Approximately Ternary Homomorphisms on 𝐶^∗-Ternary Algebras

Abstract

Gordji et al. established the Hyers-Ulam stability and the superstability of 𝐶^∗-ternary homomorphisms and 𝐶^∗-ternary derivations on 𝐶^∗-ternary algebras, associated with the following functional equation: 𝑓((𝑥₂−𝑥₁)/3)+𝑓((𝑥₁−3𝑥₃)/3)+𝑓((3𝑥₁+3𝑥₃−𝑥₂)/3)=𝑓(𝑥₁), by the direct method. Under the conditions in the main theorems, we can show that the related mappings must be zero. In this paper, we correct the conditions and prove the corrected theorems. Furthermore, we prove the Hyers-Ulam stability and the superstability of 𝐶^∗-ternary homomorphisms and 𝐶^∗-ternary derivations on 𝐶^∗-ternary algebras by using a fixed point approach.

1. Introduction

A 𝐶^∗-ternary algebra is a complex Banach space 𝐴, equipped with a ternary product (𝑥,𝑦,𝑧)↣[𝑥,𝑦,𝑧] of 𝐴³ into 𝐴, which is ℂ-linear in the outer variables, conjugate ℂ-linear in the middle variable, and associative in the sense that [𝑥,𝑦,[𝑧,𝑤,𝑣]]=[𝑥[𝑤,𝑧,𝑦],𝑣]=[[𝑥,𝑦,𝑧],𝑤,𝑣], and satisfies ‖[𝑥,𝑦,𝑧]‖≤‖𝑥‖⋅‖𝑦‖⋅‖𝑧‖ and ‖[𝑥,𝑥,𝑥]‖=‖𝑥‖³. If a 𝐶^∗-ternary algebra (𝐴[⋅,⋅,⋅]) has an identity, that is, an element 𝑒∈𝐴 such that 𝑥=[𝑥,𝑒,𝑒]=[𝑒,𝑒,𝑥] for all 𝑥∈𝐴, then it is routine to verify that 𝐴, endowed with 𝑥∘𝑦∶=[𝑥,𝑒,𝑦] and 𝑥^∗∶=[𝑒,𝑥,𝑒], is a unital 𝐶^∗-algebra. Conversely, if (𝐴,∘) is a unital 𝐶^∗-algebra, then [𝑥,𝑦,𝑧]∶=𝑥∘𝑦^∗∘𝑧 makes 𝐴 into a 𝐶^∗-ternary algebra. A ℂ-linear mapping 𝐻∶𝐴→𝐵 between 𝐶^∗-ternary algebras is called a 𝐶^∗-ternary homomorphism if 𝐻𝑥,𝑦,𝑧=𝐻(𝑥),𝐻𝑦,𝐻(𝑧) (1) for all 𝑥,𝑦,𝑧∈𝐴. A ℂ-linear mapping 𝛿∶𝐴→𝐴 is called a 𝐶^∗-ternary derivation if 𝛿=𝛿(𝑥)+𝑦+(𝑧.𝑥,𝑦,𝑧,𝑦,𝑧𝑥,𝛿,𝑧𝑥,𝑦,𝛿)𝑥,𝑦,𝑧∈𝐴 (2)

Ternary structures and their generalization, the so-called 𝑛-ary structures, raise certain hopes in view of their applications in physics (see [1–4]).

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 ‖𝑓(𝑥+𝑦)−𝑓(𝑥)−𝑓(𝑦)‖≤𝜀(‖𝑥‖^𝑝+‖𝑦‖^𝑝),(𝜀>0,𝑝∈[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 [8–12]).

Gordji et al. [13] proved the Hyers-Ulam stability and the superstability of 𝐶^∗-ternary homomorphisms and 𝐶^∗-ternary derivations on 𝐶^∗-ternary algebras, associated with the functional equation 𝑓𝑥₂−𝑥₁3𝑥+𝑓¹−3𝑥₃3+𝑓3𝑥₁+3𝑥₃−𝑥₂3=𝑓𝑥₁ (3) 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 𝐶^∗-ternary homomorphisms and 𝐶^∗-ternary derivations on 𝐶^∗-ternary algebras by employing a fixed point method. In fact, we show that some results of [13] are the special cases of our results.

2. Superstability: Direct Method

Throughout this paper, we assume that 𝐴 is a 𝐶^∗-ternary algebra with norm ‖⋅‖ and that 𝐵 is a 𝐶^∗-ternary algebra with norm ‖⋅‖. Moreover, we assume that 𝑛₀∈ℕ is a positive integer and suppose that 𝕋¹_1/𝑛0∶={𝑒^𝑖𝜃;0≤𝜃≤2𝜋/𝑛₀}.

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.

Lemma 1 (see [13]). Let 𝑓∶𝐴→𝐵 be a mapping such that 𝑓𝑥₂−𝑥₁3𝑥+𝑓¹−3𝑥₃3+𝑓3𝑥₁+3𝑥₃−𝑥₂3≤𝑓𝑥₁ (4) for all 𝑥₁,𝑥₂,𝑥₃∈𝐴. Then, 𝑓 is additive.

We correct the statements of  [13, Theorem 2.2] as follows.

Theorem 2. Let 𝑝≠1 and 𝜃 be nonnegative real numbers, and let 𝑓∶𝐴→𝐵 be a mapping such that 𝑓𝜇𝑥₂−𝑥₁3𝑥+𝑓¹−3𝜇𝑥₃3+𝜇𝑓3𝑥₁+3𝑥₃−𝑥₂3≤𝑓𝑥₁,𝑓𝑥₁,𝑥₂,𝑥₃−𝑓𝑥₁,𝑓𝑥₂,𝑓𝑥₃≤𝜃𝑥₁^3𝑝+𝑥₂^3𝑝+𝑥₃^3𝑝 (5)(6) for all 𝜇∈𝕋¹_1/𝑛0 and all 𝑥₁,𝑥₂,𝑥₃∈𝐴. Then, the mapping 𝑓∶𝐴→𝐵 is a 𝐶^∗-ternary homomorphism.

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 𝑝≠1 and 𝜃 be nonnegative real numbers, and let 𝑓∶𝐴→𝐴 be a mapping satisfying (5) and 𝑓𝑥₁,𝑥₂,𝑥₃−𝑓𝑥₁,𝑥₂,𝑥₃−𝑥₁,𝑓𝑥₂,𝑥₃−𝑥₁,𝑥₂,𝑓𝑥₃≤𝜃𝑥₁^3𝑝+𝑥₂^3𝑝+𝑥₃^3𝑝 (7) for all 𝑥₁,𝑥₂,𝑥₃∈𝐴. Then, the mapping 𝑓∶𝐴→𝐴 is a 𝐶^∗-ternary derivation.

3. Hyers-Ulam Stability: Direct Method

In this section, we prove the Hyers-Ulam stability of 𝐶^∗-ternary homomorphisms and 𝐶^∗-ternary derivations on 𝐶^∗-ternary algebras by the direct method.

Theorem 4. Let 𝑝>1 and 𝜃 be nonnegative real numbers, and let 𝑓∶𝐴→𝐵 be a mapping satisfying (6) and 𝑓𝜇𝑥₂−𝑥₁3𝑥+𝑓¹−3𝜇𝑥₃3+𝜇𝑓3𝑥₁+3𝑥₃−𝑥₂3−𝜇𝑓𝑥₁≤𝜃𝑥₁^𝑝+𝑥₂^𝑝+𝑥₃^𝑝 (8) for all 𝜇∈𝕋¹_1/𝑛0 and all 𝑥₁,𝑥₂,𝑥₃∈𝐴. Then, there exists a unique 𝐶^∗-ternary homomorphism 𝐻∶𝐴→𝐵 such that 𝐻𝑥₁−𝑓𝑥₁≤3^𝑝1+2^𝑝𝜃𝑥₁^𝑝3^𝑝−3 (9) for all 𝑥₁∈𝐴.

Proof. Letting 𝜇=1, 𝑥₂=2𝑥₁, and 𝑥₃=0 in (8), we get 𝑥3𝑓¹3−𝑓𝑥₁≤1+2^𝑝𝜃𝑥₁^𝑝 (10) for all 𝑥₁∈𝐴. By induction, we have 3^𝑛𝑓𝑥₁3^𝑛−𝑓𝑥₁≤1+2^𝑝𝜃𝑥₁^𝑝𝑛−1_𝑖=03^{𝑖(1−𝑝)} (11) for all 𝑥₁∈𝐴. Hence, 3^𝑛+𝑚𝑓𝑥₁3^𝑛+𝑚−3^𝑚𝑓𝑥₁3^𝑚≤1+2^𝑝𝜃𝑥₁^𝑝𝑛−1_𝑖=03^{(𝑖+𝑚)(1−𝑝)}≤1+2^𝑝𝜃𝑥₁^{𝑝𝑛+𝑚−1}_𝑖=𝑚3^{𝑖(1−𝑝)} (12) for all nonnegative integers 𝑚 and 𝑛 with 𝑛≥𝑚 and all 𝑥₁∈𝐴. It follows that the sequence {3^𝑛𝑓(𝑥₁/3^𝑛)} is a Cauchy sequence for all 𝑥₁∈𝐴. Since 𝐵 is complete, the sequence {3^𝑛𝑓(𝑥₁/3^𝑛)} converges. Thus, one can define the mapping 𝐻∶𝐴→𝐵 by 𝐻𝑥₁∶=lim_𝑛→∞3^𝑛𝑓𝑥₁3^𝑛 (13) for all 𝑥₁∈𝐴. Moreover, letting 𝑚=0 and passing the limit 𝑛→∞ in (12), we get (9). It follows from (8) that 𝐻𝜇𝑥₂−𝑥₁3𝑥+𝐻¹−3𝜇𝑥₃3+𝜇𝐻3𝑥₁+3𝑥₃−𝑥₂3−𝜇𝐻𝑥₁=lim_𝑛→∞3^𝑛𝑓𝜇𝑥₂−𝑥₁3^𝑛+1𝑥+𝑓¹−3𝜇𝑥₃3^𝑛+1+𝜇𝑓3𝑥₁+3𝑥₃−𝑥₂3^𝑛+1𝑥−𝜇𝑓¹3^𝑛≤lim_𝑛→∞3^𝑛𝜃3^𝑛𝑝𝑥₁^𝑝+𝑥₂^𝑝+𝑥₃^𝑝=0 (14) for all 𝜇∈𝕋¹_1/𝑛0 and all 𝑥₁,𝑥₂,𝑥₃∈𝐴. So 𝐻𝜇𝑥₂−𝑥₁3𝑥+𝐻¹−3𝜇𝑥₃3+𝜇𝐻3𝑥₁+3𝑥₃−𝑥₂3=𝜇𝐻𝑥₁ (15) for all 𝜇∈𝕋¹_1/𝑛0 and all 𝑥₁,𝑥₂,𝑥₃∈𝐴. Put 𝜇=1 in (15). Then the mapping 𝐻∶𝐴→𝐵 satisfies the inequality (4), and thus, the mapping 𝐻∶𝐴→𝐵 is additive. Letting 𝑥₁=𝑥₂=0 in (15), we get 𝐻(−3𝜇𝑥₃/3)+𝜇𝐻(3𝑥₃/3)=0 and so 𝐻(𝜇𝑥₃)=𝜇𝐻(𝑥₃) for all 𝜇∈𝕋¹_1/𝑛0 and all 𝑥₃∈𝐴. By the same reasoning as in the proof of [14, Theorem 2.2], the mapping 𝐻 is ℂ-linear. Now, let 𝐻^′∶𝐴→𝐵 be another additive mapping satisfying (9). Then, we have 𝐻𝑥₁−𝐻^′𝑥₁=3^𝑛𝐻𝑥₁3^𝑛−𝐻^′𝑥₁3^𝑛≤3^𝑛𝐻𝑥₁3^𝑛𝑥−𝑓¹3^𝑛+𝐻^′𝑥₁3^𝑛𝑥−𝑓¹3^𝑛≤2⋅3^𝑛1+2^𝑝3^𝑝3^𝑛𝑝3^𝑝𝜃‖𝑥‖−3^𝑝, (16) which tends to zero as 𝑛→∞ for all 𝑥₁∈𝐴. Thus, we can conclude that 𝐻(𝑥₁)=𝐻^′(𝑥₁) for all 𝑥₁∈𝐴. This shows the uniqueness of 𝐻. It follows from (6) that 𝐻𝑥₁,𝑥₂,𝑥₃−𝐻𝑥₁,𝐻𝑥₂,𝐻𝑥₃=lim_𝑛→∞27^𝑛𝑓𝑥₁,𝑥₂,𝑥₃27^𝑛−𝑓𝑥₁3^𝑛𝑥,𝑓²3^𝑛𝑥,𝑓³3^𝑛≤lim_𝑛→∞27^𝑛𝜃27^𝑛𝑝𝑥₁^3𝑝+𝑥₂^3𝑝+𝑥₃^3𝑝=0 (17) for all 𝑥₁,𝑥₂,𝑥₃∈𝐴. Therefore, the mapping 𝐻 is a unique 𝐶^∗-ternary homomorphism satisfying (9).

Theorem 5. Let 𝑝<1 and 𝜃 be nonnegative real numbers, and let 𝑓∶𝐴→𝐵 be a mapping satisfying (6) and (8). Then, there exists a unique 𝐶^∗-ternary homomorphism 𝐻∶𝐴→𝐵 such that 𝐻𝑥₁−𝑓𝑥₁≤3^𝑝1+2^𝑝𝜃𝑥₁^𝑝3−3^𝑝 (18) for all 𝑥₁∈𝐴.

Proof. The proof is similar to the proof of Theorem 4.

In the following theorem, we prove the Hyers-Ulam stability of derivations on 𝐶^∗-ternary algebras via the direct method.

Theorem 6. Let 𝑝>1 and 𝜃 be nonnegative real numbers, and let 𝑓∶𝐴→𝐴 be a mapping satisfying (7) and 𝑓𝜇𝑥₂−𝑥₁3𝑥+𝑓¹−3𝜇𝑥₃3+𝜇𝑓3𝑥₁+3𝑥₃−𝑥₂3−𝜇𝑓𝑥₁≤𝜃𝑥₁^𝑝+𝑥₂^𝑝+𝑥₃^𝑝 (19) for all 𝜇∈𝕋¹_1/𝑛0 and all 𝑥₁,𝑥₂,𝑥₃∈𝐴. Then, there exists a unique 𝐶^∗-ternary derivation 𝐷∶𝐴→𝐴 such that 𝐷𝑥₁−𝑓𝑥₁≤3^𝑝1+2^𝑝𝜃𝑥₁^𝑝3^𝑝−3 (20) for all 𝑥₁∈𝐴.

Proof. By the same reasoning as in the proof of Theorem 4, there exists a unique ℂ-linear mapping 𝐷∶𝐴→𝐴 satisfying (20) which is defined by 𝐷𝑥₁∶=lim_𝑛→∞3^𝑛𝑓𝑥₁3^𝑛 (21) for all 𝑥₁∈𝐴. The inequality (7) implies that 𝐷𝑥₁,𝑥₂,𝑥₃−𝐷𝑥₁,𝑥₂,𝑥₃−𝑥₁,𝐷𝑥₂,𝑥₃−𝑥₁,𝑥₂,𝐷𝑥₃=lim_𝑛→∞27^𝑛𝑓𝑥₁,𝑥₂,𝑥₃27^𝑛−𝑓𝑥₁3^𝑛,𝑥₂3^𝑛,𝑥₃3^𝑛−𝑥₁3^𝑛𝑥,𝑓²3^𝑛,𝑥₃3^𝑛−𝑥₁3^𝑛,𝑥₂3^𝑛𝑥,𝑓³3^𝑛≤lim_𝑛→∞27^𝑛𝜃27^𝑛𝑝𝑥₁^3𝑝+𝑥₂^3𝑝+𝑥₃^3𝑝=0 (22) for all 𝑥₁,𝑥₂,𝑥₃∈𝐴. So 𝐷𝑥₁,𝑥₂,𝑥₃=𝐷𝑥₁,𝑥₂,𝑥₃+𝑥₁,𝐷𝑥₂,𝑥₃+𝑥₁,𝑥₂,𝐷𝑥₃ (23) for all 𝑥₁,𝑥₂,𝑥₃∈𝐴. Consequently, the mapping 𝐷 is a unique 𝐶^∗-ternary derivation satisfying (20).

The following consequence is analogous to Theorem 4 for 𝐶^∗-ternary derivations and its proof is similar to the proof of Theorems 4 and 6.

Theorem 7. Let 𝑝<1 and 𝜃 be nonnegative real numbers, and let 𝑓∶𝐴→𝐴 be a mapping satisfying (7) and (19). Then, there exists a unique 𝐶^∗-ternary derivation 𝐷∶𝐴→𝐴 such that 𝐷𝑥₁−𝑓𝑥₁≤3^𝑝1+2^𝑝𝜃𝑥₁^𝑝3−3^𝑝 (24) for all 𝑥₁∈𝐴.

4. Superstability: A Fixed Point Approach

In this section, we prove the superstability of 𝐶^∗-ternary homomorphisms and of 𝐶^∗-ternary derivations on 𝐶^∗-ternary algebras by using the fixed point method (Theorem 8).

Let 𝑋 be a set. A function 𝑑∶𝑋×𝑋→[0,∞] is called a generalized metric on 𝑋 if 𝑑 satisfies (1)𝑑(𝑥,𝑦)=0 if and only if 𝑥=𝑦; (2)𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥) for all 𝑥,𝑦∈𝑋; (3)𝑑(𝑥,𝑧)≤𝑑(𝑥,𝑦)+𝑑(𝑦,𝑧) for all 𝑥,𝑦,𝑧∈𝑋.

We recall a fundamental result in the fixed point theory from [15] which is a useful tool to achieve our purposes in the sequel.

Theorem 8. Let (𝑋,𝑑) be a complete generalized metric space, and let 𝐽∶𝑋→𝑋 be a strictly contractive mapping with the Lipschitz constant 𝛼<1. Then, for each given element 𝑥∈𝑋, either 𝑑𝐽^𝑛𝑥,𝐽^𝑛+1𝑥=∞ (25) for all nonnegative integers 𝑛 or there exists a positive integer 𝑛₀ such that(i)𝑑(𝐽^𝑛𝑥,𝐽^𝑛+1𝑥)<∞, for all 𝑛≥𝑛₀; (ii)the sequence {𝐽^𝑛𝑥} converges to a fixed point 𝑦^∗ of 𝐽; (iii)𝑦^∗ is the unique fixed point of 𝐽 in the set 𝑌={𝑦∈𝑋∣𝑑(𝐽^𝑛0𝑥,𝑦)<∞}; (iv)𝑑(𝑦,𝑦^∗)≤(1/(1−𝛼))𝑑(𝑦,𝐽𝑦) for all 𝑦∈𝑌.

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, Cădariu 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 𝐶^∗-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 [22–28]).

From now on, we denote 𝐴×𝐴×⋯^𝑛-times×𝐴 by 𝐴^𝑛. We prove the superstability of 𝐶^∗-ternary homomorphism on 𝐶^∗-ternary algebras by employing Theorem 8 as follows.

Theorem 9. Let 𝜑∶𝐴³→[0,∞) be a function such that there exists an 𝛼<1 with 𝜑𝑥₁,𝑥₂,𝑥₃𝑥≤27𝛼𝜑¹3,𝑥₂3,𝑥₂3 (26) for all 𝑥₁,𝑥₂,𝑥₃∈𝐴. Let 𝑓∶𝐴→𝐵 be a mapping satisfying (5) and 𝑓𝑥₁,𝑥₂,𝑥₃−𝑓𝑥₁,𝑓𝑥₂,𝑓𝑥₃≤𝜑𝑥₁,𝑥₂,𝑥₃ (27) for all 𝑥₁,𝑥₂,𝑥₃∈𝐴. Then, the mapping 𝑓∶𝐴→𝐵 is a 𝐶^∗-ternary homomorphism.

Proof. Since the proof is similar to the proof of [13, Theorem 2.2], we only show some parts of it. From the proof of [13, Theorem 2.2], one can show that the mapping 𝑓∶𝐴→𝐵 is ℂ-linear. The inequality (26) implies that lim_𝑛→∞127^𝑛𝜑3^𝑛𝑥₁,3^𝑛𝑥₂,3^𝑛𝑥₃=0 (28) for all 𝑥₁,𝑥₂,𝑥₃∈𝐴. Since 𝑓 is additive, it follows from (27) and (28) that 𝑓𝑥₁,𝑥₂,𝑥₃=𝑓𝑥₁,𝑓𝑥₂,𝑓𝑥₃ (29) for all 𝑥₁,𝑥₂,𝑥₃∈𝐴. Thus, the mapping 𝑓∶𝐴→𝐵 is a 𝐶^∗-ternary homomorphism.

Theorem 10. Let 𝜑∶𝐴³→[0,∞) be a function such that there exists an 𝛼<1 with 𝜑𝑥₁,𝑥₂,𝑥₃≤𝛼3𝜑3𝑥₁,3𝑥₂,3𝑥₃ (30) for all 𝑥₁,𝑥₂,𝑥₃∈𝐴. Let 𝑓∶𝐴→𝐵 be a mapping satisfying (5) and (27). Then, the mapping 𝑓∶𝐴→𝐵 is a 𝐶^∗-ternary homomorphism.

Proof. Similar to the proof of Theorem 9, the mapping 𝑓∶𝐴→𝐵 is ℂ-linear. It also follows from (30) that lim_𝑛→∞3^𝑛𝜑𝑥₁3^𝑛,𝑥₂3^𝑛,𝑥₃3^𝑛=0 (31) for all 𝑥₁,𝑥₂,𝑥₃∈𝐴. Since 𝑓 is additive, we can deduce from (27) and (31) that 𝑓𝑥₁,𝑥₂,𝑥₃=𝑓𝑥₁,𝑓𝑥₂,𝑓𝑥₃ (32) for all 𝑥₁,𝑥₂,𝑥₃∈𝐴. Therefore, the mapping 𝑓 is a 𝐶^∗-ternary homomorphism.

Remark 11. Theorem 2 follows from Theorems 9 and 10 by taking 𝜑(𝑥₁,𝑥₂,𝑥₃)=𝜃(‖𝑥₁‖^3𝑝+‖𝑥₂‖^3𝑝+‖𝑥₃‖^3𝑝) for all 𝑥₁,𝑥₂,𝑥₃∈𝐴.

In analogy with Theorems 9 and 10, we have the following theorems for the superstability of 𝐶^∗-ternary derivations on 𝐶^∗-ternary algebras.

Theorem 12. Let 𝜑∶𝐴³→[0,∞) be a function satisfying (26). Let 𝑓∶𝐴→𝐴 be a mapping satisfying (5) and 𝑓𝑥₁,𝑥₂,𝑥₃−𝑓𝑥₁,𝑥₂,𝑥₃−𝑥₁,𝑓𝑥₂,𝑥₃−𝑥₁,𝑥₂,𝑓𝑥₃≤𝜑𝑥₁,𝑥₂,𝑥₃ (33) for all 𝑥₁,𝑥₂,𝑥₃∈𝐴. Then, the mapping 𝑓∶𝐴→𝐴 is a 𝐶^∗-ternary derivation.

Proof. The proof is similar to the proof of Theorem 9.

Theorem 13. Let 𝜑∶𝐴³→[0,∞) be a function satisfying (30). Let 𝑓∶𝐴→𝐴 be a mapping satisfying (5) and (33). Then, the mapping 𝑓∶𝐴→𝐴 is a 𝐶^∗-ternary derivation.

Proof. Refer to the proof of Theorem 10.

Note that Theorem 3 follows immediately from Theorems 12 and 13 by putting 𝜑(𝑥₁,𝑥₂,𝑥₃)=𝜃(‖𝑥₁‖^3𝑝+‖𝑥₂‖^3𝑝+‖𝑥₃‖^3𝑝) for all 𝑥₁,𝑥₂,𝑥₃∈𝐴.

5. Hyers-Ulam Stability: Fixed Point Method

In this section, we apply Theorem 8 to prove the Hyers-Ulam stability of 𝐶^∗-ternary homomorphisms and 𝐶^∗-ternary derivations on 𝐶^∗-ternary algebras.

Theorem 14. Let 𝜑∶𝐴³→[0,∞) be a function satisfying (30). Let 𝑓∶𝐴→𝐵 be a mapping satisfying (27) and 𝑓𝜇𝑥₂−𝑥₁3𝑥+𝑓¹−3𝜇𝑥₃3+𝜇𝑓3𝑥₁+3𝑥₃−𝑥₂3−𝜇𝑓𝑥₁≤𝜑𝑥₁,𝑥₂,𝑥₃ (34) for all 𝜇∈𝕋¹_1/𝑛0 and all 𝑥₁,𝑥₂,𝑥₃∈𝐴. Then, there exists a unique 𝐶^∗-ternary homomorphism 𝐻∶𝐴→𝐵 such that 𝐻𝑥₁−𝑓𝑥₁≤𝛼𝜑𝑥1−𝛼¹,2𝑥₁,0 (35) for all 𝑥₁∈𝐴.

Proof. Letting 𝜇=1, 𝑥₂=2𝑥₁, and 𝑥₃=0 in (34), we get 𝑥3𝑓¹3−𝑓𝑥₁≤𝑥₁,2𝑥₁,0 (36) for all 𝑥₁∈𝐴. Consider the set 𝑆∶={ℎ∶𝐴⟶𝐵} (37) and introduce the generalized metric on 𝑆 as follows: 𝑑𝑔,ℎ=inf𝜇∈ℝ⁺∶𝑔(𝑥)(𝑥(),−ℎ)≤𝜇𝜑𝑥,2𝑥,0,∀𝑥∈𝐴 (38) where, as usual, inf𝜙=+∞. Similar to the proof of [29, Theorem 2.2], we can show that 𝑑 is a generalized metric on 𝑆 and the metric space (𝑆,𝑑) is complete. We now define the linear mapping 𝐽∶𝑆→𝑆 via 𝐽𝑔(𝑥)∶=(1/3)𝑔(3𝑥) for all 𝑥∈𝐴. Let 𝑔,ℎ∈𝑆 be given such that 𝑑(𝑔,ℎ)=𝜀. Then 𝑔(𝑥)−ℎ(𝑥)≤𝜑(𝑥,2𝑥,0) (39) for all 𝑥∈𝐴. Hence (𝑥)(𝑥=1𝐽𝑔−𝐽ℎ)3𝑔()−13𝑥3ℎ(3𝑥)≤𝛼𝜑(𝑥,2𝑥,0) (40) for all 𝑥∈𝐴. Thus, 𝑑(𝑔,ℎ)=𝜀 implies that 𝑑(𝐽𝑔,𝐽ℎ)≤𝛼𝜀. This means that 𝑑𝐽𝑔,𝐽ℎ≤𝛼𝑑𝑔,ℎ (41) for all 𝑔,ℎ∈𝑆. It follows from (36) that 𝑓𝑥₁−13𝑓3𝑥₁≤13𝜑3𝑥₁,6𝑥₁,0≤𝛼𝜑𝑥₁,2𝑥₁,0 (42) for all 𝑥₁∈𝐴. So 𝑑(𝑓,𝐽𝑓)≤𝛼. By Theorem 8, there exists a mapping 𝐻∶𝐴→𝐵 satisfies the following:(1)𝐻 is a fixed point of 𝐽, that is, 𝐻(3𝑥)=3𝐻(𝑥) (43) for all 𝑥∈𝐴. Indeed, the mapping 𝐻 is a unique fixed point of 𝐽 in the set 𝑀={𝑔∈𝑆∶𝑑(ℎ,𝑔)<∞}. This implies that 𝐻 satisfying (43) such that there exists a 𝜇∈(0,∞) satisfying 𝑓(𝑥)−𝐻(𝑥)≤𝜇𝜑(𝑥,2𝑥,0) (44) for all 𝑥∈𝐴;(2)𝑑(𝐽^𝑛𝑓,𝐻)→0 as 𝑛→∞, and thus, we have the following equality: lim_𝑛→∞13^𝑛𝑓3^𝑛𝑥=𝐻(𝑥)(𝑥∈𝐴); (45)(3)𝑑(𝑓,𝐻)≤(1/(1−𝛼))𝑑(𝑓,𝐽𝑓), which implies the followin inequality: 𝑑≤𝛼𝑓,𝐻.1−𝛼 (46)This shows that the inequality (35) holds. The rest of the proof is similar to the proof of Theorem 4.

Theorem 15. Let 𝜑∶𝐴³→[0,∞) be a function satisfying (26). Let 𝑓∶𝐴→𝐵 be a mapping satisfying (27) and (34). Then, there exists a unique 𝐶^∗-ternary homomorphism 𝐻∶𝐴→𝐵 such that 𝐻𝑥₁−𝑓𝑥₁≤1𝜑𝑥1−𝛼¹,2𝑥₁,0 (47) for all 𝑥₁∈𝐴.

Proof. Let (𝑆,𝑑) be the generalized metric space defined in the proof of Theorem 14. Consider the linear mapping 𝐽∶𝑆→𝑆 such that (𝑥)𝑥𝐽𝑔∶=3𝑔3 (48) for all 𝑥∈𝑋. The inequality (36) implies that 𝑑(𝑓,𝐽𝑓)≤1. So 𝑑(𝑓,𝐻)≤1/(1−𝛼). Thus, we obtain the inequality (47). The rest of the proof is similar to the proofs of Theorems 4 and 14.

The following parallel results for the Hyers-Ulam stability of derivations on 𝐶^∗-ternary algebras can be proved in similar ways to the proofs of Theorems 6 and 14, and so we omit their proofs.

Theorem 16. Let 𝜑∶𝐴³→[0,∞) be a function satisfying (30). Let 𝑓∶𝐴→𝐴 be a mapping satisfying (33) and 𝑓𝜇𝑥₂−𝑥₁3𝑥+𝑓¹−3𝜇𝑥₃3+𝜇𝑓3𝑥₁+3𝑥₃−𝑥₂3−𝜇𝑓𝑥₁≤𝜑𝑥₁,𝑥₂,𝑥₃ (49) for all 𝜇∈𝕋¹_1/𝑛0 and all 𝑥₁,𝑥₂,𝑥₃∈𝐴. Then, there exists a unique 𝐶^∗-ternary derivation 𝐷∶𝐴→𝐴 such that 𝐷𝑥₁−𝑓𝑥₁≤𝛼𝜑𝑥1−𝛼¹,2𝑥₁,0 (50) for all 𝑥₁∈𝐴.

Theorem 17. Let 𝜑∶𝐴³→[0,∞) be a function satisfying (26). Let 𝑓∶𝐴→𝐴 be a mapping satisfying (33) and (49). Then, there exists a unique 𝐶^∗-ternary derivation 𝐷∶𝐴→𝐴 such that 𝐷𝑥₁−𝑓𝑥₁≤1𝜑𝑥1−𝛼¹,2𝑥₁,0 (51) for all 𝑥₁∈𝐴.

Remark 18. All results of Section 3 are the direct consequences of the results of this section as follows: (i)Theorem 4 follows from Theorem 15 by taking 𝜑(𝑥₁,𝑥₂,𝑥₃)=𝜃(‖𝑥₁‖^3𝑝+‖𝑥₂‖^3𝑝+‖𝑥₃‖^3𝑝) for all 𝑥₁,𝑥₂,𝑥₃∈𝐴, and 𝛼=3^1−𝑝; (ii)we can obtain Theorem 5 from Theorem 14 by letting 𝜑(𝑥₁,𝑥₂,𝑥₃)=𝜃(‖𝑥₁‖^3𝑝+‖𝑥₂‖^3𝑝+‖𝑥₃‖^3𝑝) for all 𝑥₁,𝑥₂,𝑥₃∈𝐴, and 𝛼=3^𝑝−1; (iii)if we put 𝜑(𝑥₁,𝑥₂,𝑥₃)=𝜃(‖𝑥₁‖^3𝑝+‖𝑥₂‖^3𝑝+‖𝑥₃‖^3𝑝) for all 𝑥₁,𝑥₂,𝑥₃∈𝐴, and 𝛼=3^1−𝑝 in Theorem 17, then we conclude Theorem 6; (iv)putting 𝜑(𝑥₁,𝑥₂,𝑥₃)=𝜃(‖𝑥₁‖^3𝑝+‖𝑥₂‖^3𝑝+‖𝑥₃‖^3𝑝) for all 𝑥₁,𝑥₂,𝑥₃∈𝐴, and 𝛼=3^𝑝−1 in Theorem 16, we get Theorem 7.

Acknowledgments

The authors would like to thank the anonymous referee for the careful reading of the paper and helpful suggestions. This work was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2012R1A1A2004299).