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

Document Type: Research Paper

Author

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 $\Bbb R$ 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
$$D([a,b,c])=[[D(a)bc]_X]_{(\sigma,\tau,\xi)}-[[D(c)ba]_X]_{(\sigma,\tau,\xi)}$$
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 $C^*$--Lie ternary $(\sigma,\tau,\xi)$--derivations on $C^*$--ternary algebras for the following Euler--Lagrange type additive mapping:
$$\sum_{i=1}^{n}f\textbf{(}\sum_{j=1}^{n}q(x_i-x_j)\textbf{)} +nf(\sum_{i=1}^{n}qx_i)=nq\sum_{i=1}^{n}f(x_i).$$

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