Linear Functional Equations and their Solutions in Lorentz Spaces

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 $f_1,\ldots,f_N \colon\Omega\to \Omega$, $g_1,\ldots, g_N\colon\Omega\to \mathbb{K}$, $h_0\colon \Omega\to V$ are given functions. We are interested in the existence and uniqueness of solutions $\varphi\colon \Omega\to V$ of the linear functional equation $\varphi=\sum_{k=1}^{N}g_k\cdot(\varphi\circ f_k)+h_0$ in Lorentz spaces.