Stability of the quadratic functional equation in Lipschitz spaces

Let G be an Abelian group with a metric d and E a normed space. For any f : G --> E we define the quadratic difference of the function f by the formula Qf (x, y) := 2f (x) + 2f (y) - f (x + y) - f (x - y) for x, y is an element of G. Under some assumptions about f and Qf we prove that if Qf is Lipschitz, then there exists a quadratic function K : G --> E such that f - K is Lipschitz. Moreover, some results concerning the stability of the quadratic functional equation in the Lipschitz norms are presented. (C) 2004 Elsevier Inc. All rights reserved.