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Abstract and Applied Analysis
VolumeΒ 2009, Article IDΒ 437931, 8 pages
http://dx.doi.org/10.1155/2009/437931
Research Article

Generalized Hyers-Ulam Stability of Generalized -Derivations

1Department of Mathematics, Semnan University, P. O. Box 35195-363, Semnan, Iran
2Section of Mathematics and Informatics, Pedagogical Department, National and Capodistrian University of Athens, 4, Agamemnonos St., Aghia Paraskevi, 15342 Athens, Greece

Received 5 February 2009; Revised 7 April 2009; Accepted 12 May 2009

Academic Editor: BruceΒ Calvert

Copyright Β© 2009 M. Eshaghi Gordji 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

Let , and be positive integers. Let be an algebra and let be an -bimodule. A -linear mapping is called a generalized -derivation if there exists a -derivation such that for all . The main purpose of this paper is to prove the generalized Hyers-Ulam stability of the generalized -derivations.

1. Introduction

It seems that the stability problem of functional equations introduced by Ulam [1]. Let be a group and let be a metric group with the metric Given does there exist a such that if a mapping satisfies the inequality for all then there exists a homomorphism with for all In other words, under what condition does there exist a homomorphism near an approximate homomorphism? The concept of stability for functional equations arises when one replaces the functional equation by an inequality which acts as a perturbation of the equation. In 1941, Hyers [2] gave the first affirmative answer to the question of Ulam for Banach spaces and . Let be a mapping between Banach spaces such that for all and for some Then there exists a unique additive mapping such that for all By the seminal paper of Th. M. Rassias [3] and work of Gadja [4], if one assumes that and are real normed spaces with complete, is a mapping such that for each fixed the mapping is continuous in real for each fixed in and that there exists and such that for all Then there exists a unique linear map such that for all

On the other hand J. M. Rassias [5] generalized the Hyers stability result by presenting a weaker condition controlled by a product of different powers of norms. If it is assumed that there exist constants and such that and is a map from a norm space into a Banach space such that the inequality for all then there exists a unique additive mapping such that for all If in addition for every is continuous in real for each fixed then is linear.

Suppose is an abelian group, is a Banach space, and that the so-called admissible control function satisfies for all If is a mapping with for all then there exists a unique mapping such that and , for all (see [6]).

Generalized derivations first appeared in the context of operator algebras [7]. Later, these were introduced in the framework of pure algebra [8, 9].

Definition 1.1. Let be an algebra and let be an -bimodule. A linear mapping is called
(i) derivation if , for all ;
(ii) generalized derivation if there exists a derivation (in the usual sense) such that , for all .

Every right multiplier (i.e., a linear map on satisfying , for all ) is a generalized derivation.

Definition 1.2. Let be positive integers. Let be an algebra and let be an -bimodule. A -linear mapping is called
(i) -derivation if for all ;
(ii) generalized -derivation if there exists a -derivation such that for all .

By Definition 1.2, we see that a generalized -derivation is a generalized derivation.

For instance, let be a Banach algebra. Then we take is an algebra equipped with the usual matrix-like operations. It is easy to check that every linear map from into is a -derivation, but there are linear maps on which are not generalized derivations.

The so-called approximate derivations were investigated by Jun and Park [10]. Recently, the stability of derivations have been investigated by some authors; see [1013] and references therein. Moslehian [14] investigated the generalized Hyers-Ulam stability of generalized derivations from a unital normed algebra to a unit linked Banach -bimodule (see also [15]).

In this paper, we investigate the generalized Hyers-Ulam stability of the generalized -derivations.

2. Main Result

In this section, we investigate the generalized Hyers-Ulam stability of the generalized -derivations from a unital Banach algebra into a unit linked Banach -bimodule. Throughout this section, assume that is a unital Banach algebra, is unit linked Banach -bimodule, and suppose that and .

We need the following lemma in the main results of the present paper.

Lemma 2.1 (see [16]). Let be linear spaces and let be an additive mapping such that , for all and all . Then the mapping is -linear.

Now we prove the generalized Hyers-Ulam stability of generalized -derivations.

Theorem 2.2. Suppose is a mapping with for which there exists a map with and a function such that for all and all . Then there exists a unique generalized -derivation such that for all .

Proof. By (2.1) we have for all and all . Setting and in (2.4), we have for all One can use induction on to show that for all and all and that for all and all . It follows from the convergence (2.2) that the sequence is Cauchy. Due to the completeness of this sequence is convergent. Set Putting and replacing by respectively, in (2.4), we get for all and all . Taking the limit as we obtain for all and all So by Lemma 2.1, the mapping is -linear.
Using (2.5), (2.2), and the above technique, we get for all and all Hence by Lemma 2.1, is -linear. Moreover, it follows from (2.7) and (2.9) that , for all It is known that the additive mapping satisfying (2.3) is unique [17]. Putting , and replacing by respectively, in (2.4), we get whence for all By (2.9), and by the convergence of series (2.2), Let tend to in (2.14). Then for all .
Next we claim that is a -derivation. Putting and replacing by respectively, in (2.5), we get whence for all Let tends to in (2.17). Then for all .
Setting in (2.18). Hence the mapping is -derivation.

Corollary 2.3. Suppose is a mapping with for which there exists constant and a map with such that for all and all Then there exists a unique generalized -derivation such that for all .

Proof. Put in Theorem 2.2.

Corollary 2.4. Suppose is a mapping with for which there exists constant and a map with such that for all Then there exists a unique generalized -derivation such that for all .

Proof. Letting in Corollary 2.3, we obtain the above result of Corollary 2.4.

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