# 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

## 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.

Proof.

Corollary 2.2.

Proof.

In Theorem 2.1, let and for all

Corollary 2.3.

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.

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.

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

## References

- Jun KW, Kim HM:
**The generalized Hyers-Ulam-Rassias stability of a cubic functional equation.***Journal of Mathematical Analysis and Applications*2002,**274**(2):267-278.MathSciNetView ArticleGoogle Scholar - Ulam SM:
*A Collection of Mathematical Problems*. Interscience, New York, NY, USA; 1960.MATHGoogle Scholar - Hyers DH:
**On the stability of the linear functional equation.***Proceedings of the National Academy of Sciences of the United States of America*1941,**27:**222-224. 10.1073/pnas.27.4.222MathSciNetView ArticleGoogle Scholar - Rassias ThM:
**On the stability of the linear mapping in Banach spaces.***Proceedings of the American Mathematical Society*1978,**72:**297-300. 10.1090/S0002-9939-1978-0507327-1MATHMathSciNetView ArticleGoogle Scholar - Faiziev VA, Rassias ThM, Sahoo PK:
**The space of****-additive mappings on semigroups.***Transactions of the American Mathematical Society*2002,**354**(11):4455-4472. 10.1090/S0002-9947-02-03036-2MATHMathSciNetView ArticleGoogle Scholar - Forti GL:
**An existence and stability theorem for a class of functional equations.***Stochastica*1980,**4:**23-30. 10.1080/17442508008833155MATHMathSciNetView ArticleGoogle Scholar - Forti GL:
**Comments on the core of the direct method for proving Hyers-Ulam stability of functional equations.***Journal of Mathematical Analysis and Applications*2004,**295:**127-133. 10.1016/j.jmaa.2004.03.011MATHMathSciNetView ArticleGoogle Scholar - Hyers DH, Isac G, Rassias ThM:
*Stability of Functional Equations in Several Variables*. Birkhäuser, Boston, Mass, USA; 1998.MATHView ArticleGoogle Scholar - Isac G, Rassias ThM:
**On the Hyers-Ulam stability of a cubic functional equation.***Journal of Approximation Theory*1993,**72**(2):131-137. 10.1006/jath.1993.1010MATHMathSciNetView ArticleGoogle Scholar - Maligranda L:
**A result of Tosio Aoki about a generalization of Hyers-Ulam stability of additive functions—a question of priority.***Aequationes Mathematicae*2008,**75:**289-296. 10.1007/s00010-007-2892-8MATHMathSciNetView ArticleGoogle Scholar - Rassias ThM, Tabor J:
*Stability of Mappings of Hyers-Ulam Type*. Hadronic Press, Palm Harbor, Fla, USA; 1994.MATHGoogle Scholar - Rassias ThM:
**On a modified Hyers-Ulam sequence.***Journal of Mathematical Analysis and Applications*1991,**158:**106-113. 10.1016/0022-247X(91)90270-AMATHMathSciNetView ArticleGoogle Scholar - Rassias ThM:
**On the stability of functional equations originated by a problem of Ulam.***Mathematica*2002,**44(67)**(1):39-75.Google Scholar - Bourgin DG:
**Classes of transformations and bordering transformations.***Bulletin of the American Mathematical Society*1951,**57:**223-237. 10.1090/S0002-9904-1951-09511-7MATHMathSciNetView ArticleGoogle Scholar - Badora R:
**On approximate ring homomorphisms.***Journal of Mathematical Analysis and Applications*2002,**276:**589-597. 10.1016/S0022-247X(02)00293-7MATHMathSciNetView ArticleGoogle Scholar - Baker J, Lawrence J, Zorzitto F:
**The stability of the equation**.*Proceedings of the American Mathematical Society*1979,**74**(2):242-246.MATHMathSciNetGoogle Scholar - Eshaghi Gordji M, Bavand Savadkouhi M: Approximation of generalized homomorphisms in quasi-Banach algebras. to appear in Analele Stiintifice ale Universitatii Ovidius ConstantaGoogle Scholar
- Eshaghi Gordji M, Karimi T, Kaboli Gharetapeh S:
**Approximately****-Jordan homomorphisms on Banach algebras.***Journal of Inequalities and Applications*2009,**2009:**-8.Google Scholar - Hyers DH, Rassias ThM:
**Approximate homomorphisms.***Aequationes Mathematicae*1992,**44:**125-153. 10.1007/BF01830975MATHMathSciNetView ArticleGoogle Scholar - Park C:
**Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras.***Bulletin des Sciences Mathématiques*2008,**132**(2):87-96.MATHView ArticleGoogle Scholar - Rassias ThM:
**The problem of S. M. Ulam for approximately multiplicative mappings.***Journal of Mathematical Analysis and Applications*2000,**246**(2):352-378. 10.1006/jmaa.2000.6788MATHMathSciNetView ArticleGoogle Scholar - Rassias ThM:
**On the stability of functional equations and a problem of Ulam.***Acta Applicandae Mathematicae*2000,**62**(1):23-130. 10.1023/A:1006499223572MATHMathSciNetView ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.