On approximation of approximately generalized quadratic functional equation via Lipschitz criteria

Abstract

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_kf(x, y) := f(x + ky) + f(x - ky) - f(x + y) - f(x - y) - 2(k² - 1) f(y)

for all x, y ∈ G and for any integer k with k ≠ 1, -1. In this paper, we achieve the general solution of equation Q_kf(x, y) = 0; after it, we show that if Q_kf 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].

Mathematics Subject Classification (2010): Primary 39B82, 39B52.

Keywords: Generalized quadratic functional equation, stability, Lipschitz space