Approximately Ternary Homomorphisms on C ∗-Ternary Algebras

and Applied Analysis 3 for all x 1 ∈ A. Moreover, letting m = 0 and passing the limit n → ∞ in (12), we get (9). It follows from (8) that 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 H( μx 2 − x 1 3 ) + H( x 1 − 3μx 3

The stability problem of functional equations is originated from the following question of Ulam [5]: under what condition does there exist an additive mapping near an approximately additive mapping?In 1941, Hyers [6] gave a partial affirmative answer to the question of Ulam in the context of Banach spaces.In 1978, Rassias [7] extended the theorem of Hyers by considering the unbounded Cauchy difference ‖(+)−()−()‖ ≤ (‖‖  +‖‖  ), ( > 0,  ∈ [0, 1)).The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [8][9][10][11][12]).
Gordji et al. [13] proved the Hyers-Ulam stability and the superstability of  * -ternary homomorphisms and  *ternary derivations on  * -ternary algebras, associated with the functional equation by applying the direct method.Under the conditions in the main theorems of [13], we can show that the related mappings must be zero.
In this paper, we change the conditions of [13] and establish the corrected theorems.Moreover, we prove the Hyers-Ulam stability and the superstability of  * -ternary homomorphisms and  * -ternary derivations on  * -ternary algebras by employing a fixed point method.In fact, we show that some results of [13] are the special cases of our results.

Superstability: Direct Method
Throughout this paper, we assume that  is a  * -ternary algebra with norm ‖ ⋅ ‖ and that  is a  * -ternary algebra with norm ‖ ⋅ ‖.Moreover, we assume that  0 ∈ N is a positive integer and suppose that T 1 1/ 0 := {  ; 0 ≤  ≤ 2/ 0 }.In this section, we modify some results of [13].Recall that a functional equation is called superstable if every approximate solution is an exact solution of it.
Proof.The proof is the same as in the proof of [13, Theorem 2.2].
In the following result, we correct Theorem 3 from [13].Since the proof is similar, it is omitted.Theorem 3. Let  ̸ = 1 and  be nonnegative real numbers, and let  :  →  be a mapping satisfying (5) for all  1 ,  2 ,  3 ∈ .Then, the mapping  :  →  is a  * -ternary derivation.

Hyers-Ulam Stability: Direct Method
In this section, we prove the Hyers-Ulam stability of  * -ternary homomorphisms and  * -ternary derivations on  * -ternary algebras by the direct method.
Proof.The proof is similar to the proof of Theorem 4.
In the following theorem, we prove the Hyers-Ulam stability of derivations on  * -ternary algebras via the direct method.Theorem 6.Let  > 1 and  be nonnegative real numbers, and let  :  →  be a mapping satisfying (7) and for all  ∈ T 1 1/ 0 and all  1 ,  2 ,  3 ∈ .Then, there exists a unique  * -ternary derivation  :  →  such that for all  1 ∈ .
Proof.By the same reasoning as in the proof of Theorem 4, there exists a unique C-linear mapping  :  →  satisfying (20) which is defined by for all  1 ∈ .The inequality (7) implies that for all  1 ,  2 ,  3 ∈ .Consequently, the mapping  is a unique  * -ternary derivation satisfying (20).
The following consequence is analogous to Theorem 4 for  * -ternary derivations and its proof is similar to the proof of Theorems 4 and 6.

Superstability: A Fixed Point Approach
In this section, we prove the superstability of  * -ternary homomorphisms and of  * -ternary derivations on  *ternary algebras by using the fixed point method (Theorem 8).
We recall a fundamental result in the fixed point theory from [15] which is a useful tool to achieve our purposes in the sequel.
In 1996, Isac and Rassias [16] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications.In 2003, Cȃdariu and Radu applied a fixed point method to the investigation of the Jensen functional equation [17].They presented a short and a simple proof for the Cauchy functional equation and the quadratic functional equation in [18,19], respectively.By using the fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors.For instance, the Hyers-Ulam stability and the superstability of a ternary quadratic derivation on ternary Banach algebras and  * -ternary rings by using Theorem 8 are investigated in [20].Recently, in [21], Park and Bodaghi proved the stability and the superstability of * -derivations associated with the Cauchy functional equation and the Jensen functional equation by the mentioned theorem (for more applications, see [22][23][24][25][26][27][28]).
Proof.Since the proof is similar to the proof of [13, Theorem for all  1 ,  2 ,  3 ∈ .Thus, the mapping  :  →  is a  * -ternary homomorphism.
Proof.Similar to the proof of Theorem 9, the mapping  :  →  is C-linear.It also follows from (30) that lim for all  1 ,  In analogy with Theorems 9 and 10, we have the following theorems for the superstability of  * -ternary derivations on  * -ternary algebras.
Proof.The proof is similar to the proof of Theorem 9.
Proof.Refer to the proof of Theorem 10.