Sequences of iterates of random-valued vector functions and continuous solutions of related equations

Abstract

Abstract. Given a probability space (Ω, A, P ), a separable metric space X, and a random-valued vector function f : X × Ω → X, we obtain some theorems on the existence and on the uniqueness of continuous solutions ϕ : X → R of the equation ϕ(x) = Ω ϕ(f (x, ω))P (dω).

On the convergence of sequences of iterates of random-valued vector functions

“…Consequently, for every x ∈ X and a.s. ω ∈ Ω ∞ the series (7) converges for every x * satisfying (ii). This ends the proof of Theorem 1 and of the first part of (6).…”

“…They are useful for instance in solving functional-integral equations (see, e.g., [3], [6]). In [7] some conditions are established which guarantee the convergence (a.s. and in L 1 ) of (f n (x, ·)) with respect to the product measure.…”

The set of probability distribution solutions of a linear functional equation

“…Measurable and nonnegative solutions of (1) have been investigated in [7], [8]. Continuous and bounded solutions of (1) have been studied in [3], [5], [9], [10]. Existence and uniqueness of solutions of (1) in a class of bounded functions have been proved in [13].…”

“…functions? Another motivation to pose the above question comes from [1], [9] and [11]. For example, we know from [1] that equation (1) may possess exactly one solution in the class of all p.d.…”

“…But the problem is: When is the unique solution continuous? According to [9] in some cases equation (1) has no continuous p.d. solutions (see Theorem B in Section 7) and the problem reads: Are there discontinuous p.d.…”