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Abstract and Applied Analysis
VolumeΒ 2009Β (2009), Article IDΒ 360432, 17 pages
http://dx.doi.org/10.1155/2009/360432
Research Article

Stability of the Jensen-Type Functional Equation in πΆβˆ—-Algebras: A Fixed Point Approach

1Department of Mathematics, Hanyang University, Seoul 133-791, South Korea
2Pedagogical Department E.E., National and Capodistrian University of Athens, 4 Agamemnonos Street, Aghia Paraskevi, Athens 15342, Greece

Received 27 November 2008; Accepted 26 January 2009

Academic Editor: BruceΒ Calvert

Copyright Β© 2009 Choonkil Park and John Michael Rassias. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Using fixed point methods, we prove the generalized Hyers-Ulam stability of homomorphisms in πΆβˆ—-algebras and Lie πΆβˆ—-algebras and also of derivations on πΆβˆ—-algebras and Lie πΆβˆ—-algebras for the Jensen-type functional equation 𝑓((π‘₯+𝑦)/2)+𝑓((π‘₯βˆ’π‘¦)/2)=𝑓(π‘₯).

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 GΔƒvruΕ£a [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 β€–π‘₯‖𝑝+‖𝑦‖𝑝 by β€–π‘₯β€–π‘β‹…β€–π‘¦β€–π‘ž for 𝑝,π‘žβˆˆβ„ with 𝑝+π‘žβ‰ 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 π‘‘βˆΆπ‘‹Γ—π‘‹β†’[0,∞] is called a generalized metric on 𝑋 if 𝑑 satisfies

(1)𝑑(π‘₯,𝑦)=0 if and only if π‘₯=𝑦;(2)𝑑(π‘₯,𝑦)=𝑑(𝑦,π‘₯) for all π‘₯,π‘¦βˆˆπ‘‹;(3)𝑑(π‘₯,𝑧)≀𝑑(π‘₯,𝑦)+𝑑(𝑦,𝑧) for all π‘₯,𝑦,π‘§βˆˆπ‘‹.

Theorem 1.1 (see [28, 29]). Let (𝑋,𝑑) be a complete generalized metric space and let π½βˆΆπ‘‹β†’π‘‹ be a strictly contractive mapping with Lipschitz constant 𝐿<1. Then for each given element π‘₯βˆˆπ‘‹, either for all nonnegative integers 𝑛 or there exists a positive integer 𝑛0 such that
(1)𝑑(𝐽𝑛π‘₯,𝐽𝑛+1π‘₯)<∞, for all 𝑛β‰₯𝑛0;(2)the sequence {𝐽𝑛π‘₯} converges to a fixed point π‘¦βˆ— of 𝐽;(3)π‘¦βˆ— is the unique fixed point of 𝐽 in the set π‘Œ={π‘¦βˆˆπ‘‹βˆ£π‘‘(𝐽𝑛0π‘₯,𝑦)<∞};(4)𝑑(𝑦,π‘¦βˆ—)≀(1/(1βˆ’πΏ))𝑑(𝑦,𝐽𝑦) for all π‘¦βˆˆπ‘Œ.

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 πΆβˆ—-algebras and of derivations on πΆβˆ—-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 πΆβˆ—-algebras and of derivations on Lie πΆβˆ—-algebras for the Jensen-type functional equation.

2. Stability of Homomorphisms in πΆβˆ—-Algebras

Throughout this section, assume that 𝐴 is a πΆβˆ—-algebra with norm ‖⋅‖𝐴 and that 𝐡 is a πΆβˆ—-algebra with norm ‖⋅‖𝐡.

For a given mapping π‘“βˆΆπ΄β†’π΅, we definefor all πœ‡βˆˆπ•‹1∢={πœˆβˆˆβ„‚βˆΆ|𝜈|=1} and all π‘₯,π‘¦βˆˆπ΄.

Note that a β„‚-linear mapping π»βˆΆπ΄β†’π΅ is called a homomorphism in πΆβˆ—-algebras if 𝐻 satisfies 𝐻(π‘₯𝑦)=𝐻(π‘₯)𝐻(𝑦) and 𝐻(π‘₯βˆ—)=𝐻(π‘₯)βˆ— for all π‘₯,π‘¦βˆˆπ΄.

We prove the generalized Hyers-Ulam stability of homomorphisms in πΆβˆ—-algebras for the functional equation π·πœ‡π‘“(π‘₯,𝑦)=0.

Theorem 2.1. Let π‘“βˆΆπ΄β†’π΅ be a mapping for which there exists a function πœ‘βˆΆπ΄2β†’[0,∞) such that for all 𝐿<1 and all πœ‘(π‘₯,𝑦)≀2πΏπœ‘(π‘₯/2,𝑦/2) If there exists an π‘₯,π‘¦βˆˆπ΄, such that πΆβˆ— for all π»βˆΆπ΄β†’π΅ then there exists a unique ‖‖‖‖𝑓(π‘₯)βˆ’π»(π‘₯)𝐡≀𝐿1βˆ’πΏπœ‘(π‘₯,0)(2.5)-algebra homomorphism π‘₯∈𝐴. such that for all limπ‘—β†’βˆž2βˆ’π‘—πœ‘ξ€·2𝑗π‘₯,2𝑗𝑦=0(2.6)Proof. It follows from π‘₯,π‘¦βˆˆπ΄. thatfor all 𝑋
Consider the setand introduce the generalized metric on (𝑋,𝑑):It is easy to show that 1𝐽𝑔(π‘₯)∢=2𝑔(2π‘₯)(2.9) is complete.
Now we consider the linear mapping π‘₯∈𝐴. such thatfor all 𝑔,β„Žβˆˆπ‘‹.
By [28, Theorem 3.1],for all 𝑦=0
Letting β€–β€–β€–ξ‚€π‘₯2𝑓2ξ‚β€–β€–β€–βˆ’π‘“(π‘₯)π΅β‰€πœ‘(π‘₯,0)(2.11) and π‘₯∈𝐴. in (2.2), we getfor all π‘₯∈𝐴. Sofor all π»βˆΆπ΄β†’π΅ Hence 𝐻
By Theorem 1.1, there exists a mapping 𝐽, such that
(1)𝐻(2π‘₯)=2𝐻(π‘₯)(2.13) is a fixed point of π‘₯∈𝐴. that is,for all 𝐽 The mapping ξ€½ξ€Ύ.π‘Œ=π‘”βˆˆπ‘‹βˆΆπ‘‘(𝑓,𝑔)<∞(2.14) is a unique fixed point of 𝐻 in the setThis implies that ‖‖‖‖𝐻(π‘₯)βˆ’π‘“(π‘₯)π΅β‰€πΆπœ‘(π‘₯,0)(2.15) is a unique mapping satisfying (2.13) such that there exists π‘₯∈𝐴. satisfyingfor all π‘›β†’βˆž.(2)limπ‘›β†’βˆžπ‘“(2𝑛π‘₯)2𝑛=𝐻(π‘₯)(2.16) as π‘₯∈𝐴. This implies the equalityfor all 𝐿𝑑(𝑓,𝐻)≀.1βˆ’πΏ(2.17)(3)‖‖‖𝐻π‘₯+𝑦2+𝐻π‘₯βˆ’π‘¦2ξ‚β€–β€–β€–βˆ’π»(π‘₯)𝐡=limπ‘›β†’βˆž12𝑛‖‖𝑓2π‘›βˆ’1ξ€Έξ€·2(π‘₯+𝑦)+π‘“π‘›βˆ’1ξ€Έξ€·2(π‘₯βˆ’π‘¦)βˆ’π‘“π‘›π‘₯‖‖𝐡≀limπ‘›β†’βˆž12π‘›πœ‘ξ€·2𝑛π‘₯,2𝑛𝑦=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 π‘₯,π‘¦βˆˆπ΄. Sofor all 𝑀=(π‘₯βˆ’π‘¦)/2 Letting 𝐻(𝑧)+𝐻(𝑀)=𝐻(𝑧+𝑀)(2.20) and 𝑧,π‘€βˆˆπ΄. in (2.19), we getfor all 𝐻(𝑧+𝑀)=𝐻(𝑧)+𝐻(𝑀) So the mapping 𝑧,π‘€βˆˆπ΄. is Cauchy additive, that is, 𝑦=π‘₯ for all πœ‡π‘“(π‘₯)=𝑓(πœ‡π‘₯)(2.21)
Letting πœ‡βˆˆπ•‹1 in (2.2), we getfor all πœ‡π»(π‘₯)=𝐻(πœ‡π‘₯)(2.22) and all πœ‡βˆˆπ•‹1 By a similar method to above, we getfor all π»βˆΆπ΄β†’π΅ and all β„‚ Thus one can show that the mapping ‖‖‖‖𝐻(π‘₯𝑦)βˆ’π»(π‘₯)𝐻(𝑦)𝐡=limπ‘›β†’βˆž14𝑛‖‖𝑓4𝑛2π‘₯π‘¦βˆ’π‘“π‘›π‘₯𝑓2𝑛𝑦‖‖𝐡≀limπ‘›β†’βˆž14π‘›πœ‘ξ€·2𝑛π‘₯,2𝑛𝑦≀limπ‘›β†’βˆž12π‘›πœ‘ξ€·2𝑛π‘₯,2𝑛𝑦=0(2.23) is π‘₯,π‘¦βˆˆπ΄.-linear.
It follows from (2.3) thatfor all π‘₯,π‘¦βˆˆπ΄. Sofor all π‘₯∈𝐴.
It follows from (2.4) thatfor all π‘₯∈𝐴. Sofor all πΆβˆ—
Thus 0<π‘Ÿ<1 is a πœƒ-algebra homomorphism satisfying (2.5), as desired.
Corollary 2.2. Let π‘“βˆΆπ΄β†’π΅ and β€–β€–π·πœ‡β€–β€–π‘“(π‘₯,𝑦)π΅ξ€·β‰€πœƒβ€–π‘₯β€–π‘Ÿπ΄+β€–π‘¦β€–π‘Ÿπ΄ξ€Έ,(2.27)‖‖‖‖𝑓(π‘₯𝑦)βˆ’π‘“(π‘₯)𝑓(𝑦)π΅ξ€·β‰€πœƒβ€–π‘₯β€–π‘Ÿπ΄+β€–π‘¦β€–π‘Ÿπ΄ξ€Έ,(2.28)‖‖𝑓π‘₯βˆ—ξ€Έβˆ’π‘“(π‘₯)βˆ—β€–β€–π΅β‰€2πœƒβ€–π‘₯β€–π‘Ÿπ΄(2.29) be nonnegative real numbers, and let πœ‡βˆˆπ•‹1 be a mapping such that for all ‖‖‖‖𝑓(π‘₯)βˆ’π»(π‘₯)𝐡≀2π‘Ÿπœƒ2βˆ’2π‘Ÿβ€–π‘₯β€–π‘Ÿπ΄(2.30) and all π‘₯∈𝐴. Then there exists a unique ξ€·πœ‘(π‘₯,𝑦)∢=πœƒβ€–π‘₯β€–π‘Ÿπ΄+β€–π‘¦β€–π‘Ÿπ΄ξ€Έ(2.31)-algebra homomorphism π‘₯,π‘¦βˆˆπ΄. such that for all π‘“βˆΆπ΄β†’π΅Proof. The proof follows from Theorem 2.1 by takingfor all 𝐿<1 Then πœ‘(π‘₯,𝑦)≀(1/2)πΏπœ‘(2π‘₯,2𝑦) and we get the desired result.

Theorem 2.3. Let π‘₯,π‘¦βˆˆπ΄, be a mapping for which there exists a function πΆβˆ— satisfying (2.2), (2.3), and (2.4). If there exists an π»βˆΆπ΄β†’π΅ such that ‖‖‖‖𝑓(π‘₯)βˆ’π»(π‘₯)𝐡≀𝐿2βˆ’2πΏπœ‘(π‘₯,0)(2.32) for all π‘₯∈𝐴. then there exists a unique π½βˆΆπ‘‹β†’π‘‹-algebra homomorphism ξ‚€π‘₯𝐽𝑔(π‘₯)∢=2𝑔2(2.33) such that for all β€–β€–β€–ξ‚€π‘₯𝑓(π‘₯)βˆ’2𝑓2‖‖‖𝐡π‘₯β‰€πœ‘2≀𝐿,02πœ‘(π‘₯,0)(2.34)Proof. We consider the linear mapping π‘₯∈𝐴. such thatfor all π»βˆΆπ΄β†’π΅
It follows from (2.11) thatfor all 𝐽, Hence 𝐻(2π‘₯)=2𝐻(π‘₯)(2.35)
By Theorem 1.1, there exists a mapping π‘₯∈𝐴. such that
(1)𝐻 is a fixed point of 𝐽 that is,for all 𝐻 The mapping 𝐢∈(0,∞) is a unique fixed point of ‖‖‖‖𝐻(π‘₯)βˆ’π‘“(π‘₯)π΅β‰€πΆπœ‘(π‘₯,0)(2.37) in the setThis implies that 𝑑(𝐽𝑛𝑓,𝐻)β†’0 is a unique mapping satisfying (2.35) such that there exists π‘›β†’βˆž. satisfyingfor all π‘₯∈𝐴.(2)𝑑(𝑓,𝐻)≀(1/(1βˆ’πΏ))𝑑(𝑓,𝐽𝑓), as 𝐿𝑑(𝑓,𝐻)≀,2βˆ’2𝐿(2.39) This implies the equalityfor all πœƒ(3)π‘“βˆΆπ΄β†’π΅ 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 π»βˆΆπ΄β†’π΅ and ‖‖‖‖𝑓(π‘₯)βˆ’π»(π‘₯)π΅β‰€πœƒ2π‘Ÿβˆ’2β€–π‘₯β€–π‘Ÿπ΄(2.40) be nonnegative real numbers, and let π‘₯∈𝐴. be a mapping satisfying (2.27), (2.28) and (2.29). Then there exists a unique ξ€·πœ‘(π‘₯,𝑦)∢=πœƒβ€–π‘₯β€–π‘Ÿπ΄+β€–π‘¦β€–π‘Ÿπ΄ξ€Έ(2.41)-algebra homomorphism π‘₯,π‘¦βˆˆπ΄. such that for all π‘“βˆΆπ΄β†’π΅Proof. The proof follows from Theorem 2.3 by takingfor all 𝐿<1 Then πœ‘(π‘₯,3π‘₯)≀2πΏπœ‘(π‘₯/2,3π‘₯/2) and we get the desired result.

Theorem 2.5. Let π‘₯∈𝐴, be an odd mapping for which there exists a function πΆβˆ— satisfying (2.2), (2.3), (2.4) and (2.6). If there exists an π»βˆΆπ΄β†’π΅ such that ‖‖‖‖𝑓(π‘₯)βˆ’π»(π‘₯)𝐡≀12βˆ’2πΏπœ‘(π‘₯,3π‘₯)(2.42) for all π‘₯∈𝐴. then there exists a unique π‘‹βˆΆ={π‘”βˆΆπ΄βŸΆπ΅}(2.43)-algebra homomorphism 𝑋 such that for all (𝑋,𝑑)Proof. Consider the setand introduce the generalized metric on 1𝐽𝑔(π‘₯)∢=2𝑔(2π‘₯)(2.45):It is easy to show that 𝑑(𝐽𝑔,π½β„Ž)≀𝐿𝑑(𝑔,β„Ž)(2.46) is complete.
Now we consider the linear mapping 𝑔,β„Žβˆˆπ‘‹. such thatfor all 𝑦
By [28, Theorem 3.1],for all ‖‖‖‖𝑓(2π‘₯)βˆ’2𝑓(π‘₯)π΅β‰€πœ‘(π‘₯,3π‘₯)(2.47)
Letting π‘₯∈𝐴. and relpacing β€–β€–β€–1𝑓(π‘₯)βˆ’2‖‖‖𝑓(2π‘₯)𝐡≀12πœ‘(π‘₯,3π‘₯)(2.48) by π‘₯∈𝐴. in (2.2), we getfor all π»βˆΆπ΄β†’π΅ Sofor all 𝐽, Hence 𝐻(2π‘₯)=2𝐻(π‘₯)(2.49)
By Theorem 1.1, there exists a mapping π‘₯∈𝐴. such that
(1)𝐻 is a fixed point of 𝐽 that is,for all 𝐻 The mapping 𝐢∈(0,∞) is a unique fixed point of ‖‖‖‖𝐻(π‘₯)βˆ’π‘“(π‘₯)π΅β‰€πΆπœ‘(π‘₯,3π‘₯)(2.51) in the setThis implies that 𝑑(𝐽𝑛𝑓,𝐻)β†’0 is a unique mapping satisfying (2.49) such that there exists π‘›β†’βˆž. satisfyingfor all π‘₯∈𝐴.(2)𝑑(𝑓,𝐻)≀(1/(1βˆ’πΏ))𝑑(𝑓,𝐽𝑓), as 1𝑑(𝑓,𝐻)≀.2βˆ’2𝐿(2.53) This implies the equalityfor all πœƒ(3)π‘“βˆΆπ΄β†’π΅ 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 πœ‡βˆˆπ•‹1 and π‘₯,π‘¦βˆˆπ΄. be nonnegative real numbers, and let πΆβˆ— be an odd mapping such that for all ‖‖‖‖𝑓(π‘₯)βˆ’π»(π‘₯)𝐡≀3π‘Ÿπœƒ2βˆ’22π‘Ÿβ€–π‘₯‖𝐴2π‘Ÿ(2.55) and all π‘₯∈𝐴. Then there exists a unique πœ‘(π‘₯,𝑦)∢=πœƒβ‹…β€–π‘₯β€–π‘Ÿπ΄β‹…β€–π‘¦β€–π‘Ÿπ΄(2.56)-algebra homomorphism π‘₯,π‘¦βˆˆπ΄. such that for all π‘“βˆΆπ΄β†’π΅Proof. The proof follows from Theorem 2.5 by takingfor all 𝐿<1 Then πœ‘(π‘₯,3π‘₯)≀(1/2)πΏπœ‘(2π‘₯,6π‘₯) and we get the desired result.

Theorem 2.7. Let π‘₯∈𝐴, be an odd mapping for which there exists a function πΆβˆ— satisfying (2.2), (2.3) and (2.4). If there exists an π»βˆΆπ΄β†’π΅ such that ‖‖‖‖𝑓(π‘₯)βˆ’π»(π‘₯)𝐡≀𝐿2βˆ’2πΏπœ‘(π‘₯,3π‘₯)(2.57) for all π‘₯∈𝐴. then there exists a unique π½βˆΆπ‘‹β†’π‘‹-algebra homomorphism ξ‚€π‘₯𝐽𝑔(π‘₯)∢=2𝑔2(2.58) such that for all β€–β€–β€–ξ‚€π‘₯𝑓(π‘₯)βˆ’2𝑓2‖‖‖𝐡π‘₯β‰€πœ‘2,3π‘₯2≀𝐿2πœ‘(π‘₯,3π‘₯)(2.59)Proof. We consider the linear mapping π‘₯∈𝐴. such thatfor all π»βˆΆπ΄β†’π΅
It follows from (2.47) thatfor all 𝐽, Hence 𝐻(2π‘₯)=2𝐻(π‘₯)(2.60)
By Theorem 1.1, there exists a mapping π‘₯∈𝐴. such that
(1)𝐻 is a fixed point of 𝐽 that is,for all 𝐻 The mapping 𝐢∈(0,∞) is a unique fixed point of ‖‖‖‖𝐻(π‘₯)βˆ’π‘“(π‘₯)π΅β‰€πΆπœ‘(π‘₯,3π‘₯)(2.62) in the setThis implies that 𝑑(𝐽𝑛𝑓,𝐻)β†’0 is a unique mapping satisfying (2.60) such that there exists π‘›β†’βˆž. satisfyingfor all π‘₯∈𝐴.(2)𝑑(𝑓,𝐻)≀(1/(1βˆ’πΏ))𝑑(𝑓,𝐽𝑓), as 𝐿𝑑(𝑓,𝐻)≀,2βˆ’2𝐿(2.64) This implies the equalityfor all πœƒ(3)π‘“βˆΆπ΄β†’π΅ 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 π»βˆΆπ΄β†’π΅ and ‖‖‖‖𝑓(π‘₯)βˆ’π»(π‘₯)π΅β‰€πœƒ22π‘Ÿβˆ’2β€–π‘₯‖𝐴2π‘Ÿ(2.65) be nonnegative real numbers, and let π‘₯∈𝐴. be an odd mapping satisfying (2.54). Then there exists a unique πœ‘(π‘₯,𝑦)∢=πœƒβ‹…β€–π‘₯β€–π‘Ÿπ΄β‹…β€–π‘¦β€–π‘Ÿπ΄(2.66)-algebra homomorphism π‘₯,π‘¦βˆˆπ΄. such that for all πΆβˆ—Proof. The proof follows from Theorem 2.7 by takingfor all πΆβˆ— Then ‖⋅‖𝐴. and we get the desired result.

3. Stability of Derivations on β„‚-Algebras

Throughout this section, assume that π›ΏβˆΆπ΄β†’π΄ is a 𝐴-algebra with norm 𝛿

Note that a 𝛿(π‘₯𝑦)=𝛿(π‘₯)𝑦+π‘₯𝛿(𝑦)-linear mapping π‘₯,π‘¦βˆˆπ΄. is called a derivation on πΆβˆ— if π·πœ‡π‘“(π‘₯,𝑦)=0. satisfies π‘“βˆΆπ΄β†’π΄ for all πœ‘βˆΆπ΄2β†’[0,∞)

We prove the generalized Hyers-Ulam stability of derivations on β€–β€–π·πœ‡β€–β€–π‘“(π‘₯,𝑦)π΄β‰€πœ‘(π‘₯,𝑦),(3.1)‖‖‖‖𝑓(π‘₯𝑦)βˆ’π‘“(π‘₯)π‘¦βˆ’π‘₯𝑓(𝑦)π΄β‰€πœ‘(π‘₯,𝑦)(3.2)-algebras for the functional equation πœ‡βˆˆπ•‹1

Theorem 3.1. Let π‘₯,π‘¦βˆˆπ΄. be a mapping for which there exists a function 𝐿<1 such that for all π›ΏβˆΆπ΄β†’π΄ and all ‖‖‖‖𝑓(π‘₯)βˆ’π›Ώ(π‘₯)𝐴≀𝐿1βˆ’πΏπœ‘(π‘₯,0)(3.3) If there exists an π‘₯∈𝐴. such that β„‚ for all π›ΏβˆΆπ΄β†’π΄ Then there exists a unique derivation π›ΏβˆΆπ΄β†’π΄ such that for all π‘₯∈𝐴.Proof. By the same reasoning as the proof of Theorem 2.1, there exists a unique involutive ‖‖‖‖𝛿(π‘₯𝑦)βˆ’π›Ώ(π‘₯)π‘¦βˆ’π‘₯𝛿(𝑦)𝐴=limπ‘›β†’βˆž14𝑛‖‖𝑓4𝑛2π‘₯π‘¦βˆ’π‘“π‘›π‘₯ξ€Έβ‹…2π‘›π‘¦βˆ’2𝑛2π‘₯𝑓𝑛𝑦‖‖𝐴≀limπ‘›β†’βˆž14π‘›πœ‘ξ€·2𝑛π‘₯,2𝑛𝑦≀limπ‘›β†’βˆž12π‘›πœ‘ξ€·2𝑛π‘₯,2𝑛𝑦=0(3.5)-linear mapping π‘₯,π‘¦βˆˆπ΄. satisfying (3.3). The mapping 𝛿(π‘₯𝑦)=𝛿(π‘₯)𝑦+π‘₯𝛿(𝑦)(3.6) is given byfor all π›ΏβˆΆπ΄β†’π΄
It follows from (3.2) thatfor all πœƒ Sofor all β€–β€–π·πœ‡β€–β€–π‘“(π‘₯,𝑦)π΄ξ€·β‰€πœƒβ€–π‘₯β€–π‘Ÿπ΄+β€–π‘¦β€–π‘Ÿπ΄ξ€Έ,(3.7)‖‖‖‖𝑓(π‘₯𝑦)βˆ’π‘“(π‘₯)π‘¦βˆ’π‘₯𝑓(𝑦)π΄ξ€·β‰€πœƒβ€–π‘₯β€–π‘Ÿπ΄+β€–π‘¦β€–π‘Ÿπ΄ξ€Έ(3.8) Thus πœ‡βˆˆπ•‹1 is a derivation satisfying (3.3).Corollary 3.2. Let π‘₯,π‘¦βˆˆπ΄. and π›ΏβˆΆπ΄β†’π΄ be nonnegative real numbers, and let ‖‖‖‖𝑓(π‘₯)βˆ’π›Ώ(π‘₯)𝐴≀2π‘Ÿπœƒ2βˆ’2π‘Ÿβ€–π‘₯β€–π‘Ÿπ΄(3.9) be a mapping such that for all π‘₯,π‘¦βˆˆπ΄. and all 𝐿=2π‘Ÿβˆ’1 Then there exists a unique derivation π‘“βˆΆπ΄β†’π΄ such that for all 𝐿<1Proof. The proof follows from Theorem 3.1 by takingfor all π‘₯,π‘¦βˆˆπ΄, Then π›ΏβˆΆπ΄β†’π΄ and we get the desired result.

Theorem 3.3. Let ‖‖‖‖𝑓(π‘₯)βˆ’π›Ώ(π‘₯)𝐴≀𝐿2βˆ’2πΏπœ‘(π‘₯,0)(3.11) be a mapping for which there exists a function π‘₯∈𝐴. satisfying (3.1) and (3.2). If there exists an π‘Ÿ>2 such that πœƒ for all π‘“βˆΆπ΄β†’π΄ then there exists a unique derivation π›ΏβˆΆπ΄β†’π΄ such that for all π‘₯∈𝐴.Proof. The proof is similar to the proofs of Theorems 2.3 and 3.1.Corollary 3.4. Let ξ€·πœ‘(π‘₯,𝑦)∢=πœƒβ€–π‘₯β€–π‘Ÿπ΄+β€–π‘¦β€–π‘Ÿπ΄ξ€Έ(3.13) and π‘₯,π‘¦βˆˆπ΄. be nonnegative real numbers, and let 𝐿=21βˆ’π‘Ÿ be a mapping satisfying (3.7) and (3.8). Then there exists a unique derivation πΆβˆ— such that for all π’ž,Proof. The proof follows from Theorem 3.3 by takingfor all π’ž, Then πΆβˆ— 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 𝐡-algebra πΆβˆ— endowed with the Lie product β„‚ on π»βˆΆπ΄β†’π΅ is called a Lie πΆβˆ—-algebra (see [13–15]).

Definition 4.1. Let 𝐻([π‘₯,𝑦])=[𝐻(π‘₯),𝐻(𝑦)] and π‘₯,π‘¦βˆˆπ΄. be Lie 𝐴-algebras. A πΆβˆ—-linear mapping ‖⋅‖𝐴 is called a Lie 𝐡-algebra homomorphism if πΆβˆ— for all ‖⋅‖𝐡.

Throughout this section, assume that πΆβˆ— is a Lie π·πœ‡π‘“(π‘₯,𝑦)=0.-algebra with norm π‘“βˆΆπ΄β†’π΅ and that πœ‘βˆΆπ΄2β†’[0,∞) is a Lie ‖‖‖‖𝑓([π‘₯,𝑦])βˆ’π‘“(π‘₯),𝑓(𝑦)π΅β‰€πœ‘(π‘₯,𝑦)(4.1)-algebra with norm π‘₯,π‘¦βˆˆπ΄.

We prove the generalized Hyers-Ulam stability of homomorphisms in Lie 𝐿<1-algebras for the functional equation πœ‘(π‘₯,𝑦)≀2πΏπœ‘(π‘₯/2,𝑦/2)

Theorem 4.2. Let π‘₯,π‘¦βˆˆπ΄, be a mapping for which there exists a function πΆβˆ— satisfying (2.2) such that for all β„‚ If there exists an π›ΏβˆΆπ΄β†’π΄ such that π»βˆΆπ΄β†’π΅ for all 𝐻(π‘₯)=limπ‘›β†’βˆžπ‘“(2𝑛π‘₯)2𝑛(4.2) then there exists a unique Lie π‘₯∈𝐴.-algebra homomorphism β€–β€–π»ξ€·ξ€Έβˆ’ξ€Ίξ€»β€–β€–[π‘₯,𝑦]𝐻(π‘₯),𝐻(𝑦)𝐡=limπ‘›β†’βˆž14𝑛‖‖𝑓4π‘›ξ€Έβˆ’ξ€Ίπ‘“ξ€·2[π‘₯,𝑦]𝑛π‘₯ξ€Έξ€·2,𝑓𝑛𝑦‖‖𝐡≀limπ‘›β†’βˆž14π‘›πœ‘ξ€·2𝑛π‘₯,2𝑛𝑦≀limπ‘›β†’βˆž12π‘›πœ‘ξ€·2𝑛π‘₯,2𝑛𝑦=0(4.3) satisfying (2.5).Proof. By the same reasoning as the proof of Theorem 2.1, there exists a unique π‘₯,π‘¦βˆˆπ΄.-linear mapping 𝐻=ξ€Ίξ€»[π‘₯,𝑦]𝐻(π‘₯),𝐻(𝑦)(4.4) satisfying (2.5). The mapping π‘₯,π‘¦βˆˆπ΄. is given byfor all πΆβˆ—
It follows from (4.1) thatfor all πœƒ Sofor all β€–β€–π‘“ξ€·ξ€Έβˆ’ξ€Ίξ€»β€–β€–[π‘₯,𝑦]𝑓(π‘₯),𝑓(𝑦)π΅ξ€·β‰€πœƒβ€–π‘₯β€–π‘Ÿπ΄+β€–π‘¦β€–π‘Ÿπ΄ξ€Έ(4.5)
Thus π‘₯,π‘¦βˆˆπ΄. is a Lie πΆβˆ—-algebra homomorphism satisfying (2.5), as desired.
Corollary 4.3. Let π»βˆΆπ΄β†’π΅ and ξ€·πœ‘(π‘₯,𝑦)∢=πœƒβ€–π‘₯β€–π‘Ÿπ΄+β€–π‘¦β€–π‘Ÿπ΄ξ€Έ(4.6) be nonnegative real numbers, and let π‘₯,π‘¦βˆˆπ΄. be a mapping satisfying (2.27) such that for all π‘“βˆΆπ΄β†’π΅ Then there exists a unique Lie πœ‘βˆΆπ΄2β†’[0,∞)-algebra homomorphism 𝐿<1 satisfying (2.30).Proof. The proof follows from Theorem 4.2 by takingfor all π‘₯,π‘¦βˆˆπ΄, Then πΆβˆ— and we get the desired result.

Theorem 4.4. Let π»βˆΆπ΄β†’π΅ be a mapping for which there exists a function π‘Ÿ>2 satisfying (2.2) and (4.1). If there exists an πœƒ such that π‘“βˆΆπ΄β†’π΅ for all πΆβˆ— then there exists a unique Lie π»βˆΆπ΄β†’π΅-algebra homomorphism ξ€·πœ‘(π‘₯,𝑦)∢=πœƒβ€–π‘₯β€–π‘Ÿπ΄+β€–π‘¦β€–π‘Ÿπ΄ξ€Έ(4.7) satisfying (2.32).Proof. The proof is similar to the proofs of Theorems 2.3 and 4.2.Corollary 4.5. Let π‘₯,π‘¦βˆˆπ΄. and 𝐿=21βˆ’π‘Ÿ be nonnegative real numbers, and let πΆβˆ— be a mapping satisfying (2.27) and (4.5). Then there exists a unique Lie 𝐴-algebra homomorphism πΆβˆ— satisfying (2.40).Proof. The proof follows from Theorem 4.4 by takingfor all π›ΏβˆΆπ΄β†’π΄ Then 𝛿([π‘₯,𝑦])=[𝛿(π‘₯),𝑦]+[π‘₯,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 π‘₯,π‘¦βˆˆπ΄.-Algebras

Definition 5.1. Let 𝐴 be a Lie πΆβˆ—-algebra. A ‖⋅‖𝐴.-linear mapping πΆβˆ— is called a Lie derivation if π·πœ‡π‘“(π‘₯,𝑦)=0. for all π‘“βˆΆπ΄β†’π΄

Throughout this section, assume that πœ‘βˆΆπ΄2β†’[0,∞) is a Lie β€–β€–π‘“ξ€·ξ€Έβˆ’ξ€Ίξ€»βˆ’ξ€Ίξ€»β€–β€–[π‘₯,𝑦]𝑓(π‘₯),𝑦π‘₯,𝑓(𝑦)π΄β‰€πœ‘(π‘₯,𝑦)(5.1)-algebra with norm π‘₯,π‘¦βˆˆπ΄.

We prove the generalized Hyers-Ulam stability of derivations on Lie 𝐿<1-algebras for the functional equation πœ‘(π‘₯,𝑦)≀2πΏπœ‘(π‘₯/2,𝑦/2)

Theorem 5.2. Let π‘₯,π‘¦βˆˆπ΄. be a mapping for which there exists a function π›ΏβˆΆπ΄β†’π΄ satisfying (3.1) such that for all π›ΏβˆΆπ΄β†’π΄ If there exists an π›ΏβˆΆπ΄β†’π΄ such that 𝛿(π‘₯)=limπ‘›β†’βˆžπ‘“(2𝑛π‘₯)2𝑛(5.2) for all π‘₯∈𝐴. Then there exists a unique Lie derivation β€–β€–π›Ώξ€·ξ€Έβˆ’ξ€Ίξ€»βˆ’ξ€Ίξ€»β€–β€–[π‘₯,𝑦]𝛿(π‘₯),𝑦π‘₯,𝛿(𝑦)𝐴=limπ‘›β†’βˆž14𝑛‖‖𝑓4π‘›ξ€Έβˆ’ξ€Ίπ‘“ξ€·2[π‘₯,𝑦]𝑛π‘₯ξ€Έ,2π‘›π‘¦ξ€»βˆ’ξ€Ί2𝑛2π‘₯,𝑓𝑛𝑦‖‖𝐴≀limπ‘›β†’βˆž14π‘›πœ‘ξ€·2𝑛π‘₯,2𝑛𝑦≀limπ‘›β†’βˆž12π‘›πœ‘ξ€·2𝑛π‘₯,2𝑛𝑦=0(5.3) satisfying (3.3).Proof. By the same reasoning as the proof of Theorem 2.1, there exists a unique involutive π‘₯,π‘¦βˆˆπ΄.-linear mapping 𝛿=ξ€Ίξ€»[π‘₯,𝑦]𝛿(π‘₯),𝑦+[π‘₯,𝛿(𝑦)](5.4) satisfying (3.3). The mapping π‘₯,π‘¦βˆˆπ΄. is given byfor all 0<π‘Ÿ<1
It follows from (5.1) thatfor all π‘“βˆΆπ΄β†’π΄ Sofor all π‘₯,π‘¦βˆˆπ΄. Thus π›ΏβˆΆπ΄β†’π΄ is a derivation satisfying (3.3).Corollary 5.3. Let ξ€·πœ‘(π‘₯,𝑦)∢=πœƒβ€–π‘₯β€–π‘Ÿπ΄+β€–π‘¦β€–π‘Ÿπ΄ξ€Έ(5.6) and π‘₯,π‘¦βˆˆπ΄. be nonnegative real numbers, and let 𝐿=2π‘Ÿβˆ’1 be a mapping satisfying (3.7) such that for all πœ‘βˆΆπ΄2β†’[0,∞) Then there exists a unique Lie derivation 𝐿<1 satisfying (3.9).Proof. The proof follows from Theorem 5.2 by takingfor all π‘₯,π‘¦βˆˆπ΄, Then π›ΏβˆΆπ΄β†’π΄ and we get the desired result.

Theorem 5.4. Let π‘Ÿ>2 be a mapping for which there exists a function πœƒ satisfying (3.1) and (5.1). If there exists an π‘“βˆΆπ΄β†’π΄ such that π›ΏβˆΆπ΄β†’π΄ for all ξ€·πœ‘(π‘₯,𝑦)∢=πœƒβ€–π‘₯β€–π‘Ÿπ΄+β€–π‘¦β€–π‘Ÿπ΄ξ€Έ(5.7) then there exists a unique Lie derivation π‘₯,π‘¦βˆˆπ΄. satisfying (3.11).Proof. The proof is similar to the proofs of Theorems 2.3 and 5.2.Corollary 5.5. Let 𝐿=21βˆ’π‘Ÿ and πΆβˆ— be nonnegative real numbers, and let πΆβˆ— be a mapping satisfying (3.7) and (5.5). Then there exists a unique Lie derivation βˆ— satisfying (3.12).Proof. The proof follows from Theorem 5.4 by takingfor all βˆ— Then πΆβˆ— 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.

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