Some Class of Linear Operators Involved in Functional Equations

Abstract

Fix $N\in \mathbb{N}$ and assume that for every $n\in \{1,\dots ,N\}$ the functions $f_n:[0,1]\to [0,1]$ and $g_n:[0,1]\to \mathbb{R}$ are Lebesgue measurable, $f_n$ is almost everywhere approximately differentiable with $|g_n(x)|<|f_n^{\prime }(x)|$ for almost all $x\in [0,1]$, there exists $K\in \mathbb{N}$ such that the set $\{x\in [0,1]:\mathrm{card}{f}_n^{-1}(x)>K\}$ 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 $P:L^1([0,1])\to L^1([0,1])$, and then we obtain results on the existence and uniqueness of solutions $\phi \in L^1([0,1])$ of the equation $\phi =P\phi +g$ with a given $g\in L^1([0,1])$.