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

TL;DR

This paper studies the stability of a specific type of derivation called Lie ternary $(\sigma, au,\xi)$-derivations on Banach ternary algebras, extending the Hyers--Ulam--Rassias stability concept.

Contribution

It introduces and analyzes the generalized Hyers--Ulam--Rassias stability for Lie ternary $(\sigma, au,\xi)$-derivations on Banach ternary algebras, a novel extension in the field.

Findings

Established stability results for Lie ternary $(\sigma, au,\xi)$-derivations.

Extended the Hyers--Ulam--Rassias stability to ternary algebra context.

Provided conditions under which stability holds.

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.