REMARKS CONNECTED WITH THE WEAK LIMIT OF ITERATES OF SOME RANDOM-VALUED FUNCTIONS AND ITERATIVE FUNCTIONAL EQUATIONS

The paper consists of two parts. At first, assuming that (Omega,A, P) is a probability space and (X, rho) is a complete and separable metric space with the sigma-algebra B of all its Borel subsets we consider the set R-c of all B circle times A-measurable and contractive in mean functions f : X x Omega -> X with finite integral integral(Omega) rho (f(x, omega), x) P(d omega) for x epsilon X, the weak limit pi(f) of the sequence of iterates of f epsilon R-c, and investigate continuity-like property of the function f ->pi(f), f epsilon R-c, and Lipschitz solutions phi that take values in a separable Banach space of the equation phi(x) = integral(Omega) phi(f(x, omega)) P(d omega) + F(x). Next, assuming that X is a real separable Hilbert space, Lambda: X -> X is linear and continuous with vertical bar vertical bar Lambda vertical bar vertical bar < 1, and mu is a probability Borel measure on X with finite first moment we examine continuous at zero solutions phi: X -> C of the equation phi(x) = <(mu)over cap>(x)phi(Lambda x) which characterizes the limit distribution pi(f) for some special f epsilon R-c.