On the structure of the solution set of evolution inclusions with Fréchet subdifferentials

Abstract

In this paper we consider a Cauchy problem in which is present an evolution inclusion driven by the Fréchet subdifferential o ∂−f of a function f:Ω→R∪{+∞} (Ω is an open subset of a real separable Hilbert space) having a φ-monotone . subdifferential of order two and a perturbation F:I×Ω→Pfc(H) with nonempty, closed and convex values.First we show that the Cauchy problem has a nonempty solution set which is an Rδ-set in C(I,H), in particular, compact and acyclic. Moreover, we obtain a Kneser-type theorem. In addition, we establish a continuity result about the solution-multifunction x→S(x). We also produce a continuous selector for the multifunction x→S(x). As an application of this result, we obtain the existence of solutions for a periodic problem.