Linear functional equations and their solutions in Lorentz spaces (Q2144422)

Assume that \(\Omega\subset\mathbb{R}^k\) is an open set, \(V\) is a (real or complex) separable Banach space, further \(f_1,\dots,f_N\colon\Omega\to\Omega\) and \(g_1,\dots,g_N\colon\Omega\to V\), \(h_0\colon\Omega\to V\) are given functions. The main result of the paper is an existence and uniqueness theorem for the solution \(\varphi\colon\Omega\to V\) of the linear functional equation \[ \varphi=\sum_{k=1}^N g_k\cdot(\varphi\circ f_k)+h_0 \] in Lorenz spaces. The approach is based on fixed point methods. In order to apply the Banach contraction principle, the change of variables formula by Hajłasz is needed.