Equilibria for set-valued maps on nonsmooth domains
type: Item
instance of: scholarly article
title: Equilibria for set-valued maps on nonsmooth domains
zbMATH Open document ID: 1188.47039
author: Hichem Ben-El-Mechaiekh
published in: Journal of Fixed Point Theory and Applications
publication date: 2009-09-18T00:00:00Z
review text: We say that a set-valued map \(\varphi: X\to 2^E\) defined on a closed subset \(X\) of a normed space \(E\) admits an equilibrium (or zero) if there exists a point \(u\in X\) such that \(0\in \varphi(u)\). In the paper under review, the following result is proved. Assume that \(X\) is a compact Lipschitzian retract of a normed space \(E\) having nontrivial Euler characteristic. Assume further that \(\varphi: X\to 2^E\) is upper semicontinuous map and belongs to the class of strongly approchable maps. If, moreover, \(\varphi\) satisfies so-called tangency condition, then \(\varphi\) admits an equilibrium. More than half of the paper consists of necessary definitions and preliminary results; only a short part of the paper is devoted to present this extension of an older result of the author for the class of nonconvex set-valued maps.
reviewed by: Stefan Czerwik
Mathematics Subject Classification ID: 47H04
Mathematics Subject Classification ID: 05C38
Mathematics Subject Classification ID: 54C15
zbMATH DE Number: 5604836
zbMATH Keywords: equilibria
zbMATH Keywords: strongly approachable maps
zbMATH Keywords: nonsmooth domains
zbMATH Keywords: approximative absolute neighborhood retracts
zbMATH Keywords: Lipschitzian retracts
zbMATH Keywords: Clarke's tangent cone
zbMATH Keywords: Euler characteristic
MaRDI profile type: Publication
full work available at URL: s11784-008-0074-5
OpenAlex ID: W2076015235
DOI: 10.1007/S11784-008-0074-5
description: scientific article; zbMATH DE number 5604836