Second Hukuhara derivative and cosine family of linear set-valued functions

Abstract

Let K be a closed convex cone with nonempty interior in a real Banach space and let cc(K) denote the family of all nonempty convex compact subsets of K. If {F_t : t ≥ 0} is a regular cosine family of continuous linear set-valued functions F_t : K → cc(K), x ∈ F_t(x) for t ≥ 0, x ∈ K and F_t ◦ F_s = F_s ◦ F_t for s, t ≥ 0, then D²F_t(x) = F_t(H(x)) for x ∈ K and t ≥ 0, where D²F_t(x) denotes the second Hukuhara derivative of F_t(x) with respect to t and H(x) is the second Hukuhara derivative of this multifunction at t = 0.