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 ∈ 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.  2004 Elsevier Inc. All rights reserved.