Existence and relaxation of solutions for a subdifferential inclusion with unbounded perturbation

An evolution inclusion is considered in a separable Hilbert space. The right-hand side of the inclusion contains the subdifferential of a time-dependent proper convex lower semicontinuous function and a multivalued perturbation with nonempty closed, not necessarily, bounded values. Along with this inclusion we consider the inclusion with the perturbation term being convexified. We prove the existence of solutions and a density of the solution set of the original inclusion in a closure of the solution set of the inclusion with the convexified perturbation. In contrast to the known results of this kind we do not suppose that the convex function has the compactness property and that the values of the perturbation are bounded sets. An example of a perturbation with closed unbounded values satisfying the conditions of the main theorems is given.