Commutativity of set-valued cosine families

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 {Ft: t ≥ 0} is a regular cosine family of continuous additive set-valued functions Ft: K → cc(K) such that x ∈ Ft(x) for t ≥ 0 and x ∈ K, then $F_t \circ F_s (x) = F_s \circ F_t (x)fors,t \geqslant 0andx \in K$.