Document Type : Research Paper

Author

• Razieh Farokhzad Rostami

Department of Mathematics, Faculty of Basic Sciences and Engineering, Gonbad Kavous University, Gonbad Kavous, Iran

Abstract

Let $A$ be a Banach ternary algebra over a scalar field $\mathbb{R}$ or $\mathbb{C}$ and $X$ be a ternary Banach $A$--module. Let $\sigma ,\tau$ and $\xi$ be linear mappings on $A$, a linear mapping $D:(A,[{]}_A)\to (X,[{]}_X)$ is called a Lie ternary $(\sigma ,\tau ,\xi )$--derivation, if
$$D([a,b,c])=[[D(a)bc{]}_X{]}_{(\sigma ,\tau ,\xi )}-[[D(c)ba{]}_X{]}_{(\sigma ,\tau ,\xi )}$$
for all $a,b,c\in A$, where $[abc{]}_{(\sigma ,\tau ,\xi )}=a\tau (b)\xi (c)-\sigma (c)\tau (b)a$ and $[a,b,c]=[abc{]}_A-[cba{]}_A$. In this paper, we prove the generalized Hyers--Ulam--Rassias stability of Lie ternary $(\sigma ,\tau ,\xi )$--derivations on Banach ternary algebras and $C^{\ast }$--Lie ternary $(\sigma ,\tau ,\xi )$--derivations on $C^{\ast }$--ternary algebras for the following Euler--Lagrange type additive mapping:
$$\sum _{i=1}^{n}f\mathbf{\text{(}}\sum _{j=1}^{n}q(x_i-x_j)\mathbf{\text{)}}+nf(\sum _{i=1}^{n}q{x}_i)=nq\sum _{i=1}^{n}f(x_i).$$

Keywords

• Banach ternary algebra
• Lie ternary $(\sigma
• \tau
• \xi)$-derivation
• Hyers-Ulam-Rassias stability