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

and Applied Analysis 3 Park and Rassias 59 proved the stability of homomorphisms in C∗-algebras and Lie C∗-algebras and also of derivations on C∗-algebras and Lie C∗-algebras for the Jensen-type functional equation μf ( x y 2 ) μf ( x − y 2 ) − fμx 0 1.6 for all μ ∈ T1 : {λ ∈ C; |λ| 1}. In this paper, by using the fixed-point methods, we establish the stability of n-Lie homomorphisms and Jordan n-Lie homomorphisms on n-Lie Banach algebras associated to the following generalized Jensen type functional equation: μf (∑n i 1 xi n ) μ n ∑ j 2 f (∑n i 1,i / j xi − n − 1 xj n ) − fμx1 ) 0 1.7 for all μ ∈ T1/no : {eiθ; 0 ≤ θ ≤ 2π/no} ∪ {1} , where n ≥ 2. Throughout this paper, assume that A, A , B, B are two n-Lie Banach algebras.


Introduction
Let n be a natural number greater or equal to 3. The notion of an n-Lie algebra was introduced by Filippov in 1985 1 . The Lie product is taken between n elements of the algebra instead of two. This new bracket is n-linear, antisymmetric and satisfies a generalization of the Jacobi identity. For n 3 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 n-Lie algebra is a natural generalization of a Lie algebra. Namely, a vector space V together with a multilinear, antisymmetric n-ary operation :Λ n V → V is called an n-Lie algebra, n ≥ 3, if the n-ary bracket is a derivation with respect to itself, that is, where x 1 , x 2 , . . . , x 2n−1 ∈ V . 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 .
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 ε > 0 and f : X → Y is a map with X a normed space, Y a Banach spaces such that for all x, y ∈ X, then there exists a unique additive map T : X → Y such that for all x ∈ X. 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 f x y f x f y 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  Park and Rassias 59 proved the stability of homomorphisms in C * -algebras and Lie C * -algebras and also of derivations on C * -algebras and Lie C * -algebras for the Jensen-type functional equation for all μ ∈ T 1 : {λ ∈ C; |λ| 1}. In this paper, by using the fixed-point methods, we establish the stability of n-Lie homomorphisms and Jordan n-Lie homomorphisms on n-Lie Banach algebras associated to the following generalized Jensen type functional equation: where n ≥ 2. Throughout this paper, assume that A, A , B, B are two n-Lie Banach algebras.

Main Results
Before proceeding to the main results, we recall a fundamental result in fixed point theory. ii the sequence {T m x} is convergent to a fixed point y * of T ; iii y * is the unique fixed point of T in Λ {y ∈ Ω : d T m 0 x, y < ∞}; iv d y, y * ≤ 1/ 1 − L d y, Ty for all y ∈ Λ.
We start our work with the main theorem of the our paper.
Proof. Let Ω be the set of all functions from A into B and let It is easy to show that Ω, d is a generalized complete metric space 61 . Now we define the mapping J : Ω → Ω by J h x 1/n h nx for all x ∈ A. Note that for all g, h ∈ Ω,

2.7
Hence we see that  Hence for all x 1 , . . . , x n ∈ A.
On the other hand, it follows from 2.2 , 2.9 , and 2.15 that for all μ ∈ T 1 and all x 1 , . . . , x n ∈ A and

2.27
for all x 1 , . . . , x n ∈ A. Then there exists a unique n-Lie homomorphism H : Proof. Put φ x 1 , x 2 , . . . , x n : θ n i 1 x i p A for all x 1 , . . . , x n ∈ A in Theorem 2.2. Then 2.9 holds for p < 1, and 2.28 holds when L 2 p−1 . for all x ∈ A.