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

Document Type : Research Paper


Department of Mathematics, Faculty of Mathematical Sciences and Statistics, Malayer University, Malayer, Iran


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.