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. A family $F^t:t\ge 0$ of continuous linear set-valued functions $F^t:K\to cc\left(K\right)$ is a differentiable iteration semigroup with F⁰(x) = x for x ∈ K if and only if the set-valued function $\Phi (t,x)=F^t\left(x\right)$ is a solution of the problem $D_t\Phi (t,x)=\Phi (t,G\left(x\right)):=\bigcup \Phi (t,y):y\in G\left(x\right)$, Φ(0,x) = x, for x ∈ K and t ≥ 0, where $D_t\Phi (t,x)$ denotes the Hukuhara derivative of Φ(t,x) with respect to t and $G\left(x\right):=li{m}_{s\to 0+}(F^s\left(x\right)-x)/s$ for x ∈ K.