Stability of Quasimonotone Variational Inequality Under Sign-Continuity

Whenever the data of a Stampacchia variational inequality, that is, the set-valued operator and/or the constraint map, are subject to perturbations, then the solution set becomes a solution map, and the study of the stability of this solution map concerns its regularity. An important literature exists on this topic, and classical assumptions, for monotone or quasimonotone set-valued operators, are some upper or lower semicontinuity. In this paper, we limit ourselves to perturbations on the constraint map, and it is proved that regularity results for the solution maps can be obtained under some very weak regularity hypothesis on the set-valued operator, namely the lower or upper sign-continuity.