Abstract and Applied Analysis

Volume 2011, Article ID 587097, 9 pages

http://dx.doi.org/10.1155/2011/587097

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

^{1}Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran^{2}Center of Excellence in Nonlinear Analysis and Applications (CENAA), Semnan University, Semnan, Iran^{3}Department of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran^{4}The 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 is * superstable * if every approximately solution of 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 , a number , and a mapping which satisfies the inequality for all in , does there exist an automorphism of and a constant , depending only on such that for all

If the answer is affirmative, we would call the equation 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 . In 1991, Gajda [4] answered the question for the case , 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 is an ablian group, is a Banach space which satisfies for all . If is a mapping with for all , then there exists a unique mapping such that and for all .

In 1949, Bourgin [8] proved the following result, which is sometimes called the superstability of ring homomorphisms: suppose that and are Banach algebras with unit. If is a surjective mapping such that for some and for all , then 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 equipped with a function (valuation) from into such that if and only if , and for all (see [20, 21]).

*Definition 2.1. *Let be a vector space over a scalar field with a non-Archimedean nontrivial valuation . A function is a non-Archimedean norm (valuation) if it satisfies the following conditions:(NA_{1}) if and only if ,(NA_{2}) for all and ,(NA_{3}) for all (the strong triangle inequality).

A sequence in a non-Archimedean space is Cauchy if and only if 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 with a linear associative multiplication, satisfying for all . A non-Archimedean complete normed algebra is called a non-Archimedean Banach algebra (see [22]).

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

*Definition 2.3. *Let be a nonempty set and satisfy the following properties: (D_{1}) if and only if ,(D_{2}) (symmetry),(D_{3}) (strong triangle in equality),for all . Then, is called a non-Archimedean generalized metric space. is called complete if every -Cauchy sequence in is -convergent.

*Definition 2.4. *Let be a non-Archimedean algebra. An additive mapping is said to be a ring derivation if for all . An additive mapping is said to be a generalized ring derivation if there exists a ring derivation such that
for all .

We need the following fixed point theorem (see [23, 24]).

Theorem 2.5 (non-Archimedean alternative Contraction Principle). * Suppose that is a non-Archimedean generalized complete metric space and is a strictly contractive mapping; that is,
**
for some . If there exists a nonnegative integer such that for some , then the followings are true: **
(a) the sequence converges to a fixed point of , **
(b) is a unique fixed point of in
**
(c) if , then
*

#### 3. Non-Archimedean Superstability of Generalized Derivations

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

Theorem 3.1. *Let be a function. Suppose that are mappings such that is additive and
**
for all . If there exists a natural number and ,
**
for all . Then, is a generalized ring derivation and is a ring derivation.*

*Proof. *By induction on , we prove that
for all and . Let in (3.1). Then,
This proves (3.4) for . Let (3.4) holds for . Replacing by and by in (3.1) for each , and for all , we get
Since
for all , it follows from induction hypothesis and (3.6) that
for all . This proves (3.4) for all . In particular,
for all where

Let be the set of all functions . We define as follows:
It is easy to see that defines a generalized complete metric on . Define by . Then, is strictly contractive on , in fact, if
then by (3.3),
It follows that
Hence, is a strictly contractive mapping with Lipschitz constant . By (3.9),
This means that . By Theorem 2.5, has a unique fixed point in the set
and for each ,

Therefore,
for all . This shows that is additive.

Replacing by in (3.2) to get
and so
for all and all . By taking , we have
for all .

Fix . By (3.21), we have
for all . Then, for all and each , and so by taking , we have . Now, we obtain , since is with unit. Replacing by in (3.2), we obtain
and hence,
for all and each . Letting tends to infinite, we have
Now, we show that is a ring derivation. By (3.25), we get
for all . Therefore, we have .

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

Theorem 3.2. *Let be a function. Suppose that are mappings such that is additive and
**
for all . If there exists a natural number and ,
**
for all . Then, is a generalized ring derivation and 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 be a non-Archimedean Banach algebra over , , and . Suppose that are mappings such that is additive and
**
for all . Then, is a generalized ring derivation and is a ring derivation.*

Corollary 3.4. *Let be a non-Archimedean Banach algebra over , and . Suppose that are mappings such that is additive and
**
for all . Then, is a generalized ring derivation and is a ring derivation.*

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