Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces
type: Item
instance of: scholarly article
title: Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces
zbMATH Open document ID: 1062.47069
author: Jong Soo Jung
published in: Journal of Mathematical Analysis and Applications
publication date: 2005-02-22T00:00:00Z
review text: Let \(E\) be a real Banach space with norm \(\|\cdot\|\), \(E^*\) its dual, \(\langle x,f\rangle\) denote the value of \(f\in E^*\) at \(x\in E\) and \(x_n\to x\) (respectively \(x_n\rightharpoonup x\), \(x_n\overset{*}\rightharpoonup x\)) denote strong (respectively weak, weak\(^*\)) convergence of the sequence \(\{x_n\}\) to \(x\). The norm of \(E\) is said to be smooth if \(\lim_{t\to 0}{\| x+ ty\|=\| x\|\over t}\) exists for each \(x\), \(y\) in its unit spere \(U= \{x\in E:\| x\|= 1\}\) and is said to be uniformly smooth if the limit is attained uniformly for \((x,y)\in U\times U\). The duality mapping \(I\) from \(E\) into the family of nonempty weak-star compact subsets of its dual \(E^*\) is defined by \(I(x)= \{f\in E^*:\langle x,f\rangle= \| x\|^2=\| f\|^2\}\) for each \(x\in E\). \(I\) is single-valued if and only if \(E\) is smooth. The single-valued \(I\) is said to be weakly sequentially continuous if for each \(\{x_n\}\in E\) with \(x_n\rightharpoonup x\), \(I(x_n)\overset{*}\rightharpoonup I(x)\). In this paper, the author establishes the strong convergence of the iteration scheme \(\{x_n\}\) defined by \(x_{n+1}= \lambda_{n+1} a+(1- \lambda_{n+1}) T_{n+1} x_n\), \(n\geq 0\), \(a\), \(x_0\) in a closed convex subset of \(E_2\) for infinitely many nonexpansive mappings \(T_n: C\to C\) in a uniformly smooth Banach space \(E\) with a weakly sequentially continuous duality maping. The main theorem (Theorem 5) extends a recent result of \textit{J. G. O'Hara}, \textit{P. Pillay} and \textit{H.-K. Xu} [Nonlinear Anal., Theory Methods Appl. 54A, No. 8, 1417--1426 (2003; Zbl 1052.47049)] to a Banach space setting. For the same iteration scheme, with finitely many mappings, a complementary result to a result of \textit{J. S. Jung} and \textit{T. H. Kim} [Bull. Korean Math. Soc. 34, No. 1, 83--102 (1997; Zbl 0885.47020)] (see also \textit{H. H. Bauschke} [J. Math. Anal. Appl. 202, No. 1, 150--159 (1996; Zbl 0956.47024)]) is obtained by imposing some other condi…
Mathematics Subject Classification ID: 47J25
Mathematics Subject Classification ID: 47H09
Mathematics Subject Classification ID: 47H10
zbMATH DE Number: 2136665
zbMATH Keywords: smooth norm
zbMATH Keywords: uniformly smooth norm
zbMATH Keywords: duality mapping
zbMATH Keywords: nonexpansive mapping
zbMATH Keywords: iteration scheme
zbMATH Keywords: weak sequential continuity
zbMATH Keywords: common fixed points
zbMATH Keywords: sunny and nonexpansive retraction
reviewed by: T. D. Narang
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.08.022
OpenAlex ID: W2063298310
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DOI: 10.1016/J.JMAA.2004.08.022
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description: scientific article; zbMATH DE number 2136665