- Research Article
- Open Access

- 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.

## Keywords

- Banach Space
- Functional Equation
- Stability Problem
- Additive Mapping
- Alternative Result

## 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

## References

- Ulam SM:
*A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8*. Interscience, New York, NY, USA; 1960:xiii+150.Google 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 - Aoki T:
**On the stability of the linear transformation in Banach spaces.***Journal of the Mathematical Society of Japan*1950,**2:**64–66. 10.2969/jmsj/00210064MathSciNetView ArticleMATHGoogle Scholar - Rassias ThM:
**On the stability of the linear mapping in Banach spaces.***Proceedings of the American Mathematical Society*1978,**72**(2):297–300. 10.1090/S0002-9939-1978-0507327-1MathSciNetView ArticleMATHGoogle Scholar - Rassias JM:
**On approximation of approximately linear mappings by linear mappings.***Journal of Functional Analysis*1982,**46**(1):126–130. 10.1016/0022-1236(82)90048-9MathSciNetView ArticleMATHGoogle Scholar - Rassias JM:
**On approximation of approximately linear mappings by linear mappings.***Bulletin des Sciences Mathématiques*1984,**108**(4):445–446.MATHGoogle Scholar - Rassias JM:
**Solution of a problem of Ulam.***Journal of Approximation Theory*1989,**57**(3):268–273. 10.1016/0021-9045(89)90041-5MathSciNetView ArticleMATHGoogle Scholar - Găvruţa P:
**A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings.***Journal of Mathematical Analysis and Applications*1994,**184**(3):431–436. 10.1006/jmaa.1994.1211MathSciNetView ArticleMATHGoogle Scholar - Skof F:
**Local properties and approximation of operators.***Rendiconti del Seminario Matematico e Fisico di Milano*1983,**53:**113–129. 10.1007/BF02924890MathSciNetView ArticleMATHGoogle Scholar - Cholewa PW:
**Remarks on the stability of functional equations.***Aequationes Mathematicae*1984,**27**(1–2):76–86.MathSciNetView ArticleMATHGoogle Scholar - Czerwik St:
**On the stability of the quadratic mapping in normed spaces.***Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg*1992,**62:**59–64. 10.1007/BF02941618MathSciNetView ArticleMATHGoogle Scholar - Rassias JM:
**On the stability of the Euler-Lagrange functional equation.***Chinese Journal of Mathematics*1992,**20**(2):185–190.MathSciNetMATHGoogle Scholar - Rassias JM:
**On the stability of the non-linear Euler-Lagrange functional equation in real normed linear spaces.***Journal of Mathematical and Physical Sciences*1994,**28**(5):231–235.MathSciNetMATHGoogle Scholar - Grabiec A:
**The generalized Hyers-Ulam stability of a class of functional equations.***Publicationes Mathematicae Debrecen*1996,**48**(3–4):217–235.MathSciNetMATHGoogle Scholar - Rassias JM:
**On the stability of the general Euler-Lagrange functional equation.***Demonstratio Mathematica*1996,**29**(4):755–766.MathSciNetMATHGoogle Scholar - Amyari M, Baak C, Moslehian MS:
**Nearly ternary derivations.***Taiwanese Journal of Mathematics*2007,**11**(5):1417–1424.MathSciNetMATHGoogle Scholar - Chou C-Y, Tzeng J-H:
**On approximate isomorphisms between Banach -algebras or -algebras.***Taiwanese Journal of Mathematics*2006,**10**(1):219–231.MathSciNetMATHGoogle Scholar - Eshaghi Gordji M, Rassias JM, Ghobadipour N:
**Generalized Hyers-Ulam stability of generalized -derivations.***Abstract and Applied Analysis*2009,**2009:**-8.Google 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 - Gao Z-X, Cao H-X, Zheng W-T, Xu L:
**Generalized Hyers-Ulam-Rassias stability of functional inequalities and functional equations.***Journal of Mathematical Inequalities*2009,**3**(1):63–77.MathSciNetView ArticleGoogle Scholar - Găvruţa P:
**On the stability of some functional equations.**In*Stability of Mappings of Hyers-Ulam Type, Hadronic Press Collection of Original Articles*. Hadronic Press, Palm Harbor, Fla, USA; 1994:93–98.Google Scholar - Găvruţa P:
**On a problem of G. Isac and Th. M. Rassias concerning the stability of mappings.***Journal of Mathematical Analysis and Applications*2001,**261**(2):543–553. 10.1006/jmaa.2001.7539MathSciNetView ArticleMATHGoogle Scholar - Găvruţa P:
**On the Hyers-Ulam-Rassias stability of the quadratic mappings.***Nonlinear Functional Analysis and Applications*2004,**9**(3):415–428.MathSciNetMATHGoogle Scholar - Hyers DH, Isac G, Rassias ThM:
*Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and Their Applications, no. 34*. Birkhäuser, Boston, Mass, USA; 1998:vi+313.View ArticleGoogle Scholar - Hyers DH, Isac G, Rassias ThM:
**On the asymptoticity aspect of Hyers-Ulam stability of mappings.***Proceedings of the American Mathematical Society*1998,**126**(2):425–430. 10.1090/S0002-9939-98-04060-XMathSciNetView ArticleMATHGoogle Scholar - Jun K-W, Kim H-M:
**On the Hyers-Ulam stability of a difference equation.***Journal of Computational Analysis and Applications*2005,**7**(4):397–407.MathSciNetMATHGoogle Scholar - Jun K-W, Kim H-M:
**Stability problem of Ulam for generalized forms of Cauchy functional equation.***Journal of Mathematical Analysis and Applications*2005,**312**(2):535–547. 10.1016/j.jmaa.2005.03.052MathSciNetView ArticleMATHGoogle Scholar - Jun K-W, Kim H-M:
**Stability problem for Jensen-type functional equations of cubic mappings.***Acta Mathematica Sinica*2006,**22**(6):1781–1788. 10.1007/s10114-005-0736-9MathSciNetView ArticleMATHGoogle Scholar - Jun K-W, Kim H-M:
**Ulam stability problem for a mixed type of cubic and additive functional equation.***Bulletin of the Belgian Mathematical Society. Simon Stevin*2006,**13**(2):271–285.MathSciNetMATHGoogle Scholar - Jun K-W, Kim H-M, Rassias JM:
**Extended Hyers-Ulam stability for Cauchy-Jensen mappings.***Journal of Difference Equations and Applications*2007,**13**(12):1139–1153. 10.1080/10236190701464590MathSciNetView ArticleMATHGoogle Scholar - Najati A:
**Hyers-Ulam stability of an -Apollonius type quadratic mapping.***Bulletin of the Belgian Mathematical Society. Simon Stevin*2007,**14**(4):755–774.MathSciNetMATHGoogle Scholar - Najati A:
**On the stability of a quartic functional equation.***Journal of Mathematical Analysis and Applications*2008,**340**(1):569–574. 10.1016/j.jmaa.2007.08.048MathSciNetView ArticleMATHGoogle Scholar - Najati A, Moghimi MB:
**Stability of a functional equation deriving from quadratic and additive functions in quasi-Banach spaces.***Journal of Mathematical Analysis and Applications*2008,**337**(1):399–415. 10.1016/j.jmaa.2007.03.104MathSciNetView ArticleMATHGoogle Scholar - Najati A, Park C:
**Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras associated to the Pexiderized Cauchy functional equation.***Journal of Mathematical Analysis and Applications*2007,**335**(2):763–778. 10.1016/j.jmaa.2007.02.009MathSciNetView ArticleMATHGoogle Scholar - Najati A, Park C:
**On the stability of an -dimensional functional equation originating from quadratic forms.***Taiwanese Journal of Mathematics*2008,**12**(7):1609–1624.MathSciNetMATHGoogle Scholar - Najati A, Park C:
**The Pexiderized Apollonius-Jensen type additive mapping and isomorphisms between -algebras.***Journal of Difference Equations and Applications*2008,**14**(5):459–479. 10.1080/10236190701466546MathSciNetView ArticleMATHGoogle Scholar - Park C-G:
**On the stability of the linear mapping in Banach modules.***Journal of Mathematical Analysis and Applications*2002,**275**(2):711–720. 10.1016/S0022-247X(02)00386-4MathSciNetView ArticleMATHGoogle Scholar - Park C-G:
**Linear functional equations in Banach modules over a -algebra.***Acta Applicandae Mathematicae*2003,**77**(2):125–161. 10.1023/A:1024014026789MathSciNetView ArticleMATHGoogle Scholar - Park CG:
**Universal Jensen's equations in Banach modules over a -algebra and its unitary group.***Acta Mathematica Sinica*2004,**20**(6):1047–1056. 10.1007/s10114-004-0409-0MathSciNetView ArticleMATHGoogle Scholar - Park C-G:
**Lie -homomorphisms between Lie -algebras and Lie -derivations on Lie -algebras.***Journal of Mathematical Analysis and Applications*2004,**293**(2):419–434. 10.1016/j.jmaa.2003.10.051MathSciNetView ArticleMATHGoogle Scholar - Park C-G:
**Homomorphisms between Lie -algebras and Cauchy-Rassias stability of Lie -algebra derivations.***Journal of Lie Theory*2005,**15**(2):393–414.MathSciNetMATHGoogle Scholar - Park C-G:
**Generalized Hyers-Ulam-Rassias stability of -sesquilinear-quadratic mappings on Banach modules over -algebras.***Journal of Computational and Applied Mathematics*2005,**180**(2):279–291. 10.1016/j.cam.2004.11.001MathSciNetView ArticleMATHGoogle Scholar - Park C-G:
**Homomorphisms between Poisson -algebras.***Bulletin of the Brazilian Mathematical Society*2005,**36**(1):79–97. 10.1007/s00574-005-0029-zMathSciNetView ArticleMATHGoogle Scholar - Park C-G, Hou J:
**Homomorphisms between -algebras associated with the Trif functional equation and linear derivations on -algebras.***Journal of the Korean Mathematical Society*2004,**41**(3):461–477.MathSciNetView ArticleMATHGoogle Scholar - Park C, Park JM:
**Generalized Hyers-Ulam stability of an Euler-Lagrange type additive mapping.***Journal of Difference Equations and Applications*2006,**12**(12):1277–1288. 10.1080/10236190600986925MathSciNetView ArticleMATHGoogle Scholar - Rassias JM, Rassias MJ:
**Asymptotic behavior of alternative Jensen and Jensen type functional equations.***Bulletin des Sciences Mathématiques*2005,**129**(7):545–558.View ArticleMATHGoogle Scholar - Rassias MJ, Rassias JM:
**Refined Hyers-Ulam superstability of approximately additive mappings.***Journal of Nonlinear Functional Analysis and Differential Equations*2007,**1**(2):175–182.MathSciNetGoogle Scholar - Rassias JM:
**Solution of the Ulam stability problem for Euler-Lagrange quadratic mappings.***Journal of Mathematical Analysis and Applications*1998,**220**(2):613–639. 10.1006/jmaa.1997.5856MathSciNetView ArticleMATHGoogle Scholar - Rassias ThM:
**On the stability of the quadratic functional equation and its applications.***Universitatis Babeş-Bolyai. Studia. Mathematica*1998,**43**(3):89–124.MATHGoogle 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.2000.6788MathSciNetView ArticleMATHGoogle Scholar - Rassias ThM:
**On the stability of functional equations in Banach spaces.***Journal of Mathematical Analysis and Applications*2000,**251**(1):264–284. 10.1006/jmaa.2000.7046MathSciNetView ArticleMATHGoogle 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:1006499223572MathSciNetView ArticleMATHGoogle Scholar - Rassias ThM, Šemrl P:
**On the Hyers-Ulam stability of linear mappings.***Journal of Mathematical Analysis and Applications*1993,**173**(2):325–338. 10.1006/jmaa.1993.1070MathSciNetView ArticleMATHGoogle Scholar - Rassias ThM, Shibata K:
**Variational problem of some quadratic functionals in complex analysis.***Journal of Mathematical Analysis and Applications*1998,**228**(1):234–253. 10.1006/jmaa.1998.6129MathSciNetView ArticleMATHGoogle Scholar - Zhang D, Cao H-X:
**Stability of group and ring homomorphisms.***Mathematical Inequalities & Applications*2006,**9**(3):521–528.MathSciNetView ArticleMATHGoogle Scholar - Gajda Z:
**On stability of additive mappings.***International Journal of Mathematics and Mathematical Sciences*1991,**14**(3):431–434. 10.1155/S016117129100056XMathSciNetView ArticleMATHGoogle Scholar - Kadison RV, Ringrose JR:
*Fundamentals of the Theory of Operator Algebras. Vol. I, Pure and Applied Mathematics*.*Volume 100*. Academic Press, New York, NY, USA; 1983:xv+398.Google 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.