Existence Of Solutions For Variational Inequalities

In this paper, we mainly study the existence of solutions toset-valued variational inequalities and the solution stability of set-valuedparametric variational inequalities problems. First, we propose a newconcept of exceptional family of elements for a finite-dimensional set-valuedvariational inequality and discuss the existence of solutions to variationalinequality provided that the set-valued mapping is upper semicontinuouswith nonempty compact convex values. Secondly, we introduce a newdefinition of exceptional family of elements for a set-valued variationalinequality in Banach space. Assuming that the mapping is an acycliccompact or is an upper semicontinuous compact with nonempty compactcontractible values, we study the solution existence of variational inequali-ties. Finally, extending the topological degree theory established in a finitedimensional space, this paper sets up a new topological degree theoryof set-valued variational inequalities in the Hilbert space with which thesolution stability of parametric variational inequality can be demonstrated.