Iterative methods for solving fixed-point problems with nonself-mappings in Banach spaces
type: Item
instance of: scholarly article
title: Iterative methods for solving fixed-point problems with nonself-mappings in Banach spaces
zbMATH Open document ID: 1028.47049
DOI: 10.1155/S1085337503203018
author: Simeon Reich
author: Jen-Chih Yao
author: Yakov Alber
published in: Abstract and Applied Analysis
publication date: 2003-03-17T00:00:00Z
full work available at URL: https://eudml.org/doc/50482
review text: Let \(G\) be a closed convex subset of a Banach space \(B\). A mapping \(A:G\to B\) is said to be (i) nonexpansive if for all \(x,y\in G\), \(\|Ax-Ay\|\leq\|x-y\|\), (ii) weakly contractive of class \(C_{\psi(t)}\) on \(G\) if there exists a continuous and increasing function \(\psi(t)\) defined on \(\mathbb{R}^+\) such that \(\psi\) is positive on \(\mathbb{R}^+\setminus \{0\}\), \(\psi(0)=0\), \(\lim_{t\to+ \infty}\psi(t) =+\infty\), and for all \(x,y\in G\), \(\|Ax-Ay\|\leq\|x-y\|-\psi(\|x-y\|)\). The authors study descent-like approximation methods and proximal methods of retraction type for solving fixed-point problems with nonself-mappings in Hilbert and Banach spaces. They prove strong and weak convergence for weakly contractive and nonexpansive maps, respectively. They also establish the stability of these methods with respect to perturbations of the operators and the constraint sets.
Mathematics Subject Classification ID: 47J25
Mathematics Subject Classification ID: 47H06
Mathematics Subject Classification ID: 47H09
Mathematics Subject Classification ID: 47H10
zbMATH DE Number: 1883745
zbMATH Keywords: weakly contraction maps
zbMATH Keywords: descent-like approximation methods
zbMATH Keywords: proximal methods
zbMATH Keywords: convergence
zbMATH Keywords: nonexpansive maps
zbMATH Keywords: stability
reviewed by: T. D. Narang
MaRDI profile type: Publication
OpenAlex ID: W2080294581
Recommended article: Iterative schemes for computing fixed points of nonexpansive mappings in Banach spaces
Recommended article: Iteration process for solving a fixed point problem of nonexpansive mappings in Banach spaces
Recommended article: Iterative Methods for Nonexpansive Mappings in Banach Spaces
Recommended article: Q5446027
Recommended article: Q3685949
Recommended article: Q3500655
Recommended article: Q5048258
Recommended article: Iterative approximation problem of fixed points for asymptotically nonexpansive mappings in Banach spaces
Recommended article: Iterative algorithms for solving variational inequalities and fixed point problems for asymptotically nonexpansive mappings in Banach spaces
Recommended article: Q4484989
description: scientific article; zbMATH DE number 1883745