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

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

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. The paper of Th. M. Rassias [4] has provided a lot of influence in the development of what we call 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]).

• J. M. Rassias [20, 21] following the spirit of the innovative approach of Th. M. Rassias [4] for the unbounded Cauchy difference proved a similar stability theorem in which he replaced the factor by for with (see also [22] for a number of other new results).

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 .

Theorem 1.1 (see [23, 24]).

Let be a complete generalized metric space and let be a strictly contractive mapping with Lipschitz constant . Then for each given element , either

(1.1)

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 a given mapping , we define

(2.1)

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.

Let be a mapping for which there exists a function such that

(2.2)

(2.3)

(2.4)

for all and all . If there exists an such that for all , then there exists a unique -algebra homomorphism such that

(2.5)

for all .

Proof.

Consider the set

(2.6)

and introduce the generalized metric on :

(2.7)

It is easy to show that is complete.

Now we consider the linear mapping such that

(2.8)

for all .

By [23, Theorem 3.1],

(2.9)

for all .

Letting and in (2.2), we get

(2.10)

for all . So

(2.11)

for all . Hence .

By Theorem 1.1, there exists a mapping such that

(1) is a fixed point of , that is,

(2.12)

for all . The mapping is a unique fixed point of in the set

(2.13)

This implies that is a unique mapping satisfying (2.12) such that there exists satisfying

(2.14)

for all .

(2) as . This implies the equality

(2.15)

for all .

(3), which implies the inequality

(2.16)

This implies that the inequality (2.5) holds.

It follows from (2.2) and (2.15) that

(2.17)

for all . So

(2.18)

for all .

Letting in (2.2), we get

(2.19)

for all and all . By a similar method to above, we get

(2.20)

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

It follows from (2.3) that

(2.21)

for all . So

(2.22)

for all .

It follows from (2.4) that

(2.23)

for all . So

(2.24)

for all .

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

Corollary 2.2.

Let and be nonnegative real numbers, and let be a mapping such that

(2.25)

(2.26)

(2.27)

for all and all . Then there exists a unique -algebra homomorphism such that

(2.28)

for all .

Proof.

The proof follows from Theorem 2.1 by taking

(2.29)

for all . Then and we get the desired result.

Theorem 2.3.

Let be a mapping for which there exists a function satisfying (2.2), (2.3), and (2.4). If there exists an such that for all , then there exists a unique -algebra homomorphism such that

(2.30)

for all .

Proof.

We consider the linear mapping such that

(2.31)

for all .

It follows from (2.10) that

(2.32)

for all . Hence, .

By Theorem 1.1, there exists a mapping such that

(1) is a fixed point of , that is,

(2.33)

for all . The mapping is a unique fixed point of in the set

(2.34)

This implies that is a unique mapping satisfying (2.33) such that there exists satisfying

(2.35)

for all .

(2) as . This implies the equality

(2.36)

for all .

(3), which implies the inequality

(2.37)

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.

Let and be nonnegative real numbers, and let be a mapping satisfying (2.25), (2.26), and (2.27). Then there exists a unique -algebra homomorphism such that

(2.38)

for all .

Proof.

The proof follows from Theorem 2.3 by taking

(2.39)

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.

Let be a mapping for which there exists a function such that

(3.1)

(3.2)

for all and all . If there exists an such that for all . Then there exists a unique derivation such that

(3.3)

for all .

Proof.

By the same reasoning as the proof of Theorem 2.1, there exists a unique involutive -linear mapping satisfying (3.3). The mapping is given by

(3.4)

for all .

It follows from (3.2) that

(3.5)

for all . So

(3.6)

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

Corollary 3.2.

Let and be nonnegative real numbers, and let be a mapping such that

(3.7)

(3.8)

for all and all . Then there exists a unique derivation such that

(3.9)

for all .

Proof.

The proof follows from Theorem 3.1 by taking

(3.10)

for all . Then and we get the desired result.

Theorem 3.3.

Let be a mapping for which there exists a function satisfying (3.1) and (3.2). If there exists an such that for all , then there exists a unique derivation such that

(3.11)

for all .

Proof.

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

Corollary 3.4.

Let and be nonnegative real numbers, and let be a mapping satisfying (3.7) and (3.8). Then there exists a unique derivation such that

(3.12)

for all .

Proof.

The proof follows from Theorem 3.3 by taking

(3.13)

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 aLie-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.

Let be a mapping for which there exists a function satisfying (2.2) such that

(4.1)

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

Proof.

By the same reasoning as the proof of Theorem 2.1, there exists a unique -linear mapping satisfying (2.5). The mapping is given by

(4.2)

for all .

It follows from (4.1) that

(4.3)

for all . So

(4.4)

for all .

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

Corollary 4.3.

Let and be nonnegative real numbers, and let be a mapping satisfying (2.25) such that

(4.5)

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

Proof.

The proof follows from Theorem 4.2 by taking

(4.6)

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.

The proof follows from Theorem 4.4 by taking

(4.7)

for all . Then and we get the desired result.

Definition 4.6.

A -algebra , endowed with the Jordan product for all , is called aJordan-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 aLie 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.

Let be a mapping for which there exists a function satisfying (3.1) such that

(5.1)

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

Proof.

By the same reasoning as the proof of Theorem 2.1, there exists a unique involutive -linear mapping satisfying (3.3). The mapping is given by

(5.2)

for all .

It follows from (5.1) that

(5.3)

for all . So

(5.4)

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

Corollary 5.3.

Let and be nonnegative real numbers, and let be a mapping satisfying (3.7) such that

(5.5)

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

Proof.

The proof follows from Theorem 5.2 by taking

(5.6)

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.

The proof follows from Theorem 5.4 by taking

(5.7)

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.