Assume that \((\Omega ,\mathcal A,\mathbb {P})\) is a probability space, \((X,\rho )\) is a compact metric space and Y is a separable Banach space. Under relevant assumptions about the given function \( f :X \times \Omega \rightarrow X \) we show that the set of all continuous functions \(F :X \rightarrow Y\) such that the equation \(\begin{aligned} \varphi (x)=\int _{\Omega }\varphi \big (f(x,\omega )\big )\mathbb {P}(d\omega )+F(x) \end{aligned}\) has a continuous solution \(\varphi :X \rightarrow Y\) is small from the points of view of both category and measure theory.
