Convex-valued selectors of a nemytskii operator with nonconvex values and their applications

The Nemytskii operator generated by a multivalued mapping whose values are compacts from a Banach space is considered. In every point this mapping is either upper semicontinuous and has convex values or it is lower semicontinuous in a neighborhood of the point. We prove that the multivalued Nemytskii operator has a multivalued selector with convex closed values which is upper semicontinuous in the weak topology of the space of integrable functions in every point of its domain. The result we obtain is applied to prove the existence of a solution to an evolution inclusion with subdifferential operators and a multivalued perturbation, the latter having different semicontinuity types at different points of the domain.