#### Abstract

In 2006, C. Park proved the stability of homomorphisms in -ternary algebras and of derivations on -ternary algebras for the following generalized Cauchy-Jensen additive mapping: . 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 be a mapping from a normed vector space into a Banach space subject to the inequality
**
for all , where and are constants with and . Then the limit
**
exists for all , and is the unique additive mapping which satisfies
**
for all . If , then inequality (1.1) holds for and (1.3) for . Also, if for each the mapping is continuous in , then 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 . 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-Rassias**stability* of functional equations (cf. the books of Czerwik [9], Hyers et al. [10]).Theorem 1.2 (Rassias [11–13]). *Let be a real normed linear space and a real Banach space. Assume that is a mapping for which there exist constants and such that and satisfies the functional inequality Cauchy-Găvruţa-Rassias inequality**
for all . Then there exists a unique additive mapping satisfying
**
for all . If, in addition, is a mapping such that the transformation is continuous in for each fixed then is linear.*

For the case 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 and by a general control function 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 with a ternary operation is called a * ternary groupoid* and is denoted by . The ternary groupoid is called * commutative* if for all and all permutations of .

If a binary operation is defined on such that for all , then we say that is derived from . We say that is a * ternary semigroup* if the operation is * associative*, that is, if holds for all (see [46]).

A -ternary algebra is a complex Banach space , equipped with a ternary product of into , which are -linear in the outer variables, conjugate -linear in the middle variable, and associative in the sense that , and satisfies and (see [45, 47]). Every left Hilbert -module is a -ternary algebra via the ternary product .

If a -ternary algebra has an identity, that is, an element such that for all , then it is routine to verify that , endowed with and , is a unital -algebra. Conversely, if is a unital -algebra, then makes into a -ternary algebra.

A -linear mapping is called a -*ternary algebra homomorphism* if
for all . If, in addition, the mapping is bijective, then the mapping is called a -*ternary algebra isomorphism*. A -linear mapping is called a -*ternary derivation* if
for all (see [23, 45, 48]).

Let be a -algebra and for all . The mapping defined by is a -ternary algebra isomorphism. Let with The mapping defined by is a -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 , are nonnegative integers with , and that and are -ternary algebras.

#### 2. Stability of Homomorphisms in -Ternary Algebras

The stability of homomorphisms in -ternary algebras has been investigated in [31] (see also [37]). In this note, we improve some results in [31]. For a given mapping we define for all and all

One can easily show that a mapping satisfies for all and all if and only if for all and all

We will use the following lemmas in this paper.

Lemma 2.1 (see [30]). *Let be an additive mapping such that for all and all Then the mapping is -linear.*

Lemma 2.2. *Let and be convergent sequences in Then the sequence is convergent in *

*Proof. *Let such that
Since
for all we get
for all So
This completes the proof.

Theorem 2.3 (see [31]). *Let and be nonnegative real numbers such that , and let be a mapping such that
**
for all and all Then there exists a unique -ternary algebra homomorphism such that
**
for all .*

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

Theorem 2.4. *Let , , and be nonnegative real numbers such that , (resp., , ), and let Suppose that is a mapping with satisfying (2.8) and
**
for all and all Then there exists a unique -ternary algebra homomorphism such that
**
for all .*

*Proof. *We prove the theorem in two cases.*Case 1. * and

Letting , and in (2.8), we get
for all If we replace by in (2.13) and divide both sides of (2.13) to we get
for all and all nonnegative integers Therefore,
for all and all nonnegative integers From this it follows that the sequence is Cauchy for all Since is complete, the sequence converges. Thus one can define the mapping by
for all Moreover, letting and passing the limit in (2.15), we get (2.12). It follows from (2.8) that
for all and all Hence
for all and all So for all and all Therefore by Lemma 2.1 the mapping is -linear.

It follows from Lemma 2.2 and (2.11) that
for all . Thus
for all . Therefore the mapping is a -ternary algebra homomorphism.

Now let be another -ternary algebra homomorphism satisfying (2.12). Then we have
for all So we can conclude that for all This proves the uniqueness of Thus the mapping is a unique -ternary algebra homomorphism satisfying (2.12), as desired.*Case 2. * and

Similar to the proof of Case 1, we conclude that the sequence is a Cauchy sequence in So we can define the mapping by
for all The rest of the proof is similar to the proof of Case 1.

Theorem 2.5 (see [31]). *Let and be nonnegative real numbers such that and let be a mapping such that
**
for all and all Then there exists a unique -ternary algebra homomorphism such that
**
for all *

The following theorem shows that the mapping in Theorem 2.5 is a -ternary algebra homomorphism when Theorem 2.6. *Let , and be nonnegative real numbers such that and for some , . **Let be a mapping satisfying
**
for all and all Then the mapping is a -ternary algebra homomorphism. We put *

*Proof. *Let for some (we have similar proof when for some ). We now assume, without loss of generality, that Letting in (2.26), we get that Letting and in (2.26) we get
for all and all Setting in (2.28), we get that for all Therefore,
for all and all If we put and and in (2.26), we get
for all and all It follows from (2.29) and (2.30) that
for all and all Therefore, by Lemma 2.1 the mapping is -linear. Let Then it follows from (2.27) that
for all Therefore,
for all Similarly, for we get (2.33).

In the rest of this section, assume that is a unital -ternary algebra with norm and unit , and that is a unital -ternary algebra with norm and unit

We investigate homomorphisms in -ternary algebras associated with the functional equation .

Theorem 2.7 (see [31]). *Let and be nonnegative real numbers, and let be a bijective mapping satisfying (2.8) such that
**
for all . If then the mapping is a -ternary algebra isomorphism.*

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

Theorem 2.8. *Let and be nonnegative real numbers, and let be a mapping satisfying (2.8) and (2.11). If there exist a real number and an element such that then the mapping is a -ternary algebra homomorphism.*

*Proof. *By using the proof of Theorem 2.4, there exists a unique -ternary algebra homomorphism satisfying (2.12). It follows from (2.12) that
for all and all real numbers Therefore, by the assumption we get that Let and It follows from (2.11) that
for all So for all Letting in the last equality, we get for all Similarly, one can shows that for all when and Therefore, the mapping is a -ternary algebra homomorphism.

#### 3. Derivations on -Ternary Algebras

Throughout this section, assume that is a -ternary algebra with norm .

Park [31] proved the Hyers-Ulam-Rassias stability and Ulam-Găvruţa-Rassias stability of derivations on -ternary algebras for the following functional equation:

For a given mapping let for all

Theorem 3.1 (see [31]). *Let and be nonnegative real numbers such that and let a mapping satisfying (2.8) and
**
for all Then there exists a unique -ternary derivation such that
**
for all *

Theorem 3.2 (see [31]). *Let and be nonnegative real numbers such that and let be a mapping satisfying (2.23) and
**
for all Then there exists a unique -ternary derivation such that
**
for all *

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

Theorem 3.3. *Let and be functions such that
**
for all where Suppose that is a mapping satisfying
**
for all and all Then the mapping is a -ternary derivation.*

*Proof. *Let us assume and in (3.10). Then we get
for all If we replace in (3.12) by and divide both sides of (3.12) to then we get
for all and all integers Hence
for all and all integers From this it follows that the sequence is Cauchy for all Since is complete, the sequence converges. Thus we can define the mapping by
for all Moreover, letting and passing the limit in (3.14), we get
for all It follows from (3.8) and (3.10) that
for all and all Hence
for all and all So for all and all Therefore, by Lemma 2.1 the mapping is -linear.

It follows from (3.9) and (3.11) that
for all Hence
for all So the mapping is a -ternary derivation.

It follows from (3.9) and (3.11)
for all Thus
for all Hence we get from (3.20) and (3.22) that
for all Letting in (3.23), we get
for all Hence for all So the mapping is a -ternary derivation, as desired.

Corollary 3.4. *Let , and be nonnegative real numbers, and let be a mapping satisfying (2.8) and
**
for all Then the mapping is a -ternary derivation.*

*Proof. *Define
for all and apply Theorem 3.3.

Corollary 3.5. *Let , and be nonnegative real numbers such that and let be a mapping satisfying (2.23) and
**
for all Then the mapping is a -ternary derivation.*

*Proof. *Define
for all and apply Theorem 3.3.Theorem 3.6. *Let and be functions such that
**
for all where Suppose that is a mapping satisfying (3.10) and (3.11). Then the mapping is a -ternary derivation.*

*Proof. *If we replace in (3.12) by and multiply both sides of (3.12) by then we get
for all and all integers Hence
for all and all integers From this it follows that the sequence is Cauchy for all Since is complete, the sequence converges. Thus we can define the mapping by
for all The rest of the proof is similar to the proof of Theorem 3.3, and we omit it.

Corollary. *Let , and be nonnegative real numbers such that and let be a mapping satisfying (2.23) and (3.27). Then the mapping is a -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.