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

Document Type : Research Paper

Authors

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

Abstract

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_{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 $x,y \in \mathcal{G}$. 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 $C :\mathcal{G} \longrightarrow \mathcal{E}$ 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.

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