Random-valued functions and iterative functional equations
Given a random-valued function \(f: [0,1] \times \Omega \to [0,1]\) on a probability space \((\Omega, A, P)\), the authors consider the bounded solutions \(\psi : [0,1] \to R\) of the inequality \[ \psi (x) \leq \int_{\Omega} \psi (f(x, \omega)) \,dP (\omega) \] A uniqueness type problem for bounded solutions \(\varphi\) of equations of the type \[ \varphi (x) = h ( x , \varphi \circ f(x, \cdot)) \] is studied. Results for \( f: R \times \Omega \to R\) of the form \( f( x, \omega) = z + \xi (\omega)\) 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.
scientific article; zbMATH DE number 2119231