Some fixed point results in \(\ell^1\) and \(c_0\)
type: Item
instance of: scholarly article
title: Some fixed point results in \(\ell^1\) and \(c_0\)
zbMATH Open document ID: 0954.47043
DOI: 10.1016/S0362-546X(98)00259-4
author: P. N. Dowling
author: C. J. Lennard
author: B. Turett
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2001-02-08T00:00:00Z
review text: \textit{R. C. James} [Ann. of Math., II. Ser. 80, 542-550 (1964; Zbl 0132.08902)] proved the necessary and sufficient conditions in order that a Banach space contains an isomorphic copy of \(\ell^1\) or \(c_0\), respectively. This result is used by the authors to construct some Lipschitzian maps defined on a closed, bounded and convex subset into itself without fixed point property. The authors introduce the idea that a Banach space contains an ``asymptotically isometric'' copy of \(\ell^1\) or \(c_0\), respectively and they proved some necessary and sufficient conditions related to this problem. Let us recall that a mapping \(T: C\to C\) is said to be asymptotically nonexpansive if \[ \|T^nx- T^ny\|\leq k_n\|x- y\|\quad\text{for all }x,y\in C\quad\text{and }n\in\mathbb{N}, \] where a sequence of real numbers \(k_n\) converges to one. The following result proved. Theorem. If a Banach space \(X\) contains an isomorphic copy of \(c_0\), then there exists a closed, bounded, convex subset \(C\) of \(X\) and an asymptotically nonexpansive mapping \(T: C\to C\) without a fixed point. The authors pointed out that it is unknown whether the corresponding result for \(\ell^1\) holds.
reviewed by: Stefan Czerwik
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 47H09
Mathematics Subject Classification ID: 46B25
zbMATH DE Number: 1419537
zbMATH Keywords: James' distorsion theorem
zbMATH Keywords: asymptotically isometric
zbMATH Keywords: Lipschitzian maps
zbMATH Keywords: fixed point property
zbMATH Keywords: asymptotically nonexpansive
MaRDI profile type: Publication
full work available at URL: s0362-546x(98)00259-4
OpenAlex ID: W1531426705
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description: scientific article; zbMATH DE number 1419537