On the weak-approximate fixed point property
type: Item
instance of: scholarly article
title: On the weak-approximate fixed point property
zbMATH Open document ID: 1188.47045
author: Cleon S. Barroso
author: Pei-Kee Lin
published in: Journal of Mathematical Analysis and Applications
publication date: 2010-02-19T00:00:00Z
review text: In [Proc.\ Am.\ Math.\ Soc.\ 93, 633--639 (1985; Zbl 0566.47039)], the second author and \textit{Y.\,Sternfeld} proved that, if \(C\) is a noncompact, closed, convex subset of a Banach space, then there exists a Lipschitz map \(f:C\to C\) without an approximate fixed point sequence, i.e., without a sequence \((x_n)\) in \(C\) such that \((x_n-f(x_n))\) converges to \(0\) in norm. It is possible, however, that there exists a sequence \((x_n)\) in \(C\) such that \((x_n-f(x_n))\) converges to \(0\) in the weak topology. A Banach space \(X\) has the \textit{weak approximate fixed point property} if, for every nonempty, closed, bounded, convex subset \(C\) of \(X\) and every continuous map \(f:C\to C\), there exists a sequence \((x_n)\) in \(C\) such that \((x_n-f(x_n))\) converges weakly to \(0\). In [Discrete Contin.\ Dyn.\ Syst.\ 25, No.\,2, 467--479 (2009; Zbl 1178.47034)], the first author proved that every reflexive space has the weak approximate fixed point property. In the article under review, the authors generalize this by showing that every Asplund space has the weak approximate fixed point property. The authors ask if every Banach space containing no isomorphic copy of \(\ell_1\) has the weak approximate fixed point property.
reviewed by: B. Turett
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 5673266
zbMATH Keywords: Asplund space
zbMATH Keywords: approximate fixed point sequence
zbMATH Keywords: asymptotic fixed point theory
zbMATH Keywords: weakly null sequences
zbMATH Keywords: Rosenthal's \(\ell _{1}\)-theorem
zbMATH Keywords: fixed point property
zbMATH Keywords: asymptotic approximation
zbMATH Keywords: weak topology
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.10.007
OpenAlex ID: W2000684458
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DOI: 10.1016/J.JMAA.2009.10.007
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description: scientific article; zbMATH DE number 5673266