Topological Structure of Solutions Sets for Semilinear Evolution Inclusions

Abstract

This paper deals with a semilinear evolution inclusion involving a nondensely defined closed linear operator satisfying the Hille–Yosida condition and source term of multivalued type in Banach spaces. The topological structure of the set of solutions is investigated in the case that semigroup is noncompact. It is shown that the solution set is nonempty, compact and an $R_{\delta }$-set. It is proved on compact intervals and then, using the inverse limit method, obtained on non-compact intervals. As a sample of application, we consider a parabolic partial differential inclusion at end of the paper.