Nonmonotone, nonlinear evolution inclusions

In this paper, we consider nonlinear evolution problems, defined on an evolution triple of spaces, driven by a nonmonotone operator, and with a perturbation term which is multivalued. We prove existence theorems for the cases of a convex and of a nonconvex valued perturbation term which is defined on all of T × H or only on T × X with values in H or even in X∗ (here X ⊂- H ⊂- X∗ is the evolution triple). Also, we prove the existence of extremal solutions, and for the “monotone” problem we have a strong relaxation theorem. Some examples of nonlinear parabolic problems are presented.