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,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=۱}^{n}f\textbf{(}\sum_{j=۱}^{n}q(x_i-x_j)\textbf{)} +nf(\sum_{i=۱}^{n}qx_i)=nq\sum_{i=۱}^{n}f(x_i).

Keywords:

Banach ternary algebra , Lie ternary (sigma , tau , xi)-derivation , Hyers-Ulam-Rassias stability