Lie ternary (σ,τ,ξ)--derivations on Banach ternary algebras

Let A be a Banach ternary algebra over a scalar field R or C and X be a ternary Banach A--module. Let σ,τ and ξ be linear mappings on A, a linear mapping
D:(A,[ ]A)→(X,[ ]X) is called a Lie ternary (σ,τ,ξ)--derivation, if
D([a,b,c])=[[D(a)bc]X](σ,τ,ξ)−[[D(c)ba]X](σ,τ,ξ)
for all
a,b,c∈A, where [abc](σ,τ,ξ)=aτ(b)ξ(c)−σ(c)τ(b)a and [a,b,c]=[abc]A−[cba]A.
In this paper, we prove the generalized Hyers--Ulam--Rassias stability of Lie ternary (σ,τ,ξ)--derivations on Banach ternary algebras and C∗--Lie ternary (σ,τ,ξ)--derivations on C∗--ternary algebras for the following Euler--Lagrange type additive mapping:$$\sum_{i=1}^{n}f\textbf{(}\sum_{j=1}^{n}q(x_i-x_j)\textbf{)}
(\sum_{i=1}^{n}qx_i)=nq\sum_{i=1}^{n}f(x_i).$$