Alienation of the quadratic and additive functional equations (Q2322512)

Let \(G\) and \(H\) be uniquely \(2\)-divisible abelian groups. The Pexider-type functional equation \(f(x+y) + f(x-y) + g(x+y) = 2f(x) + 2f(y) + g(x) + g(y)\) is constructed by summing up the quadratic functional equation and additive Cauchy functional equation side by side. Here \(f, g : G \to H\). The author studies this equation and shows that, modulo a constant, \(f\) is a quadratic function and \(g\) is an additive function. He uses results of [\textit{D. H. Hyers}, Proc. Natl. Acad. Sci. USA 27, 222--224 (1941; Zbl 0061.26403); \textit{F. Skof}, Rend. Sem. Mat. Fis. Milano 53, 113--129 (1983; Zbl 0599.39007); \textit{P. W. Cholewa}, Aequationes Math. 27, 76--86 (1984; Zbl 0549.39006)] to prove the Hyers-Ulam stability of the above equation.