Janusz Morawiec, Thomas Zürcher. Source: Annals of Functional Analysis, Volume 10, Number 3, 381--394.Abstract:
Fix $N\in\mathbb{N}$ , and assume that, for every $n\in\{1,\ldots,N\}$ , the functions $f_{n}\colon[0,1]\to[0,1]$ and $g_{n}\colon[0,1]\to\mathbb{R}$ are Lebesgue-measurable, $f_{n}$ is almost everywhere approximately differentiable with $|g_{n}(x)|\lt |f'_{n}(x)|$ for almost all $x\in[0,1]$ , there exists $K\in\mathbb{N}$ such that the set $\{x\in[0,1]:\operatorname{card}{f_{n}^{-1}(x)}\gt K\}$ is of Lebesgue measure zero, $f_{n}$ satisfy Luzin’s condition N, and the set $f_{n}^{-1}(A)$ is of Lebesgue measure zero for every set $A\subset\mathbb{R}$ of Lebesgue measure zero. We show that the formula $Ph=\sum_{n=1}^{N}g_{n}\cdot (h\circ f_{n})$ defines a linear and continuous operator $P\colon L^{1}([0,1])\to L^{1}([0,1])$ , and then we obtain results on the existence and uniqueness of solutions $\varphi\in L^{1}([0,1])$ of the equation $\varphi=P\varphi+g$ with a given $g\in L^{1}([0,1])$ .