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
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.
- Rafał Kapica
- Silesian University, Institute of Mathematics, Bankowa 14, 40-007 Katowice, Poland
- Janusz Morawiec
- Silesian University, Institute of Mathematics, Bankowa 14, 40-007 Katowice, Poland
- Received: 2008-04-22.
- Revised: 2008-11-03.
- Accepted: 2009-03-09.
