Necessary and sufficient conditions for the existence of the implicit variational inequality problem
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.
scientific article; zbMATH DE number 1078541
scientific article