Continuous solutions of iterative equations of infinite order

Abstract. Given a probability space \((\Omega,\mathcal{A}, P)\) and a complete separable metric space \(X\), we consider continuous and bounded solutions \(\varphi: X \to \mathbb{R}\) of the equations \(\varphi(x) = \int_{\Omega} \varphi(f(x,\omega))P(d\omega)\) and \(\varphi(x) = 1-\int_{\Omega} \varphi(f(x,\omega))P(d\omega)\), assuming that the given function \(f:X \times \Omega \to X\) is controlled by a random variable \(L: \Omega \to (0,\infty)\) with \(-\infty \lt \int_{\Omega} \log L(\omega)P(d\omega) \lt 0\). An application to a refinement type equation is also presented.