A Variant of a Generalized Quadratic Functional Equation on Groups

Let G be a group and the field of complex numbers. Suppose sigma : G -> G is an involutive endomorphism, that is, is an endomorphism of G and it satisfies the condition sigma(sigma(x)) = x for all x in G. In this paper, we find the solutions f, g, h, k, : G -> C of the equation f(xy) + g sigma(y)x) = h(x) + k(y)for all x, y is an element of G assuming f and g to be central functions. This equation is a variant of a generalized quadratic functional equation on groups with an involutive endomorphism. As an application, using the solutions of this equation, we find the solutions f, g, h, k, : G x G -> C of the equation f(pr, qs) + g(sp, rq) = h(p, q) + k(r, s) for all p, q, r, s is an element of G assuming f and g to be central functions.