Abstract and Applied Analysis

Abstract and Applied Analysis / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 437931 | 8 pages | https://doi.org/10.1155/2009/437931

Generalized Hyers-Ulam Stability of Generalized ( 𝑁 , 𝐾 ) -Derivations

Academic Editor: Bruce Calvert
Received05 Feb 2009
Revised07 Apr 2009
Accepted12 May 2009
Published06 Jul 2009

Abstract

Let , and be positive integers. Let be an algebra and let be an -bimodule. A -linear mapping is called a generalized -derivation if there exists a -derivation such that for all . The main purpose of this paper is to prove the generalized Hyers-Ulam stability of the generalized -derivations.

1. Introduction

It seems that the stability problem of functional equations introduced by Ulam [1]. Let be a group and let be a metric group with the metric Given does there exist a such that if a mapping satisfies the inequality for all then there exists a homomorphism with for all In other words, under what condition does there exist a homomorphism near an approximate homomorphism? The concept of stability for functional equations arises when one replaces the functional equation by an inequality which acts as a perturbation of the equation. In 1941, Hyers [2] gave the first affirmative answer to the question of Ulam for Banach spaces and . Let be a mapping between Banach spaces such that for all and for some Then there exists a unique additive mapping such that for all By the seminal paper of Th. M. Rassias [3] and work of Gadja [4], if one assumes that and are real normed spaces with complete, is a mapping such that for each fixed the mapping is continuous in real for each fixed in and that there exists and such that for all Then there exists a unique linear map such that for all

On the other hand J. M. Rassias [5] generalized the Hyers stability result by presenting a weaker condition controlled by a product of different powers of norms. If it is assumed that there exist constants and such that and is a map from a norm space into a Banach space such that the inequality for all then there exists a unique additive mapping such that for all If in addition for every is continuous in real for each fixed then is linear.

Suppose is an abelian group, is a Banach space, and that the so-called admissible control function satisfies for all If is a mapping with for all then there exists a unique mapping such that and , for all (see [6]).

Generalized derivations first appeared in the context of operator algebras [7]. Later, these were introduced in the framework of pure algebra [8, 9].

Definition 1.1. Let be an algebra and let be an -bimodule. A linear mapping is called
(i) derivation if , for all ;
(ii) generalized derivation if there exists a derivation (in the usual sense) such that , for all .

Every right multiplier (i.e., a linear map on satisfying , for all ) is a generalized derivation.

Definition 1.2. Let be positive integers. Let be an algebra and let be an -bimodule. A -linear mapping is called
(i) -derivation if for all ;
(ii) generalized -derivation if there exists a -derivation such that for all .

By Definition 1.2, we see that a generalized -derivation is a generalized derivation.

For instance, let be a Banach algebra. Then we take is an algebra equipped with the usual matrix-like operations. It is easy to check that every linear map from into is a -derivation, but there are linear maps on which are not generalized derivations.

The so-called approximate derivations were investigated by Jun and Park [10]. Recently, the stability of derivations have been investigated by some authors; see [10–13] and references therein. Moslehian [14] investigated the generalized Hyers-Ulam stability of generalized derivations from a unital normed algebra to a unit linked Banach -bimodule (see also [15]).

In this paper, we investigate the generalized Hyers-Ulam stability of the generalized -derivations.

2. Main Result

In this section, we investigate the generalized Hyers-Ulam stability of the generalized -derivations from a unital Banach algebra into a unit linked Banach -bimodule. Throughout this section, assume that is a unital Banach algebra, is unit linked Banach -bimodule, and suppose that and .

We need the following lemma in the main results of the present paper.

Lemma 2.1 (see [16]). Let be linear spaces and let be an additive mapping such that , for all and all . Then the mapping is -linear.

Now we prove the generalized Hyers-Ulam stability of generalized -derivations.

Theorem 2.2. Suppose is a mapping with for which there exists a map with and a function such that for all and all . Then there exists a unique generalized -derivation such that for all .

Proof. By (2.1) we have for all and all . Setting and in (2.4), we have for all One can use induction on to show that for all and all and that for all and all . It follows from the convergence (2.2) that the sequence is Cauchy. Due to the completeness of this sequence is convergent. Set Putting and replacing by respectively, in (2.4), we get for all and all . Taking the limit as we obtain for all and all So by Lemma 2.1, the mapping is -linear.
Using (2.5), (2.2), and the above technique, we get for all and all Hence by Lemma 2.1, is -linear. Moreover, it follows from (2.7) and (2.9) that , for all It is known that the additive mapping satisfying (2.3) is unique [17]. Putting , and replacing by respectively, in (2.4), we get whence for all By (2.9), and by the convergence of series (2.2), Let tend to in (2.14). Then for all .
Next we claim that is a -derivation. Putting and replacing by respectively, in (2.5), we get whence for all Let tends to in (2.17). Then for all .
Setting in (2.18). Hence the mapping is -derivation.

Corollary 2.3. Suppose is a mapping with for which there exists constant and a map with such that for all and all Then there exists a unique generalized -derivation such that for all .

Proof. Put in Theorem 2.2.

Corollary 2.4. Suppose is a mapping with for which there exists constant and a map with such that for all Then there exists a unique generalized -derivation such that for all .

Proof. Letting in Corollary 2.3, we obtain the above result of Corollary 2.4.

References

  1. S. M. Ulam, Problems in Modern Mathematics, Science Editions John Wiley & Sons, New York, NY, USA, 1964, (Chapter VI, Some Questions in Analysis: 1, Stability). View at: Zentralblatt MATH | MathSciNet
  2. D. H. Hyers, β€œOn the stability of the linear functional equation,” Proceedings of the National Academy of Sciences of the United States of America, vol. 27, pp. 222–224, 1941. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  3. Th. M. Rassias, β€œOn the stability of the linear mapping in Banach spaces,” Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297–300, 1978. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  4. Z. Gajda, β€œOn stability of additive mappings,” International Journal of Mathematics and Mathematical Sciences, vol. 14, no. 3, pp. 431–434, 1991. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  5. J. M. Rassias, β€œOn approximation of approximately linear mappings by linear mappings,” Journal of Functional Analysis, vol. 46, no. 1, pp. 126–130, 1982. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  6. P. Găvruţa, β€œA generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings,” Journal of Mathematical Analysis and Applications, vol. 184, no. 3, pp. 431–436, 1994. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  7. M. Mathieu, Ed., Elementary Operators & Applications, World Scientific, River Edge, NJ, USA, 1992, Proceedings of the International Workshop.
  8. F. Wei and Z. Xiao, β€œGeneralized Jordan derivations on semiprime rings,” Demonstratio Mathematica, vol. 40, no. 4, pp. 789–798, 2007. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  9. B. Hvala, β€œGeneralized derivations in rings,” Communications in Algebra, vol. 26, no. 4, pp. 1147–1166, 1998. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  10. K.-W. Jun and D.-W. Park, β€œAlmost derivations on the Banach algebra Cn[0,1],” Bulletin of the Korean Mathematical Society, vol. 33, no. 3, pp. 359–366, 1996. View at: Google Scholar | MathSciNet
  11. M. Amyari, C. Baak, and M. S. Moslehian, β€œNearly ternary derivations,” Taiwanese Journal of Mathematics, vol. 11, no. 5, pp. 1417–1424, 2007. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  12. R. Badora, β€œOn approximate derivations,” Mathematical Inequalities & Applications, vol. 9, no. 1, pp. 167–173, 2006. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  13. C.-G. Park, β€œLinear derivations on Banach algebras,” Nonlinear Functional Analysis and Applications, vol. 9, no. 3, pp. 359–368, 2004. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  14. M. S. Moslehian, β€œHyers-Ulam-Rassias stability of generalized derivations,” International Journal of Mathematics and Mathematical Sciences, vol. 2006, Article ID 93942, 8 pages, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  15. M. E. Gordji and N. Ghobadipour, β€œNearly generalized Jordan derivations,” to appear in Mathematica Slovaca. View at: Google Scholar
  16. C.-G. Park, β€œHomomorphisms between Poisson JC-algebras,” Bulletin of the Brazilian Mathematical Society, vol. 36, no. 1, pp. 79–97, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  17. C. Baak and M. S. Moslehian, β€œOn the stability of J-homomorphisms,” Nonlinear Analysis: Theory, Methods & Applications, vol. 63, no. 1, pp. 42–48, 2005. View at: Google Scholar | Zentralblatt MATH | MathSciNet

Copyright © 2009 M. Eshaghi Gordji et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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