Strong and weak convergence theorems for general mixed equilibrium problems and variational inequality problems and fixed point problems in Hilbert spaces
type: Item
instance of: scholarly article
title: Strong and weak convergence theorems for general mixed equilibrium problems and variational inequality problems and fixed point problems in Hilbert spaces
zbMATH Open document ID: 1266.65110
author: Gang Cai
author: Shangquan Bu
published in: Journal of Computational and Applied Mathematics
publication date: 2013-04-18T00:00:00Z
review text: The authors study two iterative algorithms for finding a common element of the set of solutions of finite general mixed equilibrium problems and the set of solutions of finite variational inequalities for inverse strongly monotone mappings and the set of common fixed points of an asymptotically \(K\)-strictly pseudocontractive mapping in the intermediate sense in a real Hilbert space. Here, the general mixed equilibrium problems include fixed point problems, optimization problems, variational inequality problems, Nash equilibrium problems, and equilibrium problems as special cases. Numerous problems in physics, optimization and economics reduce to find a solution of problem: \(F(x,y) + \varphi (y) - \varphi(x) + \langle Ax,y-x\rangle \geq 0\) \(\forall y \in \mathbb C\), where a mapping \(A:\mathbb C \rightarrow H\) is \(\alpha\)-inverse strongly monotone (a nonlinear mapping), \(\varphi :\mathbb C \rightarrow R\) is a real-valued function. If \(\varphi =0\), then the proposed problem reduces to the generalized equilibrium problem which is to study \(x \in \mathbb C\) such that: \(F(x,y) + \langle Ax,y-x\rangle \geq 0\) \(\forall y \in \mathbb C\). Furthermore, as \(A=0\), then the mixed equilibrium problem (find \(x \in \mathbb C\) such that \(F(x,y) + \varphi (y) - \varphi(x) \geq 0\) \(\forall y \in \mathbb C\)), is considered. The authors prove some strong and weak convergence theorems by using the proposed schemes under some appropriate conditions. The results obtained in this paper improve and extend the corresponding results announced by many others.
Mathematics Subject Classification ID: 65K15
Mathematics Subject Classification ID: 49J40
Mathematics Subject Classification ID: 49M05
zbMATH DE Number: 6155440
zbMATH Keywords: general mixed equilibrium problem
zbMATH Keywords: variational inequality
zbMATH Keywords: fixed point
zbMATH Keywords: Hilbert space
zbMATH Keywords: weak convergence
zbMATH Keywords: strong convergence
zbMATH Keywords: closed convex subset
zbMATH Keywords: pseudocontractive mapping
zbMATH Keywords: normal cone
zbMATH Keywords: iterative algorithms
zbMATH Keywords: Nash equilibrium problems
reviewed by: Jan Lovíšek
MaRDI profile type: Publication
full work available at URL: j.cam.2013.01.004
OpenAlex ID: W2011881806
DOI: 10.1016/J.CAM.2013.01.004
description: scientific article; zbMATH DE number 6155440