Measurable solutions of an alternative functional equation

Abstract

In this paper we investigate the functional equation \[ φ\left( \frac{x+y}{2} \right) \left( ψ_1(x) - ψ_2(y) \right) = 0 \hspace{20mm} \left( \mbox{ for all } x \in I_1 \mbox{ and } y \in I_2 \right) \] where $I_1,I_2$ are open intervals of $\mathbb{R}$, $J=\frac{1}{2}\left(I_1+I_2\right)$ moreover ${\psi }_1:I_1\to \mathbb{R}$, ${\psi }_2:I_2\to \mathbb{R}$ and $\phi :J\to \mathbb{R}$ are unknown functions. We describe the structure of the possible solutions assuming that $\phi$ is measurable. In the case when $\phi$ is a derivative, we give a complete characterization of the solutions. Furthermore, we present an example of a solution consisting of irregular Darboux functions. This provides the answer to an open problem proposed during the 59th International Symposium on Functional Equations.

15 pages