Integrable solutions of inhomogeneous refinement type equations on intervals

Abstract

Given a probability measure $P$ on a $\sigma$-algebra of subsets of a set $\Omega$, an interval $I\subset \mathbb{R}$, $g\in L^1(I)$, and a function $\phi :I\times \Omega \to I$ fulfilling some conditions we obtain results on the existence of solutions $f\in L^1(I)$ of the inhomogeneous refinement type equation $$f(x)={\int }_{\Omega }|{\phi }_x^{\prime }(x,\omega )|f(\phi (x,\omega ))dP(\omega )+g(x).$$