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# Superstability for Generalized Module Left Derivations and Generalized Module Derivations on a Banach Module (I)

*Journal of Inequalities and Applications*
**volume 2009**, Article number: 718020 (2009)

## Abstract

We discuss the superstability of generalized module left derivations and generalized module derivations on a Banach module. Let be a Banach algebra and a Banach -module, and . The mappings , and are defined and it is proved that if (resp., is dominated by then is a generalized (resp., linear) module- left derivation and is a (resp., linear) module- left derivation. It is also shown that if (resp., is dominated by then is a generalized (resp., linear) module- derivation and is a (resp., linear) module- derivation.

## 1. Introduction

The study of stability problems had been formulated by Ulam in [1] during a talk in 1940: under what condition does there exist a homomorphism near an approximate homomorphism? In the following year 1941, Hyers in [2] has answered affirmatively the question of Ulam for Banach spaces, which states that if and is a map with , a normed space, , a Banach space, such that

for all in , then there exists a unique additive mapping such that

for all in . In addition, if the mapping is continuous in for each fixed in , then the mapping is real linear. This stability phenomenon is called the *Hyers-Ulam stability* of the additive functional equation . A generalized version of the theorem of Hyers for approximately additive mappings was given by Aoki in [3] and for approximate linear mappings was presented by Rassias in [4] by considering the case when the left-hand side of (1.1) is controlled by a sum of powers of norms. The stability result concerning derivations between operator algebras was first obtained by Šemrl in [5], Badora in [6] gave a generalization of Bourgin's result [7]. He also discussed the Hyers-Ulam stability and the Bourgin-type superstability of derivations in [8].

Singer and Wermer in [9] obtained a fundamental result which started investigation into the ranges of linear derivations on Banach algebras. The result, which is called the Singer-Wermer theorem, states that any continuous linear derivation on a commutative Banach algebra maps into the Jacobson radical. They also made a very insightful conjecture, namely, that the assumption of continuity is unnecessary. This was known as the Singer- Wermer conjecture and was proved in 1988 by Thomas in [10]. The Singer-Wermer conjecture implies that any linear derivation on a commutative semisimple Banach algebra is identically zero [11]. After then, Hatori and Wada in [12] proved that the zero operator is the only derivation on a commutative semisimple Banach algebra with the maximal ideal space without isolated points. Based on these facts and a private communication with Watanabe [13], Miura et al. proved the Hyers-Ulam-Rassias stability and Bourgin-type superstability of derivations on Banach algebras in [13]. Various stability results on derivations and left derivations can be found in [14–20]. More results on stability and superstability of homomorphisms, special functionals, and equations can be found in [21–30].

Recently, Kang and Chang in [31] discussed the superstability of generalized left derivations and generalized derivations. Indeed, these superstabilities are the so-called "Hyers-Ulam superstabilities." In the present paper, we will discuss the superstability of generalized module left derivations and generalized module derivations on a Banach module.

To give our results, let us give some notations. Let be an algebra over the real or complex field and an -bimodule.

Definition 1.1.

A mapping is said to be *module-**additive* if

A module- additive mapping is said to be a *module-** left derivation* (resp., *module-** derivation*) if the functional equation

respectively,

holds.

Definition 1.2.

A mapping is said to be *module-** additive* if

A module- additive mapping is called a *generalized module-** left derivation* (resp., *generalized module-** derivation*) if there exists a module- left derivation (resp., module- derivation) such that

respectively,

In addition, if the mappings and are all linear, then the mapping is called a *linear generalized module-** left derivation* (resp., *linear generalized module-** derivation*).

Remark 1.3.

Let and be one of the following cases: (a) a unital algebra; (b) a Banach algebra with an approximate unit; (c) a -algebra. Then module- left derivations, module- derivations, generalized module- left derivations, and generalized module- derivations on become left derivations, derivations, generalized left derivations, and generalized derivations on discussed in [31].

## 2. Main Results

Theorem 2.1.

Let be a Banach algebra, a Banach -bimodule, and integers greater than 1, and satisfy the following conditions:

(a)

(b)

(c)

Suppose that and are mappings such that , exists for all and

for all and where

Then is a generalized module- left derivation and is a module- left derivation.

Proof.

By taking , we see from (2.1) that

for all . Letting and replacing by in (2.3) yield that

for all . From [32, Theorem 1] (analogously as in [33, the proof of Theorem 1] or [34]), one can easily deduce that the limit exists for every , and

Next, we show that the mapping is additive. To do this, let us replace by in (2.3), respectively. Then

for all . If we let in the above inequality, then the condition (a) yields that

for all . Since , taking and , respectively, we see that and for all . Now, for all , put . Then by (2.7), we get that

This shows that is additive.

Now, we are going to prove that is a generalized module- left derivation. Letting in (2.1) gives that

that is,

for all and . By replacing with in (2.10), respectively, we deduce that

for all and . Letting , the condition (b) yields that

for all and . Since is additive, is module- additive. Put . Then by (2.10) we see from the condition (a) that

for all and . Hence

for all and . It follows from (2.12) that for all and , and then for all . Since is additive, is module- additive. So, for all and by (2.12)

This shows that is a module- left derivation on and then is a generalized module- left derivation on .

Lastly, we prove that is a module- left derivation on . To do this, we compute from (2.10) that

for all . By letting , we get from the condition (a) that

for all . Now, (2.12) implies that for all and all . Hence, is a module- left derivation on . This completes the proof.

Remark 2.2.

It is easy to check that the functional satisfies the conditions (a), (b), and (c) in Theorem 2.1, where , . Especially, if has a unit and are mappings with such that for all , then is a generalized left derivation and is a left derivation.

Remark 2.3.

In Theorem 2.1, if the condition (2.1) is replaced with

for all and where

then is a generalized module- derivation and is a module- derivation. Especially, if has a unit and are mappings with such that for all and some constants , then is a generalized derivation and is a derivation.

Lemma 2.4.

Let be complex vector spaces. Then a mapping is linear if and only if

for all and all .

Proof.

It suffices to prove the sufficiency. Suppose that for all and all . Then is additive and for all and all Let be any nonzero complex number. Take a positive integer such that . Take a real number such that . Put . Then and, therefore,

for all . This shows that is linear. The proof is completed.

Theorem 2.5.

Let be a Banach algebra, a Banach -bimodule, and integers greater than , and satisfy the following conditions:

(a)

(b)

(c).

Suppose that and are mappings such that , exists for all and

for all , and all , where stands for

Then is a linear generalized module- left derivation and is a linear module- left derivation.

Proof.

Clearly, the inequality (2.1) is satisfied. Hence, Theorem 2.1 and its proof show that is a generalized left derivation and is a left derivation on with

for every . Taking in (2.22) yields that

for all and all . If we replace and with and in (2.25), respectively, then we see that

as for all and all . Hence,

for all and all . Since is additive, taking in (2.27) implies that

for all and all . Lemma 2.4 yields that is linear and so is . This completes the proof.

Remark 2.6.

It is easy to check that the functional satisfies the conditions (a), (b), and (c) in Theorem 2.5, where , are constants. Especially, if is a complex semiprime Banach algebra with unit and are mappings with such that

for all . Then is a linear generalized left derivation and is a linear derivation which maps into the intersection of the center and the Jacobson radical of .

Remark 2.7.

In Theorem 2.5, if the condition (2.22) is replaced with

for all , and where stands for

then is a linear generalized module- derivation on and is a linear module- derivation on . Especially, if is a unital commutative Banach algebra and are mappings with such that for all , all and some constants , then is a linear generalized derivation and is a linear derivation which maps into the Jacobson radical rad of .

Remark 2.8.

The controlling function

consists of the "mixed sum-product of powers of norms," introduced by Rassias (in 2007) [28] and applied afterwards by Ravi et al. (2007-2008). Moreover, it is easy to check that the functional

satisfies theconditions (a), (b), and (c) in Theorems 2.1 and 2.5, where and are all constants.

## References

- 1.
Ulam SM:

*Problems in Modern Mathematics*. John Wiley & Sons, New York, NY, USA; 1964:xvii+150. - 2.
Hyers DH:

**On the stability of the linear functional equation.***Proceedings of the National Academy of Sciences*1941,**27:**222–224. 10.1073/pnas.27.4.222 - 3.
Aoki T:

**On the stability of the linear transformation in Banach spaces.***Journal of the Mathematical Society of Japan*1950,**2:**64–66. 10.2969/jmsj/00210064 - 4.
Rassias TM:

**On the stability of the linear mapping in Banach spaces.***Proceedings of the American Mathematical Society*1978,**72**(2):297–300. 10.1090/S0002-9939-1978-0507327-1 - 5.
Šemrl P:

**The functional equation of multiplicative derivation is superstable on standard operator algebras.***Integral Equations and Operator Theory*1994,**18**(1):118–122. 10.1007/BF01225216 - 6.
Badora R:

**On approximate ring homomorphisms.***Journal of Mathematical Analysis and Applications*2002,**276**(2):589–597. 10.1016/S0022-247X(02)00293-7 - 7.
Bourgin DG:

**Approximately isometric and multiplicative transformations on continuous function rings.***Duke Mathematical Journal*1949,**16:**385–397. 10.1215/S0012-7094-49-01639-7 - 8.
Badora R:

**On approximate derivations.***Mathematical Inequalities & Applications*2006,**9**(1):167–173. - 9.
Singer IM, Wermer J:

**Derivations on commutative normed algebras.***Mathematische Annalen*1955,**129:**260–264. 10.1007/BF01362370 - 10.
Thomas MP:

**The image of a derivation is contained in the radical.***Annals of Mathematics*1988,**128**(3):435–460. 10.2307/1971432 - 11.
Johnson BE:

**Continuity of derivations on commutative algebras.***American Journal of Mathematics*1969,**91:**1–10. 10.2307/2373262 - 12.
Hatori O, Wada J:

**Ring derivations on semi-simple commutative Banach algebras.***Tokyo Journal of Mathematics*1992,**15**(1):223–229. 10.3836/tjm/1270130262 - 13.
Miura T, Hirasawa G, Takahasi S-E:

**A perturbation of ring derivations on Banach algebras.***Journal of Mathematical Analysis and Applications*2006,**319**(2):522–530. 10.1016/j.jmaa.2005.06.060 - 14.
Amyari M, Baak C, Moslehian MS:

**Nearly ternary derivations.***Taiwanese Journal of Mathematics*2007,**11**(5):1417–1424. - 15.
Moslehian MS:

**Ternary derivations, stability and physical aspects.***Acta Applicandae Mathematicae*2008,**100**(2):187–199. 10.1007/s10440-007-9179-x - 16.
Moslehian MS:

**Hyers-Ulam-Rassias stability of generalized derivations.***International Journal of Mathematics and Mathematical Sciences*2006,**2006:**-8. - 17.
Park C-G:

**Homomorphisms between**-algebras,**-derivations on a**-**algebra and the Cauchy-Rassias stability.***Nonlinear Functional Analysis and Applications*2005,**10**(5):751–776. - 18.
Park C-G:

**Linear derivations on Banach algebras.***Nonlinear Functional Analysis and Applications*2004,**9**(3):359–368. - 19.
Amyari M, Rahbarnia F, Sadeghi Gh:

**Some results on stability of extended derivations.***Journal of Mathematical Analysis and Applications*2007,**329**(2):753–758. 10.1016/j.jmaa.2006.07.027 - 20.
Brešar M, Vukman J:

**On left derivations and related mappings.***Proceedings of the American Mathematical Society*1990,**110**(1):7–16. - 21.
Găvruţa P:

**A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings.***Journal of Mathematical Analysis and Applications*1994,**184**(3):431–436. 10.1006/jmaa.1994.1211 - 22.
Ulam SM:

*Problems in Modern Mathematics*. John Wiley & Sons, New York, NY, USA; 1960:xiii+150. - 23.
Baker JA:

**The stability of the cosine equation.***Proceedings of the American Mathematical Society*1980,**80**(3):411–416. 10.1090/S0002-9939-1980-0580995-3 - 24.
Hyers DH, Rassias TM:

**Approximate homomorphisms.***Aequationes Mathematicae*1992,**44**(2–3):125–153. 10.1007/BF01830975 - 25.
Rassias TM, Tabo J (Eds):

*Stability of Mappings of Hyers-Ulam Type, Hadronic Press Collection of Original Articles*. Hadronic Press, Palm Harbor, Fla, USA; 1994:vi+173. - 26.
Hyers DH, Isac G, Rassias TM:

*Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and Their Applications, 34*. Birkhäuser, Boston, Mass, USA; 1998:vi+313. - 27.
Isac G, Rassias TM:

**On the Hyers-Ulam stability of**-**additive mappings.***Journal of Approximation Theory*1993,**72**(2):131–137. 10.1006/jath.1993.1010 - 28.
Rassias JM, Rassias MJ:

**Refined Ulam stability for Euler-Lagrange type mappings in Hilbert spaces.***International Journal of Applied Mathematics & Statistics*2007,**7:**126–132. - 29.
Rassias JM, Rassias MJ:

**The Ulam problem for 3-dimensional quadratic mappings.***Tatra Mountains Mathematical Publications*2006,**34, part 2:**333–337. - 30.
Rassias JM, Rassias MJ:

**On the Hyers-Ulam stability of quadratic mappings.***The Journal of the Indian Mathematical Society*2000,**67**(1–4):133–136. - 31.
Kang S-Y, Chang I-S:

**Approximation of generalized left derivations.***Abstract and Applied Analysis*2008, -8. - 32.
Forti GL:

**Comments on the core of the direct method for proving Hyers-Ulam stability of functional equations.***Journal of Mathematical Analysis and Applications*2004,**295**(1):127–133. 10.1016/j.jmaa.2004.03.011 - 33.
Brzdęk J, Pietrzyk A:

**A note on stability of the general linear equation.***Aequationes Mathematicae*2008,**75**(3):267–270. 10.1007/s00010-007-2894-6 - 34.
Pietrzyk A:

**Stability of the Euler-Lagrange-Rassias functional equation.***Demonstratio Mathematica*2006,**39**(3):523–530.

## Acknowledgment

This subject is supported by the NNSFs of China (no: 10571113,10871224).

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Cao, H., Lv, J. & Rassias, J.M. Superstability for Generalized Module Left Derivations and Generalized Module Derivations on a Banach Module (I).
*J Inequal Appl* **2009, **718020 (2009). https://doi.org/10.1155/2009/718020

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### Keywords

- Generalize Module
- Additive Mapping
- Banach Algebra
- Generalize Derivation
- Jacobson Radical