of Pu 1 u 2 implies that, there exist x1 , x2 , ..., xn ∈ Pu 1 u 2 such that Pu 1 u 2 ⊆ i=1 (int Bγxi (x i )). We select ū j ∈ Pu 1 u 2 as ū j = u 1 + λ j η(u, u 1 ) such that 0 = λ0 < λ1 < ... < λm = 1, j ∈ {0, 1, ..., m − 1}, and [ū j+1 , ū j ] ⊆ int Bγxi (xi j ), with some i j ∈ {1, ..., n}. j I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 66 M.T. Nadi and J. Zafarani For arbitrary v̄ j ∈ F(ū j ) where j ∈ {1, ..., m − 1} and for v̄0 = v1 and v̄m = v2 , by semi-local η − monotonicit y of F and Assumption C, we have: v̄ j+1 − v̄ j , η(ū j+1 , ū j ) = v̄ j+1 − v̄ j , η(u 1 + λ j+1 η(u, u 1 ), u 1 + λ j η(u, u 1 )) = v̄ j+1 − v̄ j , η(u 1 + λ j+1 η(u, u 1 ), u 1 + λ j+1 η(u, u 1 ) + (λ j − λ j+1 )η(u, u 1 )) = v̄ j+1 − v̄ j , η(u 1 + λ j+1 η(u, u 1 ), u 1 + λ j+1 η(u, u 1 ) λ j − λ j+1 η(u, u 1 + λ j+1 η(u, u 1 )) 1 − λ j+1 λ j+1 − λ j = v̄ j+1 − v̄ j , η(u, u 1 + λ j+1 η(u, u 1 )) 1 − λ j+1 + = (λ j+1 − λ j ) v̄ j+1 − v̄ j , η(u, u 1 ) ≥ 0. Therefore, we conclude that v̄ j+1 − v̄ j , η(u, u 1 ) ≥ 0 for any j ∈ {0, 1, ..., m − 1}. Thus, we have v2 − v1 , η(u, u 1 ) = v̄m − v̄m−1 , η(u, u 1 ) + v̄m−1 − v̄m−2 , η(u, u 1 ) + ... + v̄1 − v̄1 , η(u, u 1 ) ≥ 0. Now, by using Assumption C again, v2 − v1 , η(u 2 , u 1 ) = v2 − v1 , η(u 1 + λη(u, u 1 ), u 1 ) = v2 − v1 , η(u, u 1 ) ≥ 0. But, this means that F is invariant monotone with respect to η. Theorem 2.2. Let F : Rn → Rn be a differentiable mapping, η : Rn × Rn → Rn be skew satisfying Assumption C and η(., y) be onto for any y ∈ Rn . If ηx (x, x)u, D F(x)u ≥ 0, for any x, u ∈ Rn , then F is invariant monotone with respect to η. Proof. Suppose on the contrary that F is not η − monotone. Therefore, by Lemma 2.1, F is not semi-locally invariant monotone. So, we can find x ∈ X, a sequence tk → 0 and yk = x + tk u, which converges to x, such that F(yk ) − F(x), η(yk , x) < 0, for any k. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Characterization of Invex Functions 67 By definition of directional derivative, F(x + tu) − F(x) t→0 t F(x + tk u) − F(x) η(x + tk u, x), η(x + tk u, x) × = lim k→∞ tk η(x + tk u, x), η(x + tk u, x) F(x + tk u) − F(x), η(x + tk u, x) η(x + tk u, x) − η(x, x) × lim . = lim k→∞ k→∞ η(x + tk u, x), η(x + tk u, x) tk D F(x)u = lim Therefore, we have D F(x)u = Mηx (x, x)u, for some M ≤ 0. But, this contradicts the assumption ηx (x, x)u, D F(x)u ≥ 0, which means that F is η − monotone. Now, we give a sufficient second-order condition for invexity of a twice differentiable function f : Rn → R. Corollary 2.1. Let f : Rn → R be a twice differentiable function, f and η satisfy Assumptions A and C, η(., y) be onto for any y ∈ Rn and skew. If ηx (x, x)u, ∇ 2 f (x)u ≥ 0, for any x, u ∈ Rn , then f is invex with respect to η. Proof. By Theorem 2.2, ∇ f (x) is invariant monotone with respect to η. Now, it suffices to use Theorem 3.1 of [13] which implies that f is invex. We can also, give another second-order sufficient condition for invexity of a function f : X → R, by using the generalization of Taylor expansion for invex case. Although, the invex case of Taylor expansion in theorem 17 of [1] is given for Rn , but we can use it for a Hilber space X as bellow. Proposition 2.3. Suppose that f : X → R is a twice differentiable function which satisfies Assumption A, and η is skew and continuous. Moreover, suppose that D 2 f (x)(η(y, x), η(y, x)) ≥ 0, for any x, y ∈ X. Then f is invex in a neighborhood of x with respect to η. Proof. By Taylor-Young expansion of order 2, for an arbitrary and fixed x ∈ X and any h ∈ X, we have: f (x + h) = f (x) + D f (x)(h) + 1 2 D f (x)(h, h) + h 2 (h), 2 with limh→0 (h) = 0. By letting h = η(y, x) and using the assumption D 2 f (x)(η(y, x), η(y, x)) ≥ 0, we conclude that f (x + η(y, x)) − f (x) ≥ D f (x)(η(y, x)), for y ∈ X which η(y, x) is sufficiently closed to zero. Since η is skew and continuous in the first argument, η(y, x) is sufficiently closed to zero, when y is sufficiently closed to x. Now, by using Assumption A, we have f (x + η(y, x)) ≤ f (y) and therefore f (y) − f (x) ≥ D f (x)(η(y, x)), which means that f is invex in a neighborhood of x. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 68 M.T. Nadi and J. Zafarani In particular, for X = Rn we have the following result: Corollary 2.2. Let f : Rn → R be a twice differentiable function which satisfying Assumption A, and η be skew and continuous. Moreover, suppose that ∇ 2 f (x)(η(y, x), η(y, x)) ≥ 0, for any y ∈ Rn . Then f is invex in a neighborhood of x with respect to η. 3. APPLICATIONS IN OPTIMIZATION The notion of invexity introduced by Hanson [12], and afterwards appeared frequently for solving the optimization problems in literature. Due to the definition of invexity, it is easy to see that when K is an invex set and f : K → R is a differentiable invex function, then ∇ f (x̄) = 0 implies that x̄ is a local minimum point of f . Specially, in constrained problems, invexity plays an important role for attaining sufficient conditions of optimality. So, the above second-order sufficient conditions can be used in constrained and unconstrained optimization problems. Now, we give the following second-order conditions in optimization. Proposition 3.1. Let f : Rn → R be a twice differentiable function which satisfies Assumption A with respect to some η, and ∇ f (x̄) = 0. Moreover, suppose that one of the following holds: (i) ηx (x, x)u, ∇ 2 f (x)u ≥ 0, for any x, u ∈ Rn , where η is skew satisfies Assumption C and η(., y) is onto for any y ∈ Rn . (ii) η(y, x̄), ∇ 2 f (x̄)η(y, x̄) ≥ 0, for any y ∈ Rn . Then x is a local minimizer of f . Proof. By Corollaries 2.1 and 2.2, conditions (i) and (ii) imply the invexity of f in a neighborhood of x̄. So, ∇ f (x̄) = 0 implies that x̄ is a local minimizer of f. Consider the following constrained optimization problem: min f 0 (x) subject to f i (x) ≤ 0 (i = 1, ..., m), (3.1) which f 0 , f 1 , ..., f m are twice differentiable functions defined on Rn . Let f (x) = ( f 1 (x), ..., f m (x)). We know that the existence of a vector λ = (λ1 , ..., λm ) ∈ Rm which satisfies the following conditions, which named Kuhn-Tucker conditions are necessary for a point x̄ to be a solution of this problem: m ∇ f 0 (x̄) + i=1 λi ∇ f i (x̄) = 0 λ, f (x̄) = 0 (3.2) (3.3) λ1 , ..., λm ≥ 0. (3.4) Hanson [12] showed that the Kuhn-Tucker conditions are also sufficient for x̄ to be a solution of (3), when each f i is invex with respect to the same η. Indeed, only the invexity in a neighborhood of x̄ for each f i guarantees that the foregoing conditions are sufficient [8]. Now, we give some second-order sufficient conditions for constrained optimization problem, by using our results. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Characterization of Invex Functions 69 Proposition 3.2. Suppose the constrained optimization problem (3). If the Kuhn-Tucker conditions hold in x̄ and each f i satisfies Assumption A, and one of the following second-order conditions holds (with respect to the same η): (i) ηx (x, x)u, ∇ 2 f i (x)u ≥ 0, for any x, u ∈ Rn , where η is skew satisfies Assumption C and η(., y) is onto for any y ∈ Rn . (ii) η(y, x̄), ∇ 2 f i (x)η(y, x̄) ≥ 0 for any y ∈ Rn , then x̄ is a solution for constrained optimization problem (3). Proof. 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I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 INDIAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019, pp. 71–86 DOI: Approximation of Periodic Functions via Statistical B-summability and Its Applications to Approximation Theorems B.B. Jena1 , S.K. Paikray2∗ and U.K. Misra3 1,2 Department of Mathematics, Veer Surendra Sai University of Technology, Burla-768018, Odisha, India 3 Department of Mathematics, National Institute of Science and Technology, Pallur Hills, Golanthara-761008, Odisha, India (∗ Corresponding author) Email: ∗ skpaikray [email protected], 1 [email protected], 3 umakanta [email protected] Abstract: The A-statistical summability is stronger than A-statistical convergence which was introduced by Edely and Mursaleen [7] (see [Edely and Mursaleen, On statistical A-summability, Math. Comput. Model. 49, 672-680). In this paper, by using the concept of statistical B-summability we establish a result on Korovkin-type approximation theorem for periodic functions defined on a Banach space C ∗ (R), which is also stronger than its statistical A-summability version. Furthermore, we demonstrate a theorem for the rate of the B-statistical convergence for same set of functions with the help of the modulus of continuity. (2010) Mathematics Subject Classification. Primary 40A05, 41A36; Secondary 40G15. Keywords: Statistical B-summability; statistical A-summability; B-statistical convergence; rate of convergence; Korovkin type approximation theorem. 1. INTRODUCTION, DEFINITIONS AND MOTIVATION The notion of statistical convergence was introduced and studied by Fast [8] and Steinhaus [25]. Recently, statistical convergence has been a dynamic research area due basically to the fact that it is more general than classical convergence and such theory is discussed in the study in the areas of (for instance) Fourier Analysis, Number Theory, Functional Analysis and Approximation Theory. For more details, see the recent works [1, 2, 5, 6], [10–14] and [17–23]. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 72 B.B. Jena, S.K. Paikray and U.K. Misra Let N be the set of natural numbers and let K ⊆ N. Also let Kn = {k : k n k ∈ K} and and suppose that |Kn | be the cardinality of Kn . Then the natural density of K is defined by d(K) = lim n→∞ |Kn | 1 = lim |{k : k n n→∞ n n and k ∈ K}|, provided that the limit exists. A given sequence (xn ) is said to be statistically convergent to a number if, for each > 0, K = {k : k ∈ N and |xk − | } has zero natural density (see [8], [25]). This means that, for each > 0, we have d(K ) = lim n→∞ |K | 1 = lim |{k : k n n→∞ n n and |xk − | }| = 0. In this case, we write stat lim xn = . n→∞ We present below an example to illustrate that every convergent sequence is statistically convergent but the converse is not true. Example 1. Let us consider a sequence x = (xn ) by ⎧ 1 (n = m 2 , m ∈ N) ⎨ 2 xn = ⎩ n (otherwise). n+1 Here, the sequence (xn ) is statistically convergent to 1 even if it is not classically convergent. Let X and Y be two sequence spaces and let A = (an,k ) be a non-negative regular matrix. If for each xk ∈ X the series, An x = ∞ an,k xk k=1 converges for each n ∈ N and the sequence (An x) belongs to Y , then the matrix A maps X into Y . Here, the symbol (X, Y ) denotes the set of all matrices which map X into Y . A matrix A is said to be regular, if lim An x = L n→∞ whenever I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS lim xk = L . k→∞ Vol. 10, No. 1 (Special Issue), Jan–June 2019 Approximation of Periodic Functions via Statistical B-summability 73 The well-known Silverman-Toeplitz theorem (see, details [4]) asserts that A = (an,k ) is regular if and only if the conditions, (i) sup n→∞ ∞ |an,k | < ∞, k=1 (ii) lim an,k = 0 for each k n→∞ and (iii) lim n→∞ ∞ an,k = 1 hold true. k=1 Freedman and Sember [9] extended the definition of statistical convergence with the help of the non-negative regular matrix A = (an,k ), which he called it A-statistical convergence. Let for any non-negative regular matrix A, we say that a sequence (xn ) is A-statistical convergent to a number if, for each > 0, we have, d A (K ) = 0, where K = {k : k ∈ N This means that, for each > 0, we have lim n→∞ |xk − | }. and an,k = 0. k:|xk −| In this case, we write stat A lim xn = . Now we recall statistical A-summability for a nonnegative regular matrix A. Definition 1. Let A = (an,k ) be a non-negative regular matrix and we say that the sequence (xn ) is statistical A-summable to a number if, for each > 0, K = {k : k ∈ N and |Ak (x) − | } has zero natural density (see [7, 17]). This means that, for each > 0, we have |K | 1 = lim |{k : k n n→∞ n n→∞ n d(K ) = lim I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS and |Ak (x) − | }| = 0. Vol. 10, No. 1 (Special Issue), Jan–June 2019 74 B.B. Jena, S.K. Paikray and U.K. Misra In this case, we write stat lim An (x) = or Astat lim xn = . n→∞ n→∞ In the year 1998, the concept of A-statistical convergence was extended by Kolk [15] to B-statistical convergence with respect to FB -convergence (or B-summable) due to Steiglitz [24]. Suppose that B = (Bi ) is a sequence of infinite matrices with Bi = (bn,k (i)). Then the given sequence (xn ) ∈ ∞ is said to be B-summable to the value B- lim (xn ), if n→∞ lim (Bi x)n = lim n→∞ n→∞ ∞ bn,k (i)(x)k = B- lim (xn ) uniformly in i (n, i = 0, 1, 2, ...). n→∞ k=0 The method (Bi ) is regular if and only if the following conditions hold true (see, for details, [3] and [24]): (i) B = sup ∞ |bn,k (i)| < ∞; n,i→∞ k=0 (ii) lim bn,k (i) = 0 n→∞ (iii) lim n→∞ ∞ for each k ∈ N; bn,k (i) = 1. k=0 Let K = {ki } ⊂ N (ki < ki+1 ) for all i. Then the B-density of K is defined by dB (K) = lim ∞ n→∞ bn,k (i). k=0 Let R+ denotes the set of all regular methods B with bn,k (i) 0 for all n, k and i. Also let B ∈ R+ . We say that a sequence (xn ) is B-statistically convergent to a number if, for each > 0, we have dB (K ) = 0, where K = {k : k ∈ N This means that, for each > 0, we have, lim and n→∞ |xk − | }. bn,k (i) = 0. k:|xk −| In this case, we write statB lim xn = . I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Approximation of Periodic Functions via Statistical B-summability 75 Definition 2. Let B = (Bn,k (i)) be a non-negative regular matrix and we say that the sequence (xn ) is statistically B-summable to a number if, for each > 0, K = {k : k ∈ N and |B(x) − | } has zero natural density (see [13], [15]). This means that, for each > 0, we have d(K ) = lim n→∞ |K | 1 = lim |{k : k n n→∞ n n and |B(x) − | }| = 0. In this case, we write stat lim B(x) = or Bstat lim xn = . n→∞ n→∞ Remark 1. Upon replacing the matrices Bi by the matrix A for all i, the B-statistical convergence reduces to the A-statistical convergence. Furthermore, by choosing B = (C, 1) (the Cesàro matrix of order one), the B-statistical convergence coincides with the statistical convergence (see [8] and [25]). Finally if B = I (identity matrix), then the B-statistical convergence coincides with the classical convergence. Example 2. Let us consider the infinite matrices B = (Bi ) with Bi = (bn,k (i)) given by (see [13]) Bi = ⎧ ⎪ ⎨ (i k i + n) 1 n+1 ⎪ ⎩ 0 (otherwise) ⎧ ⎪ ⎨ 1 (n = odd) and let a sequence x = (xn ) is given by xn = ⎪ ⎩ (1.1) −1 (n = even). Then, we observe that Bi = (bn,k (i)) is a non-negative regular matrix and for the given sequence (xn ), stat lim Bn (x) = 0. n→∞ Here the sequence (xn ) is neither convergent nor statistically convergent. However, (xn ) is statistically B-summable to 0 but it is not B-statistical convergent to 0. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 76 B.B. Jena, S.K. Paikray and U.K. Misra 2. A KOROVKIN TYPE THEOREM VIA STATISTICAL B-SUMMABILITY In recent years, quite a few researchers worked toward extending or generalizing the Korovkin type theorems in many ways based on several aspects, including (for example) function spaces, abstract Banach lattices, Banach algebras, Banach spaces, and so on. This theory is extremely valuable in Real Analysis, Functional Analysis, Harmonic Analysis, Measure Theory, Probability Theory, Summability Theory and Partial Differential Equations, and many other fields. In this section, by using the concept of statistical B-summability, we prove a Korovkin type approximation theorem (see for details [16]) for periodic functions. Also by using Fejér operators, we show that our proposed method is stronger than that of classical and statistical versions of Korovkin’s theorem. We denote C ∗ (R), the space of all real valued 2π periodic continuous functions defined on R. We recall that if a function f has period 2π for all x ∈ R, f (x + 2π k) = f (x) holds for (k = 0, ±1, ±2, ...). This space is equipped with supremum norm. That is f C ∗ (R) = sup | f (x)|, f ∈ C ∗ (R). x∈R Let L be a linear operator from C ∗ (R) into C ∗ (R). Then, as usual, we say that L is a positive linear operator provided that f 0 implies L( f ) 0. Also, we denote the value of L( f ) at a point x ∈ R by L( f (u); x) or, briefly, L( f ; x). Throughout this paper, we use the following test functions f 0 (x) = 1, f 1 (x) = cos x and f 2 (x) = sin x. Theorem 1. Let B = (bn,k (i)) be a nonnegative regular matrix and let (L n ) (n ∈ N) be a sequence of positive linear operators from C ∗ (R) into itself and let f ∈ C ∗ (R). Then ∞ bn,k (i)L n ( f ; x) − f (x) = 0, f ∈ C ∗ (R) (2.1) stat − lim n C ∗ (R) n=0 if and only if ∞ bn,k (i)L n ( f j ; x) − f j (x) stat − lim n n=0 I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS = 0 ( j = 0, 1, 2.). (2.2) C ∗ (R) Vol. 10, No. 1 (Special Issue), Jan–June 2019 Approximation of Periodic Functions via Statistical B-summability 77 Proof. Since each of the functions given by f 0 (x) = 1, f 1 (x) = cos x f 2 (x) = sin x and are in C ∗ (R), the implication (2.1) =⇒ (2.2) is fairly obvious. In order to complete the proof of the Theorem 1, we first assume that (2.2) holds true. Let f ∈ C ∗ (R), ∀ x ∈ C ∗ (R) and let I be a closed subinterval of length 2π of R. Fix x ∈ I . By the continuity of f at x, for given > 0 there exists δ > 0 such that | f (t) − f (x)| < |t − x| < δ whenever (2.3) for all t, x ∈ R. Since f (x, y) is bounded on R, then there exists a constant f C ∗ (R) > 0 such that (∀ x ∈ R), | f (x)| f C ∗ (R) which implies that | f (t) − f (x)| 2 f C ∗ (R) (t, x ∈ R). (2.4) Clearly, f is a continuous function on R, and for a given > 0, there exists δ = δ() > 0; Moreover from equation (2.3) and (2.4), we get | f (t) − f (x)| < + 2 f C ∗ (R) ϕ(t), sin2 2δ (2.5) where ϕ(t) = sin2 t−x 2 . Since the function f ∈ C ∗ (R), the inequality (2.5) holds for t, x ∈ R. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 78 B.B. Jena, S.K. Paikray and U.K. Misra Now, since the operator L n ( f ; x) is monotone and linear, by applying this operator to the inequality in (2.5), we have ∞ bn,k (i)L n ( f (t); x) − f (x) = n=0 ∞ bn,k (i)L n ( f (t) − f (x); x) + f (x) n=0 ∞ ∞ 2 f C ∗ (R) + ϕ(t); x sin2 2δ bn,k (i)L n n=0 + ∞ ∞ bn,k (i)L k ( f 0 ; x) − f 0 bn,k (i)L k (1; x) − 1 n=0 + | f (x) ∞ bn,k (i)L n (1; x) − 1 n=0 bn,k (i)L n ( f 0 ; x) − f 0 (x) + f C ∗ (R) n=0 + n=0 bn,k (i)L n (| f (t) − f (x)|; x) + | f (x)| n=0 ∞ ∞ bn,k (i)L n ( f 0 ; x) − f 0 (x) n=0 ∞ 2 f C ∗ (R) bn,k (i)L n (ϕ(t); x). sin2 2δ n=0 (2.6) After some simple calculations, we get ϕ(t) = 1 (1 − cos t cos x − sin t sin x). 2 Next, we have ∞ 1 bn,k (i)L n (ϕ(t); x) 2 n=0 ∞ bn,k (i)L n ( f 0 ; x) − f 0 (x) + | cos x| n=0 ∞ bn,k (i)L n ( f 1 ; x) − f 1 (x) n=0 ∞ 1 | sin x| bn,k (i)L n ( f 2 ; x) − f 2 (x) . (2.7) + 2 n=0 Then using (2.7), we have ∞ 2 f C ∗ (R) bn,k (i)L n (ϕ(t); x) − f (x) + ϕ(t); x sin2 2δ n=0 +| cos x| ∞ ∞ bn,k (i)L n ( f 0 ; x) − f 0 (x) n=0 bn,k (i)L n ( f 1 ; x) − f 1 (x) n=0 +| sin x| ∞ bn,k (i)L n ( f 2 ; x) − f 2 (x) . (2.8) n=0 I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Approximation of Periodic Functions via Statistical B-summability 79 Further, taking supx,y∈R , in both side of (2.8), we get ∞ bn,k (i)L n (ϕ(t); x) − f (x) +M ∞ C ∗ (R) n=0 bn,k (i)L n ( f 0 ; x) − f 0 (x) C ∗ (R) n=0 ∞ +| cos x| bn,k (i)L n ( f 1 ; x) − f 1 (x) C ∗ (R) n=0 +| sin x| ∞ bn,k (i)L n ( f 2 ; x) − f 2 (x) (2.9) C ∗ (R) n=0 where, 2 f C ∗ (R) M = + ϕ(t); x . sin2 2δ Now, for a given r > 0, we choose > 0, such that 0 < < r . Then, upon setting ∞ bn,k (i)L n ( f (t); x) − f (x) r An = n : n ∈ N and n=0 and A j,n = n:n∈N and ∞ r − bn,k (i)L n ( f j (t); x) − f j (x) 3M n=0 ( j = 0, 1, 2) we easily find from (2.9) that An 3 A j,n . j=0 Thus, we have A j,n C ∗ (R) An C ∗ (R) . n n j=0 3 (2.10) Finally, by using the above assumption about the implication in (2.2) and Definition 2, the right-hand side of (2.10) is seen to tend to zero (n → ∞). Consequently, we get ∞ stat − lim bn,k (i)L n ( f j (t); x) − f j (x) = 0. n→∞ n=0 C ∗ (R) Hence, the implication (2.1) holds true. The proof of Theorem 1 is thus completed. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 80 B.B. Jena, S.K. Paikray and U.K. Misra Remark 4. If B = (I ) (identity matrix) in our Theorem 1, then we obtain classical version of Korovkin-type approximation theorem (see [16]). Also, if B = (C, 1), Cesàro matrix of order 1, in our Theorem 1, then we obtain statistical version of Korovkin type approximation theorem (see [8]. Furthermore, if B = (A), in our Theorem 1, then we obtain statistical A-summability version of Korovkin-type approximation theorem (see [17]). We now present below an illustrative example for Theorem 1 based on following Fejér operators. Example 3. Let I = [−π, π]. For a function f ∈ C ∗ (I ), the operators 1 Fn ( f ; x) = kπ π −π sin2 k2 (t − x) f (t) dt. 2 sin2 [ t−x ] 2 (2.11) Also, observe that Fn ( f 0 ; x) = 1, Fn ( f 1 ; x) = k−1 k−1 cos x, Fn ( f 2 ; x) = sin x. k k Now we consider consider the following positive linear operators L n : C ∗ (I ) → C ∗ (I ) such that L n ( f ; x) = (1 + xn )Fn ( f ; x), (2.12) where (xn ) is a sequence defined as in Example 2. It is clear that the sequence (L n ) satisfies the conditions (2.2) of our Theorem 1, thus we obtain: ∞ stat − lim bn,k (i)L n ( f j ; x) − f j (x) n→∞ = 0. C ∗ [−π,π] n=0 Therefore by Theorem 1, we have ∞ stat − lim bn,k (i)L n ( f ; x) − f (x) n→∞ n=0 = 0. C ∗ [−π,π] However, since (xn ) is not statistically weighted A-summable, so the result of Karakuş et al. ( [7], p. 162 Theorem 2.1) does not hold true for our operators defined by (2.12). Moreover, since (xn ) is statistical B-summable, therefore we conclude that our Theorem 1 works for the same operators. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Approximation of Periodic Functions via Statistical B-summability 81 3. RATE OF THE B-STATISTICAL CONVERGENCE In this section, we introduce the rate of the B-statistical convergence for a sequence of positive linear operators defined on C ∗ (R) into itself by using the modulus of continuity. We first present Definition 3 below. Definition 3. Let B ∈ R+ , and also let (u n ) is positive and non-decreasing sequence. We say that the sequence (xn ) is B-statistical convergent to a number with rate o(u n ) if, for each > 0, 1 bm,k (i) = 0, lim n→∞ u n k∈K where K = {k : k n and |xk − | }. In this case, we write xn − = statB − o(u n ). We now state and prove Lemma 1 as follows. Lemma 1. Let (u n ) and (vn ) are two positive non-decreasing sequences. Assume that B ∈ R+ , and let x = (xm ) and y = (ym ) be two sequences such that xk − 1 = statB − o(u n ) and yk − 2 = statB − o(vn ). Then each of the following assertions hold true: (i) (xk − L 1 ) ± (yk − 2 ) = statB − o(wn ); (ii) (xk − L 1 )(yk − 2 ) = statB − o(u n vn ); (iii) γ (x k − 1 ) = statB − o(u n ) (for any scalar γ ); √ (iv) |xk − 1 | = statB − o(u n ), where wn = max{u n , vn }. Proof. In order to prove the assertion (i) of Lemma 1, we choose > 0 and define the following sets Tn = k : k N and | (xk + yk ) − (1 + 2 )| , T0;n = k : k N and |xk − 1 | I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS 2 Vol. 10, No. 1 (Special Issue), Jan–June 2019 82 B.B. Jena, S.K. Paikray and U.K. Misra and T1,n = k : k N and |yk − 2 | . 2 Clearly, we have Tn ⊆ T0,n ∪ T1,n and this implies, for n ∈ N, that lim bm,k (i) lim bm,k (i) + lim bm,k (i). n→∞ n→∞ k∈Tn n→∞ k∈T0,n (3.1) k∈,T 1,n Moreover, since wn = max{u n , vn }, (3.2) by (3.1), we get lim n→∞ 1 1 1 bm,k (i) lim bm,k (i) + lim bm,k (i). n→∞ n→∞ wn k∈T u n k∈T vn k∈,T n 0,n (3.3) 1,n Also, by applying the Theorem 1, we obtain 1 bm,k (i) = 0, n→∞ wn k∈T lim (3.4) n which proves the assertion (i) of Lemma 1. The other assertion (ii) to (iv) of Lemma 1 are similar to (i), so it is not difficult to prove these assertions along similar lines. This, evidently completes the proof Lemma 1. Recalling that the modulus of continuity of a function f (x) ∈ C ∗ [π, π ] is defined by | f (t) − f (x)| : (t − x)2 δ (δ > 0), (3.5) ω( f ; δ) = sup (t,x)∈[−π,π] which implies | f (t) − f (x)| ω f ; (t − x)2 . (3.6) Now we propose a theorem to get the rates of B-statistical convergence by using the modulus of continuity in (3.5). Theorem 2. Let B ∈ R+ , and let L n : C ∗ [−π, π ] → C ∗ [−π, π ] be sequences of positive linear operators. Also let (u n ) and (vn ) be the positive non-decreasing sequences. Suppose that the following conditions are satisfied: I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Approximation of Periodic Functions via Statistical B-summability 83 (i) L n (1; x) − 1C ∗ [−π,π] = statB − o(u n ); (ii) ω( f, λn ) = statB − o(vn ) on [−π, π], where λn = L n (ϕ 2 (t, x)C ∗ [−π,π] ϕ(t) = sin2 and (t − x) . 2 Then, for all f ∈ C ∗ [−π, π], the following assertion holds true: L n ( f ; x) − f (x)C ∗ [−π,π] = statB − o(wn ), (3.7) where (wn ) is given by (3.2). Proof. Let f ∈ C ∗ [−π, π] and by using (3.6), we have |L n ( f ; x) − f (x)| L n (| f (t) − f (x)|; x) + | f (x)||L n (1; x) − 1|, (t − x)2 + 1; x ω( f, δ) + N |L n (1; x) − 1|, Ln δ L n (1; x) + 1 L n (ϕ(t); x) ω( f, δ) + N |L n (1; x) − 1|, δ2 where N = f C B (D) . Taking the supremum over (x, y) ∈ D on both sides, we have L n ( f ; x) − f (x)C B (D) ω( f, δ) Now, putting δ = λn = 1 L (ϕ(t); x) + L (1; x) − 1 + 1 n C (D) n C (D) B B δ2 +N L n (1; x) − 1C B (D) . L n (ϕ 2 ; x), we get L n ( f ; x) − f (x)C B (D) ω( f, λn ) L n (1; x) − 1C B (D) + 2 + N L n (1; x) − 1C B (D) ω( f, λn )L n (1; x) − 1C B (D) + 2ω( f, λn ) + N L n (1; x) − 1C B (D) . So, we have L m ( f ; x) − f (x)C B (D) μ ω( f, λn )L n (1; x) − 1C B (D) + ω( f, λn ) + L n (1; x) − 1C B (D) where μ = max{2, N }. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 84 B.B. Jena, S.K. Paikray and U.K. Misra For a given > 0, we consider the following sets: Hn = n : n ∈ N and L n ( f ; x) − f (x)C B (D) ; H0,n = n : n ∈ N and ω( f, λn )Tn ( f ; x) − f (x)C B (D) ; 3μ H1,n = n : n ∈ N and ω( f, λn ) 3μ and H2,n (3.8) (3.9) (3.10) . = n : n ∈ N and L n (1; x) − 1C B (D) 3μ (3.11) Finally, in view of the conditions (i) and (ii) of Theorem 2 in conjunction with Lemma 1, this last inequalities (3.8)-(3.11) leads us to the assertion (3.7) of Theorem 2. This completes the proof of Theorem 2. 4. CONCLUDING REMARKS AND OBSERVATIONS In this concluding section of our investigation, we present several further remarks and observations concerning the various results which we have proved here. Remark 5. Let (xn )n∈N be the sequence given in Example 2. Then, since C ∗ (R), Bstat lim xn → 0 on n→∞ we have ∞ stat lim bn,k (i)L n ( f j ; x) − f j (x) n = 0, f ∈ C ∗ (R). (4.1) = 0, f ∈ C ∗ (R) (4.2) C ∗ (R) n=0 Therefore, by applying Theorem 1, we write ∞ stat lim bn,k (i)L n ( f ; x) − f (x) n n=0 C ∗ (R) where f 0 = 1, f 1 = cos x and f 2 = sin x. However, since (xn ) is not ordinarily convergent and so also it does not converge uniformly in the ordinary sense. Thus, the classical Korovkin Theorem does not work here for the operators defined by (2.12). Hence, this application clearly indicates that our Theorem 1 is a non-trivial generalization of the classical Korovkin-type theorem (see [16]). I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Approximation of Periodic Functions via Statistical B-summability 85 Remark 6. Let (xn )n∈N be the sequence as given in Example 2. Then, since Bstat lim xn → 0 on C ∗ (R), n→∞ so (4.1) holds. Now by applying (4.1) and our Theorem 1, condition (4.2) holds. However, since (xn ) does not statistical A-summable, so we can say that the result of Edely and Mursaleen ( [7], p. 162, Theorem 2.1) does not hold true for our operator defined in (2.12). Thus, our Theorem 1 is also a non-trivial extension of Edely and Mursaleen ( [7], p. 162, Theorem 2.1) and [16]. Based upon the above results, it is concluded here that our proposed method has successfully worked for the operators defined in (2.12) and therefore it is stronger than the classical and statistical version of the Korovkin type approximation theorem (see [16] and [25] ) established earlier. Remark 7. Let us suppose we replace the conditions (i) and (ii) in our Theorem 2 by the following condition: |L n ( f j ; x) − f j (x)|C ∗ (R) = statB − o(u n j ) ( j = 0, 1, 2). (4.3) Now, we can write L n (ϕ 2 ; x) = N 2 L n ( f j (t); x) − f j (x)C ∗ (R) , (4.4) j=0 where N = 1 + f 1 C ∗ (R) + f 2 C ∗ (R) , ( j = 0, 1, 2). It now follows from (4.3), (4.4) and Lemma 1 that λn = L n (ϕ 2 ) = statB − o(dn ) on C ∗ (R), (4.5) where o(dn ) = max{u n 0 , u n 1 , u n 2 }. Hence, we get ω( f, δ) = statB − o(dn ) on C ∗ (R). By using (4.5) in Theorem 2, we immediately see for all f ∈ CB (D) that L n ( f ; x) − f (x) = statB − o(dn ) on C ∗ (R). (4.6) Therefore, if we use the condition (4.3) in Theorem 2 instead of conditions (i) and (ii), then we obtain the rates of the statistical B-summability of the sequence (L n ) of positive linear operators in Theorem 1. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 86 B.B. Jena, S.K. Paikray and U.K. Misra REFERENCES [1] A. Alotaibi and M. Mursaleen, Generalized statistical convergence of difference sequences, Adv. Differ. Equ. 2013 (2013), Article ID 212, 1–5. [2] C. Belen and S. A. Mohiuddine, Generalized weighted statistical convergence and application, Appl. Math. Comput., 219 (2013), 9821–9826. [3] H. T. 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I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 INDIAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019, pp. 87–108 DOI: On Semi-infinite Optimization Problem Under the Generalized Convexity Pushkar Jaisawal∗ , Vivek Laha and S.K. Mishra Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi-221005, India ∗ Corresponding author Email: [email protected] Abstract: In this paper, we study nonsmooth semi-infinite optimization problems involving infinite number of inequality constraints. We derive sufficient optimality conditions for a point to be an optimal solution of the problem under generalized convexity assumptions. We also formulate Wolfe and Mond-Weir type dual models and establish several duality results. Keywords: Sufficient optimality condition, Nonsmooth optimization, Semi-infinite programming, Wolfe type dual, Mond-Weir type dual, Generalized convexity 1. INTRODUCTION An optimization problem in which infinite (countable or uncountable) number of inequality constraints present with finite number of variables is known as Semi-Infinite Optimization problem. We examine the semi-infinite optimization programming (SIP) problem which is nonsmooth and presented in Kanzi and Nobakhtian [15] (SIP) min f (x), subject to gi (x) ≤ 0, ∀i ∈ I, x∈Rn where f : Rn → R ∪ {+∞} , gi : Rn → R ∪ {+∞}, ∀i ∈ I , all are locally Lipschitz functions and I is the index set which is infinite (countable or uncountable). Using Clarke’s subdifferential [4] Kanzi and Nobakhtian [15] collected the necessary and sufficient optimality conditions and Mishra et al. [21] established the Wolfe [27] and Mond-Weir [23] dual problems and related results like weak duality, strong duality and strict converse duality results under invexity assumptions for nonsmooth (SIP). Semi-infinite programming problems are used in engineering, optimal control theory, robust optimization, social work and different fields of mathematics etc. Because of this, many scholars are interested in working in this field, see e.g., [8–10, 12, 20, 24]. We have also seen the work of past decades, which have been done by Dinh et al. [6], Canovas et al. [2, 3], Shapiro [24, 25] and also see this [16–19] for the application of (SIP). I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 88 Pushkar Jaisawal, Vivek Laha and S.K. Mishra Sufficient optimality results play an important role in optimization problems. Wolfe duality [27] and Mond-Weir duality [23] also play a very important role for optimization. Wolfe [27] in 1961 gave the Wolfe type dual problem and Mond and Weir [23] in 1981 gave the Mond-Weir type dual problem, see e.g., Jeyakumar [13], Daldoul and Baccari [5] and Wang et al. [26]. In the above (SIP) problem Kanzi and Nobakhtian [15] and Mishra et al. [21] gave the results for invex functions. The requirement of invex function is limited to the same kernal function for all the constraints and objective functions. For that reason, in this paper, we establish the sufficient optimality conditions and duality results for (SIP) under V −invex function. The concept of V −invex function was first established in 1992 by Jeyakumar and Mond [14]. The outline of this paper is as follows: in Section 2, we give some well-known definitions and KKT condition given by Kanzi and Nobakhtian [15]. In Section 3, we give some definitions which will be used in the sequel, we derive sufficient optimality conditions for (SIP) involving V −invex function and generalized V −invex function both using Clarke’s subdifferential. In Section 4, we derive a new type of Wolfe dual model and Mod-Weir dual model for r −invex and V −invex functions, respectively, and establish weak duality, strong duality and strict converse duality results for both the dual models under V −invexity and generalized V −invexity assumptions with the use of Clarke’s subdifferential. And lastly in Section 5, we conclude the results present in this paper. 2. PRELIMINARIES Let S be a nonempty and open subset of Rn , we denote by cone(S), S̄ and conv(S), the convex cone generated by S containing the origin of Rn , closure of S and convex hull of S. The strictly negative and negative polar cones of S are denoted by S S N and S N , respectively, and defined by: S S N = {d ∈ Rn : x, d < 0, ∀x ∈ S} S N = {d ∈ Rn : x, d ≤ 0, ∀x ∈ S} Definition 1. (Contingent cone) Let x̄ ∈ S̄. The contingent cone of S at x̄ is denoted by K (S, x̄) and defined by x t − x̄ t t t K (S, x̄) := lim : λ → 0 , x ∈ S, x → x̄ . + t→∞ λt Definition 2. (Lipschitz function) Let f be a function defined by f : Rn → R, if there exist a k, where k is a positive constant such that | f (x) − f (y)| ≤ k||x − y|| then f is said to be locally Lipschitz at z ∈ S where x, y ∈ N , N is a neighbourhood of z and S be an open subset of Rn . Also, if it satisfies for every z ∈ S then f is said to be Lipschitz on S. Clarke [4] introduced the concept of Clarke’s tangent cone and Clarke’s subdifferential. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 On Semi-infinite Optimization Problem Under the Generalized Convexity 89 Definition 3 [4]. (Clarke’s subdifferential) A vector x ∗ ∈ Rn is a Clarke’s subdifferential vector of f at x̄ ∈ S if it satisfies f 0 (x̄; d) ≥ x ∗ , d , ∀d ∈ Rn where f 0 (x̄; d) is a generalized Clarke’s directional derivative of a locally Lipschitz function f at x̄ ∈ S in the direction d and it is defined by f 0 (x̄, d) = lim sup y→x̄, t↓0 f (y + td) − f (y) . t The collection of all such vectors is called the Clarke’s subdifferential at x̄ ∈ S. It is denoted by ∂ C f (x̄) and defined by ∂ C f (x̄) = {x ∗ ∈ Rn : f 0 (x̄; d) ≥ x ∗ , d , ∀d ∈ Rn }. Definition 4 [4]. (Clarke’s tangent cone) Collection of all the vectors v ∈ Rn is called the Clarke’s tangent cone of S at x̄ ∈ S which satisfies d S0 (x̄; v) = 0. It is denoted by T (S, x̄) and mathematically express by T (S, x̄) = v ∈ Rn : d S0 (x̄; v) = 0 , where d S (x) is the distance function related to S and defined by d S (x) = inf{||x − y|| : y ∈ S}. In 1981 Hanson [11] gave the concept of differential invex function. The generalization of invex function was introduced by Jeyakumar and Mond [14] which is known as V −invex function. The generalization of V −invex function for the nonsmooth case was given by Egudo and Hanson [7] using the Clarke’s generalized subdifferential [4]. Definition 5 [7]. (Clarke V −invex function) The function f defined by f := ( f1 , ..., f m ) : S ⊆ Rn → Rm is locally Lipschitz near x̄ ∈ S. The function f is said to be Clarke V −invex at x̄ ∈ S with respect to η : Rn × Rn → Rn and αi : Rn × Rn → R+ \ {0}, for all i ∈ M = {1, ..., m} over S, iff for all i ∈ M, x ∈ S and x̄i∗ ∈ ∂ C f i (x̄), we have f i (x) − f i (x̄) ≥ αi (x, x̄) x̄i∗ , η(x, x̄) . The function f is said to be Clarke V −invex on S with respect to η and αi , for i ∈ M, iff f is Clarke V −invex at x̄ ∈ S over S with respect to η and αi , for all i ∈ M and for all x̄ ∈ S. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 90 Pushkar Jaisawal, Vivek Laha and S.K. Mishra Definition 6 [7]. (Clarke V −strictly-invex function) The function f defined by f := ( f 1 , ..., f m ) : S ⊆ Rn → Rm is locally Lipschitz near x̄ ∈ S. The function f is said to be Clarke V −strictlyinvex at x̄ ∈ S with respect to η : Rn × Rn → Rn and αi : Rn × Rn → R+ \ {0}, for all i ∈ M = {1, ..., m} over S, iff for all i ∈ M, x ∈ S, x = x̄ and for all x̄i∗ ∈ ∂ C f i (x̄), we have f i (x) − f i (x̄) > αi (x, x̄) x̄i∗ , η(x, x̄) . The function f is said to be Clarke V −strictly-invex on S with respect to η and αi , for all i ∈ M, iff f is Clarke V −strictly-invex at x̄ ∈ S over S with respect to η and αi , for all i ∈ M and for all x̄ ∈ S. In 1996 Mishra and Mukherjee [22] introduced the following concept of V −pseudo-invex and V −quasi-invex functions. Definition 7 [22]. (Clarke V −pseudo-invex function) The function f defined by f := ( f1 , ..., f m ) : S ⊆ Rn → Rm is locally Lipschitz near x̄ ∈ S. The function f is said to be Clarke V −pseudoinvex at x̄ ∈ S if there exist functions η : Rn × Rn → Rn and αi : Rn × Rn → R+ \ {0}, for all i ∈ M = {1, ..., m} such that for x, x̄ ∈ S αi (x, x̄) f i (x) < αi (x, x̄) f i (x̄), i∈M ⇒ for any i∈M x̄i∗ ∈ ∂ f i (x̄), C x̄i∗ , η(x, x̄) < 0. i∈M The function f is said to be Clarke V −pseudo-invex on S with respect to η and αi , for all i ∈ M, iff f is Clarke V −pseudo-invex at x̄ ∈ S over S with respect to η and αi , for all i ∈ M and for all x̄ ∈ S. Definition 8 [22]. (Clarke V −quasi-invex function) The function f defined by f := ( f1 , ..., f m ) : S ⊆ Rn → Rm is locally Lipschitz near x̄ ∈ S. The function f is said to be Clarke V −quasi-invex at x̄ ∈ S if there exist functions η : Rn × Rn → Rn and αi : Rn × Rn → R+ \ {0}, for all i ∈ M = {1, ..., m} such that for x, x̄ ∈ S αi (x, x̄) f i (x) ≤ αi (x, x̄) f i (x̄), i∈M ⇒ for any i∈M x̄i∗ ∈ ∂ f i (x̄), C x̄i∗ , η(x, x̄) ≤ 0. i∈M The function f is said to be Clarke V −quasi-invex on S with respect to η and αi , for all i ∈ M, iff f is Clarke V −quasi-invex at x̄ ∈ S over S with respect to η and αi , for all i ∈ M and for all x̄ ∈ S. Definition 9. (Clarke V −strictly-pseudo-invex function) The function f defined by f := ( f 1 , ..., f m ) : S ⊆ Rn → Rm is locally Lipschitz near x̄ ∈ S. The function f is said to be Clarke V −strictly-pseudo-invex at x̄ ∈ S if there exist functions η : Rn × Rn → Rn and αi : Rn × Rn → I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 On Semi-infinite Optimization Problem Under the Generalized Convexity 91 R+ \ {0}, for all i ∈ M = {1, ..., m} such that for x, x̄ ∈ S αi (x, x̄) f i (x) ≤ αi (x, x̄) f i (x̄), i∈M ⇒ for any i∈M x̄i∗ ∈ ∂ f i (x̄), C x̄i∗ , η(x, x̄) < 0. i∈M The function f is said to be Clarke V −strictly-pseudo-invex on S with respect to η and αi , for all i ∈ M, iff f is Clarke V −strictly-pseudo-invex at x̄ ∈ S over S with respect to η and αi , for all i ∈ M and for all x̄ ∈ S. Consider the following constraint qualification from [15]. Definition 10. (ACQ) The Abadie constraint qualification (ACQ) holds at x̄ ∈ S if B 0 (x̄) ⊆ K (S, x̄), where B(x̄) := i∈I (x̄) ∂ C gi (x̄) and I (x̄) is the active constraints. The KKT condition established by Kanzi and Nobakhtian [15] in terms of Clarke’s subdifferentials is as follows: Theorem 1. (KKT Condition) Suppose that x̄ is an optimal solution of (SIP), A(x̄) := cone(B(x̄)) is closed, (ACQ) holds and I (x̄) = φ. Then there exist λi ≥ 0, i ∈ I (x̄), with λi = 0, for finitely many indexes such that, λi ∂ C gi (x̄). (2.1) 0 ∈ ∂ C f (x̄) + i∈I (x̄) 3. SUFFICIENT CONDITIONS We consider the following (SIP) problem (SIP) min f (x), subject to gi (x) ≤ 0, ∀i ∈ I, x∈Rn where f, gi : Rn → R ∪ {+∞}, for all i ∈ I , all are locally Lipschitz functions and I is the index set which is infinite (countable or uncountable). Let S = {x ∈ Rn : gi (x) ≤ 0} and I (x̄) = {i ∈ I : gi (x̄) = 0}, where x̄ ∈ S. We consider the following possibilities for the functions ( f, g I ). (a) The function ( f, g I ) is said to be the V −invex at x̄ ∈ S, iff there exist β, αi : Rn × Rn → R+ \ {0}, for all i ∈ I and η : Rn × Rn → Rn such that f (x) − f (x̄) ≥ β(x, x̄) x̄ ∗ , η(x, x̄) , ∀x̄ ∗ ∈ ∂ C f (x̄) (3.1) gi (x) − gi (x̄) ≥ αi (x, x̄) x̄i∗ , η(x, x̄) , ∀x̄i∗ ∈ ∂ C gi (x̄), ∀i ∈ I. (3.2) and I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 92 Pushkar Jaisawal, Vivek Laha and S.K. Mishra (b) The function ( f, g I ) is said to be the V −strictly-invex at x̄ ∈ S, iff there exist β, αi : Rn × Rn → R+ \ {0}, for all i ∈ I and η : Rn × Rn → Rn such that f (x) − f (x̄) > β(x, x̄) x̄ ∗ , η(x, x̄) , ∀x̄ ∗ ∈ ∂ C f (x̄) (3.3) gi (x) − gi (x̄) ≥ αi (x, x̄) x̄i∗ , η(x, x̄) , ∀x̄i∗ ∈ ∂ C gi (x̄), ∀i ∈ I. (3.4) and (c) The function ( f, g I ) is said to be the V −pseudo-quasi-invex at x̄ ∈ S, iff there exist β, αi : Rn × Rn → R+ \ {0}, for all i ∈ I and η : Rn × Rn → Rn such that β(x, x̄) f (x) < β(x, x̄) f (x̄) ⇒ for any x̄ ∗ ∈ ∂ C f (x̄), and αi (x, x̄)gi (x) ≤ i∈I αi (x, x̄)gi (x̄) ⇒ for any x̄i∗ ∈ ∂ C gi (x̄), i∈I x̄ ∗ , η(x, x̄) < 0 x̄i∗ , η(x, x̄) ≤ 0. (3.5) (3.6) i∈I (d) The function ( f, g I ) is said to be the V −strictly-pseudo-quasi-invex at x̄ ∈ S, iff there exist β, αi : Rn × Rn → R+ \ {0}, for all i ∈ I and η : Rn × Rn → Rn such that β(x, x̄) f (x) ≤ β(x, x̄) f (x̄) ⇒ for any x̄ ∗ ∈ ∂ C f (x̄), and i∈I αi (x, x̄)gi (x) ≤ αi (x, x̄)gi (x̄) ⇒ for any x̄i∗ ∈ ∂ C gi (x̄), i∈I x̄ ∗ , η(x, x̄) < 0 x̄i∗ , η(x, x̄) ≤ 0. (3.7) (3.8) i∈I In this section, we analyse sufficient optimality conditions of the above taken programming problem. Theorem 1. Let x̄ ∈ S and I (x̄) = φ. Suppose that there exist λi ≥ 0, i ∈ I (x̄), with λi = 0, for finitely many indexes, and 0 ∈ ∂ C f (x̄) + λi ∂ C gi (x̄) (3.9) i∈I (x̄) holds, where I (x̄) = {i ∈ I : gi (x̄) = 0}. Moreover, suppose that ( f, g I (x̄) ) are V −invex at x̄. Then x̄ is a global solution of (SIP). Proof. The given condition is 0 ∈ ∂ C f (x̄) + λi ∂ C gi (x̄), i∈I (x̄) I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 On Semi-infinite Optimization Problem Under the Generalized Convexity then there exist x̄ ∗ ∈ ∂ C f (x̄) and x̄i∗ ∈ ∂ C gi (x̄), such that 0 = x̄ ∗ + λi x̄i∗ . 93 (3.10) i∈I (x̄) Let x ∈ S, then for each i ∈ I (x̄), we have gi (x) ≤ 0 = gi (x̄), ∀i ∈ I (x̄) and ∀x ∈ S, then, gi (x) − gi (x̄) ≤ 0, ∀i ∈ I (x̄) and ∀x ∈ S. (3.11) By the V −invexity of ( f, g I (x̄) ) at x̄, we obtain, f (x) − f (x̄) ≥ β(x, x̄) x̄ ∗ , η(x, x̄) , ∀x̄ ∗ ∈ ∂ C f (x̄) and ∀x ∈ S, gi (x) − gi (x̄) ≥ αi (x, x̄) x̄i∗ , η(x, x̄) , ∀i ∈ I (x̄), ∀x ∈ S and ∀x̄i∗ ∈ ∂ C gi (x̄). (3.12) (3.13) By (3.11), we obtain 0 ≥ gi (x) − gi (x̄) ≥ αi (x, x̄) x̄i∗ , η(x, x̄) , ∀i ∈ I (x̄), ∀x ∈ S and ∀x̄i∗ ∈ ∂ C gi (x̄), (3.14) that is, 0 ≥ αi (x, x̄) x̄i∗ , η(x, x̄) , ∀x ∈ S and ∀x̄i∗ ∈ ∂ C gi (x̄), ∀i ∈ I (x̄), then, there exist λi ≥ 0, for all i ∈ I (x̄), we get, λi x̄i∗ , η(x, x̄) ≤ 0, ∀x̄i∗ ∈ ∂ C gi (x̄), ∀x ∈ S. (3.15) (3.16) i∈I (x̄) Using (3.10), we get, x̄ ∗ , η(x, x̄) ≥ 0, ∀x ∈ S and ∃ x̄ ∗ ∈ ∂ C f (x̄). (3.17) β(x, x̄) x̄ ∗ , η(x, x̄) ≥ 0, ∀x ∈ S and ∃ x̄ ∗ ∈ ∂ C f (x̄). (3.18) Then, By (3.12), we get, f (x) − f (x̄) ≥ 0, ∀x ∈ S. Hence, we obtain the result. This case is validated by the following example. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 94 Pushkar Jaisawal, Vivek Laha and S.K. Mishra Example 1. Examine the specified semi-infinite programming problem (P) min f (x) subject to gi (x) ≤ 0, ∀i ∈ N , and g j (x) ≤ 0, ∀ j ∈ N , x ∈ R2 , x1 1 1 , x2 = 0, gi (x) = x2 − x1 − and g j (x) = −x2 − . It is easy to verify that x2 i j f (x), gi (x) and g j (x), for all i, j ∈ N all are V −invex functions with respect to β(x, y) = y2 , αi (x, y) = α j (x, y) = 1, and η(x, y) = x − y, respectively, at for all x1 = x2 points in the feax2 sible set, where x = (x1 , x2 ), y = (y1 , y2 ) ∈ R2 . Using the above theorem, we can easily to see that all the feasible points x = (x1 , x2 ) which are like x1 = x2 are optimal solution of the problem. where f (x) = Theorem 2. Let x̄ ∈ S and I (x̄) = φ. Suppose that there exist λi ≥ 0, i ∈ I (x̄), with λi = 0, for finitely many indexes, such that (3.9) holds. Moreover, Suppose that ( f, g I (x̄) ) is V −pseudo-quasiinvex at x̄. Then, x̄ is a global solution of (SIP). Proof. Since (3.9) holds, then, there exist x̄ ∗ ∈ ∂ C f (x̄) and x̄i∗ ∈ ∂ C gi (x̄), for all i ∈ I (x̄), such that, λi x̄i∗ . (3.19) 0 = x̄ ∗ + i∈I (x̄) Let x ∈ S. Then, for each i ∈ I (x̄), we have, gi (x) ≤ 0 = gi (x̄), ∀x ∈ S. (3.20) Then, there exist αi (x, x̄) ∈ R+ \ {0}, such that, αi (x, x̄)gi (x) ≤ αi (x, x̄)gi (x̄), ∀i ∈ I (x̄), ∀x ∈ S, which implies that, i∈I (x̄) αi (x, x̄)gi (x) ≤ αi (x, x̄)gi (x̄), ∀x ∈ S. (3.21) i∈I (x̄) Since ( f, g I (x̄) ) is V −pseudo-quasi-invex at x̄, then, there exist β(x, x̄), αi (x, x̄) ∈ R+ \ {0}, for all i ∈ I (x̄) and for all x ∈ S, such that, β(x, x̄) f (x) < β(x, x̄) f (x̄) ⇒ for any x̄ ∗ ∈ ∂ C f (x̄), I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS x̄ ∗ , η(x, x̄) < 0 (3.22) Vol. 10, No. 1 (Special Issue), Jan–June 2019 On Semi-infinite Optimization Problem Under the Generalized Convexity 95 and αi (x, x̄)gi (x) ≤ i∈I (x̄) αi (x, x̄)gi (x̄) ⇒ for any x̄i∗ ∈ ∂ C gi (x̄), i∈I (x̄) x̄i∗ , η(x, x̄) ≤ 0. i∈I (x̄) (3.23) Using (3.21) in (3.23), we get, x̄i∗ , η(x, x̄) ≤ 0, ∀x̄i∗ ∈ ∂ C gi (x̄), ∀x ∈ S, (3.24) i∈I (x̄) then, there exist λi ≥ 0, for all i ∈ I (x̄), we get, λi x̄i∗ , η(x, x̄) ≤ 0, ∀x̄i∗ ∈ ∂ C gi (x̄), ∀x ∈ S. (3.25) i∈I (x̄) By (3.19), we get, x̄ ∗ , η(x, x̄) ≥ 0, for ∃ x̄ ∗ ∈ ∂ C f (x̄), ∀x ∈ S. (3.26) Assume f (x) < f (x̄), for all x ∈ S, and there exist β(x, x̄) ∈ R+ \ {0}, then, β(x, x̄) f (x) < β(x, x̄) f (x̄), ∀x ∈ S. (3.27) x̄ ∗ , η(x, x̄) < 0, ∀x̄ ∗ ∈ ∂ C f (x̄), ∀x ∈ S. (3.28) By (3.22), we get Which is contradiction of (3.26). Hence the result. 4. DUALITY In this section, we can develop Wolfe and Mond-Weir type dual models and establish the weak, strong and strict converse duality results for Wolfe type dual model and Mond-Weir type dual model under the r −invexity and V −invexity assumptions, respectively. Also, we can derive these results for Mond-Weir dual model in terms of generalized V −invexity assumptions. Antczak [1] introduced a class of (scalar) Lipschitz r −invex functions, which is Definition 1. [1] (r −invex functions) Let f : Rn → R be a Lipschitz function on a nonempty set S ⊆ Rn , and let r be an arbitrary real number. If, for all x ∈ S, the inequality 1. 2. ≥ r1 er f (x̄) [1 + r x̄ ∗ , η(x, x̄) ] (> and x = x̄) for r = 0, f (x) − f (x̄) ≥ x̄ ∗ , η(x, x̄) (> and x = x̄) for r = 0, 1 r f (x) e r holds, then, f is said to be (strict) r −invex with respect to η at x̄ on S. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 96 Pushkar Jaisawal, Vivek Laha and S.K. Mishra 4.1. Wolfe-type Dual In this subsection, first of all, we develop a dual model for the nonsmooth semi-infinite programming problem taken up using the Clarke’s subdifferentials, which is called the Wolfe-type dual problem (WSID). After that, we get some fundamental results, which is a weak duality, strong duality and strictly converse duality theorems. We start with Wolfe-type dual problem which is given by (WSID) λi gi (u) Maximize f (u) + i∈I subject to 0 ∈ ∂ C ( f (u) + λi gi (u)), i∈I where λi ≥ 0, for i ∈ I and λi = 0, for finitely many i ∈ I (u). The feasible solution of the (WSID) problem is denoted by W. Further, we denote by SW the following set SW = {u ∈ Rn : (u, λ) ∈ W }. Theorem 1. (Weak Duality) Let x be feasible solution of the (SIP) and (x̄, λ) be feasible for (WSID). If the Lagrangian i.e., ( f (.) + i∈I λi gi (.)) is r −invex with respect to η at x̄ over S ∪ SW . Then, we have, f (x) ≥ f (x̄) + i∈I λi gi (x̄). Proof. Let x be feasible for (SIP) and (x̄, λ) be feasible for (WSID) and λi ≥ 0, for all i ∈ I. Then, we have, gi (x) ≤ 0, ∀i ∈ I, which implies that, λi gi (x) ≤ 0, ∀i ∈ I, (4.1) and 0 ∈ ∂ C ( f (x̄) + λi gi (x̄)), i∈I then, there exist x̄ ∗ ∈ ∂ C f (x̄) and x̄i∗ ∈ ∂ C gi (x̄), for all i ∈ I, such that 0 = x̄ ∗ + λi x̄i∗ . (4.2) i∈I We start with the contradiction. Suppose that f (x) < f (x̄) + λi gi (x̄), (4.3) i∈I I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 On Semi-infinite Optimization Problem Under the Generalized Convexity 97 by (4.1) and (4.3), we get f (x) + λi gi (x) < f (x̄) + i∈I λi gi (x̄). (4.4) i∈I By assumptions, the Lagrangian is r −invex with respect to η at x̄ over S ∪ SW , then, 1 r( f + e r i∈I λi gi )(x) 1 ≥ er ( f + r i∈I λi gi )(x̄) ∗ × 1 + r x̄ + λi x̄i∗ , η(x, x̄) , ∀x ∈ S ∪ SW . (4.5) i∈I Then, by (4.4) and (4.5), we obtain x̄ ∗ + λi x̄i∗ , η(x, x̄) < 0, (4.6) i∈I holds for each x̄ ∗ ∈ ∂ C f (x̄) and x̄i∗ ∈ ∂ C gi (x̄), for all i ∈ I. Which is the contradiction. Hence our supposition is wrong. Hence, f (x) ≥ f (x̄) + λi gi (x̄). i∈I Hence the result. The following theorem gives strong duality results between (SIP) and (WSID) under the assumptions that the dual objective function is Clarke regular. Theorem 2. (Strong Duality) Let x̄ be an optimal solution for (SIP) at which a constraint qualification is satisfied. Then, there exist λ̄ = (λ̄i ), i ∈ I, such that λ̄i gi (x̄) = 0, for all i ∈ I and (x̄, λ̄) is feasible for (WSID) . If weak duality holds between (SIP) and (WSID), then, (x̄, λ̄) is optimal for (WSID) and the respective objective values are equal. Proof. Let x̄ is an optimal solution for (SIP) and a suitable constraints qualification is satisfied, hence there exist λ̄ = (λ̄i ), i ∈ I such that the KKT conditions for (SIP) are satisfied. That is, 0 ∈ ∂ C ( f (x̄) + λ̄i gi (x̄)), (4.7) i∈I therefore (x̄, λ̃) is a feasible solution for (WSID). By assumptions, λ̄i gi (x̄) = 0. (4.8) I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 i∈I 98 Pushkar Jaisawal, Vivek Laha and S.K. Mishra We proceed by contradiction. Suppose that (x̄, λ̄) is not an optimal solutions for (WSID). Then, there exist at least one element of the feasible set in (WSID), which is ( ỹ, λ̃) such that f (x̄) + λ̄i gi (x̄) < f ( ỹ) + i∈I λ̃i gi ( ỹ), (4.9) i∈I using (4.8), we get the inequality f (x̄) < f ( ỹ) + λ̃i gi ( ỹ), (4.10) i∈I which contradicts the weak duality theorem for (WSID). Hence our supposition is wrong, i.e., (x̄, λ̄) is an optimal solution for (WSID) and the respective objective values are equal. Hence the result. Theorem 3. (Strict Converse Duality) Let x̃ be an optimal solutions for (SIP) and (x̄, λ̄) be an optimal solution for (WSID). If a constraint qualification is satisfied for (SIP) and the Lagrange function is strictly r −invex at x̄ on S ∪ SW , then, x̄ = x̃. Proof. We proceed by contradiction. Assume x̃ = x̄. By strong duality theorem for (WSID) there exist λ∗ = (λi∗ ), i ∈ I and λi∗ ≥ 0 such that (x̃, λ∗ ) is optimal for (WSID). Hence, λi∗ gi (x̃) = f (x̄) + (4.11) λ̄i gi (x̄). f (x̃) = f (x̃) + i∈I i∈I Again, by 0 ∈ ∂ C ( f (x̄) + λ̄i gi (x̄)), i∈I then, there exist x̄ ∗ ∈ ∂ C f (x̄) and x̄i∗ ∈ ∂ C gi (x̄), such that 0 = x̄ ∗ + λ̄i x̄i∗ . (4.12) i∈I Now the Lagrangian function is strictly r −invex at x̄ on S ∪ SW , then, we get 1 r( f + e r i∈I λ̄i gi )(x) 1 r( f + e r ≥ i∈I λ̄i gi )(x̄) × 1 + r x̄ ∗ + λ̄i x̄i∗ , η(x, x̄) , ∀x ∈ S ∪ SW . (4.13) i∈I then, by (4.12), we get (f + i∈I λ̄i gi )(x) > ( f + λ̄i gi )(x̄), ∀x ∈ S ∪ SW , (4.14) i∈I I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 On Semi-infinite Optimization Problem Under the Generalized Convexity 99 then, for x̃ ∈ S, we get (f + λ̄i gi )(x̃) > ( f + i∈I λ̄i gi )(x̄), (4.15) i∈I which implies that, f (x̃) > ( f + λ̄i gi )(x̄), (4.16) i∈I which contradicts the equation (4.11). Hence our supposition is wrong. Hence x̃ = x̄. This completes the proof. 4.2. Mond-Weir type dual In this subsection, first of all, we develop a dual model for the nonsmooth semi-infinite programming problem taken up using the Clarke’s subdifferentials, which is called the Mond-Weir-type dual problem (MWSID). After that, we get some fundamental results, which are weak duality, strong duality and strictly converse duality theorems. We start with Mond-Weir-type dual problem which is given by (MWSID) Maximize f (u) λi ∂ C gi (u), subject to 0 ∈ ∂ C f (u) + i∈I and λi gi (u) ≥ 0. i∈I Where λi ≥ 0, for i ∈ I and λi = 0, for finitely many i ∈ I (u). Theorem 4. (Weak Duality) Let x be feasible for (SIP) and (u, λ I ) where λ I = (λi ), i ∈ I, be feasible for (MWSID). Let ( f, g I ) be V −invex at u, where u ∈ Rn with respect to β : Rn × Rn → R+ \ {0} and αi : Rn × Rn → R+ \ {0}, for every i ∈ I. Then, we have, f (x) ≥ f (u). Proof. Let x be feasible for (SIP) and (u, λ I ) be feasible for (MWSID). Then, we have gi (x) ≤ 0, ∀i ∈ I, λi ∂ C gi (u) 0 ∈ ∂ C f (u) + (4.17) (4.18) i∈I I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 100 Pushkar Jaisawal, Vivek Laha and S.K. Mishra and λi gi (u) ≥ 0. (4.19) i∈I Hence, there exist u ∗ ∈ ∂ C f (u) and u i∗ ∈ ∂ C gi (u), i ∈ I such that λi u i∗ . 0 = u∗ + (4.20) i∈I Since ( f, g I ) are V −invex at u with respect to η and (β, α I ) where β : Rn × Rn → R+ \ {0}, and αi : Rn × Rn → R+ \ {0}, with λi ≥ 0, i ∈ I, we have f (x) − f (u) ≥ β(x, u) u ∗ , η(x, u) , ∀u ∗ ∈ ∂ C f (u) and ∀x ∈ S, then, f (x) − f (u) ≥ u ∗ , η(x, u) , ∀u ∗ ∈ ∂ C f (u) and ∀x ∈ S. β(x, u) (4.21) And gi (x) − gi (u) ≥ αi (x, u) u i∗ , η(x, u) , ∀u i∗ ∈ ∂ C gi (u), ∀i ∈ I and ∀x ∈ S, then, gi (x) − gi (u) ≥ u i∗ , η(x, u) , ∀u i∗ ∈ ∂ C gi (u), ∀i ∈ I and ∀x ∈ S, αi (x, u) which implies that, i∈I λi λi gi (x) − gi (u) ≥ λi u i∗ , η(x, u) , ∀u i∗ ∈ ∂ C gi (u) and ∀x ∈ S, αi (x, u) α (x, u) i i∈I i∈I we have, λi λi gi (x) − gi (u) ≥ λi u i∗ , η(x, u) , ∀u i∗ ∈ ∂ C gi (u) and ∀x ∈ S, α (x, u) α (x, u) i i i∈I i∈I i∈I (4.22) from (4.17), we get 0≥ i∈I λi gi (u) + λi u i∗ , η(x, u) , ∀u i∗ ∈ ∂ C gi (u) and ∀x ∈ S. αi (x, u) i∈I (4.23) Adding (4.21) and (4.23), we get f (x) − f (u) λi ≥ gi (u) + u ∗ + λi u i∗ , η(x, u) , ∀u i∗ ∈ ∂ C gi (u) and ∀x ∈ S, β(x, u) α (x, u) i i∈I i∈I I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 On Semi-infinite Optimization Problem Under the Generalized Convexity 101 then, by (4.20), we get f (x) − f (u) λi ≥ gi (u), ∀x ∈ S. β(x, u) αi (x, u) i∈I With the help of (4.19), we get, f (x) − f (u) ≥ 0, ∀x ∈ S. β(x, u) Hence, we get the required results, which is f (x) ≥ f (u). This completes the proof. Theorem 5. (Strong Duality) Let x̄ be an optimal solution for (SIP) at which a constraint qualification is satisfied. Then, there exist λ̄ = (λ̄i ), i ∈ I, such that λ̄i gi (x̄) = 0, for all i ∈ I and (x̄, λ̄) is feasible for (MWSID) . If weak duality holds between (SIP) and (MWSID), then, (x̄, λ̄) is optimal for (MWSID) and the respective objective values are equal. Proof. From the KKT conditions, there exist λ̄ = (λ̄i ), i ∈ I, such that 0 ∈ ∂ C f (x̄) + λ̄i ∂ C gi (x̄), i∈I and by assumptions, λ̄i gi (x̄) = 0, i ∈ I. Therefore, (x̄, λ̄) is feasible for (MWSID). On the other hand by weak duality theorem, we have f (x̄) ≥ f (u), for any feasible solution (u, λ) for (MWSID). Thus, (x̄, λ̄) is optimal for (MWSID) and the respective objective values are equal. Hence the results. Theorem 6. (Strict Converse Duality) Let ( f, g I ) are V −invex at x̄. let x̃ be an optimal solution for (SIP) and (x̄, λ̄) be an optimal solution for (MWSID). If a constraint qualification is satisfied for (SIP), and ( f, g I ) is V −strictly-invex for (SIP) at x̄, then, x̃ = x̄. Proof. We proceed by contradiction. Assume x̃ = x̄. By strong duality theorem for (MWSID) there exist λ̄ = (λ̄i ), i ∈ I and λ̄i ≥ 0 such that (x̄, λ̄) is optimal for (MWSID). I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 102 Pushkar Jaisawal, Vivek Laha and S.K. Mishra Hence, f (x̃) = f (x̄). (4.24) Again, by (4.2) there exist x̄ ∗ ∈ ∂ C f (x̄) and x̄i∗ ∈ ∂ C gi (x̄) such that 0 = x̄ ∗ + λ̄i x̄i∗ . (4.25) i∈I Now by V −strict-invexity of ( f, g I ) at x̄, with λi ≥ 0, such that f (x̃) − f (x̄) > β(x̃, x̄) x̄ ∗ , η(x̃, x̄) , ∀x̄ ∗ ∈ ∂ C f (x̄) and x̃ ∈ S, then, f (x̄) f (x̃) − > x̄ ∗ , η(x̃, x̄) , ∀x̄ ∗ ∈ ∂ C f (x̄) and x̃ ∈ S, β(x̃, x̄) β(x̃, x̄) (4.26) and gi (x̃) − gi (x̄) ≥ αi (x̃, x̄) x̄i∗ , η(x̃, x̄) , ∀x̄i∗ ∈ ∂ C gi (x̄), ∀i ∈ I and x̃ ∈ S, then, gi (x̄) gi (x̃) − ≥ x̄i∗ , η(x̃, x̄) , ∀x̄i∗ ∈ ∂ C gi (x̄), ∀i ∈ I and x̃ ∈ S, αi (x̃, x̄) αi (x̃, x̄) since λ̄i ≥ 0, then, we get, λ̄i gi (x̃) λ̄i gi (x̄) − ≥ λ̄i x̄i∗ , η(x̃, x̄) , ∀x̄i∗ ∈ ∂ C gi (x̄), ∀i ∈ I and x̃ ∈ S, αi (x̃, x̄) αi (x̃, x̄) then, λ̄i gi (x̃) λ̄i gi (x̄) − ≥ λ̄i x̄i∗ , η(x̃, x̄) , ∀x̄i∗ ∈ ∂ C gi (x̄) and x̃ ∈ S. α ( x̃, x̄) α ( x̃, x̄) i i i∈I i∈I i∈I (4.27) As, gi (x̃) ≤ 0 and λ̄i ≥ 0, we have 0≥ λ̄i gi (x̄) + λ̄i x̄i∗ , η(x̃, x̄) , ∀x̄i∗ ∈ ∂ C gi (x̄) and x̃ ∈ S. α ( x̃, x̄) i i∈I i∈I (4.28) By Adding (4.26) and (4.28), we get, λ̄i gi (x̄) f (x̃) f (x̄) − > + x̄ ∗ + λ̄i x̄i∗ , η(x̃, x̄) , ∀x̄i∗ ∈ ∂ C gi (x̄) and x̃ ∈ S. β(x̃, x̄) β(x̃, x̄) α ( x̃, x̄) i i∈I i∈I I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 On Semi-infinite Optimization Problem Under the Generalized Convexity 103 Then, by using (4.25), we get λ̄i gi (x̄) f (x̃) f (x̄) − > , x̃ ∈ S. β(x̃, x̄) β(x̃, x̄) αi (x̃, x̄) i∈I By the feasibility of (MWSID), we get f (x̃) f (x̄) − > 0, x̃ ∈ S, β(x̃, x̄) β(x̃, x̄) (4.29) then, f (x̃) > f (x̄). Which is contradiction. Hence x̃ = x̄. This completes the proof. 4.3. Mond-Weir duality using generalized V − invexity In this subsection, we prove weak duality, strong duality and strict converse duality results under the generalized V −invex functions. Theorem 7. (Weak Duality) Let x be feasible for (SIP) and (u, λ) be feasible for (MWSID). If ( f, λ I g I ) is V −pseudo-quasi-invex at u. Then, f (x) ≥ f (u). Proof. Since x be feasible for (SIP), then, gi (x) ≤ 0, for all i ∈ I, λi ≥ 0, for i ∈ I and λi = 0, for finitely many i ∈ I (u). Then, λi gi (x) ≤ 0, then, there exist αi (x, u) ∈ R+ \ {0} such that, αi (x, u)λi gi (x) ≤ 0, ∀i ∈ I, ∀x ∈ S, then, αi (x, u)λi gi (x) ≤ 0, ∀x ∈ S. (4.30) i∈I Since, (u, λ) be feasible for (MWSID) then, λi gi (u) ≥ 0, ∀i ∈ I, then, αi (x, u)λi gi (u) ≥ 0, ∀i ∈ I, ∀x ∈ S, which implies that, αi (x, u)λi gi (u) ≥ 0, ∀x ∈ S. (4.31) i∈I I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 104 Pushkar Jaisawal, Vivek Laha and S.K. Mishra From (4.30) and (4.31), we get, αi (x, u)λi gi (x) ≤ αi (x, u)λi gi (u), ∀x ∈ S, i∈I (4.32) i∈I Since ( f, λ I g I ) is V −pseudo-quasi-invex at u, then, there exist β(x, u), αi (x, u) ∈ R+ \ {0}, for all i ∈ I and for all x ∈ S, such that, β(x, u) f (x) < β(x, u) f (u) ⇒ for any u ∗ ∈ ∂ C f (u), and αi (x, u)λi gi (x) ≤ i∈I u ∗ , η(x, u) < 0 αi (x, u)λi gi (u) ⇒ for any u i∗ ∈ ∂ C gi (u), i∈I (4.33) λi u i∗ , η(x, u) ≤ 0. i∈I (4.34) From (4.32), we get, λi u i∗ , η(x, u) ≤ 0, ∀u i∗ ∈ ∂ C gi (u), ∀x ∈ S. (4.35) i∈I Let, 0 ∈ ∂ C f (u) + λi ∂ C gi (u), i∈I then, there exist u ∗ ∈ ∂ C f (u) and u i∗ ∈ ∂ C gi (u), i ∈ I, such that, 0 = u∗ + λi u i∗ , i∈I then, 0 = u ∗ , η(x, u) + λi u i∗ , η(x, u) , ∀x ∈ S. i∈I Then, by (4.35), we get, u ∗ , η(x, u) ≥ 0, for some u ∗ ∈ ∂ C f (u), ∀x ∈ S. Assume that f (x) < f (u), since β(x, u) ∈ R+ \ {0}, we get, β(x, u) f (x) < β(x, u) f (u), ∀x ∈ S. Then, by (4.33), we get, u ∗ , η(x, u) < 0, for any u ∗ ∈ ∂ C f (u), ∀x ∈ S. Which is contradiction. Hence the results. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 On Semi-infinite Optimization Problem Under the Generalized Convexity 105 Theorem 8. (Strong Duality) Let x̄ be an optimal solution for (SIP) at which a constraint qualification is satisfied. Then, there exist λ̄ = (λ̄i ), i ∈ I, such that λ̄i gi (x̄) = 0, for all i ∈ I and (x̄, λ̄) is feasible for (MWSID). If weak duality holds between (SIP) and (MWSID), then, (x̄, λ̄) is optimal for (MWSID) and the respective objective values are equal. Proof. From the KKT conditions, there exist λ̄ = (λ̄i ), i ∈ I, such that 0 ∈ ∂ C f (x̄) + λ̄i ∂ C gi (x̄), i∈I and by assumptions, λ̄i gi (x̄) = 0, i ∈ I. Therefore, (x̄, λ̄) is feasible for (MWSID). On the other hand by weak duality theorem, we have f (x̄) ≥ f (u), for any feasible solution (u, λ) for (MWSID). Thus, (x̄, λ̄) is optimal for (MWSID) and the respective objective values are equal. Hence the results. Theorem 9. (Strict Converse Duality) Let (SIP) have an optimal solution x̃ at which a constraint qualification is satisfied. Assume that ( f, λ¯I g I ) are V −pseudo-quasi-invex function. If (x̄, λ̄) is an optimal solution for (MWSID) and ( f, λ¯I g I ) is V −strictly-pseudo-quasi-invex at x̄, then, x̄ = x̃. Proof. We prove this by the contradiction. Assume that x̃ = x̄. Then, by strong duality theorem there exist λ∗ = (λi∗ ), i ∈ I such that (x̃, λ∗ ) is optimal for (MWSID). Hence f (x̃) = f (x̄). (4.36) Since x̃ is an optimal solution for (SIP) then, gi (x̃) ≤ 0, for all i ∈ I, λ̄i ≥ 0, for i ∈ I and λ̄i = 0, for finitely many i ∈ I (x̄). Then, λ̄i gi (x̃) ≤ 0, and there exist αi (x, x̄) ∈ R+ \ {0}, such that, αi (x, x̄)λ̄i gi (x̃) ≤ 0, ∀i ∈ I, ∀x ∈ S, then, αi (x, x̄)λ̄i gi (x̃) ≤ 0, ∀x ∈ S. (4.37) i∈I I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 106 Pushkar Jaisawal, Vivek Laha and S.K. Mishra Since (x̄, λ̄) is an optimal solution for (MWSID) then, λ̄i gi (x̄) ≥ 0, ∀i ∈ I, then, αi (x, x̄)λ̄i gi (x̄) ≥ 0, ∀i ∈ I, ∀x ∈ S, which implies that, αi (x, x̄)λ̄i gi (x̄) ≥ 0, ∀x ∈ S. (4.38) i∈I From (4.37) and (4.38), we get, αi (x, x̄)λ̄i gi (x̃) ≤ αi (x, x̄)λ̄i gi (x̄), ∀x ∈ S, i∈I i∈I since x̃ ∈ S, then, αi (x̃, x̄)λ̄i gi (x̃) ≤ i∈I αi (x̃, x̄)λ̄i gi (x̄), (4.39) i∈I Since ( f, λ¯I g I ) is V −strictly-pseudo-quasi-invex at x̄, then, there exist β(x, x̄), αi (x, x̄) ∈ R+ \ {0}, for all i ∈ I and for all x ∈ S, such that, β(x, x̄) f (x) ≤ β(x, x̄) f (x̄) ⇒ for any x̄ ∗ ∈ ∂ C f (x̄), and αi (x, x̄)λ̄i gi (x) ≤ i∈I x̄ ∗ , η(x, x̄) < 0 αi (x, x̄)λ̄i gi (x̄) ⇒ for any x̄i∗ ∈ ∂ C gi (x̄), i∈I (4.40) λ̄i x̄i∗ , η(x, x̄) ≤ 0. i∈I (4.41) From (4.39), we get, λ̄i x̄i∗ , η(x̃, x̄) ≤ 0, ∀x̄i∗ ∈ ∂ C gi (x̄), x̃ ∈ S. (4.42) i∈I Let, 0 ∈ ∂ C f (x̄) + λi ∂ C gi (x̄), i∈I then, there exist x̄ ∗ ∈ ∂ C f (x̄) and x̄i∗ ∈ ∂ C gi (x̄), i ∈ I, such that, 0 = x̄ ∗ + λi x̄i∗ , i∈I I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 On Semi-infinite Optimization Problem Under the Generalized Convexity 107 then, 0 = x̄ ∗ , η(x, x̄) + λ̄i x̄i∗ , η(x, x̄) , ∀x ∈ S, i∈I which implies that, 0 = x̄ ∗ , η(x̃, x̄) + λ̄i x̄i∗ , η(x̃, x̄) , x̃ ∈ S. i∈I Then, by (4.42), we get, x̄ ∗ , η(x̃, x̄) ≥ 0, for some x̄ ∗ ∈ ∂ C f (x̄), x̃ ∈ S. (4.43) Assume f (x̃) ≤ f (x̄), then, β(x̃, x̄) f (x̃) ≤ β(x̃, x̄) f (x̄), for x̃ ∈ S, then, by (4.40), we get x̄ ∗ , η(x̃, x̄) < 0, ∀x̄ ∗ ∈ ∂ C f (x̄), for x̃ ∈ S, which is contradiction of (4.43), then, this is not possible. Hence, f (x̃) > f (x̄), which is a contradiction of (4.36), i.e., contradiction of x̃ = x̄. Hence the results. 5. CONCLUSIONS In this paper, we have taken a semi-infinite programming problem involving V −invex and generalized V −invex functions. We have derived sufficient optimality conditions under V −invexity and generalized V −invexity both using the help of Clarke’s subdifferential. We have formulated Wolfe type duality model and Mond-Weir type duality model and gave weak duality, strong duality and strict converse duality results for r −invex and V −invex functions with the use of Clarke’s subdifferential. ACKNOWLEDGEMENTS The authors are thankful to the anonymous referees of this paper for their useful suggestions which helped to modify the papers in its present form. The research of the first author is supported by the Council of Scientific and Industrial Research, New Delhi, Ministry of Human Resources Development, Government of India Grant [09/013(0672)/2017 − EMR − I]. The research of the second author is supported by UGC-BSR start up grant by University grant Commission, New Delhi, India (Letter No. F. 30 − 370/2017(BSR)) (Project No. M−14 − 40). I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 108 Pushkar Jaisawal, Vivek Laha and S.K. Mishra REFERENCES [1] Antczak, T., Multiobjective programming under d−invexity, European Journal of Operational Research, 137(1), 28– 36 (2002). [2] Canovas, M.J., Lopez, M.A., Mordukhovich, B.S., Parra, J., Variational analysis in semi-infinite and finite programming, I: Stability of linear inequality systems of feasible solutions, 1504–1526. SIAM J. Optim., Philadelphia (2009). [3] Canovas, M.J., Lopez, M.A., Mordukhovich, B.S., Parra, J., Variational analysis in semi-infinite and finite programming, II: Necessary optimality conditions (preprint). [4] Clarke, F.H., Optimization and Nonsmooth Analysis. In: Classics in Applied Mathematics, Vol. 5. 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[25] Shapiro, A., Semi-infinite programming, duality, discretization and optimality condition, Optimization, 58(2), 133– 161 (2009). [26] Wang, Q.L., Li, S.J., Teo, K.L., Higher-order optimality conditions for weakly efficient solutions in nonconvex setvalued optimization, Optim. Lett., 4(3), 425–437 (2010). [27] Wolfe, P., A duality theorem for nonlinear programming problem, Quart. Appl., 19, 239–244 (1961). I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 INDIAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019, pp. 109–120 DOI: A Strict Constraint Qualification in Vector Optimization Muskan Kapoor1∗ and C.S. Lalitha2 1 Department of Mathematics, University of Delhi, Delhi-110007, India Department of Mathematics, University of Delhi South Campus, New Delhi-110021, India (∗ Corresponding author) Email: ∗ [email protected], 2 [email protected] 2 Abstract: The main aim of this paper is to present the notion of cone-continuity property introduced by Andreani et al. (Andreani, R., Martı́nez, J.M., Ramos, A. and Silva, P.J.S., A conecontinuity constraint qualification and algorithmic consequences, SIAM J. Optim., 26(2016), 96– 110) in the context of vector optimization. We extend this property and establish it to be a strict regularity condition while dealing with efficient solutions. This property is the weakest condition which ensures that weak approximate Karush-Kuhn-Tucker conditions imply weak Karush-KuhnTucker conditions. Further, cone-continuity is established to be a strict constraint qualification while dealing with weak and proper efficient solutions. Keywords: Vector optimization, Karush-Kuhn-Tucker conditions, approximate Karush-KuhnTucker conditions, strict constraint qualification, cone-continuity property 1. INTRODUCTION The problem of identifying an optimal solution of a nonlinear programming problem led to the development of various types of optimality conditions. These conditions are also important in designing numerical methods for finding the solutions. One such optimality conditions namely Fritz John optimality conditions are satisfied at an optimal solution, but there are cases for which the corresponding multiplier of the objective function is zero. It is important to ensure the positivity of the multiplier corresponding to the objective function as this leads to the identification of the solutions of the problem. This is achieved by imposing certain assumptions on the constraints called constraint qualification (CQ). The optimality conditions with positive multipliers are referred to as the Karush-Kuhn-Tucker (KKT) conditions. The main approach to derive optimality in vector optimization has been through scalarization of the vector optimization problem. This is achieved by translating the vector optimization problem into a scalar optimization problem and by using the optimality conditions of the scalar problem to I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 110 Muskan Kapoor and C.S. Lalitha deduce KKT type conditions for the vector problem. Hence, in such cases it becomes a necessity to impose conditions on the constraints and on all the objective functions except possibly on one, while deriving KKT conditions from Fritz John conditions. In this situation one uses the term regularity condition rather than constraint qualification (see [5, 9, 13]). When at least one of the multipliers, corresponding to the objective functions, is positive we say that weak KKT conditions hold for the problem. On the other hand strong KKT conditions are said to hold when all the multipliers corresponding to the objective functions are positive. Practical methods for solving constrained optimization problems are usually iterative. At each iteration, one must decide whether it is sensible to terminate the execution of the algorithm or not. Since testing true optimality is very difficult, the obvious idea is to terminate when a necessary optimality condition is approximately satisfied. In literature there are many papers dealing with sequential optimality conditions in vector optimization (see [2–4,7,14]). Such conditions are termed as Approximate Karush-Kuhn-Tucker (AKKT) conditions in [2–4] which can be tested by most of numerical optimization solvers. Sequential optimality conditions provide adequate theoretical tools to justify stopping criteria. A local minimizer can be approximated by a sequence of AKKT points. Strict constraint qualification (SCQ) is a property of feasible points that guarantees AKKT points to satisfy KKT conditions. Andreani et al. [2,3] introduced relaxed constant positive linear dependence and constant positive generator strict constraint qualifications. Andreani et al. [4] introduced a conecontinuity property (CCP) and showed it to be weakest possible SCQ that guarantees AKKT implies KKT for constrained optimization problems. The main aim of the paper is to bring the cone-continuity property into the scenario of vector optimization both as a strict regularity and strict constraint qualification. This is done by using some well-known scalarization schemes available in literature for characterizing efficient, weak efficient and properly efficient solutions. The paper is organized as follows. In Section 2, we provide the cone-continuity notion for scalar problems and scalarization schemes for a vector optimization problem with both inequality and equality constraints. In Section 3, we introduce weak AKKT and a cone-continuity property for the vector problem and establish that cone-continuity is a strict regularity condition for efficient solutions. Also, we establish that this property is the weakest condition which ensures that weak AKKT implies weak KKT conditions. In Section 4, we establish that the cone-continuity considered in [4] is a strict constraint qualification while considering weak and proper efficient solutions. 2. PRELIMINARIES We first consider the following scalar minimization problem (CP) minimize f (x) subject to gi (x) ≤ 0, i = 1, 2, . . . , m, h i (x) = 0, i = 1, 2, . . . , l, where f : Rn → R, g : Rn → Rm , h : Rn → Rl are continuously differentiable on Rn . Let S = {x ∈ Rn : gi (x) ≤ 0, i = 1, 2, . . . , m, h i (x) = 0, i = 1, 2, . . . , l} I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 A Strict Constraint Qualification in Vector Optimization 111 be the feasible set of (CP). For a point x ∈ S we denote the set of active indices of g by I (x), that is, I (x) = {i ∈ {1, 2, . . . , m} : gi (x) = 0}. We next recall sequential optimality AKKT conditions [1, 4, 6] for problem (CP). Definition 2.1 [4, Definition 1.2]. A point x̄ ∈ S is said to satisfy AKKT conditions if there exist k l k sequences {x k } ⊆ Rn , {α k } ⊆ Rm + , {μ } ⊆ R such that x → x̄, lim k→∞ ∇ f (x k ) + m i=1 k αik ∇gi (x k ) + l μik ∇h i (x k ) = 0, (2.1) i=1 lim min{αik , −gi (x )} = 0, ∀ i = 1, 2, . . . , m. k→∞ The AKKT conditions have been rewritten in an equivalent form [1, 4] as follows. Lemma 2.1 [1]. The AKKT conditions hold at x̄ ∈ S if and only if there exist sequences {x k } ⊆ Rn , k k l / I (x̄) such that x k → x̄ and (2.1) holds. {α k } ⊆ Rm + , {μ } ⊆ R with αi = 0 for i ∈ It is known that every local minimizer of (CP) satisfies AKKT conditions. One may refer to Theorem 2.1 in [1] for more details. We next recall the notion of the cone-continuity property (CCP) introduced by Andreani et al. [4] to establish that it is the weakest possible strict constraint qualification. Definition 2.2 [4]. A point x̄ ∈ S satisfies cone-continuity property (CCP) if the set-valued mapping K : Rn → Rn defined by K (x) := ⎧ ⎨ ⎩ i∈I (x̄) αi ∇gi (x) + l i=1 ⎫ ⎬ μi ∇h i (x) : αi ∈ R+ , i ∈ I (x̄), μi ∈ R, i = 1, 2, . . . , l , ⎭ is outer semicontinuous at x̄, that is, lim sup K (x) ⊆ K (x̄) where x→x̄ lim sup K (x) := { ȳ ∈ Rn : ∃ (x k , y k ) → (x̄, ȳ) such that y k ∈ K (x k )}. x→x̄ The following theorem establishes the fact that CCP is the weakest SCQ which guarantees that AKKT implies KKT for the constrained scalar problem (CP). Theorem 2.1 [4, Definition 3.2]. The CCP condition at x̄ ∈ S is the weakest property under which AKKT conditions at x̄ implies KKT conditions at x̄, for every continuously differentiable objective function f that attains a minimum at x̄. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 112 Muskan Kapoor and C.S. Lalitha We study the role of cone-continuity property in vector optimization. We consider the following vector optimization problem (VP) minimize f (x) = ( f 1 (x), f 2 (x), . . . , f p (x)) subject to x ∈ S where f : Rn → R p , g : Rn → Rm , h : Rn → Rl , are continuously differentiable functions. We now give the following well-known solution concepts in vector optimization. Definition 2.3. A point x̄ ∈ S is said to be 1. 2. an efficient solution of (VP) if there does not exist any other x ∈ S such that fi (x) ≤ f i (x̄), for all i = 1, 2, . . . , p, fr (x) < fr (x̄), for some r ; a weak efficient solution of (VP) if there does not exist any x ∈ S such that fi (x) < f i (x̄), for all i = 1, 2, . . . , p. The optimality conditions for a vector problem are usually established through the scalar problem associated with it. The efficient solution of (VP) having p objectives can be characterized in terms of the optimal solutions of p scalar problems considered by Chankong and Haimes [10]. Lemma 2.2 [10]. A point x̄ ∈ S is an efficient solution of (VP) if and only if x̄ is an optimal solution of (Pr (x̄)) for each r = 1, 2, . . . , p, where (Pr (x̄)) is defined as (Pr (x̄)) minimize fr (x) subject to f i (x) ≤ f i (x̄), for all i = 1, 2, . . . , p, i = r, gi (x) ≤ 0, i = 1, 2, . . . , m, h i (x) = 0, i = 1, 2, . . . , l. The weak efficient solutions have been characterized by the optimal solutions of the associated weighted scalar problem. For this we first recall the notion of cone convexlike function from Illés and Kassay [12] by taking cone as the nonnegative orthant in the finite dimensional setting. Definition 2.4 [12]. A function f : Rn → R p is said to be convexlike on Rn if for there exists t ∈ [0, 1] such that for every x1 , x2 ∈ Rn , there exists x3 ∈ Rn such that p (1 − t) f (x1 ) + t f (x2 ) − f (x3 ) ∈ R+ . The following lemma is a consequence of the Alternative Theorem 3.1 in Illés and Kassay [12] for cone convexlike by taking the cone as nonegative orthant. Lemma 2.3. Let f be convexlike on Rn . A point x̄ ∈ S is a weak efficient solution of (VP) if and p only if there exists λ ∈ R+ \ {0} such that x̄ minimizes the following scalar problem (Pλ ) minimize p λi f i (x) i=1 subject to x ∈ S. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 A Strict Constraint Qualification in Vector Optimization 113 A notion of proper efficiency has been introduced by Geoffrion [11] for a vector maximization problem which is now formulated for the minimization problem (VP). Definition 2.5 [11]. A point x̄ ∈ S is a properly efficient solution of (VP) if it is efficient and if there exists a scalar M > 0 such that for each i and x ∈ S satisfying fi (x) < f i (x̄) we have f i (x̄) − f i (x) ≤M f j (x) − f j (x̄) for some j satisfying f j (x) > f j (x̄). Now we formally state the Geoffrion criteria for scalarization in the form of the following lemma. Lemma 2.4 [11]. Let S be a convex set and fi , for i = 1, 2, . . . , p be convex on S. Then x̄ ∈ S is properly efficient solution of (VP) if and only if x̄ is optimal for (Pλ ) for some λ with strictly positive components. The multipliers corresponding to all the components of the objective functions are not always positive. Two types of KKT conditions, namely weak and strong KKT conditions, are usually considered in literature while dealing with a vector optimization problem. Definition 2.6 [8]. A point x̄ ∈ S is said to satisfy 1. p l weak KKT conditions if there exist λ ∈ R+ \ {0}, α ∈ Rm + , μ ∈ R such that p i=1 λi ∇ f i (x̄) + m αi ∇gi (x̄) + i=1 l μi ∇h i (x̄) = 0 αi gi (x̄) = 0, ∀ i = 1, 2, . . . , m; 2. (2.2) i=1 (2.3) p l strong KKT conditions if there exist λ ∈ intR+ , α ∈ Rm + , μ ∈ R such that (2.2) and (2.3) hold. 3. CONE-CONTINUITY PROPERTY AS STRICT REGULARITY CONDITION In this section we establish that cone-continuity property is the weakest property under which weak AKKT conditions imply weak KKT conditions. Moreover, we prove that cone-continuity property is a strict regularity condition, that is, they guarantee that weak AKKT implies KKT optimality conditions for (VP). In view of Lemma 2.1 we define a notion of weak AKKT conditions for (VP). Definition 3.1. Let x̄ ∈ S. The sequential weak AKKTr conditions with respect to r ∈ {1, 2, . . . , p} p−1 k k l hold at x̄, if there exist sequences {x k } ⊆ Rn , λk ⊆ R+ , {α k } ⊆ Rm + , {μ } ⊆ R with αi = 0 for I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 114 Muskan Kapoor and C.S. Lalitha i∈ / I (x̄) such that x k → x̄, ⎡ ⎤ ⎢ lim ⎣∇ fr (x k ) + λik ∇ f i (x k ) + p k→∞ m αik ∇gi (x k ) + i=1 i=1 i =r l ⎥ μik ∇h i (x k )⎦ = 0. i=1 Since the cone-continuity property defined below involves all objective function except the r th objective for some r ∈ {1, 2, . . . , p} the cone-continuity property CCPr will be shown to be a strict regularity condition. Definition 3.2. A point x̄ ∈ S satisfies cone-continuity property (CCPr ) with respect to r ∈ {1, 2, . . . , p}, if the set-valued mapping K r : Rn → Rn defined by ⎧ ⎫ ⎪ ⎪ p l ⎨ ⎬ λi ∇ f i (x) + αi ∇gi (x) + μi ∇h i (x) : λi ∈ R+ , αi ∈ R+ , μi ∈ R , K r (x) := ⎪ ⎪ ⎩ i=1 ⎭ i∈I (x̄) i=1 i =r is outer semicontinuous at x̄, that is, lim sup K r (x) ⊆ K r (x̄). x→x̄ We now give an example to illustrate the cone-continuity property. Example 3.1. Consider the vector optimization where the functions are defined on R2 minimize f (x1 , x2 ) = ( f 1 (x1 , x2 ), f 2 (x1 , x2 )) 2 = (x13 , (x1 )2 e(x2 ) ) subject to g1 (x1 , x2 ) = x1 e x2 ≤ 0. The constraints are active at the feasible point x̄ = (0, 0) and K 1 (x) = {λ2 ∇ f 2 (x) + α1 ∇g1 (x) : λ2 ≥ 0, α1 ≥ 0} 2 2 = {(2λ2 x1 e(x2 ) + α1 e x2 , 2λ2 (x1 )2 x2 e(x2 ) + α1 x1 e x2 ) : λ2 ≥ 0, α1 ≥ 0}, K 2 (x) = {λ1 ∇ f 1 (x) + α1 ∇g1 (x) : λ1 ≥ 0, α1 ≥ 0} = {(3λ1 (x1 )2 + α1 e x2 , α1 x1 e x2 ) : λ1 ≥ 0, α1 ≥ 0}. Hence, K 1 (x̄) = K 2 (x̄) = R+ × {0}. We now show that x̄ satisfies CCP2 . Let z̄ = (z̄ 1 , z̄ 2 ) = lim sup K 2 (x), hence there exists a x→x̄ sequence (x1k , x2k , z 1k , z 2k ) → (0, 0, z̄ 1 , z̄ 2 ) such that k k (z 1k , z 2k ) = (3λk1 (x1k )2 + α1k e x2 , α1k x1k e x2 ) I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS (3.1) Vol. 10, No. 1 (Special Issue), Jan–June 2019 A Strict Constraint Qualification in Vector Optimization 115 for some λk1 ≥ 0, α1k ≥ 0. We next show that z̄ ∈ K 2 (x̄). Suppose on the contrary that z̄ ∈ / K 2 (x̄). Then, from (3.1) it is clear that z̄ 2 = 0 and hence for sufficiently large k we have k |z 2k | = α1k |x1k |e x2 > |z̄ 2 | > 0. 2 Thus, from (3.1) it follows that k k z 1k = 3λk1 (x1k )2 + α1k e x2 ≥ α1k e x2 > |z̄ 2 | . 2|x1k | Taking limits we obtain z 1k → ∞, which is a contradiction. Thus, x̄ satisfies CCP2 . However, in this example we show that x̄ fails to satisfy CCP1 . Define x1k = − 1k , x2k = k1 , λk2 = k k, α1 = k1 . Then for every k we have k 2 k k 2 k (y1k , y2k ) = (2λk2 x1k e(x2 ) + α1k e x2 , 2λk2 (x1k )2 x2k e(x2 ) + α1k x1k e x2 ) 1 1 2 1 2 1 1 1 2 = −2e( k ) + e k , 2 e( k ) − 2 e k ∈ K 1 (x k ) k k k but its limit (−2, 0) ∈ / K 1 (x̄). Theorem 3.1. If CCPr condition holds at x̄ ∈ S for some r ∈ {1, 2, . . . , p} then sequential AKKTr at x̄ implies weak KKT at x̄ with λr > 0. Proof. Let us first show that if x̄ satisfies CCPr , that is, lim sup K r (x) ⊆ K r (x̄), then the sequential x→x̄ AKKTr condition at x̄ implies the weak KKT condition at x̄ independently of the objective function. Let fr be the objective function such that sequential AKKTr condition holds at x̄, then there are p−1 k l k sequences {x k } ⊆ Rn , {λk } ⊆ R+ , {α k } ⊆ Rm + , {μ } ⊆ R such that x → x̄, ⎡ ⎤ ⎢ lim ⎣∇ fr (x k ) + λik ∇ f i (x k ) + p k→∞ m αik ∇gi (x k ) + i=1 i=1 i =r l ⎥ μik ∇h i (x k )⎦ = 0. i=1 Then yrk = p λik ∇ f i (x k ) + m αik ∇gi (x k ) + l i=1 i=1 i =r μik ∇h i (x k ) ∈ K r (x k ). i=1 Since x̄ satisfies CCPr we have lim yrk = lim (−∇ fr (x k )) = −∇ fr (x̄) ∈ K r (x̄) k→∞ k→∞ that is, x̄ satisfies weak KKT condition at x̄ with λr > 0. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 116 Muskan Kapoor and C.S. Lalitha Using the above theorem we show that cone-continuity property CCPr is a strict regularity condition. Theorem 3.2. If x̄ is an efficient solution of (VP), CCPr condition holds at x̄ for some r ∈ {1, 2, . . . , p} then x̄ satisfies weak KKT condition with λr > 0. Proof. By Lemma 2.2 it is clear that x̄ is an optimal solution of (Pr (x̄)) which implies that x̄ satisfies AKKTr and hence the conclusion follows from Theorem 3.1. Theorem 3.3. If x̄ is an efficient solution of (VP), CCPr condition holds at x̄ for each r = 1, 2, . . . , p then x̄ satisfies strong KKT condition. Proof. By Theorem 3.2 it follows that x̄ satisfies weak KKT condition for each r = 1, 2, . . . , p, p r l r hence there exist λr ∈ R+ \ {0}, αr ∈ Rm + , μ ∈ R such that λr > 0, p i=1 λri ∇ f i (x̄) + m αir ∇gi (x̄) + i=1 l μri ∇h i (x̄) = 0, i=1 αir gi (x̄) = 0, ∀ i = 1, 2, . . . , m. Summing the above inequalities over r and setting λi = p r =1 λri , αi = that x̄ satisfies strong KKT conditions with multipliers λ, α and μ. p r =1 αir , μi = p r =1 μri , it follows Remark 3.1. Since a properly efficient solution is an efficient solution the above two theorems also hold for properly efficient solutions. The next theorem establishes that CCPr is the weakest condition which guarantees that weak AKKT implies weak KKT conditions. Theorem 3.4. If weak AKKTr implies weak KKT condition with λr > 0 for every continuously differentiable function fr that attains a minimum at x̄ ∈ S then CCPr condition holds at x̄. Proof. Let ȳr ∈ lim sup K r (x), hence there exists a sequence (x k , yrk ) → (x̄, ȳr ) such that yrk ∈ x→x̄ K r (x k ). Define fr (x) = − ȳr , x for all x ∈ Rn . Since ∇ fr (x k ) + yrk = − ȳr + yrk → 0 it follows that sequential AKKTr holds at x̄ and hence by hypothesis we have that weak KKT condition holds at x̄ with λr > 0 which further implies that −∇ fr (x̄) ∈ K r (x̄), that is, ȳr ∈ K r (x̄). 4. CONE-CONTINUITY PROPERTY AS STRICT CONSTRAINT QUALIFICATION While dealing with weak efficient solutions it is known that the scalarization scheme of Chankong and Haimes [10] fails to hold. In order to show that cone-continuity is a strict constraint qualification p or a strict regularity condition we need to consider scalarized problem (Pλ ) for some λ ∈ R+ \ {0}. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 A Strict Constraint Qualification in Vector Optimization 117 In such situations one need to consider CCP rather than CCPr . Also, there are situations where CCP holds but CCPr fails to hold as illustrated in the following example. Example 4.1. Consider the vector optimization where the functions are defined on R2 minimize f (x1 , x2 ) = ( f 1 (x1 , x2 ), f 2 (x1 , x2 )) = (−x1 , −(x1 x2 )2 ) subject to g1 (x1 , x2 ) = x13 ≤ 0 g2 (x1 , x2 ) = x1 e x2 ≤ 0. Both the constraints are active at the feasible point x̄ = (0, 0) and K (x) = {α1 ∇g1 (x) + α2 ∇g2 (x) : α1 ≥ 0, α2 ≥ 0} = {(3α1 (x1 )2 + α2 e x2 , α2 x1 e x2 ) : α1 ≥ 0, α2 ≥ 0}, K 1 (x) = {λ2 ∇ f 2 (x) + α1 ∇g1 (x) + α2 ∇g2 (x) : λ2 ≥ 0, α1 ≥ 0, α2 ≥ 0} = {(−2λ2 x1 (x2 )2 + 3α1 (x1 )2 + α2 e x2 , −2λ2 (x1 )2 x2 + α2 x1 e x2 ) : λ2 ≥ 0, α1 ≥ 0, α2 ≥ 0}, K 2 (x) = {λ1 ∇ f 1 (x) + α1 ∇g1 (x) + α2 ∇g2 (x) : λ1 ≥ 0, α1 ≥ 0, α2 ≥ 0} = {(−λ1 + 3α1 (x1 )2 + α2 e x2 , α2 x1 e x2 ) : λ1 ≥ 0, α1 ≥ 0, α2 ≥ 0}. Hence, K (x̄) = K 1 (x̄) = R+ × {0} and K 2 (x̄) = R × {0}. We now show that x̄ satisfies CCP. Let ȳ = ( ȳ1 , ȳ2 ) ∈ lim sup K (x), hence there exists a x→x̄ sequence (x1k , x2k , y1k , y2k ) → (0, 0, ȳ1 , ȳ2 ) such that k k (y1k , y2k ) = (3α1k (x1k )2 + α2k e x2 , α2k x1k e x2 ) (4.1) for some α1k ≥ 0, α2k ≥ 0. We next show that ȳ ∈ K (x̄). Suppose on the contrary that ȳ ∈ / K (x̄). Then, from (4.1) it is clear that ȳ2 = 0 and hence for sufficiently large k we have k |y2k | = α2k |x1k |e x2 > | ȳ2 | > 0. 2 Thus, from (4.1) it follows that k k y1k = 3α1k (x1k )2 + α2k e x2 ≥ α2k e x2 > | ȳ2 | . 2|x1k | Taking limits we obtain y1k → ∞, which is a contradiction. Thus, x̄ satisfies CCP. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 118 Muskan Kapoor and C.S. Lalitha We next show that x̄ does not satisfy CCP1 . Define x1k = x2k = k1 , λk2 = α2k = 0. Then for every k we have 1 , 2(x1k )2 x2k k α1k = x2k , 3(x1k )3 k (y1k , y2k ) = (−2λk2 x1k (x2k )2 + 3α1k (x1k )2 + α2k e x2 , −2λk2 (x1k )2 x2k + α2k x1k e x2 ) = (0, −1) ∈ K 1 (x k ) but its limit (0, −1) ∈ / K 1 (x̄). Similarly, we next show that x̄ does not satisfy CCP2 . Define x1k = x2k = k1 , α1k = k, α2k = x2k 1 k x1k e x2 , λk1 = 3α1k (x1k )2 + α2k e . Then for every k we have k k (y1k , y2k ) = (−λk1 + 3α1k (x1k )2 + α2k e x2 , −α2k x1k e x2 ) = (0, 1) ∈ K 2 (x k ) but its limit (0, 1) ∈ / K 2 (x̄). The following theorem establishes that CCP is a strict constraint qualification for a weak efficient solution. Theorem 4.1. If f is convexlike on Rn , x̄ is a weak efficient solution of (VP), CCP condition holds at x̄ then x̄ satisfies weak KKT condition. Proof. Since f is convexlike on Rn and x̄ is a weak efficient solution of (VP), then by Lemma 2.3 p there exists λ ∈ R+ \ {0} such that x̄ is a minimizer of (Pλ ). Since x̄ is a minimizer of (Pλ ) there are k k k l / I (x̄) such that x k → x̄, sequences {x } ⊆ Rn , {α k } ⊆ Rm + , {μ } ⊆ R with αi = 0 for i ∈ lim k→∞ p λi ∇ f i (x ) + k i=1 m αik ∇gi (x k ) + i=1 l μik ∇h i (x k ) = 0. i=1 Then yk = m αik ∇gi (x k ) + i=1 l μik ∇h i (x k ) ∈ K (x k ). i=1 Since x̄ satisfies CCP we have lim y = lim k k→∞ k→∞ − p λi ∇ f i (x ) = − k i=1 p λi ∇ f i (x̄) ∈ K (x̄) i=1 that is, x̄ satisfies weak KKT condition. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 A Strict Constraint Qualification in Vector Optimization 119 Remark 4.1. Since every properly efficient solution is a weak efficient solution it follows from the above theorem that even for properly efficient solutions CCP is a strict constraint qualification which leads to weak KKT conditions. The following theorem establishes that strict constraint qualification CCP in fact leads to strong KKT conditions for properly efficient solutions. Theorem 4.2. If S is a convex set, fi is convex on S for i = 1, 2, . . . , p, x̄ is a properly efficient solution of (VP) and CCP condition holds at x̄ then x̄ satisfies strong KKT condition. Proof. Since S is a convex set and fi , for i = 1, 2, . . . , p, is convex on S and x̄ is a properly efficient p solution of (P), then by Lemma 2.4 there exists λ ∈ int R+ such that x̄ is a minimizer of (Pλ ). Since k k n k l x̄ is a minimizer of (Pλ ) there are sequences {x } ⊆ R , {α k } ⊆ Rm + , {μ } ⊆ R with αi = 0 for k i∈ / I (x̄) such that x → x̄, p m l λi ∇ f i (x k ) + αik ∇gi (x k ) + μik ∇h i (x k ) = 0. lim k→∞ i=1 i=1 i=1 Then yk = m αik ∇gi (x k ) + i=1 Since x̄ satisfies CCP we have lim y = lim k k→∞ k→∞ l μik ∇h i (x k ) ∈ K (x k ). i=1 − p λi ∇ f i (x ) = − k i=1 p λi ∇ f i (x̄) ∈ K (x̄) i=1 that is, x̄ satisfies strong KKT condition. REFERENCES [1] Andreani, R., Haeser, G., Martı́nez, J.M.: On sequential optimality conditions for smooth constrained optimization. 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Sahu∗ and Satyendra Kumar1 Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi-221005, India (∗ Corresponding author) Email: ∗ [email protected], 1 [email protected] Abstract: The purpose of this paper is to introduce a new class of generalized nonexpansive mappings and establish a fixed point existence result in a uniformly convex Banach space. The convergence analysis of Mann iteration and S-iteration process for this class of mappings has been studied in Banach spaces. By some examples, it is shown that S-iteration process is faster than that of Mann and Ishikawa iteration processes. Our result extends and improves various existing results in literature in the context of S-iteration process. Keywords: Generalized nonexpansive mapping, S-iteration process, Mann iteration process. Mathematics Subject Classification (2010) 47J25.47H09.47H10 1. INTRODUCTION Let C be a nonempty subset of a Banach space X and T : C → X a mapping. Let Fi x(T ) be the set of all fixed points of mapping T . The mapping T is said to be: (1) nonexpansive if T x − T y ≤ x − y for all x, y ∈ C, (2) quasi-nonexpansive if Fi x(T ) = ∅ and T x − z ≤ x − z for all x ∈ C and z ∈ Fi x(T ), I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 122 D.R. Sahu and Satyendra Kumar (3) firmly nonexpansive [1] if T x − T y ≤ r (x − y) + (1 − r )(T x − T y) for all r > 0 and x, y ∈ C. (4) Zamfirescu operator [16] if there exist real numbers, a, b, and c satisfying 0 < a < 1, 0 < b, c < 12 such that for each pair x, y ∈ X , at least one of the following is true: (Z 1 ) T x − T y ≤ ax − y, (Z 2 ) T x − T y ≤ b(x − T x + y − T y), (Z 3 ) T x − T y ≤ c(x − T y + y − T x). Clearly, every firmly nonexpansive mapping is nonexpansive mapping. More details on firmly nonexpansive mappings can be found in [2–5]. The generalization of firmly nonexpansive mappings was studied by various authors (see [6–9]). Recently, nearly firmly nonexpansive sequence is studied in Sahu, Ansari and Yao [10]. In 2010, Takahashi [11] introduced the class of hybrid mappings which is an important generalization of firmly nonexpansive mappings in Hilbert spaces. The class of λ−hybrid mappings was introduced by Aoyama et al. [12] in 2010. Aoyama et al. [12] proved some fixed point theorems and ergodic theorems for λ-hybrid mappings. The class of λ−hybrid mappings contains the classes of nonexpansive mappings, nonspreading mappings and hybrid mappings in Hilbert spaces. In 2011, Aoyama and Kohsaka [13] introduced the concept of α−nonexpansive mappings in Banach spaces. They generalized some results of Aoyama et al. [12] in Banach spaces. Let C be a nonempty convex subset of a Banach space X and T : C → C be a mapping. For x1 ∈ C, the Mann iteration process [14] {xn } is defined as follows: xn+1 = (1 − αn )xn + αn T xn for all n ∈ N, where {αn } is a sequence in [0, 1]. The Ishikawa iteration process [15] is defined as follows: xn+1 = (1 − αn )xn + αn T yn ; yn = (1 − βn )xn + βn T xn for all n ∈ N, (1.1) (1.2) where {αn } and {βn } are a sequences in [0, 1]. In 2007, Agarwal, O’Regan and Sahu [18] introduced the S-iteration process which is defined as follows: xn+1 = (1 − αn )T xn + αn T yn ; (1.3) yn = (1 − βn )xn + βn T xn for all n ∈ N, where {αn } and {βn } are sequences in (0, 1). In 2011, Sahu [19], introduced the Normal S-iteration as follows: xn+1 = T [(1 − αn )xn + αn T xn ] for all n ∈ N, (1.4) where {αn } is a sequence in (0, 1). I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Fixed Point Theorems for Generalized [a, b, c] p -nonexpansive Mappings in Banach Spaces 123 It is remarkable that S-iteration process is independent of Mann and Ishikawa iteration processes, also S-iteration process is faster than Mann and Ishikawa iteration processes for contraction mappings (see, [18, 19]) and it work well for nonexpansive mappings. The S-iteration process is faster than the Mann and Ishikawa iteration processes, and normal S-iteration process is faster than the Picard, Mann, Ishikawa iteration processes for Zamfirescu operators (see, [35, 36]). The normal S-iteration process is applicable for finding solutions of constrained minimization problems and split feasibility problems (see, [19]). In 2013, Naraghirad, Wang and Yao [22] studied some existence, weak and strong convergence theorems for a fixed point of an α-nonexpansive mapping by Ishikawa iteration process in uniformly convex Banach spaces and CAT(0) spaces. In 2016, Song et. al. [23] studied the existence of monotone α-nonexpansive mappings in uniformly convex Banach spaces by Mann iteration process, also the convergence analysis of Mann iteration process for monotone α-nonexpansive have been discussed in ordered Banach spaces. Motivated and inspired by work of Agarwal, O’Regan and Sahu [18], Aoyama and Kohsaka [13], Naraghirad, Wang and Yao [22], Song et. al. [23], in this paper we introduce a new class of generalized [a, b, c] p -nonexpansive mappings in Banach spaces and study the existence theorem and convergence analysis of Mann iteration, S−iteration and Normal S-iteration processes for generalized [a, b, c]-nonexpansive mappings. As the S-iteration process and normal S-iteration process is faster than that of Mann and Ishikawa iteration processes for various mappings, in this paper we will also compare these iteration processes for new class of generalized [a, b, c]-nonexpansive mappings with some examples. 2. PRELIMINARIES A Banach space X is said to be strictly convex [17] if x, y ∈ S X with x = y ⇒ (1 − λ)x + λy < 1 for all λ ∈ (0, 1), where S X = {x ∈ X : x = 1}. The modulus of convexity of X [17] is defined by x + y : x, y ≤ 1, x − y ≥ for all ∈ [0, 2]. δ X () = inf 1 − 2 The space X is said to be uniformly convex if δ X (0) = 0 and δ X () > 0 for all 0 < ≤ 2. The space X is said to be p-uniformly convex, if there is a constant c p > 0 such that δ X () ≥ c p p . Every Hilbert space is 2-uniformly convex and for p > 1 the space L p is max{ p, 2}-uniformly convex. Let C be a nonempty subset of a Banach space X . A function f : C → R is said to be coercive, if f (z n ) → ∞ whenever {z n } is a sequence in C such that z n → ∞. Let ∞ be the Banach space of all bounded real sequences with the supremum norm. A linear functional μ on ∞ is called a mean if μ(e) = μ = 1, where e = (1, 1, 1, ...). For x = (x1 , x2 , x3 , ...), the value μ(x) is also denoted by μn xn . A Banach limit on ∞ is an invariant mean i.e., μn (xn+1 ) = μn (xn ). If μ is a Banach limit on ∞ , then for x = (x1 , x2 , x3 , ...) ∈ ∞ , lim inf xn ≤ μn xn ≤ lim sup xn . n→∞ n→∞ Clearly, if limn→∞ xn = x, the Banach limit of {xn } is also x, i.e., μn xn = x (see, [17]). I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 124 D.R. Sahu and Satyendra Kumar Definition 2.1. Let C be a nonempty closed and convex subset of a Banach space X. Let {xn } be a sequence in C. A mapping T : C → X is said to be demiclosed, if {xn } weakly converges to x and {T xn } converges to y imply T x = y. Definition 2.2. (Opial condition) [21] A Banach space X is said to satisfy the Opial property, if for every weakly convergent sequence xn x in X, we have lim sup xn − x < lim sup xn − y n→∞ n→∞ for all y ∈ X with x = y. In above definition, lim sup can be replaced by lim inf . Remark 2.1. Every Hilbert spaces, finite dimensional Banach spaces and the Banach spaces p (1 ≤ p < ∞) satisfy the Opial property, but the uniformly convex Banach spaces L p [0, 2π ]( p = 2) do not satisfy Opial property. Lemma 2.1. [24] Let C be a nonempty closed and convex subset of a reflexive Banach space X and let f : C → R be a convex continuous and coercive function. Then there exists x ∗ ∈ C such that f (x ∗ ) = inf x∈C f (x). Lemma 2.2. [26] The Banach space X is uniformly convex iff .2 is uniformly convex on bounded convex sets i.e., for each r > 0 and ∈ (0, 1], there exist δ > 0 such that λx + (1 − λ)y2 ≤ λx2 + (1 − λ)y2 − λ(1 − λ)δ, for all λ ∈ (0, 1) and for all x, y ∈ Br [0] := {u ∈ X : u ≤ r } with x − y ≥ . Lemma 2.3. [17] Let p > 1 be a given real number and X be a Banach space. Then X is puniformly convex iff there exist a constant c p > 0 such that λx + (1 − λ)y p ≤ λx p + (1 − λ)y p − (λ p (1 − λ) + λ(1 − λ) p )c p x − y p , for all λ ∈ (0, 1) and for all x, y ∈ X. Lemma 2.4. [25] Let r > 0 be a fixed real number. If X is a uniformly convex Banach space, then there exists a continuous strictly increasing convex function g : [0, ∞) → [0, ∞) with g(0) = 0 such that λx + (1 − λ)y2 ≤ λx2 + (1 − λ)y2 − λ(1 − λ)g(x − y) for all x, y ∈ Br [0] and λ ∈ [0, 1]. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Fixed Point Theorems for Generalized [a, b, c] p -nonexpansive Mappings in Banach Spaces 125 Lemma 2.5. [17] Let X be a uniformly convex Banach space with modulus of convexity δ X . If r > 0 and x, y ∈ X with x ≤ r, y ≤ r, then x − y λx + (1 − λ)y ≤ r 1 − 2 min{λ, 1 − λ}δ X for all λ ∈ (0, 1). r Lemma 2.6. [20] If C is a closed convex subset of a strictly convex normed linear space, and T : C → C is quasi-nonexpansive, then Fi x(T ) = { p : p ∈ C and T p = p} is a nonempty closed convex set on which T is continuous. Let C be a nonempty convex subset of a normed linear space X and T : C → C a mapping with Fi x(T ) = ∅. Let {xn } be a sequence in C. Then the sequence {xn } is said to satisfy (D1 ) limit existence property (or LE property) if limn→∞ xn − z exists for all z ∈ Fi x(T ), (D2 ) approximate fixed point property (or AF point property) if limn→∞ xn − T xn = 0. (D3 ) LEAF point property if {xn } has both LE property and AF point property (see [19]). The properties (D1 ) and (D2 ) play important role in the approximation of fixed points of nonexpansive and asymptotically nonexpansive mappings by means of the Mann and S-iteration processes (see [27–34]) in Banach spaces under suitable geometric structures. 3. MAIN RESULTS 3.1. Generalized [a, b, c] p -nonexpansive mappings Definition 3.1. [12] Let C be a nonempty subset of a real Hilbert space H and λ a real number. A mapping T : C → H is said to be λ-hybrid if T x − T y2 ≤ x − y2 + 2(1 − λ) x − T x, y − T y for all x, y ∈ C. Definition 3.2. [13] Let C be a nonempty subset of a Banach space X and let α be a real number such that α < 1. A mapping T : C → X is said to be α-nonexpansive if T x − T y2 ≤ αT x − y2 + αx − T y2 + (1 − 2α)x − y2 for all x, y ∈ C. It can be noted that 0−nonexpansive mapping is nonexpansive and every firmly nonexpansive mapping is α−nonexpansive for all real number α such that 0 ≤ α ≤ 12 . Every α−nonexpansive mapping T with Fi x(T ) = ∅ is quasi-nonexpansive. Let C be a nonempty subset of a Hilbert space H and let T : C → H be a mapping. Let λ . Then T is λ−hybrid iff T is α−nonexpansive be a real number such that λ < 2 and put α = 1−λ 2−λ (see [13]). I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 126 D.R. Sahu and Satyendra Kumar We now introduced the generalized [a, b, c] p -nonexpansive mapping as follows: Definition 3.3. Let p > 1 and let C be a nonempty subset of a Banach space X . A mapping T : C → X is said to be generalized [a, b, c] p -nonexpansive mapping if there exists real numbers a, b and c with a < 1, b < 1 and a + b + c = 1 such that T x − T y p ≤ aT x − y p + bx − T y p + cx − y p for all x, y ∈ C. Remark 3.1. (i) If p = 2, then we denote generalized [a, b, c] p -nonexpansive mapping by generalized [a, b, c]-nonexpansive mapping. (ii) It follows from definition that c > −1. (iii) If a = b = α and p = 2, then generalized [a, b, c] p -nonexpansive mapping is αnonexpansive mapping. (iv) Generalized [0, 0, 1] p -nonexpansive mapping is nonexpansive mapping. Example 3.1. Let X = R with usual norm and C = [0, 1]. Let T : [0, 1] → [0, 1] be defined by x − x 2 , if x ∈ [0, 1); Tx = 1 (3.1) , if x = 1. 2 Then we have the following: (a) T is quasi-nonexpansive with Fi x(T ) = {0}. (b) T is generalized [0.1, 0.6, 0.3]-nonexpansive. (c) T is not 0.1-nonexpansive. Proof. (a) It is obvious. (b) Consider the following cases: Case I. x, y ∈ [0, 1). In this case T x − T y2 = (x − x 2 ) − (y − y 2 )2 = x − y2 1 − (x + y)2 ≤ x − y2 . Case II. x = y = 1. In this case T x − T y = 0. Case III. x ∈ [0, 1) and y = 1. Define F(x) = T x − 12 2 , f (x) = x − 12 2 , g(x) = T x − 12 and h(x) = x − 12 for all x ∈ [0, 1). From Figure 1, it is clear that F(x) ≤ 0.1 f (x) + 0.6g(x) + 0.3h(x) for all x ∈ [0, 1). I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Fixed Point Theorems for Generalized [a, b, c] p -nonexpansive Mappings in Banach Spaces 127 Figure 1. Graphical representation of F(x), 0.1 f (x) + 0.6g(x) + 0.3h(x) and 0.1 f (x) + 0.1g(x) + 0.8h(x) It means that 1 1 T x − 2 ≤ 0.1x − 2 + 0.6T x − 12 + 0.3x − 12 for all x ∈ [0, 1). 2 2 From cases I, II and III, we conclude that the mapping T is generalized [0.1, 0.6, 0.3]-nonexpansive. (c) From Figure 1, it is also clear that F(x) 0.1 f (x) + 0.1g(x) + (1 − 0.2)h(x) for all x ∈ [0, 1). It means that the mapping T is not 0.1-nonexpansive. Remark 3.2. From Remark 3.1 and Example 3.1, we observe that the class of generalized [a, b, c]nonexpansive mappings is essentially wider than class of α-nonexpansive mapping. Proposition 3.1. Let X be a Banach space with norm ., define norm on X × X × X by |(x1 , y1 , z 1 ) − (x2 , y2 , z 2 )| = (x1 − x2 2 + y1 − y2 2 + z 1 − z 2 2 )1/2 , for all (x1 , y1 , z 1 ), (x2 , y2 , z 2 ) ∈ X × X × X. Let Ti : X → X be generalized [a, b, c]nonexpansive mappings (i = 1, 2, 3). Define a mapping T : X × X × X → X × X × X by T (x1 , x2 , x3 ) = (T1 x1 , T2 x2 , T3 x3 ) for all x1 , x2 , x3 ∈ X. Then T is generalized [a, b, c]-nonexpansive. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 128 D.R. Sahu and Satyendra Kumar Proof. Let x = (x1 , y1 , z 1 ), y = (x2 , y2 , z 2 ) ∈ X × X × X. Then T x − T y2 = T (x1 , y1 , z 1 ) − T (x2 , y2 , z 2 )2 = (T1 x1 , T2 y1 , T3 z 1 ) − (T1 x2 , T2 y2 , T3 z 2 )2 = T1 x1 − T1 x2 2 + T2 y1 − T2 y2 2 + T3 z 1 − T3 z 2 2 ≤ aT1 x1 − x2 2 + bx1 − T1 x2 2 + cx1 − x2 2 +aT2 y1 − y2 2 + by1 − T2 y2 2 + cy1 − y2 2 +aT3 z 1 − z 2 2 + bz 1 − T3 z 2 2 + cz 1 − z 2 2 = a(T1 x1 − x2 , T2 y1 − y2 , T3 z 1 − z 2 )2 + b(x1 − T1 x2 , y1 − T2 y2 , z 1 − T3 z 2 )2 +c(x1 − x2 , y1 − y2 , z 1 − z 2 )2 = a(T1 x1 , T2 y1 , T3 z 1 ) − (x2 , y2 , z 2 )2 + b(x1 , y1 , z 1 ) − (T1 x2 , T2 y2 , T3 z 2 )2 +c(x1 , y1 , z 1 ) − (x2 , y2 , z 2 )2 = aT x − y2 + bx − T y2 + cx − y2 . This shows that T is generalized [a, b, c]-nonexpansive mapping. Lemma 3.1. Let p > 1 and let C be a nonempty subset of a Banach space X and T : C → X a generalized [a, b, c] p -nonexpansive mapping with Fi x(T ) = ∅. Then T is quasi-nonexpansive. Proof. For x ∈ C and z ∈ F(T ), we have T x − z p ≤ aT x − z p + bx − T z p + cx − z p = aT x − z p + (b + c)x − z p . This implies that (1 − a)T x − z p ≤ (b + c)x − z p . Thus, T is quasi-nonexpansive mapping. Proposition 3.2. Let p > 1 and let C be a nonempty closed convex subset of a p-uniformly convex Banach space X . Let T : C → C be a generalized [a, b, c] p -nonexpansive mapping. If Fi x(T ) = ∅, then Fi x(T ) is closed convex subset of C. Proof. Following Lemma 3.1 we have that if Fi x(T ) = ∅, then T is quasi-nonexpansive. By Lemma 2.6, Fi x(T ) is closed convex subset of C. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Fixed Point Theorems for Generalized [a, b, c] p -nonexpansive Mappings in Banach Spaces 129 Proposition 3.3. Let C be a nonempty subset of a Banach space X and T : C → X a generalized [a, b, c]-nonexpansive mapping. Then for all x, y ∈ C the following assertions holds: (i) If 0 ≤ a, b < 1, then x − T y2 ≤ 1+a 2 x − T x2 + (ax − y + T x − T y)x − T x + x − y2 . 1−b 1−b (ii) If a, b < 0, then x −T y2 ≤ 1−a 2 x −T x2 + ((−a)y −T x+T x −T y)x −T x+x − y2 . 1−b 1−b Proof. (i) Suppose that 0 ≤ a, b < 1. Let x, y ∈ C. Then x − T y2 = x − T x + T x − T y2 ≤ (x − T x + T x − T y)2 = x − T x2 + T x − T y2 + 2x − T xT x − T y ≤ x −T x2 +aT x − y2 + bx − T y2 + cx − y2 + 2x − T xT x − T y ≤ x − T x2 + a(T x − x + x − y)2 + bx − T y2 + cx − y2 +2x − T xT x − T y = x − T x2 + aT x − x2 + ax − y2 + 2aT x − xx − y +bx − T y2 + cx − y2 + 2x − T xT x − T y = (1 + a)x − T x2 + (a + c)x − y2 + bx − T y2 +2aT x − xx − y + 2x − T xT x − T y. This implies that (1−b)x −T y2 ≤ (1+a)x −T x2 + 2(ax − y + T x − T y)x − T x + (a + c)x − y2 . Therefore, we get x − T y2 ≤ 1+a 2 x − T x2 + (ax − y + T x − T y)x − T x + x − y2 . 1−b 1−b I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 130 D.R. Sahu and Satyendra Kumar (ii) Suppose that a, b < 0. Let x, y ∈ C. Then x − T y2 = x − T x + T x − T y2 ≤ (x − T x + T x − T y)2 = x − T x2 + T x − T y2 + 2x − T xT x − T y ≤ x − T x2 + aT x − y2 + bx − T y2 + cx − y2 +2x − T xT x − T y = x − T x2 + aT x − y2 + bx − T y2 + (1 − b)x − y2 +(−a)x − y2 + 2x − T xT x − T y ≤ x − T x2 + aT x − y2 + bx − T y2 + (1 − b)x − y2 +(−a)(x − T x + T x − y)2 + 2x − T xT x − T y = x − T x2 + aT x − y2 + bx − T y2 + (1 − b)x − y2 +(−a)(x − T x2 + T x − y2 + 2x − T xT x − y)2 +2x − T xT x − T y = (1 − a)x − T x2 + bx − T y2 +2((−a)T x − y + T x − T y)x − T x + (1 − b)x − y2 . This implies that (1−b)x −T y2 ≤ (1−a)x −T x2 +2((−a)T x − y+T x −T y)x −T x+(1 − b)x − y2 . Therefore, we get x − T y2 ≤ 1−a 2 x − T x2 + ((−a)T x − y + T x − T y)x − T x + x − y2 . 1−b 1−b Lemma 3.2. (Demicloseness Principle) Let C be a nonempty closed convex subset of a Banach space X with the Opial property. Let T : C → C be a generalized [a, b, c]-nonexpansive mapping. Then (I − T ) is demiclosed at zero. Proof. Let {xn } be a sequence in C such that it converges weakly to x ∗ and limn→∞ xn − T xn = 0. Then, sequences {xn } and {T xn } are bounded. Let M1 := sup{xn , T xn , x ∗ , T x ∗ : n ∈ N} < ∞. For 0 ≤ a, b < 1, we have xn − T x ∗ 2 ≤ ≤ 1+a 2 xn − T xn 2 + (axn − x ∗ + T xn − T x ∗ )xn − T xn 1−b 1−b +xn − x ∗ 2 1+a 4M1 (1 + a) xn − T xn 2 + xn − T xn + xn − x ∗ 2 . 1−b 1−b I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Fixed Point Theorems for Generalized [a, b, c] p -nonexpansive Mappings in Banach Spaces 131 Next, for a, b < 0, we have xn − T x ∗ 2 ≤ ≤ 1−a 2 xn − T xn 2 + ((−a)T xn − x ∗ + T xn − T x ∗ )xn − T xn 1−b 1−b +xn − x ∗ 2 1−a 4M1 (1 − a) xn − T xn 2 + xn − T xn + xn − x ∗ 2 . 1−b 1−b Therefore, in each cases, we have lim sup xn − T x ∗ ≤ lim sup xn − x ∗ . n→∞ n→∞ From Opial’s property, we get T x ∗ = x ∗ . 3.2. Existence In this section, we established the existence result of fixed points of generalized [a, b, c]nonexpansive mappings. We begin with the following proposition: Proposition 3.4. Let p > 1 and let C be a nonempty subset of a Banach space X . Let T : C → C be a generalized [a, b, c] p -nonexpansive mapping. Suppose that {T n x} is bounded for some x ∈ C. Then μn T n x − T y p ≤ μn T n x − y p for all Banach limit μ and for all y ∈ C. Proof. Let μ be a Banach limit and y ∈ C. Since T is generalized [a, b, c] p -nonexpansive mapping, we have T n+1 x − T y p ≤ aT n+1 x − y p + bT n x − T y p + cT n x − y p . Since μ is Banach limit, we get that μn T n x − T y p ≤ aμn T n x − y p + bμn T n x − T y p + cμn T n x − y p , which implies that (1 − b)μn T n x − T y p ≤ (a + c)μn T n x − y p . Theorem 3.1. Let p > 1 and let C be a nonempty closed convex subset of a p-uniformly convex Banach space X. Let T : C → C be a generalized [a, b, c] p -nonexpansive mapping. Suppose there exists x ∈ C such that {T n x} is bounded, then Fi x(T ) = ∅. Proof. Let f : C → R be a function defined by f (y) = μn T n x − y p for all y ∈ C. We claim that f is a convex, continuous and coercive function. Convexity of function f follows from Lemma 2.3. For continuity of f , let {ym } be a sequence in C such that limm→∞ ym = y. Then by mean value theorem, we have |T n x − ym p − T n x − y p | = p−1 |T n x − ym − T n x − y| | pcm,n |, I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS (3.2) Vol. 10, No. 1 (Special Issue), Jan–June 2019 132 D.R. Sahu and Satyendra Kumar for all m, n ∈ N, where min{T n x − ym , T n x − y} ≤ cm,n ≤ max{T n x − ym , T n x − y}. From (3.2), we get |T n x − ym p − T n x − y p | ≤ |T n x − ym − T n x − y| | p(T n x − ym + T n x − y) p−1 | ≤ ym − y sup{ p(T n x − ym + T n x − y) p−1 : m, n ∈ N}. This shows that the function g : C → ∞ defined by g(z) = {T 1 x − z p , T 2 x − z p , T 3 x − z p , ...} for all z ∈ C is continuous. Hence the mapping f = μog is continuous. Now to show that f is coercive, let {z m } is a sequence in C such that z m → ∞. Then we have T n x − z m p ≥ (|z m − T n x|) p . This implies that f (z m ) → ∞. Hence f is coercive. Since f is convex, continuous and coercive, from Lemma 2.1 there exists x * in C such that * f (x ) = inf x∈C f (x). By the uniform convexity of X , one can show that such a point x * is unique. From Proposition 3.4, we have f (T x * ) ≤ f (x * ). Hence x * ∈ Fi x(T ). Corollary 3.1. Let p > 1 and let C be a nonempty closed convex subset of a p-uniformly convex Banach space X. Let T : C → C be a α-nonexpansive mapping for some α < 1. Suppose there exists x ∈ C such that {T n x} is bounded, then Fi x(T ) = ∅. In view of Lemma 2.2, Proposition 3.4 and Theorem 3.1, we obtain the following result: Theorem 3.2. Let C be a nonempty closed convex subset of a uniformly convex Banach space X . Let T : C → C be a generalized [a, b, c]-nonexpansive mapping. Suppose there exist x ∈ C such that {T n x} is bounded, then Fi x(T ) = ∅. Corollary 3.2. Let C be a nonempty closed convex subset of a uniformly convex Banach space X . Let T : C → C be an α-nonexpansive mapping for some α < 1. Suppose there exist x ∈ C such that {T n x} is bounded, then Fi x(T ) = ∅. 3.3. Convergence We begin with following proposition: Proposition 3.5. Let C be a nonempty closed convex subset of a Banach space X . Let T : C → C be a generalized [a, b, c]-nonexpansive mapping such that Fi x(T ) = ∅. Let {αn } be sequences in (0, 1), and let {xn } be a sequence in C defined by Mann iteration process (1.1). Then for all z ∈ Fi x(T ), we have the following: I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Fixed Point Theorems for Generalized [a, b, c] p -nonexpansive Mappings in Banach Spaces 133 (a) xn+1 − z ≤ xn − z for all n ∈ N. (b) The sequence {xn } holds limit existence property (LE property) for T , i.e., limn→∞ xn − z exists. (c) limn→∞ d(xn , Fi x(T )) exists, where d(xn , Fi x(T )) = inf z∈Fi x(T ) xn − z. Proof. (a) Let z ∈ Fi x(T ). From (1.1) and using Lemma 3.1, we get xn+1 − z = (1 − αn )xn + αn T xn − z ≤ (1 − αn )xn − z + αn T xn − z ≤ (1 − αn )xn − z + αn xn − z = xn − z. (3.3) Therefore, we obtain xn+1 − z ≤ xn − z for all n ∈ N. (b) From (3.3), it is clear that the sequence {xn − z} is a monotonic decreasing sequence of nonnegative numbers. Therefore, it is convergent, i.e., limn→∞ xn − z exist. (c) This part follows from part (b). Theorem 3.3. Let C be a nonempty closed convex subset of a uniformly convex Banach space X. Let T : C → C be a generalized [a, b, c]-nonexpansive mapping such that Fi x(T ) = ∅. Let {αn } be sequence in (0, 1), and let {xn } be a sequence in C defined by the Mann iteration process (1.1). Then we have the following: (a) If {xn } is bounded and lim inf n→∞ xn − T xn = 0, then Fi x(T ) = ∅. (b) Assume that Fi x(T ) = ∅. Then {xn } is bounded and following holds: (i) If lim supn→∞ αn (1 − αn ) > 0, then lim inf n→∞ xn − T xn = 0. (ii) If lim inf n→∞ αn (1 − αn ) > 0, then the sequence {xn } holds approximate fixed point property (AF point property) for T , i.e., limn→∞ xn − T xn = 0. Proof. (a) Let {xn } is bounded and lim inf n→∞ xn − T xn = 0. Then there exists a subsequence {xn k } of {xn } such that limk→∞ xn k − T xn k = 0. Suppose that A(C, {xn k }) = {z}, and let M2 = supk∈N {xn k , T xn k , z, T z} < ∞. Then for 0 ≤ a, b < 1, we have 1+a 2 xn k −T xn k 2 + (axn k − z + T xn k −T z)xn k −T xn k +xn k − z2 1−b 1−b 1+a 4M2 (1 + a) xn k − T xn k 2 + xn k − T xn k + xn k − z2 . ≤ 1−b 1−b xn k − T z2 ≤ Also for a, b < 0, we have xn k − T z2 ≤ ≤ 1−a 2 xn k − T xn k 2 + ((−a)z − T xn k + T xn k − T z)xn k − T xn k 1−b 1−b +xn k − z2 1−a 2M2 (1 − a) xn k − T xn k 2 + xn k − T xn k + xn k − z2 . 1−b 1−b Therefore in each case, we get lim sup xn k − T z2 ≤ lim sup xn k − z2 . k→∞ k→∞ I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 134 D.R. Sahu and Satyendra Kumar This implies that r (T z, xn k ) = lim sup xn k − T z ≤ lim sup xn k − z = r (z, xn k ). k→∞ k→∞ This shows that T z ∈ A(C, {xn k }). Since X is uniformly convex Banach space, therefore we get that T z = z i.e., Fi x(T ) = ∅. (b) (i) Assume that Fi x(T ) = ∅ and z ∈ Fi x(T ). From Proposition 3.5, we have that limn→∞ xn − z exists and hence {xn } is bounded. Let lim xn − z = r > 0. n→∞ (3.4) In view of Lemma 2.4, there exists a continuous strictly increasing convex function g : [0, ∞) → [0, ∞) with g(0) = 0 such that xn+1 − z2 = = (1 − αn )xn + αn T xn − z2 (1 − αn )(xn − z) + αn (T xn − z)2 ≤ ≤ (1 − αn )xn − z2 + αn T xn − z2 − αn (1 − αn )g(xn − T xn ) (1 − αn )xn − z2 + αn xn − z2 − αn (1 − αn )g(xn − T xn ) = xn − z2 − αn (1 − αn )g(xn − T xn ). This implies that αn (1 − αn )g(xn − T xn ) ≤ xn − z2 − xn+1 − z2 . (3.5) Thus, if lim supn→∞ αn (1 − αn ) > 0, then we have lim inf n→∞ g(xn − T xn ) = 0. Since g(0) = 0, we get that if lim supn→∞ αn (1 − αn ) > 0, then lim inf n→∞ xn − T xn = 0. (ii) From (3.5), in same manner, we obtain that if lim inf n→∞ αn (1 − αn ) > 0, then limn→∞ xn − T xn = 0. Corollary 3.3. Let C be a nonempty closed convex subset of a uniformly convex Banach space X. Let T : C → C be an α-nonexpansive mapping for some α < 1. Let {αn } be sequence in (0, 1), and let {xn } be a sequence in C defined by the Mann iteration process (1.1). Then we have the following: (a) If {xn } is bounded and lim inf n→∞ xn − T xn = 0, then Fi x(T ) = ∅. (b) Assume that Fi x(T ) = ∅. Then {xn } is bounded and following holds: (i) If lim supn→∞ αn (1 − αn ) > 0, then lim inf n→∞ xn − T xn = 0. (ii) If lim inf n→∞ αn (1 − αn ) > 0, then limn→∞ xn − T xn = 0. Theorem 3.4. Let C be a nonempty closed convex subset of a uniformly convex Banach space X with the Opial property. Let T : C → C be a generalized [a, b, c]-nonexpansive mapping with Fi x(T ) = ∅. Let {αn } be sequence in (0, 1) such that lim inf n→∞ αn (1 − αn ) > 0, and let {xn } be a sequence in C defined by the Mann iteration process (1.1). Then {xn } converges weakly to a fixed point of T . Proof. From Theorem 3.3, it follows that the sequence {xn } is bounded and limn→∞ xn − T xn = 0. Since X uniformly convex Banach space, therefore X is reflexive. Then there exists a subsequence {xn k } of {xn } such that xn k z ∈ C as k → ∞. It follows Lemma 3.2 that z ∈ Fi x(T ). We claim I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Fixed Point Theorems for Generalized [a, b, c] p -nonexpansive Mappings in Banach Spaces 135 that xn z ∈ C as n → ∞. If possible, suppose that there exists another subsequence {xn j } of {xn } such that xn j z ∈ C as j → ∞ and z = z . From Lemma 3.2 we again conclude that z ∈ Fi x(T ). By Proposition 3.5 we have that limn→∞ xn − z and limn→∞ xn − z exists. Using Opial property, we have lim xn − z = n→∞ = < lim xn k − z < lim xn k − z k→∞ k→∞ lim xn − z = lim xn j − z n→∞ j→∞ lim xn j − z = lim xn − z, j→∞ n→∞ which is a contradiction. Thus we have z = z . This proves the theorem. Corollary 3.4. Let C be a nonempty closed convex subset of a uniformly convex Banach space X with the Opial property. Let T : C → C be an α-nonexpansive mapping for some α < 1 with Fi x(T ) = ∅. Let {αn } be sequence in (0, 1) such that lim inf n→∞ αn (1 − αn ) > 0, and let {xn } be a sequence in C defined by the Mann iteration process (1.1). Then {xn } converges weakly to a fixed point of T . Theorem 3.5. Let C be a nonempty compact convex subset of a uniformly convex Banach space X . Let T : C → C be a generalized [a, b, c]-nonexpansive mapping. Let {αn } be a sequence in (0, 1) such that lim supn→∞ αn (1 − αn ) > 0, and let {xn } be a sequence in C defined by the Mann iteration process (1.1). Then {xn } converges strongly to a fixed point of T . Proof. By compactness of C it follows that C is bounded and from Theorem 3.2 we get that Fi x(T ) = ∅. Since lim supn→∞ αn (1 − αn ) > 0, therefore from Proposition 3.3, we have that {xn } is bounded and lim inf n→∞ xn − T xn = 0. The compactness of C implies that there exists a subsequence {xn k } of {xn } such that xn k → z as k → ∞ for some z ∈ C and limk→∞ xn k − T xn k = 0. Let M3 = sup{xn k , T xn k , z, T z : k ∈ N} < ∞. For 0 ≤ a, b < 1, from Proposition 3.3, we have 1+a 2 xn k −T xn k 2 + (axn k −z+T xn k − T z)xn k − T xn k + xn k − z2 1−b 1−b 1+a 4M3 (1 + a) xn k − T xn k 2 + + xn k − z2 . ≤ 1−b 1−b xn k −T z2 ≤ Also, for a, b < 0, from Proposition 3.3, we have xn k − T z2 ≤ ≤ 1−a 2 xn k − T xn k 2 + ((−a)z − T xn k + T xn k − T z)xn k − T xn k 1−b 1−b +xn k − z2 1−a 4M3 (1 − a) xn k − T xn k 2 + xn k − T xn k + xn k − z2 . 1−b 1−b I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 136 D.R. Sahu and Satyendra Kumar Thus in each cases, we have lim sup xn k − T z2 ≤ lim sup xn k − z2 . k→∞ k→∞ This implies that limk→∞ xn k − T z = 0 and we have T z = z. By Proposition 3.5, limn→∞ xn − z exists. Thus, we have limn→∞ xn − z = 0. Corollary 3.5. Let C be a nonempty compact convex subset of a uniformly convex Banach space X . Let T : C → C be an α-nonexpansive mapping for some α < 1. Let {αn } be a sequence in (0, 1) such that lim supn→∞ αn (1 − αn ) > 0, and let {xn } be a sequence in C defined by the Mann iteration process (1.1). Then {xn } converges strongly to a fixed point of T . Proposition 3.6. Let C be a nonempty closed convex subset of a Banach space X . Let T : C → C be a generalized [a, b, c]-nonexpansive mapping such that Fi x(T ) = ∅. Let {αn } and {βn } be two sequences in (0, 1), and let {xn } be a sequence in C defined by S−iteration process (1.3). Then for all z ∈ Fi x(T ), we have the following: (a) max{xn+1 − z, yn − z} ≤ xn − z for all n ∈ N. (b) The sequence {xn } holds limit existence property (LE property) for T, i.e., limn→∞ xn − z exists. (c) limn→∞ d(xn , Fi x(T )) exists, where d(xn , Fi x(T )) = inf z∈Fi x(T ) xn − z. Proof. (a) Let z ∈ Fi x(T ). From (1.3) and using Lemma 3.1, we get yn − z = (1 − βn )xn + βn T xn − z ≤ (1 − βn )xn − z + βn T xn − z ≤ (1 − βn )xn − z + βn xn − z = xn − z. (3.6) From (1.3), (3.6) and Lemma 3.1, we get xn+1 − z = (1 − αn )T xn + αn T yn − z ≤ ≤ (1 − αn )T xn − z + αn T yn − z (1 − αn )xn − z + αn yn − z ≤ (1 − αn )xn − z + αn xn − z = xn − z. (3.7) Therefore, we obtain max{xn+1 − z, yn − z} ≤ xn − z for all n ∈ N. (b) From (3.7), it is clear that the sequence {xn − z} is a monotonic decreasing sequence of nonnegative numbers. Therefore, it is convergent, i.e., limn→∞ xn − z exists. (c) This part follows from part (b). Theorem 3.6. Let C be a nonempty closed convex subset of a uniformly convex Banach space X. Let T : C → C be a generalized [a, b, c]-nonexpansive mapping such that Fi x(T ) = ∅. Let {αn } I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Fixed Point Theorems for Generalized [a, b, c] p -nonexpansive Mappings in Banach Spaces 137 and {βn } be sequences in (0, 1) such that limn→∞ βn (1 − βn ) > 0, and let {xn } be a sequence in C defined by the S−iteration process (1.3). Then we have the following: (a) If {xn } is bounded and lim inf n→∞ xn − T xn = 0, then Fi x(T ) = ∅. (b) Assume that Fi x(T ) = ∅. Then {xn } is bounded and following holds: (i) If lim supn→∞ αn (1 − αn ) > 0, then lim inf n→∞ xn − T xn = 0. (ii) If lim inf n→∞ αn (1 − αn ) > 0, then the sequence {xn } satisfy approximate fixed point property (AF point property), i.e., limn→∞ xn − T xn = 0. Proof. (a) This part follows from Theorem 3.3. (b) (i) Assume that Fi x(T ) = ∅ and z ∈ Fi x(T ). From Proposition 3.6, we have that limn→∞ xn − z exists and hence {xn } is bounded. Let lim xn − z = r > 0. (3.8) n→∞ In view of Lemma 2.4, there exists a continuous strictly increasing convex function g : [0, ∞) → [0, ∞) with g(0) = 0 such that xn+1 − z2 = (1 − αn )T xn + αn T yn − z2 = (1 − αn )(T xn − z) + αn (T yn − z)2 ≤ (1 − αn )T xn − z2 + αn T yn − z2 − αn (1 − αn )g(T xn − T yn ) ≤ ≤ (1 − αn )xn − z2 + αn yn − z2 − αn (1 − αn )g(T xn − T yn ) (1 − αn )xn − z2 + αn xn − z2 − αn (1 − αn )g(T xn − T yn ) = xn − z2 − αn (1 − αn )g(T xn − T yn ). This implies that αn (1 − αn )g(T xn − T yn ) ≤ xn − z2 − xn+1 − z2 . (3.9) If lim supn→∞ αn (1 − αn ) > 0. Then, from (3.9), we have lim inf n→∞ g(T xn − T yn ) = 0. g(0) = 0 implies that lim inf T xn − T yn = 0. n→∞ (3.10) Now xn+1 − T xn = (1 − αn )T xn + αn T yn − T xn = αn T xn − T yn . Using (3.10), we get lim inf xn+1 − T xn = 0. n→∞ (3.11) Again xn+1 − T yn ≤ xn+1 − T xn + T xn − T yn . Therefore, from (3.10) and (3.11), we get lim inf xn+1 − T yn = 0. n→∞ I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS (3.12) Vol. 10, No. 1 (Special Issue), Jan–June 2019 138 D.R. Sahu and Satyendra Kumar Next xn+1 − z ≤ xn+1 − T yn + T yn − z ≤ xn+1 − T yn + yn − z. Using (3.8) and (3.12), we have r ≤ lim inf yn − z. (3.13) n→∞ Notice that yn − z ≤ xn − z for all n ∈ N. Therefore, from (3.8), we obtain that lim sup yn − z ≤ r. (3.14) lim yn − z = r. (3.15) n→∞ From (3.13) and (3.14), we have n→∞ Since yn − z ≤ xn − z and T xn − z ≤ xn − z. Therefore, from (3.15) and using Lemma 2.5, we get r = = ≤ lim yn − z = lim (1 − βn )xn + βn T xn − z n→∞ n→∞ lim (1 − βn )(xn − z) + βn (T xn − z) n→∞ xn − T xn . lim xn − z 1 − 2βn (1 − βn )δ X n→∞ xn − z This implies that 2 lim xn − zβn (1 − βn )δ X n→∞ xn − T xn xn − z ≤ 0. Notice that limn→∞ xn − z = r > 0 and limn→∞ βn (1 − βn ) > 0. Therefore, we get xn − T xn = 0. lim δ X n→∞ xn − z Hence, by property of δ X we conclude that if lim supn→∞ αn (1 − αn ) > 0, then lim inf n→∞ xn − T xn = 0. (ii) In the same manner as above, from (3.9), we can obtain that if lim inf n→∞ αn (1 − αn ) > 0, then limn→∞ xn − T xn = 0. Corollary 3.6. Let C be a nonempty closed convex subset of a uniformly convex Banach space X. Let T : C → C be a α-nonexpansive mapping for some α < 1. Let {αn } and {βn } be sequences in (0, 1) such that limn→∞ βn (1 − βn ) > 0, and let {xn } be a sequence in C defined by the S−iteration process (1.3). Then we have the following: (a) If {xn } is bounded and lim inf n→∞ xn − T xn = 0, then Fi x(T ) = ∅. (b) Assume that Fi x(T ) = ∅. Then {xn } is bounded and following holds: (i) If lim supn→∞ αn (1 − αn ) > 0, then lim inf n→∞ xn − T xn = 0. (ii) If lim inf n→∞ αn (1 − αn ) > 0, then limn→∞ xn − T xn = 0. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Fixed Point Theorems for Generalized [a, b, c] p -nonexpansive Mappings in Banach Spaces 139 Theorem 3.7. Let C be a nonempty closed convex subset of a uniformly convex Banach space X with the Opial property. Let T : C → C be a generalized [a, b, c]-nonexpansive mapping with Fi x(T ) = ∅. Let {αn } and {βn } be sequences in (0, 1) such that lim inf n→∞ αn (1 − αn ) > 0 and limn→∞ βn (1 − βn ) > 0, and let {xn } be a sequence in C defined by the S-iteration process (1.3). Then {xn } converges weakly to a fixed point of T . Corollary 3.7. Let C be a nonempty closed convex subset of a uniformly convex Banach space X with the Opial property. Let T : C → C be an α-nonexpansive mapping for some α < 1 with Fi x(T ) = ∅. Let {αn } and {βn } be sequences in (0, 1) such that lim inf n→∞ αn (1 − αn ) > 0 and limn→∞ βn (1 − βn ) > 0, and let {xn } be a sequence in C defined by the S-iteration process (1.3). Then {xn } converges weakly to a fixed point of T . Theorem 3.8. Let C be a nonempty compact convex subset of a uniformly convex Banach space X . Let T : C → C be a generalized [a, b, c]-nonexpansive mapping. Let {αn } and {βn } be sequences in (0, 1) such that lim supn→∞ αn (1 − αn ) > 0 and limn→∞ βn (1 − βn ) > 0, and let {xn } be a sequence in C defined by the S-iteration process (1.3). Then {xn } converges strongly to a fixed point of T . Corollary 3.8. Let C be a nonempty compact convex subset of a uniformly convex Banach space X . Let T : C → C be an α-nonexpansive mapping for some α < 1. Let {αn } and {βn } be sequences in (0, 1) such that lim supn→∞ αn (1 − αn ) > 0 and limn→∞ βn (1 − βn ) > 0, and let {xn } be a sequence in C defined by the S-iteration process (1.3). Then {xn } converges strongly to a fixed point of T . Proposition 3.7. Let C be a nonempty closed convex subset of a Banach space X . Let T : C → C be a generalized [a, b, c]-nonexpansive mapping such that Fi x(T ) = ∅. Let {αn } be a sequence in (0, 1), and let {xn } be a sequence defined by Normal S−iteration process (1.4). Then for all z ∈ Fi x(T ), we have the following: (a) xn+1 − z ≤ xn − z for all n ∈ N. (b) The sequence {xn } satisfy limit existence property (LE property), i.e., limn→∞ xn − z exists. (c) limn→∞ d(xn , Fi x(T )) exists, where d(xn , Fi x(T )) = inf z∈Fi x(T ) xn − z. Proof. (a) Let z ∈ Fi x(T ). From (1.4) and using Lemma 3.1, we get xn+1 − z = T [(1 − αn )xn + αn T xn ] − z ≤ ≤ [(1 − αn )xn + αn T xn − z (1 − αn )xn − z + αn T xn − z ≤ = (1 − αn )xn − z + αn xn − z xn − z. (3.16) Therefore, we obtain xn+1 − z ≤ xn − z for all n ∈ N. (b) From (3.7), it is clear that the sequence {xn − z} is a monotonic decreasing sequence of nonnegative numbers. Therefore, it is convergent i.e., limn→∞ xn − z exist. (c) This part follows from part (b). I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 140 D.R. Sahu and Satyendra Kumar Theorem 3.9. Let C be a nonempty closed convex subset of a uniformly convex Banach space X. Let T : C → C be a generalized [a, b, c]-nonexpansive mapping such that Fi x(T ) = ∅. Let {αn } be a sequence in (0, 1), and let {xn } be a sequence in C defined by the Normal S−iteration process (1.4). Then we have the following: (a) If {xn } is bounded and lim inf n→∞ xn − T xn = 0, then Fi x(T ) = ∅. (b) Assume that Fi x(T ) = ∅. Then {xn } is bounded and following holds: (i) If lim supn→∞ αn (1 − αn ) > 0, then lim inf n→∞ xn − T xn = 0. (ii) If lim inf n→∞ αn (1 − αn ) > 0, then the sequence {xn } satisfy approximate fixed point property (AF point property), i.e., limn→∞ xn − T xn = 0. Proof. (a) This part follows from Theorem 3.3. (b) (i) Assume that Fi x(T ) = ∅ and z ∈ Fi x(T ). From Proposition 3.7, we have that limn→∞ xn − z exists and hence {xn } is bounded. Let lim xn − z = r > 0. n→∞ (3.17) In view of Lemma 2.4, there exists a continuous strictly increasing convex function g : [0, ∞) → [0, ∞) with g(0) = 0 such that xn+1 − z2 = T [(1 − αn )xn + αn T xn ] − z2 ≤ ≤ (1 − αn )xn + αn T xn − z2 (1 − αn )(xn − z) + αn (T xn − z)2 ≤ ≤ (1 − αn )xn − z2 + αn T xn − z2 − αn (1 − αn )g(xn − T xn ) (1 − αn )xn − z2 + αn xn − z2 − αn (1 − αn )g(xn − T xn ) = xn − z2 − αn (1 − αn )g(xn − T xn ). This implies that αn (1 − αn )g(xn − T xn ) ≤ xn − z2 − xn+1 − z2 . (3.18) From (3.18), it is easy to show that if lim sup αn (1 − αn ) > 0, then lim inf xn − T xn = 0. n→∞ n→∞ (ii) From (3.18), we can show that if lim inf αn (1 − αn ) > 0, then lim xn − T xn = 0. n→∞ n→∞ Corollary 3.9. Let C be a nonempty closed convex subset of a uniformly convex Banach space X. Let T : C → C be a α-nonexpansive mapping for some α < 1. Let {αn } be a sequence in (0, 1), and let {xn } be a sequence in C defined by the Normal S−iteration process (1.4). Then we have the following: I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Fixed Point Theorems for Generalized [a, b, c] p -nonexpansive Mappings in Banach Spaces 141 (a) If {xn } is bounded and lim inf n→∞ xn − T xn = 0, then Fi x(T ) = ∅. (b) Assume that Fi x(T ) = ∅. Then {xn } is bounded and following holds: (i) If lim supn→∞ αn (1 − αn ) > 0, then lim inf n→∞ xn − T xn = 0. (ii) If lim inf n→∞ αn (1 − αn ) > 0, then limn→∞ xn − T xn = 0. Theorem 3.10. Let C be a nonempty closed convex subset of a uniformly convex Banach space X with the Opial property. Let T : C → C be a generalized [a, b, c]-nonexpansive mapping with Fi x(T ) = ∅. Let {αn } and be a sequence in (0, 1) such that lim inf n→∞ αn (1 − αn ) > 0, and let {xn } be a sequence in C defined by the Normal S-iteration process (1.4). Then {xn } converges weakly to a fixed point of T . Corollary 3.10. Let C be a nonempty closed convex subset of a uniformly convex Banach space X with the Opial property. Let T : C → C be an α-nonexpansive mapping for some α < 1 with Fi x(T ) = ∅. Let {αn } be a sequence in (0, 1) such that lim inf n→∞ αn (1 − αn ) > 0, and let {xn } be a sequence in C defined by the Normal S-iteration process (1.4). Then {xn } converges weakly to a fixed point of T . Theorem 3.11. Let C be a nonempty compact convex subset of a uniformly convex Banach space X . Let T : C → C be a generalized [a, b, c]-nonexpansive mapping. Let {αn } be a sequence in (0, 1) such that lim supn→∞ αn (1 − αn ) > 0, and let {xn } be a sequence in C defined by the Normal S-iteration process (1.4). Then {xn } converges strongly to a fixed point of T . Corollary 3.11. Let C be a nonempty compact convex subset of a uniformly convex Banach space X . Let T : C → C be an α-nonexpansive mapping for some α < 1. Let {αn } be a sequence in (0, 1) such that lim supn→∞ αn (1 − αn ) > 0, and let {xn } be a sequence in C defined by the Normal Siteration process (1.4). Then {xn } converges strongly to a fixed point of T . 4. NUMERICAL EXAMPLE Example 4.1. Let X = R with usual norm and C = [0, 1]. Let T : [0, 1] → [0, 1] be a mapping defined by (3.1). Set α = 0.1, a = 0.1, b = 0.6 and c = 0.3. It is shown in Example 3.1 that, T is not α-nonexpansive but it is generalized [a, b, c]-nonexpansive mapping. 1 1 ) and βn = ( 18 + 4n ) for all n ∈ N. For initial points 0.1, 0.4, 0.7 and 1.0, the Set αn = ( 61 + 4n convergence of Mann, Ishikawa, S-iteration and Normal S-iterations processes upto 60000 iterations are shown in Figure 2. Set stopping criteria xn − x ∗ ≤ 10−4 . The effect of parameters αn and βn on convergence of different iteration processes are given in Table 1 for the following set of parameters: (i) αn (ii) αn (iii) αn (iv) αn 1 1 = ( 16 + 4n ) and βn = ( 81 + 4n ). 1 1 1 = (1 − 2n ) and βn = ( 2 + 3n ). = (1 − 2n1 2 ) and βn = 12 . n = n+1 and βn = n−1 . n+1 I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 142 D.R. Sahu and Satyendra Kumar Figure 2. Convergence of iteration processes with different initial points Table 1. Number of iterations required to obtain fixed point with different initial points and parameters 1 1 αn = ( 61 + 4n ) and βn = ( 18 + 4n ) for all n ∈ N 0.1 0.4 0.7 59917 59960 59966 53263 53303 53309 9780 9786 9786 31 36 36 1 1 αn = (1 − 2n ) and βn = ( 21 − 3n ) for all n ∈ N Initial point 0.1 0.4 0.7 Mann iteration 9990 9996 9996 Ishikawa iteration 6664 6669 6669 S-iteration 6661 6665 6665 Normal S-iteration 6 6 7 αn = (1 − 2n12 ) and βn = 12 for all n ∈ N Initial point 0.1 0.4 0.7 Mann iteration 9985 9992 9992 Ishikawa iteration 6659 6663 6663 S-iteration 6658 6662 6662 Normal S-iteration 5 5 5 n αn = n+1 and βn = n−1 for all n ∈ N n+1 Initial point 0.1 0.4 0.7 Mann iteration 9994 10000 10000 Ishikawa iteration 5009 5012 5013 S-iteration 5005 5008 5007 Normal S-iteration 7 7 7 Initial point Mann iteration Ishikawa iteration S-iteration Normal S-iteration I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS 1.0 59969 53311 9787 36 1.0 9997 6669 6666 7 1.0 9992 6664 6663 5 1.0 10001 5013 5008 7 Vol. 10, No. 1 (Special Issue), Jan–June 2019 Fixed Point Theorems for Generalized [a, b, c] p -nonexpansive Mappings in Banach Spaces 143 Using stopping criteria xn − x ∗ ≤ 10−4 and above set of parameters, we have studied convergence of different iteration processes for initial points 0.1, 0.4, 0.7 and 1.0. We observe that the S-iteration process is faster than Mann and Ishikawa iteration processes, and the Normal Siteration process is faster than Mann, Ishikawa and S-iteration processes for generalized [a, b, c]nonexpansive mappings. Also it is stable with respect to the parameters αn and βn . REFERENCES [1] R. E. Bruck Jr., Nonexpansive projections on subsets of Banach spaces, Pacific J. Math., 47(1973), 341–355. [2] K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, in: Cambridge Studies in Advanced Mathematics, vol. 28, Cambridge University Press, Cambridge, 1990. [3] K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, in: Monographs and Textbooks in Pure and Applied Mathematics, vol. 83, Marcel Dekker Inc., New York, 1984. [4] S. Reich and I. Shafrir, The asymptotic behavior of firmly nonexpansive mappings, Proc. Amer. Math. Soc., 101(1987), 246–250. [5] R. Smarzewski, On firmly nonexpansive mappings, Proc. Amer. Math. 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Takahashi, Fixed point theorems for new nonlinear mappings in Hilbert spaces, J. Nonlinear Conve Anal., 11(2010), 79–88. [12] K. Aoyama, S. Lemoto, F. Kohsaka and W. Takahashi, Fixed point and ergodic theorems for λ-hybrid mapping in Hilbert spaces, J. Nonlinear Convex Anal., 11(2010), 335–3473. [13] K. Aoyama and F. Kohsaka, Fixed point theorem for α-nonexpansive mappings in Banach spaces, Nonlinear Anal., 74(2011), 4387–4391. [14] W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4(1953), 506–610. [15] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44(1974), 147–150. [16] T. Zamfirescu, Fix point theorems in metric spaces, Archiv der. Mathematik., 23(1992), 292–298. [17] R. P. Agarwal, Donal O’Regan and D. R. Sahu, Fixed point theory for Lipschitzian-type mappings with applications, Series: Topological Fixed Point Theory and Its Applications, Springer, New York, 6, 2009. [18] R. P. Agarwal, D. O’Regan and D. R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex Anal., 8(2007), 61–79. [19] D. R. Sahu, Applications of the S-iteration process to constrained minimization problems and split feasibility problems, Fixed Point Theory and Appl., 12(2011), 187–204. [20] W. G. Dotson, Jr., Fixed points of quasi-nonexpansive mappings, North Carolina State University at Raleigh Raleigh, North Carolina, 1969. [21] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73(1967), 595–597. [22] E. Naraghirad, N. C. Wang and J. C. Yao, Approximating fixed points of α-nonexpansive mappings in uniformly convex Banach spaces and CAT(0) spaces, Fixed Point Theory and Appl., 2013: 57(2013). [23] Y. Song, K. Promluang, P. Kumam and Y. J. Cho, Some convergence theorems of the Mann iteration for monotone α-nonexpansive mappings, Appl. Math. Comput., 287–288(2016), 74–82. [24] W. 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Krasnoselskii, Two observations about the method of successive approximations, Uspehi Math. Nauk, 10(1955), 123–127. [32] Z.Q. Liu and S. M. Kang, Weak and strong convergence for fixed points of asymptotically nonexpansive mappings, Acta Math. Sinica, 20(2004), 1009–1018. [33] B. E. Rhoades, Fixed point iterations for certain nonlinear mappings, J. Math. Anal. Appl., 183(1994), 118–120. [34] K. K. Tan and H. K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl., 178(1993), 301–308. [35] H. Hussain, A. Rafiq, B. Damjanoic and R. Lazovic, On rate of convergence of various iterative schemes, Fixed Point Theory and Appl., (2011), 1/45. [36] V. Kumar, A. Latif, A. Rafiq and N. Hussain, S-iteration process for quasi-contractive mappings, J. Ineq. and Appl., (2013), 206. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 INDIAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019, pp. 145–151 DOI: On Generalized Operator Mixed Vector Quasi-equilibrium Problem Rais Ahmad1 , Haider Abbas Rizvi2 , Mijanur Rahaman1 and Javid Iqbal3 1 Department of Mathematics, Aligarh Muslim University, Aligarh-202002, India Department of Applied Mathematics, ZHCET, Aligarh Muslim University, Aligarh-202002, India 3 Department of Mathematical Sciences, Baba Ghulam Shah Badshah University, Rajouri, India Email: 1 raisain [email protected]; 2 [email protected]; 1 [email protected]; 3 [email protected] 2 Abstract: In this paper, we study a new operator equilibrium problem which is a conjunction of two problems i.e., an operator vector quasi-equilibrium problem and an operator vector quasi-variational inequality problem. We call our problem as generalized operator mixed vector quasi-equilibrium problem. We prove some existence results for our problem in topological vector spaces by using 1-person game theorem and core of a set of operators. We do not use KKM-theory and generalized p-quasi convexity as used by other authors in related works. Some special cases of generalized operator mixed quasi-equilibrium problem are also discussed. Keywords: Existence, Game Theorem, Core, Operator, Equilibrium. AMS subject classification: 2010 49J40, 47H19, 47H10 1. INTRODUCTION The important generalizations of a variational inequality problem is a vector variational inequality problem as well as a vector equilibrium problem. It is well known in the theory of applied sciences that equilibrium problems and their generalizations provides us powerful tools to study a wide range of problems arising in nonlinear analysis, optimization, economics, finance and game theory, etc.. The equilibrium problems includes many mathematical programming problems,complementarity problems, variational inequality problems, fixed point problems, minimax inequality problem, Nash equilibrium problems in non-cooperative games, etc., see [3, 5, 8, 15, 16]. Extending the concept of scalar and vector variational inequalities, Domokos and Kolumban [7] introduced the concept of operator variational inequalities and shown that scalar and vector variational inequalities are the special cases of operator variational inequalities, also see [1, 2, 4, 9, 11, 12, 17–20] and references therein. On the other hand operator equilibrium problem for I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 146 Rais Ahmad, Haider Abbas Rizvi, Mijanur Rahaman and Javid Iqbal single-valued operators was studied by Kazmi and Raouf [10] and for multi-valued operators by Kum and Kim [19]. The above mentioned research works motivated us to introduce and study a generalized operator mixed vector quasi-equilibrium problem which involves an operator quasi vector variational inequality problem as well as an operator quasi vector equilibrium problem. The concept of core of a set [3] is extended for core of a set of operators and a related proposition is proved. We prove two existence results for generalized operator mixed vector quasi equilibrium problem by using 1-person game theorem and core of a set of operators. Our results are more general than others available in the literature. 2. PRELIMINARIES Let X and Y be the Hausdorff topological vector spaces, L(X, Y ) be the space of all continuous linear operator from X into Y equipped with the topology of pointwise convergence and K ⊂ L(X, Y ) a nonempty convex set. A nonempty subset C of X is called a convex cone if λC ⊆ C, for all λ > 0 and C + C = C. We denote by 2 K the family of all subsets of K , int X K the interior of K in X , cl X K the closure of K in X , and coK the convex hull of K. Suppose that T : K → X , F : K × K → Y are the single-valued operators such that F( f, f ) = 0, for all f ∈ K . Let A : K → 2 K and C : K → 2Y be multi-valued operators such that C( f ) is a convex cone with intC( f ) = φ and 0 ∈ / C( f ), for all f ∈ K . We consider the following problem of finding f ∈ K such that f ∈ cl K A( f ) and f − g, T ( f ) + F( f, g) ∈ / −C( f ), for all g ∈ A( f ). (2.1) We call problem 2.1 as generalized operator mixed vector quasi-equilibrium problem. Below are some special cases of problem 2.1. T = 0 and A( f ) = K , then from problem 2.1, we get an operator equilibrium problem mentioned in Kazmi and Rauf [10]. (2) If F = 0 and A( f ) = K , then problem 2.1 reduces to the operator variational inequality problem considered by Domokos and Kolumbán [7]. (3) If F = 0, A( f ) = K and T is a multi-valued operator, then problem 2.1 coincides with the operator vector variational inequality studied by Kum and Kim [13]. (4) If T = 0 and F is a multi-valued operator, then problem 2.1 reduces to the generalized operator quasi-equilibrium problem considered by Kum and Kim [14]. (1) We need the following lemma, which is a variant form of Theorem 2 of Ding, Kim and Tan [6]. Lemma 2.1. Let K be a nonempty compact convex subset of a Hausdorff topological vector space X . Suppose that A, cl X A, P : K → 2 K are multi-valued operator such that for each x ∈ K , A(x) is nonempty convex set, for each y ∈ K , A−1 (y) is open set in K , cl X A is upper semicontinuous, for each x ∈ K , x ∈ / CoP(x) and for each y ∈ K , P −1 (y) in open in K . Then there exists x ∗ ∈ K such ∗ that x ∈ cl K A(x ∗ ) and A(x ∗ ) ∩ P(x ∗ ) = φ. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 On Generalized Operator Mixed Vector Quasi-equilibrium Problem 147 The following definition is an extension of definition of Core of a set defined by Blum and Oettli [3] for operators. Definition 2.1. Let K and D be the convex subsets of L(X, Y ) and D ⊂ K . The core of D relative to K , denoted by cor e K f D, is the set defined by f ∈ cor e K f D if and only if f ∈ D and D ∩ {λ f + (1 − λ)g} = φ, λ ∈ (0, 1), for all g ∈ K \ D. Definition 2.2. An operator T : → X is said to be C( f )-convex if for any f, g ∈ K and λ ∈ [0, 1], T (λg + (1 − λ f )) ∈ λT (g) + (1 − λ)T ( f ) − C( f ). Definition 2.3. Let B be a subset of K . A multi-valued operator W : K → 2Y is said to have a closed graph with respect to B if for every net { f α }α∈τ ⊂ K and {yα }α∈τ ⊂ Y such that yα ∈ C( f α ), f α converges to f ∈ B and yα converges to y ∈ Y, then y ∈ C( f ). 3. EXISTENCE RESULTS In this section, we prove some existence results for generalized operator mixed vector quasiequilibrium problem 2.1. Theorem 3.1. Let K be a nonempty compact convex subset of L(X, Y ), where L(X, Y ) is equipped with the topology of pointwise convergence. Let T : K → X , F : K × K → Y be the single-valued operators and C : K → 2Y be the multi-valued operator such that for each f ∈ K , C( f ) is closed convex cone in Y with intC( f ) = φ and 0 ∈ / C( f ). Assume that For each f ∈ K , F( f, f ) = 0, the operator T is continuous, the operator F is continuous in the first argument and affine in the second argument, the operator W : K → 2Y defined by W ( f ) = Y \ −C( f ) has a closed graph in K × Y , the multi-valued operator A : K → 2Y such that A( f ) is nonempty convex subset of K and A−1 (g) is open in K for each f, g ∈ K , 6. the mapping cl K A( f ) : K → 2 K is upper semicontinuous. 1. 2. 3. 4. 5. Then the generalized operator mixed vector quasi-equilibrium problem 2.1 admits a solution. Proof. As L(X, Y ) is equipped with the topology of pointwise convergence, it is a locally convex Hausdorff topological vector space. We define a multi-valued operator P : K → 2 K by P( f ) = {g ∈ K : f − g, T ( f ) + F( f, g) ∈ −C( f )}, for each f ∈ K . Initially, we show that f ∈ / CoP( f ). Suppose to the contrary that there exists f ∈ K such n that f ∈ CoP( f ). Then there exists a finite subset {g1 , g2 , ....gn } of K and λi ≥ 0, such that i=1 λi = 1 n λi gi . and f = i=1 I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 148 Rais Ahmad, Haider Abbas Rizvi, Mijanur Rahaman and Javid Iqbal That is f − gi , T ( f ) + F( f, gi ) ∈ −C( f ). Hence n λi f − gi , T ( f ) + i=1 n F( f, gi ) ∈ −C( f ). i=1 It is clear from the hypothesis that f − n i=1 λi gi , T ( f ) + F( f, n λi gi ) ∈ −C( f ), i=1 i.e., f − f, T ( f ) + F( f, f ) ∈ −C( f ). Since F( f, f ) = 0, we have, 0 ∈ −C( f ), which contradicts that 0 ∈ / C( f ) and hence f ∈ / CoP( f ). Now, we show that P −1 (g) is open in K , which is equivalent to show that [P −1 (g)]c = K \ / P −1 (g) is closed. In fact, let { f λ } be a net in [P −1 (g)]c convergent to f ∈ K . Then, we have g ∈ P( f λ ) and hence f λ − g, T ( f λ ) + F( f λ , g) ∈ / −C( f λ ), thus f λ − g, T ( f λ ) + F( f λ , g) ∈ W ( f λ ). As W has a closed graph in K × Y , T is continuous and F is continuous in the first argument, we have f − g, T ( f ) + F( f, g) ∈ W ( f ), i.e., f − g, T ( f ) + F( f, g) ∈ / −C( f ). Hence f ∈ [ p −1 (g)]c and [ p −1 (g)]c is closed, i.e., P −1 is open in K . Thus all the condition of Lemma 2.1 are satisfied. Hence there exists f ∈ K such that f ∈ cl K A( f ) and A( f ) ∩ P( f ) = φ, i.e., f − g, T ( f ) + F( f, g) ∈ / −C( f ), for all g ∈ A( f ). Thus the generalized operator mixed vector quasi-equilibrium problem 2.1 admits a solution. The following corollaries can be derived from Theorem 3.1, easily. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 On Generalized Operator Mixed Vector Quasi-equilibrium Problem 149 Corollary 3.1. If T = 0 and F( f, g) = ϕ( f ) − ϕ(g), where ϕ : K → Y is a mapping, then there exists f ∈ K such that the following operator minimization problem has a solution. / −C( f ), for all g ∈ A( f ). f ∈ cl K A( f ) and ϕ( f ) − ϕ(g) ∈ Corollary 3.2. If F = 0, then there exists f ∈ K such that the following operator variational inequality problem has a solution f ∈ cl K A( f ) and f − g, T ( f ) ∈ / −C( f ), for all g ∈ A( f ). The following Lemma is an extension of Lemma 4 of Blum and Oettli [3] for operators. Lemma 3.1. Let K and D be the convex subsets of L(X, Y ) and D ⊂ K . Assume that ϕ : K → L(X, Y ) is C( f )-convex, f 0 ∈ cor e K f D, ϕ( f 0 ) ∈ −C( f ) and ϕ(g) ∈ C( f ), for all g ∈ D. Then ϕ(g) ∈ C( f ), for all g ∈ K . Proof. On the contrary, suppose that ϕ(h) ∈ / C( f ), for some h ∈ K \ D. Suppose for some λ ∈ (0, 1), f̄ = λ f 0 + (1 − λ)h. By using the C( f )-convexity of ϕ, we have ϕ(λ f 0 + (1 − λ)h) ∈ λϕ( f 0 ) + (1 − λ)ϕ(h) − C( f ) i.e., ϕ( f̄ ) ∈ λϕ( f 0 ) + (1 − λ)ϕ(h) − C( f ) ∈ −C( f ) + [Y \ C( f )] − C( f ) = [Y \ C( f )], That is ϕ( f̄ ) ∈ / C( f ). Since f 0 ∈ cor ek f D and D ∩ {λ f 0 + (1 − λh)} = φ, so we have g ∈ D ∩ {λ f 0 + (1 − λh)} such that ϕ(g) ∈ / C( f ), which is a contradiction to hypothesis. Hence ϕ(g) ∈ C( f ), ∀g ∈ K . Without using compactness assumption and employing above Lemma 3.1, we prove the following result for generalized operator mixed vector quasi-equilibrium problem 2.1. Theorem 3.2. Let K be a nonempty convex subset of L(X, Y ). Let T : K → X and F : K × K → Y be the single-valued operators and C : K → 2Y and A : K → 2 K be the multi-valued operators such that intC( f ) = φ and 0 ∈ C( f ), for each f ∈ K , satisfying all the conditions of Theorem 3.1. In addition, the following condition is satisfied : there exists a non-empty compact convex subset D of K such that for f ∈ D \ cor e K f D, h ∈ cor e K f D ∩ A( f ), such that f ∈ cl D A( f ), f − h, T ( f ) + F( f, h) ∈ −C( f ). I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 150 Rais Ahmad, Haider Abbas Rizvi, Mijanur Rahaman and Javid Iqbal Then, there exists f ∈ K such that f ∈ cl K A( f ) and f − g, T ( f ) + F( f, g) ∈ / −C( f ), for all g ∈ A( f ). Proof. By Theorem 3.1, it follows that there exists f ∈ D such that f ∈ cl D A( f ) and f − g, T ( f ) + F( f, g) ∈ / −C( f ), ∀ g ∈ A( f ). Set ϕ(g) = f − g, T ( f ) + F( f, g). Then ϕ(g) is C( f )-convex and ϕ(g) ∈ C( f ), for all g ∈ D. If f ∈ cor e K f D, then choose f 0 = f. If f ∈ D \ Cor e K f D, then choose f 0 = h, where h is same as in the hypothesis of the theorem. In both cases, f 0 ∈ Cor e K f D and φ( f 0 ) ∈ −C( f ). Applying Lemma 3.1, it follows that φ(g) ∈ C( f ), for all g ∈ K , which implies that there exist f ∈ K such that f − g, T ( f ) + F( f, g) ∈ C( f ), for all g ∈ K . As cl K A : K → 2 K is upper semicontinuous with compact values, there exists f ∈ K such that f ∈ cl K A( f ) and f − g, T ( f ) + F( f, g) ∈ / −C( f ), for all g ∈ A( f ). REFERENCES [1] Q.H. Ansari, Vector equilibrium problems and vector variational inequalities, Non-convex optimization and its Application, (F. Gianessi, Editor) (Kluwer Academic Publishers, Dordrecht, 2000), 1–14. [2] Q.H. Ansari, S. Schaible and J.C. Yao, Systems of vector equilibrium problems and applications, J. Optim. Theory Appl., 107(2000), 547–557. [3] E. Blum, W. Oettli, From optimization and variational inequalities to equlibrium problems, Math. Student, 63(1994), 123–145. [4] G.Y. Chen, Existence of solutions for vector variational inequalities: An extension of Hartmann-Stampacchia theorem, J. Optim. Theory Appl., 74(1992), 445–456. [5] X.P. 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Sehlager, Generalized vectorial equilibria and generalized monotonicity in Functional Analysis with current Application in Sciences, Technology and Industry, (M. Brokate and A.H. Siddiqi, Editors), Pitman Research Notes in Mathematical series 77 (Longman Publishing co., London 1997). [18] S. Park, Foundations of the KKM theory via coincidences of composites of upper semicontinuous maps, J. Korean Math. Soc., 31(1994), 493–519. [19] N.X. Tan and P.N. Tinh, On the existence of equilibrium points of vector functions, Numer. Funct. Anal. Optim., 19(1998), 141–156. [20] S.J. Yu, J.C. Yao, On vector variational inequalities, J. optim Theory Appl., 89(1996), 749–769. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 INDIAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019, pp. 152–164 DOI: On Approximation of Signals in the Generalized Zygmund Class Using (E, 1)(N, pn )-summability Means of Fourier Series Tejaswini Pradhan1 , Susanta Kumar Paikray2∗ and Umakanta Misra3 1,2 Department of Mathematics, Veer Surendra Sai University of Technology, Burla, Odisha-768018, India 3 Department of Mathematics, National Institute of Science and Technology, Palur Hills, Golanthara, Odisha-761008, India (∗ Corresponding author) Email: ∗ skpaikray [email protected], 1 [email protected], 3 umakanta [email protected] Abstract: In this paper, we investigate the degree of approximation of a function in the generalized Zygmund class Z r(ω) (r ≥ 1) by (E, 1)(N , pn )-summability means of trigonometric Fourier series. Keywords: Degree of approximation; generalized Zygmund class; trigonometric Fourier series; (E, 1)-summability means; (N , pn )-summability means; (E, 1)(N , pn )-summability means AMS Subject classification (2010): 41A24; 41A25; 42B05; 42B08 1. INTRODUCTION AND PRELIMINARIES Signal analysis is concerned with the reliable estimation, detection and classification of signals (functions) which are subject to random fluctuations. Signal analysis has its roots in probability theory, mathematical statistics and, more recently, approximation theory and communications theory. These approximation analysis of signals (functions) have great importance in the field of science and engineering. And it has given a new dimensions due to their wide applications in signal analysis, system design in modern telecommunications, radar and image processing system. The error estimation of functions in various function spaces such as Lipschitz, Hölder, Zygmund, Besov spaces using different summability techniques of trigonometric fourier series has been received a growing interest of several researchers in last decades. Functions in L r (r ≥ 1)-spaces assumed to be most practicable in signal analysis. Particularly, L 1 , L 2 and L ∞ spaces are used by engineers for designing digital filters. Very recently, the generalized different Lipschitz classes have been established in various summability means. For more details, see the current works [5, 6, 7, 8] and [9]. Subsequently, the generalized Zygmund class Z r(ω) (r ≥ 1) is a generalization of Z (α) , Z (α),r , Z (ω) -classes. The generalized Zygmund class Z r(ω) (r ≥ 1) is investigated by Leindler [4], Moricz [11], Moricz and Nemeth [12]. Very recently in 2017 Singh et al. [14] proved approximation of functions in the I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Approximation of Signals in the Generalized Zygmund Class 153 generalized Zygmund class using Hausdörff means. Lal and Shireen [2] proved best approximation of functions of generalized Zygmund class by matrix-Euler summability mean of Fourier series. To get best approximation and advance study in this direction, In the proposed paper, we investigate the degree of approximation of a function in the generalized Zygmund class Z r(ω) (r ≥ 1) by (E, 1)(N , pn ) means of trigonometric Fourier series. Let f be a 2π -periodic and integrable functions belonging to 2π L r [0, 2π ] = f : [0, 2π ] → R; | f (x)|r d x < ∞ . 0 The Fourier series of f (x) is given by ∞ u n (x) = n=0 ∞ 1 (an cos nx + bn sin nx) a0 + 2 n=1 (1.1) with its partial sum Sn ( f ; x) = 1 π π −π f (x + t)Dn (t)dt, where Dn (t) = sin(n + 12 )t . 2 sin 2t The L r norm of a function f (x) is defined by ⎧ 2π ⎪ 1 r ⎪ ⎪ ⎨ 2π 0 | f (x)| d x f r = ⎪ ⎪ ⎪ ⎩ess sup | f (x)| 1 r (1 ≤ r < ∞) (r = ∞). 0<x≤2π Zygmund modulus of continuity of f (x) is defined by ω( f, h) = sup | f (x + t) + f (x − t) − 2 f (x)| (see [16]). 0≤h, x∈R Let C2π denote the Banach space of all 2π -periodic continuous functions defined on [0, 2π ] under the supremum norm. For 0 < α ≤ 1, the function space Z (α) = { f ∈ C2π : | f (x + t) + f (x − t) − 2 f (x)| = O(|t|α )} is a Banach space under the norm · (α) is defined by | f (x + t) + f (x − t) − 2 f (x)| . |t|α x,t =0 f (α) = sup | f (x)| + sup 0≤x≤2π I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 154 Tejaswini Pradhan, Susanta Kumar Paikray and Umakanta Misra For f ∈ L r [0, 2π ] (r ≥ 1) the integral Zygmund modulus of continuity is defined by ωr ( f, h) = sup 0<t≤h 1 2π 2π | f (x + t) + f (x − t) − 2 f (x)| d x r1 r . 0 For f ∈ C2π and r = ∞, ω∞ ( f, h) = sup max | f (x + t) + f (x − t) − 2 f (x)|. 0<t≤h x It is known that ωr ( f, h) → 0 as r → 0. Now define Z (α),r = f ∈ L r [0, 2π ] : 2π | f (x + t) + f (x − t) − 2 f (x)| d x r r1 α = O(|t| ) . 0 The space Z (α),r , r ≥ 1, 0 < α ≤ 1 is a Banach space under the norm · (α),r . f (α),r = f r + sup t =0 f (· + t) + f (· − t) − 2 f (·)r . |t|α The class of function Z (ω) is defined as Z (ω) = { f ∈ C2π : | f (x + t) + f (x − t) − 2 f (x)| = O(ω(t))} where ω is a Zygmund modulus of continuity, that is, ω is positive, non-decreasing continuous function with the sub linearity property that is, (i) ω(0) = 0 (ii) ω(t1 + t2 ) ≤ ω(t1 ) + ω(t2 ) Let ω : [0, 2π ] → R be an arbitrary function with ω(t) > 0 for 0 ≤ t < 2π and let limt→0+ ω(t) = ω(0) = 0, define f (· + t) + f (· − t) − 2 f (·)r (ω) Z r = f ∈ L r : 1 ≤ r ≤ ∞, sup <∞ ω(t) t =0 where f r(ω) = f r + sup t =0 f (· + t) + f (· − t) − 2 f (·)r , r ≥ 1. ω(t) Clearly, · r(ω) is a norm on Z r(ω) . As we know L r (r ≥ 1) is complete, the space Z r(ω) is also complete. Hence we can say Z r(ω) is a Banach space under the norm · r(ω) . I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Approximation of Signals in the Generalized Zygmund Class Let ω(t) and v(t) denotes the Zygmund moduli of continuity such that decreasing, then ω(2π ) f r(ω) ≤ ∞. f r(v) ≤ max 1, v(2π ) Thus, we have ω(t) v(t) 155 be positive, non- (1.2) Z r(ω) ⊆ Z r(v) ⊆ L r (r ≥ 1). Remark 1. (i) If we take r → ∞ then the class Z r(ω) reduces to the Z (ω) class. (ii) If we take ω(t) = t α in Z r(ω) class, then it reduces to Z (α),r class. (iii) If we take ω(t) = t α , the Z (ω) class reduces to Z (α) class. Let u n be a given infinite series with the sequence of partial sum {sn }. Let { pk } for k = 0, 1, 2, .... be a sequence of nonnegative integers such that p0 > 0 and n Pn = pk → ∞ as n → ∞. (1.3) k=0 Let the sequence to sequence transformation n 1 τnN = pk sk , n = 0, 1, 2, ..., Pn k=0 (1.4) u n is said to be defines (N , pn ) mean of {sn } generated by the sequence { pk }. The series summable to s by (N , pn ) method if, limn→∞ τnN → s as n → ∞ and this (N , pn ) method is regular [1]. The sequence to sequence transformation, E n1 n 1 k sk , = n 2 k=0 v (1.5) defines the (E, 1) transform of the sequence {sn }. The series u n is summable to s with respect to (E, 1) summability, If E n1 → s as n → ∞. Also (E, 1) method is regular [1]. Now we define a new composite transform using the product (E, 1)(N , pn ) transform. As the (N , pn ) and (E, 1) summability methods are regular, the product (E, 1)(N , pn ) method is also regular. Let n k 1 n 1 EN p v sv , (1.6) τn = n 2 k=0 k Pk v=0 defines the (E, 1)(N , pn ) transform of the sequence {sn }. We say that series by product (E, 1)(N , pn ) transform, if τnE N → s as n → ∞. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS u n is summable to s Vol. 10, No. 1 (Special Issue), Jan–June 2019 156 Tejaswini Pradhan, Susanta Kumar Paikray and Umakanta Misra We use the following notations through out this paper; φ(x, t) = f (x + t) + f (x − t) − 2 f (x) k n 1 sin(v + )t n 1 1 2 K nE N (t) = pv . π 2n+1 k=0 k Pk v=0 sin( 2t ) 2. KNOWN THEOREMS Lal and Shireen [2] proved the following theorems using Matrix-Euler summability means of Fourier series. Theorem 1. Let the lower triangular matrix A = (an,k ) satisfying the following conditions: n an,k ≥ 0 (n = 0, 1, 2, ...; k = 0, 1, 2, ...); n−1 |an,k | = O k=0 1 n+1 an,k = 1, (2.1) k=0 and (n + 1)an,n = O(1). (2.2) if f : [0, 2π ] → R be a 2π periodic, Lebesgue integrable and belonging to generalized Zygmund class Z r(ω) (r ≥ 1); ω and v be Zygmund modulus of continuity and ω(t) be positive, non-decreasing v(t) then the best approximation of f by triangular matrix-Euler means tnE = n an,k k=0 k 1 k sv 2k v=0 v of its Fourier series (1.1) is given by E n ( f ) = inf tnE tnE − f r(v) =O 1 n+1 π 1 n+1 ω(t) . t 2 v(t) (2.3) Theorem 2. Let A = (an,k ) be lower triangular matrix satisfying conditions (2.7) and (2.8) and in ω(t) is non-increasing. For f ∈ Z r(ω) (r ≥ 1), its best approximation by addition to Theorem 1, tv(t) triangular matrix-Euler means tnE satisfies 1 ω( n+1 ) En = O log(n + 1)π . (2.4) 1 v( n+1 ) 3. MAIN RESULTS The main objective of this paper is to prove the following theorems. Theorem 1. Let f : [0, 2π ] → R be a 2π periodic, Lebesgue integrable and belonging to generalized Zygmund class Z r(ω) (r ≥ 1). Then the degree of approximation of signal (function) f, using I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Approximation of Signals in the Generalized Zygmund Class product (E, 1)(N , pn ) mean of Fourier series (1.1), is given by E n ( f ) = inf τnE N − f rv = O τnE N π 1 n+1 ω(t) , t v(t) where ω(t) and v(t) denotes the Zygmund moduli of continuity such that ing. 157 (3.1) ω(t) v(t) is positive and increas- Theorem 2. Let f : [0, 2π ] → R be a 2π periodic, Lebesgue integrable and belonging to generalized Zygmund class Z r(ω) (r ≥ 1). Then the degree of approximation of signal (function) f, using (E, 1)(N , pn ) means of Fourier series (1.1), is given by 1 ω( n+1 ) EN v E n ( f ) = inf τn − f r = O (π (n + 1) − 1) , (3.2) 1 (n + 1)2 v( n+1 ) τnE N where w(t) and v(t) denotes the Zygmund moduli of continuity such that decreasing. w(t) tv(t) is positive and To prove the above theorems, we need the following Lemmas. Lemma 1. |K nE N (t)| = O(n) for 0 ≤ t ≤ Proof. For 0 ≤ t ≤ 1 n+1 1 . n+1 and sin nt ≤ n sin t. |K nE N (t)| 1 ≤ π 2n+1 n n k 1 ≤ π 2n+1 n n k k=0 k 1 sin(v + 12 )t pv Pk v=0 sin( 2t ) k=0 k 1 (2v + 1) sin( 2t )t pv Pk v=0 sin( 2t ) n k 1 n 2k + 1 = pv n+1 k π2 P k k=0 v=0 n 2n + 1 n = k π 2n+1 k=0 2n + 1 = π 2n+1 ∵ 2n = n n k=0 = O(n). I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS k (3.3) Vol. 10, No. 1 (Special Issue), Jan–June 2019 158 Tejaswini Pradhan, Susanta Kumar Paikray and Umakanta Misra Lemma 2. |K nE N (t)| = O( 1t ) for Proof. For 1 n+1 1 n+1 ≤ t ≤ π. ≤ t ≤ π , sin nt ≤ 1 and sin 2t ≥ πt . n n 1 1 |K nE N (t)| ≤ π 2n+1 k=0 k Pk n 1 n 1 ≤ π 2n+1 k=0 k Pk n 1 π n ≤ · π 2n+1 t k=0 k n 1 n ≤ t2n+1 k=0 k 1 = O . t k sin(v + 12 )t pv sin( 2t ) v=0 k 1 pv t sin( ) 2 v=0 k 1 pv Pk v=0 (3.4) Lemma 3 (see [2. )] Let f ∈ Z r(ω) then for 0 < t ≤ π, (i) φ(·, t)r = O(ω(t)) O(ω(t)) O(ω(y)) (iii) If ω(t) and v(t) defined as in Theorem 1, then (ii) φ(· + y, t) + φ(· − y, t) − 2φ(·, t)r = ω(t) φ(· + y, t) + φ(· − y, t) − 2φ(·, t)r = O v(y) v(t) where φ(x, t) = f (x + t) + f (x − t) − 2 f (x). 4. PROOF OF THEOREM 3 Let sk ( f ; x) denotes the k th partial sum of the series (1.1) and following [15], we have π 1 sin(k + 1/2)t dt. φ(x; t) sk ( f ; x) − f (x) = 2π 0 sin(t/2) Therefore using (1.4), the (N , pn ) transform of sk ( f ; x) is given by π n n 1 1 1 sin(k + 1/2)t dt. pk (sk ( f ; x) − f (x)) = φ(x; t) pk Pn k=0 2π 0 Pn k=0 sin(t/2) I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS (4.1) (4.2) Vol. 10, No. 1 (Special Issue), Jan–June 2019 Approximation of Signals in the Generalized Zygmund Class 159 Now denoting the (E, 1)(N , pn ) transform of sk ( f ; x) by τnE N , then we write v n 1 π φ(x; t) 1 − f (x) = pv sin(v + 1/2)tdt. π 2n+1 k=0 0 sin(t/2) Pk k=0 τnE N (4.3) Let Ln (x) = τnE N − f (x) = π 0 φ(x; t)K nE N (t)dt. (4.4) Then Ln (x + y) + Ln (x − y) − 2Ln (x) = π 0 [φ(x + y; t) + φ(x − y; t) − 2φ(x; t)] K nE N (t)dt. (4.5) Using generalized Minkowski’s inequality we have Ln (· + y) + Ln (· − y) − 2Ln (·)r = = 1 2π 1 2π ≤ 0 π 1/r 2π |Ln (x + y) + Ln (x − y) − 2Ln (x)|r d x 0 π r 1/r EN [φ(x + y; t) + φ(x − y; t) − 2φ(x; t)]K n (t)dt d x 2π 0 1 2π 0 2π 0 π = 0 r 1/r dt. [φ(x + y; t) + φ(x − y; t) − 2φ(x; t)]K nE N (t)d x (|K nE N (t)|)1/r +φ(x − y; t) − π = 0 0 + π 1 n+1 0 2π [φ(x + y; t) r 1/r EN 2φ(x; t)]K n (t) d x dt φ(· + y; t) + φ(· − y; t) − 2φ(·; t)r |K nE N (t)|dt 1 n+1 = 1 2π φ(· + y; t) + φ(· − y; t) − 2φ(·; t)r |K nE N (t)|dt φ(· + y; t) + φ(· − y; t) − 2φ(·; t)r |K nE N (t)|dt = I1 + I2 (say). I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS (4.6) Vol. 10, No. 1 (Special Issue), Jan–June 2019 160 Tejaswini Pradhan, Susanta Kumar Paikray and Umakanta Misra Now, using Lemma 1, Lemma 3 and monotonicity of (ω(t)/v(t)) with respect to t, we have I1 1 n+1 = ≤ φ(· + y; t) + φ(· − y; t) − 2φ(·; t)r |K nE N (t)|dt ω(t) O(n)dt O v(y) v(t) 0 1 n+1 ω(t) dt . O n v(y) v(t) 0 = 0 1 n+1 By using 2nd mean value theorem of integral, we have 1 1 ω( n+1 ) n+1 I1 ≤ O nv(y) dt 1 v( n+1 ) 0 1 ) ω( n+1 n = O v(y) 1 n+1 v( n+1 ) 1 ) ω( n ∵ = O(1) . = O v(y) n+1 1 n+1 v( n+1 ) (4.7) Next, using Lemma 2 and Lemma 3, we get π φ(· + y; t) + φ(· − y; t) − 2φ(·; t)r |K nE N (t)|dt I2 = 1 ≤ = ω(t) 1 dt v(y) 1 v(t) t n+1 π ω(t) O v(y) dt . 1 tv(t) n+1 n+1 π (4.8) From (4.20), (4.21) and (4.22), we have Ln (· + y) + Ln (· − y) − 2Ln (·)r = O v(y) 1 ω( n+1 ) + O v(y) 1 v( n+1 ) π 1 n+1 ω(t) dt . tv(t) (4.9) Therefore, we have Ln (· + y) + Ln (· − y) − 2Ln (·)r =O sup v(y) y =0 1 ω( n+1 ) 1 v( n+1 ) +O π 1 n+1 ω(t) dt . tv(t) (4.10) Clearly, φ(x; t) = | f (x + t) + f (x − t) − 2 f (x)|. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Approximation of Signals in the Generalized Zygmund Class 161 Now applying Minkowski’s inequality, we have φ(·, t)r = = f (x + t) + f (x − t) − 2 f (x)r O(ω(t)). (4.11) Now using Lemma 1, Lemma 2 and (4.25), we obtain Ln (·)r ≤ 1 n+1 + 1 n+1 0 = = = = π φ(·, t)r |K nE N (t)|dt ω(t) dt ω(t)dt + O 1 t 0 n+1 1 π n+1 1 w(t) O nω( ) dt ω(t)dt + O 1 n+1 0 t n+1 π n 1 ω(t) O w( ) +O dt 1 n+1 n+1 t n+1 π 1 ω(t) O ω( ) +O dt . 1 n+1 t n+1 O n 1 n+1 π (4.12) From (4.24) and (4.26), we have Ln (·)rv Ln (· + y) + Ln (· − y) − 2Ln (·)r v(y) y =0 1 π π ω( n+1 ) ω(t) ω(t) 1 ) +O dt + O dt . O ω( +O 1 1 1 n+1 t tv(t) v( n+1 ) n+1 n+1 = Ln (·)r + sup = = 4 Ji . (4.13) i=1 Now we write J1 in terms of J3 and further J2 , J3 in terms of J4 . In view of monotonicity of v(t) for 0 < t ≤ π , we have ω(t) ω(t) .v(t) ≤ v(π ) .v(t) = O ω(t) = v(t) v(t) ω(t) . v(t) Therefore, we can write J1 = O(J3 ). I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS (4.14) Vol. 10, No. 1 (Special Issue), Jan–June 2019 162 Tejaswini Pradhan, Susanta Kumar Paikray and Umakanta Misra Again by using monotonicity of v(t), v π ω(t) J2 = dt 1 t n+1 1 n+1 ≤ = Using the fact that ω(t) v(t) π ω(t) v(t)dt tv(t) π ω(t) dt v(π ) 1 tv(t) n+1 = O(J4 ). (4.15) is positive and increasing, we have π 1 1 ω( n+1 ω( n+1 ) π dt ) ω(t) dt = ≥ J4 = . 1 1 1 1 tv(t) t v( n+1 ) n+1 v( n+1 ) n+1 (4.16) Therefore, we can write J3 = O(J4 ). Now combining (4.27) and (4.31), we have Ln (·)rv = O(J4 ) = O En ( f ) = ω(t) . tv(t) π 1 n+1 Hence, inf Ln (·)rv n (4.17) =O π 1 n+1 ω(t) . tv(t) (4.18) (4.19) This completes the proof of Theorem 3. PROOF OF THEOREM 4 Following the proof of Theorem 3, we have E n ( f ) = inf Ln (·)rv = O n In Theorem 4 we assumed π 1 n+1 ω(t) dt . tv(t) (4.20) ω(t) tv(t) is positive and decreasing in t. Thus, we have π 1 ω( n+1 ) v dt E n ( f ) = inf Ln (·)r = O 1 n 1 (n + 1)v( n+1 ) n+1 1 ω( n+1 ) = O [t]π 1 1 (n + 1)v( n+1 ) n+1 1 ω( n+1 ) = O (π (n + 1) − 1) . 1 (n + 1)2 v( n+1 ) (4.21) This completes the proof of Theorem 4. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Approximation of Signals in the Generalized Zygmund Class 163 5. APPLICATIONS The following corollaries can be obtained from Theorem 3. Corollary 1. If we replace (E, 1)(N , pn ) mean by (E, 1)(C, 1) mean in Theorem 3, then the degree of approximation of a function f ∈ Z r(ω) by (E, 1)(C, 1) mean τnEC n k 1 1 n = n sv (see [13]) 2 k=0 k k + 1 v=0 of Fourier series (1.1) is given by is given by En ( f ) = O π 1 n+1 ω(t) dt . tv(t) (5.1) Corollary 2. If we replace (E, 1)(N , pn ) mean by (E, q)(N , pn , qn ) mean in Theorem 3, then the degree of approximation of a function f ∈ Z r(ω) by (E, q)(N , pn , qn ) mean n k n n−k 1 1 EN τn = q pk−v qv sv (see [10]) (1 + q)n k=0 k Rk v=0 of Fourier series (1.1) is given by En ( f ) = O π 1 n+1 ω(t) dt . tv(t) (5.2) Corollary 3. If we replace (E, 1)(N , pn ) mean by Euler-Hausdörff mean in Theorem 3, then the degree of approximation of a function f ∈ Z r(ω) by Euler-Hausdörff mean τnE H = n k n n−k 1 q h k,v sv (see [3]) (1 + q)n k=0 k v=0 of Fourier series (1.1) is given by En ( f ) = O 1 n+1 π 1 n+1 ω(t) dt . t 2 v(t) (5.3) REFERENCES [1] Hardy G. H., Divergent Series, Oxford University Press, First Edition, (1949). [2] Lal S. and Shireen, Best approximation of functions of generalized Zygmund class by Matrix-Euler summability mean of Fourier series.Bull. Math. Anal. Appl., 5(4) (2013), 1–13. [3] Lal S. and Mishra A., Euler-Hausdörff matrix summability operator and trigonometric approximation of the conjugate of a function belonging to the generalized Lipschitz class, J. Inequal. Appl., 2013, 59 (2013). [4] Leindler L., Strong approximation and generalized Zygmund class.Acta Sci. Math., 43 (1981), 301–309. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 164 Tejaswini Pradhan, Susanta Kumar Paikray and Umakanta Misra [5] Mishra V. N., Khatri K. and Mishra L. N., Approximation of Functions belonging to Li p(ξ (t), r ) class by (N , pn )(E, q)-summability of Conjugate Series of Fourier series, J. Inequal. Appl., 2012 (2012), Article ID: 296. [6] Mishra L. N., Mishra V. N., Khatri K. and Deepmala, On the Trigonometric approximation of signals belonging to generalized weighted Lipschitz W (L r , ξ (t))(r ≥ 1)− class by matrix (C 1 .N p ) Operator of conjugate series of its Fourier series, Appl. Math. 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I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 INDIAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019, pp. 165–181 DOI: n-tupled Coincidence Point Results for Nonlinear Contraction in Partially Ordered Complete Metric Spaces Mohammad Imdad1 , Aftab Alam2 , Anupam Sharma3∗ and Mohammad Arif4 1,2,4 Department of Mathematics, Aligarh Muslim University, Aligarh-202 002, India Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur-208016, India (∗ Corresponding author) Email: ∗ [email protected], 1 [email protected], 2 [email protected], 4 [email protected] 3 Abstract: In this paper, we establish n-tupled coincidence point theorems for a pair of mappings satisfying a nonlinear contractivity condition in partially ordered complete metric spaces. Our results extend and generalize several existing results in the literature: [Bhaskar and Lakshmikantham: Nonlinear Anal. 65(7) (2006), 1379-1393], [Lakshmikantham, and Ćirić: Nonlinear Anal. 70 (2009), 4341-4349], [Berinde and Borcut: Nonlinear Anal. 74(15) (2011), 4889-4897], [Borcut and Berinde: App. Math. Comp. 218(10) (2012), 5929-5936], [Borcut: App. Math. Comp. 218(14) (2012), 7339-7346], [Borcut: HJMC], [Gordji and Ramezani: preprint]. 2010 Mathematics Subject Classiffication. 47H10, 54H25. Keywords: Partially ordered metric space; nonlinear -contraction; MCB property; n-tupled coincidence point. 1. INTRODUCTION The classical Banach fixed point theorem [2] and its applications are well known. In the recent past, many authors extended this theorem by considering relatively more general contractive mappings on various types of metric spaces. In 2004, Ran and Reurings [17] extended Banach fixed point theorem to partially ordered complete metric spaces. Thereafter, Nieto and López [15] modified the Ran and Reuring’s fixed point theorem. Nieto and López’s fixed point theorem further generalized by many authors, for example ( [1, 16]). The idea of coupled fixed point was initiated by Guo and Lakshmikantham [9] in 1987 which is also followed by Bhaskar and Lakshmikantham [4] wherein authors introduced the notion of mixed monotone property for a weakly linear contraction mapping F : X 2 → X, (where X is a partially I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 166 M. Imdad, A. Alam, A. Sharma and M. Arif ordered metric space) and utilized the same to prove some theorems on existence and uniqueness of coupled fixed points, which are viewed as the coupled formulation of certain results of Nieto and López [15]. In 2009, Lakshmikantham and Ćirić [14] generalized these results for nonlinear contraction mappings by introducing the notions of coupled coincidence point and mixed g-monotone property. In recent years, the notion of coupled fixed point is extended to higher dimensions by many authors. In an attempt to extend the definition from X 2 to X 3 , Berinde and Borcut [3] introduced the concept of tripled fixed point and utilize the same to prove some tripled fixed point theorems. Karapinar and Luong [13] introduced the notion of quadrupled fixed point and utilize the same to prove some quadrupled fixed point theorems for nonlinear contractions satisfying mixed monotone property. In fact, in 2010, Samet and Vetro [18] introduced the notion of fixed point of n-order (or n-tupled fixed point), n ∈ N and n ≥ 3, as natural extension of coupled as well as quadrupled fixed points and established some results for n-tupled fixed point in complete metric spaces, using a new concept of F-invariant set. Here it can be pointed out that the notion of tripled fixed point due to Berinde and Borcut [3] is different from the one defined by Samet and Vetro [18] for n = 3 in the case of partially ordered metric spaces in order to keep the mixed monotone property properly working. Afterwards, Imdad et al. [10] generalized the notion of n-tupled fixed point for a pair of mappings F and g by defining n-tupled coincidence point and introduced the general concept of mixed gmonotone property on X n , for even n. For more details see [11, 12, 19]. Also they employed the same to prove an even-tupled coincidence theorem for nonlinear contraction mappings satisfying mixed g-monotone property. Basically this result is true for only even n but not for odd ones. Recently, Gordji and Ramezani [8] introduced the concept of a new n-tupled fixed point, which is an extension of coupled as well as tripled fixed points and established some n-tupled fixed point results for linear contractions in complete metric spaces. However, the notion of n-fixed point introduced by Gordji and Ramazani [8] is different from n-tupled fixed point in the sense of Samet and Vetro [18], but it holds for a general natural number n (odd or even) in order to keep mixed monotone property properly working. Gordji and Ramezani [8] say this type n-tupled fixed point as n-fixed point. 2. PRELIMINARIES Throughout the paper, n stands for a general natural number, In := {1, 2, ..., n}, i, j ∈ In . Let X denotes a non-empty set and X n denotes the product set X × X × ... × X (n times) and {On , E n } denotes a partition of In , where n+1 , On = 2 p − 1 : p ∈ 1, 2, ..., 2 that is, the set of all odd natural numbers in In . n , E n = 2 p : p ∈ 1, 2, ..., 2 that is, the set of all even natural numbers in In . It is clear that On ∪ E n = In and On ∩ E n = ∅. Also, m ∈ N ∪ {0}, {x m } and {U m } denote sequences in X and X n respectively, where U m = I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 n-tupled Coincidence Point Results 167 (x1m , x2m , . . . , xnm ) so that {x1m }, {x2m }, . . . , {xnm } are sequences in X. Let F : X n → X and g : X → X be two mappings. Definition 2.1 [16]. A triplet (X, d, ) is called a partially ordered metric space if X is a nonempty set, d is a metric on X and is a partial order relation on X . Moreover, if d is a complete metric on X , we say that (X, d, ) is a partially ordered complete metric space. Definition 2.2 [1]. Let (X, d, ) be a partially ordered metric space. We say that (X, d, ) has MCB (Monotone-Convergence-Boundedness) property if it satisfies the following conditions: (a) every non-decreasing convergent sequence {x m } in X is bounded above by its limit (as an upper bound), i.e., xm (b) d x m+1 and x m − → x ⇒ xm x, ∀ m ∈ N ∪ {0}. every non-increasing convergent sequence {x m } in X is bounded below by its limit (as a lower bound), i.e., xm d x m+1 and x m − → x ⇒ xm x, ∀ m ∈ N ∪ {0}. Definition 2.3 [1]. Let (X, d, ) be a partially ordered metric space. We say that (X, d, ) has g-MCB property if it satisfies the following conditions: (a) g-image of every non-decreasing convergent sequence {x m } in X is bounded above by g-image of its limit (as an upper bound), i.e., xm (b) d x m+1 and x m − → x ⇒ g(x m ) g(x), ∀ m ∈ N ∪ {0}. g-image of every non-increasing convergent sequence {x m } in X is bounded below by g-image of its limit (as a lower bound), i.e., xm d x m+1 and x m − → x ⇒ g(x m ) g(x), ∀ m ∈ N ∪ {0}. If g is the identity mapping on X, then Definition 2.3 reduces to Definition 2.2. Definition 2.4 [10]. Let (X, ) be a partially ordered set. Then the mapping F : X n → X is said to have the mixed monotone property if F is monotone nondecreasing in its odd position arguments and monotone nonincreasing in its even position arguments, that is, for any x1 , x2 , ..., xi−1 , xi , xi+1 , ..., xn ∈ X and for all i ∈ In , xi , xi ∈ X, xi xi ⇒ F(x1 , x2 , ..., xi−1 , xi , xi+1 , ..., xn ) F(x1 , x2 , ..., xi−1 , xi , xi+1 , ..., xn ) ∀ i ∈ On , xi , xi ∈ X, xi xi ⇒ F(x1 , x2 , ..., xi−1 , xi , xi+1 , ..., xn ) F(x1 , x2 , ..., xi−1 , xi , xi+1 , ..., xn ) ∀ i ∈ E n . I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 168 M. Imdad, A. Alam, A. Sharma and M. Arif Definition 2.5 [10]. Let (X, ) be a partially ordered set. Then the mapping F is said to have the mixed g-monotone property if F is monotone g-nondecreasing in its odd position arguments and monotone g-nonincreasing in its even position arguments, that is, for any x1 , x2 , ..., xi−1 , xi , xi+1 , ..., xn ∈ X and for all i ∈ In , xi , xi ∈ X, gxi gxi ⇒ F(x1 , x2 , ..., xi−1 , xi , xi+1 , ..., xn ) F(x1 , x2 , ..., xi−1 , xi , xi+1 , ..., xn ) ∀ i ∈ On , xi , xi ∈ X, gxi gxi ⇒ F(x1 , x2 , ..., xi−1 , xi , xi+1 , ..., xn ) F(x1 , x2 , ..., xi−1 , xi , xi+1 , ..., xn ) ∀ i ∈ E n . Definition 2.6 [8]. Let X be a non-empty set. Then an element (x1 , x2 , ..., xn ) ∈ X n is called ntupled fixed point of the mapping F if xi = F(xi , xi−1 , ..., x2 , x1 , x2 , ..., xn−i+1 ) ∀ i ∈ In . Definition 2.7. Let X be a non-empty set. Then an element (x1 , x2 , ..., xn ) ∈ X n is called n-tupled coincidence point of the mappings F and g if gxi = F(xi , xi−1 , ..., x2 , x1 , x2 , ..., xn−i+1 ) ∀ i ∈ In . Remark 2.1. Notice that if g = I, the identity mapping on X, Definition 2.5 reduces to Definition 2.4 while Definition 2.7 reduces to Definition 2.6. Definition 2.8. Let X be a non-empty set. Then an element (x1 , x2 , ..., xn ) ∈ X n is called common n-tupled fixed point of the mappings F and g if xi = gxi = F(xi , xi−1 , ..., x2 , x1 , x2 , ..., xn−i+1 ) ∀ i ∈ In . Definition 2.9 [10]. Let X be a non-empty set. The mappings F and g are said to be commuting if g(F(x1 , x2 , ..., xn )) = F(gx1 , gx2 , ..., gxn ) ∀ x1 , x2 , ..., xn ∈ X. Definition 2.10 [5]. A function ψ : [0, ∞) → [0, ∞) is said to be a D-function if it satisfies the following conditions: 1. 2. 3. ψ is non-decreasing, ψ(t) < t, for all t > 0, lim+ ψ(r ) < t, for all t > 0. r →t The class of all functions ψ : [0, ∞) → [0, ∞) is denoted by . 3. EXISTENCE OF N-TUPLED COINCIDENCE POINT Theorem 3.1. Let (X, d, ) be a partially ordered complete metric space. Let F : X n → X and g : X → X be two mappings. Suppose that the following conditions are satisfied: I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 n-tupled Coincidence Point Results (i) (ii) (iii) (iv) (v) (vi) 169 F(X n ) ⊆ g(X ), F has mixed g-monotone property, (F, g) is a commuting pair, g is continuous, either F is continuous or (X, d, ) has g-MCB property, there exists a D-function ψ ∈ such that d(F(U ), F(V )) ≤ ψ(max d(gxi , gxi )) ∀ U = (x1 , x2 , ..., xn ), V = (y1 , y2 , ..., yn ) ∈ X n , i∈In (3.1) such that gxi gyi ∀ i ∈ On and gxi (vii) there exist x10 , x20 , ..., xn0 ∈ X such that gxi0 gxi0 gyi ∀ i ∈ E n , 0 0 F(xi0 , xi−1 , ..., x20 , x10 , x20 , ..., xn−i+1 ) ∀ i ∈ On 0 0 0 0 0 0 F(xi , xi−1 , ..., x2 , x1 , x2 , ..., xn−i+1 ) ∀ i ∈ E n . (3.2) Then F and g have an n-tupled coincidence poiint. Proof. In view of hypothesis (i), we construct the sequences {x1m }, {x2m }, ..., {xnm } in X as follows: m m g(xim+1 ) = F(xim , xi−1 , ..., x2m , x1m , x2m , ..., xn−i+1 ) ∀ i ∈ In . (3.3) By induction, we prove that for all m ∈ N ∪ {0}, gxim gxim+1 ∀ i ∈ On and gxim gxim+1 ∀ i ∈ E n . (3.4) On using (3.2) and (3.3), we obtain gxi0 0 0 F(xi0 , xi−1 , ..., x20 , x10 , x20 , ..., xn−i+1 ) = gxi1 ∀ i ∈ On , gxi0 0 0 F(xi0 , xi−1 , ..., x20 , x10 , x20 , ..., xn−i+1 ) = gxi1 ∀ i ∈ E n . So (3.4) holds for m = 0. Suppose (3.4) holds for some m = k > 0. On using mixed g-monotone property of F, for i ∈ On , we have k k gxik+1 = F(xik , xi−1 , ..., x2k , x1k , x2k , ..., xn−i+1 ) k k , ..., x2k , x1k , x2k , ..., xn−i+1 ) F(xik+1 , xi−1 k+1 k , ..., x2k , x1k , x2k , ..., xn−i+1 ) F(xik+1 , xi−1 .. . k+1 k+1 , ..., x2k+1 , x1k+1 , x2k+1 , ..., xn−i+1 ) = gxik+2 , F(xik+1 , xi−1 I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 170 M. Imdad, A. Alam, A. Sharma and M. Arif and for i ∈ E n , we have k k , ..., x2k , x1k , x2k , ..., xn−i+1 ) gxik+1 = F(xik , xi−1 k k F(xik+1 , xi−1 , ..., x2k , x1k , x2k , ..., xn−i+1 ) k+1 k , ..., x2k , x1k , x2k , ..., xn−i+1 ) F(xik+1 , xi−1 .. . k+1 k+1 , ..., x2k+1 , x1k+1 , x2k+1 , ..., xn−i+1 ) = gxik+2 . F(xik+1 , xi−1 Thus (3.4) holds for m = k + 1 and hence by induction (3.4) holds for all m ∈ N ∪ {0}. For m ∈ N ∪ {0}, let us denote δm = max d(gxim , gxim+1 ). i∈In (3.5) Let us consider the case δm 0 = 0 for some m 0 ∈ N ∪ {0}. Then, gxim 0 = gxim 0 +1 ∀ i ∈ In , which on using (3.3) implies that m0 m0 , ..., x2m 0 , x1m 0 , x2m 0 , ..., xn−i+1 ) gxim 0 = F(xim 0 , xi−1 so that (x1m 0 , x2m 0 , ..., xnm 0 ) is an n-tupled coincidence point of F and g and in this case we are done. On the other hand, assume that δm > 0, ∀ m ∈ N ∪ {0}, then we show that for each i ∈ In and for all m ∈ N ∪ {0}, d(gxim+1 , gxim+2 ) ≤ ψ(δm ). (3.6) To prove this, first we consider i ∈ On . Denote JOn ,i = {i, i − 2, . . . , 3, 1, 3, . . . , n − i (if n is even) or, n − i + 1 (if n is odd)} JO n ,i = {i − 1, i − 3, . . . , 4, 2, 4, . . . , n − i (if n is odd) or, n − i + 1 (if n is even)}. Clearly, JOn ,i ⊆ On and JO n ,i ⊆ E n . Thus by using (3.4), we get gx mj gx m+1 ∀ j ∈ JOn ,i and gx mj j gx m+1 ∀ j ∈ JO n ,i . j Hence by using (3.1), (3.2) and monotone property of ψ, m m , ..., x2m , x1m , x2m , ..., xn−i+1 ), d(gxim+1 , gxim+2 ) = d(F(xim , xi−1 m+1 m+1 , ..., x2m+1 , x1m+1 , x2m+1 , ..., xn−i+1 )) F(xim+1 , xi−1 )), ≤ ψ(max d(gx mj , gx m+1 j j∈J where J = JOn ,i ∪ JO n ,i ≤ ψ(max d(gxim , gxim+1 )) = ψ(δm ). i∈In I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 n-tupled Coincidence Point Results 171 Now, we consider i ∈ E n . Denote JEn ,i = {i, i − 2, . . . , 4, 2, 4, . . . , n − i (if n is even) or, n − i + 1 (if n is odd)} JE n ,i = {i − 1, i − 3, . . . , 3, 1, 3, . . . , n − i (if n is odd) or, n − i + 1 (if n is even)}. Clearly, JEn ,i ⊆ E n and JE n ,i ⊆ On . Thus by using (3.4), we get gx m+1 j gx mj ∀ j ∈ JE n ,i . gx mj ∀ j ∈ JEn ,i and gx m+1 j Hence by using symmetric property of d, (3.1), (3.3) and monotone property of , we get d(gxim+1 , gxim+2 ) = d(gxim+2 , gxim+1 ) m+1 m+1 = d(F(xim+1 , xi−1 , ..., x2m+1 , x1m+1 , x2m+1 , ..., xn−i+1 ), m m F(xim , xi−1 , ..., x2m , x1m , x2m , ..., xn−i+1 )) where J = JEn ,i ∪ JE n ,i , gx mj )), ≤ ψ(max d(gx m+1 j j∈J ≤ ψ(max d(gxim+1 , gxim )) = ψ(δm ). i∈In Hence (3.6) holds for each i and for all m. Now, we will show that {δm } is a monotone decreasing sequence. On taking maximum over i ∈ In on both sides of inequality (3.6), we get max d(gxim+1 , gxim+2 )) = ψ(δm ). i∈In ⇒ δm+1 ≤ ψ(δm ). (3.7) Since ψ(t) < t for all t > 0, therefore δm+1 < δm ∀ m. Hence {δm } is a monotone decreasing sequence of nonnegative real numbers. Since it is bounded below by 0, there exists some δ ≥ 0 such that lim δm = δ + . Now, we will show that δ = 0. Suppose on contrary that δ > 0. Taking m→∞ limit as m → ∞ of both sides of (3.7) and keeping in mind that lim+ ψ(r ) < t ∀ t > 0, we have r →t δ = lim δm+1 ≤ lim ψ(δm ) = lim + ψ(δm ) < δ, m→∞ m→∞ δm →δ which is a contradiction so that δ = 0, yielding thereby lim δm = lim max d(gxim , gxim+1 ) = 0. m→∞ ⇒ m→∞ i∈In (3.8) lim d(gxim , gxim+1 ) = 0 ∀ i ∈ In . m→∞ Next, we show that {gxim }, i ∈ In are Cauchy sequences. If possible, suppose that at least one of {gxim }, i ∈ In is not a Cauchy sequence. Then there exists an > 0 and sequences of positive I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 172 M. Imdad, A. Alam, A. Sharma and M. Arif integers {σ (k)} and {μ(k)} such that for all positive integers k with μ(k) > σ (k) ≥ k, tk = max d(gxiσ (k) , gxiμ(k) ) ≥ . (3.9) i∈In We may choose μ(k), corresponding to σ (k), such that it is the smallest integer satisfying (3.9) and μ(k) > σ (k) ≥ k. Hence μ(k)−1 max d(gxiσ (k) , gxi i∈In ) < . (3.10) On using the triangular inequality and (3.10), we have μ(k) d(gxiσ (k) , gxi μ(k)−1 ) ≤ d(gxiσ (k) , gxi μ(k)−1 < + d(gxi μ(k)−1 ) + d(gxi μ(k) , gxi μ(k) , gxi ) ). Taking the maximum over i ∈ In on both the sides of above inequality, we have max d(gxiσ (k) , gxiμ(k) ) < + max d(gxiμ(k)−1 , gxiμ(k) ). i∈In i∈In Using (3.5) and (3.9), we get ≤ tk < + δμ(k)−1 . Letting k → ∞ in the above inequality and using (3.8), we get lim tk = . (3.11) k→∞ On the other hand, we have μ(k) d(gxiσ (k) , gxi μ(k)+1 ) ≤ d(gxiσ (k) , gxiσ (k)+1 ) + d(gxiσ (k)+1 , gxi μ(k) ⇒ max d(gxiσ (k) , gxi i∈In μ(k)+1 ) + d(gxi μ(k) , gxi ), ) ≤ max d(gxiσ (k) , gxiσ (k)+1 ), i∈In μ(k)+1 + max d(gxiσ (k)+1 , gxi i∈In μ(k)+1 ) + max d(gxi i∈In tk ≤ δσ (k) + δμ(k) + max d(gxiσ (k)+1 , gxiμ(k)+1 ). i∈In μ(k) , gxi ). (3.12) Now, we have to show that for each i ∈ In , d(gxiσ (k)+1 , gxiμ(k)+1 ) ≤ ψ(tk ). (3.13) As σ (k) < μ(k), on using (3.4), we get gxiσ (k) μ(k) gxi ∀ i ∈ On and gxiσ (k) I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS μ(k) gxi ∀ i ∈ En . (3.14) Vol. 10, No. 1 (Special Issue), Jan–June 2019 n-tupled Coincidence Point Results 173 The proof of (3.13) is similar as the proof of (3.6). First we consider i ∈ On , then in view of (3.14), we have μ(k) gx σj (k) gx j μ(k) ∀ j ∈ JOn ,i and gx σj (k) gx j ∀ j ∈ JO n ,i . On using (3.1), (3.3) and monotone property of ψ, we have σ (k) σ (k) , ..., x2σ (k) , x1σ (k) , x2σ (k) , ..., xn−i+1 ), d(gxiσ (k)+1 , gxiμ(k)+1 ) = d(F(xiσ (k) , xi−1 μ(k) F(xi μ(k) μ(k) , xi−1 , ..., x2 μ(k) ≤ ψ(max d(gx σj (k) , gx j j∈J μ(k) ≤ ψ(max d(gxiσ (k) , gxi i∈In μ(k) , x1 μ(k) , x2 μ(k) , ..., xn−i+1 )) where J = JOn ,i ∪ JO n ,i )) )) = ψ(tk ). Next, we consider i ∈ E n . Then again in view of (3.14), we have μ(k) μ(k) gx σj (k) ∀ j ∈ JEn ,i and gx j gx j gx σj (k) ∀ j ∈ JEn ,i . Again on using (3.1), (3.3) and monotone property of ψ, we have μ(k)+1 d(gxiσ (k)+1 , gxi μ(k)+1 ) = d(gxi μ(k) = d(F(xi , gxiσ (k)+1 ) μ(k) μ(k) , xi−1 , ..., x2 μ(k) , x1 μ(k) , x2 μ(k) , ..., xn−i+1 ), σ (k) σ (k) F(xiσ (k) , xi−1 , ..., x2σ (k) , x1σ (k) , x2σ (k) , ..., xn−i+1 )) μ(k) ≤ ψ(max d(gx j j∈J , gx σj (k) )) μ(k) ≤ ψ(max d(gxiσ (k) , gxi i∈In where J = JEn ,i ∪ JE n ,i )) = ψ(tk ). Thus (3.13) holds for each i and for all m. Now, on using (3.13) in (3.12), we get tk ≤ δσ (k) + δμ(k) + ψ(tk ). Letting k → ∞ in above relation and using (3.8) & (3.11), we have ≤ lim ψ(tk ) = lim+ ψ(tk ) < , k→∞ tk → which is a contradiction. Therefore {gxim }, i ∈ In are Cauchy sequences in X. Since X is complete, there exist xi ∈ X, i ∈ In such that lim gxim = xi ∀ i ∈ In . (3.15) lim g(gxim ) = gxi ∀ i ∈ In . (3.16) m→∞ By continuity of g and (3.15), we get m→∞ I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 174 M. Imdad, A. Alam, A. Sharma and M. Arif From (3.3) and commutativity of F with g, we get m m , ..., x2m , x1m , x2m , ..., xn−i+1 )) g(gxim+1 ) = g(F(xim , xi−1 m m m m m m = F(gxi , gxi−1 , ..., gx2 , gx1 , gx2 , ..., gxn−i+1 ), ∀ i ∈ In . (3.17) Now, we will show that F and g have n-tupled coincidence point. To accomplish this we use condition (v) of our hypotheses. Firstly, we assume that F is continuous. Then on using (3.15)-(3.17) and continuity of F, we obtain gxi = lim g(gxim+1 ) m→∞ m = lim F(gxim , gxim−1 , ..., gx2m , gx1m , gx2m , ..., gxn−i+1 ) m→∞ m = F( lim gxim , lim gxim−1 , ..., lim gx2m , lim gx1m , lim gx2m , ..., lim gxn−i+1 ) m→∞ m→∞ m→∞ m→∞ m→∞ m→∞ = F(xi , xi−1 , ..., x2 , x1 , x2 , ..., xn−i+1 ) ∀ i ∈ In , that is, (x1 , x2 , ..., xn ) is an n-tupled coincidence point of F and g. Otherwise (X, d, ) has g-MCB property. Since {gxim } is monotone non-decreasing for each i ∈ On and monotone non-increasing for each i ∈ E n and gxim → xi as m → ∞, we have g(gxim ) g(xi ) ∀ i ∈ On and g(gxim ) g(xi ) ∀ i ∈ E n . Now, similar as earlier, two cases arise whenever i ∈ On and i ∈ E n . For both cases, using (3.1), (3.17), triangular inequality and monotonicity of ψ, we get d(gxi , F(xi , xi−1 , ..., x2 , x1 , x2 , ..., xn−i+1 )) ≤ d(gxi , gxim+1 ) + d(g(gxim+1 ), F(xi , xi−1 , ..., x2 , x1 , x2 , ..., xn−i+1 )) m m = d(gxi , gxim+1 ) + d(F(gxim , gxi−1 , ..., gx2m , gx1m , gx2m , ..., gxn−i+1 ), F(xi , xi−1 , ..., x2 , x1 , x2 , ..., xn−i+1 )) ≤ d(gxi , gxim+1 ) + ψ(max d((gxim , gxi )). i∈In Letting m → ∞ in above relation and using (3.16), we have gxi = F(xi , xi−1 , ..., x2 , x1 , x2 , ..., xn−i+1 ) ∀ i ∈ In . Thus (x1 , x2 , .., xn ) is an n-tupled coincidence point of F and g. This completes the proof of the theorem. Theorem 3.1 immediately yields the following Corollaries: Corollary 3.1. Theorem 3.1 remains true if we replace the contraction condition (vi) by the following condition besides retaining the rest of the hypotheses: I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 n-tupled Coincidence Point Results 175 (vi ) there exists c ∈ [0, 1) such that d(F(U ), F(V )) ≤ c max d(gxi , gyi ) ∀ U = (x1 , x2 , ..., xn ), V = (y1 , y2 , ..., yn ) ∈ X n i∈In such that gxi gyi for all i ∈ On and gxi gyi for all i ∈ E n . Proof. Taking ψ(t) = ct with c ∈ [0, 1) in Theorem 3.1. Corollary 3.2. Theorem 3.1 remains true if we replace the contraction condition (vi) by the following condition besides retaining the rest of the hypotheses: n αi < 1 such that (vi ) there exist α1 , α2 , ..., αn ∈ [0, 1) where i=1 d(F(U ), F(V )) ≤ n αi d(gxi , gyi ) ∀ U = (x1 , x2 , ..., xn ), V = (y1 , y2 , ..., yn ) ∈ X n i=1 such that gxi gyi for all i ∈ On and gxi Proof. Taking c = n gyi for all i ∈ E n . αi < 1 in Corollary 3.1. Then, we have i=1 d(F(U ), F(V )) ≤ ≤ n i=1 n αi d(gxi , gyi ) αi max d(gx j , gy j ) i=1 j∈In = c max d(gxi , gyi ). i∈In Thus (vii ) implies (vii ) and hence we obtain Corollary 3.2 from Corollary 3.1. Corollary 3.3. Theorem 3.1 remains true if we replace the contraction condition (vi) by the following condition besides retaining the rest of the hypotheses: (vi ) there exists α ∈ [0, 1) such that d(F(U ), F(V )) ≤ such that gxi n α d(gxi , gyi ) ∀ U = (x1 , x2 , ..., xn ), V = (y1 , y2 , ..., yn ) ∈ X n n i=1 gyi for all i ∈ On and gxi Proof. Taking αi = α n gyi for all i ∈ E n . for all i ∈ In , where α ∈ [0, 1) in Corollary 3.2. On setting g = I, the identity mapping on X in Theorem 3.1, Corollaries 3.1, 3.2 and 3.3, we obtain their corresponding n-tupled fixed point results which are the following (Corollaries 3.4, 3.5, 3.6 and 3.7 respectively). I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 176 M. Imdad, A. Alam, A. Sharma and M. Arif Corollary 3.4. Let (X, d, ) be a partially ordered complete metric space. Let F : X n → X be a mapping. Suppose that the following conditions are satisfied: (viii) F has mixed monotone property, (ix) either F is continuous or (X, d, ) has MCB property, (x) there exists a D-function ψ ∈ such that d(F(U ), F(V )) ≤ ψ(max d(xi , yi )) ∀ U = (x1 , x2 , ..., xn ), V = (y1 , y2 , ..., yn ) ∈ X n , i∈In such that xi yi ∀ i ∈ On and xi yi ∀ i ∈ E n , (xi) there exists x10 , x20 , ..., xn0 ∈ X such that xi0 0 0 F(xi0 , xi−1 , ..., x20 , x10 , x20 , ..., xn−i+1 ) ∀ i ∈ On xi0 0 0 F(xi0 , xi−1 , ..., x20 , x10 , x20 , ..., xn−i+1 ) ∀ i ∈ En . Then F has an n-tupled fixed point. Corollary 3.5. Corollary 3.4 remains true if we replace the contraction condition (x) by the following contraction condition besides retaining the rest of the hypotheses: (x ) there exists c ∈ [0, 1) such that d(F(U ), F(V )) ≤ c max d(xi , yi ) ∀ U = (x1 , x2 , ..., xn ), V = (y1 , y2 , ..., yn ) ∈ X n i∈In such that xi yi for all i ∈ On and xi yi for all i ∈ E n . Corollary 3.6. Corollary 3.4 remains true if we replace the contraction condition (x) by the following contraction condition besides retaining the rest of the hypotheses: n (x ) there exist α1 , α2 , ..., αn ∈ [0, 1) with αi < 1 such that i=1 d(F(U ), F(V )) ≤ n αi d(xi , yi ) ∀ U = (x1 , x2 , ..., xn ), V = (y1 , y2 , ..., yn ) ∈ X n i=1 such that xi yi for all i ∈ On and xi yi for all i ∈ E n . Corollary 3.7. Corollary 3.4 remains true if we replace the contraction condition (x) by the following contraction condition besides retaining the rest of the hypotheses: (x ) there exists α ∈ [0, 1) such that d(F(U ), F(V )) ≤ such that xi n α d(xi , yi ) ∀ U = (x1 , x2 , ..., xn ), V = (y1 , y2 , ..., yn ) ∈ X n n i=1 yi for all i ∈ On and xi yi for all i ∈ E n . I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 n-tupled Coincidence Point Results 177 Notice that Corollary 3.6 is the main result of [8]. Now, we point that previous results of this section yield several known results proved on coupled as well as tripled fixed and coincidence points as special cases. Some of them are listed below: (1) By setting n = 3 in Theorem 3.1, we get Theorem 5 contained in Borcut [5]. (2) By setting n = 3 in Corollary 3.1, we get Corollary 1 contained in Borcut [5]. (3) By setting n = 3 in Corollary 3.2, we get Theorem 4 contained in Borcut and Berinde [6]. (4) By setting n = 3 in Corollary 3.6, we get Theorems 7 and 8 contained in Berinde and Borcut [3]. (5) By setting n = 3 in Corollary 3.7, we get main results of Borcut [7]. (6) By setting n = 2 in Corollary 3.3, we get Corollary 2.1 contained in Lakshmikantham and Ćirić [14]. (7) By setting n = 2 in Corollary 3.7, we get Theorem 2.4 contained in Bhaskar and Lakshmikantham [4]. 4. UNIQUENESS OF N-TUPLED COINCIDENCE POINT Let (X, ) be a partially ordered set. Equip the product set X n with the following partial ordering n n : for all U = (x 1 , x 2 , ..., x n ), V = (y1 , y2 , ..., yn ) ∈ X U n V ⇔ xi yi ∀ i ∈ On and xi yi ∀ i ∈ E n . We say that two elements U and V of X n are comparable if either U n V or U n V. Now, we state and prove the corresponding result regarding the uniqueness of n-tupled coincidence points. Theorem 4.1. In addition to the hypotheses of Theorem 3.1, suppose for all n-tupled coincidence points U = (x1 , x2 , ..., xn ), V = (y1 , y2 , ..., yn ) of F and g, there exists W = (z 1 , z 2 , ..., z n ) such that (gz 1 , gz 2 , ..., gz n ) is comparable to (gx1 , gx2 , ..., gxn ) and (gy1 , gy2 , ..., gyn ), then F and g have a unique n-tupled coincidence point. Moreover, this unique n-tupled coincidence point is a unique common n-tupled fixed point of mappings F and g. Proof. In view of Theorem 3.1, the set of n-tupled coincidence points is nonempty. If U = (x1 , x2 , ..., xn ) and V = (y1 , y2 , ..., yn ) ∈ X n are two n-tupled coincidence points of F and g, then gxi = F(xi , xi−1 , ..., x2 , x1 , x2 , ..., xn−i+1 ) ∀ i ∈ In , (3.18) gyi = F(yi , yi−1 , ..., y2 , y1 , y2 , ..., yn−i+1 ) ∀ i ∈ In . (3.19) gxi = gyi ∀ i ∈ In . (3.20) We have to show that By assumption, there exists W = (z 1 , z 2 , ..., z n ) ∈ X n such that (gz 1 , gz 2 , ..., gz n ) is comparable to (gx1 , gx2 , ..., gxn ) and (gy1 , gy2 , ..., gyn ). Suppose that (gx1 , gx2 , ..., gxn ) n (gz 1 , gz 2 , ..., gz n ) and (gy1 , gy2 , ..., gyn ) I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS n (gz 1 , gz 2 , ..., gz n ). Vol. 10, No. 1 (Special Issue), Jan–June 2019 178 M. Imdad, A. Alam, A. Sharma and M. Arif The other cases are similar. Now, put z i0 = z i ∀ i ∈ In . Since F(X n ) ⊆ g(X ), similar as on the lines of the proof of the Theorem 3.1, we can inductively define sequences {zim }, i ∈ In such that m m , ..., z 2m , z 1m , z 2m , ..., z n−i+1 ) ∀ i ∈ In . g(z im+1 ) = F(z im , z i−1 (3.21) As F has mixed g-monotone property, by a similar region as in the proof of Theorem 3.1, we have gz im+1 ∀ i ∈ On and gz im gz im Since (gx1 , gx2 , ..., gxn ) gxi n gz im gz im+1 ∀ i ∈ E n . (gz 1 , gz 2 , ..., gz n ), we have gz im+1 ∀ i ∈ On and gxi gz im gz im+1 ∀ i ∈ E n . (3.22) Now, we claim that for each i ∈ In and for all m ≥ 1 that d(gxi , gz im+1 ) ≤ ψ(γm ), (3.23) where γm = max d(gxi , gz im+1 ). i∈In Now, similar on the lines of the proof of Theorem 3.1, two cases arise, whenever i ∈ E n as well as i ∈ On . For both cases, on using (3.1), (3.18), (3.21) and (3.22) and monotonicity of ψ, we get d(gxi , gz im+1 ) = d((xi , xi−1 , ..., x2 , x1 , x2 , ..., xn−i+1 ), m m F(z im , z i−1 , ..., z 2m , z 1m , z 2m , ..., z n−i+1 )) m ≤ ψ(max{d(gxi , gz im ), d(gxi−1 , gz i−1 ), ..., d(gx2 , gz 2m ), m d(gx1 , gz 1m ), d(gx2 , gz 2m ), ..., d(gxn−i+1 , gz n−i+1 )}) ≤ ψ(max d(gxi , gz im )) = ψ(γm ). i∈I It follows that for all m ≥ 1, γm+1 = max d(gxi , gz im+1 )) ≤ ψ(γm ) ≤ ψ 2 (γm+1 ) . . . ≤ ψ m (γ1 ) ≤ ψ m+1 (γ0 ). i∈In (3.24) Now, on using properties of ψ, we can obtain lim ψ m+1 (t) = 0 ∀ t > 0. m→∞ (3.25) Taking the limit as m → ∞ in (3.24) and using (3.25), we get lim γm = 0. m→∞ Consequently, we have lim d(gxi , gz im ) = 0 ∀ i ∈ In . m→∞ I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS (3.26) Vol. 10, No. 1 (Special Issue), Jan–June 2019 n-tupled Coincidence Point Results 179 Similarly, we can show that lim d(gyi , gz im ) = 0 ∀ i ∈ In . m→∞ (3.27) By triangular inequality, (3.26) and (3.27), we get for each i ∈ In , d(gxi , gyi ) ≤ d(gxi , gz im ) + d(gz im , gyi ) → 0 as m → ∞, which implies that gxi = gyi ∀ i ∈ In . Thus (3.20) is proved. On using (3.18) and commutativity of F and g, we get g(gxi ) = g(F(xi , xi−1 , ..., x2 , x1 , x2 , ..., xn−i+1 )) = F(gxi , gxi−1 , ..., gx2 , gx1 , gx2 , ..., gxn−i+1 ). (3.28) Write gxi = x i for each i ∈ In , then from (3.28), we get g(x i ) = F(x i , x i−1 , ..., x 2 , x 1 , x 2 , ..., x n−i+1 ). Thus (x 1 , x 2 , ..., x n ) is also n-tupled coincidence point of F and g. Now, on using (3.20), we get x i = g(x i ) = F(x i , x i−1 , ..., x 2 , x 1 , x 2 , ..., x n−i+1 ). Hence is (x 1 , x 2 , ..., x n ) is a common n-tupled fixed point of F and g. To prove uniqueness, assume that (u 1 , u 2 , ..., u n ) is another common n-tupled fixed point. Then by (3.20), for each i ∈ In , we have u i = g(u i ) = g(x i ) = x i . This completes the proof. For any U = (x1 , x2 , ..., xn ) ∈ X n and for each i ∈ In , denote by U(i) the fixed partially ordered element of X n relative to U as U(i) = (xi , xi−1 , ..., x2 , x1 , x2 , ..., xn−i+1 ). Clearly, U(1) = U. Theorem 4.2. In addition to the hypotheses of Theorem 3.1, suppose for all n-tupled coincidence points U = (x1 , x2 , ..., xn ), V = (y1 , y2 , ..., yn ) of F and g, there exists W = (z 1 , z 2 , ..., z n ) such that (F(W(1) ), F(W(2) ), ..., F(W(n) )) is comparable to (gx1 , gx2 , ..., gxn ) and (gy1 , gy2 , ..., gyn ), then F and g have a unique common n-tupled fixed point. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 180 M. Imdad, A. Alam, A. Sharma and M. Arif Proof. In this case, we observe that, gxi F(W(i) ) = F(z i , z 1−1 , . . . , z 2 , z 1 , z 2 , . . . , z n−i+1 ) 0 0 = F(z i0 , z 1−1 , . . . , z 20 , z 10 , z 20 , . . . , z n−i+1 ) = g(z i1 ) ∀ i ∈ On , gxi F(W(i) ) = F(z i , z 1−1 , . . . , z 2 , z 1 , z 2 , . . . , z n−i+1 ) 0 0 = F(z i0 , z 1−1 , . . . , z 20 , z 10 , z 20 , . . . , z n−i+1 ) = g(z i1 ) ∀ i ∈ E n . Hence by induction process, we find that gxi gz im gz im+1 ∀ i ∈ On and gxi gz im gz im+1 ∀ i ∈ E n . Thus, the proof is done on the similar lines of Theorem 4.1. REFERENCES [1] Alam, A. and Imdad, M.: Some Coincidence Theorems for Nonlinear φ−cotraction in partially ordered Metric Spaces with Applications, Preprint. [2] Banach, S.: Sur les operations dans les ensembles abstraits et leur application aux quations intgrales, Fund. Math., 3 (1922), 133–181. [3] Berinde, V. and Borcut, M.: Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces, Nonlinear Anal., 74 (15) (2011), 4889–4897. [4] Bhaskar, T. G. and Lakshmikantham, V.: Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal., 65 (7) (2006), 1379–1393. [5] Borcut, M.: Tripled coincidence theorems for contractive type mappings in partially ordered metric spaces, App. Math. Comp., 218 (14) (2012), 7339–7346. 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[12] Imdad, M., Sharma, A. and Rao, K. P. R.: Generalized n-tupled fixed point theorems for nonlinear contraction mapping, Afrika Matematika, 26 (3), (2015), 443–455. [13] Karapinar, E. and Luong, N. V.: Quadruple fixed point theorems for nonlinear contractions, Computers & Mathematics with Applications, 64 (6) 2012, 1839–1848. [14] Lakshmikantham, V. and Ćirić, L.: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal., 70 (2009), 4341–4349. [15] Nieto, J. J. and López, R. R.: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order, 22 (3) (2005), 223–239. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 n-tupled Coincidence Point Results 181 [16] O’Regan, D. and Petruşel, A.: Fixed point theorems for generalized contractions in ordered metric spaces, J. Math. Anal. Appl., 341 (2) (2008), 1241–1252. [17] Ran, A. C. M. Ran and Reurings, M. C. B.: A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc., 132 (2004), 1435–1443. [18] Samet, B. and Vetro, C.: Coupled fixed point, F-invariant set and fixed point of N -order, Ann. Funct. Anal., 1 (2) (2010), 46–56. [19] Sharma, A., Imdad, M. and Alam, A.: Shorter proofs of some recent even tupled coincidence theorems for weak contractions in ordered metric spaces, Mathematical Sciences, 8 (4), (2015), 131–138. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 INDIAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019, pp. 182–201 DOI: On Starlikeness and Convexity Properties of the Generalized Bessel-Hypergeometric Function Nusrat Raza1∗ and Eman S.A. AbuJarad2 1 Mathematics Section Women’s College Aligarh Muslim University, India Department of Mathematics Aligarh Muslim University, India (∗ Corresponding author) Email: ∗ [email protected], 2 [email protected] 2 Abstract: In this paper, we introduce the Generalized Bessel-Hypergeometric function by making convolution of the Generalized Bessel function and the Hypergeometric function. Also we obtain the necessary and sufficient conditions for the Generalized Bessel-Hypergeometric function to be β - uniformly starlike and β - uniformly convex functions of order α. Further, we introduce a function in terms of integral involving the Generalized Bessel-Hypergeomtric function and obtain its order of starlikness and convexity. Keywords: Generalized Bessel function, Hypergeometric function, univalent function, Starlike function, convex function, integral operator, β-uniformly starlike function, β-uniformly convex function. 1. INTRODUCTION AND PRELIMINARIES An important property of analytic functions, is that these functions can be locally represented as power serieses. Certain geometric properties, for example, starlikness and convexity of some analytic functions like Generalized Bessel functions and Hypergeometric functions, are studied by Baricz [3], Carlson and Schaffer [4], respectively. Let A be the class of analytic functions of the form f (z) = z + ∞ an z n , (1.1) n=2 where f is analytic in the open unit disc U = {z ∈ C : |z| < 1}. This research article is funded by University Grant commission, New Delhi, India, under UGC-BSR Research Start-Up-Grant No. F.30-129(A)/2015(BSR). I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Starlikeness and Convexity Properties of the Generalized Bessel-Hypergeometric Function 183 A function f is said to be univalent [6] in a domain D ⊂ C, if it never takes on the same values twice, that is, f (z 1 ) = f (z 2 ) for z 1 , z 2 ∈ D implies that z 1 = z 2 , in other words, f is one to one on D. Let B be the class of functions which are normalized by the condition f (0) = f (0) − 1 = 0, where f is univalent in an open unit disc U . We note that, B is a subclass of A. Let T be the class of functions of the form [17]: f (z) = z − ∞ an z n (an ≥ 0, z ∈ U ). (1.2) n=2 It is clear that T is a subclass of A. Let g ∈ A be another function, given by the following power series: g(z) = z + ∞ bn z n , (1.3) n=2 then the Hadamard product (or Convolution) of the analytic functions f (z), given by equation (1.1) and g(z), given by equation (1.3), denoted by ( f *g) (z), is defined as [9]: ( f *g) (z) = z + ∞ an bn z n . (1.4) n=2 A set ε is said to be starlike with respect to ω0 ∈ ε, if the line segment joining ω0 to every other point ω ∈ ε lies entirely in ε . If the function f (z) maps a domain D ⊂ C onto a domain that is starlike with respect to ω0 , then f (z) is said to be starlike with respect to ω0 . Also, the set ε is said to be convex, if the line segment joining any two points of ε lies entirely in ε . Equivalently t z 1 + (1 − t)z 2 ∈ ε for every z 1 , z 2 ∈ ε and 0 ≤ t ≤ 1. Schild [15] introduced the classes S(α) and K (α), which are given by the following definitions: Definition 1.1. A function f (z) in A is said to be starlike of order α (0 ≤ α < 1), if and only if z f (z) >α (z ∈ U ). Re f (z) We denote the class of such functions by S(α), we note that S(α) ⊆ S(0), where S = S(0) is the class of starlike function with respect to the origin in the open unit disc U . Definition 1.2. A function f (z) in A is said to be convex of order α (0 ≤ α < 1), if and only if z f (z) > α (z ∈ U ). Re 1 + f (z) We denote the class of such functions by K (α), we note that K (α) ⊆ K (0), where K = K (0) is the class of convex function with respect to the origin in the open unit disc U . I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 184 Nusrat Raza and Eman S.A. AbuJarad According to Alexander [1], if f is analytic in U with f (0) = f (0) − 1 = 0, then f ∈ K (0) if and only if z f ∈ S(0), and thus f ∈ K (α) if and only if z f ∈ S(α). The class β − U C V was introduced and its geometric definition, connections with the conic domains were considered by Kanas and Wisniowska [8]. The class β − U C V was defined pure geometrically as a subclass of univalent functions, that the image of every circular arc γ contained in the open unit disc U with a center ζ , |ζ | ≤ β (0 ≤ β < 1) is convex arc. The notion of β uniformly convex function is a natural extension of the classical convexity. We note that, if β = 0, then the center ζ is the origin and the class β − U C V reduces to the class of convex univalent functions K . Moreover for β = 1 corresponds to the class of uniformly convex functions U C V introduced by Goodman [7] and studied extensively by Rønning [14]. Alexander [1] proved that the relation between class β − U C V and class β − S P which is equivalent to the relation between the classes of convex K (α) and starlike S(α). The classes S P (α, β) and U C V (α, β) are defined in the following definitions: Definition 1.3. A function f ∈ A is said to be in the class β-uniformly starlike functions of order α, which is denoted by S P (α, β), for −1 < α ≤ 1 and β ≥ 0 if it satisfies the following condition: z f (z) z f (z) (z ∈ U ). (1.5) −α >β − 1 Re f (z) f (z) Definition 1.4. A function f ∈ A is said to be in the class β-uniformly convex functions of order α, which is denoted by U C V (α, β), for −1 < α ≤ 1 and β ≥ 0 if it satisfies the following condition: z f (z) z f (z) (z ∈ U ). (1.6) Re 1 + − α > β f (z) f (z) It follows from (1.5) and (1.6) that f ∈ U C V (α, β) if and only if z f ∈ S P (α, β). Remark 1.1. It is clear that S P (α, β) and U C V (α, β) are subclasses of A and U C V (α, 0) = K (α) and S P (α, 0) = S(α). Now, we define the following subclasses S P (λ, α, β) and U C V (λ, α, β) of A introdued by Murugusundaramoorthy and Magesh [11], respectively: Definition 1.5. A function f ∈ A is said to be in the class S P (λ, α, β), for 0 ≤ α < 1, 0 ≤ λ < 1 and β ≥ 0 if it satisfies the following condition: z f (z) z f (z) (z ∈ U ). (1.7) −α >β − 1 Re (1 − λ) f (z) + λz f (z) (1 − λ) f (z) + λz f (z) Definition 1.6. A function f ∈ A is said to be in the class U C V (λ, α, β), for 0 ≤ α < 1, 0 ≤ λ < 1 and β ≥ 0 if it satisfies the following condition: f (z) + z f (z) f (z) + z f (z) Re (z ∈ U ). (1.8) − α > β − 1 f (z) + λz f (z) f (z) + λz f (z) I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Starlikeness and Convexity Properties of the Generalized Bessel-Hypergeometric Function 185 Murugusundaramoorthy and Magesh [11] introduced T S P (λ, α, β) and U C T (λ, α, β) as: T S P (λ, α, β) = S P (λ, α, β) ∩ T and U C T (λ, α, β) = U C V (λ, α, β) ∩ T. Certain characterization properties for the classes S P (λ, α, β), T S P (λ, α, β), U C V (λ, α, β) and U C T (λ, α, β) are as follows [11]: Lemma 1.1. Let 0 ≤ α < 1, 0 ≤ λ < 1 and β ≥ 0, then a function f (z) of the form (1.1) is in S P (λ, α, β) if [11] ∞ [n(1 + β) − (α + β)(1 + nλ − λ)] |an | ≤ 1 − α. (1.9) n=2 Lemma 1.2. Let 0 ≤ α < 1, 0 ≤ λ < 1 and β ≥ 0, then a function f (z) of the form (1.2) is in T S P (λ, α, β) if and only if [11] ∞ [n(1 + β) − (α + β)(1 + nλ − λ)] |an | ≤ 1 − α. (1.10) n=2 Lemma 1.3. Let 0 ≤ α < 1, 0 ≤ λ < 1 and β ≥ 0, then a function f (z) of the form (1.1) is in U C V (λ, α, β) if [11] ∞ n [n(1 + β) − (α + β)(1 + nλ − λ)] |an | ≤ 1 − α. (1.11) n=2 Lemma 1.4. Let 0 ≤ α < 1, 0 ≤ λ < 1 and β ≥ 0, then a function f (z) of the form (1.2) is in U C T (λ, α, β) if and only if [11] ∞ n [n(1 + β) − (α + β)(1 + nλ − λ)] |an | ≤ 1 − α. (1.12) n=2 The Bessel functions are associated with a wide range of problems in important areas of mathematical physics and Engineering. These functions appear in the solutions of heat transfer and other problems in cylindrical and spherical coordinates. Rainville [13] discussed the properties of the Bessel function. The Generalized Bessel functions wν,b,d (z) are defined as [2]: wν,b,d (z) = ∞ n=0 z (−d)n b+1 2 n! ν+n+ 2 2n+ν , (1.13) where ν, b, d, z ∈ C. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 186 Nusrat Raza and Eman S.A. AbuJarad The Generalized Bessel function wν,b,d (z) is the solution of the following differential equation [2]: z 2 w (z) + bzw (z) + dz 2 − ν 2 + (1 − b)ν w(z) = 0. (1.14) Orhan, Deniz and Srivastava [5] defined the function ϕν,b,d (z) : U → C as: ϕν,b,d (z) = 2ν ν+ ν √ b + 1 1− z 2 wν,b,d ( z), 2 (1.15) by using the Generalized Bessel function wν,b,d (z), given by equation(1.14). The power series representation for the function ϕν,b,d (z) is as follows [5]: ϕν,b,d (z) = ∞ (−d/4)n n=0 z n+1 , (c)n n! (1.16) b+1 > 0, ν, b, d ∈ R and z ∈ U = {z ∈ C : |z| < 1}. where c = ν + 2 Now, we discuss the following special case for the function ϕν,b,d (z): For −b = d = −1, the Generalized Bessel function wν,b,d (z) reduces to the modified Bessel function. Therefore taking −b = d = −1 in equation (1.13), we get the following series definition for the modified Bessel function Hν (z) [13]: Hν (z) = ∞ n=0 z 1 n! (ν + n + 1) 2 2n+ν (z ∈ U, ν ∈ R). (1.17) Also, in view of equations (1.15) and (1.16), we get ϕν,1,−1 (z) = 2 ν (ν + 1) z 1− ν ∞ √ 2 Hν ( z) = n=0 4n 1 z n+1 . (ν + 1)n n! (1.18) The Hypergeometric functions appear as the solutions of certain problems in classical and quantum physics, engineering and applied mathematics .Rainville [13] discussed the Hypergeometric function F(a, p; q; z) and its properties. The Hypergeometric functions F(a, p; q; z) are defined by [13]: F(a, p; q; z) = ∞ (a)n ( p)n n=0 (q)n n! zn , (1.19) where a, p, q ∈ R and q > 0. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Starlikeness and Convexity Properties of the Generalized Bessel-Hypergeometric Function 187 2. MAIN RESULTS In this section, we introduce the Generalized Bessel-Hypergeometric function by making convolution of the Generalized Bessel function and the Hypergeometric function. Also we discuss the starlikness and convexity of this function. a,d, p We define the function Ic,q (z) : C → C by making convolution of the function ϕν,b,d (z), given by equation (1.16) and F(a, p; q; z), given by equation (1.19) as: a,d, p Ic,q (z) = ϕv,b,d (z)*z F(a, p; q; z) = ∞ (−d/4)n (a)n ( p)n (q)n (c)n ((n)!)2 n=0 =z+ ∞ (−d/4)n−1 (a)n−1 ( p)n−1 n=2 where c = ν + z n+1 (q)n−1 (c)n−1 ((n − 1)!)2 (2.1) zn , b+1 > 0, a, b, d, p, q, ν ∈ R and q > 0. 2 Remark 2.1. Since the series representing F(a, p, q, 1) (a, p, q ∈ R) converges [13], if q > a + a,d, p p. Therefore the series on the right hand side of equation (2.1), which represent Ic,q (1), for z = 1, also converges, if q > a + p. Now, we obtain the following result involving the Generalized Bessel-Hypergeometric function a,d, p Ic,q (z): Theorem 2.1. If a, c, d, p, q ∈ R, q > 0, c > 0 and q > a + p, then a,d, p a,d, p z Ic,q (z) − Ic,q (z) = ∞ (−d/4)n (a)n ( p)n n+1 z (z ∈ C). (q)n (c)n (n)!(n − 1)! n=1 (2.2) Also, a,d, p (z) z 2 Ic,q a,d, p a,d, p − z Ic,q (z) + Ic,q (z) = ∞ (−d/4)n (a)n ( p)n n+1 z (z ∈ C). (q)n (c)n ((n − 1)!)2 n=1 (2.3) Proof. Differentiating equation (2.1) with respect to z and then multiplying the resultant equation with z, we get a,d, p (z) = z Ic,q ∞ (n + 1)(−d/4)n (a)n ( p)n n=0 (q)n (c)n ((n)!)2 z n+1 ∞ ∞ (−d/4)n (a)n ( p)n n+1 (−d/4)n (a)n ( p)n n+1 z = + z , (q)n (c)n (n − 1)!(n)! (q)n (c)n ((n)!)2 n=1 n=0 I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 188 Nusrat Raza and Eman S.A. AbuJarad which are again using equation (2.1), gives ∞ (−d/4)n (a)n ( p)n n+1 a,d, p + Ic,q (z). z (q) (c) (n − 1)!(n)! n n n=1 a,d, p (z) = z Ic,q Hence, we obtained assertion (2.2). Again, differentiating equation (2.2) with respect to z and then multipling with z, we get a,d, p (z) z 2 Ic,q a,d, p a,d, p + z Ic,q (z) − z Ic,q (z) = ∞ (n + 1)(−d/4)n (a)n ( p)n (q)n (c)n (n − 1)!(n)! n=1 z n+1 , or, equivalently z 2 a.d, p Ic,q (z) ∞ ∞ (−d/4)n (a)n ( p)n n+1 n(−d/4)n (a)n ( p)n n+1 z , = z + (q)n (c)n ((n − 1)!)2 (q)n (c)n (n − 1)!(n)! n=1 n=1 which on again using equation (2.2), gives a,d, p (z) z 2 Ic,q = ∞ (−d/4)n (a)n ( p)n n+1 a,d, p a, p z + z Ic,q (z) − Ic,q (z). 2 (q) (c) ((n − 1)!) n n n=1 Hence, we obtained assertion (2.3). a,d, p Next, we define the function θc,q (z) : C → C as : a,d, p a,d, p θc,q (z) = 2z − Ic,q (z), which on using equation (2.1), gives a,d, p (z) = z − θc,q ∞ (−d/4)n−1 (a)n−1 ( p)n−1 n=2 (q)n−1 (c)n−1 ((n − 1)!)2 zn . (2.4) a,d, p Now, we obtain the following necessary and sufficient condition for the function θc,q (z) to be in the classes T S P (λ, α, β) and U C T (λ, α, β): a,d, p Theorem 2.2. If a, b, p, q, ν ∈ R, d < 0, q > 0, c > 0 and q > a + p, then θc,q (z) ∈ T S P (λ, α, β) if and only if a,d, p a,d, p (1) + [(α + β)(λ − 1)] Ic,q (1) ≤ 2(1 − α), [(1 + β) − λ(α + β)] Ic,q (2.5) a,d, p where 0 ≤ α < 1, 0 ≤ λ < 1, β ≥ 0 and Ic,q (z) is defined by equation (2.1). I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Starlikeness and Convexity Properties of the Generalized Bessel-Hypergeometric Function 189 a,d, p Proof. Since d < 0, therefore in view of Lemma 1.2, θc,q (z) ∈ T S P (λ, α, β) if and only if ∞ [n(1 + β) − (α + β)(1 + nλ − λ)] n=2 (−d/4)n−1 (a)n−1 ( p)n−1 ≤ (1 − α) . (q)n−1 (c)n−1 ((n − 1)!)2 (2.6) Now, ∞ [n(1 + β) − (α + β)(1 + nλ − λ)] n=2 = ∞ (−d/4)n−1 (a)n−1 ( p)n−1 (q)n−1 (c)n−1 ((n − 1)!)2 [(n − 1) [1 + β − λ(α + β)] + (1 − α)] n=2 = [1 + β − λ(α + β)] ∞ (n − 1)(−d/4)n−1 (a)n−1 ( p)n−1 n=2 = [1 + β − λ(α + β)] (−d/4)n−1 (a)n−1 ( p)n−1 (q)n−1 (c)n−1 ((n − 1)!)2 (q)n−1 (c)n−1 ((n − 1)!)2 + (1 − α) ∞ (−d/4)n−1 (a)n−1 ( p)n−1 n=2 (q)n−1 (c)n−1 ((n − 1)!)2 ∞ ∞ (−d/4)n (a)n ( p)n (−d/4)n (a)n ( p)n + (1 − α) . (q)n (c)n (n)!(n − 1)! (q)n (c)n ((n)!)2 n=1 n=1 Since q > a + p, therefore in view of Remark 2.1 and using equations (2.1) and (2.2), for z = 1, in the above equation, we get ∞ [n(1 + β) − (α + β)(1 + nλ − λ)] n=2 (−d/4)n−1 (a)n−1 ( p)n−1 (q)n−1 (c)n−1 ((n − 1)!)2 = [1 + β − λ(α + β)] a,d, p a,d, p a,d, p Ic,q (1) − Ic,q (1) + (1 − α) Ic,q (1) − 1 , or, equivalently ∞ n=2 [n(1 + β) − (α + β)(1 + nλ − λ)] (−d/4)n−1 (a)n−1 ( p)n−1 (q)n−1 (c)n−1 ((n − 1)!)2 a,d, p a,d, p = [1 + β − λ(α + β)] Ic,q (1) + [(α + β)(λ − 1)] Ic,q (1) − (1 − α). (2.7) Using equation (2.7) is condition (2.6), we get the necessary and sufficient condition (2.5) for the a,d, p function θc,q (z) to be in T S P (λ, α, β). Further, taking −b = d = −1, c = ν + (1.18), we get the following result: b+1 = ν + 1 in assertion (2.5) and using equation 2 I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 190 Nusrat Raza and Eman S.A. AbuJarad a,−1, p Corollary 2.1. If c = ν + 1 > 0, a, p, q, ν ∈ R, q > a + p and q > 0 , then θν+1,q (z) = 2z − a,−1, p Iν+1,q (z) ∈ T S P (λ, α, β) if and only if a,−1, p [(1 + β) − λ(α + β)] Iν+1,q (1) a,−1, p + [(α + β)(λ − 1)] Iν+1,q (1) ≤ 2(1 − α), (2.8) where 0 ≤ α < 1, 0 ≤ λ < 1, β ≥ 0 and a,−1, p Iν+1,q (z) = ϕν,1,−1 (z)*z F(a, p; q; z) = ∞ n=0 (a)n ( p)n z n+1 . 4n (q)n (ν + 1)n ((n)!)2 (2.9) a,d, p Next, we obtain the following necessary and sufficient conditions for θc,q (z) to be β- uniformly convex of order α: a,d, p Theorem 2.3. If a, b, p, q, c ∈ R, d < 0, q > 0, c > 0 and q > a + p, then θc,q (z) ∈ U C T (λ, α, β) if and only if a,d, p (1) [(1 + β) − λ(α + β)] Ic,q a,d, p + (1 − α) Ic,q (1) ≤ 2(1 − α), (2.10) where 0 ≤ α < 1, 0 ≤ λ < 1 and β ≥ 0. a,d, p Proof. Since d < 0, therefore, in view of Lemma 1.4, θc,q (z) ∈ U C T (λ, α, β) if and only if ∞ n [n(1 + β) − (α + β)(1 + nλ − λ)] n=2 (−d/4)n−1 (a)n−1 ( p)n−1 ≤ 1 − α. (q)n−1 (c)n−1 ((n − 1)!)2 (2.11) Now, we consider the left hand side of relation (2.11). On shifting index, we get ∞ n [n(1 + β) − (α + β)(1 + nλ − λ)] n=2 = (−d/4)n−1 (a)n−1 ( p)n−1 (q)n−1 (c)n−1 ((n − 1)!)2 ∞ (−d/4)n (a)n ( p)n (n + 1) [(n + 1)(1 + β) − (α + β)(1 + (n + 1)λ − λ)] (q)n (c)n ((n)!)2 n=1 = [1 + β − λ(α + β)] ∞ ∞ (−d/4)n (a)n ( p)n (n + 1)(−d/4)n (a)n ( p)n (n + 1)2 − (α + β)(1 − λ) . 2 (q)n (c)n ((n)!) (q)n (c)n ((n)!)2 n=1 n=1 On simplifying the above equation, we get ∞ n=2 n [n(1 + β) − (α + β)(1 + nλ − λ)] (−d/4)n−1 (a)n−1 ( p)n−1 (q)n−1 (c)n−1 ((n − 1)!)2 ∞ ∞ ∞ (−d/4)n (a)n ( p)n (−d/4)n (a)n ( p)n (−d/4)n (a)n ( p)n + = [1 + β − λ(α + β)] + 2 2 (q)n (c)n (n)!(n − 1)! n=1 (q)n (c)n ((n)!)2 (q)n (c)n ((n − 1)!) n=1 n=1 ∞ ∞ (−d/4)n (a)n ( p)n (−d/4)n (a)n ( p)n − [(α + β)(1 − λ)] + . (q)n (c)n (n − 1)!(n)! n=1 (q)n (c)n ((n)!)2 n=1 I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Starlikeness and Convexity Properties of the Generalized Bessel-Hypergeometric Function 191 Since q > a + p, therefore in view of Remark 2.1 and using equations (2.1) and (2.3), for z = 1, in the above equation, we get ∞ n [n(1 + β) − (α + β)(1 + nλ − λ)] n=2 = [1 + β − λ(α + β)] a,d, p Ic,q (1) (−d/4)n−1 (a)n−1 ( p)n−1 (q)n−1 (c)n−1 ((n − 1)!)2 a,d, p a,d, p + Ic,q (1) − 1 − [(α + β)(1 − λ)] Ic,q (1) − 1 , or, equivalently ∞ (−d/4)n−1 (a)n−1 ( p)n−1 (q)n−1 (c)n−1 ((n − 1)!)2 n [n(1 + β) − (α + β)(1 + nλ − λ)] n=2 a,d, p = [(1 + β) − λ(α + β)] Ic,q (1) (2.12) a,d, p + (1 − α) Ic,q (1) − (1 − α). Using equation (2.12) is condition (2.11), we get the necessary and sufficient condition (2.10) for a,d, p the function θc,q (z) to be in U C T (λ, α, β) Finally, taking −b = d = −1, c = ν + (1.18), we get the following result: b+1 = ν + 1 in assertion (2.10) and using equation 2 a,−1, p Corollary 2.2. If c = ν + 1 > 0, a, p, q, ν ∈ R, q > a + p and q > 0, then θν+1,q (z) = 2z − a,−1, p Iν+1,q (z) ∈ U C T (λ, α, β) if and only if a,−1, p [(1 + β) − λ(α + β)] Iν+1,q (1) a,−1, p + (1 − α) Iν+1,q (1) ≤ 2(1 − α), (2.13) a,−1, p where 0 ≤ α < 1, 0 ≤ λ < 1, β ≥ 0 and Iν+1,q (z), given by equation (2.9). 3. CONVEXITY OF MORE GENERALIZED FUNCTIONS In this section, we obtained the conditions for uniformly convexity of the convolution of the Generalized Bessel-Hypergeometric function with some analytic functions in class A. a,d, p We defined the function L c,q (z) : C → C by making convolution of the Generalized Bessela,d, p Hypergeometric function Ic,q (z), given by equation (2.1) with the function f (z), given by equation (1.1) as: p a,d, p L a,d, c,q (z) = Ic,q (z)*z f (z), I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 192 Nusrat Raza and Eman S.A. AbuJarad which on using equations (1.1) and (2.1), gives p L a,d, c,q (z) = ∞ (−d/4)n (a)n ( p)n n=0 =z+ (q)n (c)n ((n)!)2 an z n+1 ∞ (−d/4)n−1 (a)n−1 ( p)n−1 n=2 (q)n−1 (c)n−1 ((n − 1)!)2 (3.1) an z n , b+1 where c = ν + > 0, a, b, d, p, q, ν ∈ R and q > 0. 2 Pal [12], defined the class R σ (m, e) (σ ∈ C {0} , −1 ≤ e < m ≤ 1) as: Definition 3.1. A function f (z) is said to be in the class R σ (m, e) (σ ∈ C {0} , −1 ≤ e < m ≤ 1), if it satisfies the following inequality: f (z) − 1 (z ∈ U ). (3.2) (m − e)σ − e [ f (z) − 1] < 1 a,d, p To obtain the condition for the function L c,q (z) to be in the class U C T (λ, α, β), we require the following Lemma [13]: Lemma 3.1. The function f (z), given by equation (1.1) belong to the class R σ (m, e) (σ ∈ C {0} , −1 ≤ e < m ≤ 1) if it satisfies the following inequality: |an | (m − e) |σ | n (n ∈ N {1}) . (3.3) Now, we obtain the following result: Theorem 3.1. If f ∈ R σ (m, e) (σ ∈ C {0} , −1 ≤ e < m ≤ 1) and the following condition is satisfied a,d, p a,d, p (m − e) |σ | [(1 + β) − λ(α + β)] (Ic,q (1)) + [(α + β)(λ − 1)] Ic,q (1) ≤ 2(1 − α), (3.4) a,d, p where f (z) and Ic,q (z) (a, b, p, q, ∈ R, d < 0, q > 0, c > 0, q > a + p) are given by equations a,d, p (1.1) and (2.1), respectively. Then L c,q (z) ∈ U C T (λ, α, β) (0 ≤ α < 1, 0 ≤ λ < 1, β ≥ 0). a,d, p Proof. Since d < 0, therefore in view of Lemma (1.4), L c,q (z) ∈ U C T (λ, α, β) if the following inequality holds: ∞ n=2 n [n(1 + β) − (α + β)(1 + nλ − λ)] (−d/4)n−1 (a)n−1 ( p)n−1 |an | ≤ 1 − α. (q)n−1 (c)n−1 ((n − 1)!)2 I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS (3.5) Vol. 10, No. 1 (Special Issue), Jan–June 2019 Starlikeness and Convexity Properties of the Generalized Bessel-Hypergeometric Function 193 Since f ∈ R σ (m, e), then using inequality (3.3) in the left hand side of inequality (3.5), we have ∞ n [n(1 + β) − (α + β)(1 + nλ − λ)] n=2 ≤ ∞ (−d/4)n−1 (a)n−1 ( p)n−1 |an | (q)n−1 (c)n−1 ((n − 1)!)2 n [n(1 + β) − (α + β)(1 + nλ − λ)] n=2 (−d/4)n−1 (a)n−1 ( p)n−1 (q)n−1 (c)n−1 ((n − 1)!)2 (m − e) |σ | . n (3.6) Now, we consider the right hand side of inequality (3.6) and simplify it to get ∞ (−d/4)n−1 (a)n−1 ( p)n−1 n [n(1 + β) − (α + β)(1 + nλ − λ)] (q)n−1 (c)n−1 ((n − 1)!)2 n=2 = (m − e) |σ | = (m − e) |σ | ∞ n=2 ∞ [n(1 + β) − (α + β)(1 + nλ − λ)] |σ | (m − e) n (−d/4)n−1 (a)n−1 ( p)n−1 (q)n−1 (c)n−1 ((n − 1)!)2 [(n − 1) (1 + β − λ(α + β)) + (1 − α)] n=2 (−d/4)n−1 (a)n−1 ( p)n−1 . (q)n−1 (c)n−1 ((n − 1)!)2 Further, simplifying the above equation, gives ∞ n=2 n [n(1 + β) − (α + β)(1 + nλ − λ)] (−d/4)n−1 (a)n−1 ( p)n−1 (q)n−1 (c)n−1 ((n − 1)!)2 (m − e) |σ | n ∞ ∞ (−d/4)n (a)n ( p)n (−d/4)n (a)n ( p)n + (1 − α) = (m − e) |σ | (1 + β − λ(α + β)) , (q)n (c)n (n − 1)!n! (q)n (c)n ((n)!)2 n=1 n=1 Since q > a + p, therefore in view of Remark 2.1 and using the equations (2.1) and (2.2), for z = 1, in the above equation, gives ∞ n [n(1 + β) − (α + β)(1 + nλ − λ)] n=2 (−d/4)n−1 (a)n−1 ( p)n−1 |an | (q)n−1 (c)n−1 ((n − 1)!)2 (3.7) a,d, p a,d, p a,d, p ≤ (m − e) |σ | (1 + β − λ(α + β)) (Ic,q (1)) − Ic,q (1) + (1 − α) Ic,q (1) − 1 . If a,d, p a,d, p a,d, p (m − e) |σ | (1 + β − λ(α + β)) (Ic,q (1)) − Ic,q (1) + (1 − α) Ic,q (1) − 1 ≤ (1 − α), (3.8) then, in view of inequality (3.7), we get inequality (3.5). Inequality (3.8) gives assertion (3.4). Now, taking −b = d = −1, c = ν + (1.18), we get the following result: b+1 = ν + 1 in assertion (3.4) and using equation 2 I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 194 Nusrat Raza and Eman S.A. AbuJarad Corollary 3.1. If f ∈ R σ (m, e) (σ ∈ C {0} , −1 ≤ e < m ≤ 1) and the following condition holds: a,−1, p a,−1, p (m − e) |σ | [(1 + β) − λ(α + β)] (Iν+1,q (1)) + [(α + β)(λ − 1)] Iν+1,q (1) ≤ 2(1 − α), (3.9) a,−1, p where 0 ≤ α < 1, 0 ≤ λ < 1, β ≥ 0 and Iν+1,q (z) (c = ν + 1 > 0, a, p, c, q, ν ∈ R, d < 0, q > a,−1, p 0, q > a + p), given by equation (2.9), then L ν+1,q (z) ∈ U C T (λ, α, β) (0 ≤ α < 1, 0 ≤ λ < 1, β ≥ 0). a,d, p Next, we define the function Pc,q (z) : C → C as: a,d, p Pc,q (z) = z 2− 0 a,d, p Ic,q (t) dt, t (3.10) a,d, p where a, b, p, q, ∈ R, d < 0, q > 0, c > 0, q > a + p and Ic,q (z) is given by equation (2.1). Using equation (2.1), for z = t, in equation (3.10) and then integrating, we get a,d, p Pc,q (z) = z − ∞ (−d/4)n−1 (a)n−1 ( p)n−1 z n n=2 (q)n−1 (c)n−1 (n − 1)!(n)! . (3.11) a,d, p Now, we obtain the following necessary and sufficient condition for the function Pc,q (z) to be βuniformly convex of order α : a,d, p Theorem 3.2. If a, b, d, p, q ∈ R, d < 0, q > 0 and c > 0, then Pc,q (z) ∈ U C T (λ, α, β) if and only if a,d, p a,d, p (1) + [(α + β)(λ − 1)] Ic,q (1) ≤ 2(1 − α). [(1 + β) − λ(α + β)] Ic,q (3.12) a,c, p where 0 ≤ α < 1, 0 ≤ λ < 1, β ≥ 0 and Ic,q (z) is defined by equation (2.1). a,d, p Proof. Since d < 0, therefore in view of Lemma 1.4, Pc,q (z) ∈ U C T (λ, α, β), if the following inequality holds: ∞ n [n(1 + β) − (α + β)(1 + nλ − λ)] n=2 (−d/4)n−1 (a)n−1 ( p)n−1 ≤ 1 − α, (q)n−1 (c)n−1 (n − 1)!n! or, equivalently ∞ n=2 [n(1 + β) − (α + β)(1 + nλ − λ)] (−d/4)n−1 (a)n−1 ( p)n−1 ≤ 1 − α. (q)n−1 (c)n−1 ((n − 1)!)2 I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS (3.13) Vol. 10, No. 1 (Special Issue), Jan–June 2019 Starlikeness and Convexity Properties of the Generalized Bessel-Hypergeometric Function 195 Simplifying the left hand side of relation (3.13), we get ∞ n [n(1 + β) − (α + β)(1 + nλ − λ)] n=2 = [1 + β − λ(α + β)] (−d/4)n−1 (a)n−1 ( p)n−1 (q)n−1 (c)n−1 (n − 1)!n! ∞ ∞ (−d/4)n (a)n ( p)n (−d/4)n (a)n ( p)n . + (1 − α) (q)n (c)n (n)!(n − 1)! (q)n (c)n ((n)!)2 n=1 n=1 (3.14) Since q > a + p, therefore in view of Remark 2.1 and using equations (2.1) and (2.2), for z = 1, in inequality (3.14), we get ∞ n [n(1 + β) − (α + β)(1 + nλ − λ)] n=2 = [(1 + β) − λ(α + β)] (−d/4)n−1 (a)n−1 ( p)n−1 (q)n−1 (c)n−1 (n − 1)!n! a, p Ic,q (1) + [(α + β)(λ − 1)] (3.15) a, p Ic,q (1) − (1 − α). Using equation (3.15) in inequality (3.13), we get the necessary and sufficient condition (3.12) for a,d, p Pc,q (z) to be in U C T (λ, α, β). 4. AN INTEGRAL OPERATOR In this section, we discuss the starlikness and convexity properties of the function obtained by using an integral operator introduced by Seenivasagan and Breaz [16]. The integral operator for analytic functions f i ∈ A, (i = 1, 2, 3, ...) are defined as [16]: 1 ⎧ ⎫ 1 ρ ⎪ ⎪ n ⎨ z f i (t) αi ⎬ F(z) := Fα1 , α2 , ...αn , ρ (z) = ρ t ρ−1 dt , ⎪ ⎪ t ⎩ 0 ⎭ i=1 where α1 , α2 , ...αn , ρ ∈ C. We define the function G α1 , α2 , ...αn , ρ (z) in terms of the integral involving the generalized a,d, p Bessel-Hypergeometric function Ic,q (z): 1 ⎧ ⎫ 1 ρ ⎪ ⎪ a,d, p ⎪ ⎪ n ⎨ z Ic,q (t) αi ⎬ ρ−1 G(z) := G α1 , α2 , ...αn , ρ (z) = ρ , t dt ⎪ ⎪ t ⎪ ⎪ i=1 ⎩ 0 ⎭ where α1 , α2 , ...αn , ρ ∈ R, c = ν + (4.1) b+1 > 0, a, b, d, p, q, ν ∈ R and q > 0. 2 I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 196 Nusrat Raza and Eman S.A. AbuJarad a,d, p Now, according to Definition 1.2, the function Ic,q (z) is starlike function of order α (0 ≤ α < 1) if the following inequality holds: ⎧ ⎫ a,d, p ⎪ ⎨ z Ic,q ⎬ (z) ⎪ Re > α (z ∈ U, 0 ≤ α < 1). (4.2) a,d, p ⎪ ⎩ Ic,q (z) ⎪ ⎭ Since, for the function f : C → C, we have Re( f (z)) < | f (z)|, therefore from relation (4.2), we get a,d, p z Ic,q (z) (z ∈ U, 0 ≤ α < 1). (4.3) >α a,d, p Ic,q (z) We need to mention the following definitions, defined by Miller [10] to establish the starlikeness property of the function G(z), given by equation (4.1): Definition 4.1. Let u = u 1 + iu 2 and v = v1 + iv2 and let be the set of functions ψ(u, v) satisfying the following conditions: (i) (ii) ψ(u, v)is continuous in a domain D ⊂ C2 , (1, 0) ∈ D and Reψ(1, 0) > 0, (iii) Reψ(iu 2 , v1 ) ≤ 0, whenever (iu 2 , v1 ) ∈ D and v1 ≤ −1 (1 + u 22 ). 2 Definition 4.2. Let h(z) = 1 + c1 z + c2 z 2 + ... be regular in a unit disc and let ψ ∈ with corresponding domain D. We denote by P(ψ) those functions h(z) that satisfy: (i) (h(z), zh (z)) ∈ D, (ii) Reψ(h(z), zh (z)) > 0, when z ∈ . Also, we need to mention the following result obtained by Miller [10]: Lemma 4.1. If ψ ∈ , with corresponding domain D, and if (h(z), zh (z)) ∈ D, then Re(ψh(z), zh (z)) > 0 =⇒ Re(h(z)) > 0. (4.4) Next, we establish the following result: Theorem 4.1. Let zG (z) = (1 − δ) p(z) + δ, G(z) I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS (4.5) Vol. 10, No. 1 (Special Issue), Jan–June 2019 Starlikeness and Convexity Properties of the Generalized Bessel-Hypergeometric Function 197 where the function G(z) is defined by equation (4.1), 0 < δ < 1, and p(z) = 1 + p1 z + p2 z 2 + ... ( p1 , p2 , ... ∈ R). If p(z) ∈ P(ψ), where ψ(∈ ) is given by ψ(u, v) = n −(1 − δ)v 1 + (1 − α) + ρ(1 − δ)(u − 1) (1 − δ)u − δ α i=1 i (u, v) ∈ D ⊂ C2 , u = δ , 1−δ (4.6) and 1 zp (z) ≥ . 2 (4.7) Then G(z) is starlike function of order δ satisfying − 2(1 − α) n i=1 2 n 1 2(1 − α) i=1 − 2ρ + 1 + 8ρ αi 1 − 2ρ + 1 + αi 1 , 2 (4.8) where α1 , α2 , ...αn , ρ positive real numbers and α is the order of starlikness of the function b+1 a,d, p > 0, a, b, d, p, q, ν ∈ R, q > 0, q > a + p), defined by equation (2.1). Ic,q (z), (c = ν + 2 0<δ≤ 4ρ , δ= Proof. From equation (4.1), we get ρ (G(z)) = ρ z t ρ−1 0 n a,d, p Ic,q (t) 1 αi dt. t i=1 (4.9) Differentiating both sides of equation (4.9), we obtain zG (z) = G(z) z !n ρ i=1 1 a,d, p Ic,q (z) αi z (G(z))ρ . (4.10) Now, from equations (4.5) and (4.10), we get zρ (1 − δ) p(z) + δ = !n i=1 a,d, p Ic,q (z) z (G(z))ρ I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS 1 αi . (4.11) Vol. 10, No. 1 (Special Issue), Jan–June 2019 198 Nusrat Raza and Eman S.A. AbuJarad Taking Logarithm of equation (4.11), then differentiating with respect to z and multiplying the resultant equation with z, yields ⎛ ⎞ a,d, p n n (z) z I c,q (1 − δ)zp (z) 1 zG (z) ⎜ ⎟ 1 =ρ+⎝ . (4.12) − −ρ ⎠ a,d, p (1 − δ) p(z) + δ α α G(z) Ic,q (z) i=1 i i=1 i Using equation (4.5) in the right hand side of equation (4.12), we get ⎛ ⎞ a,d, p n n z Ic,q (z) 1 1 (1 − δ)zp (z) ⎜ ⎟ + = + ρ(1 − δ) [ p(z) − 1] . ⎝ ⎠ a,d, p α (1 − δ) p(z) + δ α i Ic,q (z) i=1 i=1 i (4.13) Since each αi > 0, i = 1, 2, 3, ..., n , therefore taking modulus of both the sides of equation (4.13), we have n a,d, p n z Ic,q (1 − δ)zp (z) (z) 1 1 + = + ρ(1 − δ) p(z) − 1] [ . a,d, p Ic,q (1 − δ) p(z) + δ α (z) i=1 αi i=1 i From equation (4.3), we get n n (1 − δ)zp (z) 1 1 + + ρ(1 − δ) [ p(z) − 1] > α . (1 − δ) p(z) + δ α α i i=1 i=1 i Using the inequality |z 1 | − |z 2 | ≤ |z 1 + z 2 | ≤ |z 1 | + |z 2 | (z 1 , z 2 ∈ C), we obtain n (1 − δ) zp (z) 1 + (1 − α) + ρ(1 − δ) (| p(z)| − 1) > 0. (1 − δ) | p(z)| − δ α i=1 i (4.14) (4.15) (4.15). Taking u = | p(z)| and v = − zp (z) in equation (4.6), we get the left hand side of relation Now, we proceed to verify that ψ(u, v) , for u = | p(z)| + 0i and v = − zp (z) + 0i, satisfies the conditions of Definition 4.1. Since, it is clear that ψ(u, v) is continuous in its domain D. Therefore the condition (i) is satisfied. Further, taking u = 1 and v = 0 in equation (4.6), we get ψ(1, 0) = (1 − α) n 1 . α i=1 i (4.16) Since 0 ≤ α < 1, α1 , α2 , ...αn positive real numbers, therefore in view of equation (4.16), we have Re (ψ(1, 0)) > 0. Thus condition (ii) of Definition 4.1 is satisfied. Next, we verify the last condition of Definition 4.1. Here (4.17) v1 = Re(v) = − zp (z) . I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Starlikeness and Convexity Properties of the Generalized Bessel-Hypergeometric Function 199 Using inequality (4.7) in equation (4.17), we get v1 ≤ −1 , 2 (4.18) or, equivalently v1 ≤ −1 1 + u 22 2 (u 2 = I m(u) = 0). Now, ψ(iu 2 , v1 ) = ψ(0, v1 ) = n 1 (1 − δ)v1 − ρ(1 − δ) + (1 − α) . δ α i=1 i (4.19) Since the right hand side of equation (4.19) is real, therefore we get Re (ψ(0, v1 )) = n 1 (1 − δ)v1 . − ρ(1 − δ) + (1 − α) δ α i=1 i (4.20) Using inequality (4.18) in equation (4.20), we get Re (ψ(0, v1 )) ≤ n 1 −(1 − δ) − ρ(1 − δ) + (1 − α) , 2δ α i=1 i or, equivalently Re (ψ(0, v1 )) ≤ −(1 − δ) − 2δρ(1 − δ) + 2δ (1 − α) n 2δ i=1 1 αi . (4.21) Since 0 < δ < 1, therefore, for the condition (iii) of Definition 4.1 to be satisfied, δ must satisfy the following inequality: −(1 − δ) − 2δρ(1 − δ) + 2δ (1 − α) n 1 ≤ 0, α i=1 i or, equivalently n 1 2ρδ + δ 2(1 − α) − 2ρ + 1 − 1 ≤ 0. α i=1 i 2 (4.22) On solving inequality (4.22) for δ, we get the assertion (4.8) of Theorem 4.1. Further, since assertion (4.8) holds, therefore in view of relations (4.21) and (4.22), the condition (iii) of Definition 4.1 is satisfied and hence ψ ∈ . I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 200 Nusrat Raza and Eman S.A. AbuJarad Since, the function p(z) ∈ P(ψ), therefore, it satisfies all the conditions of Definition 4.2 . Thus, in view of Lemma 4.1, we have Re( p(z)) > 0. Therefore from equation (4.5), we have zG (z) > δ, (4.23) Re G(z) which proves that, G(z) is starlike function of order δ satisfying the relation (4.8). Now, we obtain the order of convexity of the function G(z), defined by (4.1) in the following result: b+1 a,d, p > 0, a, b, d, p, q, ν ∈ R, q > 0, q > Theorem 4.2. If the function Ic,q (z), (c = ν + 2 a + p), defined by equation (2.1) is starlike function of order α, (0 ≤ α < 1) and the function G(z), defined by equation (4.1), is starlike function of order δ, given by inequality (4.8), then the function G(z) is convex of order η, given by η = (α − 1) n 1 + ρ − δ(ρ − 1), α i=1 i (4.24) where α1 , α2 , ...αn , ρ are positive numbers. Proof. Taking Logarithm of equation (4.9), then differentiating with respect to z, and multiplying the resultant equation with z, yields ⎛ ⎞ a,d, p n n (z) z I (z) c,q 1 zG zG (z) ⎜ ⎟ 1 + ρ − 1 + (z) = ⎝ − (ρ − 1) . (4.25) ⎠ a,d, p G α G(z) α Ic,q (z) i=1 i i=1 i Taking the real parts of both sides of equation (4.25), we get ⎛ ⎞ a,d, p (z) n n (z) (z) z I c,q zG 1 zG ⎜ ⎟ 1 Re 1 + (z) = Re ⎝ + ρ − − (ρ − 1)Re . ⎠ a,d, p G α G(z) α Ic,q (z) i=1 i i=1 i Using relations (4.2) and (4.23), we get n 1 zG (z) + ρ − δ(ρ − 1), Re 1 + (z) > (α − 1) G α i=1 i this shows that, G(z) is convex function of order η, given by equation (4.24). REFERENCES [1] J. W. Alexander, Functions which map the interior of the unit circle upon simple regions, The Annals of Mathematics, 17(1) (1915) 12–22. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Starlikeness and Convexity Properties of the Generalized Bessel-Hypergeometric Function 201 [2] Á. Baricz, Generalized Bessel functions of the first kind, Springer, (2010). [3] Á. Baricz, Geometric properties of generalized Bessel functions, Publ Math-Debrecen, 73(1-2) (2008) 155–178. [4] B. Carlson, D. B. Shaffer, Starlike and prestarlike hypergeometric functions, SIAM J. Numer. Anal., 15(4) (1984) 737–745. [5] E. Deniz, H. Orhan, H. Srivastava, Some sufficient conditions for univalence of certain families of integral operators involving generalized Bessel functions, Taiwan. J. Math., 15(2) (2011) 883–917. [6] P. L. Duren, Univalent functions, Vol. 259, Springer Science & Business Media, 2001. [7] A. Goodman, On uniformly convex functions, Ann. Polon. Math., 56(1) (1991) 87–92. [8] S. Kanas, A. Wisniowska, Conic regions and k-uniform convexity, J. Comput. Appl. Math., 105(1) (1999) 327–336. [9] J.-L. Liu, Certain convolution properties of multivalent analytic functions associated with a linear operator, Anal. Appl, 292 (2004) 470–483. [10] S. Miller, Differential inequalities and Caratheodory functions, Bull. Am. Math. Soc., 81(1) (1975) 79–81. [11] G. Murugusundaramoorthy, On certain subclasses of analytic functions associated with hypergeometric functions, Appl. Math. Lett., 24(4) (2011) 494–500. [12] K. D. K. Pal, On a class of univalent functions related to complex order, Indian J. Pure Appl. Math., 26(9) (1995) 889–896. [13] E. D. Rainville, Special functions, Chelsea (1971). [14] F. Rønning, Uniformly convex functions and a corresponding class of starlike functions, Proc. Am. Math. Soc., 118(1) (1993) 189–196. [15] A. Schild, On starlike functions of order α, Am. J. Math., 87(1) (1965) 65–70. [16] N. Seenivasagan, D. Breaz, Certain sufficient conditions for univalence, Gen. Math, 15(4) (2007) 7–15. [17] H. Silverman, Univalent functions with negative coefficients, Proc. Am. Math. Soc., 51(1) (1975) 109–116. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 INDIAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019, pp. 202–219 DOI: A New Iterative Class for Finding Common Fixed Points of a Finite Family of Generalized Total Asymptotically Nonexpansive and Multivalued Mappings in Hyperbolic Spaces Shamshad Husain1∗ and Nisha Singh2 Department of Applied Mathematics, Z H College of Engineering and Technology Aligarh Muslim University, Aligarh-202002, India (∗ Corresponding author) Email: ∗ [email protected], 2 [email protected] Abstract: The purpose of the paper is to introduce a general iteration process for the finite families of total asymptotically nonexpansive mappings and generalized nonexpansive multivalued mappings in hyperbolic space which include Hadamard manifold and CAT(0) space as special cases. Some important properties of the new defined iterative process are analyzed. Finally, - convergence and strong convergence for the iterative process in hyperbolic space are established. The results presented in this paper extend and improve some existing results. Keywords: common fixed point; asymptotically nonexpansive mapping; strong convergence; convergence; hyperbolic space. AMS Subject Classification: 47H10. 47H09. 49M05. 54E70. 1. INTRODUCTION In nonlinear functional analysis, one of the most productive tools is the fixed point theory, which has numerous applications in many quantitative disciplines such as biology, chemistry, computer science and in many branches of engineering. The study of metric spaces without linear structure has played a vital role in various branches of pure and applied sciences. In the past two decades, the metric fixed point theory has been investigated extensively by numerous mathematicians. Takahashi [17] introduced the concept of convexity in a metric space (X,d) as follows: I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 New Iterative Class for Finding Common Fixed Points of a Finite Family 203 A convex structure in a metric space (X, d) is a mapping W : X × X × [0, 1] → X such that, for all x, y, u ∈ X and all λ ∈ [0, 1], d(u, W (x, y, λ)) ≤ λd(u, x) + (1 − λ)d(u, y). A metric space together with a convex structure is called a convex metric space. A nonempty subset K of X is said to be convex if W (x, y, λ) ∈ K for all (x, y, λ) ∈ K × K × [0, 1]. Recently, Kohlenbach [8] introduced the concept of convex metric space by defining hyperbolic space. A hyperbolic space [8], (X, d, W ) is a metric space (X, d) together with a convexity mapping W : X × X × [0, 1] → X such that: i. ii. iii. iv. d(u, W (x, y, α)) ≤ αd(u, x) + (1 − α)d(u, y) d(W (x, y, α), W (x, y, β)) = |α − β|d(x, y) W (x, y, α) = W (y, x, (1 − α)) d(W (x, z, α), W (y, w, α)) ≤ αd(x, y) + (1 − α)d(z, w) for all x, y, z, w ∈ X and α, β ∈ [0, 1]. If (X, d, W ) satisfies only (i), then it coincides with the convex metric space which is introduced by Takahashi [17]. The concept of hyperbolic spaces defined above is more restrictive than the hyperbolic type defined by Goebel and Kirk [4]. But it is more general than the hyperbolic space defined by Reich and Shafrir [14]. All normed spaces and their subsets are the examples of hyperbolic spaces as well as convex metric spaces. It is remarked that every CAT(0) and Banach spaces are special cases of hyperbolic space. In 2008, Zhao et al. [21] introduced the generalized implicit iterative process for the common fixed points of the finite family {T j : j ∈ J }. In 2011, Khan [6] generalized the results of Zhao et al [21] for two finite families of nonexpansive mappings. In 2012, Khan et al [7] studied implicit iteration for two finite families of nonexpansive mappings in a hyperbolic space. Later on some authors discussed the convergence of the iterative process in hyperbolic spaces [1, 3, 5, 15, 19]. Remark: A hyperbolic space (X,d,W) is said to be uniformly convex if for any r > 0 and ∈ (0, 2] and for all u, x, y ∈ X , there exists δ ∈ (0, 1] such that, 1 , u ≤ (1 − δ)r, d W x, y, 2 provided max{d(x, u), d(y, u)} ≤ r and d(x, y) ≥ r [9, 16]. The class of uniformly convex hyperbolic spaces includes both uniformly convex normed spaces and CAT(0) spaces as special cases. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 204 Shamshad Husain and Nisha Singh Markin [11] and Nadler [12] introduced the study of fixed points for multivalued contractions and nonexpansive mappings using the Hausdorff metric. Shimizu and Takahashi [16] proved the existence of fixed points for multivalued nonexpansive mappings in convex metric spaces, that is, every multivalued mapping T : X → C(X ) has a fixed point in a bounded, complete and uniformly convex metric space (X,d), where C(X) is the family of all compact subsets of X. Inspired and motivated by the above work, we define a new algorithm as follows: Let K be a nonempty closed and convex subset of a hyperbolic space X . For j ∈ J , let T j : j j j K → K be a ({μn }, {ξn }, ρ j )-total asymptotically nonexpansive mapping with limn→∞ μn = 0 j and limn→∞ ξn = 0 and a strictly increasing continuous function ρ j : [0, +∞) → [0, +∞) satj j isfying ρ j (0) = 0 and let S j : K → K be a ({μ̂n }, {ξ̂n }, ρ̂ j )-total asymptotically nonexpansive j j mapping with limn→∞ μ̂n = 0 and limn→∞ ξ̂n = 0 and a strictly increasing continuous function j j ρ̂ : [0, +∞) → [0, +∞) satisfying ρ̂ (0) = 0 and Q j : K → K be an asymptotically nonexpan j sive mapping such that ∞ k < +∞ and R j : K → K be an asymptotically nonexpansive map∞ j n=1 n ping such that n=1 k̂n < ∞. Let T̂ : K → P(K ) be a multivalued mapping such that F(T̂ ) = ∅ and PT̂ is a nonexpansive mapping. We consider the following general iterative sequence {xn }: ⎧ ⎪ δr n ⎪ , β ⎨ yn = W Trn xn , W xn , Q rn u n , 1−β rn rn ⎪ θr n ⎪ , α ⎩xn+1 = W Srn yn , W u n , Rrn vn , 1−α rn , rn (1.1) where u n ∈ PT̂ (xn ) and vn ∈ PT̂ (yn ) and suppose that {αr n }, {βr n }, {δr n } and {θr n } are the real sequences in [a, b] for some a, b ∈ (0, 1) where F = nj=1 [F(T j ) F(S j ) F(R j ) F(Q j )] F(T̂ ) is the set of all common fixed points. Further, we study some results concerning -convergence as well as strong convergence for the above defined iteration process. The results presented in the paper improve and extend the corresponding results given by [6, 7, 13, 21] 2. PRELIMINARIES From now, N denotes the set of natural numbers and J = {1, 2, . . . , N }, the set of first N natural numbers. Definition 2.1. Let T : K → 2 K be a multivalued mapping. An element x ∈ K is said to be a fixed point of T , if x = T x. Denote by F(T ), the set of fixed points of T . Definition 2.2. A subset K ⊂ X = ∅ is said to be proximal, if for each x ∈ X , there exists an element y ∈ K such that, d(x, y) = dist(x, K ) := inf{d(x, z) : z ∈ K }. It is well known that each weakly compact convex subset of a Banach space is proximal as well as each closed convex subset of a uniformly convex Banach space is also proximal. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 New Iterative Class for Finding Common Fixed Points of a Finite Family 205 Definition 2.3. Let C B(K ) be the collection of all nonempty and closed bounded subsets and P(K ) be the collection of all nonempty proximal bounded and closed subsets of K , respectively. Let H (., .) be the Hausdorff distance on C B(K ) is defined by: H (A, B) := max{sup dist(x, B), sup dist(y, A)}, ∀A, B ∈ C B(X ). x∈A y∈B Definition 2.4. A mapping T : K → K is said to be: (i) (ii) (iii) semi-compact if every bounded sequence {xn } ⊂ K , satisfying d(xn , T xn ) → 0 as n → ∞, has a convergent subsequence; nonexpansive if d(T x, T y) ≤ d(x, y) for any x, y ∈ K ; asymptotically nonexpansive if there exists a sequence {kn } ⊂ [0, +∞) and limn→∞ kn = 0 such that d(T n x, T n y) ≤ (1 + kn )d(x, y), ∀x, y ∈ K , n ≥ 1; (iv) ({μn }, {ξn }, ρ)-total asymptotically nonexpansive, if there exist nonnegative sequences {μn }, {ξn } with μn → 0, ξn → 0 and a strictly increasing continuous function ρ : [0, +∞) → [0, +∞) with ρ(0) = 0 such that d(T n x, T n y) ≤ d(x, y) + μn ρ(d(x, y)) + ξn , ∀x, y ∈ K , n ≥ 1; (v) uniformly L-Lipschitzian if there exists a constant L > 0 such that d(T n x, T n y) ≤ Ld(x, y), ∀x, y ∈ K , n ≥ 1. Definition 2.5. Let {xn } be a bounded sequence in a hyperbolic space X . For x ∈ X , we define a continuous functional r (., {xn }) : X → [0, +∞) by: r (x, {xn }) = lim sup d(x, xn ). n→∞ (i) The asymptotic radius r̂ ({xn }) of {xn } is given by r̂ ({xn }) = inf{r (x, {xn }) : x ∈ X }. (ii) The asymptotic center of a bounded sequence {xn } with respect to K ⊂ X is defined as follows: A K ({xn }) = {x ∈ X : r (x, {xn }) ≤ r (y, {xn }), ∀y ∈ K }, which is the set of minimizers for (., {xn }). It is simply denoted by A({xn }) when the asymptotic center is taken with respect to X and a sequence {xn } in X is said to be -convergent to x ∈ X if x is the unique asymptotic center of {u n } for every subsequence {u n } of {xn }. In this case, we write -limn→∞ xn = x and call x the -limit of {xn }. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 206 Shamshad Husain and Nisha Singh Lemma 2.1 [10]. Let (X, d, W ) be a complete uniformly convex hyperbolic space with monotone modulus of uniform convexity. Then every bounded sequence {xn } in X has a unique asymptotic center with respect to any nonempty closed convex subset K of X. Lemma 2.2 [20]. Let (X, d, W ) be a uniformly convex hyperbolic space with monotone modulus of uniform convexity η. Let x ∈ X and {αn } be a sequence in [a, b] for some a, b ∈ (0, 1). If {xn } and {yn } are sequences in X such that for some c ≥ 0, limn→∞ sup d(xn , x) ≤ c, limn→∞ sup d(yn , x) ≤ c, limn→∞ d(W (xn , yn , αn ), x) = c. Then limn→∞ d(xn , yn ) = 0. Lemma 2.3 [20]. Let K be a nonempty closed convex subset of a uniformly convex hyperbolic space and {xn } a bounded sequence in K such that A({xn }) = {y} and r ({xn }) = ρ. If {ym } is another sequence in K such that limm→∞ r (ym , {xn }) = ρ, then limm→∞ ym = y. Lemma 2.4 [18]. Let {αn }, {βn } and {γn } be nonnegative real sequences satisfying αn+1 ≤ (1 + γn )αn + βn , ∀n ≥ 1. ∞ If ∞ n=1 γn < ∞ and n=1 βn < ∞, then the limn→∞ αn exist. If there exists a subsequence {αn i } ⊂ {αn } such that αni → 0, then limn→∞ αn = 0. 3. MAIN RESULTS Theorem 3.1. Let K be a nonempty closed convex subset of a hyperbolic space X . For j ∈ J , let T j , S j : K → K be a total asymptotically nonexpansive mapping and Q j , R j : K → K be an asymptotically nonexpansive mapping and let T̂ : K → P(K ) be a multivalued mapping. Assume that F = ∅ and for each j ∈ J , the following holds: (i) ∞ n=1 ∞ j μn < +∞; j n=1 kn (ii) < +∞; ∞ n=1 ∞ j μ̂n < +∞; j n=1 k̂n ∞ j n=1 ξn ∞ j n=1 ξ̂n < +∞; < +∞; < +∞; there exists a constant M ∗ > 0 such that ρ j (r ) ≤ M ∗r, ρ̂ j (r ) ≤ M ∗r, ∀r > 0. Let {xn } be the sequence defined in (1.1), then limn→∞ d(xn , p) exists for all p ∈ F. j j j j j j Proof. Let p ∈ F. Set μn = max j∈J {μn , μ̂n }; ξ̂n }; kn = max j∈J {k n , k̂n } and ξn = max j∈J {ξn , ∞ ∞ μ < +∞, ξ < +∞ and ρ = max j∈J {ρ j , ρ̂ j }. By (i), we know that ∞ n n n=1 n=1 n=1 kn < I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 New Iterative Class for Finding Common Fixed Points of a Finite Family +∞. Then from (1.1), we have d(yn , p) ≤ δr n ,p 1 − βr n βr n d(Trn xn , p) + δr n d(xn , p) + (1 − βr n − δr n )d(Q rn u n , p) ≤ βr n [d(xn , p) + μrn ρ r (d(xn , p)) + ξnr ] + δr n d(xn , p) ≤ βr n d(Trn xn , 207 p) + (1 − βr n )d W xn , Q rn u n , +(1 − βr n − δr n )knr d(u n , p) ≤ βr n [d(xn , p) + μn ρ(d(xn , p)) + ξn ] + δr n d(xn , p) ≤ βr n [d(xn , p) + μn M ∗ d(xn , p) + ξn ] + δr n d(xn , p) +(1 − βr n − δr n )kn dist(u n , PT̂ ( p)) +(1 − βr n − δr n )kn H (PT̂ (xn ), PT̂ ( p)) ≤ βr n [(1 + μn M ∗ )d(xn , p) + ξn ] + δr n d(xn , p) +(1 − βr n − δr n )kn d(xn , p) ≤ Next d(xn+1 , p) (1 + μn M ∗ )kn d(xn , p) + ξn . (3.1) ≤ θr n ,p 1 − αr n αr n d(Srn yn , p) + θr n d(u n , p) + (1 − αr n − θr n )d(Rrn vn , p) ≤ αr n [d(yn , p) + μ̂rn ρ̂ r (d(yn , p)) + ξ̂nr ] + θr n d(u n , p) ≤ αr n d(Srn yn , p) + (1 − αr n )d W u n , Rrn vn , +(1 − αr n − θr n )k̂nr d(vn , p) ≤ αr n [d(yn , p) + μn ρ(d(yn , p)) + ξn ] + θr n dist(u n , PT̂ ( p)) +(1 − αr n − θr n )kn dist(vn , PT̂ ( p)) ≤ αr n [d(yn , p) + μn M ∗ d(yn , p) + ξn ] + θr n H (PT̂ (xn ), PT̂ ( p)) +(1 − αr n − θr n )kn H (PT̂ (yn ), PT̂ ( p)) ≤ αr n [(1 + μn M ∗ )d(yn , p) + ξn ] + θr n d(xn , p) ≤ αr n [(1 + μn M ∗ )d(yn , p) + ξn ] + θr n d(xn , p) +(1 − αr n − θr n )kn d(yn , p) +(1 − αr n − θr n )kn [(1 + μn M ∗ )kn d(xn , p) + ξn ] ≤ αr n [(1 + μn M ∗ )2 kn d(xn , p) + ξn (1 + μn M ∗ + ξn ] + θr n d(xn , p) +(1 − αr n − θr n )kn [(1 + μn M ∗ )kn d(xn , p) + ξn ] ≤ ≤ (1 + μn M ∗ )2 kn2 d(xn , p) + (1 + μn M ∗ )ξn 2 2 μn M ∗ + d(xn , p)kn2 1 + (μn M ∗ )2 + (1 + μn M ∗ )ξn 1 2 ≤ (1 + an2 μn )d(xn , p) + (1 + μn M ∗ )ξn ≤ (1 + M1 μn )d(xn , p) + M2 ξn , I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS (3.2) Vol. 10, No. 1 (Special Issue), Jan–June 2019 208 Shamshad Husain and Nisha Singh where an2 = 21 M ∗ + 22 μn (M ∗ )2 and from (i), there exists positive constants M1 and M2 such that an2 ≤ M1 , (1 + μn M ∗ ) ≤ M2 for each n ≥ 1. Now from Lemma 2.4, we have limn→∞ d(xn , p) exists for each p ∈ F. In 1993, Bruck et al [2] introduced a notion of asymptotically nonexpansive mapping in the intermediate sense. A mapping T : K → K is said to be asymptotically nonexpansive mapping in the intermediate sense, provided that T is uniformly continuous and lim supn→∞ supx,y∈K {d(T n x, T n y) − d(x, y)} ≤ 0. The following corollaries can be obtained immediately from Theorem 3.1. Corollary 3.1. Let K be a nonempty closed convex subset of a hyperbolic space X . For j ∈ J , j let T j : K → K be a {ξn }-asymptotically nonexpansive mapping in the intermediate sense and S j : j j K → K be a {ξ̂n }-asymptotically nonexpansive mapping in the intermediate sense. If ∞ n=1 ξn < ∞ j +∞, n=1 ξ̂n < +∞ and Q j , R j : K → K be an asymptotically nonexpansive mapping and F = ∅, then for the sequence {xn } defined in (1.1), limn→∞ d(xn , p) exists for all p ∈ F. Corollary 3.2. Let K be a nonempty closed convex subset of a hyperbolic space X . For j ∈ J let j j T j : K → K be a {ln }-asymptotically nonexpansive mapping with ∞ l n=1 n < +∞ and S j : K → ∞ j j K be a {l̂n }-asymptotically nonexpansive mapping with n=1 l̂n < +∞ and Q j , R j : K → K be an asymptotically nonexpansive mapping and F = ∅, then for the sequence {xn } defined in (1.1), limn→∞ d(xn , p) exists for all p ∈ F. Theorem 3.2. Let K be a nonempty closed convex subset of a uniformly convex hyperbolic space X with monotone modulus of uniform convexity η. For j ∈ J , let T j : K → K be a uniformly L j -Lipschitzian and total asymptotically nonexpansive mapping and let S j : K → K be a uniformly L̂ j -Lipschitzian and total asymptotically nonexpansive mapping and Q j , R j : K → K be an asymptotically nonexpansive mapping and let T̂ : K → P(K ) be a multivalued mapping. Assume that F = ∅ and the conditions (i) and (ii) in Theorem 3.1 hold. Then for j ∈ J , the sequence {xn } generated by (1.1), we have lim d(xn , T j xn ) = lim d(xn , S j xn ) = lim d(xn , R j xn ) = lim d(xn , Q j xn ) = 0. n→∞ n→∞ n→∞ n→∞ Proof. From Theorem 3.1, it follows that limn→∞ d(xn , p) exists for each p ∈ F. Assume that lim d(xn , p) = c > 0. n→∞ (3.3) Since μn → 0 and ξn → 0 as n → ∞, so taking lim sup on both sides of (3.1), we have lim sup d(yn , p) ≤ c. (3.4) lim inf d(yn , p) ≥ c. (3.5) n→∞ Also taking lim inf in (3.2), we get n→∞ I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 New Iterative Class for Finding Common Fixed Points of a Finite Family 209 From (3.4) and (3.5), we get lim d(yn , p) = c. (3.6) n→∞ From (3.6), we get lim d W Trn xn , W xn , Q rn u n , n→∞ δr n , βr n , p = c. 1 − βr n (3.7) Now, d W xn , Q rn u n , δr n ,p 1 − βr n ≤ ≤ ≤ ≤ ≤ ≤ δr n d(xn , p) + 1 − βr n δr n d(xn , p) + 1 − βr n δr n d(xn , p) + 1 − βr n δr n d(xn , p) + 1 − βr n δr n d(xn , p) + 1 − βr n d(xn , p), 1− 1− 1− 1− 1− δr n 1 − βr n δr n 1 − βr n δr n 1 − βr n δr n 1 − βr n δr n 1 − βr n d(Q rn u n , p) knr d(u n , p) kn dist(u n , PT̂ ( p)) kn H (PT̂ (xn ), PT̂ ( p)) kn d(xn , p) which implies that lim sup d W xn , Q rn u n , n→∞ δr n , p ≤ c. 1 − βr n (3.8) Further, lim sup d(Trn xn , p) ≤ c. (3.9) n→∞ From (3.7)-(3.9) and Lemma 2.2, we have lim d Trn xn , W xn , Q rn u n , n→∞ δr n 1 − βr n = 0. (3.10) Next, lim d(xn+1 , p) = lim d W Srn yn , W u n , Rrn vn n→∞ n→∞ I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS θr n , αr n , p = c. 1 − αr n (3.11) Vol. 10, No. 1 (Special Issue), Jan–June 2019 210 Shamshad Husain and Nisha Singh Next from (3.1), we have d W u n , Rrn vn , θr n θr n θr n ,p ≤ d(u n , p) + 1 − d(Rrn vn , p) 1 − αr n 1 − αr n 1 − αr n θr n θr n d(u n , p) + 1 − k̂n d(vn , p) ≤ 1 − αr n 1 − αr n θr n θr n dist(u n , PT̂ ( p)) + 1 − kn dist(vn , PT̂ ( p)) ≤ 1 − αr n 1 − αr n θr n θr n H (PT̂ (xn ),PT̂ ( p))+ 1− kn H (PT̂ (yn ),PT̂ ( p)) ≤ 1−αr n 1−αr n θr n θr n ≤ d(xn , p) + 1 − kn d(yn , p) 1 − αr n 1 − αr n θr n θr n d(xn , p) + 1 − kn ≤ 1 − αr n 1 − αr n [(1 + μn M ∗ )kn d(xn , p) + ξn ] ≤ (1 + μn M ∗ )kn2 d(xn , p) + ξn , which implies that lim sup d W u n , Rrn vn , n→∞ θr n , p ≤ c. 1 − αr n (3.12) Also, lim sup d(Srn yn , p) ≤ c. (3.13) n→∞ From (3.11)- (3.13) and Lemma 2.2, we have lim d n→∞ Srn yn , W u n , Rrn vn , θr n 1 − αr n = 0. (3.14) Further, d(xn+1 , Srn yn ) θr n n , αr n , Sr yn 1 − αr n θr n n n n n , Sr yn , αr n d(Sr yn , Sr yn ) + (1 − αr n )d W u n , Rr vn , 1 − αr n n = d W Sr yn , W u n , Rrn vn , ≤ and using (3.14), we get lim d(xn+1 , Srn yn ) = 0. n→∞ I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS (3.15) Vol. 10, No. 1 (Special Issue), Jan–June 2019 New Iterative Class for Finding Common Fixed Points of a Finite Family 211 Next, we have d(xn+1 , p) ≤ ≤ ≤ θr n ,p 1 − αr n θr n n n ,p αr n d(xn+1 , p) + αr n d(xn+1 , Sr yn ) + (1 − αr n )d W u n , Rr vn , 1 − αr n αr n θr n ,p . (3.16) d(xn+1 , Srn yn ) + d W u n , Rrn vn , 1 − αr n 1 − αr n αr n d(Srn yn , p) + (1 − αr n )d W u n , Rrn vn , Taking lim inf on both sides of (3.16) and using (3.15), we have lim inf d W u n , Rrn vn , n→∞ θr n , p ≥ c. 1 − αr n (3.17) From (3.12) and (3.17), we get d W u n , Rrn vn , θr n , p = c. 1 − αr n (3.18) Using Lemma 2.2 and (3.18), we get lim d(u n , Rrn vn ) = 0. n→∞ (3.19) Further, d(yn , Trn xn ) = ≤ δr n n , βr n , Tr xn 1 − βr n δr n n n n n , Tr xn . βr n d(Tr xn , Tr xn ) + (1 − βr n )d W xn , Q r u n , 1 − βr n n d W Tr xn , W xn , Q rn u n , Using (3.10), we get lim d(yn , Trn xn ) = 0. n→∞ (3.20) Next, we have d(yn , p) ≤ ≤ ≤ δr n n ,p p) + (1 − βr n )d W xn , Q r u n , 1 − βr n δr n n n ,p βr n d(yn , p) + βr n d(Tr xn , yn ) + (1 − βr n )d W xn , Q r u n , 1 − βr n βr n δr n ,p . (3.21) d(Trn xn , yn ) + d W xn , Q rn u n , 1 − βr n 1 − βr n βr n d(Trn xn , I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 212 Shamshad Husain and Nisha Singh Taking lim inf on both sides of (3.21) and using (3.20), we get lim inf d W xn , Q rn u n , n→∞ δr n , p ≥ c. 1 − βr n (3.22) From (3.8) and (3.22), we get lim d W xn , Q rn u n , n→∞ δr n , p = c. 1 − βr n (3.23) Using Lemma 2.2 and (3.23), we get lim d(xn , Q rn u n ) = 0. n→∞ (3.24) Further, d(xn , Trn xn ) ≤ d(xn , yn ) + d(yn , Trn xn ) ≤ ≤ ≤ ≤ δr n n , xn + d(yn , Trn xn ) + (1 − βr n )d W xn , Q r u n , 1 − βr n 1 δr n n , xn + d(yn , Trn xn ) d W xn , Q r u n , 1 − βr n 1 − βr n δr n 1 δr n n d(xn , xn ) + 1 − d(Q r u n , xn ) + d(yn , Trn xn ) 1 − βr n 1 − βr n 1 − βr n 1 δr n n d(Q r u n , xn ) + d(yn , Trn xn ). 1− 1 − βr n 1 − βr n βr n d(xn , Trn xn ) From (3.24) and (3.20), we have lim d(xn , Trn xn ) = 0. n→∞ (3.25) Moreover, d(xn , yn ) ≤ ≤ δr n , xn 1 − βr n βr n d(xn , Trn xn ) + (1 − βr n − δr n )d(xn , Q rn u n ). βr n d(xn , Trn xn ) + (1 − βr n )d W xn , Q rn u n , From (3.24) and (3.25), we have lim d(xn , yn ) = 0. n→∞ I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS (3.26) Vol. 10, No. 1 (Special Issue), Jan–June 2019 New Iterative Class for Finding Common Fixed Points of a Finite Family 213 Let L = max j∈J {L j , L̂ j }, where L j and L̂ j are Lipschitz constants for T j and S j for j ∈ J , respectively. Since each T j is uniformly L-Lipschitzian for j ∈ J , we have d(xn , T jn xn ) ≤ d(xn , yn ) + d(yn , T jn xn ) ≤ d(xn , yn ) + d(yn , T jn yn ) + d(T jn yn , T jn xn ) ≤ d(xn , yn ) + d(yn , T jn yn ) + Ld(yn , xn ) ≤ (1 + L)d(xn , yn ) + d(yn , T jn yn ). (3.27) Further d(yn , T jn yn ) ≤ d(yn , T jn xn ) + d(T jn xn , T jn yn ) ≤ d(yn , T jn xn ) + Ld(xn , yn ). From (3.20) and (3.28) we get lim d(yn , T jn yn ) = 0. n→∞ (3.28) Using (3.26) and (3.28) in (3.27), we get lim d(xn , T jn xn ) = 0 ∀ j ∈ J. n→∞ (3.29) Further, d(xn , T j xn ) ≤ d(xn , T jn xn ) + d(T jn xn , T jn yn ) + d(T jn yn , T j xn ) ≤ d(xn , T jn xn ) + Ld(xn , yn ) + Ld(T jn−1 yn , xn ) ≤ d(xn , T jn xn ) + 2Ld(xn , yn ) + Ld(T jn−1 yn , yn ). From (3.26), (3.28) and (3.29), we get lim d(xn , T j xn ) = 0, ∀ j ∈ J. n→∞ Similarly, we show that lim d(xn , S j xn ) = 0, ∀ j ∈ J. n→∞ Next, we show that lim d(xn , Q j xn ) = 0. n→∞ I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 214 Shamshad Husain and Nisha Singh Now, we have d(xn , Q rn xn ) ≤ d(xn , Q rn u n ) + d(Q rn u n , Q rn xn ) ≤ d(xn , Q rn u n ) + knr d(u n , xn ) ≤ d(xn , Q rn u n ) + kn dist(u n , PT̂ (xn )) ≤ d(xn , Q rn u n ) + kn H (PT̂ (xn ), PT̂ (xn )) ≤ d(xn , Q rn u n ) + kn d(xn , xn ). So from (3.24), we have lim d(xn , Q rn xn ) = 0. n→∞ (3.30) Further, d(xn , Q j xn ) ≤ ≤ d(xn , Q nj xn ) + d(Q nj xn , Q j xn ) d(xn , Q nj xn ) + kn d(Q n−1 j x n , x n ). From (3.35), we get lim d(xn , Q j xn ) = 0. n→∞ Similarly, we show that lim d(xn , R j xn ) = 0. n→∞ Further, we have d(u n , p) = dist(u n , PT̂ ( p)) ≤ H (PT̂ (xn ), PT̂ ( p)) ≤ d(xn , p). (3.31) Taking the lim sup of both sides of (3.31), we have lim sup d(u n , p) ≤ c. (3.32) lim sup d(vn , p) ≤ c. (3.33) n→∞ Similarly, we have n→∞ Since limn→∞ d(xn+1 , p) = c, so from (3.32) and (3.33) and using Lemma 2.2, we have lim d(u n , vn ) = 0. (3.34) lim d(u n , xn ) = 0. (3.35) n→∞ By using (3.4) and Lemma 2.2, we have n→∞ I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 New Iterative Class for Finding Common Fixed Points of a Finite Family 215 From d(xn , T̂ xn ) ≤ d(xn , PT̂ (xn )) ≤ d(xn , u n ). (3.36) So, we have lim d(xn , T̂ xn ) = 0. (3.37) n→∞ Corollary 3.3. Let K be a nonempty closed convex subset of a uniformly convex hyperbolic space X with monotone modulus of uniform convexity η. For j ∈ J , let T j : K → K be a uniformly j L j -Lipschitzian and {ξn }-asymptotically nonexpansive mapping in the intermediate sense and let j S j : K → K be a uniformly L̂ j -Lipschitzian and {ξ̂n }-asymptotically nonexpansive mapping in the ∞ j ∞ j intermediate sense. If n=1 ξn < +∞ and n=1 ξ̂n < +∞ and Q j , R j : K → K be an asymptotically nonexpansive mapping and F = ∅. Then for j ∈ J , the sequence {xn } generated by (1.1), we have lim d(xn , T j xn ) = lim d(xn , S j xn ) = lim d(xn , R j xn ) = lim d(xn , Q j xn ) = 0. n→∞ n→∞ n→∞ n→∞ Corollary 3.4. Let K be a nonempty closed convex subset of a uniformly convex hyperbolic space X with monotone modulus of uniform convexity η. For j ∈ J , let T j : K → K be a uniformly L j j j Lipschitzian and {ln }-asymptotically nonexpansive mapping with ∞ Sj : K → n=1 l n < +∞ and let j j K be a uniformly L̂ j -Lipschitzian and {l̂n }-asymptotically nonexpansive mapping with ∞ n=1 l̂ n < +∞ and Q j , R j : K → K be an asymptotically nonexpansive mapping and F = ∅. Then for j ∈ J , the sequence {xn } generated by (1.1), we have lim d(xn , T j xn ) = lim d(xn , S j xn ) = lim d(xn , R j xn ) = lim d(xn , Q j xn ) = 0. n→∞ n→∞ n→∞ n→∞ 4. STRONG CONVERGENCE Now we establish -convergence and strong convergence of the iteration process (1.1). Theorem 4.1. Let K be a nonempty closed convex subset of a uniformly convex hyperbolic space X with monotone modulus of uniform convexity η. For j ∈ J , let T j : K → K be a uniformly j j L j -Lipschitzian and ({μn }, {ξn }, ρ j )-total asymptotically nonexpansive mapping and S j : K → K j j be a uniformly L̂ j -Lipschitzian and ({μ̂n }, {ξ̂n }, ρ̂ j )-total asymptotically nonexpansive mapping. Let Q j , R j : K → K be an asymptotically nonexpansive mapping and let T̂ : K → P(K ) be a multivalued mapping. Assume that F = ∅ and the conditions (i) and (ii) in Theorem 3.1 hold. Then for j ∈ J , the sequence {xn } generated by (1.1), -converges to a common fixed point of F. Proof. Since the sequence {xn } is bounded, therefore from Lemma 2.1, {xn } has a unique asymptotic center in K , i.e., A({xn }) = {x}. Let {vn } be any subsequence of {xn } such that A({vn }) = {v}. Then I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 216 Shamshad Husain and Nisha Singh from Theorem 3.2 ∀ j ∈ J , we have lim d(vn , T j vn ) = lim d(vn , S j vn ) = lim d(vn , R j vn ) = lim d(vn , Q j vn ) = 0. n→∞ n→∞ n→∞ n→∞ (4.1) Now, we claim that v is the common fixed point of {T j }rj=1 ; {S j }rj=1 ; {Q j }rj=1 and {R j }rj=1 . For each j ∈ J , define a sequence {z m } in K by z m = T jm v. Then, d(z m , vn ) ≤ ≤ d(T jm v, T jm vn ) + d(T jm vn , T jm−1 vn ) + . . . + d(T j vn , vn ) [d(v, vn ) + μmj ρ j (d(v, vn )) + ξmj ] + m−1 d(T ji+1 vn , T ji vn ). j=0 Since each T j is uniformly L j -Lipschitzian for j ∈ J , so we have d(z m , vn ) ≤ [(1 + μm M ∗ )d(v, vn ) + ξm ] + m Ld(T j vn , vn ), (4.2) where L = max j∈J {L j , L̂ j }. Taking lim sup on both sides of (4.2) and using (4.1), we have r (z m , {vn }) = lim sup d(z m , vn ) ≤ lim sup d(v, vn ) = r (v, {vn }), n→∞ n→∞ which implies that |r (z m , {vn }) − r (v, {vn })| → 0 as m → ∞. From Lemma 2.3, it follows that limm→∞ T jm v = v, by the uniform continuity of T j , we have T j (v) = T ( lim T jm v) = lim T jm+1 v = v. m→∞ m→∞ From the arbitrariness of j ∈ J , we have v is the common fixed point of {T j }rj=1 . Similarly, we show that v is the common fixed point of {S j }rj=1 ; {Q j }rj=1 and {R j }rj=1 . Hence v ∈ F. Next, we claim that v is the unique asymptotic center for each subsequence {vn } of {xn }. On the contrary, let v = x. From Theorem 3.1, limn→∞ d(xn , v) exists and by the uniqueness of asymptotic centers, we have lim sup d(vn , v) n→∞ < < lim sup d(vn , x) < lim sup d(xn , x) n→∞ n→∞ lim sup d(xn , v) = lim sup d(vn , v), n→∞ n→∞ which is a contradiction. Therefore, v = x. Since {vn } is an arbitrary subsequence of {xn }, A({vn }) = {x} for all subsequence {vn } of {xn }, so {xn }-converges to a common fixed point x of {T j }rj=1 ; {S j }rj=1 ; {Q j }rj=1 and {R j }rj=1 . Corollary 4.1. Let K be a nonempty closed convex subset of a uniformly convex hyperbolic space X with monotone modulus of uniform convexity η. For j ∈ J , let T j : K → K be a unij formly L j -Lipschitzian and {ξn }-asymptotically nonexpansive mapping in the intermediate sense j and S j : K → K be a uniformly L̂ j -Lipschitzian and {ξ̂n }-asymptotically nonexpansive mapping I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 New Iterative Class for Finding Common Fixed Points of a Finite Family 217 ∞ j j in the intermediate sense. If ∞ n=1 ξn < +∞ and n=1 ξ̂n < +∞ and if Q j , R j : K → K be an asymptotically nonexpansive mapping and F = ∅. Then for j ∈ J , the sequence {xn } generated by (1.1), -converges to a common fixed point of F. Corollary 4.2. Let K be a nonempty closed convex subset of a uniformly convex hyperbolic space X with monotone modulus of uniform convexity η. For j ∈ J , let T j : K → K be a uniformly L j j j Lipschitzian and {ln }-asymptotically nonexpansive mapping with ∞ :K →K n=1 l n < +∞ and S j j j be a uniformly L̂ j -Lipschitzian and {l̂n }-asymptotically nonexpansive mapping with ∞ n=1 l̂ n < +∞. If Q j , R j : K → K be an asymptotically nonexpansive mapping and F = ∅. Then for j ∈ J , the sequence {xn } generated by (1.1), -converges to a common fixed point of F. In order to prove strong convergence of the iteration (1.1) in the hyperbolic space we define the following condition: () There exists a nondecreasing self mapping on [0, +∞) with f (0) = 0 and f (t) > 0 for all t ∈ (0, +∞) such that d(x, T x) ≥ f (d(x, F(T ))) for all x ∈ K , where T : K → K is a nonlinear mapping with F(T ) = ∅ and d(x, F(T )) = inf{d(x, y) : y ∈ F(T )}. Lemma 4.1. Let K , X, {T j }rj=1 , {S j }rj=1 , {Q j }rj=1 and {R j }rj=1 be as in Theorem 3.1. Then {xn } converges strongly to some p ∈ F if and only if lim inf d(xn , F) = 0. n→∞ Proof. If {xn } converges strongly to p ∈ F, then limn→∞ d(xn , p) = 0. Since 0 ≤ d(xn , F) ≤ d(xn , p), we have limn→∞ inf d(xn , F) = 0. Conversely, suppose that limn→∞ inf d(xn , F) = 0. It follows from Theorem 3.1 that limn→∞ d(xn , F) exists. Now limn→∞ inf d(xn , F) = 0 implies that limn→∞ d(xn , F) = 0. Next, we show that {xn } is a Cauchy sequence. From (3.2), we get d(xn+1 , p) ≤ (1 + M1 μn )d(xn , p) + M2 ξn . Taking infimum on p ∈ F on both sides in the above inequality, we have d(xn+1 , F) ≤ (1 + M1 μn )d(xn , F) + M2 ξn . ∞ ∞ M1 ∞ n=1 μn = M. Since lim Since n→∞ d(x n , F) = 0, for any n=1 μn < ∞, n=1 ξn < ∞, set e given > 0, there exists a positive integer n 0 such that d(xn o , F) < ∞ and ξn < . 4(M + 1) 2M M2 n=n o I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 218 Shamshad Husain and Nisha Singh Above inequality implies that there exists po ∈ F such that d(xn o , po ) < n ≥ n o and m ≥ 1, we have d(xn o +m , xn o ) ≤ ≤ Hence, for any d(xn o +m , po ) + d(xn o , po ) M1 [e n o +m−1 μk k=n o +ξn o +m−3 e ≤ . 2(M+1) + 1]d(xn o , po ) + M2 [ξn o +m−1 + ξn o +m−2 e M1 μno +m−1 M1 n o +m−1 μk + . . . + ξn e ∞ (M + 1)d(xn o , po ) + M M2 ξn k=n o +m−2 M1 n o +m−1 k=n o +1 μk ] n=n o + M M2 < (M + 1) = , 2(M + 1) 2M M2 which implies that {xn } is a Cauchy sequence in X . Since K is a closed subset of a complete hyperbolic space X , so it is complete. Now we assume that limn→∞ xn = q and q ∈ K . Since limn→∞ d(xn , F) = 0, we get q ∈ F. Theorem 4.2. Suppose that K , X, {T j }rj=1 , {S j }rj=1 , {Q j }rj=1 and {R j }rj=1 be the same as in Theorem and satisfies (). Then the sequence {xn } defined in (1.1) converges strongly to some p ∈ F. Corollary 4.3. Suppose that K , X, {T j }rj=1 , {S j }rj=1 , {Q j }rj=1 and {R j }rj=1 be the same as in Corollary 3.1 and satisfies (). Then the sequence {xn } defined in (1.1) converges strongly to some p ∈ F. Corollary 4.4. Suppose that K , X, {T j }rj=1 , {S j }rj=1 , {Q j }rj=1 and {R j }rj=1 be the same as in Corollary 3.2 and satisfies (). Then the sequence {xn } defined in (1.1) converges strongly to some p ∈ F. Remarks: In order to prove -convergence and strong convergence of the iteration (1.1) in hyperbolic spaces, we gave and analyzed some important properties related to the general iterative processes (1.1) and proposed some results presented in this paper, which show that our results extend and improve the corresponding results of iterative approximation for asymptotically nonexpansive mapping, multivalued nonexpansive mapping of all normed linear spaces, Hadamard manifolds and CAT(0) spaces as special cases. It is well known that iterative processes are the main tool for approximation of fixed points of generalizations of nonexpansive mappings. Furthermore, the analysis of general iterative processes, in a more general setting, is a problem of interest in theoretical numerical analysis. REFERENCES [1] S. Akbulut and B. Gunduz, Strong and -convergence of a faster iteration process in hyperbolic space, Commun. Korean Math. Soc., 30(3), (2015), 209–219. [2] R. Bruck, T. Kuczumow and S. 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I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 INDIAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019, pp. 220–233 DOI: On a Class of Generalised ( p, q) Bernstein Operators Lakshmi Narayan Mishra1∗ , Shikha Pandey2 and Vishnu Narayan Mishra3 1 Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology (VIT) University, Vellore-632 014, Tamil Nadu, India 2 Department of Applied Mathematics & Humanities, Sardar Vallabhbhai National Institute of Technology, Ichchhanath Mahadev Dumas Road, Surat-395 007 (Gujarat), India 3 Department of Mathematics, Indira Gandhi National Tribal University, Lalpur, Amarkantak-484 887, Madhya Pradesh, India (∗ Corresponding author) Email: ∗ [email protected], 2 [email protected], 3 [email protected] Abstract: In the present paper, we introduce a generalised class of ( p, q) Bernstein operators. The study proves that this generalised operator gives better approximation results comparison to the ( p, q) Bernstein operators defined in [11]. We estimate the rate of approximation using modulus of continuity and prove Voronovskaya type theorem as well. In the end we have proved the statistical and weighted approximation results for this operator. Keywords: ( p, q)-integers; ( p, q)-Bernstein operators; modulus of continuity; linear positive operators. 2010 AMS Subject Classification: Primary 41A25, 41A36. 1. INTRODUCTION AND PRELIMINARIES For a function f (x) defined on the closed interval [0, 1], the expression n k n k n−k Bn ( f ; x) = f x (1 − x) k n k=0 (1.1) is called the Bernstein polynomial of order n of the function f (x) defined by S.N. Bernstein [1] in 1912. Since then various mathematicians have studied different generalizations of Bernstein polynomials (See [13]). Indefinite work has been done in the field of approximation of linear positive operators (See [7, 8, 12]). I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Lakshmi Narayan Mishra, Shikha Pandey and Vishnu Narayan Mishra 221 The study of (p,q) calculus has its origin in the theory of hopf algebra. In hopf algebra we study a structure named as quantum groups which are basicly symmetry group of noncommutative spaces. This is one reason they have been investigated in physics and mathematical physics (noncommutative spaces arise as quantization of commutative ones). The two parameter quantizations (or (p,q)-analogues) of the general linear group GL(n) is the quantum group with the algebraic operations that are resulted into (p,q) calculus. Let’s go through some basic definitions of ( p, q) calculus. For any n ∈ N (1) (p, q)-Integer: [0] p,q := 0 and [n] p,q = (2) pn − q n if n ≥ 1, p−q where 0 < q < p ≤ 1. (p, q)-Factorial: [0] p,q ! := 1 and [n]! p,q = [1] p,q [2] p,q · · · [n] p,q if n ≥ 1. (3) (p, q)-Binomial coefficient : [n] p,q ! n for all n, k ∈ N with n ≥ k. = k p,q [k] p,q ! [n − k] p,q ! (4) (p, q) binomial expansion: (ax + by)np,q := n k=0 n k a n−k bk x n−k y k . p,q (x + y)np,q := (x + y)( px + qy)( p 2 x + q 2 y) . . . ( p n−1 x + q n−1 y). (5) (p, q)-Derivative: Let f : R → R, then the ( p, q)-derivative of function f is defined by ∗ (D p,q f )(x) = f p,q (x) = (6) (7) f ( px) − f (q x) ∗ , x = 0, f p,q (0) := f (0) ( p − q)x provided that f is differentiable at 0. It happens clearly that (x n )∗p,q = [n] p,q x n−1 . The following product rules hold: ( f · h)∗p,q (x) = ∗ f p,q (x) · h(q x) + h ∗p,q (x) · f ( px), ( f · h)∗p,q (x) = ∗ h ∗p,q (x) · f (q x) + f p,q (x) · h( px). The definite integrals of the function f are given by 0 a k ∞ pk p f (x)d p,q x = (q − p)a f a , k+1 k+1 q q k=0 I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS p < 1, q Vol. 10, No. 1 (Special Issue), Jan–June 2019 222 On a Class of Generalised ( p, q) Bernstein Operators and a f (x)d p,q x = ( p − q)a 0 k ∞ qk q f a , k+1 p p k+1 k=0 p > 1. q Further ( p, q) analysis can be found in [2]. Recently, Mursaleen et al. [11] studied Bernstein polynomials based on ( p, q) integers defined as Bn, p,q ( f ; x) = 1 p n(n−1) 2 n−k n n−k−1 k(k−1) p [k] p,q n k s s 2 p x ( p − q x) f , k p,q [n] p,q k=0 s=0 (1.2) where f ∈ C[0, 1], x ∈ [0, 1]. In this paper we study a new generalization of ( p, q) Bernstein polynomials. Let a, b ∈ N. [n+a] p,q , For f ∈ C 0, [n+b] p,q n n−k−1 k(k−1) [n + b] p,q n 1 n k s [n + a] p,q s 2 p p x −q x n(n−1) [n + a] p,q [n + b] p,q p 2 k=0 k p,q s=0 n−k p [k] p,q [n + a] p,q (1.3) f [n] p,q [n + b] p,q a,b Bn, p,q ( f ; x) = For a = b the above operator converts into (1.2), and for p = q = 1, a = b then it converts into [n+a] p,q (1.1). With the help of MAPLE, we can plot the [n+b] p,q with respect to different value of n. So for a > b and a < b, we have I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Lakshmi Narayan Mishra, Shikha Pandey and Vishnu Narayan Mishra 223 Thus for a < b we can increase the upper bound of the approximated function. So, considering a < b throughout the paper in order to increase the interval of approximation. 2. MAIN RESULTS [n+a] p,q , a < b, n ∈ N the following Lemma 1. Assume ei = t i , i = 1, 2, 3, .... For all x ∈ 0, [n+b] p,q equalities holds 1. 2. a,b Bn, p,q (1; x) = 1, a,b Bn, p,q (e; x) = x, 3. a,b 2 2 Bn, p,q (e ; x) = x + Proof. pn−1 x([n+a] p,q −[n+b] p,q x) . [n] p,q [n+b] p,q n n−k−1 k(k−1) [n + b] p,q n 1 n k s [n + a] p,q s 2 p x − q x p n(n−1) [n + a] p,q [n + b] p,q p 2 k=0 k p,q s=0 n n [n + a] p,q [n + b] p,q = −x+x [n + a] p,q [n + b] p,q =1 a,b Bn, p,q (1; x) = a,b Bn, p,q (e; x) = = = [n + b] p,q [n + a] p,q [n + b] p,q [n + a] p,q =x = = = [n + b] p,q [n + a] p,q [n + b] p,q [n + a] p,q x [n] p,q k k=1 k−1 k p [n] p,q p n(n−5) 2 [n] p,q p n−2 p n k=0 k k−1 s=0 k(k−1) 2 n−k−2 x k+1 ps s=0 p k(k−1) 2 p n−k [k] p,q [n + a] p,q [n] p,q [n + b] p,q [n + a] p,q − qs x [n + b] p,q n−k−1 xk ps s=0 p,q p k(k−5) 2 n−1 k=0 k [n + a] p,q − qs x [n + b] p,q n−k−1 [k] p,q x k ps s=0 p,q n−1 p n−1 [n + a] p,q − qs x [n + b] p,q ps p,q k=1 (n−1)(n−2) 2 xk [n + a] p,q − qs x [n + b] p,q n−1 n−1 n 2 −5n+4 2 k(k−3) 2 p n 1 ps s=0 n n(n−1) 2 n−k−1 xk n−k−1 p,q k=0 1 k(k−1) 2 p,q p n−1 n p n−1 1 [n + b] p,q [n + a] p,q k=0 n(n−1) 2 [n + b] p,q +x−x [n + a] p,q [n + b] p,q [n + a] p,q n−2 n n−1 (n−1)(n−2) 2 n−1 n n 1 p 1 p n(n−3) 2 n−1 n−2 n 1 p a,b 2 Bn, p,q (e ; x) = n−1 [n + b] p,q [n + a] p,q =x [n + b] p,q [n + a] p,q k(k−3) 2 n−k−2 x k+1 ( p k + q[k] p,q ) n−1 n−1 k=0 k ps s=0 p,q p p,q k(k−1) 2 n−k−2 xk ps s=0 I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS p n−k [k] p,q [n + a] p,q [n] p,q [n + b] p,q [n + a] p,q − qs x [n + b] p,q p [n + a] p,q − qs x [n + b] p,q [n + a] p,q − qs x [n + b] p,q 2 Vol. 10, No. 1 (Special Issue), Jan–June 2019 224 On a Class of Generalised ( p, q) Bernstein Operators + n−1 n−k−2 k(k−3) q[n − 1] p,q n − 2 k s [n + a] p,q s 2 p p x − q x n 2 −5n+4 k − 1 p,q [n + b] p,q p 2 k=1 = x [n] p,q [n + b] p,q [n + a] p,q s=0 n−2 p n−1 [n + b] p,q +x−x [n + a] p,q n−1 n−2 n−k−3 k(k−1) q[n − 1] p,q n − 2 k+1 s [n + a] p,q s 2 p + (n−2)(n−3) p x −q x k [n + b] p,q p 2 p,q k=0 = p n−1 x [n] p,q = x2 + [n + a] p,q [n + b] p,q s=0 + qx2 [n − 1] p,q [n] p,q p n−1 x([n + a] p,q − [n + b] p,q x) [n] p,q [n + b] p,q [n+a] p,q a,b Theorem 1. limn→∞ Bn, p,q ( f ; x) = f (x), ∀ f ∈ C 0, [n+b] p,q . Proof. From Lemma 1 we have lim n→∞ a,b i i max ⎤ |Bn, p,q (e ; x) − x | = 0, i = 0, 1 [n + a] p,q ⎦ x∈⎣0, [n + b] p,q ⎡ and for i = 2, a,b 2 2 max max ⎤ |Bn, p,q (e ; x) − x | = lim ⎤ ⎡ n→∞ n→∞ [n + a] p,q [n + a] p,q ⎦ ⎦ x∈⎣0, x∈⎣0, [n + b] p,q [n + b] p,q lim ⎡ p n−1 ( p + q − 1) = lim n→∞ [2]2p,q [n] p,q p n−1 x([n + a] p,q − [n + b] p,q x) [n] p,q [n + b] p,q [n + a] p,q [n + b] p,q 2 =0 According to Bohman-Korovkin theorem [9], we obtain the desired result. [n+a] p,q , then we have the following inequality of rate of convergence Theorem 2. If f ∈ C 0, [n+b] p,q [n + a] 1 1 p,q a,b |Bn, ( p + q − 1) ω f ; . p,q ( f ; x) − f (x)| ≤ 1 + 2 [n + b] p,q [n] p,q I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS (2.1) Vol. 10, No. 1 (Special Issue), Jan–June 2019 Lakshmi Narayan Mishra, Shikha Pandey and Vishnu Narayan Mishra 225 Proof. Using the definition of modulus of continuity, for any sequence of positive numbers δn we have, |t − x| ω( f ; δn ) | f (t) − f (x)| ≤ 1 + δn Operating both sides with the linear positive operator (1.3), we get 1 a,b a,b |Bn, p,q ( f ; x) − f (x)| ≤ 1 + Bn, p,q ((t − x); x) ω( f ; δn ) δn Using Cauchy-Schwartz inequality, Lemma 1 we get 1 a,b a,b 2 |Bn, p,q ( f ; x) − f (x)| ≤ 1 + Bn, p,q ((t − x) ; x) ω( f ; δn ) δn 1 p n−1 x([n + a] p,q − [n + b] p,q x) = 1+ ω( f ; δn ) δn [n] p,q [n + b] p,q 1 p n−1 ( p + q − 1) [n + a] p,q 2 ≤ 1+ ω( f ; δn ) δn [2]2p,q [n] p,q [n + b] p,q [n + a] p,q 1 1 ≤ 1+ ( p + q − 1) ω( f ; δn ) δn [2] p,q [n] p,q [n + b] p,q Put δn = √ 1 [n] p,q , then we get the result. Theorem 3. (Voronovskaya Type Theorem) [n+a] p,q [n+a] p,q ∀ f ∈ C 2 0, [n+b] p,q such that f, D p,q ( f ), D2p,q ( f ) ∈ C 0, [n+b] p,q , we have a,b lim [n] p,q (Bn, p,q ( f ; x) − f (x)) = n→∞ x(1 − x) 2 D p,q ( f ). [2] p,q (2.2) [n+a] p,q , Define Proof. Let f ∈ C 2 0, [n+b] p,q f (y)− f (x)−(y−x)D ψ(y, x) = p,q ( f )− [2]1p,q (y−x)2p,q D 2p,q ( f ) (y−x)( py−q x) if y = x, if y = x. 0 [n+a] p,q . Taylor’s theorem states that One can check that ψ(x, x) = 0 and ψ(·, x) ∈ C 0, [n+b] p,q f (y) = f (x) + (y − x)D p,q ( f ) + 1 (y − x)2p,q D 2p,q ( f ) + (y − x)2p,q ψ(y, x), where lim ψ(y, x) = 0. y→x [2] p,q I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 226 On a Class of Generalised ( p, q) Bernstein Operators Thus, a,b [n](Bn, p,q ( f ; x) − f (x)) = + [n] a,b B ((y − x)2p,q ; x)D 2p,q ( f ) [2] n, p,q a,b 2 [n]Bn, (2.3) p,q ((y − x) p,q ψ(y, x); x). a,b [n]Bn, p,q ((y − x); x)D p,q ( f ) + Using Cauchy-Schwartz inequality we can get the last term as 1 1 a,b 2 2 a,b 4 a,b 2 2 2 [n]Bn, p,q ((y − x) p,q ψ(y, x); x) ≤ ([n] Bn, p,q ((y − x) p,q ; x)) (Bn, p,q (ψ (y, x); x)) . [n+a] p,q Let η(y, x) := ψ 2 (y, x), which implies that η(x, x) = 0 and η(·, x) ∈ C 0, [n+b] p,q . From Theorem 1 we get a,b 2 a,b lim Bn, p,q (ψ (y, x); x) = lim Bn, p,q (η(y, x); x) = η(x, x) = 0. n→∞ n→∞ Using these results in (2.3) and taking limit for large values of n, we get [n] a,b B ((y − x)2p,q ; x)D 2p,q ( f ) [2] n, p,q [n] p n−1 x([n + a] p,q − [n + b] p,q x) 2 D p,q ( f ) = lim n→∞ [2] [n] p,q [n + b] p,q x(1 − x) 2 D p,q ( f ) = [2] p,q a,b lim [n](Bn, p,q ( f ; x) − f (x)) = lim n→∞ n→∞ which is the required result. 3. STATISTICAL APPROXIMATION The statistical version of Korovkin theorem for sequence of positive linear operators has been given by Gadjiev and Orhan [5, 6]. Let K be a subset of the set N of natural numbers. Then, the asymptotic density δ(K ) of K is defined as δ(K ) = limn n1 |{k ≤ n : k ∈ K }| and | · | represents the cardinality of the enclosed set. A sequence x = (xk ) said to be statistically convergent to the number L if for each > 0, the set K () = {k ≤ n : |xk − L| > } has asymptotic density zero, i.e., 1 lim |{k ≤ n : k ∈ K }| = 0 n n In this case, we write st − lim x = L. Let us recall the following theorem: Theorem 1 [5]. Let An be the sequence of linear positive operators from C[0, 1] to C[0, 1] satisfies the conditions st − limn An (t ν ; x) − x ν C[0,1] = 0 for ν = 0, 1, 2. then for any function f ∈ C[0, 1], st − lim An ( f ) − f n C[0,1] I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS = 0. Vol. 10, No. 1 (Special Issue), Jan–June 2019 Lakshmi Narayan Mishra, Shikha Pandey and Vishnu Narayan Mishra 227 3.1. Korovkin Type statistical approximation properties Theorem 2. In order to get statistical approximation properties, consider two sequences p = pn and q = qn s.t 0 < qn < pn ≤ 1, and st − limn qn = 1, st − limn pn = 1, and st − limn→∞ pnn = [n+a] p,q A, st − limn→∞ qnn = B, with 0 < A, B ≤ 1 then for any function f ∈ C 0, [n+b] p,q a,b st − lim Bn, pn ,qn ( f, ·) − f = 0. n Proof. Clearly for ν = 0, 1 a,b a,b Bn, p,q (1; x) = 1, Bn, p,q (t; x) = x which implies a,b a,b st − lim Bn, pn ,qn (1; x) − 1 = 0 and st − lim Bn, pn ,qn (t; x) − x = 0 n n For ν = 2 [n + a] p,q [n + b] p,q [n − 1] p,q − x2 [n] p,q p n−1 [n + a] p,q q[n − 1] p,q = x+ − 1 x2 [n] p,q [n + b] p,q [n] p,q a,b 2 2 Bn, ≤ pn ,qn (t ; x) − x q[n−1] p,q [n] p,q − 1 and βn = + qx2 q[n − 1] p,q p n−1 [n + a] p,q x −1 + [n] p,q [n] p,q [n + b] p,q ≤ Choose αn = p n−1 x [n] p,q pn−1 [n+a] p,q . [n] p,q [n+b] p,q Clearly st − lim αn = st − lim βn = 0. n n Given > 0 we can define: a,b 2 2 ≥ 0, U := Bn, pn ,qn (t ; x) − x U1 := {n : αn ≥ } and U2 := {n : βn ≥ }. 2 2 One can easily check U ⊆ U1 ∪ U2 , Hence we can get a,b 2 2 δ{K ≤ n : Bn, ≥ } pn ,qn (t ; x) − x δ{K ≤ n : αn ≥ } + δ{K ≤ n : βn ≥ } 2 2 Thus by Theorem 1 we can say that the following holds true. a,b st − lim Bn, pn ,qn ( f, ·) − f = 0. n I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 228 On a Class of Generalised ( p, q) Bernstein Operators 3.2. Rate of Statistical Convergence In this section, we will obtain the rate of statistical convergence using modulus of continuity and Lipschitz type maximal functions. Theorem 3. Let the sequence p := pn and q := qn satisfy for 0 < qn < pn ≤ 1, so we have a,b |Bn, pn ,qn ( f (t), x) − f (x)| ≤ 2ω( f ; δn (x)), where 1 a,b 2 2 δn (x) = [Bn, pn ,qn ((t − x) ; x)] . (3.1) Proof. Using the definition of modulus of continuity, we have |t − x| | f (t) − f (x)| ≤ 1 + ω( f ; δ) δ Therefore 1 a,b a,b B |Bn, ( f (t); x) − f (x)| ≤ 1 + (|t − x|; x) ω( f ; δ) pn ,qn δ n, pn ,qn Using Cauchy-Schwartz inequality 1 1 1 a,b 2 a,b 2 2 [Bn, f (t); x) − f (x)| ≤ ω( f ; δ(x)) 1 + pn ,qn ((t − x) ; x)] [Bn, pn ,qn (1; x)] δ(x) 1 1 a,b 2 2 [B ≤ ω( f ; δ(x)) 1 + ((t − x) ; x)] δ(x) n, pn ,qn a,b |Bn, pn ,qn ( Choosing δ(x) = δn (x) as in the (3.1), we get the result. As by Theorem 2 st − limn δn = 0 and by the definition we have st − limn ω( f ; δ) = 0. This a,b gives pointwise rate of convergence of the operators Bn, pn ,qn ( f ; x) to f (x). Theorem 4. Let f ∈ Li p M (θ ), M > 0, 0 < θ < 1 is a continuous bounded function on [0, 1] × E then we have a,b θ |Bn, pn ,qn ( f (t); x) − f (x)| ≤ M(δn (x)) . Proof. Since f is a Lipschitz function, we have a,b a,b |Bn, pn ,qn ( f (t); x) − f (x)| ≤ Bn, pn ,qn (| f (t) − f (x)|; x) a,b θ ≤ M Bn, pn ,qn (|t − x| ; x). I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Lakshmi Narayan Mishra, Shikha Pandey and Vishnu Narayan Mishra 229 Applying Hölder inequality we get a,b θ2 a,b 2 |Bn, . pn ,qn ( f (t); x) − f (x)| ≤ M Bn, pn ,qn ((t − x) ; x) Taking δn (x) as in (3.1), we get the result. 4. WEIGHTED APPROXIMATION In this section, with the help of Korovkin type theorem proved by Gadjiev in [3, 4], we give a,b approximation properties of the operators Bn, p,q of the weighted spaces of continuous functions + on R0 = [0, ∞). For this purpose, we consider the following weighted spaces of functions which are defined on the R+ 0 . Let ρ(x) be the weight function and M f be a positive constant, we define the weighted space of functions as 1. + Bρ (R+ 0 ) be the space of functions f defined on R0 satisfying | f (x)| ≤ M f ρ(x). + 2. Cρ (R+ 0 ) be the subspace of all continuous functions in Bρ (R0 ). + 3. Cρ∗ (R+ 0 ) is the subspace of functions f ∈ C ρ (R0 ) for which limx→∞ f (x) ρ(x) is finite. Note that the space Bρ (R+ 0 ) is a normed linear space with the norm f ρ = sup x∈R+ 0 | f (x)| . ρ(x) In order to calculate the rate of convergence consider the weighted modulus of continuity defined as ( f ; δ) = sup x≥0, 0<h≤δ | f (x + h) − f (x)| ρ(x + h)2 f or f ∈ Cρ∗ (R+ 0 ). Lemma 1. The weighted modulus of continuity has following properties as defined in [10], 1. ( f ; δ) is a monotonic increasing function of δ. 2. limδ→0+ ( f ; δ) = 0. 3. For any λ ≥ 0, ( f ; λδ) ≤ (1 + λ) ( f ; δ). We consider sequences ( pn ) and (qn ) for 0 < qn < pn ≤ 1 satisfying lim pn = lim qn = 1, lim pnn = a and lim qnn = c where a, c ∈ (0, 1), n→∞ n→∞ n→∞ n→∞ Moreover st − lim pn = st − lim qn = 1, st − lim pnn = d and st − lim qnn = e n n where 0 < d, e ≤ 1. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS n→∞ n→∞ (4.1) Vol. 10, No. 1 (Special Issue), Jan–June 2019 230 On a Class of Generalised ( p, q) Bernstein Operators Lemma 2. Let ( pn ),(qn ) be two sequences with the property (4.1) and ρ(x) = 1 + x 2 be a weight function. If f ∈ Cρ (R+ 0 ), then a,b Bn, pn ,qn (ρ; x) ρ ≤ 1 + M. Proof. Using Lemma 1, we have a,b Bn, pn ,qn (ρ; x) = a,b a,b 2 Bn, pn ,qn (1; x) + Bn, pn ,qn (t ; x) [n − 1] pn ,qn pnn−1 x [n + a] pn ,qn + qn x 2 . = 1+ [n] pn ,qn [n + b] pn ,qn [n] pn ,qn Now multiplying the both-sides of the above equality by we deduce that and taking the supremum over x ≥ 0, 1 1+x 2 a,b Bn, pn ,qn (ρ; x) ρ 1 pnn−1 x [n + a] pn ,qn 2 [n − 1] pn ,qn 1 + + q = sup x n 2 [n] pn ,qn [n + b] pn ,qn [n] pn ,qn x≥0 1 + x n−1 [n + a] pn ,qn [n − 1] pn ,qn p + qn ≤ 1+ n [n] pn ,qn [n + b] pn ,qn [n] pn ,qn Since limn→∞ 1 [n] pn ,qn = 0 , there exists a positive M such that a,b Bn, pn ,qn (ρ; x) ρ ≤ 1 + M. Thus, the proof is completed. + + a,b It clearly indicates that the operator Bn, pn ,qn act from C ρ (R0 ) to Bρ (R0 ). a,b Theorem 1. Let Bn, pn ,qn be the sequence of positive linear operators defined by (1.3) and ρ(x) = 2 1 + x , then for each f ∈ Cρk (R+ 0) lim n→∞ a,b Bn, pn ,qn ( f ; x) − f (x) ρ = 0. Proof. It is sufficient to prove that the conditions of the weighted Korovkin type theorem [3, 4] are satisfied. From Lemma 1, one obtains n→∞ lim a,b Bn, pn ,qn (1; x) − 1 ρ = 0, lim a,b Bn, pn ,qn (t; x) − x ρ = 0, n→∞ I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Lakshmi Narayan Mishra, Shikha Pandey and Vishnu Narayan Mishra 231 and and in the last a,b 2 2 Bn, pn ,qn (t ; x) − x ρ [n + a] pn ,qn [n − 1] pn ,qn x 2 pnn−1 x x2 lim sup + q − n n→∞ [n] pn ,qn [n + b] pn ,qn 1 + x 2 [n] pn ,qn 1 + x 2 1 + x2 x∈R+ 0 n−1 [n + a] pn ,qn [n − 1] pn ,qn pn ≤ lim + qn −1 . n→∞ [n] p ,q [n + b] [n] pn ,qn pn ,qn n n lim n→∞ = which implies lim n→∞ a,b 2 2 Bn, pn ,qn (t ; x) − x ρ = 0. ρ = 0, Thus, as per weighted Korovkin theorem, we have lim n→∞ a,b i i Bn, pn ,qn (t ; x) − x which is the desired result. 2 Theorem 2. If f ∈ Cρ∗ (R+ 0 ), and ρ(x) = 1 + x then as n → ∞ we have a,b |Bn, pn ,qn ( f ; x) − f (x)| ≤ (1 + x 2 )(a2 + a4 x 2 + x + 1) ( f ; δ) ≤ a(1 + x 2+λ ) ( f ; δ), √ ∗ where λ ≥ 1, δn = μ (n) and a is a positive constant independent of f and n. Proof. By the definition of weighted modulus of continuity and Lemma 1, we get |t − x| ( f ; δ) | f (t) − f (x)| ≤ 1 + (x + |t − x|)2 1 + δ |t − x| ( f ; δ). ≤ 1 + (2x + t)2 1 + δ Operating with (1.3) both sides of inequality and using Cauchy-Schwarz inequality we get a,b |Bn, p,q ( f ; x) − f (x)| a,b 2 a,b 2 |t − x| ≤ Bn, p,q ( f ; x) 1 + (2x + t) ; x + Bn, p,q ( f ; x) 1 + (2x + t) ;x δ 1 a,b a,b 2 2 2;x ≤ Bn, ( f ; x) 1 + (2x + t) ; x + B ( f ; x) 1 + (2x + t) n, p,q p,q δn a,b 2 ( f ; δ). × Bn, p,q ( f ; x) (t − x) ; x I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS ( f ; δ) Vol. 10, No. 1 (Special Issue), Jan–June 2019 232 On a Class of Generalised ( p, q) Bernstein Operators For large values of n, from Lemma 1, we can obtain a,b 2 Bn, p,q ( f ; x)(1 + t ; x) 1 + x2 ≤ 1 + a1 , (4.2) where a1 is a positive constant. From (4.2) we can get a,b 2 2 Bn, p,q ( f ; x)(1 + (2x + t) ; x) ≤ a2 (1 + x ), a2 > 0. In the same way we can get a,b 4 Bn, p,q ( f ; x)(1 + t ; x) 1 + x4 ≤ 1 + a3 , a3 > 0 and hence for large values of n, a,b 2 2 2 Bn, p,q ( f ; x)((1 + (2x + t) ) ; x) ≤ a4 (1 + x ), a4 > 0. Hence, we have 1 a,b 2 ∗ (n) |Bn, ( f ; x) − f (x)| ≤ (1 + x ) a + a (1 + x) μ ( f ; δ). 2 4 pn ,qn δn √ If we choose δn = μ∗ (n), then a,b 2 ∗ |Bn, p,q ( f ; x) − f (x)| ≤ (1 + x )(a2 + a4 (1 + x)) ( f ; μ (n)) ≤ a(1 + x 2+λ ) ( f ; μ∗ (n)), where a := a2 + a4 . 5. GRAPHICAL RESULTS In this section, with the help of MAPLE, we show comparisons and some illustrative graphics for 1 j the convergence of operators (1.2) and (1.3) to the Weierstrass function f (x) = 100 j j=0 5 cos(29 π x) under different choices for the parameters. We have found it to be convenient to investigate our series only for finite sums n = 20. More powerful equipments with higher speed can easily compute the more complicated infinite series in a similar manner. Here, in our computations, we take n = 20 and p = 0.99, q = 0.98, a = 20, b = 1 which gives the interval of approximation as [0, 1.82]. Figure 1. shows that the approximated new modified polynomial follows the nature of the original function and Figure 2. shows the comparison of the errors that occur for classical Bernstein, ( p, q)-Bernstein and Modified ( p, q)- Bernstein after the interval [0, 1]. Conflict of Interest: The authors declare that they have no competing interests regarding the publication of this manuscript. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Lakshmi Narayan Mishra, Shikha Pandey and Vishnu Narayan Mishra Figure 1. Plot 233 Figure 2. Comparison Plot REFERENCES [1] S.N. Bernstein, Démostration du théoréme de Weierstrass fondée sur le calcul de probabilités, Comm. Soc. Math. Kharkow, (2), 13 (1912/1913), pp. 1–2. [2] I. M. Burban, A. U. Klimyk, ( p, q)-differentiation, ( p, q)-integration, and ( p, q)-hypergeometric functions related to quantum groups, Integral Transforms and Special Functions, 1994, Vol. 2, No. 1, pp. 15–36. [3] A.D. Gadjiev, The convergence problem for a sequence of positive linear operators on bounded sets and theorems analogous to that of P.P. Korovkin, Dokl. Akad. Nauk SSSR 218 (5) (1974). Transl. in Soviet Math. Dokl. 15 (5) (1974) 1433–1436. [4] A.D. Gadjiev, On P. P. Korovkin type theorems, Mat. Zametki 20 (1976) 781786. Transl. in Math. Notes (5-6) (1978) 995–998. [5] A. D. Gadjiv, C. Orhan, Some approximzation theorems via statistical convergence, Rocky Mount. J. Math., 32 (2002) 129–138. [6] O. Duman, C. Orhan (2004), Statistical approximation by positive linear operators. Studia Math, 161(2), 187–197. [7] G. Içöz, R.N. Mohapatra, Approximation properties by q-Durrmeyer-Stancu operators. Anal. Theory Appl., 29 (2013), no. 4, 373–383. [8] G. Içöz, R.N. Mohapatra, Weighted approximation properties of Stancu type modification of q- Szász-Durrmeyer operators, Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat., Volume 65, Number 1, (2016), pp. 87–103. [9] P.P. Korovkin, On Convergence of Linear Positive Operators in the Space of Continuous Functions, Dokl. Akad. Nauk, 90(1953) 961–964. [10] AJ, López-Moreno, Weighted simultaneous approximation with Baskakov type operators, Acta Math. Hung., 104 (1-2), 143–151 (2004). [11] M. Mursaleen, K.J. Ansari, A. Khan, On ( p, q)-analogue of Bernstein operators, Appl. Math. Comput., 266 (2015), pp. 874–882 [Erratum: Appl. Math. Comput., 278 (2016) 70–71]. [12] R.N. Mohapatra, Z. Walczak, Remarks on a class of Szász-Mirakyan type operators, East J. Approx., 15 (2) (2009) 197–206. [13] M.A. Siddique, R.R. Agarwal and N. Gupta, On a Class of New Bernstein Operators, Advanced Studies in Contemporary Mathematics, 2014. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 INDIAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019, pp. 234–246 DOI: Dhage Iteration Method for Approximating Solutions of IVPs of Nonlinear Second Order Hybrid Neutral Functional Differential Equations Bapurao C. Dhage Kasubai, Gurukul Colony, Ahmepur-413515, Dist. Latur, Maharashtra, India E-mail: [email protected] Abstract: In this paper we prove the existence and approximation result for a nonlinear initial value problem of second order hybrid functional differential equations of neutral type via construction of an algorithm. The main results rely on the Dhage iteration method embodied in a recent hybrid fixed point principle of Dhage [7]. An example is also furnished to illustrate the hypotheses and the abstract result of this paper. 2010 MSC: 34A12, 34A45, 47H07, 47H10. Keywords: Second order neutral functional differential equation; Hybrid fixed point principle; Dhage iteration method; Existence and Approximation theorem. 1. STATEMENT OF THE PROBLEM Given the real numbers r > 0 and T > 0, consider the closed and bounded intervals I0 = [−r, 0] and I = [0, T ] in R and let J = [−r, T ]. By C = C(I0 , R) we denote the space of continuous realvalued functions defined on I0 . We equip the space C with he norm · C defined by xC = sup |x(θ )|. (1.1) −r ≤θ≤0 Clearly, C is a Banach space with this supremum norm and it is called the history space of the functional differential equation in question. For any continuous function x : J → R and for any t ∈ I , we denote by xt the element of the space C defined by xt (θ ) = x(t + θ ), −r ≤ θ ≤ 0. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS (1.2) Vol. 10, No. 1 (Special Issue), Jan–June 2019 Dhage Iteration Method for Nonlinear Differential Equations 235 The differential equations involving the history of the dynamic systems are called functional differential equations and the differential equations involving the derivative of history function are called neutral functional differential equations. It has been recognized long back the importance of such problems in the theory of differential equations. Since then, several classes of nonlinear functional differential equations of neutral type have been discussed in the literature for different qualitative properties of the solutions (see Ntouyas [21] and the references therein). Recently, the study of a special class of functional differential equations involving maxima is initiated by Dhage [7], Dhage and Otrocol [16] and Dhage and Dhage [12] via a new Dhage iteration method and established the existence and approximation results along with algorithm of the solutions. Therefore, it is desirable to extend this new method to other classes of functional differential equations involving delay in the arguments. Very recently, the present author in [8] applied this new iteration method to IVPs of nonlinear first order neutral functional differential equations involving a delay. The present paper is also an attempt in this direction and extends the Dhage iteration method to ordinary second order functional differential equations of neutral type. In this paper, we consider the nonlinear second order hybrid functional differential equations (in short HFDE) of neutral type ⎫ d x (t) − f (t, x t ) = g(t, xt ), t ∈ I,⎬ dt (1.3) ⎭ x0 = φ, x (0) = η, where φ ∈ C and f, g : I × C → R are continuous functions. Definition 1.1. A function x ∈ C(J, R) is said to be a solution of the HFDE (1.3) if (i) x0 ∈ C, (ii) xt ∈ C for each t ∈ I , and (iii) the function t → [x (t) − f (t, x t )] is continuously differentiable on I and satisfies the equations in (1.3), where C(J, R) is the space of continuous real-valued functions defined on J . The neutral HFDE (1.3) is well-known and is a linear perturbation of second type of functional differential equations. See Dhage [3, 4] and the references therein) and can be handled with the hybrid operator theoretic technique involving the sum of two operators in a Banach space (see Dhage [4] and the references therein). It has been discussed in Ntouyas et al. [21] with usual known method of Lerray-Schauder fixed point principle and established the existence theorem. The special cases of it are well-known and extensively discussed in the literature for different aspects of the solutions (sSee Hale [19], Dhage [8, 9] and the references therein). There is a vast literature on nonlinear functional differential equations of neutral type for different aspects of the solutions via different approaches and methods. The method of upper and lower solution or monotone method is interesting and well-known, however it requires the existence of both the lower as well as upper solutions as well as certain inequality involving monotonicity of the nonlinearity. In this paper we prove the existence and approximation theorem for the hybrid functional differential equations I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 236 B. C. Dhage neutral type (1.3) via a new Dhage iteration method which does not require the existence of both upper and lower solution as well the related monotonic inequality and also obtain the algorithm for the solutions under some natural conditions. The rest of the paper is organized as follows. Section 2 deals with the preliminary definitions and auxiliary results that will be used in subsequent sections of the paper. The main result and an illustrative example are given in Section 3. 2. AUXILIARY RESULTS Throughout this paper, unless otherwise mentioned, let (E, , · ) denote a partially ordered normed linear space. Two elements x and y in E are said to be comparable if either the relation x y or y x holds. A non-empty subset C of E is called a chain or totally ordered if all the elements of C are comparable. It is known that E is regular if {xn } is a nondecreasing (resp. nonincreasing) sequence in E and xn → x ∗ as n → ∞, then xn x ∗ (resp. xn x ∗ ) for all n ∈ N. The conditions guaranteeing the regularity of E may be found in Guo and Lakshmikantham [18] and the references therein. Similarly a few details of a partially ordered normed linear space is given in Dhage [4] while orderings defined by different order cones are given in Deimling [1], Guo and Lakshmikantham [18], Heikkilaá and Lakshmikantham [20] and the references therein. We need the following definitions (see Dhage [3,4] and the references therein) in what follows. A mapping T : E → E is called isotone or nondecreasing if it preserves the order relation , that is, if x y implies T x T y for all x, y ∈ E. Similarly, T is called nonincreasing if x y implies T x T y for all x, y ∈ E. Finally, T is called monotonic or simply monotone if it is either nondecreasing or nonincreasing on E. A mapping T : E → E is called partially continuous at a point a ∈ E if for > 0 there exists a δ > 0 such that T x − T a < whenever x is comparable to a and x − a < δ. T called partially continuous on E if it is partially continuous at every point of it. It is clear that if T is partially continuous on E, then it is continuous on every chain C contained in E and vice-versa. A non-empty subset S of the partially ordered Banach space E is called partially bounded if every chain C in S is bounded. An operator T on a partially normed linear space E into itself is called partially bounded if T (E) is a partially bounded subset of E. T is called uniformly partially bounded if all chains C in T (E) are bounded by a unique constant. A non-empty subset S of the partially ordered Banach space E is called partially compact if every chain C in S is a compact subset of E. A mapping T : E → E is called partially compact if T (E) is a partially relatively compact subset of E. T is called uniformly partially compact if T is a uniformly partially bounded and partially compact operator on E. T is called partially totally bounded if for any bounded subset S of E, T (S) is a partially relatively compact subset of E. If T is partially continuous and partially totally bounded, then it is called partially completely continuous on E. Remark 2.1. Suppose that T is a nondecreasing operator on E into itself. Then T is a partially bounded or partially compact if T (C) is a bounded or relatively compact subset of E for each chain C in E. Definition 2.1 Dhage [4]. The order relation and the metric d on a non-empty set E are said to be D-compatible if {xn } is a monotone sequence, that is, monotone nondecreasing or monotone I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Dhage Iteration Method for Nonlinear Differential Equations 237 nonincreasing sequence in E and if a subsequence {xn k } of {xn } converges to x ∗ implies that the original sequence {xn } converges to x ∗ . Similarly, given a partially ordered normed linear space (E, , · ), the order relation and the norm · are said to be D-compatible if and the metric d defined through the norm · are D-compatible. A subset S of E is called Janhavi if the order relation and the metric d or the norm · are D-compatible in it. In particular, if S = E, then E is called a Janhavi metric or Janhavi Banach space. Definition 2.2 Dhage [4]. An upper semi-continuous and monotone nondecreasing function ψ : R+ → R+ is called a D-function provided ψ(0) = 0. An operator T : E → E is called partial nonlinear D-contraction if there exists a D-function ψ such that T x − T y ≤ ψ x − y (2.1) for all comparable elements x, y ∈ E, where 0 < ψ(r ) < r for r > 0. In particular, if ψ(r ) = k r , k > 0, T is called a partial Lipschitz operator with a Lipschitz constant k and moreover, if 0 < k < 1, T is called a partial linear contraction on E with a contraction constant k. The Dhage iteration method embodied in the following applicable hybrid fixed point principle of Dhage [7] in a partially ordered normed linear space is used as a key tool for our work contained in this paper. The details of other hybrid fixed point theorems involving the Dhage iteration principle and method are given in Dhage [3–5, 10, 11], Dhage and Dhage [13], Dhage et.al [14, 15], Dhage and Otrocol [16] and the references therein. Theorem 2.1. Let E, , · be a regular partially ordered complete normed linear and let every compact chain C of E be Janhavi. Let A, B : E → E be two nondecreasing operators such that (a) A is a partially bounded and partial nonlinear D-contraction, (b) B is partially continuous and partially compact, and (c) there exists an element α0 ∈ E such that α0 Aα0 + Bα0 or α0 Aα0 + Bα0 . Then the operator equation Ax + Bx = x has a solution x ∗ and the sequence {xn } of successive iterations defined by x0 = α0 , xn+1 = Axn + Bxn , n = 0, 1, . . . ; converges monotonically to x ∗ . Remark 2.2. The condition that every compact chain of E is Janhavi holds if every partially compact subset of E possesses the compatibility property with respect to the order relation and the norm · in it. This simple fact is used to prove the main existence results of this paper. Remark 2.3. The regularity of E in above Theorem 2.1 may be replaced with a stronger continuity condition of the operator A and B on E which is a result proved in Dhage [3]. 3. MAIN RESULTS In this section, we prove an existence and approximation result for the neutral HFDE (1.3) on a closed and bounded interval J = [a, b] under mixed partial Lipschitz and partial compactness type I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 238 B. C. Dhage conditions on the nonlinearities involved in it. We place the neutral HFDE (1.3) in the function space C(J, R) of continuous real-valued functions defined on J . We define a norm · and the order relation in C(J, R) by x = sup |x(t)| (3.1) t∈J and x y ⇐⇒ x(t) ≤ y(t) for all t ∈ J. (3.2) Clearly, C(J, R) is a Banach space with respect to above supremum norm and also partially ordered with respect to the above partially order relation . It is known that the partially ordered Banach space C(J, R) is regular and lattice so that every pair of elements of E has a lower and an upper bound in it (see Dhage [3,4] and the references therein). The following useful lemma concerning the Janhavi subsets of C(J, R) follows immediately from the Arzelá-Ascoli theorem for compactness (see Dhage [8] and Dhage and Dhage [12]). Lemma 3.1 Dhage [8] and Dhage and Dhage [12]. Let C(J, R), , · be a partially ordered Banach space with the norm · and the order relation defined by (3.1) and (3.2) respectively. Then every partially compact subset of C(J, R) is Janhavi. We introduce an order relation C in C induced by the order relation defined in C(J, R). Thus, for any x, y ∈ C, x C y implies x(θ ) y(θ ) for all θ ∈ I0 . Moreover, if x, y ∈ C(J, R) and x y, then xt C yt for all t ∈ I . We need the following definition in what follows. Definition 3.1. A function u ∈ C(J, R) is said to be a solution of the HFDE (1.3) if (i) u t ∈ C for each t ∈ I , and (ii) the function t → [u (t) − f (t, u t )] is continuously differentiable on I and satisfies ⎫ d u (t) − f (t, u t ) ≤ g(t, u t ), t ∈ I,⎬ dt ⎭ u 0 C φ, , u (0) ≤ η. (∗) Similarly, a continuously differentiable function v ∈ C(J, R) is called an upper solution of the neutral HFDE (1.3) if the above inequality is satisfied with reverse sign. We consider the following set of assumptions in what follows: (H1 ) (H2 ) There exists a constant M f > 0 such that | f (t, x)| ≤ M f for all t ∈ I and x ∈ C; There exists D-function ϕ : R+ → R+ such that 0 ≤ f (t, x) − f (t, y) ≤ ϕ(x − yC ) for all t ∈ I and x, y ∈ C, x C y. Moreover, T ϕ(r ) < r , r > 0. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Dhage Iteration Method for Nonlinear Differential Equations (H3 ) (H4 ) (H5 ) 239 The function g is bounded on I × C with bound Mg . The function g(t, x) is nondecreasing in x for each t ∈ I . The neutral HFDE (1.3) has a lower solution u ∈ C(J, R). Lemma 3.2. A function x ∈ C(J, R) is a solution of the neutral HFDE (1.3) if and only if it is a solution of the nonlinear functional integral equation ⎧ ⎪ φ(0) + η − f (0, φ) t ⎪ ⎪ ⎪ t t ⎨ f (s, xs ) ds + (t − s)g(s, x s ) ds, if t ∈ I, (3.3) x(t) = + 0 0 ⎪ ⎪ ⎪ ⎪ ⎩φ(t), if t ∈ I . 0 Theorem 3.1. Suppose that hypotheses (H1 )-(H2 ) and (H4 ) hold. Then the neutral HFDE (1.3) has a solution x * defined on J and the sequence {xn } of successive approximations defined by x0 = u, ⎧ ⎪ φ(0) + η − f (0, φ) t ⎪ ⎪ ⎪ t t ⎨ n + f (s, x ) ds + (t − s)g(s, x sn ) ds, if t ∈ I, s xn+1 (t) = 0 0 ⎪ ⎪ ⎪ ⎪ ⎩φ(t), if t ∈ I , 0 (3.4) where xsn (θ ) = xn (s + θ ), θ ∈ I0 , converges monotonically to x * . Proof. Set E = C(J, R). Then, in view of Lemma 3.1, every compact chain C in E possesses the compatibility property with respect to the norm · and the order relation so that every compact chain C is Janhavi in E. Define two operators A and B on E by ⎧ t ⎨η − f (0, φ)t + f (s, xs ) ds if t ∈ I, Ax(t) = (3.5) 0 ⎩ 0, if t ∈ I0 , and Bx(t) = ⎧ ⎨φ(0) + ⎩ t (t − s)g(s, x s ) ds, if t ∈ I, 0 (3.6) φ(t), if t ∈ I0 . From the continuity of the functions f , g and the integral, it follows that A and B define the operators A, B : E → E. Applying Lemma 3.2, the neutral HFDE (1.3) is equivalent to the operator equation Ax(t) + Bx(t) = x(t), t ∈ J. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS (3.7) Vol. 10, No. 1 (Special Issue), Jan–June 2019 240 B. C. Dhage Now, we show that the operators A and B satisfy all the conditions of Theorem 2.1 in a series of following steps. Step I: A and B are nondecreasing on E. let x, y ∈ E be such that x y. Then xt C yt for all t ∈ I and by hypothesis (H2 ), we get ⎧ ⎨η − f (0, φ)t + Ax(t) = ⎩ 0, if t ∈ I0 , ⎧ ⎨η − f (0, φ)t + ≥ ⎩ 0, if t ∈ I0 , t f (s, xs ) ds, if t ∈ I, 0 t f (s, ys ) ds, if t ∈ I, 0 = Ay(t), for all t ∈ J . This shows that the operator A is also nondecreasing on E. Similarly, by hypothesis (H4 ), we get Bx(t) = ⎧ ⎨φ(0) + ⎩ t (t − s)g(s, x s ) ds, if t ∈ I, 0 φ(t), if t ∈ I0 , ⎧ t ⎨φ(0) + (t − s)g(s, ys ) ds, if t ∈ I, ≥ 0 ⎩ φ(t), if t ∈ I0 , = By(t), for all t ∈ J . This shows that the operator B is also nondecreasing on E. Step II: A is a nonlinear partial D-contraction on E. Let x, y ∈ E be any two elements such that x y. Then, by hypothesis (H2 ), t |Ax(t) − Ay(t)| ≤ | f (s, xs ) − f (s, ys )| ds 0 ≤ T ϕ(xs − ys C ) ≤ T ϕ(x − y) (3.8) for all t ∈ J . Taking the supremum over t, we obtain Ax − Ay ≤ ψ(x − y) for all x, y ∈ E, x y, where ψ(r ) = T ϕ(r ) < r for r > 0. As a result A is a nonlinear partial D-contraction on E in view of Remark 2.2. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Dhage Iteration Method for Nonlinear Differential Equations 241 Step III: B is partially continuous on E. Let {xn }n∈N be a sequence in a chain C such that xn → x as n → ∞. Then xsn → xs as n → ∞. Since the g is continuous and hypothesis (H3 ) holds, by dominated convergence theorem, we obtain ⎧ t ⎨φ(0) + (t − s) lim g s, xsn ds, if t ∈ I, n→∞ lim Bxn (t) = 0 n→∞ ⎩ φ(t), if t ∈ I0 , ⎧ t ⎨φ(0) + (t − s)g(s, x s ) ds, if t ∈ I, = 0 ⎩ φ(t), if t ∈ I0 , = Bx(t) for all t ∈ J . This shows that Bxn converges to Bx pointwise on J . Now we show that {Bxn }n∈N is an equicontinuous sequence of functions in E. Now there are three cases: Case I: Let t1 , t2 ∈ J with t1 > t2 ≥ 0. Then we have t2 |Bxn (t2 ) − Bxn (t1 )| ≤ (t2 − s)g s, xsn ds − ≤ 0 t2 0 (t2 − s)g s, xsn ds − + ≤ t1 0 t2 t1 t1 0 t1 0 (t2 − s)g s, xsn ds − T g s, xsn ds + T 0 (t1 − s)g s, xsn ds (t2 − s)g s, xsn ds t1 0 (t1 − s)g s, xsn ds |t1 − t2 |g s, xsn ds ≤ 2Mg T |t2 − t1 | →0 as t2 → t1 , uniformly for all n ∈ N. Case II: Let t1 , t2 ∈ J with t1 < t2 ≤ 0. Then we have |Bxn (t2 ) − Bxn (t1 )| = |φ(t2 ) − φ(t1 )| → 0 as t2 → t1 , uniformly for all n ∈ N. Case III: Let t1 , t2 ∈ J with t1 < 0 < t2 . Then we have |Bxn (t2 ) − Bxn (t1 )| ≤ |Bxn (t2 ) − Bxn (0)| + |Bxn (0) − Bxn (t1 )| → 0 as t2 → t1 . uniformly for all n ∈ N. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 242 B. C. Dhage Thus in all three cases, we obtain |Bxn (t2 ) − Bxn (t1 )| → 0 as t2 → t1 , uniformly for all n ∈ N. This shows that the convergence Bxn → Bx is uniform and that B is a partially continuous operator on E into itself. Step IV: B is partially compact operator on E. Let C be an arbitrary chain in E. We show that B(C) is uniformly bounded and equicontinuous set in E. First we show that B(C) is uniformly bounded. Let y ∈ B(C) be any element. Then there is an element x ∈ C such that y = T x. By hypothesis (H2 ), |y(t)| = |Bx(t)| ⎧ t ⎨|φ(0)| + |t − s| |g(s, xs )| ds, if t ∈ I, ≤ 0 ⎩ |φ(t)|, if t ∈ I0 , ≤ φ + M f T 2 = r, for all t ∈ J . Taking the supremum over t we obtain y ≤ Bx ≤ r for all y ∈ B(C). Hence B(C) is a uniformly bounded subset of E. Next we show that B(C) is an equicontinuous set in E. Let t1 , t2 ∈ J , with t1 < t2 . Then proceeding with the arguments that given in Step II it can be shown that y(t2 ) − y(t1 ) = |Bx(t2 ) − Bx(t1 )| → 0 as t1 → t2 uniformly for all y ∈ B(C). This shows that B(C) is an equicontinuous subset of E. Now, B(C) is a uniformly bounded and equicontinuous subset of functions in E and hence it is compact in view of Arzelá-Ascoli theorem. Consequently B : E → E is a partially compact operator on E into itself. Step IV: u satisfies the operator inequality u Au + Bu. By hypothesis (H4 ), the neutral HFDE (1.3) has a lower solution u defined on J . Then we have ⎫ d u (t) − f (t, u t ) ≤ g(t, u t ), t ∈ I,⎬ dt ⎭ u 0 C φ, u (0) ≤ η. Integrating the above inequality from 0 to t, we get ⎧ φ(0) + η − f (0, φ) t ⎪ ⎪ ⎪ t t ⎨ f (s, u s ) ds + (t − s)g(s, u s ) ds, if t ∈ I, u(t) ≤ + ⎪ 0 0 ⎪ ⎪ ⎩ φ(t), if t ∈ I0 , = Au(t) + Bu(t) for all t ∈ J . As a result we have that u Au + Bu. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Dhage Iteration Method for Nonlinear Differential Equations 243 Thus, A and B satisfy all the conditions of Theorem 2.1 and so the operator equation Ax + Bx = x has a solution. Consequently the integral equation (3.3), and a fortiori the hybrid functional differential equation (1.3) has a solution x * defined on J . Furthermore, the sequence {xn }∞ n=0 of successive approximations defined by (3.5) converges monotonically to x * . This completes the proof. Remark 3.1. The conclusion of Theorem 3.1 also remains true if we replace the hypothesis (H4 ) and (H7 ) with the following ones: (H4 )The neutral HFDE (1.3) has an upper solution v ∈ C(J, R). The proof of Theorem 3.1 under this new hypothesis is similar and can be obtained by closely observing the same arguments with appropriate modifications. Example 3.1. Given the closed and bounded intervals I0 = − π2 , 0 and I = [0, 1], consider the second order neutral hybrid functional differential equation, ⎫ d x (t) − f 1 (t, xt ) = g1 (t, xt ), t ∈ I,⎬ dt ⎭ x0 = φ, x (0) = 1, (3.9) where φ ∈ C and f 1 , g1 : I × C → R are continuous functions given by π φ(t) = sin t, t ∈ − , 0 , 2 ⎧ x C ⎨ + 1, if x ≥C 0, x = 0, 1 + xC f 1 (t, x) = ⎩ 1, if x C 0, and g1 (t, x) = tanh(xC ) + 1, 1, if x ≥C 0, x = 0, if x C 0, for all t ∈ I . Clearly, f 1 is continuous and bounded on I × C with bound M f1 = 2. We show that f 1 satisfies the hypothesis (H2 ). Let x, y ∈ C be such that x C y C 0. Then xC ≥ yC > 0 and therefore, we have 0 ≤ f 1 (t, x) − f 1 (t, x) = xC yC − ≤ ϕ(x − yC ) 1 + xC 1 + yC I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 244 B. C. Dhage for all t ∈ I , where ϕ(r ) = obtain r < r , r > 0. Again, if x, y ∈ C be such that x C y C 0, then we 1+r 0 ≤ f 1 (t, x) − f 1 (t, x) ≤ ϕ(x − yC ) for all t ∈ I . This shows that the function f 1 (t, x) satisfies the hypothesis (H2 ). Next, g1 is bounded on I × C with M f1 = 2. Again, let x, y ∈ C be such that x Then xC ≥ yC > 0 and therefore, we have C y C 0. g1 (t, x) = tanh(xC ) + 1 ≥ tanh(yC ) + 1 = g1 (t, y) for all t ∈ I . Again, if x, y ∈ C be such that x C y C 0, then we obtain g1 (t, x) = 1 = g1 (t, y) for all t ∈ I0 . This shows that the function g1 (t, x) is nondecreasing in x for each t ∈ I . Finally, ⎧ t ⎪ ⎨− , if t ∈ [0, 1], 2 u(t) = ⎪ ⎩ sin t, if t ∈ − π2 , 0 , is a lower solution of the neutral HFDE (3.9) defined on J . Thus, f 1 satisfies the hypotheses (H1 ), (H2 ) and (H4 ). Hence we apply Theorem 3.1 and conclude that the neutral HFDE (3.9) has a solution x ∗ on J and the sequence {xn } of successive approximation defined by ⎧ t ⎪ ⎨− , if t ∈ [0, 1], 2 x0 (t) = ⎪ ⎩ sin t, if t ∈ − π2 , 0 ⎧ t ⎪ ⎪ [1 − f (0, φ)]t + f 1 (s, xsn ) ds ⎪ ⎪ ⎪ 0 ⎨ t xn+1 (t) = + (t − s)g1 (s, xsn ) ds, if t ∈ [0, 1], ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎩ sin t, if t ∈ − π2 , 0 , for n = 0, 1, . . . , converges monotonically to x * . Remark 3.2. The conclusion given in Example 3.1 also remains true if we replace the lower solution u with the upper solution v of the neutral HFDE (3.9) defined by ⎧ t ⎪ ⎨2t + 1 , if t ∈ [0, 1], 2 v(t) = ⎪ ⎩ sin t, if t ∈ − π2 , 0 . I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 Dhage Iteration Method for Nonlinear Differential Equations 245 Remark 3.3. We note that if the neutral HFDE (1.3) has a lower solution u as well as an upper solution v such that u v, then under the given conditions of Theorem 3.1 it has corresponding solutions x∗ and x ∗ and these solutions satisfy x∗ x ∗ . Hence they are the minimal and maximal solutions of the neutral HFDE (1.3) respectively in the vector segment [u, v] of the Banach space E = C(J, R), where the vector segment [u, v] is a set in C(J, R) defined by [u, v] = {x ∈ C(J, R) | u x v}. This is because the order relation defined by (3.2) is equivalent to the order relation defined by the order cone K = {x ∈ C(J, R) | x θ } which is a closed set in the Banach space C(J, R). The existence of extremal solutions for the neutral HFDE (1.3) may be obtained in the vector segment [u, v] via generalized iteration method given Heikkilá and Lakshmikatham [20], but in that case we do not get the algorithm for the extremal solutions. Therefore our Dhage iteration method has some advantages over the iteration method presented in Heikkilá and Lakshmikatham [20]. ACKNOWLEDGMENT The author is thankful to the referee for pointing out some misprints/typos in the previous version of this paper. REFERENCES [1] K. Deimling, Nonlinear Functional Analysis, Springer Verlag, 1985. [2] B.C. Dhage, Quadratic perturbations of periodic boundary value problems of second order ordinary differential equations, Differ. Equ. Appl., 2 (2010), 465–486. [3] B. C. Dhage, Hybrid fixed point theory in partially ordered normed linear spaces and applications to fractional integral equations, Differ. Equ. Appl. 5 (2013), 155–184. [4] B.C. Dhage, Partially condensing mappings in partially ordered normed linear spaces and applications to functional integral equations, Tamkang J. Math., 45 (4) (2014), 397–427. [5] B.C. Dhage, A new monotone iteration principle in the theory of nonlinear first order integro-differential equations, Nonlinear Studies, 22 (3) (2015), 397–417. [6] B.C. Dhage, Approximating solutions of nonlinear periodic boundary value problems with maxima, Cogent Mathematics, (2016), 3: 1206699. [7] B.C. Dhage, Two general fixed point principles and applications, J. Nonlinear Anal. Appl., Volume 2016, No. 1 (2016), 23–27. [8] B.C. Dhage, Dhage iteration method for approximating solutions of a IVP of nonlinear first order hybrid functional differential equations of neutral type, Nonlinear Analysis Forum, 22(1)(2017), 59–70. [9] B.C. Dhage, Dhage iteration method for approximating solutions of a IVP of nonlinear first order functional differential equations, Malaya J. Mat., 5(4) (1917), 680–689. [10] B.C. Dhage, Dhage iteration method in the theory of ordinary nonlinear PBVPs of first order functional differential equations, Commun. Optim. Theory, 2017 (2017), Article ID 32, pp. 22. [11] B.C. Dhage, The Dhage iteration method for initial value problems of nonlinear first order hybrid functional integrodifferential equations, In “Recent Advances in Fixed Point Theory and Applications” edited by U.C. Gairola and R. Pant, Nova Publisher, (2017), pp. 177–186. [12] B.C. Dhage, S.B. Dhage, Approximating solutions of nonlinear first order ordinary differential equations, GJMS Special Issue for Recent Advances in Mathematical Sciences and Applications-13, GJMS Vol. 2 (2) (2013), 25–35. [13] S.B. Dhage, B.C. Dhage, Dhage iteration method for Approximating positive solutions of PBVPs of nonlinear hybrid differential equations with maxima, Intern. Jour. Anal. Appl., 10(2) (2016), 101–111. [14] S.B. Dhage, B.C. Dhage, J.B. Graef, Dhage iteration method for initial value problem for nonlinear first order integrodifferential equations, J. Fixed Point Theory Appl., 18 (2016), 309–325. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 246 B. C. Dhage [15] S.B. Dhage, B.C. Dhage, J.B. Graef, Dhage iteration method for approximating the positive solutions of IVPs for nonlinear first order quadratic neutral functional differential equations with delay and maxima, Intern. J. Appl. Math., 31(6) (2018), 1–21. [16] B.C. Dhage, D. Otrocol, Dhage iteration method for approximating solutions of nonlinear differential equations with maxima, Fixed Point Theory, 19 (2) (2018), 545–556. [17] L. Erbe, W. Zhicheng, L. Longtu, Boundary value problems for second order mixed type functional differential equations, “Boundary value Problems for Functional Differential Equations”, World scientific 1996, 143–151. [18] D. Guo, V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, New York, London 1988. [19] J.K. Hale, Theory of Functional Differential equations, Springer-Verlag, New York-Berlin, 1977. [20] S. Heikkilä, V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations, Marcel Dekker inc., New York 1994. [21] S. Ntouyas, Y. Sficas, P. Tsamatos, Existence results for initial value problems for neutral functional differential equations, J. Diff. Equations, 114 (1994), 527–537. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 INDIAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019, pp. 247–254 DOI: A New Version of KKM Theorem with Geodesic Convex Hull with an Application on Hadamard Manifold J.C. Yao Research Center for Interneural Computing, China Medical University Hospital, China Medical University, Taichung, Taiwan E-mail: [email protected] Abstract: In this paper, we define a new type of the KKM mapping involving geodesic convex hull on Hadamard manifold, and we prove a related KKM theorem which we call generalized KKM theorem. A classical equilibrium problem on Hadamard manifold is considered. We prove an existence result for classical equilibrium problem by applying generalized KKM theorem. An example is also given in support of our problem. Keywords: Hadamard manifold, Geodesic convex hull, Generalized KKM theorem, Classical equilibrium problem. 2010 MSC: 49J40; 58E35; 90C33. 1. INTRODUCTION The famous KKM theorem, Browder fixed point theorem and Ky Fan’s minimax inequality are of fundamental importance in modern nonlinear analysis, and are mutually equivalent in the sense that each one can be deduced from another with or without aid of some minor results. In 1929, Knaster, Kureatowski and Mazurkiewiez [13] formulated the KKM principle (theorem) in finite dimensional spaces involving certain type of multi-valued mapping which is called the KKM mapping. In 1961, Fan [7] generalized the classical KKM theorem to infinite dimensional Hausdorff topological vector spaces and proved geometric lemma for multi-valued mappings, which is called Fan’s geometric lemma. After that, Browder [2] extended Fan’s geometric lemma to Fan-Browder’s fixed point theorem. In [11], Horvath replacing convex hull by contractible subset, gave a purely topological version of the KKM theorem. Very recently, Yang et al. [18] introduced and generalized KKM theorem without using convex hull. The theory of variational inequalities was introduced by Hartman and Stampacchia [10], has important applications, including but not limited to problems in complementarity problems, I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 248 J.C. Yao boundary value problems, optimization problems etc.. It has been studied extensively in finite and infinite dimensional linear spaces, see e.g., [6, 8, 9, 12]. Equally important is the area of equilibrium problem initiated by Blum and Oettli [1], which has emerged as an interesting and fascinating branch of applicable mathematics. They have shown that variational inequality problem can be viewed as a special realization of the abstract equilibrium problem. This theory has become a rich source of inspiration and motivation for the study of a wide class of problems arising in economics, finance, optimization, operation research in a general and unified way. In [1], it was shown that for f (x, y) = g(x, y) + h(x, y) with f, g, h : K × K → R, where K is subset of a topological vector space if g = 0, the result becomes a variant of Ky Fans theorem, whereas for h = 0 it becomes a variant of the Browder-Minty theorem for variational inequalities. In recent past, several applied problems such as constrained minimization problems, optimization problems, boundary value problems have been extended from Euclidean space to a Riemannian manifold setting in order to extend the study of convex theory, variational inequality, equilibrium and fixed point theory. Recently, much attention has been given to study the variational inequalities, equilibrium and related optimization problems on the Riemannian manifold and Hadamard manifold. Németh [15], Colao et al. [4], Zhou et al. [20] have considered the variational inequalities, equilibrium problems and optimization problems on Hadamard manifolds, respectively and studied the existence of solutions for their problems under some suitable conditions. Due to importance of the problems discussed above, in this paper, we define a new type of KKM mapping involving geodesic convex hull on Hadamard manifold, and we prove a generalized KKM theorem. Finally, classical equilibrium problem is solved by applying the generalized KKM theorem on Hadamard manifold. An example is constructed in support of our problem. 2. PRELIMINARIES The elementary and primary knowledge about differentiable manifolds can be found in many introductory books on Riemannian geometry, topology and equilibrium problems, see e.g., [5, 16, 17]. Let M be a simply connected m-dimensional manifold and x ∈ M. The tangent space of M Tx M, which is inherently a at x is denoted by Tx M and the tangent bundle of M by T M = x∈M manifold. A vector field V on M is a mapping of M into T M which relates with each point x ∈ M to a vector V (x) ∈ Tx M. We always assume that M can be equipped with a Riemannian metric to become a Riemannian manifold. We denote by ·, ·x the scalar product on Tx M with the associated norm · x , in the sequel the subscript x will be excluded. For x, y ∈ M, let γ : [a, b] −→ M be a piecewise smooth curve joining x to y (i.e., γ (a) = x and γ (b) = y). Then the arc-length of γ is defined by b L(γ ) = γ (t)dt. a Definition 2.1. The geodesic distance d(x, y) is the length of minimal geodesic segment between any two points x, y on a manifold. Let ∇ be the Levi-Civita connection associated with (M, ·, ·). Let γ be a smooth curve in M. A vector field V is said to be parallel along γ if ∇γ V = 0. If γ itself is parallel along γ , we say I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 New Version of KKM Theorem with Geodesic Convex Hull 249 that γ is a geodesic, and in this case γ is a constant. If γ = 1, then γ is said to be normalized. A geodesic joining x to y in M is said to be minimal if its length equals d(x, y). Definition 2.2. A Hadamard manifold M is a simply connected complete Riemannian manifold of non-positive sectional curvature. Definition 2.3. The exponential mapping expx : Tx M −→ M is defined by expx (v) = γv (1, x), where γv (·, x) = γ (·) is the geodesic starting at x with velocity v. It is clear that expx tv = γv (t, x), for each real number t (i.e., γ (0) = x, γ (0) = v). Definition 2.4 [3]. A set X ⊂ M is said to be geodesic convex if for any x, y ∈ X , the minimal geodesic joining x to y is contained in X , i.e., γx,y (t) = expx (t exp−1 x y) ∈ X , for every t ∈ [0, 1]. Remark 2.1. The exponential mapping and its inverse are continuous on Hadamard manifold, and exponential mapping is diffeomorphism. It is also remarked that for any x, y ∈ M, the minimal geodesic joining x and y, expx t exp−1 x y, for t ∈ [0, 1] is unique. Lemma 2.1. Let x0 ∈ M and {xn } ⊂ M such that xn → x0 . Then the following assertions hold. (i) For any y ∈ M, −1 −1 −1 exp−1 xn y → expx0 y and exp y x n → exp y x 0 ; (ii) (iii) If {vn } is a sequence such that vn ∈ Txn M and vn → v0 , then v0 ∈ Tx0 M; Given the sequences {u n } and {vn } with u n , vn ∈ Txn M, if u n → u 0 and vn → v0 with u 0 , v0 ∈ Tx0 M, then u n , vn → u 0 , v0 . Definition 2.5 [17]. A real-valued mapping g : X −→ R is said to be geodesic convex if and only if, for any geodesic γ , the composition mapping g ◦ γ : R −→ R is convex, i.e., (g ◦ γ )(t x + (1 − t))y ≤ t(g ◦ γ )(x) + (1 − t)(g ◦ γ )(y), for any x, y ∈ R and t ∈ [0, 1]. Remark 2.2. By Definition 2.4 and Remark 2.1, we can see that a real-valued mapping g : X −→ R is said to be geodesic convex if and only if, for any x, y ∈ X and t ∈ [0, 1], we have g expx t exp−1 x y ≤ tg(x) + (1 − t)g(y). Definition 2.6 [19]. Let z be any given point in M. The geodesic convex hull for a set S ⊂ M, denote by GCoS, is defined as follows: n ∀x1 , · · · , xn ∈ S; t1 , · · · , tn ∈ [0, 1] . GCoS = expz ti exp−1 z xi i=1 I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 250 J.C. Yao Lemma 2.2 [15]. If X ⊂ M is nonempty, compact and geodesic convex, then every continuous mapping h : X −→ X has a fixed point. 3. GENERALIZED KKM THEOREM We begin this section by introducing a new version of KKM mapping using geodesic convex hull. Definition 3.1. Let X be a nonempty subset of a Hadamard manifold M. A multi-valued mapping A : X −→ 2 X is called a generalized KKM mapping if, for any finite subset {x1 , · · · , xn } ⊂ M, n GCo{x1 , · · · , xn } ⊆ A(xi ). i=1 Remark 3.1. If M = Rn and A : X −→ Rn , then geodesic convex hull collapses to linear convex hull. Then, generalized KKM mapping reduces to classical KKM mapping. Now, we prove the following generalized KKM theorem. Theorem 3.1. Let X be a nonempty, compact and geodesic convex subset of a Hadamard manifold M. Suppose that A : X −→ 2 X be a closed valued generalized KKM mapping. Then A(x) = ∅. x∈X A(x) = ∅. Then Proof. On contrary, suppose that x∈X X=X\ A(x) = x∈X [X \ A(x)] . x∈X Since X is a nonempty and compact set and A(x) is closed, for all x ∈ X , then there exists a finite set {x1 , · · · , xn } of X such that n X= [X \ A(xi )] . i=1 Let {ti : i = 1, · · · , n} be the partition of unity subordinate to the open covering {X \ A(xi ) : i = 1, · · · , n} of X which implies that {ti : i = 1, · · · , n} is the set of continuous mappings with the following condition: 0 ≤ ti (x) ≤ 1, ∀x ∈ X, i = 1, · · · , n; and if x ∈ A(x j ), for some j, then t j (x) = 0. Now, consider a mapping ϕ : X −→ X by −1 ϕ(x) = expz t1 (x) exp−1 z x 1 + · · · + tn (x) expz x n , ∀x ∈ X. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 New Version of KKM Theorem with Geodesic Convex Hull 251 Then, ϕ is a continuous mapping. By Lemma 2.2, there exists x̄ ∈ X such that ϕ(x̄) = x̄. Since A is a generalized KKM mapping, the following inclusion holds n GCo{x1 , · · · , xn } ⊆ A(xi ). i=1 Then, there exists j ∈ {i ∈ {1, · · · , n} : ti (x̄) > 0} such that −1 x̄ = ϕ(x̄) = expz t1 (x) exp−1 z x 1 + · · · + tn (x) expz x n = GCo{x1 , · · · , xn } ∈ A(x j ). It implies that t j (x̄) = 0, which contradicts the fact that t j (x̄) > 0. Therefore, A(x) = ∅. x∈X 4. APPLICATION As an application of generalized KKM theorem, we prove an existence result for classical equilibrium problem, where the function involved is a sum of two functions, and the assumptions are required separately on each of these functions. Let X be a nonempty, geodesic convex and compact subset of a Hadamard manifold M. Given two real-valued mappings g, h : X × X −→ R. We consider the following classical equilibrium problem of finding x ∈ X such that g(x, y) + h(x, y) ≥ 0, ∀y ∈ X. (4.1) which was introduced by Blum and Oettli [1]. Next, we prove an existence result for classical equilibrium problem (4.1) by applying generalized KKM theorem on Hadamard manifold. Theorem 4.1. Let X ⊂ M be a nonempty, compact and geodesic convex subset of a Hadamard manifold M. Suppose that the continuous mappings g, h : X × X −→ R are geodesic convex mappings in the second arguments, respectively, such that g(x, x) = 0 and h(x, x) = 0, for all x ∈ X . Then, the classical equilibrium problem (4.1) admits a solution. Proof. For any y ∈ X , define the mapping F : X −→ 2 X by F(y) = {x ∈ X : g(x, y) + h(x, y) ≥ 0} . We claim that F(y) is closed in X . Let {xλ } be any net in X with xλ ∈ F(y) and xλ → x0 . Then, g(xλ , y) + h(xλ , y) ≥ 0. Since g and h are continuous mappings, we have g(xλ , y) + h(xλ , y) −→ g(x0 , y) + h(x0 , y). Therefore, x0 ∈ F(y), and hence F is closed in X . I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 252 J.C. Yao In order to apply generalized KKM theorem 3.1, we have to prove that for any choice of {y1 , · · · , yn } ∈ X , n GCo{y1 , · · · , yn } ⊆ F(yi ). i=1 On contrary suppose that F(y) is not a generalized KKM mapping. Then, there exists x̄ ∈ GCo{y1 , · · · , yn } such that for all ti ≥ 0, i = 1, · · · , n, we have n n −1 ti expz yi ∈ F(yi ). x̄ = expz / i=1 i=1 Therefore, we have g(x̄, yi ) + h(x̄, yi ) < 0, which implies that for any i = 1, · · · , n, we have yi ∈ {w ∈ X : g(x̄, w) + h(x̄, w) < 0} . Now, we show that the set {w ∈ X : g(x̄, w) + h(x̄, w) < 0} is geodesic convex. Let x̄ ∈ X be fixed, and let w1 , w2 ∈ {w ∈ X : g(x̄, w) + h(x̄, w) < 0}, for any λ ∈ [0, 1]. Since g and h are geodesic convex mappings in the second arguments, respectively, we have −1 g x̄, expw1 t exp−1 w1 w2 + h x̄, expw1 t expw1 w2 ≤ tg (x̄, w1 ) + (1 − t)g (x̄, w2 ) + th (x̄, w1 ) + (1 − t)h (x̄, w2 ) = t {g(x̄, w1 ) + h(x̄, w1 )} + (1 − t) {g(x̄, w2 ) + h(x̄, w2 )} < 0, which shows that {w ∈ X : g(x̄, w) + h(x̄, w) < 0} is a geodesic convex set. Therefore, by definition of the geodesic convex hull, we have x̄ ∈ GCo{y1 , · · · , yn } ⊆ {w ∈ X : g(x̄, w) + h(x̄, w) < 0} , and hence g(x̄, x̄) + h(x̄, x̄) < 0, which implies that 0 < 0, a contradiction, using the hypotheses g(x, x) = h(x, x) = 0. Thus, F(y) is a generalized KKM mapping. By applying generalized KKM theorem 3.1, there exists a point x0 ∈ X such that x0 ∈ F(y) = ∅, y∈X I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 New Version of KKM Theorem with Geodesic Convex Hull 253 i.e., g(x0 , y) + h(x0 , y) ≥ 0, ∀y ∈ X. Hence, classical equilibrium problem (4.1) admits a solution. This completes the proof. We now provide an example to illustrate the existence of solutions of classical equilibrium problem (4.1). Example 4.1. Consider the 2-dimensional unit sphere M = S 2 = (x1 , x2 , x3 ) ∈ R3 : x12 + x22 + x32 = 1 , endowed with the standard metric. Let the tangent spaces be defined by Tx M = p ∈ R3 : p, q = 0 , ∀q ∈ M. Let X = {(x1 , x2 , x3 ) ∈ M : x1 , x2 > 0, x3 = 0} . Then, it is easy to check that X is a nonempty, compact and geodesic convex subset of M. We now define g, h : X × X −→ R as follows: g(x, y) = 1 1 (x + y)2 and h(x, y) = (y − x)2 , 3 2 for all x = (x1 , x2 , x3 ), y = (y1 , y2 , y3 ) ∈ X . Then, g and h are continuous, and geodesic convex mappings in the second arguments. Hence by Theorem 4.1, the classical equilibrium problem (4.1) admits at least one solution as x0 = 23 , 1, 0 . REFERENCES [1] E. Blum and W. 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Pardalos, Equilibrium problems and variational models, Kluwer, Boston (2001). [10] G.J. Hartmann and G. Stampacchia, On some nonlinear elliptic differential functional equations, Acta Mathematica, 115, pp. 271–310 (1966). I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 254 J.C. Yao [11] C.D. Horvath, Some results on multi-valued mappings and inequalities without convexity, In: B.L. Lin and S. Simons (eds.) Nonlinear and Convex Analysis, Lecture Notes in Pure and Applied Math., pp. 99–106, Dekker, New York (1987). [12] Y.H. Hu and W. Song, Weak sharp solutions for variational inequalities in Banach spaces, J. Math. Anal. Appl., 374, pp. 118–132 (2011). [13] B. Knaster, C. Kuratowski and S. Mazurkiewicz, Ein beweis des fixpunksatzes fur n-dimensionale simplexe, Fundam. Math., 14, pp. 132–137 (1929). [14] A. Moudafi and M. Théra, Proximal and dynamical approaches to equilibrium problems, Lecture Notes in Economics and Mathematical Systems, 477, pp. 187–201, Springer, New York (1999) [15] S.Z. Németh, Variational inequalities on Hadamard manifolds, Nonlinear Anal., 52, pp. 1491–1498 (2003). [16] T. Sakai, Riemannian geometry, Transl. Math. Monogr., 149, American Mathematical Society, Providence, RI (1996). [17] C. Udriste, Convex functions and optimization methods on Riemannian manifolds, In: Mathematics and Its Applications, Vol. 297, Kluwer Academics, Dordrecht (1994). [18] Z. Yang and Y.J. Pu, Generalized Knaster-Kuratowski-Mazurkiewicz theorem without convex hull, J. Optim. Theory Appl., 154, pp. 17–29 (2012). [19] Z. Yang and Y.J. Pu, Existence and stability of solutions for maximal element theorem on Hadamard manifolds with applications, Nonlinear Anal., 75, pp. 516–525 (2012). [20] L.-W. Zhou and N.-J. Huang, Existence of solutions for vector optimization on Hadamard manifolds, J. Optim. Theory Appl., 157, pp. 44–53 (2013). I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 INDIAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019, pp. 255–268 DOI: The Relaxed Resolvent Operator For Solving Fuzzy Variational Inclusion Problem Involving Ordered RME Set-Valued Mapping Iqbal Ahmad1 , Abul H. Siddiqi2 and Rais Ahmad3∗ 1 College of Engineering, Qassim University, Buraidah-51452, Al-Qassim, Saudi Arabia School of Basic Sciences and Research, Sharda University, Greater Noida-201306, India 3 Department of Mathematics, Aligarh Muslim University, Aligarh-202002, India (∗ Corresponding author) E-mail: ∗ [email protected]; 1 [email protected]; 2 [email protected] 2 Abstract: In this paper, we consider a relaxed resolvent operator and prove some of its properties. Using the properties of relaxed resolvent operator, we obtain the solution of a fuzzy variational inclusion problem with ordered RME set-valued mapping in ordered Hilbert spaces by suggesting an iterative algorithm. Some special cases are discussed. Some examples are also constructed. Keywords: Ordered RME Mapping; Ordered Hilbert Space; Inclusion; Algorithm; Fuzzy Mappings. 2010 AMS Subject Classification: 49J40; 47H05; 47S40. 1. INTRODUCTION In mathematics, a variational inequality is an inequality involving a functional, which has to be solved for all possible values of a given variable, belonging usually to a convex set. The mathematical theory of variational inequalities has initially developed to deal with equilibrium problems, precisely the Signorini problem. The formal definition of a variational inequality is as follows: Let X be a Hilbert space, K ⊂ X a closed convex set and F : K → X be a bounded linear functional on X. The problem of finding x ∈ K such that F(x), y − x ≥ 0, ∀ y ∈ K , where ., . is the usual inner product on X. An important generalization of a variational inequality is called a variational inclusion is mainly due to Hassouni and Moudafi [10]. Using resolvent operator methods, many researchers have I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 256 Iqbal Ahmad, Abul H. Siddiqi and Rais Ahmad developed iterative algorithms to solve variational inclusions and other equivalent problems. Ansari [1] and Chang and Zhu [4] at the same time introduced and studied variational inequalities with fuzzy mappings. A vast literature is available about variational inequalities (inclusions) with fuzzy mappings, see e.g., [3,5–8,11–14,22,28] and references therein. We also refer to [2,8,20,23,25,26] for a reasonable study of variational inequalities and its equivalent problems. It is known that the fixed point theory with applications for nonlinear mappings have been intensively studied in ordered Banach spaces. A lot of work is done by Li [15–19] to approximate the solution of a general nonlinear ordered variational inequalities and ordered equations in ordered Banach spaces. On the other hand the fuzzy set theory due to Zadeh [27] specifically designed to mathematically represent uncertainty and vagueness and to provide formalized tools for dealing with the imprecision intrinsic to many problems. In this paper, we consider a relaxed resolvent operator and prove that it is single-valued, compression as well as Lipschitz continuous. Then these new results are used to solve a fuzzy variational inclusion problem involving ordered RME set-valued mapping after defining an iterative algorithm. We claim that all the results of this paper either preliminary or main are the extension of results of Li [15]. 2. PRELIMINARIES Throughout the paper, we assume that X is a real ordered Hilbert space equipped with a norm ||.|| and an inner product ., .. Let C be a normal cone, “ ≤ " is a partial ordering defined by C and K be the normal constant of C. For the set of arbitrary elements x, y ∈ X, i.e., {x, y}, we denote the least upper bound by lub{x, y} and greatest lower bound by glb{x, y}, also we suppose that they always exist. The operator ⊕ is called an XOR operator if x ⊕ y =lub{x − y, y − x}. Let F(X ) be a collection of all fuzzy sets over X. A mapping F : X → F(X ) is said to be fuzzy mapping on X . For each x ∈ X, F(x) (in the sequel, it will be denoted by Fx ) is a fuzzy set on X and Fx (y) is the membership function of y in Fx . A fuzzy mapping F : X → F(X ) is said to be closed if for each x ∈ E, the function y → Fx (y) is upper semi-continuous, that is, for any given net {yα } ⊂ X, satisfying yα → y0 ∈ X, we have lim sup Fx (yα ) ≤ Fx (y0 ). α For B ∈ F(X ) and λ ∈ [0, 1], the set (B)λ = {x ∈ X : B(x) ≥ λ} is called a λ-cut set of B. Let F : X → F(X ) be a closed fuzzy mapping satisfying the following condition: Condition (∗) : If there exists a function a : X → [0, 1] such that for each x ∈ X, the set (Fx )a(x) = {y ∈ X : Fx (y) ≥ a(x)} is a nonempty bounded subset of X. If F is a closed fuzzy mapping satisfying the condition (∗), then for each x ∈ X, (Fx )a(x) ∈ C B(X ). In fact, let {yα } ⊂ (Fx )a(x) be a net and yα → y0 ∈ X, then (Fx )a(x) ≥ a(x), for each α. Since F is a closed, we have Fx (y0 ) ≥ lim sup Fx (yα ) ≥ a(x), α which implies that y0 ∈ (Fx )a(x) and so (Fx )a(x) ∈ C B(X ). I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 The Relaxed Resolvent Operator For Solving Fuzzy Variational Inclusion Problem Involving 257 Definition 2.1 [24]. Let X be an ordered Hilbert space and C be a cone with a partial ordered relation ≤ . For any x, y ∈ X if either x ≤ y or y ≤ x hold, then x and y are said to be comparable to each other (denoted by x ∝ y). Definition 2.2 [9]. The cone C is said to be normal if and only if there exists a constant K > 0 such that for 0 ≤ x ≤ y, we have ||x|| ≤ K ||y||. Proposition 2.1. If x ∝ y, then lub{x, y} and glb{x, y} exist, x − y ∝ y − x, and 0 ≤ (x − y) ∨ (y − x). Proposition 2.2 [9]. For any natural number n, if x ∝ yn and yn → y ∗ (n → ∞), then x ∝ y ∗ . Proposition 2.3 [9]. Let X be an ordered Hilbert space and α, β be two real numbers. Let ⊕ be an XOR operator and for x, y, u, v ∈ X, the operation x y is defined by x y = (x − y) ∧ (y − x). Then the following relations hold: (i) x y = y x, x x = 0, x y = y x = −(x ⊕ y); (ii) x 0 ≤ 0, if x ∝ 0; (iii) 0 ≤ x ⊕ y, if x ∝ y; (iv) (x + y) (u + v) ≥ (x u) + (y v); (v) (x + y) (u + v) ≥ (x v) + (y u); (vi) αx ⊕ βx = |α − β|x, if x ∝ 0. Proposition 2.4 [9]. Let C be a normal cone in X with normal constant K , then for each x, y ∈ X, the following conditions hold: (i) (ii) (iii) (iv) ||0 ⊕ 0|| = ||0|| = 0; ||x ∨ y|| ≤ ||x|| ∨ ||y|| ≤ ||x|| + ||y||; ||x ⊕ y|| ≤ ||x − y|| ≤ K ||x ⊕ y||; if x ∝ y, then ||x ⊕ y|| = ||x − y||. Proposition 2.5. Let C be a normal cone in X, if for x, y ∈ X, they can be compared to each other, then the following condition holds: (x + y) ∨ ((−x) + (−y)) ≤ ((x ∨ (−x)) + (y ∨ (−y))). Definition 2.3 [9]. Let M : X → 2 X be a set-valued mapping such that M(x) is a nonempty closed subset of X. Then (i) (ii) M is said to be a comparison mapping, if for any vx ∈ M(x), x ∝ vx , and if x ∝ y, then for any vx ∈ M(x) and any v y ∝ M(y), vx ∝ v y , ∀ x, y ∈ X ; a comparison mapping M is said to be ordered rectangular, if for each x, y ∈ X, u x ∈ M(x) and u y ∈ M(y) such that u x u y , −(x ⊕ y) = 0; I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 258 Iqbal Ahmad, Abul H. Siddiqi and Rais Ahmad (iii) a comparison mapping M is said to be λ-ordered monotone, if there exists a constant λ > 0 such that λ(vx − v y ) ≥ x − y, ∀ x, y ∈ X, vx ∈ M(x) and v y ∈ M(y); (iv) a comparison mapping M is said to be β-ordered extended, if there exists a constant β > 0 such that β(x ⊕ y) ≤ vx ⊕ v y , ∀ x, y ∈ X, vx ∈ M(x), v y ∈ M(y); (v) a comparison mapping M is said to be ordered RME with respect to JM,λ , if M is ordered rectangular and λ-ordered monotone with respect to JM,λ and β-ordered extended, and (I + λM)(X ) = X, for λ, β > 0, where I stands for the identity mapping on X. Definition 2.4 [9]. Let A, B : X → X be two single-valued mappings. Then (i) (ii) A is said to be a comparison, if for each x, y ∈ X, x ∝ y then A(x) ∝ A(y), x ∝ A(x) and y ∝ A(y). A and B are said to be comparable to each other, if for each x ∈ X, A(x) ∝ B(x). Obviously, if A is a comparison mapping, then A ∝ I. Definition 2.5. Let X be a real ordered Hilbert space, C be a normal cone with a normal constant K in X. A mapping A : X × X → X is said to be β-ordered compression mapping, if A is a comparison mapping and A((x, .)) ⊕ A((y, .)) ≤ β(x ⊕ y), f or 0 < β < 1. Definition 2.6. Let X be a real ordered Hilbert space, C be a normal cone with a normal constant K in X. A mapping N : X × X → X is said to be (μ, ν)-ordered Lipschitz continuous, if x ∝ y, u ∝ v, then N (x, u) ∝ N (y, v) and there exist constants μ, ν > 0 such that N (x, u) ⊕ N (y, v) ≤ μ(x ⊕ y) + ν(u ⊕ v). Definition 2.7. A set-valued mapping A : X → C B(X ) is said to be D-Lipschitz continuous, if for each x, y ∈ X , x ∝ y, there exists a constant δ A such that D(A(x), A(y)) ≤ δ A x ⊕ y, ∀x, y ∈ X. Definition 2.8. Let M : X → 2 X be a set-valued mapping such that M(x) is a nonempty closed subset of X and let R : X → X be a single-valued mapping. Then a comparison mapping M is said (I −R) , if M is ordered rectangular and λ-ordered monotone to be ordered RME with respect to JM,λ I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 The Relaxed Resolvent Operator For Solving Fuzzy Variational Inclusion Problem Involving 259 (I −R) with respect to JM,λ and β-ordered extended, and [(I − R) + λM](X ) = X, for λ, β > 0, where I stands for the identity mapping on X. we construct the following examples. Example 2.1. Let X = [0, 1] with usual inner product and suppose (i) (ii) the set-valued mapping M : X → 2 X is defined by [0, 1], i f x = 0, M(x) = (0, 1], i f x = 0. Then, it is easy check that M is ordered rectangular mapping. the set-valued mapping M : X → 2 X is defined by { 1 }, i f x = 0, M(x) = x {0}, i f x = 0. Then, it is easy to check that M is 9 -ordered monotone and 12 -ordered extended mapping. 10 Example 2.2. Let X = R and N : X × X → X is defined by x + y + a, ∀ x, y ∈ X, a > 0. N (x, y) = sin 5 Then, N is ( 15 , 15 )-ordered Lipschitz continuous mapping. Using the Proposition 2.5., we have y + v x + u + a ⊕ sin +a N (x, u) ⊕ N (y, v) = sin 5 5 y + v y + v x + u x + u − sin ∨ sin − sin = sin 5 5 5 5 x + u y + v y + v x + u − ∨ − ≤ 5 5 5 5 x − y u − v x − y u − v + ∨ − + − = 5 5 5 5 x − y x − y u − v u − v ∨ − + ∨ − ≤ 5 5 5 5 1 1 ≤ (x ⊕ y) + (u ⊕ v). 5 5 Therefor, N (x, u) ⊕ N (y, v) ≤ 1 1 (x ⊕ y) + (u ⊕ v). 5 5 i.e., N is ( 15 , 15 )-ordered Lipschitz continuous mapping. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 260 Iqbal Ahmad, Abul H. Siddiqi and Rais Ahmad 3. FORMULATION OF THE PROBLEM AND SOME BASIC RESULTS Let A, B : X → F(X ) be the closed fuzzy mappings satisfying the following condition (∗). Then there exists mappings a, b : X → [0, 1] such that for each x ∈ X, (Ax)a(x) ∈ C B(X ) and (Bx)b(x) ∈ C B(X ). Thus, we define the set-valued mappings by Ã(x) = (Ax)a(x) and B̃(x) = (Bx)b(x) , ∀ x ∈ X, where à and B̃ are called the set-valued mappings induced by the fuzzy mappings A and B, respectively. Suppose that M : X → 2 X be an ordered RME set-valued mapping and N : X × X :→ X be a single-valued mapping. We consider the following problem: Find x ∈ X, u ∈ (Ax)a(x) , v ∈ (Bx)b(x) such that 0 ∈ N (u, v) + M(x). (3.1) Problem (3.1) is called fuzzy variational inclusion problem involving ordered RME set-valued mapping. Below we have the following special case of problem (3.1) and one can obtain many other cases for suitable choice of operators involved in the formulation of problem (3.1). If A, B : X → C B(X ) are the classical set-valued mappings, we can define the fuzzy mappings A, B by x → χ A(x) , x → χ B(x) , where χ A(x) and χ B(x) are the characteristic functions of A and B, respectively. Taking a(x) = b(x) = 1, for all x ∈ X, the problem (3.1) converted into the problem. Find x ∈ X, u ∈ A(x) and v ∈ B(x) such that 0 ∈ N (u, v) + M(x). (3.2) Problem (3.2) was studied by Verma [26] and many other researchers in different settings. Definition 3.1. Let C be a normal cone with normal constant K and M : X → 2 X be a set-valued ordered rectangular mapping. Let I : X → X be the identity mapping and R : X → X be a single(I −R) : X → X associated with I, R and M is valued mapping. The relaxed resolvent operator Jλ,M defined by (I −R) Jλ,M (x) = [(I − R) + λM]−1 (x), f or all x ∈ X and λ > 0. (3.3) If R = 0, then the relaxed resolvent operator defined by (3.3) coincides with the resolvent operator of Li [15]. Now, we show that the relaxed resolvent operator defined by (3.3) is single-valued, a comparison mapping as well as Lipschitz continuous. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 The Relaxed Resolvent Operator For Solving Fuzzy Variational Inclusion Problem Involving 261 Proposition 3.1. Let R : X → X be a comparison and γ -ordered compression mapping, and let (I −R) M : X → 2 X be the set-valued ordered rectangular mapping. Then the operator Jλ,M = [(I − −1 X R) + λM] : X → 2 is a single-valued, for all λ > 0. Proof. For any given z ∈ X and a constant λ > 0, let x, y ∈ [(I − R) + λM]−1 (z). Then 1 (z − (I − R)(x)) ∈ M(x), λ and 1 (z − (I − R)(y)) ∈ M(y). λ Using (iv) and (vi) of Proposition 2.3., we have = ≥ = = = = ≥ 1 1 (z − (I − R)(x)) (z − (I − R)(y)) λ λ 1 [(z − (I − R)(x)) (z − (I − R)(y))] λ 1 [z z + (−(I − R)(x)) (−(I − R)(y))] λ 1 [−(I − R)(x) −(I − R)(y)] λ 1 − [−(−(I − R)(x)) −(−(I − R)(y))] λ 1 − [(I − R)(x) ⊕ (I − R)(y)] λ 1 − [x − R(x) ⊕ y − R(y)] λ 1 − [x ⊕ y + R(x) ⊕ R(y)]. λ Thus, we have 1 1 1 (z − (I − R)(x)) (z − (I − R)(y)) ≥ − [x ⊕ y + R(x) ⊕ R(y)]. λ λ λ (3.4) Since R is γ -ordered compression mapping and using (3.4), we have 1 1 (z − (I − R)(x)) (z − (I − R)(y)) λ λ 1 − [x ⊕ y + γ (x ⊕ y)] λ 1 = − (1 + γ )(x ⊕ y). λ ≥ I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS (3.5) Vol. 10, No. 1 (Special Issue), Jan–June 2019 262 Iqbal Ahmad, Abul H. Siddiqi and Rais Ahmad Since M is an ordered rectangular mapping and using (3.5), we have 0 = ≥ = 1 1 (z − (I − R)(x)) (z − (I − R)(y)), −(x ⊕ y) λ λ 1 − (1 + γ )(x ⊕ y), −(x ⊕ y) λ 1 (1 + γ )||x ⊕ y||2 . λ i.e., 1 (1 + γ )||x ⊕ y||2 ≤ 0. λ (I −R) = [(I − R) + λM]−1 and consequently x ⊕ y = 0, i.e., we have x = y. Thus, the operator Jλ,M is a single-valued. Proposition 3.2. Let M : X → 2 X be a set-valued ordered rectangular, comparison and λ-ordered (I −R) . Let R : X → X be a strongly comparison mapping and monotone mapping with respect to Jλ,M (I −R) (I − R) be an strongly comparison mapping with respect to Jλ,M . Then, the resolvent operator (I −R) Jλ,M : X → X is a comparison mapping. (I −R) (I −R) Proof. Since M : X → 2 X is a comparison mapping with respect to Jλ,M so that x ∝ Jλ,M (x). For any x, y ∈ X, let x ∝ y, and let vx = 1 (I −R) (I −R) (x − (I − R)(Jλ,M (x))) ∈ M(Jλ,M (x)) λ (3.6) and 1 (I −R) (I −R) (y − (I − R)(Jλ,M (y))) ∈ M(Jλ,M (y)). λ Taking difference of (3.6) and (3.7), we have 1 1 (I −R) (I −R) (x − (I − R)(Jλ,M (y − (I − R)(Jλ,M (x))) − (y))) vx − v y = λ λ 1 (I −R) (I −R) x − y + (I − R)(Jλ,M (y)) − (I − R)(Jλ,M (x)) . = λ vy = (3.7) As M is a λ-ordered monotone, we have 0 ≤ λ(vx − v y ) − (x − y) (I −R) (I −R) (y)) − (I − R)(Jλ,M (x)) − (x − y) = (x − y) + (I − R)(Jλ,M (I −R) (I −R) = (I − R)(Jλ,M (y)) − (I − R)(Jλ,M (x)), which implies that (I −R) (I −R) 0 ≤ (I − R)(Jλ,M (y)) − (I − R)(Jλ,M (x)). I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 The Relaxed Resolvent Operator For Solving Fuzzy Variational Inclusion Problem Involving 263 (I −R) (I −R) Since (I − R) is strongly comparison mapping with respect to Jλ,M . Therefore, Jλ,M (x) ∝ (I −R) Jλ,M (y). The proof is completed. (I −R) Proposition 3.3. Let M : X → 2 X be a ordered RME set-valued mapping with respect to Jλ,M , X for βλ > γ + 1. Let R : X → 2 be a comparison and γ -ordered compression mapping with (I −R) . Then the following condition holds: respect to Jλ,M (I −R) (I −R) (x) ⊕ Jλ,M (y) ≤ Jλ,M 1 (x ⊕ y). βλ − γ − 1 (I −R) (I −R) Proof. For x, y ∈ X, set u x = Jλ,M (x), u y = Jλ,M (y), and let vx = 1 (I −R) (I −R) x − (I − R)(Jλ,M (x)) ∈ M(Jλ,M (x)) λ vy = 1 (I −R) (I −R) y − (I − R)(Jλ,M (y)) ∈ M(Jλ,M (y)). λ and (I −R) . That is, M is ordered rectanguAs M be an ordered RME set-valued mapping with respect to Jλ,M (I −R) lar, λ-ordered monotone with respect to Jλ,M and β-ordered extended mapping. From Proposition (I −R) 3.2., we know that Jλ,M is a comparison mapping, and as x ∝ y so that vx ∈ M(u x ), v y ∈ M(u y ), we have u x ∝ u y , vx ∝ v y . By (i x) of the Proposition 2.3, we have vx ⊕ v y = ≤ ≤ ≤ = 1 [(x − (I − R)(u x )) ⊕ (y − (I − R)(u y ))] λ 1 [x ⊕ y + (I − R)(u x ) ⊕ (I − R)(u y )] λ 1 [x ⊕ y + u x ⊕ u y + R(u x ) ⊕ R(u y )] λ 1 [x ⊕ y + u x ⊕ u y + γ (u x ⊕ u y )] λ 1 [x ⊕ y + (1 + γ )(u x ⊕ u y )]. λ Since M is β-ordered extended mapping, we have β(u x ⊕ u y ) ≤ vx ⊕ v y = It follows that β − (1+γ ) λ 1 [x ⊕ y + (1 + γ )(u x ⊕ u y )]. λ (u x ⊕ u y ) ≤ λ1 (x ⊕ y) and consequently, we have (I −R) (I −R) (x) ⊕ Jλ,M (y) ≤ Jλ,M 1 (x ⊕ y). βλ − γ − 1 The proof is completed. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 264 Iqbal Ahmad, Abul H. Siddiqi and Rais Ahmad 4. ITERATIVE ALGORITHM AND EXISTENCE RESULTS In this section, we define an iterative algorithm to obtain the solution of fuzzy variational inclusion problem involving ordered ordered RME set-valued mapping (3.1). Iterative Algorithm 4.1. Let R : X → X, N : X × X → X be the single-valued mappings and I : X → X is an identity mapping. Let A, B : X → F(X ) be the closed fuzzy mappings satisfying condition (∗) and Ã, B̃ be the set-valued mappings induced by the fuzzy mappings. Suppose that M : X → 2 X is the ordered RME set-valued mapping. We define the following scheme: For any given initial x0 ∈ X, u 0 ∈ (Ax0 )a(x0 ) and v0 ∈ (Bx0 )b(x0 ) , let (I −R) [(I − R)(x0 ) − λN (u 0 , v0 )]. x1 = Jλ,M Since u 0 ∈ (Ax0 )a(x0 ) ∈ C B(X ), v0 ∈ (Bx0 )b(x0 ) ∈ C B(X ), by Nadler’s theorem [21], there exist u 1 ∈ (Ax1 )a(x1 ) , v1 ∈ (Bx1 )b(x1 ) and suppose that x0 ∝ x1 , u 0 ∝ u 1 , v0 ∝ v1 such that ||u 1 ⊕ u 0 || = ||u 1 − u 0 || ≤ D(A(x1 )a(x1 ) , A(x0 )a(x0 ) ) and ||v1 ⊕ v0 || = ||v1 − v0 || ≤ D(B(x1 )b(x1 ) , B(x0 )b(x0 ) ). Continuing the above process inductively with the supposition that xn+1 ∝ xn , u n+1 ∝ u n and vn+1 ∝ vn , for all n ∈ N, we have the following algorithm: (I −R) [(I − R)(xn ) − λN (u n , vn )], xn+1 = Jλ,M (4.1) u n+1 ∈ A(xn+1 )a(xn+1 ) , ||u n+1 ⊕ u n || = ||u n+1 − u n || ≤ D(A(xn+1 )a(xn+1 ) , A(xn )a(xn ) ), (4.2) vn+1 ∈ B(xn+1 )b(xn+1 ) , ||vn+1 ⊕ vn || = ||vn+1 − vn || ≤ D(B(xn+1 )b(xn+1 ) , B(xn )b(xn ) ), (4.3) where λ > 0 is a constant. Now, we convert our problem (3.1) into a fixed point problem. Lemma 4.1. Let x ∈ X, u ∈ (Ax)a(x) and v ∈ (Bx)b(x) is a solution of fuzzy variational inclusion problem involving ordered RME set-valued mapping (3.1) if and only if (x, u, v) satisfies the equation: (I −R) [(I − R)(x) − λN (u, v)], x = Jλ,M where (I −R) Jλ,M = [(I − R) + λM]−1 , and λ > 0 is a constant. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 The Relaxed Resolvent Operator For Solving Fuzzy Variational Inclusion Problem Involving 265 (I −R) Proof. The proof directly follows from the definition of the relaxed resolvent operator Jλ,M . Now, we prove the following existence and convergence result for fuzzy variational inclusion problem involving ordered RME set-valued mapping (3.1). Theorem 4.1. Let X be a real ordered Hilbert space and C be a normal cone with normal constant K . Let R : X → X be a comparison, γ -ordered compression mapping and N : X × X → X be an (μ, ν)-ordered Lipschitz continuous mapping. Let A, B : X → F(X ) be the closed fuzzy mappings satisfying the condition (∗) and the let Ã, B̃ : X → C B(X ) be the set-valued mappings induced by the fuzzy mappings A and B such that à and B̃ are D-Lipschitz continuous mappings with constants δ A and δ B , respectively. Suppose that M : X → 2 X be an ordered RME set-valued mapping and if the following condition is satisfied K λ(μδ A + νδ B ) + (1 + K )(1 + γ ) < βλ, (4.4) then the iterative sequences {xn }, {u n } and {vn } generated by Algorithm 4.1. converges strongly to x, u and v, respectively and (x, u, v) is a solution of fuzzy variational inclusion problem involving ordered RME set-valued mapping (3.1), where x ∈ X, u ∈ (Ax)a(x) and v ∈ (Bx)b(x) . Proof. Since R is γ -ordered compression mapping and N is (μ, ν)-ordered Lipschitz continuous mapping. By Algorithm 4.1., Proposition 2.2 and Proposition 3.3, we have 0 ≤ xn+1 ⊕ xn = (I −R) (I −R) Jλ,M [(I − R)(xn ) − λN (u n , vn )] ⊕ Jλ,M [(I − R)(xn−1 ) − λN (u n−1 , vn−1 )] ≤ 1 [(I − R)(x n ) − λN (u n , vn ) ⊕ (I − R)(xn−1 ) − λN (u n−1 , vn−1 )] βλ − γ − 1 ≤ 1 [(I − R)(x n ) ⊕ (I − R)(xn−1 ) + λ(N (u n , vn ) ⊕ N (u n−1 , vn−1 ))] βλ − γ − 1 ≤ 1 [(xn ⊕ xn−1 ) + R(xn ) ⊕ R(xn−1 ) + λ(N (u n , vn ) ⊕ N (u n−1 , vn−1 ))] βλ − γ − 1 ≤ 1 [(xn ⊕ xn−1 ) + γ (xn ⊕ xn−1 ) + λ(N (u n , vn ) ⊕ N (u n−1 , vn−1 ))] βλ − γ − 1 ≤ ≤ 1 [(xn ⊕ xn−1 ) + γ (xn ⊕ xn−1 ) + λ μ(u n ⊕ u n−1 ) βλ − γ − 1 +ν(vn ⊕ vn−1 ) ] 1 [(xn ⊕ xn−1 ) + γ (xn ⊕ xn−1 ) + λμ(u n ⊕ u n−1 ) + λν(vn ⊕ vn−1 )]. βλ − γ − 1 I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 266 Iqbal Ahmad, Abul H. Siddiqi and Rais Ahmad Using Definition 2.2., Proposition 2.3. and D-Lipschitz continuous of à and B̃, we have ||xn+1 − xn || ≤ ≤ ≤ ≤ 1 [(xn ⊕ xn−1 ) + γ (xn ⊕ xn−1 ) + λμ(u n ⊕ u n−1 ) βλ − γ − 1 +λν(vn ⊕ vn−1 )]|| K [(1 + γ )||xn − xn−1 || + λμ||u n − u n−1 || βλ − γ − 1 +λν||vn − vn−1 ||] K [(1 + γ )||xn − xn−1 || + λμδ A ||xn − xn−1 || βλ − γ − 1 +λνδ B ||xn − xn−1 ||] K [(1 + γ ) + λ(μδ A + νδ B )]||xn − xn−1 ||. βλ − γ − 1 K || (4.5) i.e., ||xn+1 − xn || ≤ ||xn − xn−1 ||, where = K [(1 + γ ) + λ(μδ A + νδ B )] . βλ − γ − 1 By condition (4.4), we have 0 < < 1, thus {xn } is a cauchy sequence in X and as X is complete, there exists x ∈ X such that xn → x as n → ∞. From (4.2) and (4.3) of Algorithm 4.1. and DLipschitz continuity of à and B̃, we have ||u n+1 − u n || ≤ D(A(xn+1 )a(xn+1 ) , A(xn )a(xn ) ) ≤ δ A ||xn+1 − xn || (4.6) ||vn+1 − vn || ≤ D(B(xn+1 )b(xn+1 ) , B(xn )b(xn ) ) ≤ δ B ||xn+1 − xn ||. (4.7) It is clear from (4.6) and (4.7) that {u n } and {vn } are also cauchy sequences in X, so there exist u and v in X such that u n → u and vn → v as n → ∞. By using the continuity of the operators N , I −R and iterative Algorithm 4.1., we have Ã, B̃, Jλ,M (I −R) [(I − R)x − λN (u, v)]. x = Jλ,M By Lemma 4.1, we conclude that (x, u, v) is a solution of problem (3.1). It remains to show that u ∈ (Ax)a(x) and v ∈ (Bx)b(x) . In fact d(u, (Ax)a(x) ) ≤ u − u n + d(u n , (Ax)a(x) ) ≤ u − u n + D((Axn )a(xn ) , (Ax)a(x) ) ≤ u − u n + δ A xn − x → 0, as n → ∞. Hence u ∈ (Ax)a(x) . Similarly, we can show that v ∈ (Bx)b(x) . This completes the proof. I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS Vol. 10, No. 1 (Special Issue), Jan–June 2019 The Relaxed Resolvent Operator For Solving Fuzzy Variational Inclusion Problem Involving 267 Remark 4.1. If N , R, Ã, B̃ are zero mappings, then from Theorem 4.1. one can obtain Theorem 2.6. of Li [18] for solving the problem 0 ∈ M(x). 5. CONCLUSION The aim of this paper is to introduced a relaxed resolvent operator and we demonstrate some of its properties. The relaxed resolvent operator is used to define an iterative algorithm for solving a fuzzy variational inclusion problem involving ordered RME set-valued mapping in ordered Hilbert spaces. Some preliminary results are proved to obtain the main result. We remark that our results may be extended for other related problems involving fuzzy mappings and also one may consider some higher ordered dimensional structure. REFERENCES [1] Q.H. Ansari: Certain problems concerning variational inequalities, Ph.D Thesis, Aligarh Muslim University, Aligarh, India (1988). [2] R. Ahmad, Q.H. Ansari, S.S. Irfan: Generalized variational inclusions and generalized resolvent equations in Banach spaces, Comput. Math. Appl., 49, 1825–1835 (2005). [3] R. Ahmad, M. 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