On a connection between the fixed point property and \(L^{\infty}(\Omega,\Sigma,\mu)\)
type: Item
instance of: scholarly article
title: On a connection between the fixed point property and \(L^{\infty}(\Omega,\Sigma,\mu)\)
zbMATH Open document ID: 1121.46015
author: Cleon S. Barroso
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2007-04-13T00:00:00Z
review text: In this article, the author investigates when weakly compact, convex subsets of \(L^\infty(\Omega, \Sigma, \mu)\) have the fixed point property for certain nonexpansive mappings. In particular, if \(K\) is a subset of \(L^\infty(\Omega, \Sigma, \mu)\), a mapping \(T:K\to K\) is called pointwise nonexpansive if, for every \(f, g \in K\), \(| T(f)(s) - T(g)(s)| \leq | f(s) - g(s)| \) for almost all \(s \in \Omega\). The author shows that weakly compact, convex subsets of \(L^\infty(\Omega, \Sigma, \mu)\) have the fixed point property for pointwise nonexpansive mappings. The author then introduces a family of Banach spaces of harmonic functions so that the Banach spaces continuously embed in \(L^\infty(\Omega)\), where \(\Omega\) is a bounded domain in \(\mathbb{R}^n\) and the Banach spaces have the fixed point property for nonexpansive mappings. [Reviewer's remark: The author mistakenly writes (on pages 1722--1723) that ``\(\text{weak}^*\) compact subsets of \(L^\infty(\Omega, \Sigma, \mu)\) always have the fixed point property for nonexpansive self-mappings.'' In fact, \textit{D.\,Alspach} [Proc.\ Am.\ Math.\ Soc.\ 82, 423--424 (1981; Zbl 0468.47038)] provided an example of a weakly compact, convex subset of \(L^1[0,1]\) which fails the fixed point property for nonexpansive mappings. Since \(L^1[0,1]\) embeds isometrically in \(L^\infty[0,1]\), Alspach's example gives a weakly (and \(\text{weak}^*\)) compact, convex set in \(L^\infty[0,1]\) without the fixed point property for nonexpansive self-mappings.]
reviewed by: B. Turett
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 47H09
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 5142225
zbMATH Keywords: fixed point property
zbMATH Keywords: pointwise nonexpansive mapping
zbMATH Keywords: weakly compact convex set
zbMATH Keywords: \(L^\infty(\Omega,\Sigma, \mu)\)
MaRDI profile type: Publication
full work available at URL: j.na.2006.02.029
OpenAlex ID: W1497150193
cites work: Equivalent Norms for Sobolev Spaces
cites work: Nonstandard methods in fixed point theory
cites work: A Fixed Point Free Nonexpansive Map
cites work: Q3733192
cites work: The \(\tau\)-fixed point property for nonexpansive mappings
cites work: FIXED-POINT THEOREMS FOR NONCOMPACT MAPPINGS IN HILBERT SPACE
cites work: NONEXPANSIVE NONLINEAR OPERATORS IN A BANACH SPACE
cites work: Q3999065
cites work: The fixed point property for subsets of some classical Banach spaces
cites work: Equivalent Norms and the Fixed Point Property for Nonexpansive Mappings
cites work: Q4141722
cites work: A Fixed Point Theorem for Mappings which do not Increase Distances
cites work: Reflexivity and approximate fixed points
cites work: On the behavior of weak convergence under nonlinearities and applications
cites work: The Fixed Point Property for Non-Expansive Mappings
cites work: The Fixed Point Property for Non-Expansive Mappings, II
cites work: Existence of Fixed Points of Nonexpansive Mappings in Certain Banach Lattices
cites work: Q4767779
DOI: 10.1016/J.NA.2006.02.029
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description: scientific article; zbMATH DE number 5142225