On fixed point property under Lipschitz and uniform embeddings
type: Item
instance of: scholarly article
title: On fixed point property under Lipschitz and uniform embeddings
zbMATH Open document ID: 1416.46023
DOI: 10.1155/2018/4758546
author: Lili Su
author: Ji Chao Zhang
author: Ling Xin Bao
published in: Journal of Function Spaces
publication date: 2019-01-11T00:00:00Z
review text: In [Stud. Math. 73, 225--251 (1982; Zbl 0506.46008)], \textit{S. Heinrich} and \textit{P. Mankiewicz} proved that every Lipschitz mapping from a separable \ Banach space to the dual of a separable space is almost everywhere weak$^*$ Gâteaux differentiable. In the article under review, the authors derive a local version of this result and give applications in fixed point theory. \par A point $x$ in a convex subset $C$ of a Banach space $X$ is a support point of $C$ if there exists a nonzero $x^*$ in the dual of $X$ such that $x^*(x) = \max\{x^*(y):y\in C\}$. The set $N(C)$ of non-support points of $C$ consists of the elements in $C$ that are not support points of $C$. The authors prove that, if $C$ is a closed convex subset of a separable Banach space with $N(C) \ne \emptyset$ and $Y$ is a separable Banach space, then a Lipschitz mapping $f:C\to Y^*$ is weak$^*$ Gâteaux differentiable almost everywhere (that is, off an Aronszajn null subset of $N(C)$). One consequence of this result is that, if $C$ is a nonempty closed bounded convex subset of a Banach space and $C$ Lipschitz embeds into a superreflexive space, then every isometry mapping $C$ into itself has a fixed point.
reviewed by: B. Turett
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46G05
Mathematics Subject Classification ID: 47H10
zbMATH DE Number: 7000072
zbMATH Keywords: Lipschitz mapping
zbMATH Keywords: Lipschitz embeddings
zbMATH Keywords: uniform embeddings
zbMATH Keywords: weak$^*$ Gâteaux differentiable
zbMATH Keywords: fixed point property for isometries
zbMATH Keywords: fixed point property for continuous affine mappings
zbMATH Keywords: Tsirelson space
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2018/4758546
OpenAlex ID: W2897109130
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Wikidata QID: Q129081132
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description: scientific article; zbMATH DE number 7000072