Extremal solutions and relaxation for second-order vector differential inclusions
type: Item
instance of: scholarly article
zbMATH Open document ID: 0973.34010
author: Evgenios P. Avgerinos
author: Nikolaos S. Papageorgiou
publication date: 2001-06-10T00:00:00Z
full work available at URL: https://eudml.org/doc/232775
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34A60
zbMATH DE Number: 1602674
zbMATH Keywords: lower semicontinuous multifunctions
zbMATH Keywords: continuous embedding
zbMATH Keywords: compact embedding
zbMATH Keywords: continuous selector
zbMATH Keywords: extremal solution
zbMATH Keywords: relaxation theorem
MaRDI profile type: Publication
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title: Extremal solutions and relaxation for second-order vector differential inclusions
published in: Archivum Mathematicum
review text: Let \(F\) be a multifunction mapping the interval \([0,b]\) into the collection of all nonempty, compact and convex subsets of \(\mathbb{R}^N\). In the first part of the paper, the authors give sufficient conditions for the solvability of periodic differential inclusions NEWLINE\[NEWLINEx''(t)-x(t)\in \operatorname {ext}F(t,x(t),x'(t)) \quad \text{a.e. on }[0,b],\qquad x(0)=x(b),\quad x'(0)=x'(b),\tag{1}NEWLINE\]NEWLINE where \(\operatorname {ext}F(t,x,y)\) denotes the extreme points of \(F(t,x,y)\). In the second part of the paper, it is shown that under some assumptions on the multifunction \(F\) every solution to the Dirichlet problem NEWLINE\[NEWLINEx''(t)-x(t)\in \operatorname {ext}F(t,x(t),x'(t)) \quad \text{a.e. on }[0,b],\qquad x(0)=x(b)=0,\tag{2}NEWLINE\]NEWLINE can be obtained as the \(C^1([0,b];\mathbb{R}^N)\)-limit of a sequence of solutions to the ``extremal'' Dirichlet problem (1), (2).
reviewed by: Svatoslav Staněk
description: scientific article; zbMATH DE number 1602674