Measurable solutions of an alternative functional equation

Abstract In this paper we investigate the functional equation $$\begin{aligned} \varphi \left( \frac{x+y}{2} \right) \left( \psi _1(x) - \psi _2(y) \right) = 0 \quad \left( \hbox { for all } x \in I_1 \hbox { and } y \in I_2 \right) \end{aligned}$$ φ x + y 2 ψ 1 ( x ) - ψ 2 ( y ) = 0 for all x ∈ I 1 and y ∈ I 2 where $$ I_1 , I_2 $$ I 1 , I 2 are open intervals of $$ \mathbb {R}$$ R , $$ J = \frac{1}{2} \left( I_1 + I_2 \right) $$ J = 1 2 I 1 + I 2 moreover $$ \psi _1: I_1 \rightarrow \mathbb {R}$$ ψ 1 : I 1 → R , $$ \psi _2: I_2 \rightarrow \mathbb {R}$$ ψ 2 : I 2 → R and $$ \varphi : J \rightarrow \mathbb {R}$$ φ : J → R are unknown functions. We describe the structure of the possible solutions assuming that $$ \varphi $$ φ is measurable. In the case when