On the Inhomogeneous Sincov’s Equation

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.