Alienation of the Quadratic and Additive Functional Equations

Let G, H be uniquely 2-divisible Abelian groups. We study the solutions f, g: G → H of Pexider type functional equation f(x+ y) + f(x−y) + g(x+ y) = 2 f(x) + 2 f(y) + g(x) + g(y) , resulting from summing up the well known quadratic functional equation and additive Cauchy functional equation side by side. We show that modulo a constant equation (*) forces f to be a quadratic function, and g to be an additive one (alienation phenomenon). Moreover, some stability result for equation (*) is also presented.