On the Stability of (𝛼,𝛽,𝛾)-Derivations and Lie 𝐶^∗-Algebra Homomorphisms on Lie 𝐶^∗-Algebras: A Fixed Points Method

Abstract

We investigate new generalized Hyers-Ulam stability results for (𝛼,𝛽,𝛾)-derivations and Lie 𝐶^∗-algebra homomorphisms on Lie 𝐶^∗-algebras associated with the additive functional equation: ∑^𝑛_𝑖=1𝑓(𝑛𝑥_𝑖+∑^𝑛_{𝑗=1,𝑗≠𝑖}𝑥_𝑗)+𝑓(∑^𝑛_𝑖=1𝑥_𝑖)=2𝑓(∑^𝑛_𝑖=1𝑛𝑥_𝑖).

1. Introduction

The theory of finite dimensional complex Lie algebras is an important part of Lie theory. It has several applications in physics and connections with other parts of mathematics. With an increasing amount of theory and applications concerning Lie algebras of various dimensions, it is becoming necessary to ascertain which tools are applicable for handling them. The miscellaneous characteristics of Lie algebras constitute such tools and have also found applications: Casimir operators [1], derived, lower central and upper central sequences, and the Lie algebra of derivations, radicals, nilradicals, ideals, subalgebras [2, 3], and recently megaideals [4]. These characteristics are particularly crucial when considering possible affinities among Lie algebras. Physically motivated relations between two Lie algebras, namely, contractions and deformations, have been extensively studied [5]. Moreover, in modern industry, various analytical approaches for solving mathematical equations are widely applied in analysis of problems in packaging engineering, and so mathematical modeling and computation methods by using mathematical equations play an important role in application of packaging engineering. From now, we wish to note that mathematical equations for stability properties in this paper can have applications to Packaging Engineering.

A 𝐶^∗-algebra 𝒜 endowed with the Lie product [𝑥,𝑦]=(𝑥𝑦−𝑦𝑥)/2 on 𝒜 is called a Lie 𝐶^∗-algebra. A ℂ-linear mapping 𝛿∶𝒜→𝒜 is called a (𝛼,𝛽,𝛾)-derivation of 𝒜 if there exist 𝛼,𝛽, and 𝛾∈𝐶 such that 𝛼𝛿[𝑥,𝑦]=𝛽[𝛿(𝑥),𝑦]+𝛾[𝑥,𝛿(𝑦)] for all 𝑥,𝑦∈𝒜 [6]. Let 𝒜 and ℬ be Lie 𝐶^∗-algebras. A ℂ-linear mapping 𝐻∶𝒜→ℬ is called a Lie 𝐶^∗-algebra homomorphism if 𝐻([𝑥,𝑦])=[𝐻(𝑥),𝐻(𝑦)] for all 𝑥,𝑦∈𝒜.

The stability problem of functional equations was originated from a question of Ulam [7, 8] concerning the stability of group homomorphisms as follows. Let (𝐺₁,∗) be a group and (𝐺₂,⋄,𝑑) be a metric group with the metric 𝑑(⋅,⋅). Given 𝜖>0, does there exist a 𝛿(𝜖)>0 such that if a mapping ℎ∶𝐺₁→𝐺₂ satisfies the inequality 𝑑(ℎ(𝑥∗𝑦),ℎ(𝑥)⋄ℎ(𝑦))<𝜖 for all 𝑥,𝑦∈𝐺₁, then there is a homomorphism 𝐻∶𝐺₁→𝐺₂ with 𝑑(ℎ(𝑥),𝐻(𝑥))<𝜖 for all 𝑥∈𝐺₁?

If the answer is affirmative, we would say that the equation of a homomorphism 𝐻(𝑥∗𝑦)=𝐻(𝑥)⋄𝐻(𝑦) is stable (see also [9–14]). Cădariu and Radu [15] applied the fixed point method to investigation of the stability of a functional equation. In 2008, Novotný and Hrivnák [6] investigated generalizing the concept of Lie derivations via certain complex parameters, obtained various Lie, and established the structure and properties of (𝛼,𝛽,𝛾)-derivations of Lie algebras. Recently, the generalized Hyers-Ulam stability of problems on 𝐶^∗-algebras associated with functional equations has been investigated by using a fixed point method (see [16–20]). The following fixed point theorem will play an important role in proving our main theorem.

Theorem 1 (see [21]). Suppose that we are given a complete generalized metric space (Ω,𝑑) and a strictly contractive mapping 𝑇∶Ω→Ω with Lipschitz constant 𝐿. Then for each given 𝑥∈Ω, either 𝑑(𝑇^𝑛𝑥,𝑇^𝑛+1𝑥)=∞ for all nonnegative integers 𝑛≥0 or there exists a natural number 𝑛_𝑜 such that (1)𝑑(𝑇^𝑛𝑥,𝑇^𝑛+1𝑥)<∞ for all 𝑛≥𝑛_𝑜;(2)the sequence {𝑇^𝑛𝑥} is convergent to a fixed point 𝑦∗ of 𝑇;(3)𝑦∗ is the unique fixed point of 𝑇 in the set Λ={𝑦∈Ω∶𝑑(𝑇^𝑛𝑜𝑥,𝑦)<∞};(4)𝑑(𝑦,𝑦∗)≤(1/(1−𝐿))𝑑(𝑦,𝑇𝑦) for all 𝑦∈Λ.

Let 𝑘 be a fixed positive integer. We recall that a mapping 𝜌∶𝐴→𝐵 having a domain 𝐴 and a codomain (𝐵,≤) that are both closed under addition is called a contractively subadditive mapping if there exists a constant 𝐿 with 0<𝐿<1 such that 𝜌(𝑥+𝑦)≤𝐿(𝜌(𝑥)+𝜌(𝑦)) and an expansively superadditive mapping if there exists a constant 𝐿 with 0<𝐿<1 such that 𝜌(𝑥+𝑦)≥(1/𝐿)(𝜌(𝑥)+𝜌(𝑦)) for all 𝑥,𝑦∈𝐴. A mapping 𝜌∶𝐴→𝐵 is called a homogeneous of degree 𝑘 if 𝜌(𝜆𝑥)=𝜆^𝑘𝜌(𝑥) for all 𝑥,𝑦∈𝐴. Also, if there exists a constant 𝐿 with 0<𝐿<1 such that a mapping 𝜌∶𝐴^𝑛→𝐵 satisfies 𝜌(𝑥₁,…,𝜆^ℓ𝑥,…,𝑥_𝑛)≤𝜆^ℓ𝐿𝜌(𝑥₁,…,𝑥,…,𝑥_𝑛) for all 𝑥,𝑦∈𝐴 and positive integer 𝜆>1, then we say that 𝜌 is a 𝑛-contractively subhomogeneous mapping if ℓ=1, and 𝜌 is an expansively superhomogeneous mapping if ℓ=−1.

Now, we consider a mapping 𝑓∶𝑋→𝑌 satisfying the following functional equation [22]: ^𝑛_𝑖=1𝑓𝑛𝑥_𝑖+^𝑛_{𝑗=1,𝑗≠𝑖}𝑥_𝑗+𝑓^𝑛_𝑖=1𝑥_𝑖=2𝑓^𝑛_𝑖=1𝑛𝑥_𝑖 (1) for all 𝑥_𝑖,𝑥_𝑗∈𝑋, where 𝑛∈ℤ⁺ is a fixed integer with 𝑛≥2. Note that the mapping 𝑓(𝑥)=𝑎𝑥 is a solution of the functional equation (1). Eskandni [22] investigated the Hyers-Ulam-Rassias stability of the functional equation (1) in quasi-Banach spaces using the direct method.

In this paper, using some strategies from [15, 18, 22], we investigate new generalized Hyers-Ulam stability for (𝛼,𝛽,𝛾)-derivations and Lie 𝐶^∗-algebra homomorphisms on Lie 𝐶^∗-algebras associated with the functional equation (1) via the fixed point method.

Throughout this paper, let 𝒜 and ℬ be Lie 𝐶^∗-algebras and 𝑇¹_1/𝑛𝑜={𝑒^𝑖𝜃∶0≤𝜃≤2𝜋/𝑛_𝑜}. For any mapping 𝑓∶𝒜→ℬ, we define 𝒟_𝜇𝑓𝑥₁,…,𝑥_𝑛=^𝑛_𝑖=1𝑓𝑛𝜇𝑥_𝑖+^𝑛_{𝑗=1,𝑗≠𝑖}𝜇𝑥_𝑗+𝑓^𝑛_𝑖=1𝜇𝑥_𝑖−2𝑓^𝑛_𝑖=1𝑛𝜇𝑥_𝑖,𝒟_{𝛼,𝛽,𝛾}𝑓𝑥,𝑦=𝛼𝑓𝑥,𝑦−𝛽𝑓(𝑥),𝑦−𝛾𝑥,𝑓𝑦 (2) for all 𝑥₁,…,𝑥_𝑛,  𝑥,𝑦∈𝑋  (𝑛≥2),  𝜇∈𝑇¹_1/𝑛𝑜,  and some 𝛼,𝛽,𝛾∈ℂ.

2. Main Results

In this section, we point out the stability of the functional equation (1) on Lie 𝐶^∗-algebras using a fixed point method. Let us recall that a mapping 𝑑∶𝑋²→[0,∞) is called a generalized metric on a nonempty set 𝑋 if (i)  𝑑(𝑥,𝑦)=0 if and only if 𝑥=𝑦, (ii)  𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥), and (iii)  𝑑(𝑥,𝑧)≤𝑑(𝑥,𝑦)+𝑑(𝑦,𝑧) for all 𝑥,𝑦, and 𝑧∈𝑋. We need the following lemma to prove the main result in this paper.

Lemma 2 (see [22]). Let 𝑋,𝑌 be linear spaces and 𝑛≥2 a fixed positive integer. A mapping 𝑓∶𝑋→𝑌 satisfies the functional equation (1) if and only if 𝑓 is additive.

Theorem 3. Assume that there exist a contractively subadditive mapping 𝜙∶𝒜^𝑛→[0,∞) and a 2-contractively subhomogeneous mapping 𝜓∶𝒜²→[0,∞) with a constant 0<𝐿<1 such that a mapping 𝑓∶𝒜→𝒜 satisfies 𝒟_𝜇𝑓𝑥₁,…𝑥_𝑛≤𝜙𝑥₁,…,𝑥_𝑛,𝒟_{𝛼,𝛽,𝛾}𝑓𝑥,𝑦≤𝜓𝑥,𝑦 (3)(4) for all 𝑥₁,…,𝑥_𝑛,  𝑥,𝑦∈𝒜, 𝜇∈𝑇¹_1/𝑛𝑜, and some 𝛼,𝛽, and 𝛾∈ℂ. Then there exists a unique (𝛼,𝛽,𝛾)-derivation 𝛿∶𝒜→𝒜 which satisfies (1) and the inequality 𝑓(𝑥)(𝑥≤1−𝛿)𝑛()𝜙()1−𝐿𝑥,0,…,0 (5) for all 𝑥∈𝒜.

Proof. Let Ω be a set of all mappings from 𝒜 into 𝒜, and introduce a generalized metric on Ω as follows: 𝑑𝑔,ℎ=inf𝐶∈ℝ⁺∶𝑔(𝑥)(𝑥().−ℎ)≤𝐶𝜙𝑥,0,…,0∀𝑥∈𝒜 (6) Then (Ω,𝑑) is a generalized complete metric space. Now, we consider the mapping 𝑇∶Ω→Ω defined by (𝑥)=1𝑇𝑔𝑛𝑔(𝑛𝑥) (7) for all 𝑔∈Ω, 𝑥∈𝒜, and 𝑛∈ℤ⁺. Let 𝑔,ℎ∈Ω and 𝐶∈(0,∞) be an arbitrary constant with 𝑑(𝑔,ℎ)≤𝐶. Then we have ‖𝑔(𝑥)−ℎ(𝑥)‖≤𝐶𝜙(𝑥,0,…,0) and (𝑥)(𝑥≤𝐶𝑇𝑔−𝑇ℎ)𝑛𝜙(𝑛𝑥,0,…,0)≤𝐿𝐶𝜙(𝑥,0,…,0) (8) for all 𝑥∈𝒜, which means that 𝑑(𝑇𝑔,𝑇ℎ)≤𝐿𝑑(𝑔,ℎ) for all 𝑔,ℎ∈Ω. Thus 𝑇 is a strictly contractive self-mapping on Ω with the Lipschitz constant 𝐿.
Letting 𝑥₁=𝑥, 𝑥₂=𝑥₃=⋯=𝑥_𝑛=0, and 𝜇=1 in (3), we have 𝑓(𝑛𝑥)−𝑛𝑓(𝑥)≤𝜙(𝑥,0,…,0), (9) which gives 𝑓(𝑥)−1𝑛𝑓(≤1𝑛𝑥)𝑛𝜙(𝑥,0,…,0) (10) for all 𝑥∈𝒜 and 𝑛∈ℤ⁺ with 𝑛≥2. Then we have 𝑑(𝑇𝑓,𝑓)≤1/𝑛<∞. From Theorem 1, there is a mapping 𝛿 which is a unique fixed point of 𝑇 in the set Ω₁={𝑔∈Ω∶𝑑(𝑓,𝑔)<∞} such that 𝛿(𝑥)=lim_𝑚→∞1𝑛^𝑚𝑓𝑛^𝑚𝑥 (11) for all 𝑥∈𝒜, since lim_𝑚→∞𝑑(𝑇^𝑚𝑓,𝛿)=0. Again by Theorem 1, we have 𝑑≤1𝑓,𝛿𝑑≤11−𝐿𝑇𝑓,𝑓𝑛(1−𝐿) (12) for all 𝑥∈𝒜. Thus, inequality (5) holds.
It follows from (3), (11), and the contractively subadditive mapping of 𝜙 that 𝒟_𝜇𝛿𝑥₁,…,𝑥_𝑛≤lim_𝑚→∞1𝑛^𝑚𝒟_𝜇𝑓𝑛^𝑚𝑥₁,…,𝑛^𝑚𝑥_𝑛≤lim_𝑚→∞𝐿^𝑚𝜙𝑥₁,…,𝑥_𝑛=0, (13) which gives 𝒟_𝜇𝛿(𝑥₁,…,𝑥_𝑛)=0 for all 𝑥₁,…,𝑥_𝑛∈𝒜 and 𝜇∈𝑇¹_1/𝑛𝑜. If we put 𝜇=1 in 𝒟_𝜇𝛿(𝑥₁,…,𝑥_𝑛)=0; then 𝛿 satisfies the functional equation (1), and so 𝛿 is additive by Lemma 2. Also, we let 𝑥₁=𝑥 and 𝑥₂=⋯=𝑥_𝑛=0, then 𝛿(𝜇𝑥)=𝜇𝛿(𝑥). By the same reasoning as that of the proof of Theorem 2.1 of [23], the mapping 𝛿 is ℂ-linear. So, it follows from the 2-contractively subhomogeneous of 𝜓, (4), and (11) that 𝒟_{𝛼,𝛽,𝛾}𝛿𝑥,𝑦≤lim_𝑚→∞1𝑛^2𝑚𝜓𝑛^𝑚𝑥,𝑛^𝑚𝑦≤lim_𝑚→∞𝐿^2𝑚𝜓𝑥,𝑦=0 (14) for all 𝑥,𝑦∈𝒜 and some 𝛼,𝛽, and 𝛾∈ℂ. Then we have 𝛼𝛿𝑥,𝑦=𝛽𝛿(𝑥),𝑦+𝛾𝑥,𝛿𝑦 (15) for all 𝑥,𝑦∈𝒜 and some 𝛼,𝛽, and 𝛾∈ℂ. Thus, 𝛿 is a unique Lie (𝛼,𝛽,𝛾)-derivation on Lie 𝐶^∗-algebra 𝒜 satisfying (5). This completes the proof.

Corollary 4. Let 0<𝑟<1,  0<𝑠<2, and 𝜃 be nonnegative real numbers. Suppose that a mapping 𝑓∶𝒜→𝒜 satisfies 𝒟_𝜇𝑓𝑥₁,…𝑥_𝑛≤𝜃^𝑛_𝑖=1𝑥_𝑖^𝑟,𝒟_{𝛼,𝛽,𝛾}𝑓𝑥,𝑦≤𝜃‖𝑥‖^𝑠+𝑦^𝑠 (16) for all 𝑥₁,…,𝑥_𝑛,  𝑥,𝑦∈𝒜,  𝜇∈𝑇¹_1/𝑛𝑜, and some 𝛼,𝛽, and 𝛾∈ℂ. Then there exists a unique (𝛼,𝛽,𝛾)-derivation 𝛿∶𝒜→𝒜 such that 𝑓(𝑥)(𝑥≤𝜃−𝛿)𝑛−𝑛^𝑟‖𝑥‖^𝑟, (17) for all 𝑥∈𝒜 and 𝑛∈ℤ⁺ with 𝑛≥2.

Proof. The proof follows from Theorem 3 by taking 𝜙(𝑥₁,…,𝑥_𝑛)=𝜃∑^𝑛_𝑖=1‖𝑥_𝑖‖^𝑟,  𝜓(𝑥,𝑦)=𝜃(‖𝑥‖^𝑠+‖𝑦‖^𝑠) for all 𝑥₁,…,𝑥_𝑛,  𝑥,𝑦∈𝒜. Then we can choose 𝐿=𝑛^𝑟−1, and we obtain the desired result. This completes the proof.

Theorem 5. Assume that there exist an expansively superadditive mapping 𝜙∶𝒜^𝑛→[0,∞) and a 2-expansively superhomogeneous mapping 𝜓∶𝒜²→[0,∞) with a constant 0<𝐿<1 such that a mapping 𝑓∶𝒜→𝒜 satisfies (3) and (4). Then there exists a unique (𝛼,𝛽,𝛾)-derivation 𝛿∶𝒜→𝒜 which satisfies (1) and the inequality 𝑓(𝑥)(𝑥≤𝐿−𝛿)𝑛()𝜙()1−𝐿𝑥,0,…,0 (18) for all 𝑥∈𝒜.

Proof. Let Ω and 𝑑 be as in the proof of Theorem 3. Then (Ω,𝑑) becomes a generalized complete metric space, and we consider the mapping 𝑇∶Ω→Ω defined by (𝑇𝑔)(𝑥)=𝑛𝑔(𝑥/𝑛) for all 𝑔∈Ω and 𝑥∈𝒜. So, 𝑑(𝑇𝑔,𝑇ℎ)≤𝐿𝑑(𝑔,ℎ) for all 𝑔,ℎ∈Ω. It follows from (9) that 𝑓(𝑥)𝑥−𝑛𝑓𝑛≤𝐿𝑛𝜙(𝑥,0,…,0) (19) for all 𝑥∈𝒜 and 𝑛∈ℤ⁺ with 𝑛≥2. Then, 𝑑(𝑇𝑓,𝑓)≤𝐿/𝑛<∞. From Theorem 1, there exists a unique mapping 𝛿 which is a unique fixed point of 𝑇 in the set Ω₁={𝑔∈Ω∶𝑑(𝑓,𝑔)<∞} such that 𝛿(𝑥)=lim_𝑚→∞𝑛^𝑚𝑓(𝑥/𝑛^𝑚) for all 𝑥∈𝒜. Thus, we have 𝑑≤1𝑓,𝛿𝑑≤𝐿1−𝐿𝑇𝑓,𝑓𝑛(),1−𝐿 (20) which implies that (18) holds. The remaining assertion goes through in similar method to the corresponding part of Theorem 3. This completes the proof.

Corollary 6. Let 𝑟>1,  𝑠>2 and 𝜃 be nonnegative real numbers. Suppose that a mapping 𝑓∶𝒜→𝒜 satisfies (16). Then there exists a unique (𝛼,𝛽,𝛾)-derivation 𝛿∶𝒜→𝒜 such that 𝑓(𝑥)(𝑥≤𝜃−𝛿)𝑛^𝑟‖𝑥‖−𝑛^𝑟, (21) for all 𝑥∈𝒜 and 𝑛∈ℤ⁺ with 𝑛≥2.

Proof. The proof follows from Theorem 5 by taking 𝜙(𝑥₁,…,𝑥_𝑛)=𝜃∑^𝑛_𝑖=1‖𝑥_𝑖‖^𝑟,  𝜓(𝑥,𝑦)=𝜃(‖𝑥‖^𝑠+‖𝑦‖^𝑠) for all 𝑥₁,…,𝑥_𝑛,  𝑥,𝑦∈𝒜. Then we can choose 𝐿=𝑛^1−𝑟, and we obtain the desired result. This completes the proof.

Next, we establish another theorem about the stability of the functional equation (1).

Theorem 7. Assume that there exists a contractively subadditive mapping 𝜑∶𝒜^𝑛+2→[0,∞) with a constant 𝐿<1 such that a mapping 𝑓∶𝒜→𝒜 satisfies 𝒟_𝜇𝑓𝑥₁,…,𝑥_𝑛+𝒟_{𝛼,𝛽,𝛾}𝑓𝑥,𝑦≤𝜑𝑥₁,…,𝑥_𝑛,𝑥,𝑦 (22) for all 𝑥₁,…,𝑥_𝑛,  𝑥,𝑦∈𝒜, 𝜇∈𝑇¹_1/𝑛𝑜, and some 𝛼,𝛽, and 𝛾∈ℂ. Then there exists a unique (𝛼,𝛽,𝛾)-derivation 𝛿∶𝒜→𝒜 which satisfies (1) and the inequality 𝑓(𝑥)(𝑥≤1−𝛿)𝑛()𝜑()1−𝐿𝑥,0,…,0,0,0 (23) for all 𝑥∈𝒜.

Proof. Substituting 𝑥₁=𝑎, 𝑥₂=⋯,𝑥_𝑛=𝑥=𝑦=0, and 𝜇=1 in (22), we obtain 1𝑛𝑓()(𝑎≤1𝑛𝑎−𝑓)𝑛𝜑(𝑎,0,…,0,0,0) (24) for all 𝑎∈𝒜 and a positive integer 𝑛≥2. Let Ω and 𝑑 be as in the proof of Theorem 3 such that (Ω,𝑑) becomes a generalized complete metric space. Let a mapping 𝑇∶Ω→Ω defined by (𝑇𝑔)(𝑥)=(1/𝑛)𝑔(𝑛𝑥) for all 𝑥∈𝒜,  𝑔∈Ω. Then, we have 𝑑(𝑇𝑔,𝑇ℎ)≤𝐿𝑑(𝑔,ℎ) for all 𝑔,ℎ∈Ω. It follows from (24) that 𝑑(𝑇𝑓,𝑓)≤1/𝑛<∞. The remaining assertion goes through in a similar way to the corresponding part of Theorem 3. This completes the proof.

Corollary 8. Assume that there exists an expansively superadditive mapping 𝜑∶𝒜^𝑛+2→[0,∞) with a constant 𝐿<1 such that a mapping 𝑓∶𝒜→𝒜 satisfies (22). Then there exists a unique (𝛼,𝛽,𝛾)-derivation 𝛿∶𝒜→𝒜 which satisfies (1) and the inequality 𝑓(𝑥)(𝑥≤𝐿−𝛿)𝑛()𝜑()1−𝐿𝑥,0,…,0,0,0 (25) for all 𝑥∈𝒜.

Next, we investigate the Lie 𝐶^∗-algebra homomorphisms on Lie 𝐶^∗-algebras associated with the functional equation (1). The results in Theorems 9 and 10 are similar to those in [24].

Theorem 9. Assume that there exist a contractively subadditive mapping 𝜙∶𝒜^𝑛→[0,∞) and a 2-contractively subhomogeneous mapping 𝜓∶𝒜²→[0,∞) with a constant 0<𝐿<1 such that a mapping 𝑓∶𝒜→ℬ satisfies (3) and 𝑓𝑥,𝑦−𝑓(𝑥),𝑓𝑦≤𝜓𝑥,𝑦 (26) for all 𝑥,𝑦∈𝒜 and 𝜇∈𝑇¹_1/𝑛𝑜. Then there exists a unique Lie 𝐶^∗-algebra homomorphism 𝐻∶𝒜→ℬ satisfying 𝑓(𝑥)(𝑥≤1−𝐻)𝑛()𝜙()1−𝐿𝑥,0,…,0 (27) for all 𝑥∈𝒜.

Proof. By the same method as in Theorem 3, we obtain a ℂ-linear mapping 𝐻∶𝒜→ℬ satisfying (27). The mapping is given by 𝐻(𝑥)=lim_𝑚→∞(1/𝑛^𝑚)𝑓(𝑛^𝑚𝑥) for all 𝑥∈𝒜. It follows from (25) that 𝐻𝑥,𝑦−𝐻(𝑥),𝐻𝑦=lim_𝑚→∞1𝑛^2𝑚𝑓𝑛^2𝑚𝑥,𝑦−𝑓𝑛^𝑚𝑥,𝑓𝑛^𝑚𝑦≤lim_𝑚→∞𝐿^2𝑚𝜓𝑥,𝑦=0 (28) for all 𝑥,𝑦∈𝒜. Thus, 𝐻 is a Lie 𝐶^∗-algebra homomorphism. This completes the proof.

Theorem 10. Assume that there exist an expansively superadditive mapping 𝜙∶𝒜^𝑛→[0,∞) and a 2-expansively superhomogeneous mapping 𝜓∶𝒜²→[0,∞) with a constant 0<𝐿<1 such that a mapping 𝑓∶𝒜→ℬ satisfies (3) and (26). Then there exists a unique Lie 𝐶^∗-algebra homomorphism 𝐻∶𝒜→ℬ satisfying 𝑓(𝑥)(𝑥≤𝐿−𝐻)𝑛()𝜙()1−𝐿𝑥,0,…,0 (29) for all 𝑥∈𝒜.

Proof. The proof is similar to the proofs of Theorems 5 and 9.

Corollary 11. Let 𝑟<1,  𝑠<2, and 𝜃 be nonnegative real numbers. Suppose that a mapping 𝑓∶𝒜→ℬ satisfies 𝒟_𝜇𝑓𝑥₁,…,𝑥_𝑛≤𝜃^𝑛_𝑖=1𝑥_𝑖^𝑟,𝑓𝑥,𝑦−𝑓(𝑥),𝑓𝑦≤𝜃‖𝑥‖^𝑠+𝑦^𝑠 (30) for all 𝑥₁,…,𝑥_𝑛,  𝑥,𝑦∈𝒜,  𝜇∈𝑇¹_1/𝑛𝑜. Then there exists a unique Lie 𝐶^∗-algebra homomorphism 𝐻∶𝒜→ℬ satisfying 𝑓(𝑥)(𝑥≤𝜃−𝐻)𝑛−𝑛^𝑟‖𝑥‖^𝑟, (31) for all 𝑥∈𝒜 and 𝑛∈ℤ⁺ with 𝑛≥2.

Acknowledgments

The authors would like to thank the referee and editors for their comments that helped them improve this paper.