Existence of solutions to a class of evolution inclusions
type: Item
instance of: scholarly article
title: Existence of solutions to a class of evolution inclusions
zbMATH Open document ID: 1012.34060
DOI: 10.1016/S0362-546X(01)00799-4
author: Aurelian Cernea
author: Vasile Staicu
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2002-08-25T00:00:00Z
review text: The authors study the existence of local solutions to the following evolution system \[ x'\in -\partial _{F}V(x)+F(x)+f(t,x),\quad x(0)=x_{0},\tag{1} \] where \(\partial _{F}V\) is the Fréchet subdifferential of a function \(V\) with a \(\psi -\)monotone subdifferential of order \(2\), \(F\) is an upper semicontinuous, cyclically monotone multifunction, and \(f:[0,+\infty)\times \Omega \rightarrow \mathbb{R}^{n}\) is a Carathéodory function, where \(\Omega \) is an open subset of \(\mathbb{R}^{n}.\) Their result extends previous results by \textit{A. Cellina} and \textit{V. Staicu} [J. Differ. Equations 90, No. 1, 71-80 (1991; Zbl 0719.34030)], \textit{V. Staicu} [Rend. Semin. Mat., Torino 48, No. 2, 137-148 (1990; Zbl 0756.34022)], and \textit{F. Papalini} [J. Math. Anal. Appl. 193, No. 2, 406-418 (1995; Zbl 0835.34022)] who considered, respectively, the following special cases of (1): \(x'\in -\partial V(x)+F(x),\) \(x(0)=x_{0}\); \((2)\) \(x'\in -\partial V(x)+F(x)+f(t,x)\), \(x(0)=x_{0}\); \((3)\) \(x'\in -\partial _{F}V(x)+F(x),\) \(x(0)=x_{0}\); and \((4)\) where \(\partial V\) is the subdifferential of a proper convex lower semicontinuous function \(V,\) and \(F\) and \(f\) are as in (1).
reviewed by: Behzad Djafari Rouhani
Mathematics Subject Classification ID: 34G25
Mathematics Subject Classification ID: 34A60
zbMATH DE Number: 1788054
zbMATH Keywords: evolution inclusion
zbMATH Keywords: Fréchet subdifferential
zbMATH Keywords: cyclically monotone
zbMATH Keywords: Carathéodory function
zbMATH Keywords: lower and upper semicontinuous
zbMATH Keywords: psi-monotone subdifferential of order 2
MaRDI profile type: Publication
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full work available at URL: s0362-546x(01)00799-4
OpenAlex ID: W2013556656
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description: scientific article; zbMATH DE number 1788054