A Viable Result for Nonconvex Differential Inclusions with Memory

Let X be a separable Banach space, σ > 0 and Cσ := C([−σ, 0],X) the Banach space of the continuous functions from [−σ, 0] into X, K a locally closed set in X and F : [a, b)×Cσ→ 2 X a closed valued and locally integrable bounded multifunction, withF (., φ) measurable and F (t, .) Lipschitz continuous in theHausdorff–Pompeiumetric. In this paper we establish some sufficient conditions in order that, for each τ ∈ [a, b) and for each φ ∈ Cσ with φ(0) ∈ K, there exist at least one solution u : [τ−σ, T ] → X of the differential inclusion u(t) ∈ F (t, ut), such that uτ = φ on [−σ, 0] and u(t) ∈ K for every t ∈ [τ, T ].