Comparison of two regulatory results on functional equations with few parameters

Abstract

Two methods to prove regularity properties of the linear functional equation

$$f(x)=h_0(x,y)+\sum_{j=1}^n h_j(x,y)f(x+g_j(y)), $$

where ${(x,y) \in D \subset \mathbb{R}^r \times \mathbb{R}^s}$, ${x \in \mathbb{R}^r}$ and ${y \in \mathbb{R}^s}$, with few parameters i.e. allowing 1 ≤ s < r are examined. It is proved that—under certain conditions, for some class of equations and in some sense—they are equivalent.