Nonlocal Cauchy problems governed by compact operator families
type: Item
instance of: scholarly article
title: Nonlocal Cauchy problems governed by compact operator families
zbMATH Open document ID: 1083.34045
author: Jin Liang
author: J. Liu
author: Ti-Jun Xiao
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2004-09-23T00:00:00Z
review text: Let \(A\) be the infinitesimal generator of a compact semigroup of linear operators on a Banach space \(X\). The authors establish the existence of mild solutions to the nonlocal Cauchy problem \[ u'(t)=Au(t)+f(t,u(t)), \quad t\in[t_0,t_0+T], \quad u(t_0)+g(u)=u_0, \] under some conditions on \(f\) and \(g\), where \(f:[t_0,t_0+T]\times X\to X\) and \(g:C([t_0,t_0+T];X)\to X\) are given functions. They assume a Lipschitz condition on \(f\) with respect to \(u\), but they do not require any compactness assumption on \(g\), opposed to \textit{S. Aizicovici} and \textit{M. McKibben} [Nonlinear Anal., Theory Methods Appl. 39, No. 5(A), 649--668 (2000; Zbl 0954.34055)] and \textit{L. Byszewski} and \textit{H. Akca} [Nonlinear Anal., Theory Methods Appl. 34, No. 1, 65--72 (1998; Zbl 0934.34068)], where the authors assume a compactness property for \(g\), but do not require any Lipschitz condition on \(f\).
Mathematics Subject Classification ID: 34G20
Mathematics Subject Classification ID: 47D06
zbMATH DE Number: 2102565
zbMATH Keywords: nonlocal Cauchy problem
zbMATH Keywords: mild solution
zbMATH Keywords: compact operator families
zbMATH Keywords: equicontinuous family of functions
zbMATH Keywords: integrodifferential equation
reviewed by: Behzad Djafari Rouhani
MaRDI profile type: Publication
full work available at URL: j.na.2004.02.007
OpenAlex ID: W1970688750
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DOI: 10.1016/J.NA.2004.02.007
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description: scientific article; zbMATH DE number 2102565