Hybrid iteration method for common fixed points of a finite family of nonexpansive mappings in Banach spaces
type: Item
instance of: scholarly article
title: Hybrid iteration method for common fixed points of a finite family of nonexpansive mappings in Banach spaces
zbMATH Open document ID: 1184.47052
DOI: 10.1155/2009/678519
author: Lin Wang
author: Yi-Juan Chen
author: Rong-Chuan Du
published in: Mathematical Problems in Engineering
publication date: 2009-11-13T00:00:00Z
full work available at URL: https://eudml.org/doc/45879
review text: Let \(E\) be a real uniformly convex Banach space, and let \(\{T_i:i\in I\}\) be \(N\) nonexpansive mappings from \(E\) into itself with \(F=\{x\in E:T_ix=x\), \(i\in I\}\neq\varnothing\), where \(I=\{1,2,\dots,N\}\). For an arbitrary initial point \(x_1\in E\), the hybrid iteration scheme \(\{x_n\}\) is defined as follows: \(x_{n+1}=\alpha_nx_n+(1-\alpha_n)(T_nx_n-\lambda_{n+1}\mu A(T_nx_n))\), \(n\geq 1\), where \(A:E\to E\) is an \(L\)-Lipschitzian mapping, \(T_n=T_i\), \(i=n\pmod N\), \(1\leq i\leq N\), \(\mu >0\), \(\{\lambda_n\}\subset [0,1)\), and \(\{\alpha_n\}\subset [a,b]\) for some \(a,b\in (0,1)\). Under some suitable conditions, strong and weak convergence theorems of \(\{x_n\}\) to a common fixed point of the mappings \(\{T_i:i\to I\}\) are obtained. The results presented in this paper extend and improve the results of \textit{L.\,Wang} [Fixed Point Theory Appl.\ 2007, Article ID 28619 (2007; Zbl 1159.47052)] and partially improve the results of \textit{M.\,O.\thinspace Osilike, F.\,O.\thinspace Isiogugu} and \textit{P.\,U.\thinspace Nwokoro} [ibid., Article ID 64306 (2007; Zbl 1156.47311)].
Mathematics Subject Classification ID: 47J25
Mathematics Subject Classification ID: 47H09
Mathematics Subject Classification ID: 47H10
zbMATH DE Number: 5632521
zbMATH Keywords: uniformly convex Banach space
zbMATH Keywords: nonexpansive mappings
zbMATH Keywords: hybrid iteration scheme
zbMATH Keywords: \(L\)-Lipschitzian mapping
zbMATH Keywords: strong convergence
Wikidata QID: Q58649433
MaRDI profile type: Publication
OpenAlex ID: W2033647046
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description: scientific article; zbMATH DE number 5632521