On concave iteration semigroups of linear set-valued functions

Let K be a closed convex cone in a real Banach space and let cc(K) denote the family of all nonempty convex compact subsets of K. Let I(x) = {x} for x is an element of K. Suppose that G : K -> cc(K) is a given continuous linear multivalued map such that 0 is an element of G(x) for x is an element of K. It is proved that a family {F(t) : t >= 0} of linear continuous set-valued functions F(t), where F(t)(x) = Sigma(infinity)(i=0) t(i)/i!C(i)(x), (a) is an iteration semigroup if and only if the equality G(x) + tG(2)(x) - (I+tG)(G(x)) (b) holds true. It is also proved that a concave iteration semigroup of continuous linear set-valued functions with the infinitesimal generator G fulfilling (b) and such that 0 is an element of G(x) is of the form (a).