On the topological structure of the solution set for a semilinear ffunctional-differential inclusion in a Banach space

Abstract

In this paper we show that the set of all mild solutions of the Cauchy problem for a functional-differential inclusion in a separable Banach space E of the form x′(t) ∈ A(t)x(t) + F (t, xt) is an Rδ-set. Here {A(t)} is a family of linear operators and F is a Caratheodory type multifunction. We use the existence result proved by V. V. Obukhovskĭi [22] and extend theorems on the structure of solutions sets obtained by N. S. Papageorgiou [23] and Ya. I. Umanskĭi [32]. Introduction. Beginning in the seventies the multivalued Cauchy problem in abstract spaces has been studied by many authors; we mention the existence theorems obtained by Chow and Schuur [6], Muhsinov [20], De Blasi [8], Anichini and Zecca [3], Sentis [26], Pavel and Vrabie [25], Tostonogov [27] and [28] and Kisielewicz [16]. The first approach to the structure of the solution set was by Davy [7] in the finite dimensional case. He proved that the set S of solution is a continuum in C([0, T ],R). Later Lasry and Robert [18] showed that S is acyclic whenever F has compact and convex values and is Hausdorff 1991 Mathematics Subject Classification: Primary 34A60; Secondary 34K30, 93B52, 93C20. The paper is in final form and no version of it will be published elsewhere.