On concave iteration semigroups of linear set-valued functions

Summary.

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 ∈ K. Suppose that G : K → cc(K) is a given continuous linear multivalued map such that 0 ∈ G(x) for x ∈ K. It is proved that a family {F ^t : t ≥ 0} of linear continuous set-valued functions F ^t, where

$$F^t(x) = \sum\limits^\infty_{i=0}\frac{t^i}{i^!}G^i(x),\,(\rm{a})$$

is an iteration semigroup if and only if the equality

$$G(x) + tG^2(x) = (I + tG)(G(x))\,(\rm{b})$$

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 ∈ G(x) is of the form (a).