Comment on "Approximate ternary Jordan derivations on Banach ternary algebras" [Bavand Savadkouhi et al. J. Math. Phys. 50, 042303, (2009)]

Abstract

Let A be a Banach ternary algebra over C and X a ternary Banach A-module. A C-linear mapping D: (A, [ ](A)) -> (X, [ ](X)) is called a ternary Jordan derivation if D([xxx](A)) = [D(x)xx](X) + [xD(x)x](X) + [xxD(x)](X) for all x is an element of A. [Bavand Savadkouhi et al., J. Math. Phys. 50, 042303 (2009)] investigated ternary Jordan derivations 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), and proved the generalized Ulam-Hyers stability of ternary Jordan derivations on Banach ternary algebras. The mapping f in Lemma 2.2 of Bavand Savadkouhi et al. is identically zero and all of the results are trivial. In this note, we correct the statements of the results and the proofs.