# Approximaten-Lie Homomorphisms and Jordan n-Lie Homomorphisms on n-Lie Algebras

Authors: M. Eshaghi Gordji; G. H. Kim
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

Using fixed point methods, we establish the stability ofn-Lie homomorphisms and Jordan n-Lie homomorphisms on n-Lie algebras associated to the following generalized Jensen functional equation μf(∑i=1nxi/n)+μ∑j=2nf(∑i=1,i≠jnxi-(n-1)xj/n)=f(μx1)(n≥2).


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## Body

## 1. Introduction

Letn 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.Ann-Lie algebra is a natural generalization of a Lie algebra. Namely, a vector space V together with a multilinear, antisymmetric n-ary operation []:ΛnV→V is called an n-Lie algebra, n≥3, if the n-ary bracket is a derivation with respect to itself, that is,(1.1)[[x1,…,xn],xn+1,…,x2n-1]=∑i=1n[x1,…,xi-1[xi,xn+1,…,x2n-1],…,xn],
where x1,x2,…,x2n-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).In [1] and several subsequent papers, [3–5] a structure theory of finite-dimensional n-Lie algebras over a field 𝔽 of characteristic 0 was developed.n-ary algebras have been considered in physics in the context of Nambu mechanics [2, 6] and, recently (for n=3), in the search for the effective action of coincident M2-branes in M-theory initiated by the Bagger-Lambert-Gustavsson (BLG) model [7, 8] (further references on the physical applications of n-ary algebras are given in [9]).From now on, we only considern-Lie algebras over the field of complex numbers. An n-Lie algebra A is a normed n-Lie algebra if there exists a norm ∥∥ on A such that ∥[x1,x2,…,xn]∥≤∥x1∥∥x2∥⋯∥xn∥ for all x1,x2,…,xn∈A. A normed n-Lie algebra A is called a Banach n-Lie algebra, if (A,∥∥) is a Banach space.Let(A,[]A) and (B,[]B) be two Banach n-Lie algebras. A ℂ-linear mapping H:(A,[]A)→(B,[]B) is called an n-Lie homomorphism if(1.2)H([x1x2⋯xn]A)=[H(x1)H(x2)⋯H(xn)]B
for all x1,x2,…,xn∈A. A ℂ-linear mapping H:(A,[]A)→(B,[]B) is called a Jordan n-Lie homomorphism if(1.3)H([xx⋯x]A)=[H(x)H(x)⋯H(x)]B
for all x∈A.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(1.4)‖f(x+y)-f(x)-f(y)‖≤ε
for all x,y∈X, then there exists a unique additive map T:X→Y such that(1.5)‖f(x)-T(x)‖≤ε
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 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 C*-algebras and Lie C*-algebras and also of derivations on C*-algebras and Lie C*-algebras for the Jensen-type functional equation(1.6)μf(x+y2)+μf(x-y2)-f(μx)=0
for all μ∈𝕋1:={λ∈ℂ;|λ|=1}.In this paper, by using the fixed-point methods, we establish the stability ofn-Lie homomorphisms and Jordan n-Lie homomorphisms on n-Lie Banach algebras associated to the following generalized Jensen type functional equation:(1.7)μf(∑i=1nxin)+μ∑j=2nf(∑i=1,i≠jnxi-(n-1)xjn)-f(μx1)=0
for all μ∈(𝕋1/no1:={eiθ;0≤θ≤2π/no}∪{1}), where n≥2.Throughout this paper, assume that(A,[]A),(B,[]B) are two n-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(Ω,d) be a complete generalized metric space, and let T:Ω→Ω be a strictly contractive function with Lipschitz constant L. Then for each given x∈Ω, either
(2.1)d(Tmx,Tm+1x)=∞∀m≥0,
or other exists a natural number m0 such that(i)
d(Tmx,Tm+1x)<∞ for all m≥m0;(ii)
the sequence{Tmx} is convergent to a fixed point y* of T;(iii)
y* is the unique fixed point of T in Λ={y∈Ω:d(Tm0x,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.Theorem 2.2.
Letn0∈ℕ be a fixed positive integer number. Let f:A→B be a function for which there exists a function ϕ:An→[0,∞) such that
(2.2)‖μf(∑i=1nxin)+μ∑j=2nf(∑i=1,i≠jnxi-(n-1)xjn)-f(μx1)‖B≤ϕ(x1,x2,…,xn)
for all μ∈(T1/no1:={eiθ;0≤θ≤2π/no}∪{1}) and all x1,…,xn∈A, and that
(2.3)‖f([x1x2⋯xn]A)-[f(x1)f(x2)⋯f(xn)]B‖B≤ϕ(x1,x2,…,xn)
for all x1,…,xn∈A. If there exists an L<1 such that
(2.4)ϕ(x1,x2,…,xn)≤nLϕ(x1n,x2n,…,xnn)
for all x1,…,xn∈A, then there exists a unique n-Lie homomorphism H:A→B such that
(2.5)‖f(x)-H(x)‖≤L1-Lϕ(x,0,0,…,0)
for all x∈A.Proof.
LetΩ be the set of all functions from A into B and let
(2.6)d(g,h):=inf⁡{C∈R+:‖g(x)-h(x)‖B≤Cϕ(x,0,…,0),∀x∈A}.
It is easy to show that (Ω,d) is a generalized complete metric space [61].
Now we define the mappingJ:Ω→Ω by J(h)(x)=(1/n)h(nx) for all x∈A.
Note that for allg,h∈Ω,
(2.7)d(g,h)<C⟹‖g(x)-h(x)‖≤Cϕ(x,0,…,0),∀x∈A,⟹‖1ng(nx)-1nh(nx)‖≤1|n|lCϕ(nx,0,…,0),∀x∈A,⟹‖1ng(nx)-1nh(nx)‖≤LCϕ(x,0,…,0),∀x∈A,⟹d(J(g),J(h))≤LC.
Hence we see that
(2.8)d(J(g),J(h))≤Ld(g,h)
for all g,h∈Ω. It follows from (2.4) that
(2.9)lim⁡m→∞1nmϕ(nmx1,nmx2,…,nmxn)=0
for all x1,…,xn∈A. Putting μ=1,x1=x, and xj=0 (j=2,…,n) in (2.2), we obtain
(2.10)‖nf(xn)-f(x)‖B≤ϕ(x,0,…,0)
for all x∈A. Thus by using (2.4), we obtain that
(2.11)‖1nf(nx)-f(x)‖B≤1nϕ(nx,0,…,0)≤Lϕ(x,0,…,0)
for all x∈A, that is,
(2.12)d(f,J(f))≤L<∞.
By Theorem 2.1, J has a unique fixed point in the set X1:={h∈Ω:d(f,h)<∞}. Let H be the fixed point of J. H is the unique mapping with
(2.13)H(nx)=nH(x)
for all x∈A, such that there exists C∈(0,∞) satisfying
(2.14)‖f(x)-H(x)‖B≤Cϕ(x,0,…,0)
for all x∈A. On the other hand we have limm→∞d(Jm(f),H)=0, so
(2.15)lim⁡m→∞1nmf(nmx)=H(x)
for all x∈A. Also by Theorem 2.1, we have
(2.16)d(f,H)≤11-Ld(f,J(f)).
It follows from (2.12) and (2.16) that
(2.17)d(f,H)≤L1-L.
This implies the inequality (2.5). By (2.21), we have
(2.18)‖H([x1x2⋯xn]A)-[H(x1)H(x2)H(x3)⋯H(xn)]B‖B=lim⁡m→∞‖1nnmH([nmx1nmx2⋯nmxn]A)-1nnm([H(nmx1)H(nmx2)H(nmx3)⋯H(nmxn)]B)‖≤lim⁡m→∞1nnmϕ(nmx1,nmx2,…,nmxn)=0
for all x1,…,xn∈A. Hence
(2.19)H([x1x2⋯xn]A)=[H(x1)H(x2)H(x3)⋯H(xn)]B
for all x1,…,xn∈A.
On the other hand, it follows from (2.2), (2.9), and (2.15) that
(2.20)‖H(∑i=1nxin)+∑j=2nH(∑i=1,i≠jnxi-(n-1)xjn)-H(x1)‖B=lim⁡m→∞1nm‖f(nm-1∑i=1nxi)+∑j=2n(f(nm-1(∑i=1,i≠jnxi-(n-1)xj)))-f(nmx1)‖B≤lim⁡m→∞1nmϕ(nmx1,nmx2,…,nmxn)=0
for all x1,…,xn∈A. Then
(2.21)H(∑i=1nxin)+∑j=2nH(∑i=1,i≠jnxi-(n-1)xjn)=H(x1)
for all x1,…,xn∈A. Putting s1=∑i=1nxi/n and sj=∑i=1,i≠jnxi-(n-1)xj/n (j=2,3,…,n) in (2.21), we obtain
(2.22)H(∑j=1nsj)=∑j=1nH(sj)
for all s1,…,sn∈A. Setting sj=0(j=3,4,…,n) in (2.22) to get
(2.23)H(s1+s2)=H(s1)+H(s2)
hence H is cauchy additive. Letting xi=x for all i=1,2,…,n in (2.2), we obtain
(2.24)‖μf(x)-f(μx)‖B≤ϕ(x,x,…,x)
for all x∈A. It follows that
(2.25)‖H(μx)-μH(x)‖=lim⁡m→∞1nm‖f(μnmx)-μf(nmx)‖B≤lim⁡m→∞1nmϕ(nmx,nmx,…,nmx)=0
for all μ∈𝕋1/no1, and all x∈A. One can show that the mapping H:A→B is ℂ-linear. Hence, H:A→B is an n-Lie homomorphism satisfying (2.5), as desired.Corollary 2.3.
Letθ and p be nonnegative real numbers such that p<1. Suppose that a function f:A→B satisfies
(2.26)‖μf(∑i=1nxin)+μ∑j=2nf(∑i=1,i≠jnxi-(n-1)xjn)-f(μx1)‖B≤θ∑i=1n(‖xi‖Ap)
for all μ∈𝕋1 and all x1,…,xn∈A and
(2.27)‖f([x1x2⋯xn]A)-[f(x1)f(x2)⋯f(xn)]B‖B≤θ∑i=1n(‖xi‖Ap)
for all x1,…,xn∈A. Then there exists a unique n-Lie homomorphism H:A→B such that
(2.28)‖f(x)-H(x)‖B≤2pl(2-2p)θ‖x‖Ap
for all x∈A.Proof.
Putϕ(x1,x2,…,xn):=θ∑i=1n(∥xi∥Ap) for all x1,…,xn∈A in Theorem 2.2. Then (2.9) holds for p<1, and (2.28) holds when L=2(p-1).Theorem 2.4.
Letn0∈ℕ be a fixed positive integer number. Let f:A→B be a function for which there exists a function ϕ:An→[0,∞) such that
(2.29)‖μf(∑i=1nxin)+μ∑j=2nf(∑i=1,i≠jnxi-(n-1)xjn)-f(μx1)‖B≤ϕ(x1,x2,…,xn)
for all μ∈(𝕋1/no1:={eiθ;0≤θ≤2π/no}∪{1}) and all x1,…,xn∈A, and that
(2.30)‖f([xx⋯x]A)-[f(x)f(x)⋯f(x)]B‖B≤ϕ(x,x,…,x)
for all x∈A. If there exists an L<1 such that
(2.31)ϕ(x1,x2,…,xn)≤nLϕ(x1n,x2n,…,xnn)
for all x1,…,xn∈A, then there exists a unique Jordan n-Lie homomorphism H:A→B such that
(2.32)‖f(x)-H(x)‖≤L1-Lϕ(x,0,0,…,0)
for all x∈A.Proof.
By the same reasoning as the proof of Theorem2.2, we can define the mapping
(2.33)H(x)=lim⁡m→∞1nmlf(nmlx)
for all x∈A. Moreover, we can show that H is ℂ-linear. The inequality (2.30) follows that
(2.34)‖H([xx⋯x]A)-[H(x)H(x)⋯H(x)]B‖B=lim⁡m→∞‖1nnmH([nmx⋯nmx]A)-1nnm([H(nmx)H(nmx)⋯H(nmx)]B)‖B≤lim⁡m→∞1nnmϕ(nmx,nmx,…,nmx)=0
for all x∈A. So
(2.35)H([xx⋯x]A)=[H(x)H(x)⋯H(x)]B
for all x∈A. Hence H:A→B is a Jordan n-Lie homomorphism satisfying (2.32).Corollary 2.5.
Letθ and p be nonnegative real numbers such that p<1. Suppose that a function f:A→B satisfies
(2.36)‖μf(∑i=1nxin)+μ∑j=2nf(∑i=1,i≠jnxi-(n-1)xjn)-f(μx1)‖B≤θ∑i=1n(‖xi‖Ap)
for all μ∈𝕋1 and all x1,…,xn∈A and
(2.37)‖f([xx⋯x]A)-[f(x)f(x)⋯f(x)]B‖B≤nθ(‖x‖Ap)
for all x∈A. Then there exists a unique Jordan n-Lie homomorphism H:A→B such that
(2.38)‖f(x)-H(x)‖B≤2pl(2-2p)θ‖x‖Ap
for all x∈A.Proof.
It follows by Theorem2.4 by putting ϕ(x1,x2,…,xn):=θ∑i=1n(∥xi∥Ap) for all x1,…,xn∈A and L=2(p-1).



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*Source: 279632-2012-03-15.xml*