Iterative methods for nonlinear Lipschitz pseudocontractive operators
type: Item
instance of: scholarly article
title: Iterative methods for nonlinear Lipschitz pseudocontractive operators
zbMATH Open document ID: 0984.47046
published in: Journal of Mathematical Analysis and Applications
publication date: 2002-05-05T00:00:00Z
review text: Considerable research efforts in the last 15 years have been devoted to iterative methods for approximating the solution of \(Au= 0\) when \(A\) is accretive (or, correspondingly, fixed points of pseudocontractions \(T= I-A\), since \(A\) is accretive if and only if \(T\) is pseudocontractive). The problem is relevant due to its relation to the initial-value evolution system \[ du/dt+ Au=0,\quad u(0)= w \] [\textit{F. E. Browder}, Proc. Natl. Acad. Sci., USA 61, 388-393 (1968; Zbl 0167.15205)]. In the special case the operator \(T\) is Lipschitz and strongly pseudocontractive, the iterative Mann process [\textit{W. R. Mann}, Proc. Am. Math. Soc. 4, 506-510 (1953; Zbl 0050.11603)] has been successfully employed in approximating fixed points of such mappings. It is a long standing question whether or not the Mann process converges when \(T\) is a Lipschitz pseudocontractions even in optimal cases (i.e. \(T\) defined on a compact convex subset of a Hilbert space) [\textit{T. L. Hicks} and \textit{J. D. Kubicek}, J. Math. Anal. Appl. 59, 498-504 (1977; Zbl 0361.65057)]. The author exhibits a simple example, constructed by the author himself and Mutangadura, answering negatively to the question. Other results about different iterative methods are summarized. The concept of acceptably paired sequences, due in the real case to R. E. Bruck, is extended to the context of real normed spaces. A couple of questions of interest are raised at the conclusion of the paper.
Mathematics Subject Classification ID: 47J25
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 47H06
zbMATH DE Number: 1545036
zbMATH Keywords: iterative methods
zbMATH Keywords: accretive
zbMATH Keywords: fixed points of pseudocontractions
zbMATH Keywords: initial-value evolution system
zbMATH Keywords: iterative Mann process
zbMATH Keywords: acceptably paired sequences
author: C. E. Chidume
MaRDI profile type: Publication
full work available at URL: jmaa.2000.7021
OpenAlex ID: W2021462305
cites work: Fixed point iterations for real functions
cites work: NONLINEAR MONOTONE AND ACCRETIVE OPERATORS IN BANACH SPACES
cites work: A strongly convergent iterative solution of \(0 \in U(x)\) for a maximal monotone operator U in Hilbert space
cites work: Iterative approximation of fixed points of Lipschitz pseudocontractive maps
cites work: Iterative solution of \(0\in Ax\) for an \(m\)-accretive operator \(A\) in certain Banach spaces
cites work: Fixed points of nonexpanding maps
cites work: On the Mann iteration process in a Hilbert space
cites work: Fixed Points by a New Iteration Method
cites work: Mean Value Methods in Iteration
cites work: An example on the Mann iteration method for Lipschitz pseudocontractions
cites work: Q3134801
cites work: Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces
DOI: 10.1006/JMAA.2000.7021
description: scientific article; zbMATH DE number 1545036