On some fundamental integral inequalities and their discrete analogues.
We present some extension of the concept of an invariant mean to a space of vector-valued mappings defined on a semigroup. Next, we apply it to the study of the stability of some functional equation.
Let be a disjoint iteration semigroup of diffeomorphisms mapping a real open interval onto . It is proved that if has a dense orbit possesing a subset of the second category with the Baire property, then for some diffeomorphism of onto the set of all reals . The paper generalizes some results of J.A.Baker and G.Blanton [3].