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

- Choonkil Park
^{1}Email author

**2009**:809232

**DOI: **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.

## 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.

*generalized metric*on :

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

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