Single-Directional Property of Multivalued Maps and Variational Systems

In [8], Dontchev and Hager have shown that a monotone set-valued map defined from a Banach space to its dual which satisfies the Aubin property around a point (x, y) of its graph is actually single-valued in a neighbourhood of x. We prove a result which is the counterpart of the above for quasimonotone set-valued maps, based on the concept of single-directional property. As applications, we provide sufficient conditions for this single-valued property to hold for the solution map of general variational systems and quasivariational inequalities. We also investigate the singledirectionality property for the normal operator to the sublevel sets of a quasiconvex function.