Approximately Lie ternary $(\sigma ,\tau ,\xi )-$derivations on Banach ternary algebras

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 $$D([abc{]}_A)=[[D(a)bc{]}_X{]}_{(\sigma ,\tau ,\xi )}+[[D(b)ac{]}_X{]}_{(\sigma ,\tau ,\xi )}+[[D(c)ba{]}_X{]}_{(\sigma ,\tau ,\xi )},$$ for all $a,b,c\in A$, where $[abc{]}_{(\sigma ,\tau ,\xi )}=a\tau (b)\xi (c)-\sigma (c)\tau (b)a.$ In this paper, we investigate the generalized Hyers--Ulam--Rassias stability of Lie ternary $(\sigma ,\tau ,\xi )-$derivations on Banach ternary algebras.

8 pages