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 .
1. Introduction
The wellknown 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 .
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
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 twosided 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.
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 twosided 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 .
Acknowledgment
The authors express their thanks to the referees for their helpful suggestions to improve the paper.
References

Ulam, SM: Problems in Modern Mathematics,p. xvii+150. John Wiley & Sons, New York, NY, USA (1964)

Hyers, DH: On the stability of the linear functional equation. Proceedings of the National Academy of Sciences of the United States of America. 27, 222–224 (1941). PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Aoki, T: On the stability of the linear transformation in Banach spaces. Journal of the Mathematical Society of Japan. 2, 64–66 (1950). Publisher Full Text

Rassias, ThM: On the stability of the linear mapping in Banach spaces. Proceedings of the American Mathematical Society. 72(2), 297–300 (1978). Publisher Full Text

Găvruţa, P: A generalization of the HyersUlamRassias stability of approximately additive mappings. Journal of Mathematical Analysis and Applications. 184(3), 431–436 (1994). Publisher Full Text

Czerwik S (ed.): Stability of Functional Equations of UlamHyers Rassias Type, Hadronic Press, Palm Harbor, Fla, USA (2003)

Hyers, DH, Isac, G, Rassias, ThM: Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and Their Applications,p. vi+313. Birkhäuser, Boston, Mass, USA (1998)

Rassias ThM (ed.): Functional Equations, Inequalities and Applications,p. x+224. Kluwer Academic Publishers, Dordrecht, The Netherlands (2003)

Amyari, M, Rahbarnia, F, Sadeghi, G: Some results on stability of extended derivations. Journal of Mathematical Analysis and Applications. 329(2), 753–758 (2007). Publisher Full Text

Badora, R: On approximate derivations. Mathematical Inequalities & Applications. 9(1), 167–173 (2006). PubMed Abstract  Publisher Full Text

Jung, YS: On the generalized HyersUlam stability of module left derivations. Journal of Mathematical Analysis and Applications. 339(1), 108–114 (2008). Publisher Full Text

Miura, T, Hirasawa, G, Takahasi, SE: A perturbation of ring derivations on Banach algebras. Journal of Mathematical Analysis and Applications. 319(2), 522–530 (2006). Publisher Full Text

Moslehian, MS: Almost derivations on ternary rings. Bulletin of the Belgian Mathematical Society. Simon Stevin. 14(1), 135–142 (2007)

Moslehian, MS: HyersUlamRassias stability of generalized derivations. International Journal of Mathematics and Mathematical Sciences. 2006, (2006)

Moslehian, MS: Superstability of higher derivations in multiBanach algebras. Tamsui Oxford Journal of Mathematical Sciences. 24(4), 417–427 (2008)

Moslehian, MS: Ternary derivations, stability and physical aspects. Acta Applicandae Mathematicae. 100(2), 187–199 (2008). Publisher Full Text

Park, CG: Homomorphisms between algebras, derivations on a algebra and the CauchyRassias stability. Nonlinear Functional Analysis and Applications. 10(5), 751–776 (2005)

Park, CG: Linear derivations on Banach algebras. Nonlinear Functional Analysis and Applications. 9(3), 359–368 (2004)

Bourgin, DG: Approximately isometric and multiplicative transformations on continuous function rings. Duke Mathematical Journal. 16, 385–397 (1949). Publisher Full Text

Šemrl, P: The functional equation of multiplicative derivation is superstable on standard operator algebras. Integral Equations and Operator Theory. 18(1), 118–122 (1994). Publisher Full Text

Cao, HX, Lv, JR, Rassias, JM: Superstability for generalized module left derivations and generalized module derivations on a Banach module (I). Journal of Inequalities and Applications. 2009, (2009)

Kang, SY, Chang, IS: Approximation of generalized left derivations. Abstract and Applied Analysis. 2008, (2008)

Grabiec, A: The generalized HyersUlam stability of a class of functional equations. Publicationes Mathematicae Debrecen. 48(34), 217–235 (1996)

Jung, SM: On modified HyersUlamRassias stability of a generalized Cauchy functional equation. Nonlinear Studies. 5(1), 59–67 (1998)

Forti, GL: Comments on the core of the direct method for proving HyersUlam stability of functional equations. Journal of Mathematical Analysis and Applications. 295(1), 127–133 (2004). Publisher Full Text

Brzdęk, J, Pietrzyk, A: A note on stability of the general linear equation. Aequationes Mathematicae. 75(3), 267–270 (2008). Publisher Full Text