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Abstract and Applied Analysis
Volume 2008, Article ID 410437, 12 pages
http://dx.doi.org/10.1155/2008/410437
Research Article

Jordan -Derivations on -Algebras and -Algebras

1Department of Mathematics Education, Pusan National University, Pusan 609-735, South Korea
2Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
3Department of Mathematics, Hanyang University, Seoul 133-791, South Korea

Received 19 September 2008; Revised 20 October 2008; Accepted 31 October 2008

Academic Editor: Ferhan Merdivenci Atici

Copyright © 2008 Jong Su An et al. 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.

Abstract

We investigate Jordan -derivations on -algebras and Jordan -derivations on -algebras associated with the following functional inequality for some integer greater than 1. We moreover prove the generalized Hyers-Ulam stability of Jordan -derivations on -algebras and of Jordan -derivations on -algebras associated with the following functional equation for some integer greater than 1.

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. In 1982–1994, a generalization of Hyers-Ulam stability result was proved by J. M. Rassias [59]. This author assumed that Cauchy-Găvruţa-Rassias inequalityis controlled by a product of different powers of norms, where and such that , and retained the condition of continuity of in for each fixed . In 1999, Găvruţa [10] studied the singular case , by constructing a nice counterexample to the above pertinent Ulam stability problem. Also J. M. Rassias [59, 1113] investigated other conditions and still obtained stability results. In all these cases, the approach to the existence question was to prove asymptotic type formulas of the formTheorem 1.1 (see [57, 9]). Let be a real normed vector space and a real complete normed vector space. Assume in addition that is an approximately additive mapping for which there exist constants and such that and satisfiesfor all Then, there exists a unique additive mapping satisfyingfor all If, in addition, is a mapping such that the transformation is continuous in for each fixed , then is an -linear mapping.Theorem 1.2 (see [4]). Let be a mapping from a normed vector space into a Banach space subject to the inequalityfor all where and are constants with and Then, the limitexists for all and is the unique additive mapping which satisfiesfor all Also, if for each the mapping is continuous in , then is linear.

Th. M. Rassias [14], during the 27th International Symposium on Functional Equations, asked the question whether such a theorem can also be proved for . Gajda [15] following the same approach as in Th. M. Rassias [4] gave an affirmative solution to this question for . It was shown by Gajda [15] as well as by Th. M. Rassias and Šemrl [16] that one cannot prove a Th. M. Rassias’ type theorem when . The counterexamples of Gajda [15] as well as of Th. M. Rassias and Šemrl [16] have stimulated several mathematicians to invent new definitions of approximately additive or approximately linear mappings, compare Găvruţa [17] and Jung [18], who among others studied the Hyers-Ulam stability of functional equations. Theorem 1.2 that was introduced for the first time by Th. M. Rassias [4] provided a lot of influence in the development of a generalization of the Hyers-Ulam stability concept. This new concept is known as generalized Hyers-Ulam stability of functional equations (cf. the books of Czerwik [19], Hyers et al. [20]).

Găvruţa [17] provided a further generalization of Th. M. Rassias’ theorem. Isac and Th. M. Rassias [21] applied the Hyers-Ulam stability theory to prove fixed point theorems and study some new applications in nonlinear analysis. In [22], Hyers et al. studied the asymptoticity aspect of Hyers-Ulam stability of mappings. Beginning around the year 1980, the topic of approximate homomorphisms and their stability theory in the field of functional equations and inequalities was taken up by several mathematicians (see [318, 2151]).

Gilányi [26] showed that if satisfies the functional inequalitythen satisfies the Jordan-von Neumann functional equationSee also [52]. Fechner [53] and Gilányi [27] proved the generalized Hyers-Ulam stability of the functional inequality (1.8). Park et al. [42] introduced and investigated 3-variable Cauchy-Jensen functional inequalities and proved the generalized Hyers-Ulam stability of the 3-variable Cauchy-Jensen functional inequalities.

Definition 1.3. Let be a -algebra. A -linear mapping is called a Jordan -derivation if for all

A -algebra , endowed with the Jordan product on , is called a -algebra (see [38, 39]).

Definition 1.4. Let be a -algebra. A -linear mapping is called a Jordan -derivation if for all

This paper is organized as follows. In Section 2, we investigate Jordan -derivations on -algebras associated with the functional inequalityand prove the generalized Hyers-Ulam stability of Jordan -derivations on -algebras associated with the functional equation

In Section 3, we investigate Jordan -derivations on -algebras associated with the functional inequality (1.12), and prove the generalized Hyers-Ulam stability of Jordan -derivations on -algebras associated with the functional equation (1.13).

Throughout this paper, let be an integer greater than 1.

2. Jordan -derivations on -algebras

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

Lemma 2.1. Let be a mapping such that for all Then, is Cauchy additive, that is, .

Proof. Letting in (2.1), we getSo .
Letting and in (2.1), we getfor all Hence for all
Letting in (2.1), we getfor all Thusfor all Letting in (2.5), we get for all Sofor all as desired.

Theorem 2.2. Let and be a nonnegative real number, and let be a mapping such that for all and all . Then, the mapping is a Jordan -derivation.Proof. Let in (2.7). By Lemma 2.1, the mapping is Cauchy additive. So and for all
Letting and , we getfor all and all Sofor all and all . Hence, for all and all By the same reasoning as in the proof of [39, Theorem 2.1], the mapping is -linear.
It follows from (2.8) thatfor all Thusfor all
Hence the mapping is a Jordan -derivation.

Theorem 2.3. Let and be a nonnegative real number, and let be a mapping satisfying (2.7) and (2.8). Then, the mapping is a Jordan -derivation.Proof. The proof is similar to the proof of Theorem 2.2.

We prove the generalized Hyers-Ulam stability of Jordan -derivations on -algebras.

Theorem 2.4. Suppose that is a mapping with for which there exists a function such thatfor all and all Then, there exists a unique Jordan -derivation such thatfor all Proof. Putting and , and replacing by in (2.15), we get
Using the induction method, we havefor all and all It follows that for every , the sequence is Cauchy, and hence it is convergent since is complete. SetLet and replace and by and , respectively, in (2.15), we getTaking the limit as , we obtainfor all and all Letting and in (2.21), we getfor all Hence,for all in particular, is additive. Now similar to the discussion in [54, Theorem 2.1], we show that is -linear. Letting in (2.18), we getTaking the limit as , we havefor all It follows from [55] that is unique.
Letting , and replacing by in (2.15), we obtainSofor all Letting tend to infinity, we havefor all Hence is a Jordan -derivation.
Corollary 2.5. Suppose that is a mapping with for which there exist constant and such thatfor all and all Then, there exists a unique Jordan -derivation such thatfor all Proof. Letting in Theorem 2.4, we obtain the result.Theorem 2.6. Suppose that is a mapping with for which there exists a function satisfying (2.15) such thatfor all Then, there exists a unique Jordan -derivation such thatfor all Proof. Letting and in (2.15), we getOne can apply the induction method to prove thatfor all and It follows that for every , the sequence is Cauchy, and hence it is convergent since is complete. SetLetting and replacing and by and , respectively, in (2.15), we getfor all Taking the limit as , we obtainfor all Hence is additive.
The rest of the proof is similar to the proof of Theorem 2.4.
Corollary 2.7. Suppose that is a mapping with for which there exist constant and such thatfor all and all Then, there exists a unique Jordan -derivation such thatfor all Proof. Letting in Theorem 2.6, we obtain the result.

3. Jordan -derivations on -algebras

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

Theorem 3.1. Let and be a nonnegative real number, and let be a mapping satisfying (2.7) such thatfor all Then, the mapping is a Jordan -derivation.Proof. By the same reasoning as in the proof of Theorem 2.2, the mapping is -linear.
It follows from (3.1) thatfor all Thus,for all
Hence, the mapping is a Jordan -derivation.

Theorem 3.2. Let and be a nonnegative real number, and let be a mapping satisfying (2.7) and (3.1). Then, the mapping is a Jordan -derivation.Proof. The proof is similar to the proofs of Theorems 2.2 and 3.1.

We prove the generalized Hyers-Ulam stability of Jordan -derivations on -algebras.

Theorem 3.3. Suppose that is a mapping with for which there exists a function satisfying (2.13) and (2.14) such thatfor all and all Then, there exists a unique Jordan -derivation such thatfor all Proof. By the same reasoning as in the proof of Theorem 2.4, there exists a unique -linear mapping such thatfor all The mapping is given by
Letting , and replacing by in (3.4), we obtainSofor all Letting tend to infinity, we havefor all Hence, is a Jordan -derivation.
Corollary 3.4. Suppose that is a mapping with for which there exist constant and such thatfor all and all Then, there exists a unique Jordan -derivation such thatfor all Proof. Letting in Theorem 3.3, we obtain the result.Theorem 3.5. Suppose that is a mapping with for which there exists a function satisfying (2.31) and (3.4). Then, there exists a unique Jordan -derivation such thatfor all Proof. The rest of the proof is similar to the proofs of Theorems 2.4 and 3.3.Corollary 3.6. Suppose that is a mapping with for which there exist constant and such thatfor all and all Then, there exists a unique Jordan -derivation such thatfor all Proof. Letting in Theorem 3.5, we obtain the result.

Acknowledgment

This work was supported by Korea Research Foundation Grant KRF-2007-313-C00033.

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