Nonlocal \(m\)-dissipative evolution inclusions in general Banach spaces
type: Item
instance of: scholarly article
title: Nonlocal \(m\)-dissipative evolution inclusions in general Banach spaces
zbMATH Open document ID: 1381.34082
author: Peng Zhang
published in: Mediterranean Journal of Mathematics
publication date: 2017-11-09T00:00:00Z
review text: Existence of solutions is proven for the nonlocal nonlinear evolution inclusion \[ \dot{x}\in Ax+F(t,x), \;t\in I\equiv \left[ t_{0},T\right], \] \[ x\left( t_{0}\right) =g\left( x\left( \cdot \right) \right) \] in a general Banach space \(X\), where \(A:D(A)\subseteq X\rightrightarrows X\) is an \(m\)-dissipative operator, the perturbation \(F:I\times X\rightrightarrows X\) has nonempty closed bounded values and \(g:C\left( I,X\right) \rightarrow \overline{D(A)}\). It is also assumed that \(F\left( \cdot ,x\right) \) has a strongly measurable selection for each \(x\in \overline{D(A)}\), \(F\) is Lebesgue integrably bounded and satisfies a weak Lipschitz type condition, and \(g\) is Lipschitz. The proof makes use of a variant of the Filippov-Pliss lemma, which concerns dependence on initial conditions. This result extends one found in [\textit{L. Zhu} et al., J. Math. Anal. Appl. 390, No. 2, 523--534 (2012; Zbl 1246.34059)]. A detailed example is also given.
reviewed by: Daniel C. Biles
Mathematics Subject Classification ID: 34G25
Mathematics Subject Classification ID: 34A45
Mathematics Subject Classification ID: 34B10
zbMATH DE Number: 6804311
zbMATH Keywords: nonlocal initial conditions
zbMATH Keywords: evolution inclusion
zbMATH Keywords: multivalued perturbation
zbMATH Keywords: existence of solutions
zbMATH Keywords: \(m\)-dissipative
zbMATH Keywords: successive approximations
MaRDI profile type: Publication
full work available at URL: s00009-017-1016-5
OpenAlex ID: W2757829911
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DOI: 10.1007/S00009-017-1016-5
description: scientific article; zbMATH DE number 6804311