Linear Functional Equations and their Solutions in Lorentz Spaces

Abstract

Assume that $\Omega \subset {\mathbb{R}}^k$ is an open set, $V$ is a separable Banach space over a field $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$ and $f_1,\dots ,f_N:\Omega \to \Omega$, $g_1,\dots ,g_N:\Omega \to \mathbb{K}$, $h_0:\Omega \to V$ are given functions. We are interested in the existence and uniqueness of solutions $\phi :\Omega \to V$ of the linear functional equation $\phi ={\sum }_{k=1}^N{g}_k\cdot (\phi \circ f_k)+h_0$ in Lorentz spaces.