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

Authors

1 Department of Mathematics, Semnan University, P. O. Box 35195-363, Semnan, Iran.

2 Department of Mathematics, Shahid Beheshti University, Tehran, Iran.

3 Department of Mathematical Sciences, University of Arkansas, Fayetteville, Arkansas 72701, USA.

Abstract

Let A be a Banach ternary algebra over a scalar eld R or C and X be a Banach ternary A-module.
Let ;  and  be linear mappings on A, a linear mapping D : (A; [ ]A) ! (X; [ ]X) is called a ternary
(; ; )-derivation, if
D([xyz]A) = [D(x) (y)(z)]X + [(x)D(y)(z)]X + [(x) (y)D(z)]X
for all x; y; z 2 A.
In this paper, we investigate ternary (; ; )-derivation on Banach ternary algebras, associated
with the following functional equation
f(
x + y + z
4
) + f(
3x 􀀀 y 􀀀 4z
4
) + f(
4x + 3z
4
) = 2f(x) :
Moreover, we prove the generalized Ulam{Hyers stability of ternary (; ; )-derivations on Banach
ternary algebras.