Approximate ternary quadratic derivation on ternary Banach algebras and C^∗-ternary rings: revisited

Recently, Shagholi et al. [S. Shagholi, M. Eshaghi Gordji, M. B. Savadkouhi, J. Comput. Anal. Appl., 13 (2011), 1097–1105] defined ternary quadratic derivations on ternary Banach algebras and proved the Hyers-Ulam stability of ternary quadratic derivations on ternary Banach algebras. But the definition was not well-defined. Using the fixed point method, Bodaghi and Alias [A. Bodaghi, I. A. Alias, Adv. Difference Equ., 2012 (2012), 9 pages] proved the Hyers-Ulam stability and the superstability of ternary quadratic derivations on ternary Banach algebras and C^∗-ternary rings. There are approximate C-quadraticity conditions in the statements of the theorems and the corollaries, but the proofs for the C-quadraticity were not completed. In this paper, we correct the definition of ternary quadratic derivation and complete the proofs of the theorems and the corollaries.