Let G be an abelian group with a metric d, E be a normed space and f : G → E be a given function. We define difference C 3,1 f by the formula C 3,1 f(x,y) = 3f(x + y) + 3f(x − y) + 48f(x) − f(3x + y) − f(3x − y) for every x,y ∈ G. Under some assumptions about f and C 3,1 f, we show that if C 3,1 f is Lipschitz, then there exists a cubic function C : G → E such that f − C is Lipschitz with the same constant. Moreover, we study the approximation of the equality C 3,1 f(x,y) = 0 in the Lipschitz norms.
