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

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

$${\displaystyle \sum _{i=1}^m\mu }f(x_i)=\frac{1}{2m}\left[{\displaystyle \sum _{i=1}^{m}f}\left(\mu m{x}_i+{\displaystyle \sum _{j=1,i\ne j}^m{x}_j}\right)+f\left({\displaystyle \sum _{i=1}^m\mu }{x}_i\right)\right]$$

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.,

$$[[x_1,\cdots ,x_n],x_{n+1},\cdots ,x_{2n-1}]={\displaystyle \sum _{i=1}^n[x_1,\cdots ,x_{i-1}[x_i,x_{n+1},\cdots ,x_{2n-1}],\cdots ,x_n]}$$

(1.1)

where x₁, x₂, · · · , x_2n-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 [3–5], a structure theory of finite-dimensional n-Lie algebras over a field $\mathrm{ F}$ 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 ℂ -linear mapping H: (A, [ ] _A ) → (B, [ ] _B ) is called an n-Lie homomorphism if

$$H\left({\left[x_1{x}_2\cdots x_n\right]}_A\right)={\left[H\left(x_1\right)H\left(x_2\right)\cdots H\left(x_n\right)\right]}_B$$

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

$$H\left({\left[xx\cdots x\right]}_A\right)={\left[H\left(x\right)H\left(x\right)\cdots H\left(x\right)\right]}_B$$

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

$$∥f\left(x+y\right)-f\left(x\right)-f\left(y\right)∥\le \epsilon$$

(1.2)

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

$$∥f\left(x\right)-T\left(x\right)∥\le \epsilon$$

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 [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 in C*-algebras and Lie C*-algebras and also of derivations on C*-algebras and Lie C*-algebras for the Jensen-type functional equation

$$\mu f\left(\frac{x+y}{2}\right)+\mu f\left(\frac{x-y}{2}\right)-f\left(\mu x\right)=0$$

for all $\mu \in {\mathbb{T}}^1:=\left\{\lambda \in ℂ:\left|\lambda \right|=1\right\}$.

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

$${\displaystyle \sum _{i=1}^m\mu }f(x_i)-\frac{1}{2m}\left[{\displaystyle \sum _{i=1}^{m}f}\left(\mu m{x}_i+{\displaystyle \sum _{j=1,i\ne j}^m\mu }{x}_j\right)+f\left({\displaystyle \sum _{i=1}^m\mu }{x}_i\right)\right]=0$$

for all

$$\mu \in T_{\frac{1}{n_o}}^1:=\left\{e^{i\theta }:0\le \theta \le \frac{2\pi }{n_o}\right\}\cup \left\{1\right\},$$

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

$$d\left(T^{m}x,T^{m+1}x\right)=\infty forallm\ge 0,$$

or other exists a natural number m ₀ such 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 in Λ = {y ∈ Ω: d(T^{m 0}x, y) < ∞};

• $d\left(y,y^{*}\right)\le \frac{1}{1-L}d\left(y,Ty\right)$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

$${\displaystyle \sum _{i=1}^{m}f}(x_i)=\frac{1}{2m}\left[{\displaystyle \sum _{i=1}^{m}f}\left(m{x}_i+{\displaystyle \sum _{j=1,i\ne j}^m{x}_j}\right)+f\left({\displaystyle \sum _{i=1}^m{x}_i}\right)\right]$$

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 ∈ ℕ be a fixed positive integer. Let f: A → B be a mapping for which there exists a function ϕ: A^m → [0, ∞) such that

$$\begin{array}{c}\Vert \mu {\displaystyle \sum _{i=1}^m\mu }f(x_i)-\frac{1}{2m}\left[{\displaystyle \sum _{i=1}^{m}f}\left(\mu m{x}_i+{\displaystyle \sum _{j=1,i\ne j}^m\mu }{x}_j\right)+f\left({\displaystyle \sum _{i=1}^m\mu }{x}_i\right)\right]\Vert \\ \le \phi (x_1,x_2,\cdots ,x_m),\end{array}$$

(2.1)

$$\parallel f({[x_1{x}_2\cdots x_n]}_A)-{[f(x_1)f(x_2)\cdots f(x_m)]}_B\parallel \le \phi (x_1,x_2,\cdots ,x_m)$$

(2.2)

for all $\mu \in {\mathbb{T}}_{\frac{1}{n_0}}^1$ and all x₁, · · · , x _m ∈ A. If there exists an L < 1 such that

$$\phi \left(x_1,x_2,\cdots ,x_m\right)\le mL\phi \left(\frac{x_1}{m},\frac{x_2}{m},\cdots ,\frac{x_m}{m}\right)$$

(2.3)

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

$$\left|\right|f\left(x\right)-H\left(x\right)\left|\right|\le \frac{\phi \left(x,0,0,\cdots ,0\right)}{m-mL}$$

(2.4)

for all x ∈ A.

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

$$d\left(g,h\right):=\mathsf{\text{inf}}\left\{C\in {ℝ}^{+}:\left|\right|g\left(x\right)-h\left(x\right)|{|}_B\le C\varphi \left(x,0,\cdots ,0\right),\forall x\in A\right\}.$$

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

Now we define the mapping J: Ω → Ω by

$$J\left(h\right)\left(x\right)=\frac{1}{m}h\left(mx\right)$$

for all x ∈ A.

Note that for all g, h ∈ Ω,

$$\begin{array}{cc}\hfill d\left(g,h\right)<C& \Rightarrow \parallel g\left(x\right)-h\left(x\right)\parallel \le C\varphi \left(x,0,\cdots ,0\right)\hfill \\ \Rightarrow ∥\frac{1}{m}g\left(mx\right)-\frac{1}{m}h\left(mx\right)∥\le \frac{C\phi \left(mx,0,\cdots ,0\right)}{|m{|}^{\ell }}\hfill \\ \Rightarrow ∥\frac{1}{m}g\left(mx\right)-\frac{1}{m}h\left(mx\right)∥\le L\mathrm{ C}\phi \left(x,0,\cdots ,0\right)\hfill \\ \Rightarrow d\left(J\left(g\right),J\left(h\right)\right)\le L\mathrm{ C}\hfill \end{array}$$

for all x ∈ A. Hence we see that

$$d\left(J\left(g\right),J\left(h\right)\right)\le Ld\left(g,h\right)$$

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

$$\underset{k\to \infty }{\mathsf{\text{lim}}}\frac{\phi \left(m^k{x}_1,m^k{x}_2,\cdots ,m^k{x}_m\right)}{m^k}\le \underset{k\to \infty }{\mathsf{\text{lim}}}{L}^k\phi \left(x_1,\cdots ,x_m\right)=0$$

(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

$$∥\frac{f\left(mx\right)}{m}-f\left(x\right)∥\le \frac{\phi \left(x,0,\cdots ,0\right)}{m}$$

for all x ∈ A. Therefore,

$$d\left(f,J\left(f\right)\right)\le \frac{1}{m}<\infty .$$

(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

$$H\left(mx\right)=mH\left(x\right)$$

such that there exists C ∈ (0, ∞) satisfying

$$\parallel f\left(x\right)-H\left(x\right)\parallel \le C\phi \left(x,0,\cdots ,0\right)$$

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

$$\underset{k\to \infty }{\mathsf{\text{lim}}}\frac{1}{m^k}f\left(m^{k}x\right)=H\left(x\right)$$

(2.7)

for all x ∈ A. By Theorem 2.1, we have

$$d\left(f,H\right)\le \frac{1}{1-L}d\left(f,J\left(f\right)\right).$$

(2.8)

It follows from (2.6) and (2.8) that

$$d\left(f,H\right)\le \frac{1}{m-mL}.$$

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

$$\begin{array}{c}\parallel H\left({\left[x_1{x}_2\cdots x_m\right]}_A\right)-{\left[H\left(x_1\right)H\left(x_2\right)H\left(x_3\right)\cdots H\left(x_m\right)\right]}_B\parallel \hfill \\ =\underset{k\to \infty }{\mathsf{\text{lim}}}∥\frac{H\left({\left[m^k{x}_1{m}^k{x}_2\cdots m^k{x}_m\right]}_A\right)}{m^{mk}}-\frac{\left({\left[H\left(m^k{x}_1\right)H\left(m^k{x}_2\right)H\left(m^k{x}_3\right)\cdots H\left(m^k{x}_m\right)\right]}_B\right)}{m^{mk}}∥\hfill \\ \le \underset{m\to \infty }{\mathsf{\text{lim}}}\frac{\phi \left(m^k{x}_1,m^k{x}_2,\cdots ,m^k{x}_m\right)}{m^{mk}}=0\hfill \end{array}$$

for all x₁, · · · , x _m ∈ A. Hence

$$H\left({\left[x_1{x}_2\cdots x_m\right]}_A\right)={\left[H\left(x_1\right)H\left(x_2\right)H\left(x_3\right)\cdots H\left(x_m\right)\right]}_B$$

for all x₁, · · · , x _m ∈ A.

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

$$\begin{array}{l}{\Vert {\displaystyle \sum _{i=1}^{m}H}(x_i)-\frac{1}{2m}\left[{\displaystyle \sum _{i=1}^{m}H}\left(m{x}_i+{\displaystyle \sum _{j=1,i\ne j}^m{x}_j}\right)+H\left({\displaystyle \sum _{i=1}^m{x}_i}\right)\right]\Vert }_B\\ =\underset{k\to \infty }{\lim }\frac{1}{m^k}\Vert {\displaystyle \sum _{i=1}^{m}f}(m^k{x}_i)-\frac{1}{2m}\left[{\displaystyle \sum _{i=1}^{m}f}\left(m^{k+1}{x}_i+{\displaystyle \sum _{j=1,i\ne j}^m{m}^k}{x}_j\right)+f\left({\displaystyle \sum _{i=1}^m{m}^k}{x}_i\right)\right]\Vert \\ \le \underset{m\to \infty }{\lim }\frac{\phi (m^k{x}_1,m^k{x}_2,\cdots ,m^k{x}_m)}{m^k}=0\end{array}$$

for all x₁, · · · , x _m ∈ A. Then

$${\displaystyle \sum _{i=1}^{m}H}(x_i)=\frac{1}{2m}\left[{\displaystyle \sum _{i=1}^{m}H}\left(m{x}_i+{\displaystyle \sum _{j=1,i\ne j}^m{x}_j}\right)+H\left({\displaystyle \sum _{i=1}^m{x}_i}\right)\right]$$

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

$$\parallel \mu f\left(x\right)-f\left(\mu x\right)\parallel \le \phi \left(x,x,\cdots ,x\right)$$

for all x ∈ A. It follows that

$$\begin{array}{cc}\hfill \parallel H\left(\mu x\right)-\mu H\left(x\right)\parallel & =\underset{k\to \infty }{\mathsf{\text{lim}}}\frac{\parallel f\left(\mu m^{k}x\right)-\mu f\left(m^{k}x\right)\parallel }{m^k}\hfill \\ \le \underset{k\to \infty }{\mathsf{\text{lim}}}\frac{\phi \left(m^{k}x,m^{k}x,\cdots ,m^{k}x\right)}{m^k}=0\hfill \end{array}$$

for all $\mu \in {\mathbb{T}}_{\frac{1}{n_0}}^1$ and all x ∈ A. One can show that the mapping H: A → B is ℂ -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

$$\Vert \mu {\displaystyle \sum _{i=1}^m\mu }f(x_i)-\frac{1}{2m}\left[{\displaystyle \sum _{i=1}^{m}f}\left(\mu m{x}_i+{\displaystyle \sum _{j=1,i\ne j}^m\mu }{x}_j\right)+f\left({\displaystyle \sum _{i=1}^m\mu }{x}_i\right)\right]\Vert \le \theta {\displaystyle \sum _{i=1}^m(}\parallel x_i{\parallel }^p),$$

(2.9)

$$\parallel f({[x_1{x}_2\cdots x_n]}_A)-{[f(x_1)f(x_2)\cdots f(x_m)]}_B\parallel \le \theta {\displaystyle \sum _{i=1}^m(}\parallel x_i{\parallel }^p)$$

(2.10)

for all $\mu \in {\mathbb{T}}_{\frac{1}{n_0}}^1$ and all x₁, · · · , x _m ∈ A. Then there exists a unique m-Lie homomorphism H: A → B such that

$$\parallel f\left(x\right)-H\left(x\right)\parallel \le \frac{\theta \parallel x{\parallel }^p}{\left(m-m^p\right)}$$

(2.11)

for all x ∈ A.

Proof. Putting $\phi \left(x_1,x_2,\cdots ,x_m\right):=\mathit{\theta }{\sum }_{i=1}^m\left(\left|\right|x_i|{|}^p\right)$ for all x₁, · · · , x _n ∈ A and letting L = 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 φ: A^m → [0, ∞) satisfying (2.1) and (2.2). If there exists an L < 1 such that

$$\phi \left(\frac{x_1}{m},\frac{x_2}{m},\cdots ,\frac{x_m}{m}\right)\le \frac{L}{m}\phi \left(x_1,x_2,\cdots ,x_m\right)$$

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

$$\parallel f\left(x\right)-H\left(x\right)\parallel \le \frac{L\phi \left(x,0,0,\cdots ,0\right)}{m-mL}$$

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

$$\parallel f\left(x\right)-H\left(x\right)\parallel \le \frac{m\theta \parallel x{\parallel }^p}{m^{p+1}-m^2}$$

(2.12)

for all x ∈ A.

Proof. Putting $\phi \left(x_1,x_2,\cdots ,x_m\right):=\mathit{\theta }{\sum }_{i=1}^m\left(\left|\right|x_i|{|}^p\right)$ for all x₁, · · · , x _n ∈ A and letting L = m^1-pin Theorem 2.5, we obtain (2.12). □

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

$$\begin{array}{c}\Vert \mu {\displaystyle \sum _{i=1}^m\mu }f(x_i)-\frac{1}{2m}\left[{\displaystyle \sum _{i=1}^{m}f}\left(\mu m{x}_i+{\displaystyle \sum _{j=1,i\ne j}^m\mu }{x}_j\right)+f\left({\displaystyle \sum _{i=1}^m\mu }{x}_i\right)\right]\Vert \\ \le \phi (x_1,x_2,\cdots ,x_m),\end{array}$$

(2.13)

$$\parallel f({[xx\cdots x]}_A)-{[f(x)f(x)\cdots f(x)]}_B\parallel \le \phi (x,x,\cdots ,x)$$

(2.14)

for all $\mu \in {\mathbb{T}}_{\frac{1}{n_0}}^1$ and all x₁, · · · , x _m ∈ A. If there exists an L < 1 such that

$$\phi \left(x_1,x_2,\cdots ,x_m\right)\le mL\phi \left(\frac{x_1}{m},\frac{x_2}{m},\cdots ,\frac{x_m}{m}\right)$$

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

$$\parallel f\left(x\right)-H\left(x\right)\parallel \le \frac{\phi \left(x,0,\cdots ,0\right)}{m-mL}$$

(2.15)

for all x ∈ A.

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

$$H\left(x\right)=\underset{k\to \infty }{\mathsf{\text{lim}}}\frac{1}{m^k}f\left(m^{k}x\right)$$

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

$$\begin{array}{c}\parallel H\left({\left[xx\cdots x\right]}_A\right)-{\left[H\left(x\right)H\left(x\right)\cdots H\left(x\right)\right]}_B\parallel \hfill \\ =\underset{k\to \infty }{\mathsf{\text{lim}}}∥\frac{1}{m^{mk}}H\left({\left[m^{k}x\cdots m^{k}x\right]}_A\right)-\frac{1}{m^{mk}}({\left[H\left(m^{k}x\right)H\left(m^{k}x\right)\cdots H\left(m^{k}x\right)\right]}_B∥\hfill \\ \le \underset{k\to \infty }{\mathsf{\text{lim}}}\frac{1}{m^{mk}}\phi \left(m^{k}x,m^{k}x,\dots ,m^{k}x\right)=0\hfill \end{array}$$

for all x ∈ A. So

$$H\left({\left[xx\cdots x\right]}_A\right)={\left[H\left(x\right)H\left(x\right)\cdots H\left(x\right)\right]}_B$$

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

$$\Vert \mu {\displaystyle \sum _{i=1}^m\mu }f(x_i)-\frac{1}{2m}\left[{\displaystyle \sum _{i=1}^{m}f}\left(\mu m{x}_i+{\displaystyle \sum _{j=1,i\ne j}^m\mu }{x}_j\right)+f\left({\displaystyle \sum _{i=1}^m\mu }{x}_i\right)\right]\Vert \le \theta {\displaystyle \sum _{i=1}^n(}\parallel x_i{\parallel }^p),$$

(2.16)

$$\parallel f\left({\left[xx\cdots x\right]}_A\right)-{\left[f\left(x\right)f\left(x\right)\cdots f\left(x\right)\right]}_B\parallel \le n\theta \left(\parallel x{\parallel }^p\right)$$

(2.17)

for all $\mu \in {\mathbb{T}}_{\frac{1}{n_0}}^1$ and all x₁, · · · , x _m ∈ A. Then there exists a unique Jordan m-Lie homomorphism H: A → B such that

$$\parallel f\left(x\right)-H\left(x\right)\parallel \le \frac{\theta \parallel x{\parallel }^p}{m-m^p}$$

for all x ∈ A.

Proof. The proof follows from Theorem 2.7 by putting $\phi \left(x_1,x_2,\cdots ,x_m\right):=\mathit{\theta }{\sum }_{i=1}^m\left(\left|\right|x_i|{|}^p\right)$ for all x₁, · · · , x _m ∈ A and letting L = 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 φ: A^m → [0, ∞) satisfying (2.13) and (2.14). If there exists an L < 1 such that

$$\phi \left(\frac{x_1}{m},\frac{x_2}{m},\cdots ,\frac{x_m}{m}\right)\le \frac{L}{m}\phi \left(x_1,x_2,\cdots ,x_m\right)$$

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

$$\parallel f\left(x\right)-H\left(x\right)\parallel \le \frac{L\phi \left(x,0,0,\cdots ,0\right)}{m-mL}$$

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

$$\parallel f\left(x\right)-H\left(x\right){\parallel }_B\le \frac{\theta \parallel x{\parallel }^p}{m^p-m}$$

(2.18)

for all x ∈ A.

Proof. Putting $\phi \left(x_1,x_2,\cdots ,x_m\right):=\mathit{\theta }{\sum }_{i=1}^m\left(\left|\right|x_i|{|}^p\right)$ for all x₁, · · · , x _n ∈ A and letting L = m^1-pin Theorem 2.9, we obtain (2.18).