Some families of sublinear correspondences

Let K be a closed convex cone in a real Banach space, {H\colon K\to\operatorname{cc}(K)} a continuous sublinear correspondence with nonempty, convex and compact values in K, and let {f\colon\mathbb{R}\to\mathbb{R}} be defined by {f(t)=\sum_{n=0}^{\infty}a_{n}t^{n}} , where {t\in\mathbb{R}} , {a_{n}\geq 0} , {n\in\mathbb{N}} . We show that the correspondence {F^{t}(x)\mathrel{\mathop{:}}=\sum_{n=0}^{\infty}a_{n}t^{n}H^{n}(x),(x\in K)} is continuous and sublinear for every {t\geq 0} and {F^{t}\circ F^{s}(x)\subseteq\sum_{n=0}^{\infty}c_{n}H^{n}(x)} , {x\in K} , where {c_{n}=\sum_{k=0}^{n}a_{k}a_{n-k}t^{k}s^{n-k}} , {t,s\geq 0} .