Homomorphisms and Derivations in 𝐶^∗-Ternary Algebras

Abstract

In 2006, C. Park proved the stability of homomorphisms in 𝐶^∗-ternary algebras and of derivations on 𝐶^∗-ternary algebras for the following generalized Cauchy-Jensen additive mapping: 2𝑓((∑^𝑝_𝑗=1𝑥_𝑗/2)+∑^𝑑_𝑗=1𝑦_𝑗)=∑^𝑝_𝑗=1𝑓(𝑥_𝑗)+2∑^𝑑_𝑗=1𝑓(𝑦_𝑗). In this note, we improve and generalize some results concerning this functional equation.

1. Introduction and Preliminaries

The stability problem of functional equations is originated from a question of Ulam [1] concerning the stability of group homomorphisms. Hyers [2] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ theorem was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference.

Theorem 1.1 (Th. M. Rassias). Let 𝑓∶𝐸→𝐸^′ be a mapping from a normed vector space 𝐸 into a Banach space 𝐸^′ subject to the inequality ‖𝑓(𝑥+𝑦)−𝑓(𝑥)−𝑓(𝑦)‖≤𝜀(‖𝑥‖^𝑝+‖𝑦‖^𝑝) (1.1) for all 𝑥,𝑦∈𝐸, where 𝜀 and 𝑝 are constants with 𝜀>0 and 𝑝<1. Then the limit 𝐿(𝑥)=lim_𝑛→∞𝑓(2^𝑛𝑥)2^𝑛 (1.2) exists for all 𝑥∈𝐸, and 𝐿∶𝐸→𝐸^′ is the unique additive mapping which satisfies ‖𝑓(𝑥)−𝐿(𝑥)‖≤2𝜀2−2^𝑝‖𝑥‖^𝑝 (1.3) for all 𝑥∈𝐸. If 𝑝<0, then inequality (1.1) holds for 𝑥,𝑦≠0 and (1.3) for 𝑥≠0. Also, if for each 𝑥∈𝐸 the mapping 𝑓(𝑡𝑥) is continuous in 𝑡∈ℝ, then 𝐿 is linear.It was shown by Gajda [5] as well as by Rassias and Šemrl [6] that one cannot prove a Rassias’s type theorem when 𝑝=1. The counter examples of Gajda [5] as well as of Rassias and Šemrl [6] have stimulated several mathematicians to invent new definitions of approximately additive or approximately linear mappings; compare Găvruţa [7] and Jung [8], who among others studied the stability of functional equations. Theorem 1.1 provided a lot of influence in the development of a generalization of the Hyers-Ulam stability concept. This new concept is known as Hyers-Ulam-Rassiasstability of functional equations (cf. the books of Czerwik [9], Hyers et al. [10]).Theorem 1.2 (Rassias [11–13]). Let 𝑋 be a real normed linear space and 𝑌 a real Banach space. Assume that 𝑓∶𝑋→𝑌 is a mapping for which there exist constants 𝜃≥0 and 𝑝,𝑞∈ℝ such that 𝑟=𝑝+𝑞≠1 and 𝑓 satisfies the functional inequality (Cauchy-Găvruţa-Rassias inequality)‖𝑓(𝑥+𝑦)−𝑓(𝑥)−𝑓(𝑦)‖≤𝜃‖𝑥‖^𝑝‖𝑦‖^𝑞 (1.4) for all 𝑥,𝑦∈𝑋. Then there exists a unique additive mapping 𝐿∶𝑋→𝑌 satisfying 𝜃‖𝑓(𝑥)−𝐿(𝑥)‖≤|2^𝑟−2|‖𝑥‖^𝑟 (1.5) for all 𝑥∈𝑋. If, in addition, 𝑓∶𝑋→𝑌 is a mapping such that the transformation 𝑡→𝑓(𝑡𝑥) is continuous in 𝑡∈ℝ for each fixed 𝑥∈𝑋, then 𝐿 is linear.

For the case 𝑟=1, a counter example has been given by Găvruţa [14]. The stability in Theorem 1.2 involving a product of different powers of norms is called Ulam-Găvruţa-Rassias stability (see [15–17]). In 1994, a generalization of Theorems 1.1 and 1.2 was obtained by Găvruţa [7], who replaced the bounds 𝜀(‖𝑥‖^𝑝+‖𝑦‖^𝑝) and 𝜃‖𝑥‖^𝑝‖𝑦‖^𝑞 by a general control function 𝜑(𝑥,𝑦). During past few years several mathematicians have published on various generalizations and applications of generalized Hyers-Ulam stability to a number of functional equations and mappings (see [16–44]).

Following the terminology of [45], a nonempty set 𝐺 with a ternary operation [⋅,⋅,⋅]∶𝐺×𝐺×𝐺→𝐺 is called a ternary groupoid and is denoted by (𝐺,[⋅,⋅,⋅]). The ternary groupoid (𝐺,[⋅,⋅,⋅]) is called commutative if [𝑥₁,𝑥₂,𝑥₃]=[𝑥_𝜎(1),𝑥_𝜎(2),𝑥_𝜎(3)] for all 𝑥₁,𝑥₂,𝑥₃∈𝐺 and all permutations 𝜎 of {1,2,3}.

If a binary operation ∘ is defined on 𝐺 such that [𝑥,𝑦,𝑧]=(𝑥∘𝑦)∘𝑧 for all 𝑥,𝑦,𝑧∈𝐺, then we say that [⋅,⋅,⋅] is derived from ∘. We say that (𝐺,[⋅,⋅,⋅]) is a ternary semigroup if the operation [⋅,⋅,⋅] is associative, that is, if [[𝑥,𝑦,𝑧],𝑢,𝑣]=[𝑥,[𝑦,𝑧,𝑢],𝑣]=[𝑥,𝑦,[𝑧,𝑢,𝑣]] holds for all 𝑥,𝑦,𝑧,𝑢,𝑣∈𝐺 (see [46]).

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 are 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 (see [45, 47]). Every left Hilbert 𝐶^∗-module is a 𝐶^∗-ternary algebra via the ternary product [𝑥,𝑦,𝑧]∶=⟨𝑥,𝑦⟩𝑧.

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 is called a 𝐶^∗-ternary algebra homomorphism if 𝐻([𝑥,𝑦,𝑧])=[𝐻(𝑥),𝐻(𝑦),𝐻(𝑧)] (1.6) for all 𝑥,𝑦,𝑧∈𝐴. If, in addition, the mapping 𝐻 is bijective, then the mapping 𝐻∶𝐴→𝐵 is called a 𝐶^∗-ternary algebra isomorphism. A ℂ-linear mapping 𝛿∶𝐴→𝐴 is called a 𝐶^∗-ternary derivation if 𝛿([𝑥,𝑦,𝑧])=[𝛿(𝑥),𝑦,𝑧]+[𝑥,𝛿(𝑦),𝑧]+[𝑥,𝑦,𝛿(𝑧)] (1.7) for all 𝑥,𝑦,𝑧∈𝐴 (see [23, 45, 48]).

Let (𝐴,∘) be a 𝐶^∗-algebra and [𝑥,𝑦,𝑧]∶=𝑥∘𝑦^∗∘𝑧 for all 𝑥,𝑦,𝑧∈𝐴. The mapping 𝐻∶𝐴→𝐴 defined by 𝐻(𝑥)=−𝑖𝑥 is a 𝐶^∗-ternary algebra isomorphism. Let 𝑎∈𝐴 with 𝑎^∗=𝑎. The mapping 𝛿_𝑎∶𝐴→𝐴 defined by 𝛿_𝑎(𝑥)=𝑖(𝑎𝑥−𝑥𝑎) is a 𝐶^∗-ternary derivation. There are some applications, although still hypothetical, in the fractional quantum Hall effect, the nonstandard statistics, supersymmetric theory, and Yang-Baxter equation (cf. [49–51]).

Throughout this paper, assume that 𝑝, 𝑑 are nonnegative integers with 𝑝+𝑑≥3, and that 𝐴 and 𝐵 are 𝐶^∗-ternary algebras.

2. Stability of Homomorphisms in 𝐶^∗-Ternary Algebras

The stability of homomorphisms in 𝐶^∗-ternary algebras has been investigated in [31] (see also [37]). In this note, we improve some results in [31]. For a given mapping 𝑓∶𝐴→𝐵, we define 𝐶_𝜇𝑓(𝑥₁,…,𝑥_𝑝,𝑦₁,…,𝑦_𝑑∑)∶=2𝑓(^𝑝𝑗=1𝜇𝑥_𝑗2+^𝑑_𝑗=1𝜇𝑦_𝑗)−^𝑝_𝑗=1𝜇𝑓(𝑥_𝑗)−2^𝑑_𝑗=1𝜇𝑓(𝑦_𝑗) (2.1) for all 𝜇∈𝕋¹∶={𝜆∈ℂ∶|𝜆|=1} and all 𝑥₁,…,𝑥_𝑝,𝑦₁,…,𝑦_𝑑∈𝐴.

One can easily show that a mapping 𝑓∶𝐴→𝐵 satisfies 𝐶_𝜇𝑓(𝑥₁,…,𝑥_𝑝,𝑦₁,…,𝑦_𝑑)=0 (2.2) for all 𝜇∈𝕋¹ and all 𝑥₁,…,𝑥_𝑝,𝑦₁,…,𝑦_𝑑∈𝐴 if and only if 𝑓(𝜇𝑥+𝜆𝑦)=𝜇𝑓(𝑥)+𝜆𝑓(𝑦) (2.3) for all 𝜇,𝜆∈𝕋¹ and all 𝑥,𝑦∈𝐴.

We will use the following lemmas in this paper.

Lemma 2.1 (see [30]). Let 𝑓∶𝐴→𝐵 be an additive mapping such that 𝑓(𝜇𝑥)=𝜇𝑓(𝑥) for all 𝑥∈𝐴 and all 𝜇∈𝕋¹. Then the mapping 𝑓 is ℂ-linear.

Lemma 2.2. Let {𝑥_𝑛}_𝑛,{𝑦_𝑛}_𝑛 and {𝑧_𝑛}_𝑛 be convergent sequences in 𝐴. Then the sequence {[𝑥_𝑛,𝑦_𝑛,𝑧_𝑛]}_𝑛 is convergent in 𝐴.

Proof. Let 𝑥,𝑦,𝑧∈𝐴 such that lim_𝑛→∞𝑥_𝑛=𝑥,lim_𝑛→∞𝑦_𝑛=𝑦,lim_𝑛→∞𝑧_𝑛=𝑧. (2.4) Since [𝑥_𝑛,𝑦_𝑛,𝑧_𝑛]−[𝑥,𝑦,𝑧]=[𝑥_𝑛−𝑥,𝑦_𝑛−𝑦,𝑧_𝑛−𝑧]+[𝑥_𝑛−𝑥,𝑦_𝑛,𝑧]+[𝑥,𝑦_𝑛−𝑦,𝑧_𝑛]+[𝑥_𝑛,𝑦,𝑧_𝑛−𝑧] (2.5) for all 𝑛, we get ‖[𝑥_𝑛,𝑦_𝑛,𝑧_𝑛]−[𝑥,𝑦,𝑧]‖≤‖𝑥_𝑛−𝑥‖‖𝑦_𝑛−𝑦‖‖𝑧_𝑛−𝑧‖+‖𝑥_𝑛−𝑥‖‖𝑦_𝑛‖‖𝑧‖+‖𝑥‖‖𝑦_𝑛−𝑦‖‖𝑧_𝑛‖+‖𝑥_𝑛‖‖𝑦‖‖𝑧_𝑛−𝑧‖ (2.6) for all 𝑛. So lim_𝑛→∞[𝑥_𝑛,𝑦_𝑛,𝑧_𝑛]=[𝑥,𝑦,𝑧]. (2.7) This completes the proof.

Theorem 2.3 (see [31]). Let 𝑟 and 𝜃 be nonnegative real numbers such that 𝑟∉[1,3], and let 𝑓∶𝐴→𝐵 be a mapping such that ‖𝐶_𝜇𝑓(𝑥₁,…,𝑥_𝑝,𝑦₁,…,𝑦_𝑑)‖_𝐵≤𝜃(^𝑝_𝑗=1‖𝑥_𝑗‖^𝑟_𝐴+^𝑑_𝑗=1‖𝑦_𝑗‖^𝑟_𝐴),‖𝑓([𝑥,𝑦,𝑧])−[𝑓(𝑥),𝑓(𝑦),𝑓(𝑧)]‖_𝐵≤𝜃(‖𝑥‖^𝑟_𝐴+‖𝑦‖^𝑟_𝐴+‖𝑧‖^𝑟_𝐴) (2.8)(2.9) for all 𝜇∈𝕋¹ and all 𝑥,𝑦,𝑧,𝑥₁,…,𝑥_𝑝,𝑦₁,…,𝑦_𝑑∈𝐴. Then there exists a unique 𝐶^∗-ternary algebra homomorphism 𝐻∶𝐴→𝐵 such that ‖𝑓(𝑥)−𝐻(𝑥)‖_𝐵≤2^𝑟(𝑝+𝑑)𝜃|2(𝑝+2𝑑)^𝑟−(𝑝+2𝑑)2^𝑟|‖𝑥‖^𝑟_𝐴 (2.10) for all 𝑥∈𝐴.

In the following theorem we have an alternative result of Theorem 2.3.

Theorem 2.4. Let 𝑟, 𝑠, and 𝜃 be nonnegative real numbers such that 0<𝑟<1, 0<𝑠<3 (resp., 𝑟>1, 𝑠>3), and let 𝑑≥2. Suppose that 𝑓∶𝐴→𝐵 is a mapping with 𝑓(0)=0, satisfying (2.8) and ‖𝑓([𝑥,𝑦,𝑧])−[𝑓(𝑥),𝑓(𝑦),𝑓(𝑧)]‖_𝐵≤𝜃(‖𝑥‖^𝑠_𝐴+‖𝑦‖^𝑠_𝐴+‖𝑧‖^𝑠_𝐴) (2.11) for all 𝜇∈𝕋¹ and all 𝑥,𝑦,𝑧∈𝐴. Then there exists a unique 𝐶^∗-ternary algebra homomorphism 𝐻∶𝐴→𝐵 such that ‖𝑓(𝑥)−𝐻(𝑥)‖_𝐵≤𝑑𝜃2|𝑑−𝑑^𝑟|‖𝑥‖^𝑟_𝐴 (2.12) for all 𝑥∈𝐴.

Proof. We prove the theorem in two cases.Case 1. 0<𝑟<1 and 0<𝑠<3.
Letting 𝜇=1, 𝑥₁=⋯=𝑥_𝑝=0 and 𝑦₁=⋯=𝑦_𝑑=𝑥 in (2.8), we get ‖𝑓(𝑑𝑥)−𝑑𝑓(𝑥)‖_𝐵≤𝑑𝜃2‖𝑥‖^𝑟_𝐴 (2.13) for all 𝑥∈𝐴. If we replace 𝑥 by 𝑑^𝑛𝑥 in (2.13) and divide both sides of (2.13) to 𝑑^𝑛+1, we get ‖1𝑑^𝑛+1𝑓(𝑑^𝑛+11𝑥)−𝑑^𝑛𝑓(𝑑^𝑛𝑥)‖_𝐵≤𝜃2𝑑^{(𝑟−1)𝑛}‖𝑥‖^𝑟_𝐴 (2.14) for all 𝑥∈𝐴 and all nonnegative integers 𝑛. Therefore, ‖1𝑑^𝑛+1𝑓(𝑑^𝑛+11𝑥)−𝑑^𝑚𝑓(𝑑^𝑚𝑥)‖_𝐵≤𝜃2^𝑛_𝑖=𝑚𝑑^{(𝑟−1)𝑖}‖𝑥‖^𝑟_𝐴 (2.15) for all 𝑥∈𝐴 and all nonnegative integers 𝑛≥𝑚. From this it follows that the sequence {(1/𝑑^𝑛)𝑓(𝑑^𝑛𝑥)} is Cauchy for all 𝑥∈𝐴. Since 𝐵 is complete, the sequence {(1/𝑑^𝑛)𝑓(𝑑^𝑛𝑥)} converges. Thus one can define the mapping 𝐻∶𝐴→𝐵 by 𝐻(𝑥)∶=lim_𝑛→∞1𝑑^𝑛𝑓(𝑑^𝑛𝑥) (2.16) for all 𝑥∈𝐴. Moreover, letting 𝑚=0 and passing the limit 𝑛→∞ in (2.15), we get (2.12). It follows from (2.8) that ∑‖2𝐻(^𝑝𝑗=1𝜇𝑥_𝑗2+^𝑑_𝑗=1𝜇𝑦_𝑗)−^𝑝_𝑗=1𝜇𝐻(𝑥_𝑗)−2^𝑑_𝑗=1𝜇𝐻(𝑦_𝑗)‖_𝐵=lim_𝑛→∞1𝑑^𝑛‖2𝑓(𝑑^𝑛∑^𝑝_𝑗=1𝜇𝑥_𝑗2+𝑑^𝑛𝑑_𝑗=1𝜇𝑦_𝑗)−^𝑝_𝑗=1𝜇𝑓(𝑑^𝑛𝑥_𝑗)−2^𝑑_𝑗=1𝜇𝑓(𝑑^𝑛𝑦_𝑗)‖_𝐵≤lim_𝑛→∞𝑑^𝑛𝑟𝑑^𝑛𝜃(^𝑝_𝑗=1‖𝑥_𝑗‖^𝑟_𝐴+^𝑑_𝑗=1‖𝑦_𝑗‖^𝑟_𝐴)=0 (2.17) for all 𝜇∈𝕋¹ and all 𝑥₁,…,𝑥_𝑝,𝑦₁,…,𝑦_𝑑∈𝐴. Hence ∑2𝐻(^𝑝𝑗=1𝜇𝑥_𝑗2+^𝑑_𝑗=1𝜇𝑦_𝑗)=^𝑝_𝑗=1𝜇𝐻(𝑥_𝑗)+2^𝑑_𝑗=1𝜇𝐻(𝑦_𝑗) (2.18) for all 𝜇∈𝕋¹ and all 𝑥₁,…,𝑥_𝑝,𝑦₁,…𝑦_𝑑∈𝐴. So 𝐻(𝜆𝑥+𝜇𝑦)=𝜆𝐻(𝑥)+𝜇𝐻(𝑦) for all 𝜆,𝜇∈𝕋¹ and all 𝑥,𝑦∈𝐴. Therefore by Lemma 2.1 the mapping 𝐻∶𝐴→𝐵 is ℂ-linear.
It follows from Lemma 2.2 and (2.11) that ‖𝐻([𝑥,𝑦,𝑧])−[𝐻(𝑥),𝐻(𝑦),𝐻(𝑧)]‖_𝐵=lim_𝑛→∞1𝑑^3𝑛‖𝑓([𝑑^𝑛𝑥,𝑑^𝑛𝑦,𝑑^𝑛𝑧])−[𝑓(𝑑^𝑛𝑥),𝑓(𝑑^𝑛𝑦),𝑓(𝑑^𝑛𝑧)]‖_𝐵=𝜃lim_𝑛→∞𝑑^𝑛𝑠𝑑^3𝑛(‖𝑥‖^𝑠_𝐴+‖𝑦‖^𝑠_𝐴+‖𝑧‖^𝑠_𝐴)=0 (2.19) for all 𝑥,𝑦,𝑧∈𝐴. Thus 𝐻([𝑥,𝑦,𝑧])=[𝐻(𝑥),𝐻(𝑦),𝐻(𝑧)] (2.20) for all 𝑥,𝑦,𝑧∈𝐴. Therefore the mapping 𝐻 is a 𝐶^∗-ternary algebra homomorphism.
Now let 𝑇∶𝐴→𝐵 be another 𝐶^∗-ternary algebra homomorphism satisfying (2.12). Then we have ‖𝐻(𝑥)−𝑇(𝑥)‖_𝐵=lim_𝑛→∞1𝑑^𝑛‖𝑓(𝑑^𝑛𝑥)−𝑇(𝑑^𝑛𝑥)‖_𝐵≤𝑑𝜃2|𝑑−𝑑^𝑟|lim_𝑛→∞𝑑^𝑛𝑟𝑑^𝑛‖𝑥‖^𝑟_𝐴=0 (2.21) for all 𝑥∈𝐴. So we can conclude that 𝐻(𝑥)=𝑇(𝑥) for all 𝑥∈𝐴. This proves the uniqueness of 𝐻. Thus the mapping 𝐻∶𝐴→𝐵 is a unique 𝐶^∗-ternary algebra homomorphism satisfying (2.12), as desired.Case 2. 𝑟>1 and 𝑠>3.
Similar to the proof of Case 1, we conclude that the sequence {𝑑^𝑛𝑓(𝑑^−𝑛𝑥)} is a Cauchy sequence in 𝐵. So we can define the mapping 𝐻∶𝐴→𝐵 by 𝐻(𝑥)∶=lim_𝑛→∞𝑑^𝑛𝑓(𝑑^−𝑛𝑥) (2.22) for all 𝑥∈𝐴. The rest of the proof is similar to the proof of Case 1.

Theorem 2.5 (see [31]). Let 𝑟 and 𝜃 be nonnegative real numbers such that 𝑟∉[1/(𝑝+𝑑),1], and let 𝑓∶𝐴→𝐵 be a mapping such that ‖𝐶_𝜇𝑓(𝑥₁,…,𝑥_𝑝,𝑦₁,…,𝑦_𝑑)‖_𝐵≤𝜃^𝑝_𝑗=1‖𝑥_𝑗‖^𝑟_𝐴⋅^𝑑_𝑗=1‖𝑦_𝑗‖^𝑟_𝐴,‖𝑓([𝑥,𝑦,𝑧])−[𝑓(𝑥),𝑓(𝑦),𝑓(𝑧)]‖_𝐵≤𝜃‖𝑥‖^𝑟_𝐴‖𝑦‖^𝑟_𝐴‖𝑧‖^𝑟_𝐴 (2.23)(2.24) for all 𝜇∈𝕋¹ and all 𝑥,𝑦,𝑧,𝑥₁,…,𝑥_𝑝,𝑦₁,…,𝑦_𝑑∈𝐴. Then there exists a unique 𝐶^∗-ternary algebra homomorphism 𝐻∶𝐴→𝐵 such that ‖𝑓(𝑥)−𝐻(𝑥)‖_𝐵≤2^{(𝑝+𝑑)𝑟}𝜃|2(𝑝+2𝑑)^{(𝑝+𝑑)𝑟}−2^{(𝑝+𝑑)𝑟}(𝑝+2𝑑)|‖𝑥‖_𝐴^{(𝑝+𝑑)𝑟} (2.25) for all 𝑥∈𝐴.

The following theorem shows that the mapping 𝑓∶𝐴→𝐵 in Theorem 2.5 is a 𝐶^∗-ternary algebra homomorphism when 𝑟>0.Theorem 2.6. Let 𝑟,𝑠,𝑞,𝑟₁,…,𝑟_𝑝,𝑠₁,…,𝑠_𝑑, and 𝜃 be nonnegative real numbers such that 𝑟+𝑠+𝑞≠3 and 𝑟_𝑘>0(𝑠_𝑘>0) for some 1≤𝑘≤𝑝, 𝑝≥2(1≤𝑘≤𝑑,𝑑≥2).
Let 𝑓∶𝐴→𝐵 be a mapping satisfying ‖𝐶_𝜇𝑓(𝑥₁,…,𝑥_𝑝,𝑦₁,…,𝑦_𝑑)‖_𝐵≤𝜃^𝑝_𝑗=1‖𝑥_𝑗‖^𝑟𝑗_𝐴⋅^𝑑_𝑗=1‖𝑦_𝑗‖^𝑠𝑗_𝐴,‖𝑓([𝑥,𝑦,𝑧])−[𝑓(𝑥),𝑓(𝑦),𝑓(𝑧)]‖_𝐵≤𝜃‖𝑥‖^𝑟_𝐴‖𝑦‖^𝑠_𝐴‖𝑧‖^𝑞_𝐴 (2.26)(2.27) for all 𝜇∈𝕋¹ and all 𝑥,𝑦,𝑧,𝑥₁,…,𝑥_𝑝,𝑦₁,…,𝑦_𝑑∈𝐴. Then the mapping 𝑓∶𝐴→𝐵 is a 𝐶^∗-ternary algebra homomorphism. (We put ‖⋅‖⁰_𝐴=1).

Proof. Let 𝑟_𝑘>0 for some 1≤𝑘≤𝑝 (we have similar proof when 𝑠_𝑘>0 for some 1≤𝑘≤𝑑). We now assume, without loss of generality, that 𝑟₁>0. Letting 𝑥₁=⋯=𝑥_𝑝=𝑦₁=⋯=𝑦_𝑑=0 in (2.26), we get that 𝑓(0)=0. Letting 𝑥₂=2𝑥 and 𝑥₁=𝑥₃=⋯=𝑥_𝑝=𝑦₁=⋯=𝑦_𝑑=0 in (2.26), we get 𝜇𝑓(2𝑥)=2𝑓(𝜇𝑥) (2.28) for all 𝜇∈𝕋¹ and all 𝑥∈𝐴. Setting 𝜇=1 in (2.28), we get that 𝑓(2𝑥)=2𝑓(𝑥) for all 𝑥∈𝐴. Therefore, 𝑓(𝜇𝑥)=𝜇𝑓(𝑥),𝑓(2𝜇𝑥)=2𝜇𝑓(𝑥) (2.29) for all 𝜇∈𝕋¹ and all 𝑥∈𝐴. If we put 𝑥₂=2𝑥 and 𝑦₁=𝑦 and 𝑥₁=𝑥₃=⋯=𝑥_𝑝=𝑦₂=⋯=𝑦_𝑑=0 in (2.26), we get 2𝑓(𝜇𝑥+𝜇𝑦)=𝜇𝑓(2𝑥)+2𝜇𝑓(𝑦) (2.30) for all 𝜇∈𝕋¹ and all 𝑥∈𝐴. It follows from (2.29) and (2.30) that 𝑓(𝜇𝑥+𝜆𝑦)=𝜇𝑓(𝑥)+𝜆𝑓(𝑦) (2.31) for all 𝜆,𝜇∈𝕋¹ and all 𝑥,𝑦∈𝐴. Therefore, by Lemma 2.1 the mapping 𝑓∶𝐴→𝐵 is ℂ-linear. Let 𝑟+𝑠+𝑞>3. Then it follows from (2.27) that ‖𝑓([𝑥,𝑦,𝑧])−[𝑓(𝑥),𝑓(𝑦),𝑓(𝑧)]‖_𝐵=lim_𝑛→∞8^𝑛𝑥‖𝑓([2^𝑛,𝑦2^𝑛,𝑧2^𝑛𝑥])−[𝑓(2^𝑛𝑦),𝑓(2^𝑛𝑧),𝑓(2^𝑛)]‖_𝐵≤𝜃‖𝑥‖^𝑟_𝐴‖𝑦‖^𝑠_𝐴‖𝑧‖^𝑞_𝐴lim_𝑛→∞(82^{𝑟+𝑠+𝑞})^𝑛=0 (2.32) for all 𝑥,𝑦,𝑧∈𝐴. Therefore, 𝑓([𝑥,𝑦,𝑧])=[𝑓(𝑥),𝑓(𝑦),𝑓(𝑧)] (2.33) for all 𝑥,𝑦,𝑧∈𝐴. Similarly, for 𝑟+𝑠+𝑞<3, we get (2.33).

In the rest of this section, assume that 𝐴 is a unital 𝐶^∗-ternary algebra with norm ‖⋅‖_𝐴 and unit 𝑒, and that 𝐵 is a unital 𝐶^∗-ternary algebra with norm ‖⋅‖_𝐵 and unit 𝑒^′.

We investigate homomorphisms in 𝐶^∗-ternary algebras associated with the functional equation 𝐶_𝜇𝑓(𝑥₁,…,𝑥_𝑝,𝑦₁,…,𝑦_𝑑)=0.

Theorem 2.7 (see [31]). Let 𝑟>1(𝑟<1) and 𝜃 be nonnegative real numbers, and let 𝑓∶𝐴→𝐵 be a bijective mapping satisfying (2.8) such that 𝑓([𝑥,𝑦,𝑧])=[𝑓(𝑥),𝑓(𝑦),𝑓(𝑧)] (2.34) for all 𝑥,𝑦,𝑧∈𝐴. If lim_𝑛→∞((𝑝+2𝑑)^𝑛/2^𝑛)𝑓(2^𝑛𝑒/(𝑝+2𝑑)^𝑛)=𝑒^′(lim_𝑛→∞(2^𝑛/(𝑝+2𝑑)^𝑛)𝑓((𝑝+2𝑑)^𝑛/2^𝑛)𝑒=𝑒^′), then the mapping 𝑓∶𝐴→𝐵 is a 𝐶^∗-ternary algebra isomorphism.

In the following theorems we have alternative results of Theorem 2.7.

Theorem 2.8. Let 𝑟<1,𝑠<2 and 𝜃 be nonnegative real numbers, and let 𝑓∶𝐴→𝐵 be a mapping satisfying (2.8) and (2.11). If there exist a real number 𝜆>1(0<𝜆<1) and an element 𝑥₀∈𝐴 such that lim_𝑛→∞(1/𝜆^𝑛)𝑓(𝜆^𝑛𝑥₀)=𝑒^′(lim_𝑛→∞𝜆^𝑛𝑓(𝑥₀/𝜆^𝑛)=𝑒^′), then the mapping 𝑓∶𝐴→𝐵 is a 𝐶^∗-ternary algebra homomorphism.

Proof. By using the proof of Theorem 2.4, there exists a unique 𝐶^∗-ternary algebra homomorphism 𝐻∶𝐴→𝐵 satisfying (2.12). It follows from (2.12) that 𝐻(𝑥)=lim_𝑛→∞1𝜆^𝑛𝑓(𝜆^𝑛𝑥),(𝐻(𝑥)=lim_𝑛→∞𝜆^𝑛𝑥𝑓(𝜆^𝑛)) (2.35) for all 𝑥∈𝐴 and all real numbers 𝜆>1(0<𝜆<1). Therefore, by the assumption we get that 𝐻(𝑥₀)=𝑒^′. Let 𝜆>1 and lim_𝑛→∞(1/𝜆^𝑛)𝑓(𝜆^𝑛𝑥₀)=𝑒^′. It follows from (2.11) that ‖[𝐻(𝑥),𝐻(𝑦),𝐻(𝑧)]−[𝐻(𝑥),𝐻(𝑦),𝑓(𝑧)]‖_𝐵=‖𝐻[𝑥,𝑦,𝑧]−[𝐻(𝑥),𝐻(𝑦),𝑓(𝑧)]‖_𝐵=lim_𝑛→∞1𝜆^2𝑛‖𝑓([𝜆^𝑛𝑥,𝜆^𝑛𝑦,𝑧])−[𝑓(𝜆^𝑛𝑥),𝑓(𝜆^𝑛𝑦),𝑓(𝑧)]‖_𝐵≤𝜃lim_𝑛→∞1𝜆^2𝑛(𝜆^𝑛𝑠‖𝑥‖^𝑠_𝐴+𝜆^𝑛𝑠‖𝑦‖^𝑠_𝐴+‖𝑧‖^𝑠_𝐴)=0 (2.36) for all 𝑥∈𝐴. So [𝐻(𝑥),𝐻(𝑦),𝐻(𝑧)]=[𝐻(𝑥),𝐻(𝑦),𝑓(𝑧)] for all 𝑥,𝑦,𝑧∈𝐴. Letting 𝑥=𝑦=𝑥₀ in the last equality, we get 𝑓(𝑧)=𝐻(𝑧) for all 𝑧∈𝐴. Similarly, one can shows that 𝐻(𝑥)=𝑓(𝑥) for all 𝑥∈𝐴 when 0<𝜆<1 and lim_𝑛→∞𝜆^𝑛𝑓(𝑥₀/𝜆^𝑛)=𝑒^′. Therefore, the mapping 𝑓∶𝐴→𝐵 is a 𝐶^∗-ternary algebra homomorphism.

3. Derivations on 𝐶^∗-Ternary Algebras

Throughout this section, assume that 𝐴 is a 𝐶^∗-ternary algebra with norm ‖⋅‖_𝐴.

Park [31] proved the Hyers-Ulam-Rassias stability and Ulam-Găvruţa-Rassias stability of derivations on 𝐶^∗-ternary algebras for the following functional equation: 𝐶_𝜇𝑓(𝑥₁,…,𝑥_𝑝,𝑦₁,…,𝑦_𝑑)=0. (3.1)

For a given mapping 𝑓∶𝐴→𝐴, let 𝐃𝑓(𝑥,𝑦,𝑧)=𝑓([𝑥,𝑦,𝑧])−[𝑓(𝑥),𝑦,𝑧]−[𝑥,𝑓(𝑦),𝑧]−[𝑥,𝑦,𝑓(𝑧)] (3.2) for all 𝑥,𝑦,𝑧∈𝐴.

Theorem 3.1 (see [31]). Let 𝑟 and 𝜃 be nonnegative real numbers such that 𝑟∉[1,3], and let 𝑓∶𝐴→𝐴 a mapping satisfying (2.8) and ‖𝐃𝑓(𝑥,𝑦,𝑧)‖_𝐴≤𝜃(‖𝑥‖^𝑟_𝐴+‖𝑦‖^𝑟_𝐴+‖𝑧‖^𝑟_𝐴) (3.3) for all 𝑥,𝑦,𝑧∈𝐴. Then there exists a unique 𝐶^∗-ternary derivation 𝛿∶𝐴→𝐴 such that ‖𝑓(𝑥)−𝛿(𝑥)‖_𝐴≤2^𝑟(𝑝+𝑑)|2(𝑝+2𝑑)^𝑟−(𝑝+2𝑑)2^𝑟|𝜃‖𝑥‖^𝑟_𝐴 (3.4) for all 𝑥∈𝐴.

Theorem 3.2 (see [31]). Let 𝑟 and 𝜃 be nonnegative real numbers such that 𝑟∉[1/(𝑝+𝑑),1], and let 𝑓∶𝐴→𝐴 be a mapping satisfying (2.23) and ‖𝐃𝑓(𝑥,𝑦,𝑧)‖_𝐴≤𝜃‖𝑥‖^𝑟_𝐴‖𝑦‖^𝑟_𝐴‖𝑧‖^𝑟_𝐴 (3.5) for all 𝑥,𝑦,𝑧∈𝐴. Then there exists a unique 𝐶^∗-ternary derivation 𝛿∶𝐴→𝐴 such that ‖𝑓(𝑥)−𝛿(𝑥)‖_𝐴≤2^{(𝑝+𝑑)𝑟}|2(𝑝+2𝑑)^{(𝑝+𝑑)𝑟}−(𝑝+2𝑑)2^{(𝑝+𝑑)𝑟}|𝜃‖𝑥‖_𝐴^{(𝑝+𝑑)𝑟} (3.6) for all 𝑥∈𝐴.

In the following theorems we generalize and improve the results in Theorems 3.1 and 3.2.

Theorem 3.3. Let 𝜑∶𝐴^𝑝+𝑑→[0,∞) and 𝜓∶𝐴³→[0,∞) be functions such that 𝜑(𝑥)∶=^∞_𝑛=0𝛾^−𝑛𝜑(𝛾^𝑛𝑥,…,𝛾^𝑛𝑥)<∞,lim_𝑛→∞𝛾^−𝑛𝜑(𝛾^𝑛𝑥₁,…,𝛾^𝑛𝑥_𝑝,𝛾^𝑛𝑦₁,…,𝛾^𝑛𝑦_𝑑)=0,lim_𝑛→∞𝛾^−3𝑛𝜓(𝛾^𝑛𝑥,𝛾^𝑛𝑦,𝛾^𝑛𝑧)=0,lim_𝑛→∞𝛾^−2𝑛𝜓(𝛾^𝑛𝑥,𝛾^𝑛𝑦,𝑧)=0 (3.7)(3.8)(3.9) for all 𝑥,𝑦,𝑧,𝑥₁,…,𝑥_𝑝,𝑦₁,…,𝑦_𝑑∈𝐴 where 𝛾=(𝑝+2𝑑)/2. Suppose that 𝑓∶𝐴→𝐴 is a mapping satisfying ‖𝐶_𝜇𝑓(𝑥₁,…,𝑥_𝑝,𝑦₁,…,𝑦_𝑑)‖_𝐴≤𝜑(𝑥₁,…,𝑥_𝑝,𝑦₁,…,𝑦_𝑑),‖𝐃𝑓(𝑥,𝑦,𝑧)‖_𝐴≤𝜓(𝑥,𝑦,𝑧) (3.10)(3.11) for all 𝜇∈𝕋¹ and all 𝑥,𝑦,𝑧,𝑥₁,…,𝑥_𝑝,𝑦₁,…,𝑦_𝑑∈𝐴. Then the mapping 𝑓∶𝐴→𝐴 is a 𝐶^∗-ternary derivation.

Proof. Let us assume 𝜇=1 and 𝑥₁=⋯=𝑥_𝑝=𝑦₁=⋯=𝑦_𝑑=𝑥 in (3.10). Then we get ‖2𝑓(𝑝+2𝑑2𝑥)−(𝑝+2𝑑)𝑓(𝑥)‖_𝐴≤𝜑(𝑥,…,𝑥) (3.12) for all 𝑥∈𝐴. If we replace 𝑥 in (3.12) by 𝛾^𝑛𝑥 and divide both sides of (3.12) to 𝛾^𝑛+1, then we get ‖1𝛾^𝑛+1𝑓(𝛾^𝑛+11𝑥)−𝛾^𝑛𝑓(𝛾^𝑛𝑥)‖_𝐴≤12𝛾^𝑛+1𝜑(𝛾^𝑛𝑥,…,𝛾^𝑛𝑥) (3.13) for all 𝑥∈𝐴 and all integers 𝑛≥0. Hence ‖1𝛾^𝑛+1𝑓(𝛾^𝑛+11𝑥)−𝛾^𝑚𝑓(𝛾^𝑚𝑥)‖_𝐴≤12𝛾^𝑛_𝑖=𝑚1𝛾^𝑖𝜑(𝛾^𝑖𝑥,…,𝛾^𝑖𝑥) (3.14) for all 𝑥∈𝐴 and all integers 𝑛≥𝑚≥0. From this it follows that the sequence {(1/𝛾^𝑛)𝑓(𝛾^𝑛𝑥)} is Cauchy for all 𝑥∈𝐴. Since 𝐴 is complete, the sequence {(1/𝛾^𝑛)𝑓(𝛾^𝑛𝑥)} converges. Thus we can define the mapping 𝛿∶𝐴→𝐴 by 𝛿(𝑥)∶=lim_𝑛→∞1𝛾^𝑛𝑓(𝛾^𝑛𝑥) (3.15) for all 𝑥∈𝐴. Moreover, letting 𝑚=0 and passing the limit 𝑛→∞ in (3.14), we get ‖𝛿(𝑥)−𝑓(𝑥)‖_𝐴≤12𝛾𝜑(𝑥) (3.16) for all 𝑥∈𝐴. It follows from (3.8) and (3.10) that ‖𝐶_𝜇𝛿(𝑥₁,…,𝑥_𝑝,𝑦₁,…,𝑦_𝑑)‖_𝐴=lim_𝑛→∞1𝛾^𝑛‖𝐶_𝜇𝑓(𝛾^𝑛𝑥₁,…,𝛾^𝑛𝑥_𝑝,𝛾^𝑛𝑦₁,…,𝛾^𝑛𝑦_𝑑)‖_𝐴≤lim_𝑛→∞1𝛾^𝑛𝜑(𝛾^𝑛𝑥₁,…,𝛾^𝑛𝑥_𝑝,𝛾^𝑛𝑦₁,…,𝛾^𝑛𝑦_𝑑)=0 (3.17) for all 𝜇∈𝕋¹ and all 𝑥,𝑦,𝑧,𝑥₁,…,𝑥_𝑝,𝑦₁,…,𝑦_𝑑∈𝐴. Hence ∑2𝛿(^𝑝𝑗=1𝜇𝑥_𝑗2+^𝑑_𝑗=1𝜇𝑦_𝑗)=^𝑝_𝑗=1𝜇𝛿(𝑥_𝑗)+2^𝑑_𝑗=1𝜇𝛿(𝑦_𝑗) (3.18) for all 𝜇∈𝕋¹ and all 𝑥₁,…,𝑥_𝑝,𝑦₁,…,𝑦_𝑑∈𝐴. So 𝛿(𝜆𝑥+𝜇𝑦)=𝜆𝛿(𝑥)+𝜇𝛿(𝑦) for all 𝜆,𝜇∈𝕋¹ and all 𝑥,𝑦∈𝐴. Therefore, by Lemma 2.1 the mapping 𝛿∶𝐴→𝐴 is ℂ-linear.
It follows from (3.9) and (3.11) that ‖𝐃𝛿(𝑥,𝑦,𝑧)‖_𝐴=lim_𝑛→∞1𝛾^3𝑛‖𝐃𝑓(𝛾^𝑛𝑥,𝛾^𝑛𝑦,𝛾^𝑛𝑧)‖_𝐴≤lim_𝑛→∞1𝛾^3𝑛𝜓(𝛾^𝑛𝑥,𝛾^𝑛𝑦,𝛾^𝑛𝑧)=0 (3.19) for all 𝑥,𝑦,𝑧∈𝐴. Hence 𝛿([𝑥,𝑦,𝑧])=[𝛿(𝑥),𝑦,𝑧]+[𝑥,𝛿(𝑦),𝑧]+[𝑥,𝑦,𝛿(𝑧)] (3.20) for all 𝑥,𝑦,𝑧∈𝐴. So the mapping 𝛿∶𝐴→𝐴 is a 𝐶^∗-ternary derivation.
It follows from (3.9) and (3.11) ‖𝛿[𝑥,𝑦,𝑧]−[𝛿(𝑥),𝑦,𝑧]−[𝑥,𝛿(𝑦),𝑧]−[𝑥,𝑦,𝑓(𝑧)]‖_𝐴=lim_𝑛→∞1𝛾^2𝑛‖𝑓[𝛾^𝑛𝑥,𝛾^𝑛𝑦,𝑧]−[𝑓(𝛾^𝑛𝑥),𝛾^𝑛𝑦,𝑧]−[𝛾^𝑛𝑥,𝑓(𝛾^𝑛𝑦),𝑧]−[𝛾^𝑛𝑥,𝛾^𝑛𝑦,𝑓(𝑧)]‖_𝐴≤lim_𝑛→∞1𝛾^2𝑛𝜓(𝛾^𝑛𝑥,𝛾^𝑛𝑦,𝑧)=0 (3.21) for all 𝑥,𝑦,𝑧∈𝐴. Thus 𝛿[𝑥,𝑦,𝑧]=[𝛿(𝑥),𝑦,𝑧]+[𝑥,𝛿(𝑦),𝑧]+[𝑥,𝑦,𝑓(𝑧)] (3.22) for all 𝑥,𝑦,𝑧∈𝐴. Hence we get from (3.20) and (3.22) that [𝑥,𝑦,𝛿(𝑧)]=[𝑥,𝑦,𝑓(𝑧)] (3.23) for all 𝑥,𝑦,𝑧∈𝐴. Letting 𝑥=𝑦=𝑓(𝑧)−𝛿(𝑧) in (3.23), we get ‖𝑓(𝑧)−𝛿(𝑧)‖³_𝐴=‖[𝑓(𝑧)−𝛿(𝑧),𝑓(𝑧)−𝛿(𝑧),𝑓(𝑧)−𝛿(𝑧)]‖_𝐴=0 (3.24) for all 𝑧∈𝐴. Hence 𝑓(𝑧)=𝛿(𝑧) for all 𝑧∈𝐴. So the mapping 𝑓∶𝐴→𝐴 is a 𝐶^∗-ternary derivation, as desired.

Corollary 3.4. Let 𝑟<1,𝑠<2, and 𝜃 be nonnegative real numbers, and let 𝑓∶𝐴→𝐴 be a mapping satisfying (2.8) and ‖𝐃𝑓(𝑥,𝑦,𝑧)‖_𝐴≤𝜃(‖𝑥‖^𝑠_𝐴+‖𝑦‖^𝑠_𝐴+‖𝑧‖^𝑠_𝐴) (3.25) for all 𝑥,𝑦,𝑧∈𝐴. Then the mapping 𝑓∶𝐴→𝐴 is a 𝐶^∗-ternary derivation.

Proof. Define 𝜑(𝑥₁,…,𝑥_𝑝,𝑦₁,…,𝑦_𝑑)=𝜃(^𝑝_𝑗=1‖𝑥_𝑗‖^𝑟_𝐴+^𝑑_𝑗=1‖𝑦_𝑗‖^𝑟_𝐴),𝜓(𝑥,𝑦,𝑧)=𝜃(‖𝑥‖^𝑠_𝐴+‖𝑦‖^𝑠_𝐴+‖𝑧‖^𝑠_𝐴) (3.26) for all 𝑥,𝑦,𝑧,𝑥₁,…,𝑥_𝑝,𝑦₁,…,𝑦_𝑑∈𝐴, and apply Theorem 3.3.

Corollary 3.5. Let 𝑟,𝑠, and 𝜃 be nonnegative real numbers such that 𝑠,𝑟(𝑝+𝑑)<1, and let 𝑓∶𝐴→𝐴 be a mapping satisfying (2.23) and ‖𝐃𝑓(𝑥,𝑦,𝑧)‖_𝐴≤𝜃‖𝑥‖^𝑠_𝐴‖𝑦‖^𝑠_𝐴‖𝑧‖^𝑠_𝐴 (3.27) for all 𝑥,𝑦,𝑧∈𝐴. Then the mapping 𝑓∶𝐴→𝐴 is a 𝐶^∗-ternary derivation.

Proof. Define 𝜑(𝑥₁,…,𝑥_𝑝,𝑦₁,…,𝑦_𝑑)=𝜃^𝑝_𝑗=1‖𝑥_𝑗‖^𝑟_𝐴^𝑑_𝑗=1‖𝑦_𝑗‖^𝑟_𝐴,𝜓(𝑥,𝑦,𝑧)=𝜃‖𝑥‖^𝑠_𝐴‖𝑦‖^𝑠_𝐴‖𝑧‖^𝑠_𝐴 (3.28) for all 𝑥,𝑦,𝑧,𝑥₁,…,𝑥_𝑝,𝑦₁,…,𝑦_𝑑∈𝐴, and apply Theorem 3.3.Theorem 3.6. Let 𝜑∶𝐴^𝑝+𝑑→[0,∞) and 𝜓∶𝐴³→[0,∞) be functions such that 𝜑(𝑥)∶=^∞_𝑛=1𝛾^𝑛𝑥𝜑(𝛾^𝑛𝑥,…,𝛾^𝑛)<∞,lim_𝑛→∞𝛾^𝑛𝑥𝜑(¹𝛾^𝑛𝑥,…,^𝑝𝛾^𝑛,𝑦₁𝛾^𝑛𝑦,…,^𝑑𝛾^𝑛)=0,lim_𝑛→∞𝛾^3𝑛𝑥𝜓(𝛾^𝑛,𝑦𝛾^𝑛,𝑧𝛾^𝑛)=0,lim_𝑛→∞𝛾^2𝑛𝑥𝜓(𝛾^𝑛,𝑦𝛾^𝑛,𝑧)=0 (3.29) for all 𝑥,𝑦,𝑧,𝑥₁,…,𝑥_𝑝,𝑦₁,…,𝑦_𝑑∈𝐴 where 𝛾=(𝑝+2𝑑)/2. Suppose that 𝑓∶𝐴→𝐴 is a mapping satisfying (3.10) and (3.11). Then the mapping 𝑓∶𝐴→𝐴 is a 𝐶^∗-ternary derivation.

Proof. If we replace 𝑥 in (3.12) by 𝑥/𝛾^𝑛+1 and multiply both sides of (3.12) by 𝛾^𝑛, then we get ‖𝛾^𝑛+1𝑥𝑓(𝛾^𝑛+1)−𝛾^𝑛𝑥𝑓(𝛾^𝑛)‖_𝐴≤𝛾^𝑛2𝑥𝜑(𝛾^𝑛+1𝑥,…,𝛾^𝑛+1) (3.30) for all 𝑥∈𝐴 and all integers 𝑛≥0. Hence ‖𝛾^𝑛+1𝑥𝑓(𝛾^𝑛+1)−𝛾^𝑚𝑥𝑓(𝛾^𝑚)‖_𝐴≤12𝛾^𝑛+1_𝑖=𝑚+1𝛾^𝑖𝑥𝜑(𝛾^𝑖𝑥,…,𝛾^𝑖) (3.31) for all 𝑥∈𝐴 and all integers 𝑛≥𝑚≥0. From this it follows that the sequence {𝛾^𝑛𝑓(𝑥/𝛾^𝑛)} is Cauchy for all 𝑥∈𝐴. Since 𝐴 is complete, the sequence {𝛾^𝑛𝑓(𝑥/𝛾^𝑛)} converges. Thus we can define the mapping 𝛿∶𝐴→𝐴 by 𝛿(𝑥)∶=lim_𝑛→∞𝛾^𝑛𝑥𝑓(𝛾^𝑛) (3.32) for all 𝑥∈𝐴. The rest of the proof is similar to the proof of Theorem 3.3, and we omit it.

Corollary. Let 𝑟,𝑠, and 𝜃 be nonnegative real numbers such that 𝑠,𝑟(𝑝+𝑑)>1, and let 𝑓∶𝐴→𝐴 be a mapping satisfying (2.23) and (3.27). Then the mapping 𝑓∶𝐴→𝐴 is a 𝐶^∗-ternary derivation.

Acknowledgment

The authors would like to thank the referees for their useful comments and suggestions. The corresponding author was supported by Daejin University Research Grant in 2009.