Existence of positive solutions for the iterative system of the generalized impulsive boundary value problem

Project no. 32489 · Status: Kapanmış (Yayın bekleniyor) · 2024

Summary

<p>In&nbsp;this&nbsp;project,&nbsp;we&nbsp;investigate&nbsp;the&nbsp;existence&nbsp;of&nbsp;positive&nbsp;solutions&nbsp;for&nbsp;a&nbsp;system&nbsp;of&nbsp;nonlinear&nbsp;iterative&nbsp;impulsive&nbsp;differential&nbsp;equations&nbsp;of&nbsp;order&nbsp;$n&nbsp;\ge&nbsp;3$&nbsp;with&nbsp;multi-point&nbsp;boundary&nbsp;conditions.&nbsp;The&nbsp;equations&nbsp;in&nbsp;our&nbsp;problem&nbsp;exhibit&nbsp;a&nbsp;cyclic&nbsp;dependence&nbsp;($\kappa_{r+1}&nbsp;=&nbsp;\kappa_1$),&nbsp;where&nbsp;each&nbsp;equation&nbsp;relies&nbsp;on&nbsp;the&nbsp;solution&nbsp;of&nbsp;the&nbsp;next.&nbsp;Additionally,&nbsp;impulses&nbsp;occur&nbsp;in&nbsp;the&nbsp;$(n-2)$-th&nbsp;and&nbsp;$(n-1)$-th&nbsp;derivatives&nbsp;at&nbsp;specific&nbsp;points&nbsp;$t_p$.&nbsp;To&nbsp;overcome&nbsp;the&nbsp;analytical&nbsp;difficulties&nbsp;caused&nbsp;by&nbsp;the&nbsp;high&nbsp;order,&nbsp;we&nbsp;use&nbsp;the&nbsp;transformation&nbsp;$\kappa_i^{(n-2)}&nbsp;=&nbsp;y_i$&nbsp;to&nbsp;reduce&nbsp;the&nbsp;problem&nbsp;to&nbsp;an&nbsp;equivalent&nbsp;second-order&nbsp;impulsive&nbsp;system.&nbsp;We&nbsp;then&nbsp;construct&nbsp;a&nbsp;Green's&nbsp;function&nbsp;for&nbsp;this&nbsp;reduced&nbsp;problem&nbsp;and&nbsp;examine&nbsp;its&nbsp;basic&nbsp;properties.&nbsp;Using&nbsp;this&nbsp;integral&nbsp;representation,&nbsp;we&nbsp;define&nbsp;a&nbsp;composite&nbsp;operator&nbsp;$T&nbsp;=&nbsp;T_r&nbsp;\circ&nbsp;\dots&nbsp;\circ&nbsp;T_1$&nbsp;acting&nbsp;on&nbsp;a&nbsp;cone&nbsp;in&nbsp;the&nbsp;piecewise&nbsp;continuous&nbsp;function&nbsp;space&nbsp;$PC(J)$.&nbsp;We&nbsp;show&nbsp;that&nbsp;this&nbsp;operator&nbsp;is&nbsp;completely&nbsp;continuous&nbsp;via&nbsp;the&nbsp;PC-Arzelà-Ascoli&nbsp;theorem,&nbsp;and&nbsp;prove&nbsp;that&nbsp;it&nbsp;preserves&nbsp;an&nbsp;invariant&nbsp;ball&nbsp;under&nbsp;specific&nbsp;growth&nbsp;conditions&nbsp;on&nbsp;the&nbsp;nonlinear&nbsp;terms.&nbsp;Finally,&nbsp;assuming&nbsp;hypotheses&nbsp;(K1)-(K8)&nbsp;hold,&nbsp;we&nbsp;apply&nbsp;the&nbsp;Schauder&nbsp;fixed&nbsp;point&nbsp;theorem&nbsp;to&nbsp;establish&nbsp;the&nbsp;existence&nbsp;of&nbsp;at&nbsp;least&nbsp;one&nbsp;positive&nbsp;$r$-tuple&nbsp;solution.&nbsp;We&nbsp;conclude&nbsp;by&nbsp;providing&nbsp;a&nbsp;third-order,&nbsp;two-dimensional&nbsp;example&nbsp;to&nbsp;demonstrate&nbsp;the&nbsp;applicability&nbsp;of&nbsp;our&nbsp;theoretical&nbsp;results.</p>

View details