Remarks connected with the weak limit of iterates of some randomvalued functions and iterative functional equations

Abstract

The paper consists of two parts. At first, assuming that ( ;A; P) is a probability space and (X; %) is a complete and separable metric space with the -algebra B of all its Borel subsets we consider the set Rc of all B A- measurable and contractive in mean functions f : X ! X with finite integral R

% (f(x; !); x) P(d!) for x 2 X, the weak limit f of the sequence of iterates of f 2 Rc, and investigate continuity-like property of the function f 7! f ; f 2 Rc, and Lipschitz solutions ' that take values in a separable Banach space of the equation