On approximation of approximately generalized quadratic functional equation via Lipschitz criteria

Let G be an Abelian group with a metric d and E ba a normed space. For any f : G -> E we define the generalized quadratic difference of the function f by the formula Q(k) f (x, y) := f (x + ky) + f (x - ky) - f (x + y) - f (x - y) - 2(k(2) - 1)f (y) for all x, y is an element of G and for any integer k with k not equal 1, -1. In this paper, we achieve the general solution of equation Q(k) f (x, y) = 0, after it, we show that if Q(k) f is Lipschitz, then there exists a quadratic function K : G -> E such that f - K is Lipschitz with the same constant. Moreover, some results concerning the stability of the generalized quadratic functional equation in the Lipschitz norms are presented. In the particular case, if k = 0 we obtain the main result that is in [7].