A new type of approximation for cubic functional equations in Lipschitz spaces

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 c3,1f by the formula c3,1 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 c3,1f, we show that if c3,1f 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 c3,1f(x, y) = 0 in the lipschitz norms.