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

Abstract

Let $A$ be a Banach ternary algebra over a scalar field $\Bbb R$ or $\Bbb C$ 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.