On the existence of mild solutions for neutral functional differential inclusions in Banach space

A theorem on existence of mild solutions for partial neutral functional differential inclusions with unbounded linear part generating a noncompact semigroup in Banach space is established. Semilinear neutral functional differential inclusion has been the object of many stud- ies by many researchers in the recent years. The method which consists in defining an integral multioperator for wich fixed points set coincides whith the solutions set of differential inclusion has been often applied to existence problems. In the case of inclu- sions on infinite dimensional spaces its direct application is complicated by the fact that the integral multioperator is noncompact except if one impose a severe compactness assumption. In this paper using the method of condensing integral multioperators and fractional power of closed operators theory, we study the existence of mild solutions for initial value problems for first order semilinear neutral functional differential inclusions in a separable Banach space E for the form: d dt (x(t) h(t,xt)) 2 Ax(t) + F(t,xt),a.e. t 2 (0,T) (1.1) x(t) = '(t), t 2 ( r,0), (1.2) where A : D(A) E ! E is the infinitesimal generator of an uniformly bounded analytic semigroup of linear operators, {e At }t0 on a separable Banach space E; the multimap F : (0,T) × C(( r,0),E) ! P(E) and h : (0,T) × C(( r,0),E) ! E, are given functions, 0 < r < 1,' 2 C(( r,0),E), where P(E) denotes the class of all nonempty subsets of E, and C(( r,0),E) denotes the space of continuous functions from ( r,0) to E. For any continuous function x defined on ( r,T) and any t 2 (0,T), we denote by xt the element of C(( r,0),E) defined by