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

We consider the problem \(\dot{x}(t) \in A(t)x(t) + F (t, θ_t x))\) 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, \(θ_t x(s) = x(t + s)\) for all \(s \in [-d, 0]\) and \(\{A(t) : 0 \leq t \leq b\}\) 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.