- Research Article
- Open Access

# Fixed Points and Stability of the Cauchy Functional Equation in -Algebras

- 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

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

## Keywords

- Linear Mapping
- Functional Equation
- Unique Mapping
- Stability Problem
- Contractive Mapping

## 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

(1) if and only if ;

(2) for all ;

(3) for all .

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

(1) ;

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

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

(4) for all .

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 .

for all and all .

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.

for all .

Proof.

It is easy to show that is complete.

for all .

for all .

for all . Hence .

By Theorem 1.1, there exists a mapping such that

for all .

for all .

This implies that the inequality (2.5) holds.

for all .

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

for all .

for all .

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

Corollary 2.2.

for all .

Proof.

for all . Then and we get the desired result.

Theorem 2.3.

for all .

Proof.

for all .

for all . Hence, .

By Theorem 1.1, there exists a mapping such that

for all .

for all .

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.

for all .

Proof.

for all . Then and we get the desired result.

## 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.

for all .

Proof.

for all .

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

Corollary 3.2.

for all .

Proof.

for all . Then and we get the desired result.

Theorem 3.3.

for all .

Proof.

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

Corollary 3.4.

for all .

Proof.

for all . Then and we get the desired result.

## 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.

for all .

for all .

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.

Let and be Jordan -algebras.

(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 .

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

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