Properties of a class of approximately shrinking operators and their applications
type: Item
instance of: scholarly article
zbMATH Open document ID: 1314.47103
author: Andrzej Cegielski
author: Rafał Zalas
publication date: 2014-10-27T00:00:00Z
full work available at URL: http://www.math.ubbcluj.ro/~nodeacj/download.php
Mathematics Subject Classification ID: 47J25
Mathematics Subject Classification ID: 47H09
Mathematics Subject Classification ID: 47J20
Mathematics Subject Classification ID: 47H05
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 65K15
zbMATH DE Number: 6361922
zbMATH Keywords: Hilbert space
zbMATH Keywords: variational inequality problem
zbMATH Keywords: fixed point
zbMATH Keywords: quasi-nonexpansive operator
zbMATH Keywords: hybrid steepest descent method
zbMATH Keywords: generalised hybrid steepest descent method
zbMATH Keywords: convergence theorem
zbMATH Keywords: common fixed point
MaRDI profile type: Publication
title: Properties of a class of approximately shrinking operators and their applications
published in: Fixed Point Theory
review text: Let \(H\) be a Hilbert space, \(C\) a closed convex subset of \(H\), and \(F: H\rightarrow H\) a strongly monotone and Lipschitz continuous operator. In order to solve the variational inequality problem \(\mathrm{VIP}(F,C)\): NEWLINE\[NEWLINE \text{ find } \overline{u}\in C \text{ such that } \langle F(\overline{u}), x-\overline{u}\rangle \geq 0 \text{ for all } x\in C, NEWLINE\]NEWLINE the authors introduce a class of approximately shrinking operators, prove the closedness of this class with respect to compositions and convex combinations, and then use these operators to construct a generalised hybrid steepest descent method (GHSD) for solving the \(\mathrm{VIP}(F,C)\) above.NEWLINENEWLINESufficient conditions for the convergence of the GHSD method as well as several examples of methods which satisfy these conditions are also presented.NEWLINENEWLINEIn particular, the results are used in the case \(C=\bigcap_{i=1}^{m} \mathrm{Fix}\,(U_i)\), where \(U_i: H\rightarrow H\), \(i=1,2,\dots,m\), are quasi-nonexpansive operators having a common fixed point.
reviewed by: Vasile Berinde
description: scientific article; zbMATH DE number 6361922