Summary.
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.
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Manuscript received: April 19, 2002 and, in final form, February 26, 2003.
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Baron, K., Jarczyk, W. Random-valued functions and iterative functional equations . Aequ. Math. 67, 140–153 (2004). https://doi.org/10.1007/s00010-003-2717-3
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DOI: https://doi.org/10.1007/s00010-003-2717-3