Superstability of Generalized Derivations
© E. Ansari-Piri and E. Anjidani. 2010
Received: 23 February 2010
Accepted: 15 April 2010
Published: 24 May 2010
The well-known problem of stability of functional equations started with a question of Ulam  in 1940. In 1941, Ulam's problem was solved by Hyers  for Banach spaces. Aoki  provided a generalization of Hyers' theorem for approximately additive mappings. In 1978, Rassias  generalized Hyers' theorem by obtaining a unique linear mapping near an approximate additive mapping.
In 1994, G vruţa  provided a generalization of Rassias' theorem in which he replaced the bound in (1.1) by a general control function .
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  in 1949.
where . Our results in this section generalize some results of Moslehian's paper . 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 .
3. Superstability of the Generalized Derivations
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]).
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 .
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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