Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 954749, 5 pages

http://dx.doi.org/10.1155/2013/954749

Research Article

On the Stability of -Derivations and Lie -Algebra Homomorphisms on Lie -Algebras: A Fixed Points Method

Seong Sik Kim,¹ Ga Ya Kim,² and Soo Hwan Kim¹

¹Department of Mathematics, Dong-eui University, Busan 614-714, Republic of Korea

²Department of Urban Engineering, Dong-eui University, Busan 614-714, Republic of Korea

Received 25 February 2013; Revised 24 May 2013; Accepted 30 May 2013

Academic Editor: Jun Wang

Copyright © 2013 Seong Sik Kim 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.

Abstract.

We investigate new generalized Hyers-Ulam stability results for -derivations and Lie -algebra homomorphisms on Lie -algebras associated with the additive functional equation:

1. Introduction

The theory of finite dimensional complex Lie algebras is an important part of Lie theory. It has several applications in physics and connections with other parts of mathematics. With an increasing amount of theory and applications concerning Lie algebras of various dimensions, it is becoming necessary to ascertain which tools are applicable for handling them. The miscellaneous characteristics of Lie algebras constitute such tools and have also found applications: Casimir operators [1], derived, lower central and upper central sequences, and the Lie algebra of derivations, radicals, nilradicals, ideals, subalgebras [2, 3], and recently megaideals [4]. These characteristics are particularly crucial when considering possible affinities among Lie algebras. Physically motivated relations between two Lie algebras, namely, contractions and deformations, have been extensively studied [5]. Moreover, in modern industry, various analytical approaches for solving mathematical equations are widely applied in analysis of problems in packaging engineering, and so mathematical modeling and computation methods by using mathematical equations play an important role in application of packaging engineering. From now, we wish to note that mathematical equations for stability properties in this paper can have applications to Packaging Engineering.

A -algebra endowed with the Lie product on is called a Lie -algebra. A -linear mapping is called a -derivation of if there exist , and such that for all [6]. Let and be Lie -algebras. A -linear mapping is called a Lie -algebra homomorphism if for all .

The stability problem of functional equations was originated from a question of Ulam [7, 8] concerning the stability of group homomorphisms as follows.

 Let be a group and be a metric group with the metric . Given , does there exist a such that if a mapping satisfies the inequality for all , then there is a homomorphism with for all ?

If the answer is affirmative, we would say that the equation of a homomorphism is stable (see also [9–14]). Cădariu and Radu [15] applied the fixed point method to investigation of the stability of a functional equation. In 2008, Novotný and Hrivnák [6] investigated generalizing the concept of Lie derivations via certain complex parameters, obtained various Lie, and established the structure and properties of -derivations of Lie algebras. Recently, the generalized Hyers-Ulam stability of problems on -algebras associated with functional equations has been investigated by using a fixed point method (see [16–20]). The following fixed point theorem will play an important role in proving our main theorem.

Theorem 1 (see [21]). Suppose that we are given a complete generalized metric space and a strictly contractive mapping with Lipschitz constant . Then for each given , either for all nonnegative integers or there exists a natural number such that

(1) for all ;(2)the sequence is convergent to a fixed point of ;(3) is the unique fixed point of in the set ;(4) for all .

Let be a fixed positive integer. We recall that a mapping having a domain and a codomain that are both closed under addition is called a contractively subadditive mapping if there exists a constant with such that and an expansively superadditive mapping if there exists a constant with such that for all . A mapping is called a homogeneous of degree if for all . Also, if there exists a constant with such that a mapping satisfies for all and positive integer , then we say that is a -contractively subhomogeneous mapping if , and is an expansively superhomogeneous mapping if .

Now, we consider a mapping satisfying the following functional equation [22]:

for all , where is a fixed integer with . Note that the mapping is a solution of the functional equation (1). Eskandni [22] investigated the Hyers-Ulam-Rassias stability of the functional equation (1) in quasi-Banach spaces using the direct method.

In this paper, using some strategies from [15, 18, 22], we investigate new generalized Hyers-Ulam stability for -derivations and Lie -algebra homomorphisms on Lie -algebras associated with the functional equation (1) via the fixed point method.

Throughout this paper, let and be Lie -algebras and . For any mapping , we define

for all ,    ,  ,  and some .

2. Main Results

In this section, we point out the stability of the functional equation (1) on Lie -algebras using a fixed point method. Let us recall that a mapping is called a generalized metric on a nonempty set if (i)   if and only if , (ii)  , and (iii)   for all , and . We need the following lemma to prove the main result in this paper.

Lemma 2 (see [22]). Let be linear spaces and a fixed positive integer. A mapping satisfies the functional equation (1) if and only if is additive.

Theorem 3. Assume that there exist a contractively subadditive mapping and a 2-contractively subhomogeneous mapping with a constant such that a mapping satisfies

for all ,  , , and some , and . Then there exists a unique -derivation which satisfies (1) and the inequality

for all .

Proof. Let be a set of all mappings from into , and introduce a generalized metric on as follows:

Then is a generalized complete metric space. Now, we consider the mapping defined by

for all , , and . Let and be an arbitrary constant with . Then we have and

for all , which means that for all . Thus is a strictly contractive self-mapping on with the Lipschitz constant .

Letting , , and in (3), we have

which gives

for all and with . Then we have . From Theorem 1, there is a mapping which is a unique fixed point of in the set such that

for all , since . Again by Theorem 1, we have

for all . Thus, inequality (5) holds.

It follows from (3), (11), and the contractively subadditive mapping of that

which gives for all and . If we put in ; then satisfies the functional equation (1), and so is additive by Lemma 2. Also, we let and , then . By the same reasoning as that of the proof of Theorem 2.1 of [23], the mapping is -linear. So, it follows from the 2-contractively subhomogeneous of , (4), and (11) that

for all and some , and . Then we have

for all and some , and . Thus, is a unique Lie -derivation on Lie -algebra satisfying (5). This completes the proof.

Corollary 4. Let ,  , and be nonnegative real numbers. Suppose that a mapping satisfies

for all ,  ,  , and some , and . Then there exists a unique -derivation such that

for all and with .

Proof. The proof follows from Theorem 3 by taking ,   for all ,  . Then we can choose , and we obtain the desired result. This completes the proof.

Theorem 5. Assume that there exist an expansively superadditive mapping and a 2-expansively superhomogeneous mapping with a constant such that a mapping satisfies (3) and (4). Then there exists a unique -derivation which satisfies (1) and the inequality

for all .

Proof. Let and be as in the proof of Theorem 3. Then becomes a generalized complete metric space, and we consider the mapping defined by for all and . So, for all . It follows from (9) that

for all and with . Then, . From Theorem 1, there exists a unique mapping which is a unique fixed point of in the set such that for all . Thus, we have

which implies that (18) holds. The remaining assertion goes through in similar method to the corresponding part of Theorem 3. This completes the proof.

Corollary 6. Let ,   and be nonnegative real numbers. Suppose that a mapping satisfies (16). Then there exists a unique -derivation such that

for all and with .

Proof. The proof follows from Theorem 5 by taking ,   for all ,  . Then we can choose , and we obtain the desired result. This completes the proof.

Next, we establish another theorem about the stability of the functional equation (1).

Theorem 7. Assume that there exists a contractively subadditive mapping with a constant such that a mapping satisfies

for all ,  , , and some , and . Then there exists a unique -derivation which satisfies (1) and the inequality

for all .

Proof. Substituting , , and in (22), we obtain

for all and a positive integer . Let and be as in the proof of Theorem 3 such that becomes a generalized complete metric space. Let a mapping defined by for all ,  . Then, we have for all . It follows from (24) that . The remaining assertion goes through in a similar way to the corresponding part of Theorem 3. This completes the proof.

Corollary 8. Assume that there exists an expansively superadditive mapping with a constant such that a mapping satisfies (22). Then there exists a unique -derivation which satisfies (1) and the inequality

for all .

Next, we investigate the Lie -algebra homomorphisms on Lie -algebras associated with the functional equation (1). The results in Theorems 9 and 10 are similar to those in [24].

Theorem 9. Assume that there exist a contractively subadditive mapping and a 2-contractively subhomogeneous mapping with a constant such that a mapping satisfies (3) and

for all and . Then there exists a unique Lie -algebra homomorphism satisfying

for all .

Proof. By the same method as in Theorem 3, we obtain a -linear mapping satisfying (27). The mapping is given by for all . It follows from (25) that

for all . Thus, is a Lie -algebra homomorphism. This completes the proof.

Theorem 10. Assume that there exist an expansively superadditive mapping and a 2-expansively superhomogeneous mapping with a constant such that a mapping satisfies (3) and (26). Then there exists a unique Lie -algebra homomorphism satisfying

for all .

Proof. The proof is similar to the proofs of Theorems 5 and 9.

Corollary 11. Let ,  , and be nonnegative real numbers. Suppose that a mapping satisfies

for all ,  ,  . Then there exists a unique Lie -algebra homomorphism satisfying

for all and with .

Acknowledgments

The authors would like to thank the referee and editors for their comments that helped them improve this paper.

References

L. Abellanas and L. Martínez Alonso, “A general setting for Casimir invariants,” Journal of Mathematical Physics, vol. 16, pp. 1580–1584, 1975.

N. Jacobson, Lie Algebras, Dover Publications, New York, NY, USA, 1979.

D. Rand, P. Winternitz, and H. Zassenhaus, “On the identification of a Lie algebra given by its structure constants. I. Direct decompositions, Levi decompositions, and nilradicals,” Linear Algebra and Its Applications, vol. 109, pp. 197–246, 1988.

R. O. Popovych, V. M. Boyko, M. O. Nesterenko, and M. W. Lutfullin, “Realizations of real low-dimensional Lie algebras,” Journal of Physics A, vol. 36, no. 26, pp. 7337–7360, 2003.

M. de Montigny and J. Patera, “Discrete and continuous graded contractions of Lie algebras and superalgebras,” Journal of Physics A, vol. 24, no. 3, pp. 525–547, 1991.

P. Novotný and J. Hrivnák, “On -derivations of Lie algebras and corresponding invariant functions,” Journal of Geometry and Physics, vol. 58, no. 2, pp. 208–217, 2008.

S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience Publishers, New York, NY, USA, 1960.

S. M. Ulam, Problems in Modern Mathematics, John Wiley & Sons, New York, NY, USA, 1964.

D. H. Hyers, “On 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.

N. Brillouët-Belluot, J. Brzdęk, and K. Ciepliński, “On some recent developments in Ulam's type stability,” Abstract and Applied Analysis, vol. 2012, Article ID 716936, 41 pages, 2012.

T. Aoki, “On the stability of the linear transformation in Banach spaces,” Journal of the Mathematical Society of Japan, vol. 2, pp. 64–66, 1950.

P. Găvruţa, “A 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.

S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, vol. 48 of Springer Optimization and Its Applications, Springer, New York, NY, USA, 2011.

Th. M. Rassias, “On the stability of the linear mapping in Banach spaces,” Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297–300, 1978.

L. C. Cădariu and V. Radu, “Fixed points and the stability of Jensen's functional equations,” Journal of Inequalities in Pure and Applied Mathematics, vol. 4, article 4, 2003.

Y. J. Cho, R. Saadati, and J. Vahidi, “Approximation of homomorphisms and derivations on non-Archimedean Lie -algebras via fixed point method,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 373904, 9 pages, 2012.

J. I. Kang and R. Saadati, “Approximation of homomorphisms and derivations on non-Archimedean random Lie -algebras via fixed point method,” Journal of Inequalities and Applications, vol. 2012, article 251, 2012.

M. E. Gordji and H. Khodaei, “A fixed point technique for investigating the stability of -derivations on Lie -algebras,” Nonlinear Analysis: Theory, Methods & Applications, vol. 76, pp. 52–57, 2013.

K. Ciepliński, “Applications of fixed point theorems to the Hyers-Ulam stability of functional equations–-a survey,” Annals of Functional Analysis, vol. 3, no. 1, pp. 151–164, 2012.

M. E. Gordji, Z. Alizadeh, H. Khodaei, and C. Park, “On approximate homomorphisms: a fixed point approach,” Mathematical Sciences, vol. 6, p. 59, 2012.

J. B. Diaz and B. Margolis, “A fixed point theorem of the alternative, for contractions on a generalized complete metric space,” Bulletin of the American Mathematical Society, vol. 74, pp. 305–309, 1968.

G. Z. Eskandani, “On the Hyers-Ulam-Rassias stability of an additive functional equation in quasi-Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 345, no. 1, pp. 405–409, 2008.

C.-G. Park, “Homomorphisms between Poisson J -algebras,” Bulletin of the Brazilian Mathematical Society, vol. 36, no. 1, pp. 79–97, 2005.

J. Brzdęk and A. Fošner, “Remarks on the stability of Lie homomorphisms,” Journal of Mathematical Analysis and Applications, vol. 400, no. 2, pp. 585–596, 2013.