Projection algorithms for the system of generalized mixed variational inequalities in Banach spaces
type: Item
instance of: scholarly article
title: Projection algorithms for the system of generalized mixed variational inequalities in Banach spaces
zbMATH Open document ID: 1223.65048
author: Qing-Bang Zhang
author: Ru-Liang Deng
published in: Applied Mathematics and Computation
publication date: 2011-06-14T00:00:00Z
review text: The essential aim of the present paper is to solve a system of generalized mixed variational inequalities without any monotonicity assumption for the mappings in Banach spaces. Formulation: Find \(x^*, y^* \in K\) such that there exist \(u^* \in T_1(y^*)\) and \(v^* \in T_2(x^*)\) satisfying \[ \begin{aligned} \langle u^* + Jx^* - Jy^*, y-x^* \rangle + f_1(y)-f_1 (x^*) \geq 0, \forall y \in K\\ \langle v^* + Jy^* - Jx^*, y-y^* \rangle + f_2(y)-f_2 (y^*) \geq 0, \forall y \in K \end{aligned}\tag{1} \] It is also pointed out, that: if \(f_i (x)=0\) \((i=1,2)\) for all \(x \in K\) (a nonempty closed convex of Banach space \(X\)), the problem (1) is equivalent to the system of generalized variational inequalities: Find \(x^*, y^* \in K\) such that there exist \(u^* \in T_1(y^*)\) and \(v^* \in T_2(x^*)\) satisfying \[ \begin{aligned} \langle u^* + Jx^* - Jy^*, y-x^* \rangle \geq 0, \forall y \in K\\ \langle v^* + Jy^* - Jx^*, y-y^* \rangle \geq 0, \forall y \in K\\ \end{aligned}\tag{2} \] If for any \(\rho, \nu > 0\), \(X=H\) is a Hilbert space, \(T_1y^*=\{\rho g_1 (y^*)\}\) and \(T_2x^*=\{\rho g_2 (x^*)\}\) for all \(x^*, y^* \in K\), where \(g_1\) and \(g_2: K \rightarrow X^*\) are single-valued mappings, then problem (1) reduces to find \(x^*, y^*\) such that \[ \begin{aligned} \langle \rho g_1(y^*)+ x^* - y^*, y-x^* \rangle + f_1(y)-f_1 (x^*) \geq 0, \forall y \in K\\ \langle \eta g_2(x^*)+ y^* - x^*, y-y^* \rangle + f_2(y)-f_2 (y^*) \geq 0, \forall y \in K\end{aligned}\tag{3} \] Main result: If the upper semi-continuous set values mappings \(T_1, T_2: K \rightarrow 2^{x^*}\) with closed values, and the proper convex lower semi-continuous mapping \(f_i: K \rightarrow\mathbb R \cup \{+ \infty \}\) satisfy the assumptions: \(f_i(x) \geq0\) for all \(x \in K\) and \(f_i(0)=0\), for any \(x \in K\) and any \(z_i \in T_i(x)\) (\(i=1,2\)), \(\langle z_i, J^* (Jx-z_i) \rangle \geq0\), the set-valued mappings \(J-T_1\) and \(J-T_2\) are compact and \( \sum^\infty_{n=0} \alp…
reviewed by: Jan Lovíšek
Mathematics Subject Classification ID: 65K15
Mathematics Subject Classification ID: 49J40
Mathematics Subject Classification ID: 49J27
Mathematics Subject Classification ID: 49M25
zbMATH DE Number: 5907619
zbMATH Keywords: System of generalized mixed variational inequalities
zbMATH Keywords: generalized \(f\)-projection operator
zbMATH Keywords: iterative algorithm
zbMATH Keywords: convergence
zbMATH Keywords: Banach space
zbMATH Keywords: monotonicity
MaRDI profile type: Publication
full work available at URL: j.amc.2011.02.074
OpenAlex ID: W1964105348
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DOI: 10.1016/J.AMC.2011.02.074
description: scientific article; zbMATH DE number 5907619