Opuscula Math. 29, no. 2 (2009), 147-155
http://dx.doi.org/10.7494/OpMath.2009.29.2.147

Opuscula Mathematica

# Continuous solutions of iterative equations of infinite order

Rafał Kapica
Janusz Morawiec

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.

Keywords: random-valued vector functions, sequences of iterates, iterative equations, continuous solutions.

Mathematics Subject Classification: 45A05, 39B12, 39B52, 60B12.

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