Hukuhara’s Derivative and Concave Iteration Semigroups of Linear Set-valued Functions

Let K be a closed convex cone with the nonempty interior in a real Banach space and cc(K) denote the family of all nonempty convex compact subsets of K. If {F t : t ≥ 0} is a concave iteration semigroup of continuous linear set-valued functions F t : K → cc(K) with F (x) = {x} for x ∈ K, then DtF (x) = F (G(x)) for x ∈ K and t ≥ 0, where DtF (x) denotes the Hukuhara derivative of F (x) with respect to t and G(x) := lim s→0+ F (x)− x s for x ∈ K. 1. Let A and B be two subsets of a real vector space X. We define the sum of A and B by the formula A+B = {a+ b : a ∈ A, b ∈ B}. A subset K of a real vector space is called a cone iff tK := {tx : x ∈ K} ⊂ K 2000 Mathematics Subject Classification. 39B12, 39B52, 26E25.