Document Type : Research Paper

Authors

• Mahshid Dashti
• Hamid Khodaei

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 $f:\mathcal{G}⟶\mathcal{E}$ 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 $C:\mathcal{G}⟶\mathcal{E}$ 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.

Keywords

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