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, [3–5] 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 [13–32].
In 1982–1994, Rassias (see [26–28]) 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 [33–57].
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 .