# Random-valued functions and iterative functional equations

Authors: Baron, Karol1; Jarczyk, Witold2

Source: aequationes mathematicae, Volume 67, Numbers 1-2, March 2004 , pp. 140-153(14)

Publisher: Springer

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Abstract:

Given a random-valued function $$f : [0, 1] \times \Omega \to [0, 1]$$ on a probability space $$(\Omega, {\mathcal A}, P)$$ , we consider bounded solutions $$\psi : [0, 1] \to \mathbb{R}$$ of the inequality

$$\psi(x) \leq \int\limits_{\Omega} \psi(f(x, \omega)) dP (\omega)$$

and a uniqueness-type problem for bounded solutions $$\varphi$$ of equations of the type

$$\varphi(x) = h(x, \varphi \circ f(x, \cdot)).$$

Analogues for $$f : \mathbb{R} \times \Omega \to \mathbb{R}$$ of the form $$f : f(x, \omega) = x + \xi (\omega)$$ are proved. Some particular cases are studied in more details, especially those where the probability space under considerations is simply the set of positive integers.

Document Type: Research Article

Affiliations: 1: Instytut Matematyki, Uniwersytet Śląski, Bankowa 14, 40-007, Katowice, Poland, 2: Instytut Matematyki, Uniwersytet Zielonogórski, Szafrana 4a, 65-516, Zielona Góra, Poland,

Publication date: March 1, 2004

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