The parametric problem of Minty-type quasivariational inequalities under consideration is to find \(u_0\in K(x,u_0)\bigcap A\) such that \( \langle t, u-u_0\rangle\leq 0 \) for all \(v\in K(x,u_0)\) and \(t\in T(x,v)\). Here, \((x,y)\in X\times Y\) where the set \(X\subset\mathbb{R}^n\) is nonempty and closed, \(A\) is a closed convex subset of \(Y=\mathbb{R}^m\), and \(K\) and \(T\) are set-valued mappings defined on \( X\times Y\) with values in \(2^Y\). The map \(K\) is supposed to be closed-valued. The solution set, \(M(x)\), is assumed to be non-empty in a neighborhood of \(x=x_0\).
The authors present a number of results showing that the mapping \(M(x)\) is upper-semicontinuous at \(x=x_0\) under rather general assumptions. A number of theorems characterize sufficient conditions for lower-semicontinuity of the solution set \(M(x)\).
The results are applied to a problem of minimization of \(f(x,u)\) subject to the constraint \(u\in M(x)\), where \(M(x)\) is the set of solutions to a quasivariational inequality problem described above. The minimization problem is perturbed by allowing for solutions only approximately satisfying the constraints, and satisfying the minimality condition up to a tolerance \(\delta>0\). The authors obtain a number of theorems asserting that, given the tolerance parameter \(\delta\), the perturbed set is upper-semicontinuous with respect to the perturbation of the variational constraint.