Теорема2. Если существует такое число k > ||A n ||, что для

ISSN 1810-0198. Âåñòíèê ÒÃÓ, ò. 16, âûï. 4, 2011
Êàê è ðàíüøå, îáîçíà÷èì N (C, f ) ìíîæåñòâî ðåøåíèé ýòîãî óðàâíåíèÿ.
−1
−1
Ò å î ð å ì à 2. Åñëè ñóùåñòâóåò òàêîå ÷èñëî k > ||A−1
1 || · ||A2 || · ... · ||An ||, ÷òî äëÿ
ëþáîé òî÷êè
x ∈ BR [x0 ]
ñïðàâåäëèâî íåðàâåíñòâî
||C(x0 ) − f (x)|| <
òî N (C, f ) = Ø. Åñëè æå êðîìå
dim(N (C, f )) dim(Ker(C)).
ýòîãî
R
,
k
òî
dim(Ker(C)) > 0,
N (C, f ) ∩ ∂BR [x0 ] = Ø
è
Äîêàçàòåëüñòâî äàííîé òåîðåìû îñíîâûâàåòñÿ íà òåîðåìå 1. Èç òåîðåìû 2 âûòåêàþò
ñëåäóþùèå óòâåðæäåíèÿ.
Ñ ë å ä ñ ò â è å 1. Ïóñòü C : D(C) ⊂ E1 → En+1 ëèíåéíûé ñþðüåêòèâíûé îïå-
ðàòîð, óäîâëåòâîðÿþùèé óñëîâèÿì òåîðåìû 2, è f : E1 → En+1 âïîëíå íåïðåðûâíîå
îòîáðàæåíèå. Åñëè ñóùåñòâóþò ÷èñëà α 0 è β 0 òàêèå, ÷òî:
1) ||f (x)|| α||x|| + β äëÿ ëþáîãî x ∈ E1 ;
−1
−1
2) α · ||A−1
1 || · ||A2 || · ... · ||An || < 1.
Òîãäà óðàâíåíèå C(x) = f (x) èìååò ðåøåíèå.
òî dim(N (C, f )) dim(Ker(C)) è äëÿ ëþáîãî
R>
ãäå
Åñëè æå êðîìå ýòîãî
βk
,
1−k·α
−1
−1
||A−1
1 || · ||A2 || · ... · ||An || < k <
ñóùåñòâóåò òî÷êà
dim(Ker(C)) > 0,
1
,
α
òàêàÿ, ÷òî ||x|| = R.
C : D(C) ⊂ E1 → En+1 ëèíåéíûé ñþðüåêòèâíûé îïåóñëîâèÿì òåîðåìû 2, è B : E1 → En+1 ëèíåéíûé âïîëíå
x ∈ N (C, f )
Ñ ë å ä ñ ò â è å 2. Ïóñòü
ðàòîð, óäîâëåòâîðÿþùèé
íåïðåðûâíûé îïåðàòîð. Åñëè
−1
−1
||B|| · ||A−1
1 || · ||A2 || · ... · ||An || < 1,
òî
dim(Ker(C + B) dim(Ker(C)).
Äîêàçàòåëüñòâî ýòîãî ñëåäñòâèÿ âûòåêàåò èç ñëåäñòâèÿ 1.
ËÈÒÅÐÀÒÓÐÀ
Êðàñíîñåëüñêèé Ì.À., Çàáðåéêî Ï.Ï. Ãåîìåòðè÷åñêèå ìåòîäû íåëèíåéíîãî àíàëèçà. Ì: Íàóê, 1975.
Ãåëüìàí Á.Ä. Òîïîëîãè÷åñêèå ñâîéñòâà ìíîæåñòâà íåïîäâèæíûõ òî÷åê ìíîãîçíà÷íûõ îòîáðàæåíèé //
1.
2.
Ìàòåìàòè÷åñêèé ñáîðíèê. 1997. Ò. 188,  12. Ñ. 3356.
Ïîñòóïèëà â ðåäàêöèþ 10 àïðåëÿ 2011 ã.
Rydanova S. S. On one class of operator equations. In this paper we study the operator
equation with a linear surjective operator A, which may be not closed, but posesses continuous
mapping of right inverse mapping. We are interested in the existence of solutions and topological
dimension of the set of solutions.
Key words: quasireversible operator; surjective operator; topological degree of maps.
Ðûäàíîâà Ñâåòëàíà Ñåðãååâíà, Âîðîíåæñêèé ãîñóäàðñòâåííûé ïåäàãîãè÷åñêèé óíèâåðñèòåò, ã. Âîðîíåæ, Ðîññèéñêàÿ Ôåäåðàöèÿ, àñïèðàíò êàôåäðû âûñøåé ìàòåìàòèêè, e-mail:
[email protected].
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ÓÄÊ 517.929
Î ËÎÊÀËÜÍÎÉ ÓÑÒÎÉ×ÈÂÎÑÒÈ ÍÅÊÎÒÎÐÛÕ ÁÈÎËÎÃÈ×ÅÑÊÈÕ
ÌÎÄÅËÅÉ Ñ ÐÀÑÏÐÅÄÅËÅÍÍÛÌ ÇÀÏÀÇÄÛÂÀÍÈÅÌ
c Ò.Ë. Ñàáàòóëèíà
Êëþ÷åâûå ñëîâà
: ôóíêöèîíàëüíî-äèôôåðåíöèàëüíîå óðàâíåíèå; ðàñïðåäåëåííîå çàïàç-
äûâàíèå; ýêñïîíåíöèàëüíàÿ óñòîé÷èâîñòü; ôóíêöèÿ Êîøè.
 ðàáîòå ðàññìàòðèâàþòñÿ íåñêîëüêî íåëèíåéíûõ óðàâíåíèé ñ ðàñïðåäåëåííûì çàïàçäûâàíèåì, ÿâëÿþùèåñÿ ìîäåëÿìè äèíàìèêè ïîïóëÿöèé è êðîâåòâîðåíèÿ. Óñòîé÷èâîñòü
ðåøåíèé äàííûõ óðàâíåíèé èññëåäóåòñÿ ïî èõ ëèíåéíîìó ïðèáëèæåíèþ, ïðåäñòàâëÿþùåìó ñîáîé ôóíêöèîíàëüíî-äèôôåðåíöèàëüíîå óðàâíåíèå çàïàçäûâàþùåãî òèïà.
Ïóñòü R = (−∞, +∞) , R+ = [0, +∞) , Δ = {(t, s) ∈ R+ 2 : t s}.
Ìàòåìàòè÷åñêàÿ áèîëîãèÿ èíòåíñèâíî ðàçâèâàþùàÿñÿ îáëàñòü ïðèëîæåíèé ôóíêöèîíàëüíî-äèôôåðåíöèàëüíûõ óðàâíåíèé (ÔÄÓ).  äàííîé ðàáîòå èññëåäóåòñÿ ëîêàëüíàÿ
óñòîé÷èâîñòü îáîáùåííûõ ìîäåëåé Õàò÷èíñîíà, Íèêîëñîíà, ËàñîòûÂàæåâñêè è Ìýêêè
Ãëàññà [1], ïðåäñòàâëÿþùèõ ñîáîé íåëèíåéíûå ÔÄÓ. Ïåðâûå äâå ìîäåëè èñïîëüçóþòñÿ äëÿ
îïèñàíèÿ äèíàìèêè ïîïóëÿöèé, âòîðûå äâå äëÿ îïèñàíèÿ ïðîöåññîâ êðîâåòâîðåíèÿ. Íàñ
áóäóò èíòåðåñîâàòü óñëîâèÿ ñòàáèëèçàöèè ÷èñëåííîñòè ïîïóëÿöèè (êîëè÷åñòâà ýðèòðîöèòîâ â êðîâè) íà äîñòàòî÷íî áîëüøèõ âðåìåííûõ èíòåðâàëàõ, ò. å. ñâîéñòâà àñèìïòîòè÷åñêîé
óñòîé÷èâîñòè ñîîòâåòñòâóþùèõ óðàâíåíèé. Íåñìîòðÿ íà ñóùåñòâåííûå áèîëîãè÷åñêèå ðàçëè÷èÿ ìîäåëåé, èññëåäîâàíèå àñèìïòîòèêè ðàññìàòðèâàåìûõ íåëèíåéíûõ óðàâíåíèé ñâîäèòñÿ ê èçó÷åíèþ ëèíåéíîãî ÔÄÓ âèäà:
ẋ(t) + ax(t) +
0
t
x(s) ds r(t, s) = f (t),
t ∈ R+ .
(1)
ñóììèðóåìà, ôóíêöèÿ r(t, ·) íå óáûâàåò ïðè
Çäåñü r : Δ → R+ , ôóíêöèÿ r(·, s) ëîêàëüíî
êàæäîì ôèêñèðîâàííîì t , r(·, 0) = 0 , 0t ds r(t, s) = k ( k ∈ R+ ), f ëîêàëüíî ñóììèðóåìàÿ ôóíêöèÿ. Èíòåãðàë ïîíèìàåòñÿ â ñìûñëå ÐèìàíàÑòèëòüåñà. Áóäåì ñ÷èòàòü, ÷òî ïðè
îòðèöàòåëüíûõ çíà÷åíèÿõ àðãóìåíòà ôóíêöèÿ x ðàâíà íóëþ.
Îáîçíà÷èì H(t) = sup{s ∈ [0, t] : r(t , s ) ≡ 0 ∀t t, ∀s s} è
lim sup(t − H(t)) ω.
t→∞
(2)
Ïîä ðåøåíèåì óðàâíåíèÿ (1) ïîíèìàåòñÿ [2, c. 9] àáñîëþòíî íåïðåðûâíàÿ ôóíêöèÿ, óäîâëåòâîðÿþùàÿ äàííîìó óðàâíåíèþ ïî÷òè âñþäó.
Äëÿ ðåøåíèÿ óðàâíåíèÿ (1) ñïðàâåäëèâî ïðåäñòàâëåíèå [2, ñ. 84, òåîðåìà 1.1]
x(t) = C(t, 0)x(0) +
0
t
C(t, s)f (s) ds.
(3)
Ôóíêöèÿ C íàçûâàåòñÿ ôóíêöèåé Êîøè; â ñèëó ôîðìóëû (2) îíà ÿâëÿåòñÿ îñíîâíûì îáúåêòîì èññëåäîâàíèÿ ïðè èçó÷åíèè óðàâíåíèÿ (1).
Áóäåì ãîâîðèòü, ÷òî óðàâíåíèå (1) ýêñïîíåíöèàëüíî óñòîé÷èâî, åñëè ïðè íåêîòîðûõ
ïîëîæèòåëüíûõ N è γ äëÿ ëþáîãî t è ïî÷òè âñåõ s, òàêèõ, ÷òî (t, s) ∈ Δ , ñïðàâåäëèâà
îöåíêà |C(t, s)| N e−γ(t−s) .
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Äëÿ èññëåäîâàíèÿ ýêñïîíåíöèàëüíîé óñòîé÷èâîñòè âîñïîëüçóåìñÿ ò. í. ìåòîäîì testóðàâíåíèé. Ñóòü ìåòîäà çàêëþ÷àåòñÿ â ñëåäóþùåì. Ïî çàäàííûì ïàðàìåòðàì a , k , ω
óðàâíåíèþ (1) ñòàâèòñÿ â ñîîòâåòñòâèå test-óðàâíåíèå:
ẏ(t) = −ay(t) − ky(t − ω),
t ∈ R+ ,
ñ íà÷àëüíûì óñëîâèåì y(ξ) = 1 ïðè ξ 0. Îêàçàëîñü, ÷òî ýêñïîíåíöèàëüíàÿ óñòîé÷èâîñòü
öåëîãî êëàññà óðàâíåíèé âèäà (1) îïðåäåëÿåòñÿ ðàñïîëîæåíèåì òî÷êè ïåðâîãî ìèíèìóìà
ðåøåíèÿ test-óðàâíåíèÿ. Îá ýòîì ôàêòå ãîâîðèò òåîðåìà 1.
Ò å î ð å ì à 1. Ïóñòü a + k > 0 , l òî÷êà ïåðâîãî ìèíèìóìà ðåøåíèÿ test-óðàâíåíèÿ.
Òîãäà åñëè y(l) > −1 , òî óðàâíåíèå (1) ýêñïîíåíöèàëüíî óñòîé÷èâî ïðè âñåõ H(t) , óäîâëåòâîðÿþùèõ óñëîâèþ (2).
Âîïðîñ îá îöåíêå ïåðâîãî ìèíèìóìà ðåøåíèÿ test-óðàâíåíèÿ áûë ðåøåí Â.Â. Ìàëûãèíîé
â ðàáîòå [3] ïðè èññëåäîâàíèè óðàâíåíèé ñ ñîñðåäîòî÷åííûì çàïàçäûâàíèåì. Ïðèìåíÿÿ
ðåçóëüòàòû ýòîé ðàáîòû, ìîæíî ïîëó÷èòü ïðèçíàêè óñòîé÷èâîñòè äëÿ óðàâíåíèÿ (1) â âèäå
îáëàñòè íà ïëîñêîñòè â êîîðäèíàòàõ {aω, kω}.
 îáîáùåííûõ ìîäåëÿõ Õàò÷èíñîíà, Íèêîëñîíà, ËàñîòûÂàæåâñêè è ÌýêêèÃëàññà èç
áèîëîãè÷åñêîãî ñìûñëà ïàðàìåòðîâ ñëåäóåò, ÷òî a > 0 è k > 0.  ýòîì ñëó÷àå ãðàíèöà
îáëàñòè ýêñïîíåíöèàëüíîé óñòîé÷èâîñòè èìååò íàèáîëåå ïðîñòîé âèä. Ïðèâåäåì ñîîòâåòñòâóþùèé ðåçóëüòàò.
Ââåäåì ôóíêöèþ ρ ñëåäóþùèì îáðàçîì:
0,
s ∈ [0, 1],
ξ = ρ(s) =
s(s+1)
s ln s2 +1 , s ∈ (1, ∞).
Ò å î ð å ì à 2. Ïóñòü a > 0, k > 0 , e−aω > ρ( ka ). Òîãäà óðàâíåíèå (1) ýêñïîíåíöèàëüíî
óñòîé÷èâî.
Êàê ïîêàçàíî â ðàáîòå [3], äëÿ óðàâíåíèé ñ ñîñðåäîòî÷åííûì çàïàçäûâàíèåì ãðàíèöû
îáëàñòè ýêñïîíåíöèàëüíîé óñòîé÷èâîñòè ÿâëÿþòñÿ òî÷íûìè. Ïîñêîëüêó óðàâíåíèÿ ñ ñîñðåäîòî÷åííûì çàïàçäûâàíèåì ÿâëÿþòñÿ ÷àñòíûì ñëó÷àåì óðàâíåíèÿ (1), òî ãðàíèöû îáëàñòè
ÿâëÿþòñÿ òî÷íûìè è äëÿ óðàâíåíèÿ (1).
Ïîñòðîåííàÿ îáëàñòü ýêñïîíåíöèàëüíîé óñòîé÷èâîñòè óðàâíåíèÿ (1) ÿâëÿåòñÿ îáëàñòüþ
ëîêàëüíîé ýêñïîíåíöèàëüíîé óñòîé÷èâîñòè äëÿ íåëèíåéíûõ óðàâíåíèé, ÿâëÿþùèõñÿ ìîäåëÿìè Õàò÷èíñîíà, Íèêîëñîíà, ËàñîòûÂàæåâñêè è ÌýêêèÃëàññà.
ËÈÒÅÐÀÒÓÐÀ
1.
Berezansky L., Braverman E., Idels L. Nicholson's blowies dierential equations revisited: Main results
and open problems // Appl. Math. Model. 2010. V. 34.  6. P. 1405-1417.
2.
Àçáåëåâ Í.Â., Ìàêñèìîâ Â.Ï., Ðàõìàòóëëèíà Ë.Ô. Ââåäåíèå â òåîðèþ ôóíêöèîíàëüíî-äèôôåðåíöè-
àëüíûõ óðàâíåíèé. Ì.: Íàóêà, 1991.
3.
Ìàëûãèíà Â.Â. Îá óñòîé÷èâîñòè ðåøåíèé íåêîòîðûõ ëèíåéíûõ äèôôåðåíöèàëüíûõ óðàâíåíèé ñ ïî-
ñëåäåéñòâèåì // Èçâ. âóçîâ. Ìàòåìàòèêà. 1993.  5. Ñ. 72-85.
Ïîñòóïèëà â ðåäàêöèþ 10 àïðåëÿ 2011 ã.
Sabatulina T.L. On local stability of some biological models with distributed delay. In this
paper some nonlinear equations with distributed delay are considered. The equations are models
of population dynamics and hematopoiesis. Stability of solutions of the equations is studied by
means of linear approximation that is a delayed funcional-dierential equation.
Key words: functional dierential equation; distributed delay; exponential stability; the Cauchy function.
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Ñàáàòóëèíà Òàòüÿíà Ëåîíèäîâíà, Ïåðìñêèé ãîñóäàðñòâåííûé òåõíè÷åñêèé óíèâåðñèòåò,
ã. Ïåðìü, Ðîññèéñêàÿ Ôåäåðàöèÿ, êàíäèäàò ôèçèêî-ìàòåìàòè÷åñêèõ íàóê, àññèñòåíò êàôåäðû âû÷èñëèòåëüíîé ìàòåìàòèêè è ìåõàíèêè, e-mail: [email protected].
ÓÄÊ 517.958
ÎÁ ÎÄÍÎÉ ÑÈÍÃÓËßÐÍÎÉ ÝËËÈÏÒÈ×ÅÑÊÎÉ ÊÐÀÅÂÎÉ ÇÀÄÀ×Å Â
ÍÅÎÃÐÀÍÈ×ÅÍÍÎÉ ÎÁËÀÑÒÈ
c À. Þ. Ñàçîíîâ, Þ. Ã. Ôîìè÷åâà
Êëþ÷åâûå ñëîâà
: çàäà÷à Äèðèõëå; Â-ýëëèïòè÷åñêèé ñèíãóëÿðíûé îïåðàòîð; ôóíäàìåí-
òàëüíîå ðåøåíèå.
 ðàáîòå ðàñìîòðåíà çàäà÷à Äèðèõëå äëÿ Â-ýëëèïòè÷åñêîãî îïåðàòîðà ñ êðàåâûìè
óñëîâèÿìè íà ãèïåðïëîñêîñòè. Ïîëó÷åíî ðåøåíèå ýòîé çàäà÷è â ÿâíîì âèäå, îïðåäåëÿåìîå âåñîâûì ïîòåíöèàëîì äâîéíîãî ñëîÿ è âûðàæåííîå èíòåãðàëîì òèïà Ïóàññîíà.
Ïóñòü Rn+1 äåéñòâèòåëüíîå åâêëèäîâî ïðîñòðàíñòâî òî÷åê
Ðàññìàòðèâàåòñÿ çàäà÷à Äèðèõëå âèäà:
Bu = 0
â îáëàñòè
xn > 0, y > 0,
u|xn =0 = ϕ(x1 , ..., xn−1 , y),
x = (x1 , ..., xn , y) = (x , y).
(1)
∂u
|y=0 = 0,
∂y
(2)
ãäå B = ni,j=1 aij ∂x∂∂x + yb ∂y∂ y∂y∂ , b > 0 , k > 0 , aij óäîâëåòâîðÿþò îïðåäåëåííîìó â
[1] óñëîâèþ B - ýëëèïòè÷íîñòè.
Îáîçíà÷èì ÷åðåç A = det(aij ) , Aij àëãåáðàè÷åñêîå äîïîëíåíèå ýëåìåíòà aij ,
ξ = (ξ1 , ..., ξn , η) = (ξ , η). Ôóíäàìåíòàëüíîå ðåøåíèå H(x , ξ) óðàâíåíèÿ (1) èìååò ñëåäóþùèé âèä:
−
η2 ,
ïðè y = 0 H(x , ξ) = ρ1−n−k , ãäå ρ2 = ni,j=1 A−1Aij (ξi − xi )(ξ
j − xj ) + b
à â îáëàñòè y > 0 H(x, ξ) = Tηy H(x , ξ) , ãäå Tηy f = Ck 0π sink−1 αf ( η2 + y2 − 2ηy cos α)dα ,
2
i
j
k
k
1
2
Γ k+1
n+1
2
Ck = √ k , ξ ∈ R+
.
πΓ 2
Ðåøåíèå çàäà÷è (1)(2) îïðåäåëÿåòñÿ âåñîâûì ïîòåíöèàëîì äâîéíîãî ñëîÿ, ðàññìîòðåííûì â [3]. Ïëîòíîñòü âåñîâîãî ïîòåíöèàëà óäîâëåòâîðÿåò èíòåãðàëüíîìó óðàâíåíèþ, ÿäðî
êîòîðîãî èìååò ñëàáóþ îñîáåííîñòü è âûðàæàåòñÿ èíòåãðàëîì òèïà Ïóàññîíà:
n
2 Ain xi Bk
u(x) =
A
i=1
+∞
−∞
···
+∞
−∞
ϕ(x)Tηy ρ−n−k−1 η k dξ1 ...dξn−1 dη.
ËÈÒÅÐÀÒÓÐÀ
1.
Êèïðèÿíîâ È.À. Î êðàåâûõ çàäà÷àõ äëÿ óðàâíåíèé â ÷àñòíûõ ïðîèçâîäíûõ ñ äèôôåðåíöèàëüíûì
îïåðàòîðîì Áåññåëÿ // ÄÀÍ ÑÑÑÐ. 1964. Ò 158.  2.
1177