Fixed point sets through iteration schemes
type: Item
instance of: scholarly article
title: Fixed point sets through iteration schemes
zbMATH Open document ID: 1228.54034
published in: Journal of Mathematical Analysis and Applications
publication date: 2011-10-21T00:00:00Z
review text: Let \((f_n)\) be a sequence of selfmaps on a Hausdorff space \(X\), \(s_n=\prod_{i=1}^nf_i\) and \(\mathcal S=(s_n)\). The sequence \(\mathcal S\) is called a fixed point iteration scheme (or simply a scheme) on \(X\) if \(\emptyset\neq F(\mathcal S)=r(\mathcal C(\mathcal S))\), where \(F(\mathcal S)=\bigcap_{f\in\mathcal S}F(f)\), \(F(f)\) is the fixed point set of \(f\), \(\mathcal C(\mathcal S)=\{x\in X:(s_n(x))\text{ converges}\}\), and \(r:\mathcal C(\mathcal S)\to F(\mathcal S)\) is defined by \(r(x)=\lim_{n\to\infty}s_n(x)\) for all \(x\in\mathcal C(\mathcal S)\). The scheme \(\mathcal S\) is said to have a continuous subsequence if there is a subsequence \((s_{c_n})\) of \(\mathcal S\) consisting of continuous maps. The authors call the scheme \(\mathcal S\) virtually stable if every point \(p\in F(\mathcal S)\) is virtually stable, i.e., for each neighborhood \(U\) of \(p\), there exist a neighborhood \(V\) of \(p\) and an increasing sequence \((k_n)\) of positive integers such that \(\prod_{i=j}^{k_n}f_i(V)\subseteq U\) for all \(n\in\mathbb{N}\) and \(j\leq k_n\). They prove the following: if \(X\) is a regular space and \(\mathcal S=(s_n)=(\prod_{i=1}^nf_i)\) is a virtually stable scheme having a continuous subsequence, then the map \(r:\mathcal C(\mathcal S)\to F(\mathcal S)\) defined above is continuous and hence \(F(\mathcal S)\) is a retract of \(\mathcal C(\mathcal S)\). The result is applied to obtain a new criterion for contractibility of the fixed point set of a Suzuki-type generalized nonexpansive selfmap on a compact convex space. Examples are provided to illustrate these results.
reviewed by: Zoran Kadelburg
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 47H10
zbMATH DE Number: 5960928
zbMATH Keywords: iteration scheme
zbMATH Keywords: fixed point set
zbMATH Keywords: virtually stable
author: P. Chaoha
author: Parunyou Chanthorn
MaRDI profile type: Publication
full work available at URL: j.jmaa.2011.07.065
OpenAlex ID: W2033206711
cites work: Structure of the fixed point set and common fixed points of asymptotically nonexpansive mappings
cites work: Properties of Fixed-Point Sets of Nonexpansive Mappings in Banach Spaces
cites work: Virtually nonexpansive maps and their convergence sets
cites work: Virtually stable maps and their fixed point sets
cites work: A note on fixed point sets in CAT(0) spaces
cites work: Q3998720
cites work: Fixed point theorems and convergence theorems for some generalized nonexpansive mappings
cites work: Iterative approximation of fixed points
DOI: 10.1016/J.JMAA.2011.07.065
Recommended article: Fixed points by certain iterative schemes with applications
Recommended article: Q4287226
Recommended article: Fixed point theory and iteration procedures
Recommended article: Construction of fixed points by some iterative schemes
Recommended article: Q3123518
Recommended article: Q4254998
Recommended article: Some fixed point iteration procedures
Recommended article: Iterative construction of fixed points
Recommended article: Iterative approximation of fixed points
Recommended article: Q4423213
description: scientific article; zbMATH DE number 5960928