Projected solutions for generalized quasivariational inequalities without compact constraint multimap

Abstract

In this paper we study the existence of projected solutions for a class of generalized quasivariational inequalities in Banach spaces. Generalized quasivariational inequalities are defined by a principal operator and a constrained set determined by a multivalued map. We obtain existence results without assuming that either the constraint multimap or the principal operator are compact. We present two types of existence results: one is obtained considering the principal operator of class S+ and in the other we assume that the constrained multivalued map is condensing. Furthermore, for both these types of results, first the upper semicontinuity associated to pseudomonotonicity on the principal operator are required, then much weaker hypotheses of regularity and monotonicity, such us the upper sign continuity and the properly quasimonotonicity, are taken into account.