General iterative methods for a one-parameter nonexpansive semigroup in Hilbert space
type: Item
instance of: scholarly article
title: General iterative methods for a one-parameter nonexpansive semigroup in Hilbert space
zbMATH Open document ID: 1177.47075
author: Suhong Li
author: Yongfu Su
author: Lihua Li
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2009-04-02T00:00:00Z
review text: Let \(H\) be a Hilbert space, \(f\) a fixed contractive mapping with coefficient \(0<\alpha<1\), \(A\) a strongly positive linear bounded operator with coefficient \(\overline\gamma>0\). Consider two iterative methods that generate the sequences \(\{x_n\}\), \(\{y_n\}\) by \[ x_n=(1-\alpha_nA)\frac{1} {t_n}\int^{t_n}_0T(s)x_n\, ds+\alpha_n\gamma f(x_n),\tag{I} \] \[ y_{n+1}=(I-\alpha_n A)\frac{1} {t_n}\int^{t_n}_0T(s)y_n\,ds+\alpha_n\gamma f(y_n),\tag{II} \] where \(\{\alpha_n\}\) and \(\{t_n\}\) are two sequences satisfying certain conditions, and \({\mathfrak I}=\{T(s):s\geq 0\}\) is a one-parameter nonexpansive semigroup on \(H\). It is proved that the sequences \(\{x_n\}\), \(\{y_n\}\) generated by the iterative methods (I) and (II), respectively, converge strongly to a common fixed point \(x^*\in F({\mathfrak I})\) which solves the variational inequality \[ \langle(A-\gamma f)x^*,x^*-z\rangle\leq 0\;z\in F({\mathfrak I}). \]
reviewed by: Jürgen Appell
Mathematics Subject Classification ID: 47J25
Mathematics Subject Classification ID: 47H20
Mathematics Subject Classification ID: 47J20
Mathematics Subject Classification ID: 65J15
zbMATH DE Number: 5539183
zbMATH Keywords: nonexpansive semigroup
zbMATH Keywords: iterative method
zbMATH Keywords: variational inequality
zbMATH Keywords: viscosity approximation
MaRDI profile type: Publication
full work available at URL: j.na.2008.04.007
OpenAlex ID: W2011513175
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DOI: 10.1016/J.NA.2008.04.007
description: scientific article; zbMATH DE number 5539183