Error bounds for approximation solutions to nonlinear equations of strongly accretive operators in uniformly smooth Banach spaces
type: Item
instance of: scholarly article
title: Error bounds for approximation solutions to nonlinear equations of strongly accretive operators in uniformly smooth Banach spaces
zbMATH Open document ID: 0884.47029
author: Luchuan Zeng
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-03-10T00:00:00Z
review text: Let \(X\) be a real Banach space with norm \(|\cdot |\) and dual \(X^*\). The normalized duality mapping from \(X\) to \(X^*\) is defined by \(J(x)=\{x^*\in X^*: \langle x,x^*\rangle=|x|^2=|x^*|^2\}\), \(x\in X\). A mapping \(T\) with domain \(D(T)\) and range \(R(T)\) in \(X\) is said to be strongly accretive, if for each \(x,y\) in \(D(T)\), there exists a positive number \(k\) such that \(\langle Tx-Ty,j\rangle\geq k|x-y|^2\) holds for some \(j\in J(x-y)\). In this paper \(x\) is a uniformly smooth Banach space and \(T: X\to X\) is a strongly accretive operator. The author gives the error bounds for approximation solutions of the nonlinear equation \(Tx=f\) generated by both Mann and Ishikawa iteration processes. The related results deal with the error bounds for the iterative approximation of the fixed points of \(T\), where \(T: K\to K\), is a strictly pseudocontractive mapping and \(K\) is a nonempty convex subset of \(X\).
Mathematics Subject Classification ID: 47H06
Mathematics Subject Classification ID: 47H05
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 47J25
zbMATH DE Number: 1031553
zbMATH Keywords: normalized duality mapping
zbMATH Keywords: uniformly smooth Banach space
zbMATH Keywords: strongly accretive operator
zbMATH Keywords: approximation solutions
zbMATH Keywords: nonlinear equation
zbMATH Keywords: Mann and Ishikawa iteration processes
zbMATH Keywords: iterative approximation of fixed points
zbMATH Keywords: strictly pseudocontractive mapping
zbMATH Keywords: error bounds
reviewed by: Alexandra Rodkina
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5229
OpenAlex ID: W2034588555
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DOI: 10.1006/JMAA.1997.5229
description: scientific article; zbMATH DE number 1031553