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.

Assume that 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

(1.1)

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

(1.2)

for all .

In 1994, Gvruţa [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.

Suppose that and are Banach algebras and is with unit. If is surjective mapping and there exist and such that

(1.3)

for all , then is a ring homomorphism, that is,

(1.4)

The first superstability result concerning derivations between operator algebras was obtained by emrl in [20]. In [10], Badora proved the superstability of functional equation where is a mapping on normed algebra with unit. In Section 2, we generalize Badora's result [10, Theorem ] for functional equations

(1.5)
(1.6)

where and are mappings on algebra with an approximate identity.

In [21, 22], the superstability of generalized derivations on Banach algebras associated to the following Jensen type functional equation:

(1.7)

where is an integer is considered. Several authors have studied the stability of the general linear functional equation

(1.8)

where , , , and are constants in the field and is a mapping between two Banach spaces (see [23, 24]). In Section 3, we investigate the superstability of generalized (ring) derivations associated to the linear functional equation

(1.9)

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.

Let be a normed algebra with a central approximate identity and . Suppose that and are mappings for which there exists such that

(2.1)
(2.2)

for all . Then for all .

Proof.

Replacing by in (2.2), we get

(2.3)

and so

(2.4)

for all and . By taking the limit as , we have

(2.5)

for all . Similarly, we have

(2.6)

for all .

Let and . Then we have

(2.7)

Since , we get

(2.8)

By taking the limit as , we get

(2.9)

Therefore, for all .

Theorem 2.2.

Let be a normed algebra with a left approximate identity and . Let and be the mappings satisfying

(2.10)

for all , where is a mapping such that

(2.11)

for all . Then for all .

Proof.

Let . We have

(2.12)

Replacing by , we get

(2.13)

and so

(2.14)

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.

Let be a Banach algebra with a two-sided approximate identity and . Let and be mappings such that exists for all and

(2.15)

for all , where is a function such that

(2.16)

for all . Then , , and .

Proof.

Replacing by in (2.15), we get

(2.17)

and so

(2.18)

for all and . By taking the limit as , we have

(2.19)

for all .

Fix By (2.19), we have

(2.20)

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

Replacing by in (2.15), we obtain

(2.21)

and hence

(2.22)

for all and all . Sending to infinity, we have

(2.23)

By (2.23), we get

(2.24)

for all . Therefore, we have .

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

Theorem 2.4.

Let be a Banach space and . Suppose that is a mapping for which there exists a function such that

(2.25)

for all . Then exists and for all .

Proof.

See [25, Theorem ] or [26, Proposition ].

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.

Let such that . Suppose that is a mapping with for which there exist a map and a function such that

(3.1)
(3.2)
(3.3)

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 .

Put in (3.3). We get

(3.4)

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

It suffices to show that and are additive.

Replacing by and by and putting in (3.3), we obtain

(3.5)

and so

(3.6)

for all and .

By taking the limit as , we get , and so

(3.7)

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.

Let such that . Suppose that is a mapping with for which there exist a map and a function such that

(3.8)
(3.9)
(3.10)

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

Proof.

Replacing , by and putting in (3.10), we get

(3.11)

for all . Since

(3.12)

it follows from Theorem 2.4 that exists for all . By (3.8), we have

(3.13)

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

Replacing by and by , putting in (3.10), and multiplying both sides of the inequality by , we obtain

(3.14)

for all and . By taking the limit as , we get

(3.15)

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.

Let or , , and with . Suppose that is a mapping with for which there exist a map and such that

(3.16)

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.

Let and let be a function satisfying either (3.1), (3.2) or (3.8), (3.9). Suppose that is a mapping with for which there exists a map such that

(3.17)

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

Proof.

Let in (3.17). We have

(3.18)

for all .

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

Replacing by and putting in (3.17), we get

(3.19)

for all , , and . Since , we obtain

(3.20)

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

Let with . Then , and so

(3.21)

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.

Let and with . Suppose that is a mapping with for which there exist a map and such that

(3.22)

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

Proof.

Define and apply Theorem 3.5.

Theorem 3.7.

Let and let be a function satisfying either (3.1), (3.2) or (3.8), (3.9). Suppose that is a mapping with for which there exists a map such that

(3.23)

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 .

Let . The mapping , defined by , is continuous in . Also, the mapping is additive, since is additive. Hence is -linear, and so

(3.24)

for all . Therefore, is -linear.

Now let . Since , there exist such that . So

(3.25)

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

Corollary 3.8.

Let or . Suppose that is a mapping with for which there exists a map such that

(3.26)

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.

Let be a mapping with for which there exist a map and a function such that

(3.27)
(3.28)
(3.29)

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).

Let . We have

(3.30)

Then , and so . Hence satisfies (3.1).

Let . By (3.28), we get

(3.31)

Hence

(3.32)

and so satisfies (3.2).

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 .