Some problems in metric fixed point theory
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 commuting and Lipschitzian. If the Lipschitz constants of \(f\) and \(g\) are sufficiently near 1, do \(f\) and \(g\) have a common fixed point?
scientific article; zbMATH DE number 5508652