The closed, convex hull of every ai \(c_{0}\)-summing basic sequence fails the FPP for affine nonexpansive mappings
type: Item
instance of: scholarly article
title: The closed, convex hull of every ai \(c_{0}\)-summing basic sequence fails the FPP for affine nonexpansive mappings
zbMATH Open document ID: 1236.47052
author: Chris Lennard
author: Veysel Nezir
published in: Journal of Mathematical Analysis and Applications
publication date: 2011-06-20T00:00:00Z
review text: In [Proc.\ Am.\ Math.\ Soc.\ 132, No.\ 6, 1659--1666 (2004; Zbl 1057.47064)], \textit{P.\ N.\ Dowling}, \textit{C.\ J.\ Lennard} and the reviewer showed that, if \(C\) is a nonempty, closed, bounded, convex subset of \(c_0\) that is not weakly compact, there exists a nonexpansive mapping of \(C\) into itself that fails to have a fixed point. Whether the fixed point free nonexpansive mapping can also be taken to be affine is an open question, and it is this question that the authors of the article under review investigate. The authors identify several non-weakly compact, closed, bounded, convex subsets of \(c_0\) which admit affine, nonexpansive, fixed point free self-mappings. For example, if, for some \(L>0\), \((x_n)\) is an \(L\)-scaled asymptotically isometric \(c_0\)-summing basis, in the sense that there exists \(0\leq \varepsilon_n\to 0\) such that \[ L \sup_{n\geq 1} \frac{1}{1+\varepsilon_n} \left| \sum_{j=n}^\infty t_j\right| \leq \left\| \sum_{j=1}^\infty t_j x_j\right\| \leq L \sup_{n\geq 1} (1+\varepsilon_n) \left| \sum_{j=n}^\infty t_j\right|\,, \] the authors construct an affine, nonexpansive mapping of \(\overline{\text{co}} \{ x_n:n\in \mathbb{N}\}\) into itself without a fixed point. The authors also prove that, if \((e_n)\) is the canonical vector basis in \(c_0\) and \((b_n)\) is a sequence with \(0<\inf_{n\in\mathbb{N}} b_n\leq \sup_{n\in\mathbb{N}} b_n<\infty\), then there exists an affine, nonexpansive self-mapping of the set \(\{\,\sum_{n=1}^\infty t_n b_n e_n: 1=t_1\geq t_2\geq \cdots \geq t_n\downarrow 0\}\) that fails to have a fixed point. Moreover, in both of these results, the authors note that the affine, fixed point free mappings can be constructed to be contractive.
reviewed by: B. Turett
Mathematics Subject Classification ID: 47H09
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 5909848
zbMATH Keywords: nonexpansive mapping
zbMATH Keywords: affine mapping
zbMATH Keywords: contractive mapping
zbMATH Keywords: fixed point property
zbMATH Keywords: asymptotically isometric \(c_0\)-summing basis
MaRDI profile type: Publication
full work available at URL: j.jmaa.2011.03.038
OpenAlex ID: W2088323501
cites work: Weak compactness and fixed point property for affine mappings.
cites work: Characterizations of weakly compact sets and new fixed point free maps in c0
cites work: Weak compactness is equivalent to the fixed point property in $c_0$
cites work: The Fixed Point Property in c0
cites work: Q3933449
DOI: 10.1016/J.JMAA.2011.03.038
description: scientific article; zbMATH DE number 5909848