Evolutionary problems driven by variational inequalities

In this paper we introduce the differential system obtained by mixing an evolution equation and a variational inequality ((EEVI), for short). First, by using KKM theorem and monotonicity arguments, we prove the superpositional measurability and upper semicontinuity for the solution set of a general variational inequality. Then we establish that the solution set of (EEVI) is nonempty and compact. Our approach is based on the theory of semigroups, Filippov implicit function lemma and fixed point theory for set-valued mappings.