Abstract and Applied Analysis

Volume 2012 (2012), Article ID 347478, 11 pages

doi:10.1155/2012/347478

Research Article

The Hyers-Ulam-Rassias Stability of -Derivations on Normed Algebras

Ajda Fošner

Faculty of Management, University of Primorska, Cankarjeva 5, 6104 Koper, Slovenia

Received 8 April 2012; Accepted 31 May 2012

Academic Editor: Nicole Brillouet-Belluot

Copyright © 2012 Ajda Fošner. 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 study the Hyers-Ulam-Rassias stability of -derivations on normed algebras.

1. Introduction

A classical question in the theory of functional equations is as follows. Under what conditions is it true that a mapping which approximately satisfies a functional equation must be somehow close to an exact solution of This problem was formulated by Ulam in 1940 (see [1, 2]). He investigated the stability of group homomorphisms. Let    be a group, and let   be a metric group with a metric . Suppose that   is a map and     a fixed scalar. Does there exists   such that if satisfies the inequality

for all , then there exists a group homomorphism   with the property

for all ?

One year later, Ulam's problem was affirmatively solved by Hyers [3] for the Cauchy  functional  equation   .: Let be a normed space, a Banach space, and a fixed scalar. Suppose that f  : → is a map with the property

for all . Then there exists a unique additive mapping   such that

for all . This gave rise to the stability theory of functional equations.

The famous Hyers stability result has been generalized in the stability of additive mappings involving a sum of powers of norms by Aoki [4] which allowed the Cauchy difference to be unbounded. In 1978, Rassias [5] proved the stability of linear mappings in the following way. Let be a real normed space and a real Banach space. If there exist scalars and such that

for all ,  then there exists a unique additive mapping with the property

for all . Moreover, if the map is continuous on for each , then is linear. This result has provided a lot of influence in the development of what we now call the Hyers-Ulam-Rassias stability of functional equations.

Later, Găvruţa [6] generalized the Rassias' theorem as follows: Let be an Abelian group and a Banach space. Suppose that the so-called admissible control function satisfies

for all . If   is a mapping with the property

for all , then there exists a unique additive mapping such that

for all .

In the last few decades, various approaches to the problem have been introduced by several authors. Moreover, it is surprising that in some cases the approximate mapping is actually a true mapping. In such cases we call the equation superstable. For the history and various aspects of this theory we refer the reader to monographs [7–9].

As we are aware, the stability of derivations was first investigated by Jun and Park [10]. During the past few years, approximate derivations were studied by a number of mathematicians (see [11–18] and references therein).

Moslehian [19] studied the stability of -derivations and generalized some results obtained in [18]. He also established the generalized Hyers-Ulam-Rassias stability of -derivations on normed algebras into Banach bimodules. This motivated us to investigate approximate -derivations on normed algebras. The aim of this paper is to study the stability of -derivations and to generalize some results given in [19].

2. Preliminaries

Throughout, will be a normed algebra and a Banach -bimodule. Let and be two linear operators on . An additive mapping is called an -derivation if

holds for all . Ordinary derivations from to and maps defined by , where is a fixed element and are endomorphisms on , are natural examples of -derivations on . Moreover, if is an endomorphism on , then is a -derivation on . We refer the reader to [20], where further information about -derivations can be found.

In [19] Moslehian studied stability of -derivations. The natural question here is, whether the analogue results hold true for -derivations. Theorem 3.1 answers this question in the affirmative.

Let and be nonnegative integers with . An additive mapping is called a -derivation if

holds for all . Clearly, -derivations are one of the natural generalizations of -derivations (the case ). If , where denotes the identity map on , and an additive mapping satisfies (2.2), then is called a -derivation. In the last few decades a lot of work has been done on the field of -derivations on rings and algebras (see, e.g, [21–25]). This motivated us to study the Hyers-Ulam-Rassias stability of functional inequalities associated with -derivations.

In the following, we will assume that and are nonnegative integers with . We will use the same symbol in order to represent the norms on a normed algebra and a Banach -bimodule . For a given (admissible control) function we will use the following abbreviation:

Let us start with one well-known lemma.

Lemma 2.1 (see [6]). Suppose that a function satisfies , . If is a mapping with

for all , then there exists a unique additive mapping such that

for all .

We say that an additive mapping is -linear if for all and all scalars . In the following, will denote the set of all complex units, that is,

For a given additive mapping , Park [26] obtained the next result.

Lemma 2.2. If for all and all , then is -linear.

3. The Results

Our first result is a generalization of [19, Theorem 2.1] (the case ). We use the direct method to construct a unique -linear mapping from an approximate one and prove that this mapping is an appropriate -derivation on . This method was first devised by Hyers [3]. The idea is taken from [19].

Theorem 3.1. Let and be mappings with . Suppose that there exists a function such that for all and

for all and . Then there exist unique -linear mappings satisfying

for all , and a unique -linear -derivation such that

for all .

Proof. Taking in (3.1) and using Lemma 2.1, it follows that there exists a unique additive mapping such that holds for all . More precisely, using the induction, it is easy to see that

for all , all positive integers , and all . According to the assumptions on , it follows that the sequence is Cauchy. Thus, by the completeness of , this sequence is convergent and we can define a map as

Using (3.1), we get

This yields that

for all and . Using Lemma 2.2, it follows that the map is -linear. Moreover, according to inequality (3.7), we have

for all .

Next, we have to show the uniqueness of . So, suppose that there exists another -linear mapping such that for all . Then

Therefore, for all , as desired.

Similarly we can show that there exist unique -linear mappings defined by

Furthermore,

for all .

It remains to prove that is an -derivation. Writing in the place of and in the place of in (3.4), we obtain

This yields that

for all . Thus, mappings and satisfy (2.2). The proof is completed.

Remark 3.2. If there exists such that and the map are continuous at point , then is continuous on . Namely, if was not continuous, then there would exist an integer and a sequence such that and , . Let . Then

since is continuous at point . Thus, there exists an integer such that for every we have

Therefore,

for every . Letting and using the continuity of the map at point , we get a contradiction.

Let and . Applying Theorem 3.1 for the case

Corollary 3.3. Let and be mappings with . Suppose that (3.1), (3.2), (3.3), and (3.4) hold true for all and , where a function is defined as above. Then there exist unique -linear mappings satisfying

for all and a unique -linear -derivation such that

Proof . Note that for all and

Remark 3.4. Recall that we can actually take any map in the form

where . In this case we have

Before stating our next result, let us write one well-known lemma about the continuity of measurable functions (see, e.g., [27]).

Lemma 3.5. If a measurable function satisfies for all , then is continuous.

Now we are in the position to state a result for normed algebras which are spanned by a subset of . For example, can be a -algebra spanned by the unitary group of or the positive part of

Theorem 3.6. Let be a normed algebra which is spanned by a subset of and , mappings with . Suppose that there exists a function such that for all and (3.1), (3.2), (3.3) holds true for all and . Moreover, suppose that (3.4) holds true for all . If for all the functions , , and are continuous on , then there exist unique -linear mappings satisfying

for all and a unique -linear -derivation such that

for all .

We will give just a sketch of the proof since most of the steps are the same as in the proof of Theorem 3.1.

Proof. As in the proof of Theorem 3.1, we can show that there exists a unique additive mapping defined by , . Moreover, for all .

Writing , in (3.1), we get

Therefore,

This yields that

for all . In the next step we will show that is -linear, that is,

for all and all .

Since is additive, we have for every and all rational numbers . Let us fix elements and , where denotes the dual space of . Then we can define a function by

Firstly, we would like to prove that is continuous. Recall that

for all . Furthermore,

for all . Set

Obviously, is a sequence of continuous functions and is its pointwise limit. This yields that is a Borel function and, by Lemma 3.5 it is continuous. Therefore, we have for all . This implies . Since was an arbitrary element from , we proved that is -linear.

Now, let . Then for some real numbers . Using (3.31), we have

for all . This means that is -linear.

Similarly we can show that there exist unique -linear mappings satisfying

for all . Moreover, (2.2) holds true for all . Since is linearly generated by , we conclude that is an -derivation on . The proof is completed.

Remark 3.7. As above, we can apply Theorem 3.6 for the case

where and .

Remark 3.8. If and , then we can use in Theorem 3.1 as well as in Theorem 3.6 a function given by

In this case

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