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.

Keywords: 39B12; 39B62; 60G42; 60K99; Bounded continuous solutions; Iterative functional equations and inequalities; Primary 39B22; Random-valued functions; Secondary 60F15; Uniqueness-type problem

Document Type: Research Article

DOI: http://dx.doi.org/10.1007/s00010-003-2717-3

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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