Integrated solutions of non-densely defined semilinear integro-differential inclusions: existence, topology and applications

Abstract

Given a linear closed but not necessarily densely defined operator on a Banach space with nonempty resolvent set and a multivalued map with weakly sequentially closed graph, we consider the integro-differential inclusion We focus on the case when generates an integrated semigroup and obtain existence of integrated solutions if is weakly compactly generated and satisfies where and denotes the De Blasi measure of noncompactness. When is separable, we are able to show that the set of all integrated solutions is a compact -subset of the space endowed with the weak topology. We use this result to investigate a nonlocal Cauchy problem described by means of a nonconvex-valued boundary condition operator. We also include some applications to partial differential equations with multivalued terms are. Bibliography: 26 titles.