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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 587097, 9 pages
http://dx.doi.org/10.1155/2011/587097
Research Article

A Fixed Point Approach to Superstability of Generalized Derivations on Non-Archimedean Banach Algebras

1Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran
2Center of Excellence in Nonlinear Analysis and Applications (CENAA), Semnan University, Semnan, Iran
3Department of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran
4The Holy Prophet Higher Education Complex, Tabriz College of Technology, P.O. Box 51745-135, Tabriz, Iran

Received 27 February 2011; Revised 6 July 2011; Accepted 18 July 2011

Academic Editor: Ngai-Ching Wong

Copyright © 2011 M. Eshaghi Gordji 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

We investigate the superstability of generalized derivations in non-Archimedean algebras by using a version of fixed point theorem via Cauchy functional equation.

1. Introduction

A functional equation (𝜉) is superstable if every approximately solution of (𝜉) is an exact solution of it.

The stability of functional equations was first introduced by Ulam [1] during his talk before a Mathematical Colloquium at the University of Wisconsin in 1940.

Given a metric group 𝐺(,𝜌), a number 𝜀>0, and a mapping 𝑓𝐺𝐺 which satisfies the inequality 𝜌(𝑓(𝑥𝑦),𝑓(𝑥)𝑓(𝑦))𝜀 for all 𝑥,𝑦 in 𝐺, does there exist an automorphism 𝑎 of 𝐺 and a constant 𝑘>0, depending only on 𝐺 such that 𝜌(𝑎(𝑥),𝑓(𝑥))𝑘𝜀 for all 𝑥𝐺?

If the answer is affirmative, we would call the equation 𝑎(𝑥𝑦)=𝑎(𝑥)𝑎(𝑦) of automorphism is stable. In 1941, Hyers [2] gave a first affirmative answer to the question of Ulam for Banach spaces. In 1978, Rassias [3] generalized the theorem of Hyers by considering the stability problem with unbounded Cauchy differences 𝑓(𝑥+𝑦)𝑓(𝑥)𝑓(𝑦)𝜖(𝑥𝑝+𝑦𝑝),(𝜖>0,𝑝[0,1)). In 1991, Gajda [4] answered the question for the case 𝑝>1, which was raised by Rassias. This new concept is known as Hyers-Ulam-Rassias or generalized Hyers-Ulam stability of functional equations [5, 6].

In 1992, Găvruţa [7] generalized the Th. M. Rassias Theorem as follows.

Suppose that (𝐺,+) is an ablian group, 𝑋 is a Banach space 𝜑𝐺×𝐺[0,) which satisfies 1𝜑(𝑥,𝑦)=2𝑛=02𝑛𝜑(2𝑛𝑥,2𝑛𝑦)<,(1.1) for all 𝑥,𝑦𝐺. If 𝑓𝐺𝑋 is a mapping with 𝑓(𝑥+𝑦)𝑓(𝑥)𝑓(𝑦)𝜑(𝑥,𝑦),(1.2) for all 𝑥,𝑦𝐺, then there exists a unique mapping 𝑇𝐺𝑋 such that 𝑇(𝑥+𝑦)=𝑇(𝑥)+𝑇(𝑦) and 𝑓(𝑥)𝑇(𝑥)𝜑(𝑥,𝑥) for all 𝑥,𝑦𝐺.

In 1949, Bourgin [8] proved the following result, which is sometimes called the superstability of ring homomorphisms: suppose that 𝐴 and 𝐵 are Banach algebras with unit. If 𝑓𝐴𝐵 is a surjective mapping such that 𝑓(𝑥+𝑦)𝑓(𝑥)𝑓(𝑦)𝜖,𝑓(𝑥𝑦)𝑓(𝑥)𝑓(𝑦)𝛿,(1.3) for some 𝜖0,𝛿0 and for all 𝑥,𝑦𝐴, then 𝑓 is a ring homomorphism.

Badora [9] and Miura et al. [10] proved the Ulam-Hyers stability and the Isac and Rassias-type stability of derivations [11] (see also [12, 13]); Savadkouhi et al. [14] have contributed works regarding the stability of ternary Jordan derivations. Jung and Chang [15] investigated the stability and superstability of higher derivations on rings. Recently, Ansari-Piri and Anjidani [16] discussed the superstability of generalized derivations on Banach algebras. In this paper, we investigate the superstability of generalized derivations on non-Archimedean Banach algebras by using the fixed point methods.

2. Preliminaries

In 1897, Hensel [17] has introduced a normed space which does not have the Archimedean property. It turned out that non-Archimedean spaces have many nice applications [18, 19].

A non-Archimedean field is a field 𝕂 equipped with a function (valuation) || from 𝕂 into [0,) such that |𝑟|=0 if and only if 𝑟=0,|𝑟𝑠|=|𝑟𝑠|, and |𝑟+𝑠|max{|𝑟|,|𝑠|} for all 𝑟,𝑠𝕂 (see [20, 21]).

Definition 2.1. Let 𝑋 be a vector space over a scalar field 𝕂 with a non-Archimedean nontrivial valuation ||. A function 𝑋 is a non-Archimedean norm (valuation) if it satisfies the following conditions:(NA1)𝑥=0 if and only if 𝑥=0,(NA2)𝑟𝑥=|𝑟|𝑥 for all 𝑟𝕂 and 𝑥𝑋,(NA3)𝑥+𝑦max{𝑥,𝑦} for all 𝑥,𝑦𝑋 (the strong triangle inequality).
A sequence {𝑥𝑚} in a non-Archimedean space is Cauchy if and only if {𝑥𝑚+1𝑥𝑚} converges to zero. By a complete non-Archimedean space, we mean one in which every Cauchy sequence is convergent. A non-Archimedean normed algebra is a non-Archimedean normed space 𝐴 with a linear associative multiplication, satisfying 𝑥𝑦𝑥𝑦 for all 𝑥,𝑦𝐴. A non-Archimedean complete normed algebra is called a non-Archimedean Banach algebra (see [22]).

Example 2.2. Let 𝑝 be a prime number. For any nonzero rational number 𝑥=(𝑎/𝑏)𝑝𝑛𝑥 such that 𝑎 and 𝑏 are integers not divisible by 𝑝, define the 𝑝–adic absolute value |𝑥|𝑝=𝑝𝑛𝑥. Then, || is a non-Archimedean norm on . The completion of with respect to || is denoted by 𝑝 which is called the 𝑝-adic number field.

Definition 2.3. Let 𝑋 be a nonempty set and 𝑑𝑋×𝑋[0,] satisfy the following properties: (D1)𝑑(𝑥,𝑦)=0 if and only if 𝑥=𝑦,(D2)𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥) (symmetry),(D3)𝑑(𝑥,𝑧)max{𝑑(𝑥,𝑦),𝑑(𝑦,𝑧)} (strong triangle in equality),for all 𝑥,𝑦,𝑧𝑋. Then, (𝑋,𝑑) is called a non-Archimedean generalized metric space. (𝑋,𝑑) is called complete if every 𝑑-Cauchy sequence in 𝑋 is 𝑑-convergent.

Definition 2.4. Let 𝐴 be a non-Archimedean algebra. An additive mapping 𝐷𝐴𝐴 is said to be a ring derivation if 𝐷(𝑥𝑦)=𝐷(𝑥)𝑦+𝑥𝐷(𝑦) for all 𝑥,𝑦𝐴. An additive mapping 𝐻𝐴𝐴 is said to be a generalized ring derivation if there exists a ring derivation 𝐷𝐴𝐴 such that 𝐻(𝑥𝑦)=𝑥𝐻(𝑦)+𝐷(𝑥)𝑦,(2.1) for all 𝑥,𝑦𝐴.
We need the following fixed point theorem (see [23, 24]).

Theorem 2.5 (non-Archimedean alternative Contraction Principle). Suppose that (𝑋,𝑑) is a non-Archimedean generalized complete metric space and Λ𝑋𝑋 is a strictly contractive mapping; that is, 𝑑(Λ𝑥,Λ𝑦)𝐿𝑑(𝑥,𝑦),(𝑥,𝑦𝑋),(2.2) for some 𝐿<1. If there exists a nonnegative integer 𝑘 such that 𝑑(Λ𝑘+1𝑥,Λ𝑘𝑥)< for some 𝑥𝑋, then the followings are true:
(a) the sequence {Λ𝑛𝑥} converges to a fixed point 𝑥 of Λ,
(b) 𝑥 is a unique fixed point of Λ in 𝑋=Λ𝑦𝑋𝑑𝑘𝑥,𝑦<,(2.3)
(c) if 𝑦𝑋, then 𝑑𝑦,𝑥𝑑(Λ𝑦,𝑦).(2.4)

3. Non-Archimedean Superstability of Generalized Derivations

Hereafter, we will assume that 𝐴 is a non-Archimedean Banach algebra with unit over a non-Archimedean field 𝕂.

Theorem 3.1. Let 𝜑𝐴×𝐴[0,) be a function. Suppose that 𝑓,𝑔𝐴𝐴 are mappings such that 𝑔 is additive and 𝑓(𝑥+𝑦)𝑓(𝑥)𝑓(𝑦)𝜑(𝑥,𝑦),(3.1)𝑓(𝑥𝑦)𝑥𝑓(𝑦)𝑔(𝑥)𝑦𝜑(𝑥,𝑦),(3.2) for all 𝑥,𝑦𝐴. If there exists a natural number 𝑘𝕂 and 0<𝐿<1, ||𝑘||1||𝑘||𝜑(𝑘𝑥,𝑘𝑦),1||𝑘||𝜑(𝑘𝑥,𝑦),1𝜑(𝑥,𝑘𝑦)𝐿𝜑(𝑥,𝑦),(3.3) for all 𝑥,𝑦𝐴. Then, 𝑓 is a generalized ring derivation and 𝑔 is a ring derivation.

Proof. By induction on 𝑖, we prove that 𝑓(𝑖𝑥)𝑖𝑓(𝑥)max{𝜑(0,0),𝜑(𝑥,𝑥),𝜑(2𝑥,𝑥),,𝜑((𝑖1)𝑥,𝑥)},(3.4) for all 𝑥𝐴 and 𝑖2. Let 𝑥=𝑦 in (3.1). Then, 𝑓(2𝑥)2𝑓(𝑥)max{𝜑(0,0),𝜑(𝑥,𝑥)},𝑛0,𝑥𝐴.(3.5) This proves (3.4) for 𝑖=2. Let (3.4) holds for 𝑖=1,2,,𝑗. Replacing 𝑥 by 𝑗𝑥 and 𝑦 by 𝑥 in (3.1) for each 𝑛0, and for all 𝑥𝐴, we get 𝑓((𝑗+1)𝑥)𝑓(𝑗𝑥)𝑓(𝑥)max{𝜑(0,0),𝜑(𝑗𝑥,𝑥)}.(3.6) Since 𝑓((𝑗+1)𝑥)𝑓(𝑗𝑥)𝑓(𝑥)=𝑓((𝑗+1)𝑥)(𝑗+1)𝑓(𝑥)+(𝑗+1)𝑓(𝑥)𝑓(𝑗𝑥)𝑓(𝑥)=𝑓((𝑗+1)𝑥)(𝑗+1)𝑓(𝑥)+𝑗𝑓(𝑥)𝑓(𝑗𝑥),(3.7) for all 𝑥𝐴, it follows from induction hypothesis and (3.6) that {}𝑓((𝑗+1)𝑥)(𝑗+1)𝑓(𝑥)max𝑓((𝑗+1)𝑥)𝑓(𝑗𝑥)𝑓(𝑥),𝑗𝑓(𝑥)𝑓(𝑗𝑥)max{𝜑(0,0),𝜑(𝑥,𝑥),𝜑(2𝑥,𝑥),,𝜑((𝑗)𝑥,𝑥)},(3.8) for all 𝑥𝐴. This proves (3.4) for all 𝑖2. In particular, 𝑓(𝑘𝑥)𝑘𝑓(𝑥)𝜓(𝑥),(3.9) for all 𝑥𝐴 where 𝜓(𝑥)=max{𝜑(0,0),𝜑(𝑥,𝑥),𝜑(2𝑥,𝑥),,𝜑((𝑘1)𝑥,𝑥)}(𝑥𝐴).(3.10)
Let 𝑋 be the set of all functions 𝑟𝐴𝐴. We define 𝑑𝑋×𝑋[0,] as follows:𝑑(𝑟,𝑠)=inf{𝛼>0𝑟(𝑥)𝑠(𝑥)𝛼𝜓(𝑥)𝑥𝐴}.(3.11) It is easy to see that 𝑑 defines a generalized complete metric on 𝑋. Define 𝐽𝑋𝑋 by 𝐽(𝑟)(𝑥)=𝑘1𝑟(𝑘𝑥). Then, 𝐽 is strictly contractive on 𝑋, in fact, if 𝑟(𝑥)𝑠(𝑥)𝛼𝜓(𝑥),(𝑥𝐴),(3.12) then by (3.3), ||𝑘||𝐽(𝑟)(𝑥)𝐽(𝑠)(𝑥)=1||𝑘||𝑟(𝑘𝑥)𝑠(𝑘𝑥)𝛼1𝜓(𝑘𝑥)𝐿𝛼𝜓(𝑥),(𝑥𝐴).(3.13) It follows that 𝑑(𝐽(𝑟),𝐽(𝑠))𝐿𝑑(𝑟,𝑠)(𝑟,𝑠𝑋).(3.14) Hence, 𝐽 is a strictly contractive mapping with Lipschitz constant 𝐿. By (3.9), 𝑘(𝐽𝑓)(𝑥)𝑓(𝑥)=1𝑓,||𝑘||(𝑘𝑥)𝑓(𝑥)1||𝑘||𝑓(𝑘𝑥)𝑘𝑓(𝑥)1𝜓(𝑥)(𝑥𝐴).(3.15) This means that 𝑑(𝐽(𝑓),𝑓)1/|𝑘|. By Theorem 2.5, 𝐽 has a unique fixed point 𝐴𝐴 in the set 𝑈={𝑟𝑋𝑑(𝑟,𝐽(𝑓))<},(3.16) and for each 𝑥𝐴, (𝑥)=lim𝑚𝐽𝑚(𝑓(𝑥))=lim𝑘𝑚𝑓(𝑘𝑚𝑥).(3.17)
Therefore,(𝑥+𝑦)(𝑥)(𝑦)=lim𝑚||𝑘||𝑚𝑓(𝑘𝑚(𝑥+𝑦))𝑓(𝑘𝑚𝑥)𝑓(𝑘𝑚𝑦)lim𝑚||𝑘||𝑚max{𝜑(0,0),𝜑(𝑘𝑛𝑥,𝑘𝑛𝑦)}lim𝑚𝐿𝑚𝜑(𝑥,𝑦)=0,(3.18) for all 𝑥,𝑦𝐴. This shows that is additive.
Replacing 𝑥 by 𝑘𝑛𝑥 in (3.2) to get𝑓(𝑘𝑛𝑥𝑦)𝑘𝑛𝑥𝑓(𝑦)𝑔(𝑘𝑛𝑥)𝑦𝜑(𝑘𝑛𝑥,𝑦),(3.19) and so 𝑓(𝑘𝑛𝑥𝑦)𝑘𝑛𝑥𝑓(𝑦)𝑔(𝑘𝑛𝑥)𝑘𝑛𝑦1||𝑘||𝑛𝜑(𝑘𝑛𝑥,𝑦)𝐿𝑛𝜑(𝑥,𝑦),(3.20) for all 𝑥,𝑦𝐴 and all 𝑛. By taking 𝑛, we have (𝑥𝑦)=𝑥𝑓(𝑦)+lim𝑛𝑔(𝑘𝑛𝑥)𝑘𝑛𝑦,(3.21) for all 𝑥,𝑦𝐴.
Fix 𝑚. By (3.21), we have𝑥𝑓(𝑘𝑚𝑦)=(𝑘𝑚𝑥𝑦)lim𝑛𝑔(𝑘𝑛𝑥)𝑘𝑛(𝑘𝑚𝑦)=𝑘𝑚𝑥𝑓(𝑦)+lim𝑛𝑔(𝑘𝑛𝑘𝑚𝑥)𝑘𝑛𝑦𝑘𝑚lim𝑛𝑔(𝑘𝑛𝑥)𝑘𝑛𝑦=𝑘𝑚𝑥𝑓(𝑦)+𝑘𝑚lim𝑛𝑔𝑘𝑛+𝑚𝑥𝑘𝑛+𝑚𝑦𝑘𝑚lim𝑛𝑔(𝑘𝑛𝑥)𝑘𝑛𝑦=𝑘𝑚𝑥𝑓(𝑦),(3.22) for all 𝑥,𝑦𝐴. Then, 𝑥𝑓(𝑦)=𝑥(𝑓(𝑘𝑚𝑦)/𝑘𝑚) for all 𝑥,𝑦𝐴 and each 𝑚, and so by taking 𝑚, we have 𝑥𝑓(𝑦)=𝑥(𝑦). Now, we obtain =𝑓, since 𝐴 is with unit. Replacing 𝑦 by 𝑘𝑛𝑦 in (3.2), we obtain 𝑓(𝑘𝑛(𝑥𝑦))𝑥𝑓(𝑘𝑛𝑦)𝑘𝑛𝑔(𝑥)𝑦𝜑(𝑥,𝑘𝑛𝑦),(3.23) and hence, 𝑓(𝑘𝑛𝑥𝑦)𝑘𝑛𝑥𝑓(𝑘𝑛𝑦)𝑘𝑛1𝑔(𝑥)𝑦||𝑘||𝑛𝜑(𝑥,𝑘𝑛𝑦)𝐿𝑛𝜑(𝑥,𝑦),(3.24) for all 𝑥,𝑦𝐴 and each 𝑛. Letting 𝑛 tends to infinite, we have 𝑓(𝑥𝑦)=𝑥𝑓(𝑦)+𝑔(𝑥)𝑦.(3.25) Now, we show that 𝑔 is a ring derivation. By (3.25), we get 𝑔(𝑥𝑦)𝑧=𝑓(𝑥𝑦𝑧)𝑥𝑦𝑓(𝑧)=𝑥𝑓(𝑦𝑧)+𝑔(𝑥)𝑦𝑧𝑥𝑦𝑓(𝑧)=(𝑥𝑔(𝑦)+𝑔(𝑥)𝑦)𝑧,(3.26) for all 𝑥,𝑦,𝑧𝐴. Therefore, we have 𝑔(𝑥𝑦)=𝑥𝑔(𝑦)+𝑔(𝑥)𝑦.

The proof of following theorem is similar to that in Theorem 3.1, hence it is omitted.

Theorem 3.2. Let 𝜑𝐴×𝐴[0,) be a function. Suppose that 𝑓,𝑔𝐴𝐴 are mappings such that 𝑔 is additive and 𝑓(𝑥+𝑦)𝑓(𝑥)𝑓(𝑦)𝜑(𝑥,𝑦),𝑓(𝑥𝑦)𝑥𝑓(𝑦)𝑔(𝑥)𝑦𝜑(𝑥,𝑦),(3.27) for all 𝑥,𝑦𝐴. If there exists a natural number 𝑘𝕂 and 0<𝐿<1, ||𝑘||𝜑𝑘1𝑥,𝑘1𝑦,||𝑘||𝜑𝑘1,||𝑘||𝜑𝑥,𝑦𝑥,𝑘1𝑦𝐿𝜑(𝑥,𝑦),(3.28) for all 𝑥,𝑦𝐴. Then, 𝑓 is a generalized ring derivation and 𝑔 is a ring derivation.

The following results are immediate corollaries of Theorems 3.1 and 3.2 and Example 2.3.

Corollary 3.3. Let 𝐴 be a non-Archimedean Banach algebra over 𝑝, 𝜀>0, and 𝑝1,𝑝2(1,). Suppose that 𝑓,𝑔𝐴𝐴 are mappings such that 𝑔 is additive and 𝑓(𝑥+𝑦)𝑓(𝑥)𝑓(𝑦)𝜀𝑥𝑝1𝑦𝑝2,(𝑓𝑥y)𝑥𝑓(𝑦)𝑔(𝑥)𝑦𝜀𝑥𝑝1𝑦𝑝2,(3.29) for all 𝑥,𝑦𝐴. Then, 𝑓 is a generalized ring derivation and 𝑔 is a ring derivation.

Corollary 3.4. Let 𝐴 be a non-Archimedean Banach algebra over 𝑝, 𝜀>0 and 𝑝1,𝑝2,𝑝1+𝑝2(,1). Suppose that 𝑓,𝑔𝐴𝐴 are mappings such that 𝑔 is additive and 𝑓(𝑥+𝑦)𝑓(𝑥)𝑓(𝑦)𝜀𝑥𝑝1𝑦𝑝2,(𝑓𝑥𝑦)𝑥𝑓(𝑦)𝑔(𝑥)𝑦𝜀𝑥𝑝1𝑦𝑝2,(3.30) for all 𝑥,𝑦𝐴. Then, 𝑓 is a generalized ring derivation and 𝑔 is a ring derivation.

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