A VIABLE RESULT FOR NONCONVEX DIFFERENTIAL INCLUSIONS WITH MEMORY

Abstract: 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].