#### Abstract

We establish the hyperstability of -Jordan homomorphisms from a normed algebra to a Banach algebra, and also we show that an -Jordan homomorphism between two commutative Banach algebras is an -ring homomorphism.

#### 1. Introduction

Let be two rings (algebras) and a positive integer greater than 1. An additive mapping is called an -Jordan homomorphism if for all and an additive mapping is called an -ring homomorphism if for all .

In 2009, Gordji et al. [1] showed the following theorems.

Theorem 1. Let be fixed. Suppose that are two commutative algebras. Let be an -Jordan homomorphism. Then is an -ring homomorphism.

Theorem 2. Let be fixed. Suppose that are commutative Banach algebras. Let and be nonnegative real numbers, and let be real numbers such that , or , , and . Assume that satisfies the system of functional inequalities: for all . Then, there exists a unique -ring homomorphism such that for all .

The stability problem of group homomorphisms was formulated by Ulam [2] in 1940. Bourgin [3] and Badora [4] solved the stability problem of ring homomorphisms (see [5]). The term hyperstability was used for the first time in [6]. Some recent results on hyperstability of Cauchy or linear equation can be founded in [5, 7, 8].

In this paper, we improve Theorems 1 and 2 into Theorems 4 and 8, respectively. In particular, we prove the hyperstability of -Jordan homomorphisms between two commutative Banach algebras.

#### 2. Generalization of Theorem 1

Lemma 3. Let be fixed natural numbers with . Let be two commutative algebras, and let be an additive mapping. Assume that satisfies the following equality: for all . Then one gets for all .

Proof. Replacing by in (3), we obtain for all . In particular, the equality (3) implies that for all . Recall that the equality, holds for all . Replacing by in (3), we obtain for all . From (5), (6), and the above equality, we get the desired equality: for all .

The following theorem is the generalization of Theorem 1.

Theorem 4. Let be two commutative algebras, and let be an -Jordan homomorphism. Then is an -ring homomorphism.

Proof. Since is an -Jordan homomorphism, together with the additivity of , we get for all . It is clear that and , so we obtain for all . If , then by (11) we have . Now let . Together with Lemma 3 and (11), we can say that the equality (4) holds for ; that is, holds for all . Notice that implies , , , , and so Therefore we get the desired equality: for all .

#### 3. Generalization of Theorem 2

We need the following lemmas to prove the generalization of Theorem 2.

Lemma 5 (see [9, Corollaries 2.5 and 3.5]). Let be a normed space, and let be a Banach space. Assume that are mappings such that for all , where and . Then there exists a unique additive mapping such that for all . In particular, is given by for all , where .

Lemma 6. Let be as in Lemma 5. If and , then is an additive mapping.

Proof. Let be the additive mapping satisfying (17). Then we have for all and . Taking the limit as , we get as desired.

The following result has already been proved in [7] (see also [8]). We show that it can also be derived from Lemma 6.

Lemma 7. Let be as in Lemma 5 and . If is a mapping such that then is an additive mapping.

Proof. By Lemma 5, we can take an additive mapping satisfying (17). Observe that for all and for all . Taking the limit as , we get . By Lemma 6, is an additive mapping.

Now we can prove the following theorem which is the generalization of Theorem 2.

Theorem 8. Let be a commutative normed algebra and a commutative Banach algebra. Assume that satisfy (16) and for all , where and . If , then there exists a unique -ring homomorphism satisfying (17).

Proof. By Lemma 5, there exists a unique additive mapping   satisfying (17). By Theorem 4, it suffices to show that . Put . From the equality below (17) in Lemma 5, we have for all . It follows from (22) that for all . Hence is an -Jordan homomorphism. By Theorem 4, is an -ring homomorphism.

The following two corollaries give results on the hyperstability of -ring homomorphisms between Banach algebras.

Corollary 9. Let be as in Theorem 8. If and , then is an -ring homomorphism.

Proof. Let be the unique -ring homomorphism satisfying (17) in Theorem 8. By Lemma 6, is the unique additive mapping satisfying (17). So is the unique -ring homomorphism.

Corollary 10. Let be as in Corollary 9. Assume that satisfies the system of functional inequalities (20) and (22) for all . Then is an -ring homomorphism.

Proof. The proof is analogous as for Corollary 9, with Lemma 6 replaced by Lemma 7.

#### Acknowledgments

The author thanks the reviewers and the editor for their very helpful comments that improve the paper. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (no. 2012R1A1A4A01002971).