- Choonkil Park
^{1}Email author

**2009**:809232

https://doi.org/10.1155/2009/809232

© Choonkil Park. 2009

**Received: **8 December 2008

**Accepted: **9 February 2009

**Published: **8 March 2009

## Abstract

## 1. Introduction and Preliminaries

*generalized Hyers-Ulam stability*of functional equations. A generalization of the Th. M. Rassias theorem was obtained by Găvruţa [5] by replacing the unbounded Cauchy difference by a general control function in the spirit of Th. M. Rassias' approach. 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 [6–19]).

We recall a fundamental result in fixed point theory.

Let
be a set. A function
is called a *generalized metric* on
if
satisfies

for all nonnegative integers or there exists a positive integer such that

(2)the sequence converges to a fixed point of ;

(3) is the unique fixed point of in the set ;

This paper is organized as follows. In Sections 2 and 3, using the fixed point method, we prove the generalized Hyers-Ulam stability of homomorphisms in -algebras and of derivations on -algebras for the Cauchy functional equation.

In Sections 4 and 5, using the fixed point method, we prove the generalized Hyers-Ulam stability of homomorphisms in Lie -algebras and of derivations on Lie -algebras for the Cauchy functional equation.

## 2. Stability of Homomorphisms in -Algebras

Throughout this section, assume that is a -algebra with norm and that is a -algebra with norm .

Note that a
-linear mapping
is called a *homomorphism* in
-algebras if
satisfies
and
for all
.

We prove the generalized Hyers-Ulam stability of homomorphisms in -algebras for the functional equation .

Theorem 2.1.

Proof.

It is easy to show that is complete.

By Theorem 1.1, there exists a mapping such that

This implies that the inequality (2.5) holds.

for all and all . Thus one can show that the mapping is -linear.

Thus is a -algebra homomorphism satisfying (2.5), as desired.

Corollary 2.2.

Proof.

for all . Then and we get the desired result.

Theorem 2.3.

Proof.

By Theorem 1.1, there exists a mapping such that

which implies that the inequality (2.30) holds.

The rest of the proof is similar to the proof of Theorem 2.1.

Corollary 2.4.

Proof.

## 3. Stability of Derivations on -Algebras

Throughout this section, assume that is a -algebra with norm .

Note that a
-linear mapping
is called a *derivation* on
if
satisfies
for all
.

We prove the generalized Hyers-Ulam stability of derivations on -algebras for the functional equation .

Theorem 3.1.

Proof.

for all . Thus is a derivation satisfying (3.3).

Corollary 3.2.

Proof.

for all . Then and we get the desired result.

Theorem 3.3.

Proof.

The proof is similar to the proofs of Theorems 2.3 and 3.1.

Corollary 3.4.

Proof.

## 4. Stability of Homomorphisms in Lie -Algebras

A
-algebra
, endowed with the Lie product
on
, is called a *Lie*
-*algebra* (see [9–11]).

Definition 4.1.

Let
and
be Lie
-algebras. A
-linear mapping
is called a*Lie*
-*algebra homomorphism* if
for all
.

Throughout this section, assume that is a Lie -algebra with norm and that is a -algebra with norm .

We prove the generalized Hyers-Ulam stability of homomorphisms in Lie -algebras for the functional equation .

Theorem 4.2.

for all . If there exists an such that for all , then there exists a unique Lie -algebra homomorphism satisfying (2.5).

Proof.

Thus is a Lie -algebra homomorphism satisfying (2.5), as desired.

Corollary 4.3.

for all . Then there exists a unique Lie -algebra homomorphism satisfying (2.28).

Proof.

for all . Then and we get the desired result.

Theorem 4.4.

Let be a mapping for which there exists a function satisfying (2.2) and (4.1). If there exists an such that for all , then there exists a unique Lie -algebra homomorphism satisfying (2.30).

Proof.

The proof is similar to the proofs of Theorems 2.3 and 4.2.

Corollary 4.5.

Let and be nonnegative real numbers, and let be a mapping satisfying (2.25) and (4.5). Then there exists a unique Lie -algebra homomorphism satisfying (2.38).

Proof.

for all . Then and we get the desired result.

Definition 4.6.

A
-algebra
, endowed with the Jordan product
for all
, is called a*Jordan*
-*algebra* (see [25]).

Definition 4.7.

(i)A
-linear mapping
is called a *Jordan*
*-algebra homomorphism* if
for all
.

(ii)A
-linear mapping
is called a *Jordan derivation* if
for all
.

Remark 4.8.

If the Lie products in the statements of the theorems in this section are replaced by the Jordan products , then one obtains Jordan -algebra homomorphisms instead of Lie -algebra homomorphisms.

## 5. Stability of Lie Derivations on -Algebras

Definition 5.1.

Let
be a Lie
-algebra. A
-linear mapping
is called a*Lie derivation* if
for all
.

Throughout this section, assume that is a Lie -algebra with norm .

We prove the generalized Hyers-Ulam stability of derivations on Lie -algebras for the functional equation .

Theorem 5.2.

for all . If there exists an such that for all . Then there exists a unique Lie derivation satisfying (3.3).

Proof.

for all . Thus is a derivation satisfying (3.3).

Corollary 5.3.

for all . Then there exists a unique Lie derivation satisfying (3.9).

Proof.

for all . Then and we get the desired result.

Theorem 5.4.

Let be a mapping for which there exists a function satisfying (3.1) and (5.1). If there exists an such that for all , then there exists a unique Lie derivation satisfying (3.11).

Proof.

The proof is similar to the proofs of Theorems 2.3 and 5.2.

Corollary 5.5.

Let and be nonnegative real numbers, and let be a mapping satisfying (3.7) and (5.5). Then there exists a unique Lie derivation satisfying (3.12).

Proof.

for all . Then and we get the desired result.

Remark 5.6.

If the Lie products in the statements of the theorems in this section are replaced by the Jordan products , then one obtains Jordan derivations instead of Lie derivations.

## Declarations

### Acknowledgment

This work was supported by Korea Research Foundation Grant KRF-2008-313-C00041.

## 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**(4):222–224. 10.1073/pnas.27.4.222MathSciNetView ArticleMATHGoogle 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 - 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 - 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: Modified Trif's functional equations in Banach modules over a
-algebra and approximate algebra homomorphisms
*Journal of Mathematical Analysis and Applications*2003,**278**(1):93–108. 10.1016/S0022-247X(02)00573-5MathSciNetView ArticleMATHGoogle Scholar - Park C-G: On an approximate automorphism on a
-algebra
*Proceedings of the American Mathematical Society*2004,**132**(6):1739–1745. 10.1090/S0002-9939-03-07252-6MathSciNetView 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: 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: Hyers-Ulam-Rassias stability of a generalized Euler-Lagrange type additive mapping and isomorphisms between
-algebras
*Bulletin of the Belgian Mathematical Society. Simon Stevin*2006,**13**(4):619–632.MathSciNetMATHGoogle Scholar - Park C:
**Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras.***Fixed Point Theory and Applications*2007,**2007:**-15.Google Scholar - Park C, Cho YS, Han M-H:
**Functional inequalities associated with Jordan-von Neumann-type additive functional equations.***Journal of Inequalities and Applications*2007,**2007:**-13.Google Scholar - Park C, Cui J:
**Generalized stability of -ternary quadratic mappings.***Abstract and Applied Analysis*2007,**2007:**-6.Google 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 - Rassias ThM:
**Problem 16; 2, Report of the 27th International Symposium on Functional Equations.***Aequationes Mathematicae*1990,**39**(2–3):292–293, 309.Google 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 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.MathSciNetMATHGoogle 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 - Cădariu L, Radu V:
**Fixed points and the stability of Jensen's functional equation.***Journal of Inequalities in Pure and Applied Mathematics*2003,**4**(1, article 4):1–7.MATHGoogle Scholar - Diaz JB, Margolis B:
**A fixed point theorem of the alternative, for contractions on a generalized complete metric space.***Bulletin of the American Mathematical Society*1968,**74**(2):305–309. 10.1090/S0002-9904-1968-11933-0MathSciNetView ArticleMATHGoogle Scholar - Fleming RJ, Jamison JE:
*Isometries on Banach Spaces: Function Spaces, Chapman & Hall/CRC Monographs and Surveys in Pure & Applied Mathematics*.*Volume 129*. Chapman & Hall/CRC, Boca Raton, Fla, USA; 2003:x+197.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.