Some problems in metric fixed point theory
type: Item
instance of: scholarly article
title: Some problems in metric fixed point theory
zbMATH Open document ID: 1166.47053
author: Kazimierz Goebel
author: W. A. Kirk
published in: Journal of Fixed Point Theory and Applications
publication date: 2009-02-17T00:00:00Z
review text: This is a survey paper in which the authors highlight the following open problems. (I) Does either of the following implications hold? \(X\) is reflexive if and only if \(X\) has the FPP. (II) Does every superreflexive Banach space have the FPP? (III) Does \(g(C,T)=0\) imply that the nonexpansive mapping \(T\) has a fixed point in \(C\)? Does the condition \(g(C)=0\) imply that every nonexpansive self-mapping of \(C\) has a fixed point? (IV) Is there a class of spaces that properly includes the uniformly convex spaces for which Browder's demiclosedness principle holds? In particular, does the demiclosedness principle hold in spaces which are \(k\)-uniformly rotund (with \(k>1\)) in the sense of \textit{F.\,Sullivan} [Can.\ J.\ Math.\ 31, 628--636 (1979; Zbl 0422.46011)]? (V) If \(K\) is a nonempty bounded closed convex subset of a Banach space and if \(T, G\) are self commuting nonexpansive mappings, is \(F_{\epsilon}(T) \cap F_{\epsilon}(G)\) nonempty for every \(\epsilon>0\)? (VI) Specialises (V) by assuming that \(T,G\) are commuting and either \(T\) or \(G\) or both are strictly contractive. (VII) Is it possible to give a characterization of all multi-indices \(\alpha\) for which which the conclusions of the theorems in [\textit{K.\,Goebel} and \textit{M.\,J.\thinspace Pineda}, ``On a type of generalized nonexpansiveness'' (Preprint, 2007), to appear in: Proceedings ICFPTA 2007, Chiang Mai, 16--22 July 2007, Yokohama Publishers, 71--82 (2008)] about \(\alpha\)-nonexpansive mappings remains true? What about \(\alpha\)'s of infinite length? (VIII) In general, is Fix(\(\tilde T\)) a nonexpansive retract of \(\tilde K\)? (IX) Let X be a compact or, more generally, a geodesically bounded \(\mathbb{R}\)-tree, and let \(f,g:X\to X\) be commuting and Lipschitzian. If the Lipschitz constants of \(f\) and \(g\) are sufficiently near 1, do \(f\) and \(g\) have a common fixed point? (X) Let \(T\) be a finite tree in which each edge has length 1, and let \(f,g: T \to T\) be …
reviewed by: Hemant Kumar Nashine
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 05C05
Mathematics Subject Classification ID: 47H09
Mathematics Subject Classification ID: 47-02
zbMATH DE Number: 5508652
zbMATH Keywords: nonexpansive mappings
zbMATH Keywords: fixed points
zbMATH Keywords: approximate fixed points
zbMATH Keywords: \({\mathbb{R}}\)-trees
MaRDI profile type: Publication
full work available at URL: s11784-008-0076-3
OpenAlex ID: W1969051080
DOI: 10.1007/S11784-008-0076-3
description: scientific article; zbMATH DE number 5508652