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Abstract and Applied Analysis
VolumeΒ 2008, Article IDΒ 672618, 13 pages
http://dx.doi.org/10.1155/2008/672618
Research Article

Fixed Points and Stability of an Additive Functional Equation of 𝑛-Apollonius Type in πΆβˆ—-Algebras

1Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran
2Department of Mathematics, Hanyang University, Seoul 133–791, South Korea

Received 22 April 2008; Revised 11 June 2008; Accepted 16 July 2008

Academic Editor: JohnΒ Rassias

Copyright Β© 2008 Fridoun Moradlou et al. 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 the fixed point method, we prove the generalized Hyers-Ulam stability of πΆβˆ—-algebra homomorphisms and of generalized derivations on πΆβˆ—-algebras for the following functional equation of Apollonius type _𝑛𝑖=1_𝑓(𝑧_π‘₯𝑖)=_(1/𝑛)___1_𝑖<𝑗_𝑛_𝑓(π‘₯𝑖+π‘₯𝑗)+𝑛𝑓(𝑧_(1/𝑛2)_𝑛𝑖=1_π‘₯𝑖).

1. Introduction and Preliminaries

A classical question in the theory of functional equations is the following: β€œwhen is it true that a function, which approximately satisfies a functional equation β„°, must be close to an exact solution of β„°?” If the problem accepts a solution, we say that the equation β„° is stable. Such a problem was formulated by Ulam [1] in 1940 and solved in the next year for the Cauchy functional equation by Hyers [2]. It gave rise to the stability theory for functional equations. The result of Hyers was extended by Aoki [3] in 1950 by considering the unbounded Cauchy differences. In 1978, Rassias [4] proved that the additive mapping 𝑇, obtained by Hyers or Aoki, is linear if, in addition, for each π‘₯∈𝐸, the mapping 𝑓(𝑑π‘₯) is continuous in π‘‘βˆˆβ„. GΔƒvruΕ£ a [5] generalized the Rassias' result. Following the techniques of the proof of the corollary of Hyers [2], we observed that Hyers introduced (in 1941) the following Hyers continuity condition about the continuity of the mapping for each fixed point and then he proved homogeneity of degree one and, therefore, the famous linearity. This condition has been assumed further till now, through the complete Hyers direct method, in order to prove linearity for generalized Hyers-Ulam stability problem forms (see [6]). Beginning around 1980, the stability problems of several functional equations and approximate homomorphisms have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [7–21]).

Rassias [22], following the spirit of the innovative approach of Hyers [2], Aoki [3], and Rassias [4] for the unbounded Cauchy difference, proved a similar stability theorem 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 (see also [23, 24] for a number of other new results).

In 2003, CΔƒdariu and Radu applied the fixed-point method to the investigation of the Jensen functional equation [25] (see also [8, 26–30]). They could present a short and a simple proof (different of the β€œdirect method,” initiated by Hyers in 1941) for the generalized Hyers-Ulam stability of Jensen functional equation [25], for Cauchy functional equation [8], and for quadratic functional equation [26].

The following functional equation:𝑄(π‘₯+𝑦)+𝑄(π‘₯βˆ’π‘¦)=2𝑄(π‘₯)+2𝑄(𝑦)(1.1)is called a quadratic functional equation, and every solution of (1.1) is said to be a quadratic mapping. Skof [31] proved the Hyers-Ulam stability of the quadratic functional equation (1.1) for mappings π‘“βˆΆπΈ1→𝐸2, where 𝐸1 is a normed space and 𝐸2 is a Banach space. In [32], Czerwik proved the Hyers-Ulam stability of the quadratic functional equation (1.1). Borelli and Forti [33] generalized the stability result of the quadratic functional equation (1.1). Jun and Lee [34] proved the Hyers-Ulam stability of the Pexiderized quadratic equation𝑓(π‘₯+𝑦)+𝑔(π‘₯βˆ’π‘¦)=2β„Ž(π‘₯)+2π‘˜(𝑦)(1.2)for mappings 𝑓,𝑔,β„Ž, and π‘˜. The stability problem of the quadratic equation has been extensively investigated by some mathematicians [35].

In an inner product space, the equalityβ€–π‘§βˆ’π‘₯β€–2+β€–π‘§βˆ’π‘¦β€–2=12β€–π‘₯βˆ’π‘¦β€–2+2β€–β€–β€–π‘§βˆ’π‘₯+𝑦2β€–β€–β€–2(1.3)holds, then it is called the Apollonius' identity. The following functional equation, which was motivated by this equation,𝑄(π‘§βˆ’π‘₯)+𝑄(π‘§βˆ’π‘¦)=12𝑄(π‘₯βˆ’π‘¦)+2π‘„ξ‚€π‘§βˆ’π‘₯+𝑦2,(1.4)holds, then it is called quadratic (see [36]). For this reason, the functional equation (1.4) is called a quadratic functional equation of Apollonius type, and each solution of the functional equation (1.4) is said to be a quadratic mapping of Apollonius type. The quadratic functional equation and several other functional equations are useful to characterize inner product spaces [37].

In [36], Park and Rassias introduced and investigated a functional equation, which is called a generalized Apollonius type quadratic functional equation. In [38], Najati introduced and investigated a functional equation, which is called a quadratic functional equation of 𝑛-Apollonius type. Recently in [39], Park and Rassias introduced and investigated the following functional equation:𝑓(π‘§βˆ’π‘₯)+𝑓(π‘§βˆ’π‘¦)=βˆ’12𝑓(π‘₯+𝑦)+2π‘“ξ‚€π‘§βˆ’π‘₯+𝑦4(1.5)which is called an Apollonius type additive functional equation, and whose solution is called an Apollonius type additive mapping. In [40], Park introduced and investigated a functional equation, which is called a generalized Apollonius-Jensen type additive functional equation and whose solution is said to be a generalized Apollonius-Jensen type additive mapping.

In this paper, employing the above equality (1.5), for a fixed positive integer 𝑛β‰₯2, we introduce a new functional equation, which is called an additive functional equation of 𝑛-Apollonius type and whose solution is said to be an additive mapping of 𝑛-Apollonius type; 𝑛𝑖=1𝑓(π‘§βˆ’π‘₯𝑖)=βˆ’1𝑛1≀𝑖<𝑗≀𝑛𝑓(π‘₯𝑖+π‘₯𝑗)+π‘›π‘“ξ‚΅π‘§βˆ’1𝑛2𝑛𝑖=1π‘₯𝑖.(1.6)

We will adopt the idea of CΔƒdariu and Radu [8, 25, 28] to prove the generalized Hyers-Ulam stability results of πΆβˆ—-algebra homomorphisms as well as to prove the generalized Ulam-Hyers stability of generalized derivations on πΆβˆ—-algebra for additive functional equation of 𝑛-Apollonius type.

We recall two fundamental results in fixed-point theory.

Theorem 1.1 (see [25]). Let (𝑋,𝑑) be a complete metric space and let π½βˆΆπ‘‹β†’π‘‹ be strictly contractive, that is, 𝑑(𝐽π‘₯,𝐽𝑦)≀𝐿𝑓(π‘₯,𝑦),βˆ€π‘₯,π‘¦βˆˆπ‘‹(1.7)for some Lipschitz constant 𝐿<1. Then, the following hold:
(1)the mapping 𝐽 has a unique fixed point π‘₯βˆ—=𝐽π‘₯βˆ—;(2)the fixed point π‘₯βˆ— is globally attractive, that is, limπ‘›β†’βˆžπ½π‘›π‘₯=π‘₯βˆ—(1.8) for any starting point π‘₯βˆˆπ‘‹;(3)one has the following estimation inequalities: 𝑑(𝐽𝑛π‘₯,π‘₯βˆ—)≀𝐿𝑛𝑑(π‘₯,π‘₯βˆ—),𝑑(𝐽𝑛π‘₯,π‘₯βˆ—)≀11βˆ’πΏπ‘‘(𝐽𝑛π‘₯,𝐽𝑛+1π‘₯),𝑑(π‘₯,π‘₯βˆ—)≀11βˆ’πΏπ‘‘(π‘₯,𝐽π‘₯)(1.9) for all nonnegative integers 𝑛 and all π‘₯βˆˆπ‘‹.

Let 𝑋 be a set. A function π‘‘βˆΆπ‘‹Γ—π‘‹β†’[0,∞] is called a generalized metric on 𝑋 if 𝑑 satisfies the following:

(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 1.2 (see [41]). 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 𝑑(𝐽𝑛π‘₯,𝐽𝑛+1π‘₯)=∞(1.10)for all nonnegative integers 𝑛 or there exists a positive integer 𝑛0 such that the following hold:
(1)𝑑(𝐽𝑛π‘₯,𝐽𝑛+1π‘₯)<∞ for all 𝑛β‰₯𝑛0;(2)he 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.

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

2. Stability of πΆβˆ—-Algebra Homomorphisms

Lemma 2.1. Let 𝑋 and π‘Œ be real-vector spaces. A mapping π‘“βˆΆπ‘‹β†’π‘Œ satisfies (1.6) for all π‘₯1,…,π‘₯𝑛,𝑧 if and only if the mapping 𝑓 is additive.

Proof. Letting π‘₯1=β‹―=π‘₯𝑛=𝑧=0 in (1.6), we get that 𝑓(0)=0. Let 𝑗 and π‘˜ be fixed integers with 1≀𝑗<π‘˜β‰€π‘›. Setting π‘₯𝑖=0 for all 1≀𝑖≀𝑛,𝑖≠𝑗,π‘˜ in (1.6), we have𝑓(π‘§βˆ’π‘₯𝑗)+𝑓(π‘§βˆ’π‘₯π‘˜)+(π‘›βˆ’2)𝑓(𝑧)=βˆ’1𝑛𝑓(π‘₯𝑗+π‘₯π‘˜)βˆ’π‘›βˆ’2𝑛(𝑓(π‘₯𝑗)+𝑓(π‘₯π‘˜))+π‘›π‘“ξ‚€π‘§βˆ’1𝑛2(π‘₯𝑗+π‘₯π‘˜)(2.1)for all π‘₯𝑗,π‘₯π‘˜,π‘§βˆˆπ‘‹. Replacing π‘₯𝑗 by βˆ’π‘₯𝑗 and π‘₯π‘˜ by π‘₯𝑗 in (2.1), respectively, we get𝑓(𝑧+π‘₯𝑗)+𝑓(π‘§βˆ’π‘₯𝑗)=βˆ’π‘›βˆ’2𝑛(𝑓(βˆ’π‘₯𝑗)+𝑓(π‘₯𝑗))+2𝑓(𝑧)(2.2)for all π‘₯𝑗,π‘§βˆˆπ‘‹. Putting 𝑧=0 in (2.2), we conclude that 𝑓(βˆ’π‘₯𝑗)=βˆ’π‘“(π‘₯𝑗) for all π‘₯π‘—βˆˆπ‘‹. This means that 𝑓 is an odd function. Letting π‘₯π‘˜=𝑧=0 in (2.1) and using the oddness of 𝑓, we obtain that𝑓1𝑛2π‘₯𝑗=1𝑛2𝑓(π‘₯𝑗),𝑓(𝑛2π‘₯𝑗)=𝑛2𝑓(π‘₯𝑗)(2.3)for all π‘₯π‘—βˆˆπ‘‹. Letting 𝑧=0 in (2.1), using the oddness of 𝑓 and (2.3), we have𝑓(π‘₯𝑗+π‘₯π‘˜)=𝑓(π‘₯𝑗)+𝑓(π‘₯π‘˜)(2.4)for all π‘₯𝑗,π‘₯π‘˜βˆˆπ‘‹. Therefore, π‘“βˆΆπ‘‹β†’π‘Œ is an additive mapping.
The converse is obviously true.

For a given mapping π‘“βˆΆπ΄β†’π΅ and for a fixed positive integer 𝑛β‰₯2, we defineπΆπœ‡π‘“(𝑧,π‘₯1,…,π‘₯𝑛)∢=𝑛𝑖=1πœ‡π‘“(π‘§βˆ’π‘₯𝑖)+1𝑛1≀𝑖<𝑗≀𝑛𝑓(πœ‡π‘₯𝑖+πœ‡π‘₯𝑗)βˆ’π‘›π‘“ξ‚΅πœ‡π‘§βˆ’1𝑛2𝑛𝑖=1πœ‡π‘₯𝑖(2.5)for all πœ‡βˆˆπ•‹1∢={πœˆβˆˆβ„‚βˆΆ|𝜈|=1} and all 𝑧,π‘₯1,…,π‘₯π‘›βˆˆπ΄.

We prove the generalized Hyers-Ulam stability of πΆβˆ—-algebra homomorphisms for the functional equation πΆπœ‡π‘“(𝑧,π‘₯1,…,π‘₯𝑛)=0.

Theorem 2.2. Let π‘“βˆΆπ΄β†’π΅ be a mapping satisfying 𝑓(0)=0 for which there exists a function πœ‘βˆΆπ΄π‘›+1β†’[0,∞) such that βˆžξ“π‘—=0𝑛2𝑛2βˆ’12π‘—πœ‘ξ‚€ξ‚€π‘›2βˆ’1𝑛2𝑗𝑧,𝑛2βˆ’1𝑛2𝑗π‘₯1,…,𝑛2βˆ’1𝑛2𝑗π‘₯𝑛<∞,(2.6)β€–πΆπœ‡π‘“(𝑧,π‘₯1,…,π‘₯𝑛)β€–π΅β‰€πœ‘(𝑧,π‘₯1,…,π‘₯𝑛),(2.7)‖𝑓(π‘₯𝑦)βˆ’π‘“(π‘₯)𝑓(𝑦)β€–π΅β‰€πœ‘(π‘₯,𝑦,0,…,0ξ„Ώξ…€ξ…€ξ…ƒξ…€ξ…€ξ…Œπ‘›βˆ’1times),(2.8)‖𝑓(π‘₯βˆ—)βˆ’π‘“(π‘₯)βˆ—β€–π΅β‰€πœ‘(π‘₯,…,π‘₯ξ„Ώξ…€ξ…€ξ…ƒξ…€ξ…€ξ…Œπ‘›+1times)(2.9) for all πœ‡βˆˆπ•‹1 and all π‘₯,𝑦,𝑧,π‘₯1,…,π‘₯π‘›βˆˆπ΄. If for some 1≀𝑗≀𝑛 there exists a Lipschitz constant 𝐿<1 such that πœ‘(π‘₯,0,…,0,π‘₯𝑗th,0,…,0)≀𝑛2βˆ’1𝑛2πΏπœ‘βŽ›βŽœβŽœβŽœβŽπ‘›2𝑛2βˆ’1π‘₯,0,…,0,𝑛2𝑛2βˆ’1π‘₯ξ„Ώξ…€ξ…€ξ…€ξ…ƒξ…€ξ…€ξ…€ξ…Œπ‘—th,0,…,0⎞⎟⎟⎟⎠(2.10) for all π‘₯∈𝐴, then there exists a unique πΆβˆ—-algebra homomorphism π»βˆΆπ΄β†’π΅ such that ‖𝑓(π‘₯)βˆ’π»(π‘₯)‖𝐡≀𝑛(𝑛2βˆ’1)Γ—(1βˆ’πΏ)πœ‘(π‘₯,0,…,0,π‘₯𝑗th,0,…,0)(2.11)for all π‘₯∈𝐴.

Proof. Consider the setπ‘‹βˆΆ={π‘”βˆΆπ΄βŸΆπ΅,𝑔(0)=0}(2.12)and introduce the generalized metric on 𝑋:𝑑(𝑔,β„Ž)=infξ‚†πΆβˆˆβ„+βˆΆβ€–π‘”(π‘₯)βˆ’β„Ž(π‘₯)β€–π΅β‰€πΆπœ‘(π‘₯,0,…,0,π‘₯𝑗th,0,…,0)βˆ€π‘₯βˆˆπ΄ξ‚‡.(2.13)It is easy to show that (𝑋,𝑑) is complete.
For convenience, setπœ‘π‘—(π‘₯,𝑦)∢=πœ‘(π‘₯,0,…,0,𝑦𝑗th,0,…,0)(2.14)for all π‘₯,π‘¦βˆˆπ΄ and all 1≀𝑗≀𝑛.
Now we consider the linear mapping π½βˆΆπ‘‹β†’π‘‹ such that𝐽𝑔(π‘₯)∢=𝑛𝛼𝑔𝛼𝑛π‘₯(2.15)for all π‘₯∈𝐴, where 𝛼=(𝑛2βˆ’1)/𝑛.
For any 𝑔,β„Žβˆˆπ‘‹, we have𝑑(𝑔,β„Ž)<πΆβŸΉβ€–π‘”(π‘₯)βˆ’β„Ž(π‘₯)β€–π΅β‰€πΆπœ‘π‘—(π‘₯,π‘₯)βˆ€π‘₯βˆˆπ΄βŸΉβ€–β€–β€–π‘›π›Όπ‘”ξ‚€π›Όπ‘›π‘₯ξ‚βˆ’π‘›π›Όβ„Žξ‚€π›Όπ‘›π‘₯ξ‚β€–β€–β€–π΅β‰€π‘›π›ΌπΆπœ‘π‘—ξ‚€π›Όπ‘›π‘₯,𝛼𝑛π‘₯ξ‚βŸΉβ€–β€–β€–π‘›π›Όπ‘”ξ‚€π›Όπ‘›π‘₯ξ‚βˆ’π‘›π›Όβ„Žξ‚€π›Όπ‘›π‘₯ξ‚β€–β€–β€–π΅β‰€πΏπΆπœ‘π‘—(π‘₯,π‘₯)βŸΉπ‘‘(𝐽𝑔,π½β„Ž)≀𝐿𝐢.(2.16)
Therefore, we see that𝑑(𝐽𝑔,π½β„Ž)≀𝐿𝑑(𝑔,β„Ž),βˆ€π‘”,β„Žβˆˆπ΄.(2.17)This means 𝐽 is a strictly contractive self-mapping of 𝑋, with the Lipschitz constant 𝐿.
Letting πœ‡=1,𝑧=π‘₯𝑗=π‘₯, and for each 1β‰€π‘˜β‰€π‘› with π‘˜β‰ π‘—,π‘₯π‘˜=0 in (2.7), we get‖‖‖𝛼𝑓(π‘₯)βˆ’π‘›π‘“ξ‚€π›Όπ‘›π‘₯ξ‚β€–β€–β€–π΅β‰€πœ‘π‘—(π‘₯,π‘₯)(2.18)for all π‘₯∈𝐴. So‖‖‖𝑓(π‘₯)βˆ’π‘›π›Όπ‘“ξ‚€π›Όπ‘›π‘₯‖‖‖𝐡≀1π›Όπœ‘π‘—(π‘₯,π‘₯)(2.19)for all π‘₯∈𝐴. Hence 𝑑(𝑓,𝐽𝑓)≀1/𝛼.
By Theorem 1.2, there exists a mapping π»βˆΆπ΄β†’π΅ such that the following hold:
(1) 𝐻 is a fixed point of 𝐽, that is,𝐻𝛼𝑛π‘₯=𝛼𝑛𝐻(π‘₯)(2.20)for all π‘₯∈𝐴; the mapping 𝐻 is a unique fixed point of 𝐽 in the setπ‘Œ={π‘”βˆˆπ‘‹βˆΆπ‘‘(𝑓,𝑔)<∞};(2.21)and this implies that 𝐻 is a unique mapping satisfying (2.20) such that there exists 𝐢∈(0,∞) satisfying‖𝐻(π‘₯)βˆ’π‘“(π‘₯)β€–π΅β‰€πΆπœ‘π‘—(π‘₯,π‘₯)(2.22)for all π‘₯∈𝐴.
(2) 𝑑(π½π‘šπ‘“,𝐻)β†’0 as π‘šβ†’βˆž; and this implies the equalitylimπ‘šβ†’βˆžξ‚€π‘›π›Όξ‚π‘šπ‘“ξ‚€ξ‚€π›Όπ‘›ξ‚π‘šπ‘₯=𝐻(π‘₯)(2.23)for all π‘₯∈𝐴;
(3) 𝑑(𝑓,𝐻)≀(1/(1βˆ’πΏ))𝑑(𝑓,𝐽𝑓), which implies the inequality𝑑(𝑓,𝐻)≀1π›Όβˆ’π›ΌπΏ;(2.24)and this implies that the inequality (2.11) holds.
It follows from (2.6), (2.7), and (2.23) that‖‖‖𝑛𝑖=1𝐻(π‘§βˆ’π‘₯𝑖)+1𝑛1≀𝑖<𝑗≀𝑛𝐻(π‘₯𝑖+π‘₯𝑗)βˆ’π‘›π»ξ‚΅π‘§βˆ’1𝑛2𝑛𝑖=1π‘₯𝑖‖‖‖𝐡=limπ‘šβ†’βˆžξ‚€π‘›π›Όξ‚π‘šβ€–β€–β€–π‘›ξ“π‘–=1π‘“ξ‚€ξ‚€π›Όπ‘›ξ‚π‘š(π‘§βˆ’π‘₯𝑖)+1𝑛1≀𝑖<π‘—β‰€π‘›π‘“ξ‚€ξ‚€π›Όπ‘›ξ‚π‘š(π‘₯𝑖+π‘₯𝑗)ξ‚βˆ’π‘›π‘“ξ‚΅ξ‚€π›Όπ‘›ξ‚π‘šπ‘§βˆ’ξ‚€π›Όπ‘›ξ‚π‘šΓ—1𝑛2𝑛𝑖=1π‘₯𝑖‖‖‖𝐡≀limπ‘šβ†’βˆžξ‚€π‘›π›Όξ‚π‘šπœ‘ξ‚€ξ‚€π›Όπ‘›ξ‚π‘šπ‘§,ξ‚€π›Όπ‘›ξ‚π‘šπ‘₯1,…,ξ‚€π›Όπ‘›ξ‚π‘šπ‘₯𝑛≀limπ‘šβ†’βˆžξ‚€π‘›π›Όξ‚2π‘šπœ‘ξ‚€ξ‚€π›Όπ‘›ξ‚π‘šπ‘§,ξ‚€π›Όπ‘›ξ‚π‘šπ‘₯1,…,ξ‚€π›Όπ‘›ξ‚π‘šπ‘₯𝑛=0(2.25)for all π‘₯1,…,π‘₯𝑛,π‘§βˆˆπ΄. So𝑛𝑖=1𝐻(π‘§βˆ’π‘₯𝑖)=βˆ’1𝑛1≀𝑖<𝑗≀𝑛𝐻(π‘₯𝑖+π‘₯𝑗)+π‘›π»ξ‚΅π‘§βˆ’1𝑛2𝑛𝑖=1π‘₯𝑖(2.26)for all π‘₯1,…,π‘₯𝑛,π‘§βˆˆπ΄. By Lemma 2.1, the mapping π»βˆΆπ΄β†’π΅ is Cauchy additive, that is, 𝐻(π‘₯+𝑦)=𝐻(π‘₯)+𝐻(𝑦) for all π‘₯,π‘¦βˆˆπ΄.
By a similar method to the proof of [14], one can show that the mapping π»βˆΆπ΄β†’π΅ is β„‚-linear.
It follows from (2.8) that‖𝐻(π‘₯𝑦)βˆ’π»(π‘₯)𝐻(𝑦)‖𝐡=limπ‘šβ†’βˆžξ‚€π‘›π›Όξ‚2π‘šβ€–β€–β€–π‘“ξ‚€ξ‚€π›Όπ‘›ξ‚2π‘šπ‘₯π‘¦ξ‚βˆ’π‘“ξ‚€ξ‚€π›Όπ‘›ξ‚π‘šπ‘₯ξ‚π‘“ξ‚€ξ‚€π›Όπ‘›ξ‚π‘šπ‘¦ξ‚β€–β€–β€–π΅β‰€limπ‘šβ†’βˆžξ‚€π‘›π›Όξ‚2π‘šπœ‘ξ‚€ξ‚€π›Όπ‘›ξ‚π‘šπ‘₯,ξ‚€π›Όπ‘›ξ‚π‘šπ‘¦,0,…,0ξ„Ώξ…€ξ…€ξ…ƒξ…€ξ…€ξ…Œπ‘›βˆ’1times=0(2.27)for all π‘₯,π‘¦βˆˆπ΄. So𝐻(π‘₯𝑦)=𝐻(π‘₯)𝐻(𝑦)(2.28)for all π‘₯,π‘¦βˆˆπ΄.
It follows from (2.9) that‖𝐻(π‘₯βˆ—)βˆ’π»(π‘₯)βˆ—β€–π΅=limπ‘šβ†’βˆžξ‚€π‘›π›Όξ‚π‘šβ€–β€–β€–π‘“ξ‚€ξ‚€π›Όπ‘›ξ‚π‘šπ‘₯βˆ—ξ‚βˆ’π‘“ξ‚€ξ‚€π›Όπ‘›ξ‚π‘šπ‘₯ξ‚βˆ—β€–β€–β€–π΅β‰€limπ‘šβ†’βˆžξ‚€π‘›π›Όξ‚π‘šπœ‘ξ‚€ξ‚€π›Όπ‘›ξ‚π‘šπ‘₯,…,ξ‚€π›Όπ‘›ξ‚π‘šπ‘₯ξ„Ώξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…ƒξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…Œπ‘›+1times≀limπ‘šβ†’βˆžξ‚€π‘›π›Όξ‚2π‘šπœ‘ξ‚€ξ‚€π›Όπ‘›ξ‚π‘šπ‘₯,…,ξ‚€π›Όπ‘›ξ‚π‘šπ‘₯ξ„Ώξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…ƒξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…Œπ‘›+1times=0(2.29)for all π‘₯∈𝐴. So𝐻(π‘₯βˆ—)=𝐻(π‘₯)βˆ—(2.30)for all π‘₯∈𝐴.
Thus π»βˆΆπ΄β†’π΅ is a πΆβˆ—-algebra homomorphism satisfying (2.11) as desired.

Corollary 2.3. Let π‘Ÿ>2 and πœƒ be nonnegative real numbers, and let π‘“βˆΆπ΄β†’π΅ be a mapping such that β€–πΆπœ‡π‘“(𝑧,π‘₯1,…,π‘₯𝑛)β€–π΅β‰€πœƒξ‚΅β€–π‘§β€–π‘Ÿπ΄+𝑛𝑖=1β€–π‘₯π‘–β€–π‘Ÿπ΄ξ‚Ά,(2.31)‖𝑓(π‘₯𝑦)βˆ’π‘“(π‘₯)𝑓(𝑦)β€–π΅β‰€πœƒ(β€–π‘₯β€–π‘Ÿπ΄+β€–π‘¦β€–π‘Ÿπ΄),(2.32)
‖𝑓(π‘₯βˆ—)βˆ’π‘“(π‘₯)βˆ—β€–π΅β‰€(𝑛+1)πœƒβ€–π‘₯β€–π‘Ÿπ΄(2.33) for all πœ‡βˆˆπ•‹1 and all π‘₯,𝑦,π‘§βˆˆπ΄. Then there exists a unique πΆβˆ—-algebra homomorphism π»βˆΆπ΄β†’π΅ such that ‖𝑓(π‘₯)βˆ’π»(π‘₯)‖𝐡≀2𝑛(𝑛2βˆ’1)βˆ’π‘Ÿπœƒ(𝑛2βˆ’1)1βˆ’π‘Ÿβˆ’π‘›2(1βˆ’π‘Ÿ)β€–π‘₯β€–π‘Ÿπ΄(2.34)for all π‘₯∈𝐴.

Proof. The proof follows from Theorem 2.2 by takingπœ‘(𝑧,π‘₯1,…,π‘₯𝑛)∢=πœƒξ‚΅β€–π‘§β€–π‘Ÿπ΄+𝑛𝑖=1β€–π‘₯π‘–β€–π‘Ÿπ΄ξ‚Ά(2.35)for all π‘₯,𝑦,π‘§βˆˆπ΄. It follows from (2.31) that 𝑓(0)=0. We can choose 𝐿=(𝑛2/(𝑛2βˆ’1))1βˆ’π‘Ÿ to get the desired result.

Theorem 2.4. Let π‘“βˆΆπ΄β†’π΅ be a mapping satisfying 𝑓(0)=0 for which there exists a function πœ‘βˆΆπ΄π‘›+1β†’[0,∞) satisfying (2.7), (2.8), and (2.9) such that βˆžξ“π‘—=0𝑛2βˆ’1𝑛2ξ‚π‘—πœ‘ξ‚€ξ‚€π‘›2𝑛2βˆ’1𝑗𝑧,𝑛2𝑛2βˆ’1𝑗π‘₯1,…,𝑛2𝑛2βˆ’1𝑗π‘₯𝑛<∞(2.36)for all 𝑧,π‘₯1,…,π‘₯π‘›βˆˆπ΄. If for some 1≀𝑗≀𝑛 there exists a Lipschitz constant 𝐿<1 such that πœ‘(π‘₯,0,…,0,π‘₯𝑗th,0,…,0)≀𝑛2𝑛2βˆ’1πΏπœ‘ξ‚€π‘›2βˆ’1𝑛2π‘₯,0,…,0,𝑛2βˆ’1𝑛2π‘₯ξ„Ώξ…€ξ…€ξ…€ξ…ƒξ…€ξ…€ξ…€ξ…Œπ‘—th,0,…,0(2.37)for all π‘₯∈𝐴, then there exists a unique πΆβˆ—-algebra homomorphism π»βˆΆπ΄β†’π΅ such that ‖𝑓(π‘₯)βˆ’π»(π‘₯)‖𝐡≀𝑛𝐿(𝑛2βˆ’1)Γ—(1βˆ’πΏ)πœ‘(π‘₯,0,…,0,π‘₯𝑗th,0,…,0)(2.38)for all π‘₯∈𝐴.

Proof. Similar to proof of Theorem (2.2), we consider the linear mapping π½βˆΆπ‘‹β†’π‘‹ such that𝐽𝑔(π‘₯)∢=𝛼𝑛𝑔𝑛𝛼π‘₯(2.39)for all π‘₯∈𝐴, where 𝛼=(𝑛2βˆ’1)/𝑛. We can conclude that 𝐽 is a strictly contractive self mapping of 𝑋 with the Lipschitz constant 𝐿.
It follows from (2.18) that‖‖‖𝑓(π‘₯)βˆ’π›Όπ‘›π‘“ξ‚€π‘›π›Όπ‘₯‖‖‖𝐡≀1π‘›πœ‘π‘—ξ‚€π‘›π›Όπ‘₯,𝑛𝛼π‘₯ξ‚β‰€πΏπ›Όπœ‘π‘—(π‘₯,π‘₯)(2.40)for all π‘₯∈𝐴. Hence, 𝑑(𝑓,𝐽𝑓)≀(𝐿/𝛼).
By Theorem 1.2, there exists a mapping π»βˆΆπ΄β†’π΅ such that the following hold:
(1) 𝐻 is a fixed point of 𝐽, that is,𝐻𝑛𝛼π‘₯=𝑛𝛼𝐻(π‘₯)(2.41)for all π‘₯∈𝐴; the mapping 𝐻 is a unique fixed point of 𝐽 in the setπ‘Œ={π‘”βˆˆπ‘‹βˆΆπ‘‘(𝑓,𝑔)<∞};(2.42)and this implies that 𝐻 is a unique mapping satisfying (2.41) such that there exists 𝐢∈(0,∞) satisfying‖𝐻(π‘₯)βˆ’π‘“(π‘₯)β€–π΅β‰€πΆπœ‘π‘—(π‘₯,π‘₯)(2.43)for all π‘₯∈𝐴;
(2) 𝑑(π½π‘šπ‘“,𝐻)β†’0 as π‘šβ†’βˆž; and this implies the equalitylimπ‘šβ†’βˆžξ‚€π›Όπ‘›ξ‚π‘šπ‘“ξ‚€ξ‚€π›Όπ‘›ξ‚π‘šπ‘₯=𝐻(π‘₯)(2.44)for all π‘₯∈𝐴;
(3) 𝑑(𝑓,𝐻)≀(1/(1βˆ’πΏ))𝑑(𝑓,𝐽𝑓), which implies the inequality𝑑(𝑓,𝐻)β‰€πΏπ›Όβˆ’π›ΌπΏ,(2.45)which implies that the inequality (2.38) holds.
The rest of the proof is similar to the proof of Theorem 2.2.

Corollary 2.5. Let π‘Ÿ<1 and πœƒ be nonnegative real numbers, and let π‘“βˆΆπ΄β†’π΅ be a mapping satisfying (2.31), (2.32), and (2.33). Then there exists a unique πΆβˆ—-algebra homomorphism π»βˆΆπ΄β†’π΅ such that ‖𝑓(π‘₯)βˆ’π»(π‘₯)‖𝐡≀2𝑛(𝑛2βˆ’1)π‘Ÿβˆ’2πΏπœƒ(𝑛2βˆ’1)π‘Ÿβˆ’1βˆ’π‘›2(π‘Ÿβˆ’1)β€–π‘₯β€–π‘Ÿπ΄(2.46)for all π‘₯∈𝐴 and 𝐿=(𝑛2/(𝑛2βˆ’1))π‘Ÿβˆ’1.

Proof. The proof follows from Theorem 2.4 by takingπœ‘(𝑧,π‘₯1,…,π‘₯𝑛)∢=πœƒξ‚΅β€–π‘§β€–π‘Ÿπ΄+𝑛𝑖=1β€–π‘₯π‘–β€–π‘Ÿπ΄ξ‚Ά(2.47)for all 𝑧,π‘₯1,…,π‘₯π‘›βˆˆπ΄. It follows from (2.31) that 𝑓(0)=0. We can choose 𝐿=(𝑛2/(𝑛2βˆ’1))π‘Ÿβˆ’1 to get the desired result.

3. Stability of Generalized Derivations on πΆβˆ—-Algebras

For a given mapping π‘“βˆΆπ΄β†’π΄ and for a fixed positive integer 𝑛β‰₯2, we defineπΆπœ‡π‘“(𝑧,π‘₯1,…,π‘₯𝑛)∢=𝑛𝑖=1πœ‡π‘“(π‘§βˆ’π‘₯𝑖)+1𝑛1≀𝑖<𝑗≀𝑛𝑓(πœ‡π‘₯𝑖+πœ‡π‘₯𝑗)βˆ’π‘›π‘“ξ‚΅πœ‡π‘§βˆ’1𝑛2𝑛𝑖=1πœ‡π‘₯𝑖(3.1)for all πœ‡βˆˆπ•‹1 and all 𝑧,π‘₯1,…,π‘₯π‘›βˆˆπ΄.

Definition 3.1 (see [42]). A generalized derivation π›ΏβˆΆπ΄β†’π΄ is involutive β„‚-linear and fulfills
𝛿(π‘₯𝑦𝑧)=𝛿(π‘₯𝑦)π‘§βˆ’π‘₯𝛿(𝑦)𝑧+π‘₯𝛿(𝑦𝑧)(1)for all π‘₯,𝑦,π‘§βˆˆπ΄.
We prove the generalized Hyers-Ulam stability of derivations on πΆβˆ—-algebras for the functional equation πΆπœ‡π‘“(𝑧,π‘₯1,…,π‘₯𝑛)=0.

Theorem 3.2. Let π‘“βˆΆπ΄β†’π΄ be a mapping satisfying 𝑓(0)=0 for which there exists a function πœ‘βˆΆπ΄π‘›+1β†’[0,∞) such that βˆžξ“π‘—=0𝑛2𝑛2βˆ’13π‘—πœ‘ξ‚€ξ‚€π‘›2βˆ’1𝑛2𝑗𝑧,𝑛2βˆ’1𝑛2𝑗π‘₯1,…,𝑛2βˆ’1𝑛2𝑗π‘₯𝑛<∞,(3.2)β€–πΆπœ‡π‘“(π‘₯1,…,π‘₯𝑛,𝑧)β€–π΄β‰€πœ‘(𝑧,π‘₯1,…,π‘₯𝑛),(3.3)‖𝑓(π‘₯𝑦𝑧)βˆ’π‘“(π‘₯𝑦)𝑧+π‘₯𝑓(𝑦)π‘§βˆ’π‘₯𝑓(𝑦𝑧)β€–π΄β‰€πœ‘(π‘₯,𝑦,𝑧,0,…,0ξ„Ώξ…€ξ…€ξ…ƒξ…€ξ…€ξ…Œπ‘›βˆ’2times),(3.4)‖𝑓(π‘₯βˆ—)βˆ’π‘“(π‘₯)βˆ—β€–π΄β‰€πœ‘(π‘₯,…,π‘₯ξ„Ώξ…€ξ…€ξ…ƒξ…€ξ…€ξ…Œπ‘›+1times)(3.5) for all πœ‡βˆˆπ•‹1 and all π‘₯,𝑦,𝑧,π‘₯1,…,π‘₯π‘›βˆˆπ΄. If for some 1≀𝑗≀𝑛 there exists a Lipschitz constant 𝐿<1 such that πœ‘(π‘₯,0,…,0,π‘₯𝑗th,0,…,0)≀𝑛2βˆ’1𝑛2πΏπœ‘ξ‚€π‘›2𝑛2βˆ’1π‘₯,0,…,0,𝑛2𝑛2βˆ’1π‘₯ξ„Ώξ…€ξ…€ξ…€ξ…ƒξ…€ξ…€ξ…€ξ…Œπ‘—th,0,…,0(3.6)for all π‘₯∈𝐴, then there exists a unique generalized derivation π›ΏβˆΆπ΄β†’π΄ such that ‖𝑓(π‘₯)βˆ’π›Ώ(π‘₯)‖𝐴≀𝑛(𝑛2βˆ’1)Γ—(1βˆ’πΏ)πœ‘(π‘₯,0,…,0,π‘₯𝑗th,0,…,0)(3.7)for all π‘₯∈𝐴.

Proof. By the same reasoning as in the proof of Theorem 2.2, there exists a unique involutive β„‚-linear mapping π›ΏβˆΆπ΄β†’π΄ satisfying (3.7). The mapping π›ΏβˆΆπ΄β†’π΄ is given by𝛿(π‘₯)=ξ‚€π‘›π›Όξ‚π‘šπ‘“ξ‚€ξ‚€π‘›π›Όξ‚π‘šπ‘₯(3.8)for all π‘₯∈𝐴.
It follows from (3.4) that‖𝛿(π‘₯𝑦𝑧)βˆ’π›Ώ(π‘₯𝑦)𝑧+π‘₯𝛿(𝑦)π‘§βˆ’π‘₯𝛿(𝑦𝑧)‖𝐴=limπ‘šβ†’βˆžξ‚€π‘›π›Όξ‚3π‘šβ€–β€–β€–π‘“ξ‚€ξ‚€π›Όπ‘›ξ‚3π‘šπ‘₯π‘¦π‘§ξ‚βˆ’π‘“ξ‚€ξ‚€π›Όπ‘›ξ‚2π‘šπ‘₯π‘¦ξ‚β‹…ξ‚€π›Όπ‘›ξ‚π‘šπ‘§+ξ‚€π›Όπ‘›ξ‚π‘šπ‘₯π‘“ξ‚€ξ‚€π›Όπ‘›ξ‚π‘šπ‘¦ξ‚β‹…ξ‚€π›Όπ‘›ξ‚π‘šπ‘§βˆ’ξ‚€π›Όπ‘›ξ‚π‘šπ‘₯𝑓𝛼𝑛2π‘šπ‘¦π‘§ξ‚β€–β€–β€–π΄β‰€limπ‘šβ†’βˆžξ‚€π‘›π›Όξ‚3π‘šπœ‘ξ‚€ξ‚€π›Όπ‘›ξ‚π‘šπ‘₯,ξ‚€π›Όπ‘›ξ‚π‘šπ‘¦,ξ‚€π›Όπ‘›ξ‚π‘šπ‘§,0,…,0ξ„Ώξ…€ξ…€ξ…ƒξ…€ξ…€ξ…Œπ‘›βˆ’2times=0(3.9)for all π‘₯,𝑦,π‘§βˆˆπ΄. So 𝛿(π‘₯𝑦𝑧)=𝛿(π‘₯𝑦)π‘§βˆ’π‘₯𝛿(𝑦)𝑧+π‘₯𝛿(𝑦𝑧)(3.10) for all π‘₯,𝑦,π‘§βˆˆπ΄. Thus π›ΏβˆΆπ΄β†’π΄ is a generalized derivation satisfying (3.7).

Theorem 3.3. Let π‘“βˆΆπ΄β†’π΄ be a mapping satisfying 𝑓(0)=0 for which there exists a function πœ‘βˆΆπ΄π‘›+1β†’[0,∞) satisfying (2.36),(3.3), (3.4) and (3.5) for all π‘₯,𝑦,𝑧,π‘₯1,…,π‘₯π‘›βˆˆπ΄. If for some 1≀𝑗≀𝑛 there exists a Lipschitz constant 𝐿<1 such that πœ‘(π‘₯,0,…,0,π‘₯𝑗th,0,…,0)≀𝑛2𝑛2βˆ’1πΏπœ‘ξ‚€π‘›2βˆ’1𝑛2π‘₯,0,…,0,𝑛2βˆ’1𝑛2π‘₯ξ„Ώξ…€ξ…€ξ…€ξ…ƒξ…€ξ…€ξ…€ξ…Œπ‘—th,0,…,0(3.11)for all π‘₯∈𝐴, then there exists a unique generalized derivation π›ΏβˆΆπ΄β†’π΄ such that ‖𝑓(π‘₯)βˆ’π›Ώ(π‘₯)‖𝐡≀𝑛𝐿(𝑛2βˆ’1)Γ—(1βˆ’πΏ)πœ‘(π‘₯,0,…,0,π‘₯𝑗th,0,…,0)(3.12)for all π‘₯∈𝐴.

Proof. The proof is similar to the proofs of Theorems 2.4 and 3.2.

Acknowledgments

This paper is based on final report of the research project of the Ph.D. thesis in University of Tabriz and the third author was supported by Grant no. F01-2006-000-10111-0 from the Korea Science and Engineering Foundation. The authors would like to thank the referees for a number of valuable suggestions regarding a previous version of this paper.

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