Virtually stable maps and their fixed point sets
type: Item
instance of: scholarly article
title: Virtually stable maps and their fixed point sets
zbMATH Open document ID: 1181.54047
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-08-20T00:00:00Z
review text: Let \(X\) be a Hausdorff topological space and \(f: X \to X\) a continuous map whose fixed point set \(F(f)\) is nonempty. A point \(x \in F(f)\) is called virtually stable if for each of its neighborhoods \(U\) there exist a neighborhood \(V\) of \(x\) and an increasing sequence \(\{k_n\}\) of positive integers such that \(f^{k_n}(V) \subseteq U\) for all \(n \in \mathbb{N}.\) A map \(f\) is called virtually stable if all of its fixed points are virtually stable. The authors study topological properties of the convergence set \(C(f) = \{x \in X\): the sequence \(\{f^n(x)\}\}\) is convergent. In particular, it is shown that if \(X\) is regular and \(f\) is virtually stable, then \(F(f)\) is a retract of \(C(f).\) It is demonstrated also that under some additional assumption of generalized homogeneity of a virtually stable map \(f\) the set \(F(f)\) is star-convex.
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 47H09
Mathematics Subject Classification ID: 47H10
zbMATH DE Number: 5597529
zbMATH Keywords: fixed point
zbMATH Keywords: virtually stable map
zbMATH Keywords: nonexpansive map
zbMATH Keywords: fixed point set
zbMATH Keywords: convergence set
author: P. Chaoha
author: Watchareepan Atiponrat
reviewed by: Valerii V. Obukhovskij
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.06.015
OpenAlex ID: W2028854722
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DOI: 10.1016/J.JMAA.2009.06.015
description: scientific article; zbMATH DE number 5597529