Stability of the Jensen-Type Functional Equation in -Algebras: A Fixed Point Approach

Abstract

Using fixed point methods, we prove the generalized Hyers-Ulam stability of homomorphisms in -algebras and Lie -algebras and also of derivations on -algebras and Lie -algebras for the Jensen-type 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 Rassias [4] for linear mappings by considering an unbounded Cauchy difference. The paper of 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 Rassias theorem was obtained by Găvruţa [5] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias' approach. In 1982, Rassias [6] followed the innovative approach of the Rassias' theorem [4] in which he replaced the factor by for with 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, 7–27]).

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 [28, 29]). Let be a complete generalized metric space and let be a strictly contractive mapping with Lipschitz constant Then for each given element either for all nonnegative integers or there exists a positive integer such that
(1) for all (2)the sequence converges to a fixed point of (3) is the unique fixed point of in the set (4) for all

By the using fixed point method, the stability problems of several functional equations have been extensively investigated by a number of authors (see [17, 30–33]).

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 Jensen-type 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 Jensen-type 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 definefor 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 for all and all If there exists an such that for all then there exists a unique -algebra homomorphism such that for all Proof. It follows from thatfor all
Consider the setand introduce the generalized metric on :It is easy to show that is complete.
Now we consider the linear mapping such thatfor all
By [28, Theorem 3.1],for all
Letting and in (2.2), we getfor all Sofor all Hence
By Theorem 1.1, there exists a mapping such that
(1) is a fixed point of that is,for all The mapping is a unique fixed point of in the setThis implies that is a unique mapping satisfying (2.13) such that there exists satisfyingfor all (2) as This implies the equalityfor all (3) which implies the inequalityThis implies that the inequality (2.5) holds.
It follows from(2.2),(2.6), and (2.16) thatfor all Sofor all Letting and in (2.19), we getfor all So the mapping is Cauchy additive, that is, for all
Letting in (2.2), we getfor all and all By a similar method to above, we getfor all and all Thus one can show that the mapping is -linear.
It follows from (2.3) thatfor all Sofor all
It follows from (2.4) thatfor all Sofor 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 for all and all Then there exists a unique -algebra homomorphism such that for all Proof. The proof follows from Theorem 2.1 by takingfor 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 for all Proof. We consider the linear mapping such thatfor all
It follows from (2.11) thatfor all Hence
By Theorem 1.1, there exists a mapping such that
(1) is a fixed point of that is,for all The mapping is a unique fixed point of in the setThis implies that is a unique mapping satisfying (2.35) such that there exists satisfyingfor all (2) as This implies the equalityfor all (3) which implies the inequalitywhich implies that the inequality (2.32) 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 (2.27), (2.28) and (2.29). Then there exists a unique -algebra homomorphism such that for all Proof. The proof follows from Theorem 2.3 by takingfor all Then and we get the desired result.

Theorem 2.5. Let be an odd mapping for which there exists a function satisfying (2.2), (2.3), (2.4) and (2.6). If there exists an such that for all then there exists a unique -algebra homomorphism such that for all Proof. Consider the setand introduce the generalized metric on :It is easy to show that is complete.
Now we consider the linear mapping such thatfor all
By [28, Theorem 3.1],for all
Letting and relpacing by in (2.2), we getfor all Sofor all Hence
By Theorem 1.1, there exists a mapping such that
(1) is a fixed point of that is,for all The mapping is a unique fixed point of in the setThis implies that is a unique mapping satisfying (2.49) such that there exists satisfyingfor all (2) as This implies the equalityfor all (3) which implies the inequalityThis implies that the inequality(2.42) holds.
The rest of the proof is similar to the proof of Theorem 2.1.Corollary 2.6. Let and be nonnegative real numbers, and let be an odd mapping such that for all and all Then there exists a unique -algebra homomorphism such that for all Proof. The proof follows from Theorem 2.5 by takingfor all Then and we get the desired result.

Theorem 2.7. Let be an odd 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 for all Proof. We consider the linear mapping such thatfor all
It follows from (2.47) thatfor all Hence
By Theorem 1.1, there exists a mapping such that
(1) is a fixed point of that is,for all The mapping is a unique fixed point of in the setThis implies that is a unique mapping satisfying (2.60) such that there exists satisfyingfor all (2) as This implies the equalityfor all (3) which implies the inequalitywhich implies that the inequality (2.57) holds.
The rest of the proof is similar to the proof of Theorem 2.1.Corollary 2.8. Let and be nonnegative real numbers, and let be an odd mapping satisfying (2.54). Then there exists a unique -algebra homomorphism such that for all Proof. The proof follows from Theorem 2.7 by takingfor 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 for all and all If there exists an such that for all Then there exists a unique derivation such that 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 byfor all
It follows from (3.2) thatfor all Sofor all Thus is a derivation satisfying (3.3).Corollary 3.2. Let and be nonnegative real numbers, and let be a mapping such that for all and all Then there exists a unique derivation such that for all Proof. The proof follows from Theorem 3.1 by takingfor 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 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 for all Proof. The proof follows from Theorem 3.3 by takingfor all Then and we get the desired result.Remark 3.5. For inequalities controlled by the product of powers of norms, one can obtain similar results to Theorems 2.5 and 2.7 and Corollaries 2.6 and 2.8.

4. Stability of Homomorphisms in Lie -Algebras

A -algebra endowed with the Lie product on is called a Lie -algebra (see [13–15]).

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 Lie -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 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 byfor all
It follows from (4.1) thatfor all Sofor 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 (2.27) such that for all Then there exists a unique Lie -algebra homomorphism satisfying (2.30).Proof. The proof follows from Theorem 4.2 by takingfor 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 (2.32).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 (2.27) and (4.5). Then there exists a unique Lie -algebra homomorphism satisfying (2.40).Proof. The proof follows from Theorem 4.4 by takingfor all Then and we get the desired result.Remark 4.6. For inequalities controlled by the product of powers of norms, one can obtain similar results to Theorems 2.5 and 2.7 and Corollaries 2.6 and 2.8.

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. Let be a mapping for which there exists a function satisfying (3.1) such that 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 byfor all
It follows from (5.1) thatfor all Sofor 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 for all Then there exists a unique Lie derivation satisfying (3.9).Proof. The proof follows from Theorem 5.2 by takingfor 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 takingfor all Then and we get the desired result.Remark 5.6. For inequalities controlled by the product of powers of norms, one can obtain similar results to Theorems 2.5 and 2.7 and Corollaries 2.6 and 2.8.

Acknowledgment

The first author was supported by the R & E program of KOSEF in 2008.

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Copyright

Copyright © 2009 Choonkil Park and John Michael Rassias. 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.