On a class of generalized variational inequalities
type: Item
instance of: scholarly article
title: On a class of generalized variational inequalities
zbMATH Open document ID: 0883.49011
author: Khalil Ahmad
author: Z. A. Siddiqui
author: K. R. Kazmi
published in: Indian Journal of Pure \& Applied Mathematics
publication date: 1998-01-25T00:00:00Z
review text: Let \(H\) be a real Hilbert space, \(2^H\) a power set of \(H\), \(\langle\;,\;\rangle\) denote the inner product of \(H\), \(T,A:H\to 2^H\), \(g:H\to H\) a nonlinear mapping and \(j:H\to R\cup\{\infty\}\) a proper convex and lower semicontinuous function with \(I_mg\cap\text{dom }\partial j\neq\emptyset\), where \(\partial j\) denotes the subdifferential of \(j\). The authors consider the following generalized variational inequality problem (GVIP): GVIP: Find \(u\in H\), \(x\in T(u)\), \(y\in A(u)\) such that \(g(u)\cap\text{dom }\partial j\neq\emptyset\) and \(\langle x-y,v-g(u)\rangle\geq j(g(u))- j(\nu)\) for all \(\nu\in H\). If \(T\) and \(A\) are single-valued mappings, then GVIP reduces to the problem which was considered and studied by Hassouni and Moudafi. If \(j=\delta_k\), the indicator function of the closed convex set \(K\) of \(H\), then GVIP reduces to the problem considered by Ding. If \(j=\delta_k\), the indicator function of the closed convex set \(K\) of \(H\), \(T\) and \(A\) are single-valued mapping, then GVIP reduces to the problem considered by Noor. If \(j=\delta_k+m(u)\), where \(m:H\to H\) is a single-valued map, and \(T\) and \(A\) are two single-valued maps, then GVIP reduces to the problem considered by Siddiqi and Ansari. The authors develop some iterative algorithms for finding the approximating solution of GVIP, which are more general and powerful than the iterative algorithms developed by Hassouni and Moudafi, and Ding. The authors prove some existence results for GVIP and discuss the convergence criteria for sequence generalized by iterative algorithms.
Mathematics Subject Classification ID: 49J40
zbMATH DE Number: 1014154
zbMATH Keywords: nonexpansive map
zbMATH Keywords: \(\alpha\)-strongly
zbMATH Keywords: monotone map
zbMATH Keywords: GVIP
zbMATH Keywords: lower semicontinuous function
zbMATH Keywords: subdifferential
zbMATH Keywords: generalized variational inequality problem
zbMATH Keywords: iterative algorithms
reviewed by: Lai-Jiu Lin
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1014154