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Abstract and Applied Analysis
Volume 2012, Article ID 279632, 11 pages
http://dx.doi.org/10.1155/2012/279632
Research Article

Approximate -Lie Homomorphisms and Jordan -Lie Homomorphisms on -Lie Algebras

1Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran
2Department of Mathematics, Kangnam University, Yongin, Gyeonggi 446-702, Republic of Korea

Received 18 October 2011; Accepted 5 January 2012

Academic Editor: Marcia Federson

Copyright © 2012 M. Eshaghi Gordji and G. H. Kim. 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

Using fixed point methods, we establish the stability of -Lie homomorphisms and Jordan -Lie homomorphisms on -Lie algebras associated to the following generalized Jensen functional equation .

1. Introduction

Let be a natural number greater or equal to 3. The notion of an -Lie algebra was introduced by Filippov in 1985 [1]. The Lie product is taken between elements of the algebra instead of two. This new bracket is -linear, antisymmetric and satisfies a generalization of the Jacobi identity. For this product is a special case of the Nambu bracket, well known in physics, which was introduced by Nambu [2] in 1973, as a generalization of the Poisson bracket in Hamiltonian mechanics.

An -Lie algebra is a natural generalization of a Lie algebra. Namely, a vector space together with a multilinear, antisymmetric -ary operation is called an -Lie algebra, , if the -ary bracket is a derivation with respect to itself, that is, where . Equation (1.1) is called the generalized Jacobi identity. The meaning of this identity is similar to that of the usual Jacobi identity for a Lie algebra (which is a 2-Lie algebra).

In [1] and several subsequent papers, [35] a structure theory of finite-dimensional -Lie algebras over a field of characteristic 0 was developed.

-ary algebras have been considered in physics in the context of Nambu mechanics [2, 6] and, recently (for ), in the search for the effective action of coincident -branes in -theory initiated by the Bagger-Lambert-Gustavsson (BLG) model [7, 8] (further references on the physical applications of -ary algebras are given in [9]).

From now on, we only consider -Lie algebras over the field of complex numbers. An -Lie algebra is a normed -Lie algebra if there exists a norm on such that for all . A normed -Lie algebra is called a Banach -Lie algebra, if is a Banach space.

Let and be two Banach -Lie algebras. A -linear mapping is called an -Lie homomorphism if for all . A -linear mapping is called a Jordan -Lie homomorphism if for all .

The study of stability problems had been formulated by Ulam [10] during a talk in 1940. Under what condition does there exist a homomorphism near an approximate homomorphism? In the following year, Hyers [11] answered affirmatively the question of Ulam for Banach spaces, which states that if and is a map with a normed space, a Banach spaces such that for all , then there exists a unique additive map such that for all . A generalized version of the theorem of Hyers for approximately linear mappings was presented by Rassias [12] in 1978 by considering the case when inequality (1.4) is unbounded. Due to that fact, the additive functional equation is said to have the generalized Hyers-Ulam-Rassias stability property. A large list of references concerning the stability of functional equations can be found in [1332].

In 1982–1994, Rassias (see [2628]) solved the Ulam problem for different mappings and for many Euler-Lagrange type quadratic mappings, by involving a product of different powers of norms. In addition, Rassias considered the mixed product sum of powers of norms control function. For more details see [3357].

In 2003 Cădariu and Radu applied the fixed-point method to the investigation of the Jensen functional equation [58]. They could present a short and a simple proof (different of the “direct method”, initiated by Hyers in 1941) for the generalized Hyers-Ulam stability of Jensen functional equation [58] and for quadratic functional equation.

Park and Rassias [59] proved the stability of homomorphisms in -algebras and Lie -algebras and also of derivations on -algebras and Lie -algebras for the Jensen-type functional equation for all .

In this paper, by using the fixed-point methods, we establish the stability of -Lie homomorphisms and Jordan -Lie homomorphisms on -Lie Banach algebras associated to the following generalized Jensen type functional equation: for all , where .

Throughout this paper, assume that are two -Lie Banach algebras.

2. Main Results

Before proceeding to the main results, we recall a fundamental result in fixed point theory.

Theorem 2.1 (see [60]). Let be a complete generalized metric space, and let be a strictly contractive function with Lipschitz constant . Then for each given , either or other exists a natural number such that(i) for all ;(ii)the sequence is convergent to a fixed point of ;(iii) is the unique fixed point of in ;(iv) for all .

We start our work with the main theorem of the our paper.

Theorem 2.2. Let be a fixed positive integer number. Let be a function for which there exists a function such that for all and all , and that for all . If there exists an such that for all , then there exists a unique -Lie homomorphism such that for all .

Proof. Let be the set of all functions from into and let It is easy to show that is a generalized complete metric space [61].
Now we define the mapping by for all .
Note that for all , Hence we see that for all . It follows from (2.4) that for all . Putting , and () in (2.2), we obtain for all . Thus by using (2.4), we obtain that for all , that is, By Theorem 2.1, has a unique fixed point in the set . Let be the fixed point of . is the unique mapping with for all , such that there exists satisfying for all . On the other hand we have , so for all . Also by Theorem 2.1, we have It follows from (2.12) and (2.16) that This implies the inequality (2.5). By (2.21), we have for all . Hence for all .
On the other hand, it follows from (2.2), (2.9), and (2.15) that for all . Then for all . Putting and () in (2.21), we obtain for all . Setting in (2.22) to get hence is cauchy additive. Letting for all in (2.2), we obtain for all . It follows that for all , and all . One can show that the mapping is -linear. Hence, is an -Lie homomorphism satisfying (2.5), as desired.

Corollary 2.3. Let and be nonnegative real numbers such that . Suppose that a function satisfies for all and all and for all . Then there exists a unique -Lie homomorphism such that for all .

Proof. Put for all in Theorem 2.2. Then (2.9) holds for , and (2.28) holds when .

Theorem 2.4. Let be a fixed positive integer number. Let be a function for which there exists a function such that for all and all , and that for all . If there exists an such that for all , then there exists a unique Jordan -Lie homomorphism such that for all .

Proof. By the same reasoning as the proof of Theorem 2.2, we can define the mapping for all . Moreover, we can show that is -linear. The inequality (2.30) follows that for all . So for all . Hence is a Jordan -Lie homomorphism satisfying (2.32).

Corollary 2.5. Let and be nonnegative real numbers such that . Suppose that a function satisfies for all and all and for all . Then there exists a unique Jordan -Lie homomorphism such that for all .

Proof. It follows by Theorem 2.4 by putting for all and .

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