# On Approximate Cubic Homomorphisms

- M. Eshaghi Gordji
^{1}Email author and - M. Bavand Savadkouhi
^{1}

**2009**:618463

**DOI: **10.1155/2009/618463

© M. Eshaghi Gordji and M. Bavand Savadkouhi 2009

**Received: **22 October 2008

**Accepted: **2 July 2009

**Published: **16 August 2009

## Abstract

We investigate the generalized Hyers-Ulam-Rassias stability of the system of functional equations: , , on Banach algebras. Indeed we establish the superstability of this system by suitable control functions.

## 1. Introduction

for all . Moreover, if is continuous in for each fixed , then the mapping is linear. Rassias [4] succeeded in extending the result of Hyers' theorem by weakening the condition for the Cauchy difference controlled by , to be unbounded. This condition has been assumed further till now, through the complete Hyers direct method, in order to prove linearity for generalized Hyers-Ulam stability problem forms. A number of mathematicians were attracted to the pertinent stability results of Rassias [4], and stimulated to investigate the stability problems of functional equations. The stability phenomenon that was introduced and proved by Rassias is called Hyers-Ulam-Rassias stability. Then 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 [5–13].

Bourgin [14] is the first mathematician dealing with stability of (ring) homomorphism . The topic of approximate homomorphisms was studied by a number of mathematicians, see [15–22] and references therein.

and they established the general solution and generalized Hyers-Ulam-Rassias stability problem for this functional equation. It is easy to see that the function is a solution of the functional equation (1.3) Thus, it is natural that (1.3) is called a cubic functional equation and every solution of the cubic functional equation is said to be a cubic function.

for all For instance, let be commutative, then the mapping defined by is a cubic homomorphism. It is easy to see that a cubic homomorphism is a ring homomorphism if and only if it is zero function. In this paper, we study the stability of cubic homomorphisms on Banach algebras. Indeed, we investigate the generalized Hyers-Ulam-Rassias stability of the system of functional equations:

on Banach algebras. To this end, we need two control functions for our stability. One control function for (1.3) and an other control function for (1.4). So this is the main difference between our hypothesis (where two-degree freedom appears in the election for two control functions and in Theorem 2.1 in what follows), and the conditions (with one control function) that appear, for example, in [1, Theorem 3.1].

## 2. Main Results

In the following we suppose that is a normed algebra, is a Banach algebra, and is a mapping from into , and are maps from into . Also, we put for

Theorem 2.1.

for all .

Proof.

By taking we get

Corollary 2.2.

for all .

Proof.

In Theorem 2.1, let and for all

Corollary 2.3.

for all .

Proof.

The proof follows from Corollary 2.2.

Corollary 2.4.

for all Then is a cubic homomorphism.

Proof.

is a cubic homomorphism. Therefore it follows from (2.21) that Hence it is a cubic homomorphism.

Corollary 2.5.

for all Then is a cubic homomorphism.

Proof.

If , then by Corollary 2.4 we get the result. If the following results from Theorem 2.1, by putting and for all

Corollary 2.6.

for all Then is a cubic homomorphism.

Proof.

Let Then by Corollary 2.4, we get the result.

Theorem 2.7.

for all .

Proof.

for all positive integers. Hence by the Cauchy criterion the limit exists for each . By taking the limit as in (2.34), we see that and (2.31) holds for all . The rest of proof is similar to the proof of Theorem 2.1.

Corollary 2.8.

for all Then is a cubic homomorphism.

Proof.

is a cubic homomorphism. Therefore, it follows from (2.39) that Hence is a cubic homomorphism.

Example 2.9.

Hence is cubic homomorphism.

Also from this example, it is clear that the superstability of the system of functional equations

with the control functions in Corollaries 2.4, 2.5 and 2.6 does not hold.

## Declarations

### Acknowledgments

The authors would like to thank the referees for their valuable suggestions. Also, M. B. Savadkouhi would like to thank the Office of Gifted Students at Semnan University for its financial support.

## Authors’ Affiliations

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