Mann and Ishikawa iterations with errors for asymptotically nonexpansive mappings
type: Item
instance of: scholarly article
title: Mann and Ishikawa iterations with errors for asymptotically nonexpansive mappings
zbMATH Open document ID: 0942.47046
DOI: 10.1016/S0898-1221(99)00040-1
publication date: 2000-01-20T00:00:00Z
review text: The paper concerns some generalizations of convergence of special type of iterations of asymptotically nonexpansive mappings in uniformly convex Banach spaces to a fixed point. A map \(T: E\to E\) defined on a subset of a Banach space is said to be asymptotically nonexpansive iff \[ \|T^nx- T^nq\|\leq k_n\|x-y\| \] for all \(x,y\in E\) and \(n\in\mathbb{N}\), where \((k_n)\) is a sequence of real numbers such that \[ k_n\geq 1\quad\text{and}\quad \lim_{n\to\infty} k_n= 1. \] The author considers the convergence of the following iteration process with errors \[ x_{n+1}= (1-\alpha_n) x_n+ \alpha_n T^n y_n+ u_n, \] \[ y_n= (1- \beta_n) x_n+ \beta_n T^nx_n+ v_n, \] where \(\{u_n\}\) and \(\{v_n\}\) are sequences in \(E\) satisfying \[ \sum^\infty_{n= 1}\|u_n\|<\infty\quad\text{and} \quad \sum^\infty_{n=1}\|v_n\|< \infty, \] and \(\{\alpha_n\}\) and \(\{\beta_n\}\) are sequences of real numbers in \([0,1]\). Under some additional assumptions it has been proved that the sequence of iterations \(\{x_n\}\) converges strongly to a fixed point of \(T\). The results presented here are some generalizations of the results obtained in 1994 by B. E. Rhoades. Let's note that some facts (see e.g. Lemma 1, Lemma 6) proved in the paper are very obvious.
reviewed by: Stefan Czerwik
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 47J25
Mathematics Subject Classification ID: 47H09
zbMATH DE Number: 1391502
zbMATH Keywords: iterations of asymptotically nonexpansive mappings
zbMATH Keywords: uniformly convex Banach spaces
zbMATH Keywords: fixed point
zbMATH Keywords: iteration process with errors
author: Zhenyu Huang
MaRDI profile type: Publication
cites work: Fixed point iterations for certain nonlinear mappings
cites work: Iterative construction of fixed points of asymptotically nonexpansive mappings
cites work: Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces
cites work: Fixed Points by a New Iteration Method
cites work: Mean Value Methods in Iteration
cites work: Inequalities in Banach spaces with applications
cites work: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process
cites work: Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings
cites work: A Fixed Point Theorem for Asymptotically Nonexpansive Mappings
published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 1391502