# Superstability of Generalized Derivations

- Esmaeil Ansari-Piri
^{1}Email author and - Ehsan Anjidani
^{1}

**2010**:740156

https://doi.org/10.1155/2010/740156

© E. Ansari-Piri and E. Anjidani. 2010

**Received: **23 February 2010

**Accepted: **15 April 2010

**Published: **24 May 2010

## Abstract

## 1. Introduction

The well-known problem of stability of functional equations started with a question of Ulam [1] in 1940. In 1941, Ulam's problem was solved by Hyers [2] for Banach spaces. Aoki [3] provided a generalization of Hyers' theorem for approximately additive mappings. In 1978, Rassias [4] generalized Hyers' theorem by obtaining a unique linear mapping near an approximate additive mapping.

*and*are real normed spaces with complete, is a mapping such that for each fixed the mapping is continuous on , and there exist and such that

In 1994, G vruţa [5] provided a generalization of Rassias' theorem in which he replaced the bound in (1.1) by a general control function .

Since then several stability problems for various functional equations have been investigated by many mathematicians (see [6–8]).

The various problems of the stability of derivations and generalized derivations have been studied during the last few years (see, e.g., [9–18]). The purpose of this paper is to prove the superstability of generalized (ring) derivations on Banach algebras.

The following result which is called the superstability of ring homomorphisms was proved by Bourgin [19] in 1949.

where and are mappings on algebra with an approximate identity.

where . Our results in this section generalize some results of Moslehian's paper [14]. It has been shown by Moslehian [14, Corollary ] that for an approximate generalized derivation on a Banach algebra , there exists a unique generalized derivation near . We show that the approximate generalized derivation is a generalized derivation (see Corollary 3.6).

Let be an algebra. An additive map is said to be ring derivation on if for all . Moreover, if for all , then is a derivation. An additive mapping (resp., linear mapping) is called a generalized ring derivation (resp., generalized derivation) if there exists a ring derivation (resp., derivation) such that for all .

## 2. Superstability of (1.5) and (1.6)

Here we show the superstability of the functional equations (1.5) and (1.6). We prove the superstability of (1.6) without any additional conditions on the mapping .

Theorem 2.1.

Proof.

Theorem 2.2.

Proof.

By taking the limit as , we have . Since has a left approximate identity, we have .

In the next theorem, we prove the superstability of (1.5) with no additional functional inequality on the mapping .

Theorem 2.3.

Proof.

for all . Then for all and all , and so by taking the limit as , we have . Now we obtain , since has an approximate identity.

for all . Therefore, we have .

The following theorem states the conditions on the mapping under which the sequence converges for all .

Theorem 2.4.

for all . Then exists and for all .

Proof.

## 3. Superstability of the Generalized Derivations

Our purpose is to prove the superstability of generalized ring derivations and generalized derivations. Throughout this section, is a Banach algebra with a two-sided approximate identity.

Theorem 3.1.

for all . Then is a generalized ring derivation and is a ring derivation. Moreover, for all .

Proof.

Put and in (3.3). We have , and so for all .

Then by (3.2) and applying Theorem 2.4, we have and for all .

for all . It follows from (3.1) and Theorem 2.3 that , , and for all .

It suffices to show that and are additive.

Putting and replacing by in (3.7), we have . Similarly, .

Replacing by and by in (3.7), we obtain for all . Therefore is an additive mapping.

Since , is additive, and has an approximate identity, is additive.

Theorem 3.2.

for all . Then is a generalized ring derivation and is a ring derivation. Moreover, for all .

Proof.

for all . Putting in (3.10), it follows from Theorem 2.3 that and for all and for all .

for all . Hence, by the same reasoning as in the proof of Theorem 3.1, and are additive mappings. Therefore, is a generalized ring derivation and is a ring derivation.

Remark 3.3.

We note that Theorems 3.1 and 3.2 and all that following results are obtained with no special conditions on the mapping (see [21, Theorems 2.1 and 2.5]).

Corollary 3.4.

for all . Then is a generalized ring derivation and is a ring derivation.

Proof.

Let . For , if , then satisfies (3.1), (3.2), and we apply Theorem 3.1, and if , then we apply Theorem 3.2 since has conditions (3.8), (3.9) in this case.

For , apply Theorem 3.2 if and apply Theorem 3.1 if .

Theorem 3.5.

for all and all . Then is a generalized derivation and is a derivation.

Proof.

Suppose that satisfies (3.1), (3.2). By Theorem 3.1, is a generalized ring derivation and is a ring derivation. Moreover, for all .

Hence, by taking the limit as , we get for all and .

for all . Now by [21, Lemma 2.4], is a linear mapping and hence is a linear mapping.

The following result generalizes Corollary and Theorem of [14].

Corollary 3.6.

for all and all . Then is a generalized derivation and is a derivation.

Proof.

Theorem 3.7.

for all . If is continuous in for each fixed , then is a generalized derivation and is a derivation.

Proof.

Suppose that satisfies (3.1), (3.2). By Theorem 3.1, is a generalized ring derivation, is a ring derivation, and for all .

for all . Therefore, is -linear.

for all . Therefore, the mapping is linear and it follows that is linear.

Corollary 3.8.

for all . Suppose that is continuous in for each fixed . Then is a generalized derivation and is a derivation.

Proof.

Let , define , and apply Theorem 3.7.

Theorem 3.9.

for and all . If is continuous in for each fixed , then is a generalized derivation and is a derivation.

Proof.

Let . By Theorem 3.7, it suffices to prove that satisfies (3.1), (3.2).

Then , and so . Hence satisfies (3.1).

The theorems similar to Theorem 3.9 have been proved by the assumption that the relations similar to (3.29) are true for , (see, e.g., [9, 14]). We proved Theorem 3.9, under condition that inequality (3.29) is true for .

## Declarations

### Acknowledgment

The authors express their thanks to the referees for their helpful suggestions to improve the paper.

## Authors’ Affiliations

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