Superstability of Generalized Derivations

We investigate the superstability of the functional equation , where and are the mappings on Banach algebra . We have also proved the superstability of generalized derivations associated to the linear functional equation , where .

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

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

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

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

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

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:

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

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

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 .

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

for all . Then for all .

Proof.

Replacing by in (2.2), we get

and so

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

for all . Similarly, we have

for all .

Let and . Then we have

Since , we get

By taking the limit as , we get

Therefore, for all .

Theorem 2.2.

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

for all , where is a mapping such that

for all . Then for all .

Proof.

Let . We have

Replacing by , we get

and so

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

for all , where is a function such that

for all . Then , , and .

Proof.

Replacing by in (2.15), we get

and so

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

for all .

Fix By (2.19), we have

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

and hence

for all and all . Sending to infinity, we have

By (2.23), we get

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

for all . Then exists and for all .

Proof.

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

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

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

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

and so

for all and .

By taking the limit as , we get , and so

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

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

for all . Since

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

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

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

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

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

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

Proof.

Let in (3.17). We have

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

for all , , and . Since , we obtain

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

Let with . Then , and so

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

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

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

for all . Therefore, is -linear.

Now let . Since , there exist such that . So

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

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

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

Then , and so . Hence satisfies (3.1).

Let . By (3.28), we get

Hence

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 .

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

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