Remarks on the structure of the fixed-point sets of uniformly Lipschitzian mappings in uniformly convex Banach spaces
type: Item
instance of: scholarly article
title: Remarks on the structure of the fixed-point sets of uniformly Lipschitzian mappings in uniformly convex Banach spaces
zbMATH Open document ID: 1172.47037
author: Jarosław Górnicki
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-05-19T00:00:00Z
review text: Let \(C\) be a nonempty, weakly compact, convex subset of a Banach space \(X\) and let \(T:C\to C\) be nonexpansive. If \(T\) has a fixed point in every nonempty, closed, bounded, convex set that it leaves invariant, \textit{R.\ E.\thinspace Bruck} [Trans.\ Am.\ Math.\ Soc.\ 179, 251--262 (1973; Zbl 0265.47043)] proved that the set \(F(T)\) of fixed points of \(T\) is a nonexpansive retract of \(C\). In the article under review, the author is able to relax the restriction on mapping \(T\), but at the cost of further restricting the Banach space. Recall that a map \(T:C\to C\) is uniformly \(k\)-Lipschitzian if \(\| T^nx -T^ny\| \leq k \| x-y \|\) for every \(x, y \in C\) and \(n\in \mathbb{N}\). The author proves that the fixed point set of a uniformly \(k\)-Lipschitzian self-map \(T\) of a nonempty, closed, bounded, convex set \(C\) in a uniform convex space is a nonexpansive retract of \(C\) if \(k<\alpha\), where \(\alpha\) is the unique solution of the equation \(\alpha^2\cdot \delta^{-1}_X (1 - 1/\alpha) \cdot \frac{1}{N(X)} = 1\) involving the modulus of convexity \(\delta_X\) and the normal structure coefficient \(N(X)\) of \(X\). In the setting that the uniformly convex space has a modulus of convexity of power type \(p\), the author derives another relationship involving \(p\) and \(N(X)\) that ensures that the set of fixed points of \(T\) is a nonexpansive retract of \(C\).
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 47H09
zbMATH DE Number: 5555253
zbMATH Keywords: uniformly Lipschitzian mapping
zbMATH Keywords: retraction
zbMATH Keywords: asymptotic center
zbMATH Keywords: fixed point
zbMATH Keywords: uniformly convex Banach space
reviewed by: B. Turett
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.02.003
OpenAlex ID: W1988506103
cites work: Measures of noncompactness in metric fixed point theory
cites work: Properties of Fixed-Point Sets of Nonexpansive Mappings in Banach Spaces
cites work: Normal structure coefficients for Banach spaces
cites work: Structure of the fixed point set and common fixed points of asymptotically nonexpansive mappings
cites work: A fixed point theorem for transformations whose iterates have uniform Lipschitz constant
cites work: Q3998720
cites work: On the Normal Structure Coefficient and the Bounded Sequence Coefficient
cites work: Q4000219
cites work: Geometrical background of metric fixed point theory
cites work: On the structure of fixed-point sets of uniformly Lipschitzian mappings
cites work: Fixed point theorems for Lipschitzian semigroups in Banach spaces
cites work: Inequalities in Banach spaces with applications
DOI: 10.1016/J.JMAA.2009.02.003
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description: scientific article; zbMATH DE number 5555253