Necessary and sufficient conditions for the existence of the implicit variational inequality problem
type: Item
instance of: scholarly article
title: Necessary and sufficient conditions for the existence of the implicit variational inequality problem
zbMATH Open document ID: 0897.49005
DOI: 10.1016/S0893-9659(96)00116-4
published in: Applied Mathematics Letters
publication date: 1998-09-28T00:00:00Z
review text: The paper deals with the implicit variational inequality problem denoted by IVIP\((K,C,X,F,g)\), which is to find \(x\in X(x)\) and \(y\in F(x)\) such that \[ g(x,y,u)\geq g(x,y,x) \qquad \text{for all } u\in X(x), \] where \(K\subset R^n\), \(C\subset R^m\), \(X:K\to 2^K\), \(F:K\to 2^C\), and \(g:K\times C\times K\to R\). One of the main results of the paper states the existence of solutions for IVIP\((K,C,X,F,g)\) if \(K\) is nonempty compact, \(C\) is nonempty closed and convex, \(X\) is continuous with compact convex values, \(F\) is upper semicontinuous with acyclic values, \(g\) is continuous, and the set \[ V(x,y)=\left\{u\in X(x)\mid g(x,y,u)=\min_{s\in X(x)}gx,y,s)\right\} \] is acyclic for each \((x,y)\in K\times C\). The paper also offers conditions for the existence of solutions in the case if \(K\) is not bounded.
Mathematics Subject Classification ID: 49J40
Mathematics Subject Classification ID: 58E35
zbMATH DE Number: 1078541
zbMATH Keywords: implicit variational inequality problem
author: Paolo Cubiotti
author: Jen-Chih Yao
reviewed by: Zsolt Páles
MaRDI profile type: Publication
cites work: Variational inequalities, complementarity problems, and duality theorems
cites work: Nonlinear monotone operators and convex sets in Banach spaces
cites work: Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications
cites work: On some non-linear elliptic differential functional equations
cites work: Q3953614
cites work: Q4115720
cites work: Variational inequalities
cites work: Q3787016
cites work: Quasi-variational inequalities with non-compact sets
cites work: The generalized quasi-variational inequality problem with applications
cites work: Variational inequality
cites work: Q3905599
cites work: On the general variational inequality
cites work: Set-valued analysis
cites work: Q3260984
cites work: Fixed Point Theorems for Multi-Valued Transformations
cites work: An existence theorem for generalized quasi-variational inequalities
cites work: The generalized quasi-variational inequality problem over non-compact sets
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description: scientific article; zbMATH DE number 1078541