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 . 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  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  and several subsequent papers , a structure theory of finite-dimensional n-Lie algebras over a field 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 ).

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  during a talk in 1940: Under what condition does there exist a homomorphism near an approximate homomorphism? In the following year, Hyers  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  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  (see also ). 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  and for quadratic functional equation .

Park and Rassias  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 $μ∈ T 1 := { λ ∈ ℂ : | λ | = 1 }$.

In this paper, by using fixed point method, we establish the Hyers-Ulam stability of n-Lie homomorphisms and Jordan n-Lie homomorphisms 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.  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

$d ( T m x , T m + 1 x ) =∞forallm≥0,$

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

$d ( g , h ) : = inf { C ∈ ℝ + : | | g ( x ) - h ( x ) | | B ≤ C ϕ ( x , 0 , ⋯ , 0 ) , ∀ x ∈ A } .$

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

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,

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 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 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 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 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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The authors declare that they have no competing interests.

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.

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