Generic existence of fixed points for set-valued mappings
type: Item
instance of: scholarly article
title: Generic existence of fixed points for set-valued mappings
zbMATH Open document ID: 1010.47029
DOI: 10.1023/A:1020602030873
author: Simeon Reich
author: Alexander J. Zaslavski
published in: Set-Valued Analysis
publication date: 2003-03-12T00:00:00Z
review text: The authors consider set-valued mappings acting on a closed convex subset of a Banach space which are nonexpansive with respect to the Hausdorff metric \(H\), and they study situations where, generically, such maps will have fixed points. More specifically, let \(K\) be a closed convex set with nonempty interior in a Banach space \(X\) and call \(S(K)\) the set of nonempty closed convex subsets of \(K\). Choose a point \(\theta\in K\) and endow the set \(M\) of nonexpansive mappings \(K\to S(K)\) with the uniformity \[ M(n)=\bigcap_{x\in K\cap B(\theta;n)}\{(T_1,T_2)\in M\times M\mid H(T_1x,T_2x)\leq 1/n\} \] for \(n\geq 1\). The authors show that there is an open dense set \(F\subset M\) such that for each \(T\in F\) there is an \(x\in K\) and a neighbourhood \(U\) of \(T\) in \(F\) such that \(x\) is a fixed point for each \(S\in U\). If \(K\) is bounded there is a similar result for mappings with convex graph.
Mathematics Subject Classification ID: 47H04
Mathematics Subject Classification ID: 47H09
zbMATH DE Number: 1880351
zbMATH Keywords: fixed point
zbMATH Keywords: set-valued mapping
zbMATH Keywords: generic property
zbMATH Keywords: nonexpansive map
reviewed by: Christian Fenske
MaRDI profile type: Publication
full work available at URL: a:1020602030873
OpenAlex ID: W150284419
description: scientific article; zbMATH DE number 1880351