A Pexider equation containing the aritmetic mean

Abstract In this paper we determine the solutions $$(\varphi ,f_1,f_2)$$ ( φ , f 1 , f 2 ) of the Pexider functional equation $$\begin{aligned} \varphi \Big (\frac{x+y}{2}\Big )\big (f_1(x)-f_2(y)\big )=0,\qquad (x,y)\in I_1\times I_2, \end{aligned}$$ φ ( x + y 2 ) ( f 1 ( x ) - f 2 ( y ) ) = 0 , ( x , y ) ∈ I 1 × I 2 , where $$I_1$$ I 1 and $$I_2$$ I 2 are nonempty open subintervals. Special cases of the above equation regularly arise in problems with two-variable means. We show that, under a rather weak regularity condition, the coordinate-functions of a typical solution of the equation are constant over several subintervals of their domain. The regularity condition in question will be that the set of zeros of $$\varphi $$ φ is closed. We also discuss particular solutions where this condition is not met.