Nonlinear evolutionary systems driven by mixed variational inequalities and its applications

In this paper we investigate the system obtained by mixing a nonlinear evolutionary equation and a mixed variational inequality ((EEVI), for short) on Banach spaces in the case where the set of constraints is not necessarily compact and the problem is driven by a ϕ-pseudomonotone operator which is not necessarily monotone. In this way, we extend the recent results in Liu-Zeng-Motranu, (2016). First, it is shown that the solution set for the mixed variational inequality associated to problem (EEVI) is nonempty, closed, convex and bounded. Upper semicontinuity and measurability properties are also established. Then, relying on these results, we prove the existence of solutions for problem (EEVI) as well as a compactness property for the solution set. Finally, as an application, we study a new class of partial differential complementarity problems.