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

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 $a,b,cin A$, where $[abc]_{(sigma,tau,xi)}=atau(b)xi(c)-sigma(c)tau(b)a$ 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: $$sum_{i=1}^{n}ftextbf{(}sum_{j=1}^{n}q(x_i-x_j)textbf{)} +nf(sum_{i=1}^{n}qx_i)=nqsum_{i=1}^{n}f(x_i).$$