Directionally continuous selections and nonconvex evolution inclusions
type: Item
instance of: scholarly article
title: Directionally continuous selections and nonconvex evolution inclusions
zbMATH Open document ID: 1029.34051
DOI: 10.1023/A:1021930626923
author: Aurelian Cernea
author: Vasile Staicu
published in: Set-Valued Analysis
publication date: 2003-04-27T00:00:00Z
review text: In [\textit{A. Bressan} and \textit{V. Staicu}, Set-Valued Anal. 2, No. 3, 415-437 (1994; Zbl 0820.47072)], the evolution inclusion in a Banach space \[ \dot x\in Ax+ F(t,x)\tag{1} \] is considered, in which \(A\) is assumed to generate a contractive semigroup on \(\text{cl}(D(A))\), \(F\) is assumed to be bounded, lower semicontinuous and closed-valued (but not necessarily convex-valued) and the domain of \(F\), denoted by \(\Omega\subseteq\mathbb{R}\times \text{cl}(D(A))\), is assumed to be compact. There is introduced a special topology \(\tau\) (stronger than the metric one) on \(\mathbb{R}\times \text{cl}(D(A))\) and then, a theorem due to Bressan and Cortes is applied to obtain a selection \(f\) of \(F\) which is directionally continuous with respect to the topology \(\tau\). Then, a set-valued function \(G\), which is upper semicontinuous and compact- and convex-valued, is constructed from \(f\). Finally, it is shown that the solutions to the evolution inclusion \[ \dot x\in Ax+ G(t,x)\tag{2} \] are solutions to (1). In this way, existence of solutions and qualitative properties of solutions to (1) can be deduced from those for (2). In the article under review, the authors show that the topology \(\tau\) satisfies a certain property which allows them to apply a different theorem to obtain the solution \(f\), namely the one due to \textit{A. Bressan} and \textit{G. Colombo} [Stud. Math. 102, No. 3, 987-992 (1992; Zbl 0807.54020)]. In this way, the assumption of compactness of \(\Omega\) can be avoided. The authors actually deal here with a Scorza-Dragoni lower semicontinuous \(F\), a weaker assumption than that of lower semicontinuity.
Mathematics Subject Classification ID: 34G25
Mathematics Subject Classification ID: 47H20
Mathematics Subject Classification ID: 54C65
Mathematics Subject Classification ID: 54C60
zbMATH DE Number: 1901528
zbMATH Keywords: directionally continuous selections
zbMATH Keywords: evolution inclusions
zbMATH Keywords: semigroup of contractions
zbMATH Keywords: semicontinuous multifunctions
reviewed by: Daniel C. Biles
MaRDI profile type: Publication
full work available at URL: a:1021930626923
OpenAlex ID: W2297574836
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description: scientific article; zbMATH DE number 1901528