Impulsive semilinear differential inclusions: topological structure of the solution set and solutions on non-compact domains
type: Item
instance of: scholarly article
title: Impulsive semilinear differential inclusions: topological structure of the solution set and solutions on non-compact domains
zbMATH Open document ID: 1147.34045
author: Tiziana Cardinali
author: Paola Rubbioni
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2008-06-18T00:00:00Z
review text: The following impulsive Cauchy problem is considered: \[ \begin{aligned} &x'(t) \in A(t)x(t) + F(t,x(t))\quad \text{a.e.} \,\, t\in [0,b],\,\,\, t \neq t_{k}, k=1,2,\dots ,m,\\ &x(t_{k}^{+}) = x(t_{k}) + I_{k}(x(t_{k})),\quad k=1,2,\dots ,m, \\ &x(0) = a_0 \in E, \end{aligned} \] where \(\{A(t)\}_{t\in [0,b]}\) is a family of linear operators (not necessarily bounded) in a Banach space \(E\) generating an evolution operator, \(F\) is a multifunction of Carathéodory type, \(0<t_0<t_1<\ldots<t_m<t_{m+1}=b,\) \(I_k:E\to E, k=1,\ldots,m,\) are impulse functions and \(x(t^+)=\lim_{s\to t^+}x(s).\) First a theorem on the compactness of the set of all mild solutions for the problem is given. Then this result is applied to obtain the existence of mild solutions for the impulsive Cauchy problem defined on noncompact domains. An example illustrating the abstract results is also presented.
Mathematics Subject Classification ID: 34G25
Mathematics Subject Classification ID: 34A37
zbMATH DE Number: 5290575
zbMATH Keywords: impulsive semilinear evolution differential inclusion
zbMATH Keywords: evolution system
zbMATH Keywords: mild solution
zbMATH Keywords: measure of non-compactness
zbMATH Keywords: upper semicontinuous multifunction
reviewed by: Sotiris K. Ntouyas
MaRDI profile type: Publication
full work available at URL: j.na.2007.05.001
OpenAlex ID: W2001178619
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DOI: 10.1016/J.NA.2007.05.001
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description: scientific article; zbMATH DE number 5290575