On a problem related to the generalized Schilling equation

Abstract

In this note it is shown that the solution $\varphi : \Bbb {R} \rightarrow \Bbb {R}$ of the functional equation¶¶>$ A \varphi (g(x)) = B \varphi (h(x)) + D \varphi (x) $ \qquad for $ x \in \Bbb {R} $¶$(g, h : \Bbb {R} \rightarrow \Bbb {R}$ are given functions, $A, B, D \in \Bbb {R} \setminus \{0\}$ are given constants) is uniquely defined by the restriction $\varphi_{\mid [h^{-1}(x_{0}), x_{0}]}$ satisfying the condition¶¶$ A \varphi (g (h^{-1} (x_{0}))) = B \varphi (x_{0}) + D \varphi(h^{-1}(x_{0})). $¶