Ternary (\sigma,\tau,\xi)-derivations on Banach ternary algebras

Let A be a Banach ternary algebra over a scalar field \mathbb{R} or \mathbb{C} and X be a Banach ternary A-module. Let \sigma, \tau and \xi be linear mappings on A, a linear mapping D : (A,[ ]_A) \to (X, [ ]_X) is called a ternary (\sigma,\tau,\xi)-derivation, ifD([xyz]_A) = [D(x)\tau(y)\xi(z)]_X + [\sigma(x)D(y)\xi(z)]_X + [\sigma(x)\tau(y)D(z)]_Xfor all x,y, z \in A. In this paper, we investigate ternary (\sigma,\tau,\xi)-derivation on Banach ternary algebras, associated with the following functional equationf(\frac{x + y + z}{۴}) + f(\frac{۳x - y - ۴z}{۴}) + f(\frac{۴x + ۳z}{۴}) = ۲f(x).Moreover, we prove the generalized Ulam-Hyers stability of ternary (\sigma,\tau,\xi)-derivations on Banach ternary algebras.

Keywords:

Banach ternary algebra , Banach ternary A-module , Ternary (sigma , tau , xi)-derivation