Some properties of probability distribution solutions of linear functional equations (Q1271583)

The author deals with the functional equation \(\varphi (x) = \int_T \varphi [ f(x,t)] d \mu (t) \) showing first that under some natural conditions, if \( f(\cdot, t)| _{[0,1]} \) is increasing for every \( t \in T\) then there exists exactly one solution \( \varphi : R \to R \) such that \( \varphi \) is bounded, \( \varphi (x) = 0 \) when \( x \leq 0 \), \( \varphi (x) = 1\) when \( x \geq 1 \), \( \varphi \) is continuous, increasing, and either absolutely continuous or singular. Then the case \( \varphi (x) = \sum_{n\in T} p_n \varphi [ f_n (x)]\), where \( \{ 1, 2\} \subset T \subset N \), is studied with detail and properties of the unique bounded solution \( \varphi \) such that \( \varphi (x) = 0\), if \( x \leq 0 \) and \( \varphi (x) = 1 \) when \( x \geq 1\), are given.