Approximating the zeros of accretive operators by the Ishikawa iteration process
type: Item
instance of: scholarly article
title: Approximating the zeros of accretive operators by the Ishikawa iteration process
zbMATH Open document ID: 0945.47044
DOI: 10.1155/S1085337596000073
author: Yuting Jia
author: Haiyun Zhou
published in: Abstract and Applied Analysis
publication date: 2000-10-04T00:00:00Z
full work available at URL: https://eudml.org/doc/48529
full work available at URL: issue-2.html
review text: Given a Banach space \(X\) and an operator \(A:X\to X\), the Ishikawa iteration \(\{x_n\}\) is defined by \(x_{n+1}=x_n-\alpha_n Ay_n-\alpha_n\beta_n Ax_n\), \(y_n=x_n-\beta_n Ax_n\). A main result states, if \(X\) is uniformly smooth, \(A\) is bounded and accretive, and the equation \(Ax=0\) has at least a solution, then the iteration process converges to a solution to the equation. Given a convex, closed subset \(C\) of \(X\) and an operator \(T:C\to C\), the Ishikawa iteration is rephrased to \(x_{n+1}=(1-\alpha_n)x_n+\alpha_n Ty_n\), \(y_n=(1-\beta_n)x_n+\beta_n Tx_n\). Another main result states, if \(X\) is uniformly smooth, \(C\) is bounded, \(T\) is nonexpansive, and the equation \(x=Tx\) has at least a solution, then the iteration process converges to a solution to the equation.
Mathematics Subject Classification ID: 47J25
Mathematics Subject Classification ID: 47H06
Mathematics Subject Classification ID: 47H09
zbMATH DE Number: 1370618
zbMATH Keywords: Ishikawa iteration
zbMATH Keywords: accretive operator
zbMATH Keywords: nonexpansive operator
reviewed by: Corneliu Ursescu
MaRDI profile type: Publication
OpenAlex ID: W2052527455
description: scientific article; zbMATH DE number 1370618