Strongly nonlinear multivalued boundary value problems
type: Item
instance of: scholarly article
title: Strongly nonlinear multivalued boundary value problems
zbMATH Open document ID: 1034.34017
DOI: 10.1016/S0362-546X(02)00162-1
author: Leszek Gasiński
author: Nikolaos S. Papageorgiou
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2003-03-11T00:00:00Z
review text: The authors study the periodic boundary value problem \[ \begin{aligned} &(a(x(t))\| x'(t)\| ^{p-2}x'(t))' \in A(x(t)) + F(t,x(t)) \quad \text{a.e. on } [0,T],\\ &x(0) = x(T), x'(0)=x'(T); \end{aligned} \] where \(A : \mathbb R^N \to \mathbb R^N\) is a maximal monotone operator, \(a : \mathbb R^N \to [c,d]\) is continuous, and \(F\) is a multivalued map with compact values satisfying a Hartman-type condition: ``\(\exists M> 0\) such that \(\langle x, y \rangle \geq 0\) \(\forall y \in F(t,x)\), for a.e. \(t \in [0,1]\) and all \(x \in \mathbb R^N\) with \(\| x\| = M\).'' The existence of a solution is established, respectively, for \(x \mapsto F(t,x)\) u.s.c. with convex values, and for \(x \mapsto F(t,x)\) l.s.c. The proofs rely on the theory of monotone and generalized pseudo-monotone operators, the theory of multivalued maps, and on the Fryskowski-Bressan-Colombo-selection theorem.
Mathematics Subject Classification ID: 34A60
zbMATH DE Number: 1880005
zbMATH Keywords: maximal monotone operator
zbMATH Keywords: differential inclusions
zbMATH Keywords: vector \(p\)-Laplacian
Wikidata QID: Q112251860
reviewed by: Marlène Frigon
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1880005