#### 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*ββββ*be a group, and let *ββ*be a metric group with a metric *. *Suppose that *ββ*is a map and*ββ ββ*a fixed scalar. Does there exists *ββ*such that if **satisfies the inequality**for all *, *then there exists a group homomorphism *ββ*with the property**for all *?

One year later, Ulam's problem was affirmatively solved by Hyers [3] for the Cauchyββfunctionalββequationβ*β *.*: Let ** be a normed space, ** a Banach space, and ** a fixed scalar. Suppose that fββ: **β ** is a map with the property **for all *. *Then there exists a unique additive mapping *ββ*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 *,ββ*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, GΔvruΕ£a [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 ββ*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 [7β9].

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 [11β18] 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, [21β25]). 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