Stability of the Jensen-Type Functional Equation in 𝐶^∗-Algebras: A Fixed Point Approach

Abstract

Using fixed point methods, we prove the generalized 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-type functional equation 𝑓((𝑥+𝑦)/2)+𝑓((𝑥−𝑦)/2)=𝑓(𝑥).

1. Introduction and Preliminaries

The stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms. Hyers [2] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers' theorem was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference. The paper of Rassias [4] has provided a lot of influence in the development of what we call generalized Hyers-Ulam stability of functional equations. A generalization of the Rassias theorem was obtained by Găvruţa [5] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias' approach. In 1982, Rassias [6] followed the innovative approach of the Rassias' theorem [4] in which he replaced the factor ‖𝑥‖^𝑝+‖𝑦‖^𝑝 by ‖𝑥‖^𝑝⋅‖𝑦‖^𝑞 for 𝑝,𝑞∈ℝ with 𝑝+𝑞≠1. 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 (see [4, 7–27]).

We recall a fundamental result in fixed point theory.

Let 𝑋 be a set. A function 𝑑∶𝑋×𝑋→[0,∞] is called a generalized metric on 𝑋 if 𝑑 satisfies

(1)𝑑(𝑥,𝑦)=0 if and only if 𝑥=𝑦;(2)𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥) for all 𝑥,𝑦∈𝑋;(3)𝑑(𝑥,𝑧)≤𝑑(𝑥,𝑦)+𝑑(𝑦,𝑧) for all 𝑥,𝑦,𝑧∈𝑋.

Theorem 1.1 (see [28, 29]). Let (𝑋,𝑑) be a complete generalized metric space and let 𝐽∶𝑋→𝑋 be a strictly contractive mapping with Lipschitz constant 𝐿<1. Then for each given element 𝑥∈𝑋, either for all nonnegative integers 𝑛 or there exists a positive integer 𝑛₀ such that
(1)𝑑(𝐽^𝑛𝑥,𝐽^𝑛+1𝑥)<∞, for all 𝑛≥𝑛₀;(2)the sequence {𝐽^𝑛𝑥} converges to a fixed point 𝑦^∗ of 𝐽;(3)𝑦^∗ is the unique fixed point of 𝐽 in the set 𝑌={𝑦∈𝑋∣𝑑(𝐽^𝑛0𝑥,𝑦)<∞};(4)𝑑(𝑦,𝑦^∗)≤(1/(1−𝐿))𝑑(𝑦,𝐽𝑦) for all 𝑦∈𝑌.

By the using fixed point method, the stability problems of several functional equations have been extensively investigated by a number of authors (see [17, 30–33]).

This paper is organized as follows: in Sections 2 and 3, using the fixed point method, we prove the generalized Hyers-Ulam stability of homomorphisms in 𝐶^∗-algebras and of derivations on 𝐶^∗-algebras for the Jensen-type functional equation.

In Sections 4 and 5, using the fixed point method, we prove the generalized Hyers-Ulam stability of homomorphisms in Lie 𝐶^∗-algebras and of derivations on Lie 𝐶^∗-algebras for the Jensen-type functional equation.

2. Stability of Homomorphisms in 𝐶^∗-Algebras

Throughout this section, assume that 𝐴 is a 𝐶^∗-algebra with norm ‖⋅‖_𝐴 and that 𝐵 is a 𝐶^∗-algebra with norm ‖⋅‖_𝐵.

For a given mapping 𝑓∶𝐴→𝐵, we definefor all 𝜇∈𝕋¹∶={𝜈∈ℂ∶|𝜈|=1} 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,𝑦∈𝐴.

We prove the generalized Hyers-Ulam stability of homomorphisms in 𝐶^∗-algebras for the functional equation 𝐷_𝜇𝑓(𝑥,𝑦)=0.

Theorem 2.1. Let 𝑓∶𝐴→𝐵 be a mapping for which there exists a function 𝜑∶𝐴²→[0,∞) such that for all 𝐿<1 and all 𝜑(𝑥,𝑦)≤2𝐿𝜑(𝑥/2,𝑦/2) If there exists an 𝑥,𝑦∈𝐴, such that 𝐶^∗ for all 𝐻∶𝐴→𝐵 then there exists a unique ‖‖𝑓(𝑥)−𝐻(𝑥)‖‖_𝐵≤𝐿1−𝐿𝜑(𝑥,0)(2.5)-algebra homomorphism 𝑥∈𝐴. such that for all lim_𝑗→∞2^−𝑗𝜑2^𝑗𝑥,2^𝑗𝑦=0(2.6)Proof. It follows from 𝑥,𝑦∈𝐴. thatfor all 𝑋
Consider the setand introduce the generalized metric on (𝑋,𝑑):It is easy to show that 𝐽𝑔(𝑥)∶=12𝑔(2𝑥)(2.9) is complete.
Now we consider the linear mapping 𝑥∈𝐴. such thatfor all 𝑔,ℎ∈𝑋.
By [28, Theorem 3.1],for all 𝑦=0
Letting ‖‖‖2𝑓𝑥2−𝑓(𝑥)‖‖‖_𝐵≤𝜑(𝑥,0)(2.11) and 𝑥∈𝐴. in (2.2), we getfor all 𝑥∈𝐴. Sofor all 𝐻∶𝐴→𝐵 Hence 𝐻
By Theorem 1.1, there exists a mapping 𝐽, such that
(1)𝐻(2𝑥)=2𝐻(𝑥)(2.13) is a fixed point of 𝑥∈𝐴. that is,for all 𝐽 The mapping 𝑌=𝑔∈𝑋∶𝑑(𝑓,𝑔)<∞.(2.14) is a unique fixed point of 𝐻 in the setThis implies that ‖‖𝐻(𝑥)−𝑓(𝑥)‖‖_𝐵≤𝐶𝜑(𝑥,0)(2.15) is a unique mapping satisfying (2.13) such that there exists 𝑥∈𝐴. satisfyingfor all 𝑛→∞.(2)lim_𝑛→∞𝑓(2^𝑛𝑥)2^𝑛=𝐻(𝑥)(2.16) as 𝑥∈𝐴. This implies the equalityfor all 𝑑(𝑓,𝐻)≤𝐿1−𝐿.(2.17)(3)‖‖‖𝐻𝑥+𝑦2+𝐻𝑥−2𝑦−𝐻(𝑥)‖‖‖_𝐵=lim_𝑛→∞12^𝑛‖‖𝑓2^𝑛−1(𝑥+𝑦)+𝑓2^𝑛−1(𝑥−𝑦)−𝑓2^𝑛𝑥‖‖_𝐵≤lim_𝑛→∞12^𝑛𝜑2^𝑛𝑥,2^𝑛𝑦=0(2.18) which implies the inequalityThis implies that the inequality (2.5) holds.
It follows from(2.2),(2.6), and (2.16) thatfor all 𝑥,𝑦∈𝐴. Sofor all 𝑤=(𝑥−𝑦)/2 Letting 𝐻(𝑧)+𝐻(𝑤)=𝐻(𝑧+𝑤)(2.20) and 𝑧,𝑤∈𝐴. in (2.19), we getfor all 𝐻(𝑧+𝑤)=𝐻(𝑧)+𝐻(𝑤) So the mapping 𝑧,𝑤∈𝐴. is Cauchy additive, that is, 𝑦=𝑥 for all 𝜇𝑓(𝑥)=𝑓(𝜇𝑥)(2.21)
Letting 𝜇∈𝕋¹ in (2.2), we getfor all 𝜇𝐻(𝑥)=𝐻(𝜇𝑥)(2.22) and all 𝜇∈𝕋¹ By a similar method to above, we getfor all 𝐻∶𝐴→𝐵 and all ℂ Thus one can show that the mapping ‖‖𝐻(𝑥𝑦)−𝐻(𝑥)𝐻(𝑦)‖‖_𝐵=lim_𝑛→∞14^𝑛‖‖𝑓4^𝑛𝑥𝑦−𝑓2^𝑛𝑥𝑓2^𝑛𝑦‖‖_𝐵≤lim_𝑛→∞14^𝑛𝜑2^𝑛𝑥,2^𝑛𝑦≤lim_𝑛→∞12^𝑛𝜑2^𝑛𝑥,2^𝑛𝑦=0(2.23) is 𝑥,𝑦∈𝐴.-linear.
It follows from (2.3) thatfor all 𝑥,𝑦∈𝐴. Sofor all 𝑥∈𝐴.
It follows from (2.4) thatfor all 𝑥∈𝐴. Sofor all 𝐶^∗
Thus 0<𝑟<1 is a 𝜃-algebra homomorphism satisfying (2.5), as desired.Corollary 2.2. Let 𝑓∶𝐴→𝐵 and ‖‖𝐷_𝜇𝑓(𝑥,𝑦)‖‖_𝐵≤𝜃‖𝑥‖^𝑟𝐴+‖𝑦‖^𝑟𝐴,(2.27)‖‖𝑓(𝑥𝑦)−𝑓(𝑥)𝑓(𝑦)‖‖_𝐵≤𝜃‖𝑥‖^𝑟𝐴+‖𝑦‖^𝑟𝐴,(2.28)‖‖𝑓𝑥^∗−𝑓(𝑥)^∗‖‖_𝐵≤2𝜃‖𝑥‖^𝑟𝐴(2.29) be nonnegative real numbers, and let 𝜇∈𝕋¹ be a mapping such that for all ‖‖𝑓(𝑥)−𝐻(𝑥)‖‖_𝐵≤2^𝑟𝜃2−2^𝑟‖𝑥‖^𝑟𝐴(2.30) and all 𝑥∈𝐴. Then there exists a unique 𝜑(𝑥,𝑦)∶=𝜃‖𝑥‖^𝑟𝐴+‖𝑦‖^𝑟𝐴(2.31)-algebra homomorphism 𝑥,𝑦∈𝐴. such that for all 𝑓∶𝐴→𝐵Proof. The proof follows from Theorem 2.1 by takingfor all 𝐿<1 Then 𝜑(𝑥,𝑦)≤(1/2)𝐿𝜑(2𝑥,2𝑦) and we get the desired result.

Theorem 2.3. Let 𝑥,𝑦∈𝐴, be a mapping for which there exists a function 𝐶^∗ satisfying (2.2), (2.3), and (2.4). If there exists an 𝐻∶𝐴→𝐵 such that ‖‖𝑓(𝑥)−𝐻(𝑥)‖‖_𝐵≤𝐿2−2𝐿𝜑(𝑥,0)(2.32) for all 𝑥∈𝐴. then there exists a unique 𝐽∶𝑋→𝑋-algebra homomorphism 𝐽𝑔(𝑥)∶=2𝑔𝑥2(2.33) such that for all ‖‖‖𝑓(𝑥)−2𝑓𝑥2‖‖‖_𝐵≤𝜑𝑥2,0≤𝐿2𝜑(𝑥,0)(2.34)Proof. We consider the linear mapping 𝑥∈𝐴. such thatfor all 𝐻∶𝐴→𝐵
It follows from (2.11) thatfor all 𝐽, Hence 𝐻(2𝑥)=2𝐻(𝑥)(2.35)
By Theorem 1.1, there exists a mapping 𝑥∈𝐴. such that
(1)𝐻 is a fixed point of 𝐽 that is,for all 𝐻 The mapping 𝐶∈(0,∞) is a unique fixed point of ‖‖𝐻(𝑥)−𝑓(𝑥)‖‖_𝐵≤𝐶𝜑(𝑥,0)(2.37) in the setThis implies that 𝑑(𝐽^𝑛𝑓,𝐻)→0 is a unique mapping satisfying (2.35) such that there exists 𝑛→∞. satisfyingfor all 𝑥∈𝐴.(2)𝑑(𝑓,𝐻)≤(1/(1−𝐿))𝑑(𝑓,𝐽𝑓), as 𝑑(𝑓,𝐻)≤𝐿2−2𝐿,(2.39) This implies the equalityfor all 𝜃(3)𝑓∶𝐴→𝐵 which implies the inequalitywhich implies that the inequality (2.32) holds.
The rest of the proof is similar to the proof of Theorem 2.1.Corollary 2.4. Let 𝐻∶𝐴→𝐵 and ‖‖𝑓(𝑥)−𝐻(𝑥)‖‖_𝐵≤𝜃2^𝑟−2‖𝑥‖^𝑟𝐴(2.40) be nonnegative real numbers, and let 𝑥∈𝐴. be a mapping satisfying (2.27), (2.28) and (2.29). Then there exists a unique 𝜑(𝑥,𝑦)∶=𝜃‖𝑥‖^𝑟𝐴+‖𝑦‖^𝑟𝐴(2.41)-algebra homomorphism 𝑥,𝑦∈𝐴. such that for all 𝑓∶𝐴→𝐵Proof. The proof follows from Theorem 2.3 by takingfor all 𝐿<1 Then 𝜑(𝑥,3𝑥)≤2𝐿𝜑(𝑥/2,3𝑥/2) and we get the desired result.

Theorem 2.5. Let 𝑥∈𝐴, be an odd mapping for which there exists a function 𝐶^∗ satisfying (2.2), (2.3), (2.4) and (2.6). If there exists an 𝐻∶𝐴→𝐵 such that ‖‖𝑓(𝑥)−𝐻(𝑥)‖‖_𝐵≤12−2𝐿𝜑(𝑥,3𝑥)(2.42) for all 𝑥∈𝐴. then there exists a unique 𝑋∶={𝑔∶𝐴⟶𝐵}(2.43)-algebra homomorphism 𝑋 such that for all (𝑋,𝑑)Proof. Consider the setand introduce the generalized metric on 𝐽𝑔(𝑥)∶=12𝑔(2𝑥)(2.45):It is easy to show that 𝑑(𝐽𝑔,𝐽ℎ)≤𝐿𝑑(𝑔,ℎ)(2.46) is complete.
Now we consider the linear mapping 𝑔,ℎ∈𝑋. such thatfor all 𝑦
By [28, Theorem 3.1],for all ‖‖𝑓(2𝑥)−2𝑓(𝑥)‖‖_𝐵≤𝜑(𝑥,3𝑥)(2.47)
Letting 𝑥∈𝐴. and relpacing ‖‖‖𝑓(𝑥)−12𝑓(2𝑥)‖‖‖_𝐵≤12𝜑(𝑥,3𝑥)(2.48) by 𝑥∈𝐴. in (2.2), we getfor all 𝐻∶𝐴→𝐵 Sofor all 𝐽, Hence 𝐻(2𝑥)=2𝐻(𝑥)(2.49)
By Theorem 1.1, there exists a mapping 𝑥∈𝐴. such that
(1)𝐻 is a fixed point of 𝐽 that is,for all 𝐻 The mapping 𝐶∈(0,∞) is a unique fixed point of ‖‖𝐻(𝑥)−𝑓(𝑥)‖‖_𝐵≤𝐶𝜑(𝑥,3𝑥)(2.51) in the setThis implies that 𝑑(𝐽^𝑛𝑓,𝐻)→0 is a unique mapping satisfying (2.49) such that there exists 𝑛→∞. satisfyingfor all 𝑥∈𝐴.(2)𝑑(𝑓,𝐻)≤(1/(1−𝐿))𝑑(𝑓,𝐽𝑓), as 𝑑(𝑓,𝐻)≤12−2𝐿.(2.53) This implies the equalityfor all 𝜃(3)𝑓∶𝐴→𝐵 which implies the inequalityThis implies that the inequality(2.42) holds.
The rest of the proof is similar to the proof of Theorem 2.1.Corollary 2.6. Let 𝜇∈𝕋¹ and 𝑥,𝑦∈𝐴. be nonnegative real numbers, and let 𝐶^∗ be an odd mapping such that for all ‖‖𝑓(𝑥)−𝐻(𝑥)‖‖_𝐵≤3^𝑟𝜃2−2^2𝑟‖𝑥‖^𝐴2𝑟(2.55) and all 𝑥∈𝐴. Then there exists a unique 𝜑(𝑥,𝑦)∶=𝜃⋅‖𝑥‖^𝑟𝐴⋅‖𝑦‖^𝑟𝐴(2.56)-algebra homomorphism 𝑥,𝑦∈𝐴. such that for all 𝑓∶𝐴→𝐵Proof. The proof follows from Theorem 2.5 by takingfor all 𝐿<1 Then 𝜑(𝑥,3𝑥)≤(1/2)𝐿𝜑(2𝑥,6𝑥) and we get the desired result.

Theorem 2.7. Let 𝑥∈𝐴, be an odd mapping for which there exists a function 𝐶^∗ satisfying (2.2), (2.3) and (2.4). If there exists an 𝐻∶𝐴→𝐵 such that ‖‖𝑓(𝑥)−𝐻(𝑥)‖‖_𝐵≤𝐿2−2𝐿𝜑(𝑥,3𝑥)(2.57) for all 𝑥∈𝐴. then there exists a unique 𝐽∶𝑋→𝑋-algebra homomorphism 𝐽𝑔(𝑥)∶=2𝑔𝑥2(2.58) such that for all ‖‖‖𝑓(𝑥)−2𝑓𝑥2‖‖‖_𝐵≤𝜑𝑥2,32𝑥≤𝐿2𝜑(𝑥,3𝑥)(2.59)Proof. We consider the linear mapping 𝑥∈𝐴. such thatfor all 𝐻∶𝐴→𝐵
It follows from (2.47) thatfor all 𝐽, Hence 𝐻(2𝑥)=2𝐻(𝑥)(2.60)
By Theorem 1.1, there exists a mapping 𝑥∈𝐴. such that
(1)𝐻 is a fixed point of 𝐽 that is,for all 𝐻 The mapping 𝐶∈(0,∞) is a unique fixed point of ‖‖𝐻(𝑥)−𝑓(𝑥)‖‖_𝐵≤𝐶𝜑(𝑥,3𝑥)(2.62) in the setThis implies that 𝑑(𝐽^𝑛𝑓,𝐻)→0 is a unique mapping satisfying (2.60) such that there exists 𝑛→∞. satisfyingfor all 𝑥∈𝐴.(2)𝑑(𝑓,𝐻)≤(1/(1−𝐿))𝑑(𝑓,𝐽𝑓), as 𝑑(𝑓,𝐻)≤𝐿2−2𝐿,(2.64) This implies the equalityfor all 𝜃(3)𝑓∶𝐴→𝐵 which implies the inequalitywhich implies that the inequality (2.57) holds.
The rest of the proof is similar to the proof of Theorem 2.1.Corollary 2.8. Let 𝐻∶𝐴→𝐵 and ‖‖𝑓(𝑥)−𝐻(𝑥)‖‖_𝐵≤𝜃2^2𝑟−2‖𝑥‖^𝐴2𝑟(2.65) be nonnegative real numbers, and let 𝑥∈𝐴. be an odd mapping satisfying (2.54). Then there exists a unique 𝜑(𝑥,𝑦)∶=𝜃⋅‖𝑥‖^𝑟𝐴⋅‖𝑦‖^𝑟𝐴(2.66)-algebra homomorphism 𝑥,𝑦∈𝐴. such that for all 𝐶^∗Proof. The proof follows from Theorem 2.7 by takingfor all 𝐶^∗ Then ‖⋅‖_𝐴. and we get the desired result.

3. Stability of Derivations on ℂ-Algebras

Throughout this section, assume that 𝛿∶𝐴→𝐴 is a 𝐴-algebra with norm 𝛿

Note that a 𝛿(𝑥𝑦)=𝛿(𝑥)𝑦+𝑥𝛿(𝑦)-linear mapping 𝑥,𝑦∈𝐴. is called a derivation on 𝐶^∗ if 𝐷_𝜇𝑓(𝑥,𝑦)=0. satisfies 𝑓∶𝐴→𝐴 for all 𝜑∶𝐴²→[0,∞)

We prove the generalized Hyers-Ulam stability of derivations on ‖‖𝐷_𝜇𝑓(𝑥,𝑦)‖‖_𝐴≤𝜑(𝑥,𝑦),(3.1)‖‖𝑓(𝑥𝑦)−𝑓(𝑥)𝑦−𝑥𝑓(𝑦)‖‖_𝐴≤𝜑(𝑥,𝑦)(3.2)-algebras for the functional equation 𝜇∈𝕋¹

Theorem 3.1. Let 𝑥,𝑦∈𝐴. be a mapping for which there exists a function 𝐿<1 such that for all 𝛿∶𝐴→𝐴 and all ‖‖𝑓(𝑥)−𝛿(𝑥)‖‖_𝐴≤𝐿1−𝐿𝜑(𝑥,0)(3.3) If there exists an 𝑥∈𝐴. such that ℂ for all 𝛿∶𝐴→𝐴 Then there exists a unique derivation 𝛿∶𝐴→𝐴 such that for all 𝑥∈𝐴.Proof. By the same reasoning as the proof of Theorem 2.1, there exists a unique involutive ‖‖𝛿(𝑥𝑦)−𝛿(𝑥)𝑦−𝑥𝛿(𝑦)‖‖_𝐴=lim_𝑛→∞14^𝑛‖‖𝑓4^𝑛𝑥𝑦−𝑓2^𝑛𝑥⋅2^𝑛𝑦−2^𝑛𝑥𝑓2^𝑛𝑦‖‖_𝐴≤lim_𝑛→∞14^𝑛𝜑2^𝑛𝑥,2^𝑛𝑦≤lim_𝑛→∞12^𝑛𝜑2^𝑛𝑥,2^𝑛𝑦=0(3.5)-linear mapping 𝑥,𝑦∈𝐴. satisfying (3.3). The mapping 𝛿(𝑥𝑦)=𝛿(𝑥)𝑦+𝑥𝛿(𝑦)(3.6) is given byfor all 𝛿∶𝐴→𝐴
It follows from (3.2) thatfor all 𝜃 Sofor all ‖‖𝐷_𝜇𝑓(𝑥,𝑦)‖‖_𝐴≤𝜃‖𝑥‖^𝑟𝐴+‖𝑦‖^𝑟𝐴,(3.7)‖‖𝑓(𝑥𝑦)−𝑓(𝑥)𝑦−𝑥𝑓(𝑦)‖‖_𝐴≤𝜃‖𝑥‖^𝑟𝐴+‖𝑦‖^𝑟𝐴(3.8) Thus 𝜇∈𝕋¹ is a derivation satisfying (3.3).Corollary 3.2. Let 𝑥,𝑦∈𝐴. and 𝛿∶𝐴→𝐴 be nonnegative real numbers, and let ‖‖𝑓(𝑥)−𝛿(𝑥)‖‖_𝐴≤2^𝑟𝜃2−2^𝑟‖𝑥‖^𝑟𝐴(3.9) be a mapping such that for all 𝑥,𝑦∈𝐴. and all 𝐿=2^𝑟−1 Then there exists a unique derivation 𝑓∶𝐴→𝐴 such that for all 𝐿<1Proof. The proof follows from Theorem 3.1 by takingfor all 𝑥,𝑦∈𝐴, Then 𝛿∶𝐴→𝐴 and we get the desired result.

Theorem 3.3. Let ‖‖𝑓(𝑥)−𝛿(𝑥)‖‖_𝐴≤𝐿2−2𝐿𝜑(𝑥,0)(3.11) be a mapping for which there exists a function 𝑥∈𝐴. satisfying (3.1) and (3.2). If there exists an 𝑟>2 such that 𝜃 for all 𝑓∶𝐴→𝐴 then there exists a unique derivation 𝛿∶𝐴→𝐴 such that for all 𝑥∈𝐴.Proof. The proof is similar to the proofs of Theorems 2.3 and 3.1.Corollary 3.4. Let 𝜑(𝑥,𝑦)∶=𝜃‖𝑥‖^𝑟𝐴+‖𝑦‖^𝑟𝐴(3.13) and 𝑥,𝑦∈𝐴. be nonnegative real numbers, and let 𝐿=2^1−𝑟 be a mapping satisfying (3.7) and (3.8). Then there exists a unique derivation 𝐶^∗ such that for all 𝒞,Proof. The proof follows from Theorem 3.3 by takingfor all 𝒞, Then 𝐶^∗ and we get the desired result.Remark 3.5. For inequalities controlled by the product of powers of norms, one can obtain similar results to Theorems 2.5 and 2.7 and Corollaries 2.6 and 2.8.

4. Stability of Homomorphisms in Lie 𝐴-Algebras

A 𝐵-algebra 𝐶^∗ endowed with the Lie product ℂ on 𝐻∶𝐴→𝐵 is called a Lie 𝐶^∗-algebra (see [13–15]).

Definition 4.1. Let 𝐻([𝑥,𝑦])=[𝐻(𝑥),𝐻(𝑦)] and 𝑥,𝑦∈𝐴. be Lie 𝐴-algebras. A 𝐶^∗-linear mapping ‖⋅‖_𝐴 is called a Lie 𝐵-algebra homomorphism if 𝐶^∗ for all ‖⋅‖_𝐵.

Throughout this section, assume that 𝐶^∗ is a Lie 𝐷_𝜇𝑓(𝑥,𝑦)=0.-algebra with norm 𝑓∶𝐴→𝐵 and that 𝜑∶𝐴²→[0,∞) is a Lie ‖‖𝑓([𝑥,𝑦])−𝑓(𝑥),𝑓(𝑦)‖‖_𝐵≤𝜑(𝑥,𝑦)(4.1)-algebra with norm 𝑥,𝑦∈𝐴.

We prove the generalized Hyers-Ulam stability of homomorphisms in Lie 𝐿<1-algebras for the functional equation 𝜑(𝑥,𝑦)≤2𝐿𝜑(𝑥/2,𝑦/2)

Theorem 4.2. Let 𝑥,𝑦∈𝐴, be a mapping for which there exists a function 𝐶^∗ satisfying (2.2) such that for all ℂ If there exists an 𝛿∶𝐴→𝐴 such that 𝐻∶𝐴→𝐵 for all 𝐻(𝑥)=lim_𝑛→∞𝑓(2^𝑛𝑥)2^𝑛(4.2) then there exists a unique Lie 𝑥∈𝐴.-algebra homomorphism ‖‖𝐻[𝑥,𝑦]−𝐻(𝑥),𝐻(𝑦)‖‖_𝐵=lim_𝑛→∞14^𝑛‖‖𝑓4^𝑛[𝑥,𝑦]−𝑓2^𝑛𝑥,𝑓2^𝑛𝑦‖‖_𝐵≤lim_𝑛→∞14^𝑛𝜑2^𝑛𝑥,2^𝑛𝑦≤lim_𝑛→∞12^𝑛𝜑2^𝑛𝑥,2^𝑛𝑦=0(4.3) satisfying (2.5).Proof. By the same reasoning as the proof of Theorem 2.1, there exists a unique 𝑥,𝑦∈𝐴.-linear mapping 𝐻[𝑥,𝑦]=𝐻(𝑥),𝐻(𝑦)(4.4) satisfying (2.5). The mapping 𝑥,𝑦∈𝐴. is given byfor all 𝐶^∗
It follows from (4.1) thatfor all 𝜃 Sofor all ‖‖𝑓[𝑥,𝑦]−𝑓(𝑥),𝑓(𝑦)‖‖_𝐵≤𝜃‖𝑥‖^𝑟𝐴+‖𝑦‖^𝑟𝐴(4.5)
Thus 𝑥,𝑦∈𝐴. is a Lie 𝐶^∗-algebra homomorphism satisfying (2.5), as desired.Corollary 4.3. Let 𝐻∶𝐴→𝐵 and 𝜑(𝑥,𝑦)∶=𝜃‖𝑥‖^𝑟𝐴+‖𝑦‖^𝑟𝐴(4.6) be nonnegative real numbers, and let 𝑥,𝑦∈𝐴. be a mapping satisfying (2.27) such that for all 𝑓∶𝐴→𝐵 Then there exists a unique Lie 𝜑∶𝐴²→[0,∞)-algebra homomorphism 𝐿<1 satisfying (2.30).Proof. The proof follows from Theorem 4.2 by takingfor all 𝑥,𝑦∈𝐴, Then 𝐶^∗ and we get the desired result.

Theorem 4.4. Let 𝐻∶𝐴→𝐵 be a mapping for which there exists a function 𝑟>2 satisfying (2.2) and (4.1). If there exists an 𝜃 such that 𝑓∶𝐴→𝐵 for all 𝐶^∗ then there exists a unique Lie 𝐻∶𝐴→𝐵-algebra homomorphism 𝜑(𝑥,𝑦)∶=𝜃‖𝑥‖^𝑟𝐴+‖𝑦‖^𝑟𝐴(4.7) satisfying (2.32).Proof. The proof is similar to the proofs of Theorems 2.3 and 4.2.Corollary 4.5. Let 𝑥,𝑦∈𝐴. and 𝐿=2^1−𝑟 be nonnegative real numbers, and let 𝐶^∗ be a mapping satisfying (2.27) and (4.5). Then there exists a unique Lie 𝐴-algebra homomorphism 𝐶^∗ satisfying (2.40).Proof. The proof follows from Theorem 4.4 by takingfor all 𝛿∶𝐴→𝐴 Then 𝛿([𝑥,𝑦])=[𝛿(𝑥),𝑦]+[𝑥,𝛿(𝑦)] and we get the desired result.Remark 4.6. For inequalities controlled by the product of powers of norms, one can obtain similar results to Theorems 2.5 and 2.7 and Corollaries 2.6 and 2.8.

5. Stability of Lie Derivations on 𝑥,𝑦∈𝐴.-Algebras

Definition 5.1. Let 𝐴 be a Lie 𝐶^∗-algebra. A ‖⋅‖_𝐴.-linear mapping 𝐶^∗ is called a Lie derivation if 𝐷_𝜇𝑓(𝑥,𝑦)=0. for all 𝑓∶𝐴→𝐴

Throughout this section, assume that 𝜑∶𝐴²→[0,∞) is a Lie ‖‖𝑓[𝑥,𝑦]−𝑓(𝑥),𝑦−𝑥,𝑓(𝑦)‖‖_𝐴≤𝜑(𝑥,𝑦)(5.1)-algebra with norm 𝑥,𝑦∈𝐴.

We prove the generalized Hyers-Ulam stability of derivations on Lie 𝐿<1-algebras for the functional equation 𝜑(𝑥,𝑦)≤2𝐿𝜑(𝑥/2,𝑦/2)

Theorem 5.2. Let 𝑥,𝑦∈𝐴. be a mapping for which there exists a function 𝛿∶𝐴→𝐴 satisfying (3.1) such that for all 𝛿∶𝐴→𝐴 If there exists an 𝛿∶𝐴→𝐴 such that 𝛿(𝑥)=lim_𝑛→∞𝑓(2^𝑛𝑥)2^𝑛(5.2) for all 𝑥∈𝐴. Then there exists a unique Lie derivation ‖‖𝛿[𝑥,𝑦]−𝛿(𝑥),𝑦−𝑥,𝛿(𝑦)‖‖_𝐴=lim_𝑛→∞14^𝑛‖‖𝑓4^𝑛[𝑥,𝑦]−𝑓2^𝑛𝑥,2^𝑛𝑦−2^𝑛𝑥,𝑓2^𝑛𝑦‖‖_𝐴≤lim_𝑛→∞14^𝑛𝜑2^𝑛𝑥,2^𝑛𝑦≤lim_𝑛→∞12^𝑛𝜑2^𝑛𝑥,2^𝑛𝑦=0(5.3) satisfying (3.3).Proof. By the same reasoning as the proof of Theorem 2.1, there exists a unique involutive 𝑥,𝑦∈𝐴.-linear mapping 𝛿[𝑥,𝑦]=𝛿(𝑥),𝑦+[𝑥,𝛿(𝑦)](5.4) satisfying (3.3). The mapping 𝑥,𝑦∈𝐴. is given byfor all 0<𝑟<1
It follows from (5.1) thatfor all 𝑓∶𝐴→𝐴 Sofor all 𝑥,𝑦∈𝐴. Thus 𝛿∶𝐴→𝐴 is a derivation satisfying (3.3).Corollary 5.3. Let 𝜑(𝑥,𝑦)∶=𝜃‖𝑥‖^𝑟𝐴+‖𝑦‖^𝑟𝐴(5.6) and 𝑥,𝑦∈𝐴. be nonnegative real numbers, and let 𝐿=2^𝑟−1 be a mapping satisfying (3.7) such that for all 𝜑∶𝐴²→[0,∞) Then there exists a unique Lie derivation 𝐿<1 satisfying (3.9).Proof. The proof follows from Theorem 5.2 by takingfor all 𝑥,𝑦∈𝐴, Then 𝛿∶𝐴→𝐴 and we get the desired result.

Theorem 5.4. Let 𝑟>2 be a mapping for which there exists a function 𝜃 satisfying (3.1) and (5.1). If there exists an 𝑓∶𝐴→𝐴 such that 𝛿∶𝐴→𝐴 for all 𝜑(𝑥,𝑦)∶=𝜃‖𝑥‖^𝑟𝐴+‖𝑦‖^𝑟𝐴(5.7) then there exists a unique Lie derivation 𝑥,𝑦∈𝐴. satisfying (3.11).Proof. The proof is similar to the proofs of Theorems 2.3 and 5.2.Corollary 5.5. Let 𝐿=2^1−𝑟 and 𝐶^∗ be nonnegative real numbers, and let 𝐶^∗ be a mapping satisfying (3.7) and (5.5). Then there exists a unique Lie derivation ∗ satisfying (3.12).Proof. The proof follows from Theorem 5.4 by takingfor all ∗ Then 𝐶^∗ and we get the desired result.Remark 5.6. For inequalities controlled by the product of powers of norms, one can obtain similar results to Theorems 2.5 and 2.7 and Corollaries 2.6 and 2.8.

Acknowledgment

The first author was supported by the R & E program of KOSEF in 2008.