On some equation for set-valued functions

In his paper Smajdor (Aequat. Math. 75 149–162, 2008) showed that the equation H + tH ² = (I + tH) ○ H, t ≥ 0 is a necessary and sufficient condition under which the family {F ^t , t ≥ 0} of set-valued functions ${F^t(x):=sum_{n=0}^{infty} frac{t^n}{n!}H^n(x), x in K}$ is an iteration semigroup. We present a simple proof of a generalization of this result, independent of the coefficients of the series.