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

Let \mathcal{G} be an abelian group with a metric d, \mathcal{E} be a normed space and f :\mathcal{G} \longrightarrow \mathcal{E} be a given function. We define difference C_{۳,۱} f by the formulaC_{۳,۱} f(x,y) = ۳f(x + y) + ۳f(x − y) + ۴۸f(x) − f(۳x + y) − f(۳x − y)for every x,y \in \mathcal{G}. Under some assumptions about f and C_{۳,۱} f , we show that if C_{۳,۱} f is Lipschitz, then there exists a cubic function C :\mathcal{G} \longrightarrow \mathcal{E} such that f − C is Lipschitz with the same constant. Moreover, we study the approximation of the equality C_{۳,۱} f(x,y) = ۰ in the Lipschitz norms.

Keywords:

Approximation , d-Lipschitz , Left invariant mean , Cubic difference , Lipschitz norm