Mann iteration process for asymptotic pointwise nonexpansive mappings in metric spaces
type: Item
instance of: scholarly article
title: Mann iteration process for asymptotic pointwise nonexpansive mappings in metric spaces
zbMATH Open document ID: 1264.54060
author: A. R. Khan
author: Buthinah A. Bin Dehaish
author: M. A. Khamsi
published in: Journal of Mathematical Analysis and Applications
publication date: 2012-11-22T00:00:00Z
review text: Let \(C\) be a nonempty subset of \((M,d)\). A mapping \(T: C\to C\) is said to be asymptotic pointwise nonexpansive if for any \(x\in C\), there exists a sequence numbers \(k_n(x)\) for all \(n\geq 1\) such that \[ d(t^n(x),T^n(y))\leq k_n(x)\cdot d(x, y) \] for any \(y\in C\), and \(\lim_{n\to\infty} k_n(x)= 1\). In the present paper the authors prove the following results: Theorem. Let \((M,d)\) be a complete hyperbolic metric space which is 2-uniformly convex, \(C\subset M\) be nonempty closed convex bounded and \(T: C\to C\) be asymptotic pointwise nonexpansive. Then the fixed point set \(\text{Fix}(T)\) is nonempty closed and convex. Additionaly, if \(\sum_n (k_n(x)- 1)< \infty\) for any \(x\in C\) and there exist two real numbers \(a\), \(b\) such that \(0< a\leq t_n\leq b< 1\), \(n\geq 1\), then for any \(p\in\text{Fix}(T)\) and the Mann iteration process: \(x_1\in C\), \(x_{n+1}= t_nT^n(x_n)\oplus(1- t_n) x_n\), \(n\geq 1\), \(\lim_{n\to\infty} d(p,x_n)\) exists.
reviewed by: Jarosław Górnicki
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 47H09
zbMATH DE Number: 6110006
zbMATH Keywords: asymptotically nonexpansive mapping
zbMATH Keywords: fixed point
zbMATH Keywords: Mann iteration process
zbMATH Keywords: uniformly convex metric space
MaRDI profile type: Publication
full work available at URL: j.jmaa.2012.08.013
OpenAlex ID: W1990704124
DOI: 10.1016/J.JMAA.2012.08.013
description: scientific article; zbMATH DE number 6110006