Linear Functional Equations and their Solutions in Lorentz Spaces

Assume that $Ω\subset \mathbb{R}^k$ is an open set, $V$ is a separable Banach space over a field $\mathbb K\in\{\mathbb R,\mathbb C\}$ and $f_1,\ldots,f_N \colonΩ\to Ω$, $g_1,\ldots, g_N\colonΩ\to \mathbb{K}$, $h_0\colon Ω\to V$ are given functions. We are interested in the existence and uniqueness of solutions $φ\colon Ω\to V$ of the linear functional equation $φ=\sum_{k=1}^{N}g_k\cdot(φ\circ f_k)+h_0$ in Lorentz spaces.

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