Approximate m -Lie homomorphisms and approximate Jordan m -Lie homomorphisms associated to a parametric additive functional equation

R E S E A R C H Open Access

Approximate m -Lie homomorphisms and approximate Jordan m -Lie homomorphisms associated to a parametric additive functional equation

Hassan Azadi Kenary¹, Hamid Rezaei², Madjid Eshaghi Gordji³, Choonkil Park^4* and Sang Og Kim⁵

* Correspondence: baak@hanyang.

ac.kr

4Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea

Full list of author information is available at the end of the article

Abstract

Using fixed point method, we establish the Hyers-Ulam stability ofm-Lie

homomorphisms and Jordanm-Lie homomorphisms inm-Lie algebras associated to the following generalized Jensen functional equation

m i=1

μf(xi) = 1 2m

m i=1

f

μmxi+ m j=1,i=jxj

+f

_m

i=1

μxi

for a fixed positive integermwithm≥2.

Mathematics Subject Classification (2010): Primary 17A42, 39B82, 39B52.

Keywords:m-Lie algebra, homomorphism, Jordan homomorphism, Hyers-Ulam stability, fixed point approach, Jensen-type functional equation

1. Introduction

Letnbe a natural number greater or equal to 3. The notion of an n-Lie algebra was introduced by V.T. Filippov in 1985 [1]. The Lie product is taken betweennelements of the algebra instead of two. This new bracket isn-linear, anti-symmetric and satisfies a generalization of the Jacobi identity. Forn = 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 spaceVtogether with a multi-linear, antisymmetricn-ary operation [ ]:ΛⁿV®V is called ann-Lie algebra,n≥3, if then-ary bracket is a derivation with respect to itself, i.e.,

[[x₁, · · · , x_n],x_n+1,· · · , x_2n−1] =ⁿ

i=1

[x₁, · · ·, x_i−1[x_i, x_n+1,· · · , x_2n−1], · · ·, x_n] (1:1) wherex1,x2,· · ·,x2n-1ÎV. The equation (1.1) is called the generalized Jacobi iden- tity. 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.

© 2012 Kenary et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 inM-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 algebraAis a normed n-Lie algebra if there exists a norm || || onA such that ||[x1, x2,· · · ,xn]||≤||x1||||x2|| · · · ||xn|| for all x1,x2,· · · ,xnÎ A. A normed n-Lie algebra Ais called a Banachn-Lie algebra if (A, || ||) is a Banach space.

Let (A, [ ]A) and (B, [ ]B) be two Banach n-Lie algebras. A ℂ-linear mappingH: (A, [ ]A)® (B, [ ]B) is called ann-Lie homomorphism if

H([x1x2· · ·xn]A) = [H(x1)H(x2)· · ·H(xn)]B

for all x1,x2,· · ·,xnÎ A. Aℂ-linear mappingH: (A, [ ]A)®(B, [ ]B) is called a Jor- dan n-Lie homomorphism if

H([xx· · ·x]A) = [H(x)H(x)· · ·H(x)]B

for all xÎA.

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] was answered affirmatively the ques- tion of Ulam for Banach spaces, which states that if ε> 0 andf:X ® Yis a mapping withXa normed space and Ya Banach spaces such that

f(x+y)−f(x)−f(y) ≤ ε (1:2) for all x,yÎX, then there exists a unique additive mapT: X®Ysuch that

f(x)−T(x) ≤ ε

for allxÎX. A generalized version of the theorem of Hyers for approximately linear mappings was presented by Rassias [12] in 1978 by considering the case when inequal- ity (1.2) is unbounded.

In 2003, Cădariu and Radu applied the fixed point method to the investigation of the Jensen functional equation [13] (see also [14-16]). They could present a short and a simple proof (different of the “direct method “, initiated by Hyers in 1941) for the Hyers-Ulam stability of Jensen functional equation [13] and for quadratic functional equation [14].

Park and Rassias [17] proved the stability of homomorphisms inC*-algebras and LieC*- algebras and also of derivations onC*-algebras and LieC*-algebras for the Jensen-type functional equation

μfx+y 2

+μf

x−y 2

−f(μx) = 0 for all μ∈T¹:={λ∈C : |λ|= 1}.

In this paper, by using fixed point method, we establish the Hyers-Ulam stability of n-Lie homomorphisms and Jordan n-Lie homomorphisms inn-Lie Banach algebras associated to the following generalized Jensen-type functional equation

m i=1

μf(xi)− 1 2m

m i=1

f

μmxi+ m j=1,i=jμxj

+f

_m

i=1

μxi

= 0

for all μ∈T¹₁

no

:=

e^iθ: 0≤θ ≤ 2π n_o

∪ {1}, wherem≥2.

Throughout this paper, assume that (A, [ ]A) and (B, [ ]B) are two m-Lie Banach algebras.

2. Main results

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

Theorem 2.1. [18]Let(Ω,d)be a complete generalized metric space and T: Ω®Ω be a strictly contractive function with Lipschitz constant L. Then for each given x ÎΩ, either

d(T^mx, T^m+1x) =∞for all m≥0,

or other exists a natural number m0such that

•d(T^mx,T ^m+1x) <∞for all m≥ m0;

•the sequence{T ^mx}is convergent to a fixed point y*of T;

•y*is the unique fixed point of T inΛ= {yÎΩ:d(T^m0x, y) <∞};

• d(y, y^∗)≤ _1−L¹ d(y, Ty)for all yÎΛ.

Theorem 2.2.Let V and W be real vector spaces. A mapping f: V® W satisfies the following functional equation

m i=1

f(xi) = 1 2m

m i=1

f

mxi+ m j=1,i=j

xj

+f

_m

i=1

xi

if and only f f is additive.

Proof. It is easy to prove the theorem.□

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

Theorem 2.3.Let n0Î Nbe a fixed positive integer.Let f: A ®B be a mapping for which there exists a functionj: A^m®[0,∞)such that

μ

m i=1

μf(xi)− 1 2m

⎡

⎣^m

i=1

f

⎛

⎝μmxi+ m j=1, i=j

μxj

⎞

⎠+f _m

i=1

μxi

⎤

⎦

≤ϕ(x1, x2, · · ·, xm),

(2:1)

f([x1x2· · ·xn]A)−[f(x1)f(x2)· · ·f(xm)]B ≤ ϕ(x1, x2, · · ·, xm) (2:2)

for all μ∈T¹₁

n0

and all x1,· · ·,xmÎA. If there exists an L <1such that ϕ(x1, x2, · · ·, xm)≤mLϕx1

m,x2

m,· · ·,xm

m

(2:3)

for all x1,· · · , xmÎ A,then there exists a unique m-Lie homomorphism H: A ® B such that

||f(x)−H(x)|| ≤ ϕ(x, 0, 0,· · ·, 0)

m−mL (2:4)

for all xÎ A.

Proof. LetΩbe the set of all functions fromAintoBand let

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 [19].

Now we define the mappingJ: Ω®Ωby J(h)(x) = 1

mh(mx) for all xÎA.

Note that for allg,hÎ Ω,

d(g, h)<C⇒ g(x)−h(x) ≤Cφ(x, 0, · · ·, 0)

⇒ 1

mg(mx)− 1

mh(mx)

≤ Cϕ(mx, 0,· · ·, 0)

|m|

⇒ 1

mg(mx)− 1

mh(mx)

≤ L Cϕ(x, 0, · · ·, 0)

⇒d(J(g), J(h))≤L C for all xÎA. Hence we see that

d(J(g), J(h))≤Ld(g, h)

for all g,hÎΩ. It follows from (2.3) that

klim→∞

ϕ(m^kx1,m^kx2,· · ·,m^kxm)

m^k ≤ lim

k→∞L^kϕ(x1, · · ·, xm) = 0 (2:5) for allx1, · · ·, xmÎ A. Putting μ= 1,x1 =xandxj = 0 (j= 2,· · · ,n) in (2.1), we obtain

f(mx)

m −f(x)

≤ ϕ(x, 0,· · ·, 0) m

for all xÎA. Therefore, d(f, J(f))≤ 1

m <∞. (2:6)

By Theorem 2.1, Jhas a unique fixed point in the set X1: = {hÎ Ω:d(f, h) <∞}. Let Hbe the fixed point ofJ. H is the unique mapping with

H(mx) =mH(x)

such that there exists CÎ (0,∞) satisfying f(x)−H(x) ≤Cϕ(x, 0, · · ·, 0)

for all xÎA. On the other hand, we have lim_k®∞d(J^k(f),H) = 0 and so

k→∞lim 1

m^kf(m^kx) =H(x) (2:7)

for all xÎA. By Theorem 2.1, we have d(f, H)≤ 1

1−Ld(f, J(f)). (2:8)

It follows from (2.6) and (2.8) that d(f, H)≤ 1

m−mL.

This implies the inequality (2.4). By (2.2), we have H([x1x2· · ·xm]A)−[H(x1)H(x2)H(x3)· · ·H(xm)]B

= lim

k→∞

H([m^kx1m^kx2· · ·m^kxm]_A)

m^mk −([H(m^kx1)H(m^kx2)H(m^kx3)· · ·H(m^kxm)]_B) m^mk

≤ lim

m→∞

ϕ(m^kx1,m^kx2,· · ·,m^kxm)

m^mk = 0

for all x1,· · · ,xmÎA. Hence

H([x1x2· · ·xm]A) = [H(x1)H(x2)H(x3)· · ·H(xm)]B

for all x1,· · · ,xmÎA.

On the other hand, it follows from (2.1), (2.5) and (2.7) that

m

i=1

H(x_i)− 1 2m

⎡

⎣^m

i=1

H

⎛

⎝mx_i+ m j=1,i=j

x_j

⎞

⎠+H _m

i=1

x_i

⎤

⎦

B

= lim

k→∞

1 m^k

m i=1

f(m^kx_i)− 1 2m

⎡

⎣^m

i=1

f

⎛

⎝m^k+1x_i+ m j=1,i=j

m^kx_j

⎞

⎠+f _m

i=1

m^kx_i

⎤

⎦

≤ lim

m→∞

ϕ(m^kx₁,m^kx₂,· · ·,m^kx_m)

m^k = 0

for all x1,· · · ,xmÎA. Then m

i=1

H(xi) = 1 2m

m i=1

H

mxi+ m j=1, i=j

xj

+H

_m

i=1

xi

for allx1,· · ·,xmÎ A. So by Theorem 2.1,His additive. Lettingxi=xfor alli= 1, 2,· · ·,nin (2.1), we obtain

μf(x)−f(μx) ≤ϕ(x, x, · · ·, x)

for all xÎA. It follows that H(μx)−μH(x) = lim

k→∞

f(μm^kx)−μf(m^kx) m^k

≤ lim

k→∞

ϕ(m^kx,m^kx,· · ·,m^kx)

m^k = 0

for all μ∈T¹1 n0

and all x Î A. One can show that the mapping H: A ® B is ℂ -linear.

HenceH:A®Bis anm-Lie homomorphism satisfying (2.4), as desired.□

Corollary 2.4.Letθand p be nonnegative real numbers such that p <1.Suppose that a mapping f: A ®B satisfies

μ

m i=1

μf(xi)− 1 2m

⎡

⎣^m

i=1

f

⎛

⎝μmxi+ m j=1,i=j

μxj

⎞

⎠+f _m

i=1

μxi

⎤

⎦ ≤θ

m i=1

( xi p), (2:9)

f([x1x2· · ·xn]A)−[f(x1)f(x2)· · ·f(xm)]B ≤θ m

i=1

( xi p) (2:10)

for all μ∈T¹₁

n0

and all x1, · · · , xm Î A. Then there exists a unique m-Lie homo- morphism H: A ®B such that

f(x)−H(x) ≤ θ x ^p

(m−m^p) (2:11)

for all xÎA.

Proof. Putting ϕ(x1, x2, · · ·, xm) :=θm

i=1(||xi||^p) for allx1, · · ·,xnÎ Aand let- tingL=m^p-1in Theorem 2.3, we obtain (2.11). □

Similarly, we have the following and we will omit the proof.

Theorem 2.5.Let f:A® B be a mapping for which there exists a function:A^m® [0, ∞)satisfying (2.1) and (2.2). If there exists an L <1such that

ϕx1

m, x2

m, · · ·, xm

m ≤ L

mϕ(x1, x2, · · ·, xm)

for all x1,· · · , xmÎ A,then there exists a unique m-Lie homomorphism H: A ® B such that

f(x)−H(x) ≤ Lϕ(x, 0, 0,· · ·, 0) m−mL for all xÎA.

Corollary 2.6.Let θ and p be nonnegative real numbers such that p > 1. Suppose that a mapping f: A ® B satisfies (2.9) and (2.10). Then there exists a unique m-Lie homomorphism H: A®B such that

f(x)−H(x) ≤ mθ x ^p

m^p+1−m² (2:12)

for all xÎA.

Proof. Putting ϕ(x₁, x₂, · · ·, x_m) :=θm

i=1(||x_i||^p) for allx1, · · ·,xnÎ Aand let- tingL=m^1-pin Theorem 2.5, we obtain (2.12). □

Theorem 2.7.Let n0Î Nbe a fixed positive integer.Let f: A ®B be a mapping for which there exists a function: Aⁿ®[0,∞)such that

μ

m i=1

μf(xi)− 1 2m

⎡

⎣^m

i=1

f

⎛

⎝μmxi+ m j=1,i=j

μxj

⎞

⎠+f _m

i=1

μxi

⎤

⎦

≤ϕ(x1, x2, · · ·, xm),

(2:13)

f([xx· · ·x]A)−[f(x)f(x)· · ·f(x)]B ≤ϕ(x, x, · · · , x) (2:14) for all μ∈T¹1

n0

and all x1,· · ·,xmÎA. If there exists an L <1such that ϕ(x1, x2, · · ·, xm)≤mLϕx1

m, x2

m, · · ·, xm

m

for all x1, · · ·,xmÎ A,then there exists a unique Jordan m-Lie homomorphism H: A

® B such that

f(x)−H(x) ≤ ϕ(x, 0,· · ·, 0)

m−mL (2:15)

for all xÎA.

Proof. By the same reasoning as in the proof of Theorem 2.3, we can define the map- ping

H(x) = lim

k→∞

1 m^kf(m^kx)

for all xÎA. Moreover, we can show thatHisℂ-linear. By (2.14), we get that H([xx· · ·x]A)−[H(x)H(x)· · ·H(x)]B

= lim

k→∞

1

m^mkH([m^kx· · ·m^kx]_A)− 1

m^mk([H(m^kx)H(m^kx)· · ·H(m^kx)]_B

≤ lim

k→∞

1

m^mkϕ(m^kx, m^kx, . . ., m^kx) = 0 for all xÎA. So

H([xx· · ·x]A) = [H(x)H(x)· · ·H(x)]B

for all xÎA. HenceH:A®Bis a Jordanm-Lie homomorphism satisfying (2.15).□ Corollary 2.8.Letθand p be nonnegative real numbers such that p <1.Suppose that a mapping f: A ®B satisfies

μ

m i=1

μf(xi)− 1 2m

⎡

⎣^m

i=1

f

⎛

⎝μmxi+ m j=1,i=j

μxj

⎞

⎠+f _m

i=1

μxi

⎤

⎦ ≤θ

n i=1

( xi p), (2:16)

f([xx· · ·x]A)−[f(x)f(x)· · ·f(x)]B ≤nθ( x ^p) (2:17)

for all μ∈T¹₁

n0

and all x1, · · · , xm Î A. Then there exists a unique Jordan m-Lie homomorphism H: A®B such that

f(x)−H(x) ≤ θ x ^p m−m^p

for all xÎA.

Proof. The proof follows from Theorem 2.7 by putting ϕ(x1, x2,· · ·, xm) :=θm i=1(||xi||^p) for allx1,· · ·,xmÎAand lettingL=m^p-1.□

Similarly, we have the following and we will omit the proof.

Theorem 2.9.Let f:A® B be a mapping for which there exists a function:A^m® [0, ∞)satisfying (2.13) and (2.14). If there exists an L <1such that

ϕx1

m, x2

m, · · ·, xm

m ≤ L

mϕ(x1, x2, · · ·, xm)

for all x1,· · ·,xmÎA,then there exists a unique Jordan m-Lie homomorphism H: A®B such that

f(x)−H(x) ≤ Lϕ(x, 0, 0,· · ·, 0) m−mL for all xÎA.

Corollary 2.10. Letθ and p be nonnegative real numbers such that p> 1.Suppose that a mapping f: A® B satisfies (2.16) and (2.17). Then there exists a unique Jordan m-Lie homomorphism H: A®B such that

f(x)−H(x) B ≤ θ x ^p

m^p−m (2:18)

for all xÎA.

Proof. Putting ϕ(x1, x2, · · ·, xm) :=θm

i=1(||xi||^p) for allx1, · · ·,xnÎ Aand let- tingL=m^1-pin Theorem 2.9, we obtain (2.18).

Author details

1Department of Mathematics, College of Sciences, Yasouj University, Yasouj 75914-353, Iran²Department of Mathematics, College of Sciences, Yasouj University, Yasouj 75914-353, Iran³Department of Mathematics, Semnan University, P. O. Box 35195-363, Semnan, Iran⁴Research Institute for Natural Sciences, Hanyang University, Seoul 133- 791, Korea⁵Department of Mathematics, Hallym University, Chunchen 200-702, Korea

Authors’contributions

All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Received: 2 March 2012 Accepted: 24 July 2012 Published: 24 July 2012

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Cite this article as:Kenaryet al.:Approximatem-Lie homomorphisms and approximate Jordanm-Lie homomorphisms associated to a parametric additive functional equation.Advances in Difference Equations2012 2012:125.

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