On the Inhomogeneous Sincov’s Equation

Abstract In this paper we obtain some results on the inhomogeneous Sincov’s equation aligned f (x, z) -f (x, y) -f (y, z) =I (x, y, z) aligned f (x, z) - f (x, y) - f (y, z) = I (x, y, z) where f: S² G f: S 2 → G is the unknown function and I: S³ G I: S 3 → G is a given function (S is a nonempty set and (G, +) (G, +) is an Abelian group). We give necessary and sufficient conditions on I for the existence of a solution for the inhomogeneous Sincov’s equation. Moreover, we give a result on Ulam stability for the Sincov’s equation and obtain the best Ulam constant of it. In this way we give an answer to a problem formulated by L. Reich at the 61st International Symposium on Functional Equations.