- Abbas Najati
^{1}and - Choonkil Park
^{2}Email author

**2009**:273165

https://doi.org/10.1155/2009/273165

© A. Najati and C. Park. 2009

**Received: **17 June 2009

**Accepted: **4 August 2009

**Published: **19 August 2009

## Abstract

Let be Banach modules over a -algebra and let be given. We prove the generalized Hyers-Ulam stability of the following functional equation in Banach modules over a unital -algebra: . We show that if , for some and a mapping satisfies the functional equation mentioned above then the mapping is Cauchy additive. As an application, we investigate homomorphisms in unital -algebras.

## 1. Introduction and Preliminaries

The stability problem of functional equations 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 Th. M. Rassias [4] for linear mappings by considering an unbounded Cauchy difference.

Theorem 1.1 (Th. M. Rassias [4]).

for all . If , then (1.1) holds for and (1.3) for . Also, if for each the mapping is continuous in , then is -linear.

Theorem 1.2 (J. M. Rassias [5–7]).

for all . If, in addition, is a mapping such that the transformation is continuous in for each fixed then is linear.

The paper of Th. M. Rassias [4] has provided a lot of influence in the development of what we call the *generalized Hyers-Ulam stability* of functional equations. In 1994, a generalization of Theorems 1.1 and 1.2 was obtained by G
vruţa [8], who replaced the bounds
and
by a general control function
.

*quadratic functional equation*. In particular, every solution of the quadratic functional equation is said to be a

*quadratic mapping*. The generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [9] for mappings , where is a normed space and is a Banach space. Cholewa [10] noticed that the theorem of Skof is still true if the relevant domain is replaced by an Abelian group. Czerwik [11] proved the generalized Hyers-Ulam stability of the quadratic functional equation. J. M. Rassias [12, 13] introduced and investigated the stability problem of Ulam for the Euler-Lagrange quadratic mappings (1.6) and

Grabiec [14] has generalized these results mentioned above. In addition, J. M. Rassias [15] generalized the Euler-Lagrange quadratic mapping (1.7) and investigated its stability problem. Thus these Euler-Lagrange type equations (mappings) are called as Euler-Lagrange-Rassias functional equations (mappings).

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 [4–8, 12, 13, 15–55]).

whose solution is said to be a *generalized additive mapping of Euler-Lagrange type.*

In this paper, we introduce the following additive functional equation of Euler-Lagrange type which is somewhat different from (1.8):

where
Every solution of the functional equation (1.9) is said to be a *generalized Euler-Lagrange type additive mapping.*

We investigate the generalized Hyers-Ulam stability of the functional equation (1.9) in Banach modules over a -algebra. These results are applied to investigate -algebra homomorphisms in unital -algebras.

Throughout this paper, assume that is a unital -algebra with norm and unit that is a unital -algebra with norm , and that and are left Banach modules over a unital -algebra with norms and respectively. Let be the group of unitary elements in and let For a given mapping and a given we define and by

## 2. Generalized Hyers-Ulam Stability of the Functional Equation (1.9) in Banach Modules Over a -Algebra

Lemma 2.1.

Let and be linear spaces and let be real numbers with and for some Assume that a mapping satisfies the functional equation (1.9) for all Then the mapping is Cauchy additive. Moreover, for all and all

Proof.

for all Letting in (2.5), we get that for all So the mapping is odd. Therefore, it follows from (2.5) that the mapping is additive. Moreover, let and Setting and for all in (1.9) and using the oddness of we get that

Using the same method as in the proof of Lemma 2.1, we have an alternative result of Lemma 2.1 when

Lemma 2.2.

Let and be linear spaces and let be real numbers with for some Assume that a mapping with satisfies the functional equation (1.9) for all Then the mapping is Cauchy additive. Moreover, for all and all

We investigate the generalized Hyers-Ulam stability of a generalized Euler-Lagrange type additive mapping in Banach spaces.

*Throughout this paper,*
*will be real numbers such that*
*for fixed*

Theorem 2.3.

for all Moreover, for all and all

Proof.

for all Letting in (2.22) and taking the limit as in (2.22), we obtain the desired inequality (2.9).

It follows from (2.7) and (2.8) that

for all Therefore, the mapping satisfies (1.9) and Hence by Lemma 2.2, is a generalized Euler-Lagrange type additive mapping and for all and all

To prove the uniqueness, let be another generalized Euler-Lagrange type additive mapping with satisfying (2.9). By Lemma 2.2, the mapping is additive. Therefore, it follows from (2.9) and (2.20) that

Theorem 2.4.

for all and all Then there exists a unique -linear generalized Euler-Lagrange type additive mapping satisfying (2.9) for all Moreover, for all and all

Proof.

By Theorem 2.3, there exists a unique generalized Euler-Lagrange type additive mapping satisfying (2.9) and moreover for all and all

By the assumption, for each , we get

By the same reasoning as in the proofs of [41, 43],

for all and all Since for all the unique generalized Euler-Lagrange type additive mapping is an -linear mapping.

Corollary 2.5.

Proof.

Corollary 2.6.

for all Moreover, for all and all

Proof.

Define Applying Theorem 2.4, we obtain the desired result.

Theorem 2.7.

for all Moreover, for all and all

Proof.

for all Letting in (2.44) and taking the limit as in (2.44), we obtain the desired inequality (2.39).

The rest of the proof is similar to the proof of Theorem 2.3.

Theorem 2.8.

for all and all Then there exists a unique -linear generalized Euler-Lagrange type additive mapping satisfying (2.39) for all Moreover, for all and all

Proof.

The proof is similar to the proof of Theorem 2.4.

Corollary 2.9.

Proof.

Define Applying Theorem 2.8, we obtain the desired result.

Corollary 2.10.

for all Moreover, for all and all

Proof.

Define Applying Theorem 2.8, we obtain the desired result.

Remark 2.11.

In Theorems 2.7 and 2.8 and Corollaries 2.9 and 2.10 one can assume that instead of

For the case in Corollaries 2.5 and 2.9, using an idea from the example of Gajda [56], we have the following counterexample.

Example 2.12.

It is clear that is continuous and bounded by 2 on . We prove that

which contradicts with (2.61).

## 3. Homomorphisms in Unital -Algebras

In this section, we investigate -algebra homomorphisms in unital -algebras.

We will use the following lemma in the proof of the next theorem.

Lemma 3.1 (see [43]).

Let be an additive mapping such that for all and all Then the mapping is -linear.

Theorem 3.2.

for all for all all and all Then the mapping is a -algebra homomorphism.

Proof.

for all Therefore, the mapping is a -algebra homomorphism, as desired.

The following theorem is an alternative result of Theorem 3.2.

Theorem 3.3.

for all for all all and all . Then the mapping is a -algebra homomorphism.

Remark 3.4.

In Theorems 3.2 and 3.3, one can assume that instead of

Theorem 3.5.

for all and all . Assume that is invertible. Then the mapping is a -algebra homomorphism.

Proof.

for all for all therefore, the mapping is a -algebra homomorphism.

The following theorem is an alternative result of Theorem 3.5.

Theorem 3.6.

for all and all . Assume that is invertible. Then the mapping is a -algebra homomorphism.

Corollary 3.7.

for all all all and all . Assume that is invertible. Then the mapping is a -algebra homomorphism.

Proof.

The result follows from Theorem 3.6 (resp., Theorem 3.5).

Remark 3.8.

In Theorem 3.6 and Corollary 3.7, one can assume that instead of

Theorem 3.9.

for and all . Assume that is invertible and for each fixed the mapping is continuous in . Then the mapping is a -algebra homomorphism.

Proof.

for all By the same reasoning as in the proof of [4], the generalized Euler-Lagrange type additive mapping is -linear.

By the same method as in the proof of Theorem 2.4, we have

For each element we have where . Thus

for all and all Hence the generalized Euler-Lagrange type additive mapping is -linear. The rest of the proof is the same as in the proof of Theorem 3.5.

The following theorem is an alternative result of Theorem 3.9.

Theorem 3.10.

for and all . Assume that is invertible and for each fixed the mapping is continuous in . Then the mapping is a -algebra homomorphism.

Remark 3.11.

## Declarations

### Acknowledgments

The authors would like to thank the referees for a number of valuable suggestions regarding a previous version of this paper. C. Park author was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2009-0070788).

## Authors’ Affiliations

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