Extensions of Browder's demiclosedness principle and Reich's lemma and their applications
type: Item
instance of: scholarly article
zbMATH Open document ID: 1384.47023
author: Rieko Kubota
author: Wataru Takahashi
author: Yukio Takeuchi
publication date: 2016-09-22T00:00:00Z
full work available at URL: p63.html
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 47H09
zbMATH DE Number: 6629189
zbMATH Keywords: Banach space
zbMATH Keywords: nonexpansive mapping
zbMATH Keywords: Browder's demiclosedness principle
zbMATH Keywords: weak convergence
zbMATH Keywords: Mann iteration
zbMATH Keywords: attractive point
zbMATH Keywords: condition (C)
MaRDI profile type: Publication
title: Extensions of Browder's demiclosedness principle and Reich's lemma and their applications
published in: Pure and Applied Functional Analysis
review text: Let \(E\) be a real Banach space and \(C\) be a closed subset of \(E\). A mapping \(T : C\rightarrow E\) is said to be demi-closed (at \(y\)) if, for any sequence \(\{x_n\}\) in \(C\), the conditions \(x_n \rightarrow \overline{x}\) weakly and \(Tx_n \rightarrow y\) strongly imply that \(T\overline{x} = y\).NEWLINENEWLINENEWLINEOne of the fundamental results in the theory of nonexpansive mappings is \textit{F. E. Browder}'s demiclosedness principle [Bull. Am. Math. Soc. 74, 660--665 (1968; Zbl 0164.44801)] which states that, if \(E\) is a uniformly convex Banach space, \(C\) is a closed convex subset of \(E\) and \(T : C\rightarrow E\) is nonexpansive, then \(I-T\) is demiclosed at \(0\).NEWLINENEWLINENEWLINEIn the paper under review, the authors study some fundamental properties of nonexpansive mappings and obtain extensions of Browder's demiclosedness principle and \textit{S. Reich}'s lemma [J. Math. Anal. Appl. 67, 274--276 (1979; Zbl 0423.47026)]. Using these results, they also obtain extensions of some weak convergence theorems due to Reich as well as \textit{S. H. Khan} and \textit{T. Suzuki} [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 80, 211--215 (2013; Zbl 1258.47069)].
reviewed by: Vasile Berinde
description: scientific article; zbMATH DE number 6629189