A new method for solving a system of generalized nonlinear variational inequalities in Banach spaces
type: Item
instance of: scholarly article
title: A new method for solving a system of generalized nonlinear variational inequalities in Banach spaces
zbMATH Open document ID: 1215.65119
author: H. W. Joseph Lee
author: Chi Kin Chan
author: J. A. Liu
author: Shihsen Chang
published in: Applied Mathematics and Computation
publication date: 2011-04-19T00:00:00Z
review text: The essential aim of the present paper is to consider a new iterative scheme to study the approximate solvability problem for a system of generalized nonlinear variational inequalities in the framework of uniformly smooth and strictly convex Banach spaces by using the generalized projection approach. Here the authors consider some special cases of the problem (the so-called system of generalized nonlinear variational inequalities problem, SGNIVP): find \(x^*, y^*, z^* \in K\) (a nonempty closed convex subset of real strictly convex Banach space) such that for all \(x \in K\) (\(S,T,U: K^3 \rightarrow X^*\) are the nonlinear mappings) (1)\(\langle S(y^*, z^*, x^*), x-x^*\rangle \geq 0\), \(\langle T(z^*, x^*, y^*), x-y^*\rangle \geq 0\), \(\langle U(x^*, y^*, z^*), x-z^*\rangle \geq 0\). (I) If \(U=0\) and \((S,T)\) are bifurcations from \(K^2\) to \(X^*\) (the dual space) with \(S(y,x) = \varrho T_1(y,x) + Jx - Jy\) (\(J: E \rightarrow 2 ^{E^*}\) is the normalized duality mapping) and \(T(x,y) = \eta T_2 (x,y) + Jy - Jx\), where \(T_i: K^2 \rightarrow X^*\) is a mapping, \(i = 1,2\), then the problem (1) reduces to find \(x^*, y^* \in K\) such that for all \(x \in X\): (2) \(\langle \varrho T_1 (y^*, x^*)+Jx^* - Jy^*, x-x^* \rangle \geq 0\), \(\langle \eta T_2 (x^*, y^*)+Jy^* - Jx^*, x-y^* \rangle \geq 0\), \(\forall x \in K\); where \(\varrho\) and \(\eta\) are two positive constants. (II) If \(S\) and \(T\) both are univariate mappings then the problem (2) reduces to the (SGNVIP). Find \(x^*, y^* \in K\) such that for all \(x \in X\); (3) \(\langle \varrho T_1 (y^*) + Jx^* - Jy^*, x-x^*\rangle \geq 0\), \(\langle \eta T_2 (x^*) + Jy^* - Jx^*, x-y^*\rangle \geq 0\). Main result: If \(X\) is a real smooth and strictly convex Banach space with Kadec-Klee property, \(K\) is a nonempty closed and convex subset of \(X\) with \(\theta \in K\) and \(S,T,U,: K^3 \rightarrow X^*\) be continuous mappings satisfying the condition: There exist a compact subset \(C \subset X^*\…
Mathematics Subject Classification ID: 65K15
Mathematics Subject Classification ID: 49J40
Mathematics Subject Classification ID: 49M25
zbMATH DE Number: 5880121
zbMATH Keywords: generalized projection mapping
zbMATH Keywords: system of generalized nonlinear variational inequalities
zbMATH Keywords: uniformly smooth Banach space
zbMATH Keywords: Lyapunov functional
zbMATH Keywords: normalized duality mapping
reviewed by: Jan Lovíšek
MaRDI profile type: Publication
full work available at URL: j.amc.2011.01.021
OpenAlex ID: W2062668032
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DOI: 10.1016/J.AMC.2011.01.021
description: scientific article; zbMATH DE number 5880121