A characterization of the minimal invariant sets of Alspach's mapping
type: Item
instance of: scholarly article
title: A characterization of the minimal invariant sets of Alspach's mapping
zbMATH Open document ID: 1195.46028
author: Jerry B. Day
author: Chris Lennard
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2010-06-09T00:00:00Z
review text: If \(C\) is a nonempty, weakly compact, convex subset of a Banach space and \(T:C\to C\) is a nonexpansive mapping without a fixed point, it is well known that Zorn's Lemma ensures the existence of a \textit{minimal} nonempty, weakly compact, convex subset \(M\) of \(C\) such that \(T(M)\subset M\). Although the existence of minimal \(T\)-invariant sets is frequently used in proofs that particular classes of weakly compact, convex sets have the fixed point property, until now, no one had given an example of a minimal \(T\)-invariant set that could be explicitly described. In the article under review, the authors consider Alspach's example of a fixed point free, nonexpansive self-map \(T\) of the weakly compact, convex set \(C = \{f\in L^{1}[0,1]: 0\leq f\leq 1, \int f = 1/2\}\) [see \textit{D.\,E.\thinspace Alspach}, Proc.\ Am.\ Math.\ Soc.\ 82, 423--424 (1981; Zbl 0468.47036)]. The authors derive a formula for the \(n\)-th iterate \(T^n\) of Alspach's map and they use it to prove that, for each \(f \in C\), the sequence \((T^nf)\) converges to \(\frac12 \chi_{[0,1]}\). From this, the authors prove that the set \(C\) contains a unique minimal \(T\)-invariant subset and they give an explicit characterization of the minimal set. Finally, the authors provide an alternative proof of their result using ergodic theory.
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 47H09
zbMATH DE Number: 5718404
zbMATH Keywords: Alspach's mapping
zbMATH Keywords: nonexpansive mapping
zbMATH Keywords: fixed point free
zbMATH Keywords: weakly compact convex set
zbMATH Keywords: minimal invariant sets
zbMATH Keywords: Baker's transform
zbMATH Keywords: strongly mixing
reviewed by: B. Turett
MaRDI profile type: Publication
full work available at URL: j.na.2010.03.015
OpenAlex ID: W2022926610
cites work: A Fixed Point Free Nonexpansive Map
cites work: New fixed point free nonexpansive maps on weakly compact, convex subsets of L1[0,1]
cites work: On Commuting Families of Nonexpansive Operators
cites work: Q3310460
cites work: More on minimal invariant sets for nonexpansive mappings.
cites work: Q4423214
cites work: A Fixed Point Theorem for Mappings which do not Increase Distances
cites work: Handbook of metric fixed point theory
cites work: Q4744504
DOI: 10.1016/J.NA.2010.03.015
Recommended article: Properties of minimal invariant sets for nonexpansive mappings
Recommended article: Q4215640
Recommended article: Examples of fixed point free mappings
Recommended article: Separable Determination of the Fixed Point Property of Convex Sets in Banach Spaces
Recommended article: On a connection between the fixed point property and \(L^{\infty}(\Omega,\Sigma,\mu)\)
Recommended article: On generic convergence of successive approximations of mappings with convex and compact point images
Recommended article: Q3217169
Recommended article: Characterizations of fixed points of nonexpansive mappings
Recommended article: Q4469627
Recommended article: Approximating fixed points of nonexpansive mappings
description: scientific article; zbMATH DE number 5718404