Existence and Topological Properties of Solution Sets for Differential Inclusions with Delay

We consider the problem $\dot{x}(t)\in A(t)x(t)+F(t,{\theta }_{t}x))$ a.e. on $[0,b]$, $x=\kappa$ on $[-d,0]$ in a Banach space $E$, where $\kappa$ belongs to the Banach space, $C_E([-d,0])$, of all continuous functions from $[-d,0]$ into $E$. A multifunction $F$ from $[0,b]\times C_E([-d,0])$ into the set, $P_{f_c}(E)$, of all nonempty closed convex subsets of $E$ is weakly sequentially hemi-continuous, ${\theta }_{t}x(s)=x(t+s)$ for all $s\in [-d,0]$ and $\{A(t):0\le t\le b\}$ is a family of densely defined closed linear operators generating a continuous evolution operator $S(t,s)$. Under a generalization of the compactness assumptions, we prove an existence result and give some topological properties of our solution sets that generalizes earlier theorems by Papageorgiou, Rolewicz, Deimling, Frankowska and Cichoń.