Open Access

On Approximate Cubic Homomorphisms

Advances in Difference Equations20092009:618463

https://doi.org/10.1155/2009/618463

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

A definition of stability in the case of homomorphisms between metric groups was suggested by a problem by Ulam [2] in 1940. Let be a group and let be a metric group with the metric . Given , does there exist a such that if a mapping satisfies the inequality for all , then there exists a homomorphism with for all ? In this case, the equation of homomorphism is called stable. On the other hand, we are looking for situations when the homomorphisms are stable, that is, if a mapping is an approximate homomorphism, then there exists an exact homomorphism near it. The concept of stability for a functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. In 1941, Hyers [3] gave a positive answer to the question of Ulam for Banach spaces. Let be a mapping between Banach spaces such that
(1.1)
for all and for some . Then there exists a unique additive mapping satisfying
(1.2)

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 [513].

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 [1522] and references therein.

Jun and Kim [1] introduced the following functional equation:
(1.3)

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.

Let be a ring. Then a mapping is called a cubic homomorphism if is a cubic function satisfying
(1.4)

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:

(1.5)

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.

Let
(2.1)
(2.2)
for all Assume that the series
(2.3)
converges, and that
(2.4)
for all . Then there exists a unique cubic homomorphism such that
(2.5)

for all .

Proof.

Setting in (2.2) yields
(2.6)
and then dividing by in (2.6), we obtain
(2.7)
for all . Now by induction we have
(2.8)
In order to show that the functions are a convergent sequence, we use the Cauchy convergence criterion. Indeed, replace by and divide by in (2.8), where is an arbitrary positive integer. We find that
(2.9)
for all positive integers . Hence by the Cauchy criterion, the limit exists for each . By taking the limit as in (2.8), we see that and (2.5) holds for all . If we replace by and by respectively, in (2.2) and divide by , we see that
(2.10)
Taking the limit as , we find that satisfies (1.3) [1, Theorem  3.1]. On the other hand we have
(2.11)
for all We find that satisfies (1.4). To prove the uniqueness property of , let be a function satisfing and Since are cubic, then we have
(2.12)
for all , hence,
(2.13)

By taking we get

Corollary 2.2.

Let and be nonnegative real numbers, and let . Suppose that
(2.14)
for all . Then there exists a unique cubic homomorphism such that
(2.15)

for all .

Proof.

In Theorem 2.1, let and for all

Corollary 2.3.

Let and be nonnegative real numbers. Suppose that
(2.16)
for all . Then there exists a unique cubic homomorphism such that
(2.17)

for all .

Proof.

The proof follows from Corollary 2.2.

Corollary 2.4.

Let and let be a positive real number. Suppose that
(2.18)
for all Moreover, suppose that
(2.19)
and that
(2.20)

for all Then is a cubic homomorphism.

Proof.

Letting in (2.20), we get that So by , in (2.20) we get for all By using induction we have
(2.21)
for all and On the other hand, by Theorem 2.1, the mapping defined by
(2.22)

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

Corollary 2.5.

Let and . Let
(2.23)
for all Moreover, suppose that
(2.24)
and that
(2.25)

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.

Let and be a positive real number. Let
(2.26)

for all Then is a cubic homomorphism.

Proof.

Let Then by Corollary 2.4, we get the result.

Theorem 2.7.

Let
(2.27)
(2.28)
for all . Assume that the series
(2.29)
converges and that
(2.30)
for all . Then there exists a unique cubic homomorphism such that
(2.31)

for all .

Proof.

Setting in (2.28) yields
(2.32)
Replacing by in (2.32), we get
(2.33)
for all . By (2.33) we use iterative methods and induction on to prove the following relation
(2.34)
In order to show that the functions are a convergent sequence, replace by in (2.34), and then multiply by , where is an arbitrary positive integer. We find that
(2.35)

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.

Let and be a positive real number. Let
(2.36)
for all Moreover, suppose that
(2.37)
(2.38)

for all Then is a cubic homomorphism.

Proof.

Letting in (2.38), we get that So by , in (2.38) we get for all By using induction, we have
(2.39)
for all and On the other hand, by Theorem  2.8, the mapping defined by
(2.40)

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

Example 2.9.

Let
(2.41)
then is a Banach algebra equipped with the usual matrix-like operations and the following norm:
(2.42)
Let
(2.43)
and we define by and
(2.44)
for all Then we have
(2.45)
Thus the limit exists. Also,
(2.46)
Furthermore,
(2.47)

Hence is cubic homomorphism.

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

(2.48)

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

(1)
Department of Mathematics, Semnan University

References

  1. 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
  2. Ulam SM: A Collection of Mathematical Problems. Interscience, New York, NY, USA; 1960.MATHGoogle Scholar
  3. 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
  4. 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
  5. 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
  6. Forti GL: An existence and stability theorem for a class of functional equations. Stochastica 1980, 4: 23-30. 10.1080/17442508008833155MATHMathSciNetView ArticleGoogle Scholar
  7. 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
  8. Hyers DH, Isac G, Rassias ThM: Stability of Functional Equations in Several Variables. Birkhäuser, Boston, Mass, USA; 1998.MATHView ArticleGoogle Scholar
  9. 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
  10. 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
  11. Rassias ThM, Tabor J: Stability of Mappings of Hyers-Ulam Type. Hadronic Press, Palm Harbor, Fla, USA; 1994.MATHGoogle Scholar
  12. 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
  13. Rassias ThM: On the stability of functional equations originated by a problem of Ulam. Mathematica 2002,44(67)(1):39-75.Google Scholar
  14. 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
  15. 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
  16. Baker J, Lawrence J, Zorzitto F:The stability of the equation . Proceedings of the American Mathematical Society 1979,74(2):242-246.MATHMathSciNetGoogle Scholar
  17. Eshaghi Gordji M, Bavand Savadkouhi M: Approximation of generalized homomorphisms in quasi-Banach algebras. to appear in Analele Stiintifice ale Universitatii Ovidius ConstantaGoogle Scholar
  18. Eshaghi Gordji M, Karimi T, Kaboli Gharetapeh S:Approximately -Jordan homomorphisms on Banach algebras. Journal of Inequalities and Applications 2009, 2009:-8.Google Scholar
  19. Hyers DH, Rassias ThM: Approximate homomorphisms. Aequationes Mathematicae 1992, 44: 125-153. 10.1007/BF01830975MATHMathSciNetView ArticleGoogle Scholar
  20. Park C: Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras. Bulletin des Sciences Mathématiques 2008,132(2):87-96.MATHView ArticleGoogle Scholar
  21. 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
  22. 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

© M. Eshaghi Gordji and M. Bavand Savadkouhi 2009

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.