On some set-valued iteration semigroups generated by interval-valued functions

Let X be an arbitrary set. We characterize all interval-valued functions ({A:Xto 2^mathbb{R}}) for which a multifunction ({F:(0,infty)times Xto 2^X}) of the form ({F(t,x)=A^{-}big(A(x)+min {t,q-inf A(x)}big)}), where ({q=sup A(X)}), is an iteration semigroup. The multifunction F is the set-valued counterpart of the fundamental form of continuous iteration semigroups of single-valued functions on an interval.