Global stability result for the generalized quasivariational inequality problem (Q1176847)

The paper studies some stability property for the generalized quasivariational problem (GQVI). Let \(X(\cdot,\omega)\) and \(F(\cdot,\omega)\) both be point-to-set mappings from the set \(\Gamma\subset\mathbb{R}^ n\) into the subsets of \(\mathbb{R}^ n\), where \(\omega\) is a vector parameter in \(W\subset\mathbb{R}^ r\). For each \(\omega\), the problem GQVI\((X,F)\) is to find vectors \(x^*\in X(x^*,\omega)\) and \(y^*\in F(x^*,\omega)\) such that \(\langle x- x^*, y^*\rangle \geq 0\) for any \(x\in X(x^*,\omega)\). The purpose of the paper is to show upper semicontinuity of the solution set \(I(\omega)\) of the above GQVI\((X,F)\). It is to show that, if \(S\) is any open set with \(I(\omega)\subset S\), then there exists \(\delta >0\) such that, for all \(\Delta\omega\in B(\omega,\delta)\), we have \(I(\omega+\Delta\omega)\subset S\). A basic assumption is that \(X(x,\omega)\neq \emptyset\), \(X(x,\omega)\subset\Gamma\), and \(F(x,\omega)\neq \emptyset\) for all \(x\) and \(\omega\). The above assertion is obtained under additional assumptions on compactness of the domain set and some semicontinuity properties of \(X\) and \(F\) relative to \(x\) and \(\omega\).