TY  - JOUR
ID  - Kapica2009
LB  - Kapica2009
AU  - Kapica, Rafał
AU  - Morawiec, Janusz
TI  - Continuous solutions of iterative equations of infinite order
ST  - Continuous solutions of iterative equations of infinite order
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2009
VL  - 29
IS  - 2
SP  - 147
EP  - 155
DO  - http://dx.doi.org/10.7494/OpMath.2009.29.2.147
UR  - http://dx.doi.org/10.7494/OpMath.2009.29.2.147
AB  - 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.
KW  - random-valued vector functions
KW  - sequences of iterates
KW  - iterative equations
KW  - continuous solutions
SN  - 1232-9274
ER  -