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 $A$ on a Banach space $E$ with nonempty resolvent set and a multivalued map $F\colon I\times E\map E$ with weakly sequentially closed graph, we consider the integro-differential inclusion \begin{center} $\dot{u}\in Au+F(t,\int u)\;\;\text{on }I,\;\;u(0)=x_0.$ \end{center} We focus on the case when $A$ generates an integrated semigroup and obtain existence of integrated solutions in the sense of \cite[Def.6.4.]{thieme} if $E$ is weakly compactly generated and $F$ satisfies \[\beta(F(t,\Omega))\<\eta(t)\beta(\Omega)\;\;\text{for all bounded }\Omega\subset E,\] where $\eta\in L^1(I)$ and $\beta$ denotes the De Blasi measure of noncompactness. When $E$ is separable, we are able to show that the set of all integrated solutions is a compact $R_\delta$-subset of the space $C(I,E)$ 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. Some applications to partial differential equations with multivalued terms are also included.