Monotone iterative technique for periodic boundary value problems with causal operators

Drici, Z.; McRae, F.A.; Devi, J. Vasundhara

doi:10.1016/j.na.2005.06.033

Cited by 45 publications

(25 citation statements)

References 2 publications

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“…Ðàçëè÷íûå çàäà÷è äëÿ ôóíêöèîíàëüíî-äèôôåðåíöèàëüíûõ óðàâíåíèé ñ êàóçàëüíûìè îïåðàòîðàìè áûëè ðàññìîòðåíû â ðàáîòàõ [410]. Â ÷àñòíîñòè, ãðàíè÷íàÿ è ïåðèîäè÷åñêàÿ ïðîáëåìû ÌÀÒÅÌÀÒÈ×ÅÑÊÎÅ ÌÎÄÅËÈÐÎÂÀÍÈÅ èçó÷àëèñü â [5,7]. Â íàñòîÿùåé ðàáîòå ïðèìåíÿåòñÿ ìåòîä íåãëàäêèõ èíòåãðàëüíûõ íàïðàâëÿþùèõ ôóíêöèé â èññëåäîâàíèè ïåðèîäè÷åñêîé çàäà÷è äëÿ äèôôåðåíöèàëü-íîãî âêëþ÷åíèÿ ñ ìíîãîçíà÷íûì êàóçàëüíûì îïåðàòîðîì.…”

Section: ââåäåíèåunclassified

Method of Nonsmooth Integral Guiding Functions in Periodic Solutions Problem for Inclusions with Causal Multioperators

Корнев¹

2016

Bulletin of the SUSU. MMP

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Êàê èçâåñòíî, äèôôåðåíöèàëüíûå âêëþ÷åíèÿ ÿâëÿþòñÿ î÷åíü óäîáíûì ìàòåìà-òè÷åñêèì àïïàðàòîì, ìîäåëèðóþùèì íåëèíåéíûå óïðàâëÿåìûå ñèñòåìû ñ îáðàòíîé ñâÿçüþ, ñèñòåìû àâòîìàòè÷åñêîãî ðåãóëèðîâàíèÿ, ñèñòåìû ñ ðàçðûâíûìè è èìïóëüñ-íûìè õàðàêòåðèñòèêàìè è äðóãèå îáúåêòû ñîâðåìåííîé èíaeåíåðèè, ìåõàíèêè, ôèçè-êè. Â íàñòîÿùåé ðàáîòå ïðåäëàãàþòñÿ íîâûå ìåòîäû ðåøåíèÿ çàäà÷è î ïåðèîäè÷åñêèõ êîëåáàíèÿõ óïðàâëÿåìûõ îáúåêòîâ, îïèñûâàåìûõ äèôôåðåíöèàëüíûì âêëþ÷åíèåì ñ êàóçàëüíûì îïåðàòîðîì. Âïåðâûå äèôôåðåíöèàëüíûå óðàâíåíèÿ ñ êàóçàëüíûì îïåðà-òîðîì, èëè óðàâíåíèÿ òèïà Âîëüòåððà, áûëè ðàññìîòðåíû Ë. Òîíåëëè è À.Í. Òèõîíî-âûì. À.Í. Òèõîíîâ èñïîëüçîâàë èõ â êà÷åñòâå ìîäåëè ïðè èçó÷åíèè ðÿäà çàäà÷ òåï-ëîïðîâîäíîñòè, â ÷àñòíîñòè, çàäà÷è îá îñòûâàíèè òåëà ïðè ëó÷åèñïóñêàíèè ñ ïîâåðõ-íîñòè. Â ïåðâîé ÷àñòè ðàáîòû ïðåäïîëàãàåòñÿ, ÷òî ïðàâàÿ ÷àñòü âêëþ÷åíèÿ ÿâëÿåòñÿ ìíîãîçíà÷íûì îòîáðàaeåíèåì, èìåþùèì âûïóêëûå çàìêíóòûå çíà÷åíèÿ. Äàëåå ïðåä-ïîëàãàåòñÿ, ÷òî ïðàâàÿ ÷àñòü âêëþ÷åíèÿ íåâûïóêëîçíà÷íà è ïîëóíåïðåðûâíà ñíèçó. Â ñèëó ñïåöèôèêè ðàññìàòðèâàåìîãî îáúåêòà â êà÷åñòâå îñíîâíîãî èíñòðóìåíòà èññëå-äîâàíèÿ ðàññìàòðèâàåìîé çàäà÷è â îáîèõ ñëó÷àÿõ èñïîëüçóåòñÿ ìîäèôèöèðîâàííûé ìåòîä êëàññè÷åñêîé íàïðàâëÿþùåé ôóíêöèè. À èìåííî, ìåòîä íåãëàäêîé èíòåãðàëü-íîé íàïðàâëÿþùåé ôóíêöèè. Ïðèìåíåíèå òåîðèè òîïîëîãè÷åñêîé ñòåïåíè è óêàçàí-íîãî ìåòîäà ïîçâîëÿåò óñòàíîâèòü ðàçðåøèìîñòü ïåðèîäè÷åñêîé çàäà÷è â êàaeäîì èç ðàññìàòðèâàåìûõ ñëó÷àåâ.Êëþ÷åâûå ñëîâà: âêëþ÷åíèå; êàóçàëüíûé îïåðàòîð; íåãëàäêàÿ èíòåãðàëüíàÿ íà-ïðàâëÿþùàÿ ôóíêöèÿ; ïåðèîäè÷åñêèå ðåøåíèÿ; òîïîëîãè÷åñêàÿ ñòåïåíü ñîâïàäåíèÿ. ÂâåäåíèåÈçó÷åíèå ñèñòåì, îïèñûâàåìûõ äèôôåðåíöèàëüíûìè è ôóíêöèîíàëüíûìè óðàâ-íåíèÿìè ñ êàóçàëüíûìè îïåðàòîðàìè, ââåäåííûìè Ë. Òîíåëëè [1] è À.Í. Òèõîíî-âûì [2], ïðèâëåêàåò âíèìàíèå ìíîãèõ èññëåäîâàòåëåé. Îòìåòèì, ÷òî â ñâîå âðåìÿ äëÿ èññëåäîâàíèÿ âëèÿíèÿ èçëó÷åíèÿ íà òåìïåðàòóðíûé ðåaeèì çåìíîé êîðû À.Í. Òèõî-íîâûì áûëè èçó÷åíû çàäà÷è äëÿ óðàâíåíèÿ òåïëîïðîâîäíîñòè ïðè íåëèíåéíûõ êðà-åâûõ óñëîâèÿõ. Èì áûëà ïðåäëîaeåíà ðåäóêöèÿ, ñâîäÿùàÿ ýòè çàäà÷è ê íåëèíåéíûì èíòåãðàëüíûì óðàâíåíèÿì òèïà Âîëüòåððà. Â êà÷åñòâå ïðèëîaeåíèé ïîëó÷åííûõ ðå-çóëüòàòîâ ê çàäà÷àì ìàòåìàòè÷åñêîé ôèçèêè áûë ðàññìîòðåí ðÿä çàäà÷ òåïëîïðîâîä-íîñòè, â ÷àñòíîñòè, çàäà÷à îá îñòûâàíèè òåëà ïðè ëó÷åèñïóñêàíèè ñ ïîâåðõíîñòè.Ïîíÿòèå êàóçàëüíîãî îïåðàòîðà îêàçàëîñü ìîùíûì èíñòðóìåíòîì äëÿ óíè-ôèêàöèè çàäà÷ â òåîðèè îáûêíîâåííûõ äèôôåðåíöèàëüíûõ óðàâíåíèé, èíòåãðî-äèôôåðåíöèàëüíûõ óðàâíåíèé, ôóíêöèîíàëüíî-äèôôåðåíöèàëüíûõ óðàâíåíèé ñ êî-íå÷íûì èëè áåñêîíå÷íûì çàïàçäûâàíèåì, èíòåãðàëüíûõ óðàâíåíèé Âîëüòåððà, ôóíêöèîíàëüíûõ óðàâíåíèé íåéòðàëüíîãî òèïà è äð. [3]. Ðàçëè÷íûå çàäà÷è äëÿ ôóíêöèîíàëüíî-äèôôåðåíöèàëüíûõ óðàâíåíèé ñ êàóçàëüíûìè îïåðàòîðàìè áûëè ðàññìîòðåíû â ðàáîòàõ [410]. Â ÷àñòíîñòè, ãðàíè÷íàÿ è ïåðèîäè÷åñêàÿ ïðîáëåìû 46

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Section: ââåäåíèåunclassified

Method of Nonsmooth Integral Guiding Functions in Periodic Solutions Problem for Inclusions with Causal Multioperators

Корнев¹

2016

Bulletin of the SUSU. MMP

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“…The details of the monotone iterative method can be found in [20][21][22][23][24][25][26][27] and for the abstract monotone iterative method, see [28,29]. To the best of the authors' knowledge, this is the first paper dealing with the integrodifferential equations in the present configuration.…”

Section: Boundary Value Problemsmentioning

confidence: 99%

The Monotone Iterative Technique for Three-Point Second-Order Integrodifferential Boundary Value Problems with p-Laplacian

Ahmad

Nieto

2007

Boundary Value Problems

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“…Its theory has the powerful quality of unifying ordinary differential equations, integro differential equations, differential equations with finite or infinite delay, Volterra integral equations and neutral functional equations. Readers can refer to the monograph [10], papers [2,3,4,6] for more details.…”

Section: Introductionmentioning

confidence: 99%

Difference equations involving causal operators with nonlinear boundary conditions

Wang¹,

Tian²

2016

J. Nonlinear Sci. Appl.

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Monotone iterative technique for periodic boundary value problems with causal operators

Cited by 45 publications

References 2 publications

Method of Nonsmooth Integral Guiding Functions in Periodic Solutions Problem for Inclusions with Causal Multioperators

Method of Nonsmooth Integral Guiding Functions in Periodic Solutions Problem for Inclusions with Causal Multioperators

The Monotone Iterative Technique for Three-Point Second-Order Integrodifferential Boundary Value Problems with p-Laplacian

Difference equations involving causal operators with nonlinear boundary conditions

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