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

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Abstract

Using fixed point method, we establish the Hyers-Ulam stability of m-Lie homomorphisms and Jordan m-Lie homomorphisms in m-Lie algebras associated to the following generalized Jensen functional equation

i = 1 m μ f ( x i ) = 1 2 m [ i = 1 m f ( μ m x i + j = 1 , i j m x j ) + f ( i = 1 m μ x i ) ]

for a fixed positive integer m with m ≥ 2.

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

1. Introduction

Let n be 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 between n elements of the algebra instead of two. This new bracket is n-linear, anti-symmetric 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 multi-linear, 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, i.e.,

[ [ x 1 , , x n ] , x n + 1 , , x 2 n 1 ] = i = 1 n [ x 1 , , x i 1 [ x i , x n + 1 , , x 2 n 1 ] , , x n ]
(1.1)

where x1, x2, · · · , x2n-1 V. The 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 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 M 2-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 consider n-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, · · · , x n ]|| ||x1||||x2|| · · · ||x n || for all x1, x2, · · · , x n 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 -linear mapping H: (A, [ ] A ) (B, [ ] B ) is called an n-Lie homomorphism if

H ( [ x 1 x 2 x n ] A ) = [ H ( x 1 ) H ( x 2 ) H ( x n ) ] B

for all x1, x2, · · · , x n A. A -linear mapping H: (A, [ ] A ) (B, [ ] B ) is called a Jordan n-Lie homomorphism if

H ( [ x x 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] was answered affirmatively the question of Ulam for Banach spaces, which states that if ε > 0 and f: X → Y is a mapping with X a normed space and Y a Banach spaces such that

f ( x + y ) - f ( x ) - f ( y ) ε
(1.2)

for all x, y X, then there exists a unique additive map T: X → Y such that

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 (1.2) is unbounded.

In 2003, Cădariu and Radu applied the fixed point method to the investigation of the Jensen functional equation [13] (see also [1416]). 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 in C*-algebras and Lie C*-algebras and also of derivations on C*-algebras and Lie C*-algebras for the Jensen-type functional equation

μ f x + y 2 + μ f x - y 2 - f ( μ x ) = 0

for all μ T 1 := { λ : | 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 in n-Lie Banach algebras associated to the following generalized Jensen-type functional equation

i = 1 m μ f ( x i ) 1 2 m [ i = 1 m f ( μ m x i + j = 1 , i j m μ x j ) + f ( i = 1 m μ x i ) ] = 0

for all

μ T 1 n o 1 := e i θ : 0 θ 2 π n o { 1 } ,

where m ≥ 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 Ω, either

d ( T m x , T m + 1 x ) =forallm0,

or other exists a natural number m 0 such that

  • d (T mx, Tm+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 in Λ = {y Ω: d(Tm 0x, y) < };

  • d ( y , y * ) 1 1 - L d ( y , T y ) for all y Λ.

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

i = 1 m f ( x i ) = 1 2 m [ i = 1 m f ( m x i + j = 1 , i j m x j ) + f ( i = 1 m x i ) ]

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 n0 be a fixed positive integer. Let f: A → B be a mapping for which there exists a function ϕ: Am [0, ) such that

μ i = 1 m μ f ( x i ) 1 2 m [ i = 1 m f ( μ m x i + j = 1 , i j m μ x j ) + f ( i = 1 m μ x i ) ] φ ( x 1 , x 2 , , x m ) ,
(2.1)
f ( [ x 1 x 2 x n ] A ) [ f ( x 1 ) f ( x 2 ) f ( x m ) ] B φ ( x 1 , x 2 , , x m )
(2.2)

for all μ T 1 n 0 1 and all x1, · · · , x m A. If there exists an L < 1 such that

φ ( x 1 , x 2 , , x m ) m L φ x 1 m , x 2 m , , x m m
(2.3)

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

| | f ( x ) - H ( x ) | | φ ( x , 0 , 0 , , 0 ) m - m L
(2.4)

for all x A.

Proof. Let Ω be the set of all functions from A into B and let

d ( g , h ) : = inf { C + : | | g ( x ) - h ( x ) | | B C ϕ ( x , 0 , , 0 ) , x A } .

It is easy to show that (Ω, d) is a generalized complete metric space [19].

Now we define the mapping J: Ω Ω by

J ( h ) ( x ) = 1 m h ( m x )

for all x A.

Note that for all g, h Ω,

d ( g , h ) < C g ( x ) - h ( x ) C ϕ ( x , 0 , , 0 ) 1 m g ( m x ) - 1 m h ( m x ) C φ ( m x , 0 , , 0 ) | m | 1 m g ( m x ) - 1 m h ( m x ) L  C φ ( x , 0 , , 0 ) d ( J ( g ) , J ( h ) ) L  C

for all x A. Hence we see that

d ( J ( g ) , J ( h ) ) Ld ( g , h )

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

lim k φ ( m k x 1 , m k x 2 , , m k x m ) m k lim k L k φ ( x 1 , , x m ) = 0
(2.5)

for all x1, · · · , x m A. Putting μ = 1, x1 = x and x j = 0 (j = 2, · · · , n) in (2.1), we obtain

f ( m x ) m - f ( x ) φ ( x , 0 , , 0 ) m

for all x A. Therefore,

d ( f , J ( f ) ) 1 m < .
(2.6)

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

H ( m x ) =mH ( x )

such that there exists C (0, ) satisfying

f ( x ) -H ( x ) Cφ ( x , 0 , , 0 )

for all x A. On the other hand, we have lim k→∞ d(J k(f), H) = 0 and so

lim k 1 m k f ( m k x ) = H ( x )
(2.7)

for all x A. By Theorem 2.1, we have

d ( f , H ) 1 1 - L d ( f , J ( f ) ) .
(2.8)

It follows from (2.6) and (2.8) that

d ( f , H ) 1 m - m L .

This implies the inequality (2.4). By (2.2), we have

H ( [ x 1 x 2 x m ] A ) - [ H ( x 1 ) H ( x 2 ) H ( x 3 ) H ( x m ) ] B = lim k H ( [ m k x 1 m k x 2 m k x m ] A ) m m k - ( [ H ( m k x 1 ) H ( m k x 2 ) H ( m k x 3 ) H ( m k x m ) ] B ) m m k lim m φ ( m k x 1 , m k x 2 , , m k x m ) m m k = 0

for all x1, · · · , x m A. Hence

H ( [ x 1 x 2 x m ] A ) = [ H ( x 1 ) H ( x 2 ) H ( x 3 ) H ( x m ) ] B

for all x1, · · · , x m A.

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

i = 1 m H ( x i ) 1 2 m [ i = 1 m H ( m x i + j = 1 , i j m x j ) + H ( i = 1 m x i ) ] B = lim k 1 m k i = 1 m f ( m k x i ) 1 2 m [ i = 1 m f ( m k + 1 x i + j = 1 , i j m m k x j ) + f ( i = 1 m m k x i ) ] lim m φ ( m k x 1 , m k x 2 , , m k x m ) m k = 0

for all x1, · · · , x m A. Then

i = 1 m H ( x i ) = 1 2 m [ i = 1 m H ( m x i + j = 1 , i j m x j ) + H ( i = 1 m x i ) ]

for all x1, · · · , x m A. So by Theorem 2.1, H is additive. Letting x i = x for all i = 1, 2, · · · , n in (2.1), we obtain

μf ( x ) -f ( μ x ) φ ( x , x , , x )

for all x A. It follows that

H ( μ x ) - μ H ( x ) = lim k f ( μ m k x ) - μ f ( m k x ) m k lim k φ ( m k x , m k x , , m k x ) m k = 0

for all μ T 1 n 0 1 and all x A. One can show that the mapping H: A → B is -linear.

Hence H: A → B is an m-Lie homomorphism satisfying (2.4), as desired. □

Corollary 2.4. Let θ and p be nonnegative real numbers such that p < 1. Suppose that a mapping f: A → B satisfies

μ i = 1 m μ f ( x i ) 1 2 m [ i = 1 m f ( μ m x i + j = 1 , i j m μ x j ) + f ( i = 1 m μ x i ) ] θ i = 1 m ( x i p ) ,
(2.9)
f ( [ x 1 x 2 x n ] A ) [ f ( x 1 ) f ( x 2 ) f ( x m ) ] B θ i = 1 m ( x i p )
(2.10)

for all μ T 1 n 0 1 and all x1, · · · , x m A. Then there exists a unique m-Lie homomorphism H: A → B such that

f ( x ) - H ( x ) θ x p ( m - m p )
(2.11)

for all x A.

Proof. Putting φ ( x 1 , x 2 , , x m ) : = θ  i = 1 m ( | | x i | | p ) for all x1, · · · , x n A and letting L = mp-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 φ: Am [0, ) satisfying (2.1) and (2.2). If there exists an L < 1 such that

φ x 1 m , x 2 m , , x m m L m φ ( x 1 , x 2 , , x m )

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

f ( x ) -H ( x ) L φ ( x , 0 , 0 , , 0 ) m - m L

for all x A.

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 ) m θ x p m p + 1 - m 2
(2.12)

for all x A.

Proof. Putting φ ( x 1 , x 2 , , x m ) : = θ  i = 1 m ( | | x i | | p ) for all x1, · · · , x n A and letting L = m1-pin Theorem 2.5, we obtain (2.12). □

Theorem 2.7. Let n0 be a fixed positive integer. Let f: A → B be a mapping for which there exists a function φ: An [0, ) such that

μ i = 1 m μ f ( x i ) 1 2 m [ i = 1 m f ( μ m x i + j = 1 , i j m μ x j ) + f ( i = 1 m μ x i ) ] φ ( x 1 , x 2 , , x m ) ,
(2.13)
f ( [ x x x ] A ) [ f ( x ) f ( x ) f ( x ) ] B φ ( x , x , , x )
(2.14)

for all μ T 1 n 0 1 and all x1, · · · , x m A. If there exists an L < 1 such that

φ ( x 1 , x 2 , , x m ) mLφ x 1 m , x 2 m , , x m m

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

f ( x ) - H ( x ) φ ( x , 0 , , 0 ) m - m L
(2.15)

for all x A.

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

H ( x ) = lim k 1 m k f ( m k x )

for all x A. Moreover, we can show that H is -linear. By (2.14), we get that

H ( [ x x x ] A ) - [ H ( x ) H ( x ) H ( x ) ] B = lim k 1 m m k H ( [ m k x m k x ] A ) - 1 m m k ( [ H ( m k x ) H ( m k x ) H ( m k x ) ] B lim k 1 m m k φ ( m k x , m k x , , m k x ) = 0

for all x A. So

H ( [ x x x ] A ) = [ H ( x ) H ( x ) H ( x ) ] B

for all x A. Hence H: A → B is a Jordan m-Lie homomorphism satisfying (2.15). □

Corollary 2.8. Let θ and p be nonnegative real numbers such that p < 1. Suppose that a mapping f: A → B satisfies

μ i = 1 m μ f ( x i ) 1 2 m [ i = 1 m f ( μ m x i + j = 1 , i j m μ x j ) + f ( i = 1 m μ x i ) ] θ i = 1 n ( x i p ) ,
(2.16)
f ( [ x x x ] A ) - [ f ( x ) f ( x ) f ( x ) ] B nθ ( x p )
(2.17)

for all μ T 1 n 0 1 and all x1, · · · , x m 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 x A.

Proof. The proof follows from Theorem 2.7 by putting φ ( x 1 , x 2 , , x m ) : = θ  i = 1 m ( | | x i | | p ) for all x1, · · · , x m A and letting L = mp- 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 φ: Am [0, ) satisfying (2.13) and (2.14). If there exists an L < 1 such that

φ x 1 m , x 2 m , , x m m L m φ ( x 1 , x 2 , , x m )

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

f ( x ) -H ( x ) L φ ( x , 0 , 0 , , 0 ) m - m L

for all x A.

Corollary 2.10. Let θ 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 x A.

Proof. Putting φ ( x 1 , x 2 , , x m ) : = θ  i = 1 m ( | | x i | | p ) for all x1, · · · , x n A and letting L = m1-pin Theorem 2.9, we obtain (2.18).

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Correspondence to Choonkil Park.

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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.

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Keywords

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