The existence of solutions of infinite boundary value problems for first-order impulsive differential systems in Banach spaces
type: Item
instance of: scholarly article
title: The existence of solutions of infinite boundary value problems for first-order impulsive differential systems in Banach spaces
zbMATH Open document ID: 1189.34061
author: B. E. Eshmatov
published in: Journal of Computational and Applied Mathematics
publication date: 2008-11-18T00:00:00Z
review text: Consider in a Banach space the boundary value problem \[ \begin{cases} x'(t) & =f(t,x(t),y(t)),\\ y'(t) & =g(t,x(t),y(t)),\;t\in J',\\ \Delta x|_{t=t_k} & =I_k(x(t_k),y(t_k)),\\ \Delta y|_{t=t_k} & =J_k(x(t_k), y(t_k)),\;k=1,2,\dots,\\ x(\infty) &=\beta x(0),\quad y(\infty)= \delta y(0),\end{cases} \tag{*} \] where \(J=[0,\infty)\), \(0<t_1<\cdots<t_k <\cdots\), \(\lim_{k\to\infty}t_k= \infty\), \(J'=J\setminus\{t_1,\dots,t_k, \dots\}\), \(x(\infty)=\lim_{t\to\infty} x(t),y(\infty)=\lim_{t\to\infty} y(t)\), \(\beta>1\), \(\delta>1\), \(\Delta x|_{t=t_k}\) and \(\Delta y|_{t=t_k}\) denote the jumps of \(x(t)\) and \(y(t)\) at \(t=t_k\), respectively, i.e., \[ \Delta x|_{t=t_k}=x(t_k^+)-x(t_k^-),\quad \Delta y|_{t=t_k}=y(t_k^+)-y(t_k^-), \] here \(x(t_k^+)\), \(x(t_k^-)\) and \(y(t_k^+)\), \(y(t_k^-)\) represent the right and left limits of \(x(t)\) and \(y(t)\) at \(t=t_k\), respectively. The authors derive conditions such that (*) has at least one nonnegative solution. The proof is based on Schauder's fixed point theorem.
reviewed by: Klaus R. Schneider
Mathematics Subject Classification ID: 34B37
Mathematics Subject Classification ID: 34B40
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34B18
Mathematics Subject Classification ID: 47N20
Mathematics Subject Classification ID: 34G20
Mathematics Subject Classification ID: 34A37
zbMATH DE Number: 5368387
zbMATH Keywords: infinite boundary value
zbMATH Keywords: impulsive differential equation
zbMATH Keywords: Schauder fixed point theorem
MaRDI profile type: Publication
full work available at URL: j.cam.2007.11.020
OpenAlex ID: W2008713960
cites work: Q3994493
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DOI: 10.1016/J.CAM.2007.11.020
description: scientific article; zbMATH DE number 5368387