Stability of the Jensen equation in C*-algebras: a fixed point approach

Abstract

Using fixed point method, we prove the Hyers-Ulam stability of homomorphisms in C*-algebras and Lie C*-algebras and also of derivations on C*-algebras and Lie C*-algebras for the Jensen equation.

2010 Mathematics Subject Classification: 39B82; 47H10; 46L05; 39B52.

1 Introduction

A classical question in the theory of functional equations is the following: "When is it true that a function which approximately satisfies a functional equation must be close to an exact solution of the equation?". If the problem accepts a solution, we say that the equation is stable. The first stability problem concerning group homomorphisms was raised by Ulam [1]. In the next year, Hyers [2] gave a positive answer to the above question for additive groups under the assumption that the groups are Banach spaces. In [3], Rassias proved a generalization of the Hyers' theorem for additive mappings.

The result of Rassias has provided a lot of influence during the last three decades in the development of a generalization of the Hyers-Ulam stability concept. Furthermore, in 1994, a generalization of the Rassias' theorem was obtained by Gǎvruta [4] by replacing the bound ϵ(||x||^p+ ||y||^p) by a general control function φ(x, y).

The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem [5–22].

Theorem 1.1. Let (X, d) be a complete generalized metric space and J: X → X be a strictly contractive mapping with Lipschitz constant L < 1. Then, for all x ∈ X, either

$$d\left(J^{n}x,J^{n+1}x\right)=\infty$$

for all nonnegative integers n or there exists a positive integer n ₀ such that

(a) d(Jⁿx, Jⁿ⁺¹x) < ∞ for all n₀ ≥ n₀;

(b) the sequence {Jⁿx} converges to a fixed point y* of J;

(c) y* is the unique fixed point of J in the set $Y=\left\{y\in X:d\left(J^{n_0}x,y\right)<\infty \right\}$;

(d) $d\left(y,y^{*}\right)\le \frac{1}{1-L}d\left(y,Jy\right)$ for all y ∈ Y.

In [20], Park proved the Hyers-Ulam stability of the following functional equation:

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

(1.1)

in fuzzy Banach spaces. In this article, using the fixed point alternative approach, we prove the Hyers-Ulam stability of homomorphisms in C*-algebras and Lie C*-algebras and also of derivations on C*-algebras and Lie C*-algebras for the Jensen Equation (1.1).

2 Stability of homomorphisms in C*-algebras

Throughout this section, assume that A is a C*-algebra with the norm ||.|| _A and that B is a C*- algebra with the norm ||.|| _B .

For a given mapping f: A → B, we define

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

for all $\mu \in T^1:=\left\{\nu \in ℂ:\left|\nu \right|=1\right\}$ and all x, y ∈ A. Note that a ℂ-linear mapping H: A → B is called a homomorphism in C*-algebras, if H satisfies H(xy) = H(x)H(y) and H(x*) = H(x)* for all x ∈ A. Throughout this section, we prove the Hyers-Ulam stability of homomorphisms in C*-algebras for the functional equation C _μ f(x, y) = 0.

Theorem 2.1. Let f: A → B be a mapping with f(0) = 0 for which there exists a function φ: A² → [0, ∞) such that

$${∥C_{\mu }f\left(x,y\right)∥}_B\le \phi \left(x,y\right),$$

(2.2)

$${∥f\left(xy\right)-f\left(x\right)f\left(y\right)∥}_B\le \phi \left(x,y\right),$$

(2.3)

$${∥f\left(x^{*}\right)-f{\left(x\right)}^{*}∥}_B\le \phi \left(x,x\right)$$

(2.4)

for all $\mu \in T^1$ and all x, y ∈ A. If there exists an $L<\frac{1}{2}$ such that

$$\phi \left(x,y\right)\le \frac{L\phi \left(2x,2y\right)}{2}$$

(2.5)

for all x, y ∈ A, then there exists a unique C*-algebra homomorphism H: A → B such that

$${∥f\left(x\right)-H\left(x\right)∥}_B\le \frac{\phi \left(x,0\right)}{1-L}.$$

(2.6)

Proof. It follows from (2.5) that

$$\underset{n\to \infty }{\text{lim}}{2}^n\phi \left(\frac{x}{2^n},\frac{y}{2^n}\right)=\underset{n\to \infty }{\text{lim}}{L}^n\phi \left(x,y\right)=0.$$

Consider the set X := {g: A → B;g(0) = 0} and the generalized metric d in X defined by

$$d\left(f,g\right)=\text{inf}\left\{C\in {ℝ}^{+}:{∥g\left(x\right)-h\left(x\right)∥}_B\le C\phi \left(x,0\right),\forall x\in A\right\}$$

It is easy to show that (X, d) is complete. Now, we consider a linear mapping J : A → A such that

$$Jh\left(x\right):=2h\left(\frac{x}{2}\right)$$

for all x ∈ A. By [[7], Theorem 3.1], d(Jg, Jh) ≤ Ld(g, h) for all g, h ∈ X. Letting μ = 1 and y = 0 in (2.2), we have

$${∥2f\left(\frac{x}{2}\right)-f\left(x\right)∥}_B\le \phi \left(x,0\right)$$

(2.7)

for all x ∈ A. It follows from (2.7) that d(f, Jf) ≤ 1. By Theorem 1.1, there exists a mapping H: A → B satisfying the following:

• (1) H is a fixed point of J, that is,

  $$H\left(\frac{x}{2}\right)=\frac{1}{2}H\left(x\right)$$

  (2.8)

for all x ∈ A. The mapping H is a unique fixed point of J in the set Ω = {g ∈ X : d(f, g) < ∞}. This implies that H is a unique mapping satisfying (2.8) such that there exists C ∈ (0, ∞) satisfying ||f(x) - H(x)|| _B ≤ Cφ(x, 0) for all x ∈ A.

• (2) d(Jⁿf, H) → 0 as n → ∞. This implies the equality

  $$\underset{n\to \infty }{\text{lim}}{2}^{n}f\left(\frac{x}{2^n}\right)=H\left(x\right)$$

  (2.9)

for all x ∈ A.

• (3) $d\left(f,H\right)\le \frac{d\left(f,Jf\right)}{1-L}$, which implies the inequality $d\left(f,H\right)\le \frac{1}{1-L}$. This implies that the inequality (2.6) holds. It follows from (2.2) and (2.9) that

  $$\begin{array}{ll}\hfill {∥2H\left(\frac{x+y}{2}\right)-H\left(x\right)-H\left(y\right)∥}_B& =\underset{n\to \infty }{\text{lim}}{∥2^{n+1}f\left(\frac{x+y}{2^{n+1}}\right)-2^{n}f\left(\frac{x}{2^n}\right)-2^{n}f\left(\frac{y}{2^n}\right)∥}_B\\ \le \underset{n\to \infty }{\text{lim}}{2}^n\phi \left(\frac{x}{2^n},\frac{y}{2^n}\right)\le \underset{n\to \infty }{\text{lim}}{L}^n\phi \left(x,y\right)=0\end{array}$$

for all x, y ∈ A. So $2H\left(\frac{x+y}{2}\right)=H\left(x\right)+H\left(y\right)$ for all x, y ∈ X. Therefore, the mapping H: A → B is Jensen additive.

Letting y = x in (2.2), we get μf(x) = f(μx) for all $\mu \in T^1$ and all x ∈ A So, we get

$${∥\mu H\left(x\right)-H\left(\mu x\right)∥}_B=\underset{n\to \infty }{\text{lim}}{∥2^n\mu f\left(\frac{x}{2^n}\right)-2^{n}f\left(\frac{\mu x}{2^n}\right)∥}_B=0.$$

So, μH(x) = H(μx) for all $\mu \in T^1$ and all x ∈ A Thus one can show that the mapping H: A → B is ℂ-linear. It follows from (2.3) that

$$\begin{array}{ll}\hfill {∥H\left(xy\right)-H\left(x\right)H\left(y\right)∥}_B& =\underset{n\to \infty }{\text{lim}}{4}^n{∥f\left(\frac{xy}{4^n}\right)-f\left(\frac{x}{2^n}\right)f\left(\frac{y}{2^n}\right)∥}_B\\ \le \underset{n\to \infty }{\text{lim}}{4}^n\phi \left(\frac{x}{2^n},\frac{y}{2^n}\right)\le \underset{n\to \infty }{\text{lim}}{\left(2L\right)}^n\phi \left(x,y\right)=0\end{array}$$

for all x ∈ A. Furthermore, By (2.4), we have

$$\begin{array}{ll}\hfill {∥H\left(x^{*}\right)-H{\left(x\right)}^{*}∥}_B& =\underset{n\to \infty }{\text{lim}}{2}^n{∥f\left(\frac{x^{*}}{2^n}\right)-f{\left(\frac{x}{2^n}\right)}^{*}∥}_B\\ \le \underset{n\to \infty }{\text{lim}}{2}^n\phi \left(\frac{x}{2^n},\frac{y}{2^n}\right)\le \underset{n\to \infty }{\text{lim}}{L}^n\phi \left(x,y\right)=0\end{array}$$

for all x ∈ A. Thus H: A → B is a C*-algebra homomorphism satisfying (2.6), as desired.

Corollary 2.1. Let 0 < r < 1 and θ be nonnegative real numbers and f: A → B be a mapping with f(0) = 0 such that

$$\begin{array}{c}{∥2\mu f\left(\frac{x+y}{2}\right)-f\left(\mu x\right)-f\left(\mu y\right)∥}_B\le \theta \left({∥x∥}_A^r+{∥y∥}_A^r\right),\\ {∥f\left(xy\right)-f\left(x\right)f\left(y\right)∥}_B\le \theta \left({∥x∥}_A^r+{∥y∥}_A^r\right)\\ {∥f\left(x^{*}\right)-f{\left(x\right)}^{*}∥}_B\le 2\theta {∥x∥}_A^r\end{array}$$

(2.10)

for all $\mu \in T^1$ and all x, y ∈ A. Then the limit $H\left(x\right)={\text{lim}}_{n\to \infty }{2}^{n}f\left(\frac{x}{2^n}\right)$ exists for all x ∈ A and H: A → B is a unique C*-algebra homomorphism such that

$${∥f\left(x\right)-H\left(x\right)∥}_B\le \frac{2\theta {∥x∥}_A^r}{2-2^r}$$

(2.11)

for all x ∈ A.

Proof. The proof follows from Theorem 2.1, if we take $\phi \left(x,y\right)=\theta \left({∥x∥}_A^r+{∥y∥}_A^r\right)$ for all x, y ∈ A. In fact, if we choose L = 2^r-1, then we get the desired result.

Theorem 2.2. Let f: A → B be a mapping with f(0) = 0 for which there exists a function φ: A² → [0, ∞) satisfying (2.2), (2.3), and (2.4). If there exists an L < 1 such that $\phi \left(x,y\right)\le 2L\phi \left(\frac{x}{2},\frac{y}{2}\right)$ for all x, y ∈ A, then there exists a unique C*-algebra homomorphism H: A → B such that

$${∥f\left(x\right)-H\left(x\right)∥}_B\le \frac{L\phi \left(x,0\right)}{1-L}.$$

(2.12)

for all x ∈ A.

Proof. We consider the linear mapping J: A → A such that $Jg\left(x\right)=\frac{1}{2}g\left(2x\right)$ for all x ∈ A. It follows from (2.7) that

$$∥f\left(x\right)-\frac{1}{2}f\left(2x\right)∥\le \frac{\phi \left(2x,0\right)}{2}\le L\phi \left(x,0\right)$$

for all x ∈ X. Hence d(f, Jf) ≤ L. By Theorem 1.1, there exists a mapping H: A → B satisfying the following:

• (1) H is a fixed point of J, that is,

  $$H\left(2x\right)=2H\left(x\right)$$

  (2.13)

for all x ∈ A. The mapping H is a unique fixed point of J in the set Ω = {g ∈ X: d(f, g) < ∞}. This implies that H is a unique mapping satisfying (2.13) such that there exists C ∈ (0, ∞) satisfying ||f(x) - H(x)|| _B ≤ Cφ(x, 0) for all x ∈ A.

• (2) d(Jⁿf, H) → 0 as n → ∞. This implies the equality ${\text{lim}}_{n\to \infty }\frac{f\left(2^{n}x\right)}{2^n}=H\left(x\right)$ for all x ∈ A.

• (3) $d\left(f,H\right)\le \frac{d\left(f,Jf\right)}{1-L}$, which implies the inequality $d\left(f,H\right)\le \frac{1}{1-L}$. which implies that the inequality (2.12). The rest of the proof is similar to the proof of Theorem 2.1.

Corollary 2.2. Let r > 1 and θ be nonnegative real numbers and f: A → B be a mapping satisfying f(0) = 0 and (2.10). Then the limit $H\left(x\right)={\text{lim}}_{n\to \infty }\frac{f\left(2^{n}x\right)}{2^n}$ exists for all x ∈ A and H: A → B is a unique C*-algebra homomorphism such that

$${∥f\left(x\right)-H\left(x\right)∥}_B\le \frac{2\theta {∥x∥}_A^r}{2^r-2}$$

(2.14)

for all x ∈ A.

Proof. The proof follows from Theorem 2.2 if we take $\phi \left(x,y\right)=\theta \left({∥x∥}_A^r+{∥y∥}_A^r\right)$ for all x, y ∈ A. In fact, if we choose L = 2^1-r, then we get the desired result.

3 Stability of derivations on C*-algebras

Throughout this section, assume that A is a C*-algebra with the norm ||.| _A . Note that a ℂ-linear mapping δ: A → A is called a derivation on A if δ satisfies δ(xy) = δ(x)y + xδ(y) for all x, y ∈ A.

Throughout this section, using the fixed point alternative approach, We prove the Hyers-Ulam stability of derivations on C*-algebras for the functional equation (1.1).

Theorem 3.1. Let f: A → A be a mapping with f(0) = 0 for which there exists a function φ: A² → [0, ∞) such that

$${∥2\mu f\left(\frac{x+y}{2}\right)-f\left(\mu x\right)-f\left(\mu y\right)∥}_A\le \phi \left(x,y\right)$$

(3.15)

$${∥f\left(xy\right)-f\left(x\right)y-xf\left(y\right)∥}_A\le \phi \left(x,y\right)$$

(3.16)

for all $\mu \in T^1$ and all x, y ∈ A. If there exists an $L<\frac{1}{2}$ such that $\phi \left(x,y\right)\le \frac{L\phi \left(2x,2y\right)}{2}$ for all x, y ∈ A, then there exists a unique derivation δ: A → A such that

$${∥f\left(x\right)-\delta \left(x\right)∥}_A\le \frac{\phi \left(x,0\right)}{1-L}.$$

(3.17)

Proof. By the same reasoning as in the proof of Theorem 2.1, there exists a unique ℂ-linear mapping δ: A → A satisfying (3.17). The mapping δ: A → A is given by

$$\delta \left(x\right)=\underset{n\to \infty }{\text{lim}}{2}^{n}f\left(\frac{x}{2^n}\right)$$

for all x ∈ A. It follows from (3.2) that

$$\begin{array}{ll}\hfill {∥\delta \left(xy\right)-\delta \left(x\right)y-x\delta \left(y\right)∥}_B& =\underset{n\to \infty }{\text{lim}}{4}^n{∥f\left(\frac{xy}{4^n}\right)-f\left(\frac{x}{2^n}\right)\frac{y}{2^n}-\frac{x}{2^n}f\left(\frac{y}{2^n}\right)∥}_B\\ \le \underset{n\to \infty }{\text{lim}}{4}^n\phi \left(\frac{x}{2^n},\frac{y}{2^n}\right)\le \underset{n\to \infty }{\text{lim}}{\left(2L\right)}^n\phi \left(x,y\right)=0\end{array}$$

for all x, y ∈ A. So

$$\delta \left(xy\right)-\delta \left(x\right)y-x\delta \left(y\right)$$

for all x, y ∈ A. Thus δ: A → A is a derivation satisfying (3.17).

Corollary 3.1. Let 0 < r < 1 and θ be nonnegative real numbers and f: A → A be a mapping with f(0) = 0 such that

$${∥2\mu f\left(\frac{x+y}{2}\right)-f\left(\mu x\right)-f\left(\mu y\right)∥}_A\le \theta \left({∥x∥}_A^r+{∥y∥}_A^r\right),$$

(3.18)

$${∥f\left(xy\right)-f\left(x\right)y-xf\left(y\right)∥}_A\le \theta \left({∥x∥}_A^r+{∥y∥}_A^r\right)$$

(3.19)

for all $\mu \in T^1$ and all x, y ∈ A. Then the limit $H\left(x\right)={\text{lim}}_{n\to \infty }{2}^{n}f\left(\frac{x}{2^n}\right)$ exists for all x ∈ A and δ: A → A is a unique derivation such that

$$∥f\left(x\right)-\delta \left(x\right)∥\le \frac{2\theta {∥x∥}_A^r}{2-2^r}$$

(3.20)

for all x ∈ A.

Proof. The proof follows from Theorem 3.1 if we take $\phi \left(x,y\right)=\theta \left({∥x∥}_A^r+{∥y∥}_A^r\right)$ for all x, y ∈ A. In fact, if we choose L = 2^r-1, then we get the desired result.

Theorem 3.2. Let f: A → A be a mapping with f(0) = 0 for which there exists a function φ: A² → [0, ∞) satisfying (3.15) and (3.2). If there exists an L < 1 such that $\phi \left(x,y\right)\le 2L\phi \left(\frac{x}{2},\frac{y}{2}\right)$ for all x, y ∈ A, then there exists a unique derivation δ: A → A such that

$${∥f\left(x\right)-\delta \left(x\right)∥}_A\le \frac{L\phi \left(x,0\right)}{1-L}.$$

(3.21)

Proof. The proof is similar to the proofs of Theorems 2.2 and 3.1.

Corollary 3.2. Let r > 1 and θ be nonnegative real numbers and f: A → A be a mapping satisfying f(0) = 0, (3.4) and (3.5). Then the limit $H\left(x\right)={\text{lim}}_{n\to \infty }\frac{f\left(2^{n}x\right)}{2^n}$ exists for all x ∈ A and δ: A → A is a unique derivation such that

$${∥f\left(x\right)-\delta \left(x\right)∥}_A\le \frac{2\theta {∥x∥}_A^r}{2^r-2}$$

(3.22)

for all x ∈ A.

Proof. The proof follows from Theorem 3.2 if we take $\phi \left(x,y\right)=\theta \left({∥x∥}_A^r+{∥y∥}_A^r\right)$ for all x, y ∈ A. In fact, if we choose L = 2^1-r, then we get the desired result.

4 Stability of homomorphisms in Lie C*-algebras

A C*-algebra $\mathcal{C}$, endowed with the Lie product $\left[x,y\right]:=\frac{xy-yx}{2}$ on $\mathcal{C}$, is called a Lie C*-algebra (see, [17–19]).

Definition 4.1. Let A and B be Lie C*-algebras, A ℂ-linear mapping H: A → B is called a Lie C*-algebra homomorphism if $H\left(\left[x,y\right]\right)=\left[H\left(x\right),H\left(y\right)\right]=\frac{H\left(x\right)H\left(y\right)-H\left(y\right)H\left(x\right)}{2}$ for all x, y ∈ A.

Throughout this section, assume that A is a Lie C*-algebra with the norm ||.|| _A and B is a Lie C*-algebra with the norm ||.|| _B .

We prove the Hyers-Ulam stability of homomorphisms in Lie C*-algebras for the functional Equation (1.1).

Theorem 4.1. Let f: A → B be a mapping with f(0) = 0 for which there exists a function φ: A² → [0, ∞) satisfying (2.2) such that

$${∥f\left(\left[x,y\right]\right)-\left[f\left(x\right),f\left(y\right)\right]∥}_B\le \phi \left(x,y\right)$$

(4.23)

for all x, y ∈ A. If there exists an $L<\frac{1}{2}$ such that $\phi \left(x,y\right)\le \frac{L}{2}\phi \left(2x,2y\right)$ for all x, y ∈ A, then there exists a unique Lie C*-algebra homomorphism H: A → B satisfying (2.6).

Proof. By the same reasoning as in the proof of Theorem 2.1, there exists a unique ℂ-linear mapping H: A → B satisfying (2.6). The mapping H: A → B is given by $H\left(x\right)={\text{lim}}_{n\to \infty }{2}^{n}f\left(\frac{x}{2^n}\right)$ for all x ∈ A. It follows from (4.23) that

$$\begin{array}{ll}\hfill {∥H\left(\left[x,y\right]\right)-\left[H\left(x\right),H\left(y\right)\right]∥}_B& =\underset{n\to \infty }{\text{lim}}{4}^n{∥f\left(\frac{\left[x,y\right]}{4^n}\right)-\left[f\left(\frac{x}{2^n}\right),f\left(\frac{y}{2^n}\right)\right]∥}_B\\ \le \underset{n\to \infty }{\text{lim}}{4}^n\phi \left(\frac{x}{2^n},\frac{y}{2^n}\right)=0\end{array}$$

for all x, y ∈ A. So H([x, y]) = [H(x), H(y)] for all x, y ∈ A. Thus H: A → B is a Lie C*-algebra homomorphism satisfying (2.6), as desired.

Corollary 4.1. Let 0 < r < 1 and θ be nonnegative real numbers, and let f: A → B be a mapping satisfying f(0) = 0 such that

$${∥C_{\mu }f\left(x,y\right)∥}_B\le \theta \left({∥x∥}_A^r+{∥y∥}_A^r\right),$$

(4.24)

$${∥f\left(\left[x,y\right]\right)-\left[f\left(x\right),f\left(y\right)\right]∥}_B\le \theta \left({∥x∥}_A^r+{∥y∥}_A^r\right)$$

(4.25)

for all $\mu \in T^1$ and all x, y ∈ A. Then there exists a unique Lie C*-algebra homomorphism H: A → B satisfying (2.11).

Proof. The proof follows from Theorem 4.1 by taking $\phi \left(x,y\right)=\theta \left({∥x∥}_A^r+{∥y∥}_A^r\right)$ for all x, y ∈ A. Then L = 2^r-1and we get the desired result.

Theorem 4.2. Let f: A → B be a mapping with f(0) = 0 for which there exists a function φ: A² → [0, ∞) satisfying (2.2) and (4.23). If there exists an L < 1 such that $\phi \left(x,y\right)\le 2L\phi \left(\frac{x}{2},\frac{y}{2}\right)$ for all x, y ∈ A, then there exists a unique Lie C*-algebra homomorphism H: A → B satisfying (2.12).

Corollary 4.2. Let r > 1 and θ be nonnegative real numbers, and let f: A → B be a mapping satisfying f(0) = 0, (4.2) and (4.3). Then there exists a unique Lie C*-algebra homomorphism H: A → B satisfying (2.14).

Proof. The proof follows from Theorem 4.2 by taking $\phi \left(x,y\right)=\theta \left({∥x∥}_A^r+{∥y∥}_A^r\right)$ for all x, y ∈ A. Then L = 2^1-rand we get the desired result.

5 Stability of Lie derivations on C*-algebras

Definition 5.1. Lat A be a Lie C*-algebras, A ℂ-linear mapping δ: A → A is called a Lie derivation if δ([x, y]) = [δ(x),y] + [x, δ(y)] for all x, y ∈ A.

Throughout this section, assume that A is a Lie C*-algebra with the norm ||.|| _A . In this section, we prove the Hyers-Ulam stability of derivations on Lie C*-algebras for the functional Equation (1.1).

Theorem 5.1. Let f: A → B be a mapping with f(0) = 0 for which there exists a function φ: A² → [0, ∞) satisfying (3.15) such that

$${∥f\left(\left[x,y\right]\right)-\left[f\left(x\right),y\right]-\left[x,f\left(y\right)\right]∥}_B\le \phi \left(x,y\right)$$

(5.26)

for all x, y ∈ A. If there exists an $L<\frac{1}{2}$ such that $\phi \left(x,y\right)\le \frac{L}{2}\phi \left(2x,2y\right)$ for all x, y ∈ A, then there exists a unique Lie derivation δ: A → A satisfying (3.17).

Proof. By the same reasoning as in the proof of Theorem 2.1, there exists a unique ℂ-linear mapping δ: A → A satisfying (3.17). The mapping δ: A → A is given by $\delta \left(x\right)={\text{lim}}_{n\to \infty }{2}^{n}f\left(\frac{x}{2^n}\right)$ for all x ∈ A. It follows from (5.26) that

$$\begin{array}{ll}\hfill {∥\delta \left(\left[x,y\right]\right)-\left[\delta \left(x\right),y\right]-\left[x,\delta \left(y\right)\right]∥}_A& =\underset{n\to \infty }{\text{lim}}{4}^n{∥f\left(\frac{\left[x,y\right]}{4^n}\right)-\left[f\left(\frac{x}{2^n}\right),\frac{y}{2^n}\right]-\left[\frac{x}{2^n},f\left(\frac{y}{2^n}\right)\right]∥}_A\\ \le \underset{n\to \infty }{\text{lim}}{4}^n\phi \left(\frac{x}{2^n},\frac{y}{2^n}\right)\le \underset{n\to \infty }{\text{lim}}{\left(2L\right)}^n\phi \left(x,y\right)=0\end{array}$$

for all x, y ∈ A. So δ([x, y]) = [δ(x),y] + [x, δ(y)] for all x, y ∈ A. Thus δ: A → A is a Lie derivation satisfying (3.17), as desired.

Corollary 5.1. Let 0 < r < 1 and θ be nonnegative real numbers, and let f: A → B be a mapping satisfying f(0) = 0 and (3.4) such that

$${∥f\left(\left[x,y\right]\right)-\left[f\left(x\right),y\right]-\left[x,f\left(y\right)\right]∥}_B\le \theta \left({∥x∥}_A^r+{∥y∥}_A^r\right)$$

(5.27)

for all x, y ∈ A. Then there exists a unique Lie derivation δ: A → A satisfying (3.20).

Proof. The proof follows from Theorem 5.1 by taking $\phi \left(x,y\right)=\theta \left({∥x∥}_A^r+{∥y∥}_A^r\right)$ for all x, y ∈ A. Then L = 2^r-1and we get the desired result.

Theorem 5.2. Let f: A → B be a mapping with f(0) = 0 for which there exists a function φ: A² → [0, ∞) satisfying (3.15) and (5.26). If there exists an L < 1 such that $\phi \left(x,y\right)\le 2L\phi \left(\frac{x}{2},\frac{y}{2}\right)$ for all x, y ∈ A, then there exists a unique Lie derivation δ: A → A satisfying (3.21).

Corollary 5.2. Let r > 1 and θ be nonnegative real numbers, and let f: A → B be a mapping satisfying f(0) = 0, (3.4) and (5.27). Then there exists a unique Lie derivation δ: A → A satisfying (3.22).

Proof. The proof follows from Theorem 5.2 by taking $\phi \left(x,y\right)=\theta \left({∥x∥}_A^r+{∥y∥}_A^r\right)$ for all x, y ∈ A. Then L = 2^1-rand we get the desired result.