Discrete Dynamics in Nature and SocietyVolume 2012 (2012), Article ID 961642, 10 pagesdoi:10.1155/2012/961642

Research Article

Nearly Quadratic n-Derivations on Non-Archimedean Banach Algebras

Madjid Eshaghi Gordji,¹ Badrkhan Alizadeh,² Young Whan Lee,³ and Gwang Hui Kim⁴

¹Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran²Technical and Vocational University of Iran, Technical and Vocational Faculty of Tabriz, P.O. Box 51745-135, Tabriz, Iran³Department of Computer Hacking and Information Security, Daejeon University, Dong-gu, Daejeon 300-716, Republic of Korea⁴Department of Mathematics, Kangnam University, Yongin, Gyeonggi 446-702, Republic of Korea

Received 27 January 2012; Revised 18 March 2012; Accepted 19 March 2012

Academic Editor: John Rassias

Copyright © 2012 Madjid 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.

Let be an integer, let be an algebra, and be an -module. A quadratic function is called a quadratic -derivation if for all ,...,. We investigate the Hyers-Ulam stability of quadratic -derivations from non-Archimedean Banach algebras into non-Archimedean Banach modules by using the Banach fixed point theorem.

1. Introduction

A functional equation is stable if any function satisfying the equation approximately is near to a true solution of .

The stability of functional equations was first introduced by Ulam [1] in 1964. In 1941, Hyers [2] gave a first affirmative answer to the question of Ulam for Banach spaces. In 1978, Th. M. Rassias [3] generalized the theorem of Hyers by considering the stability problem with unbounded Cauchy differences . In 1994, a generalization of Th. M. Rassias theorem was obtained by Gvruţa [4], who replaced the bound by a general control function (see also [5–7]).

Every solution of the following functional equation

is said to be a quadratic function [8]. It is well known that a mapping between real vector spaces is quadratic mapping if and only if there exists a unique symmetric biadditive mapping such that for all . The biadditive mapping is given by .

The stability problem of the quadratic functional equation was proved by Skof [9] for mappings , where is a normed space and is a Banach space (see also [10, 11]). Let be an algebra and let be a -bimodule. A quadratic function is called a quadratic -derivation if

for all . Recently, Gordji and Ghobadipour [12] introduced the quadratic derivations on Banach algebras. Indeed, they investigated the Hyers-Ulam-Aoki-Rassias stability and Ulam-Gavruta-Rassias type stability of quadratic derivations on Banach algebras.

More recently, Gordji et al. [13] investigated the Hyers-Ulam stability and the superstability of higher ring derivations on non-Archimedean Banach algebras (see also [12–32]). In this paper we investigate the Hyers-Ulam stability of quadratic -derivations from non-Archimedean Banach algebras into non-Archimedean Banach modules by using the weighted space method (see [33]).

2. Preliminaries

Let us recall that a non-Archimedean field is a field equipped with a function (valuation) from into such that if and only if , and for all . An example of a non-Archimedean valuation is the mapping taking everything but 0 into 1 and . This valuation is called trivial (see [34]).

Definition 2.1. Let be a vector space over a scalar field with a non-Archimedean non-trivial valuation . A function is a non-Archimedean norm (valuation) if it satisfies the following conditions: (NA₁) if and only if ;(NA₂) for all and ;(NA₃) for all (the strong triangle inequality).

In 1897, Hensel [35] introduced a normed space which does not have the Archimedean property. It turned out that non-Archimedean spaces have many nice applications. The most important examples of non-Archimedean spaces are -adic numbers. 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 2.2. Let be a nonempty set and let satisfy the following properties:(D₁) if and only if ,(D₂) (symmetry),(D₃) (strong triangle inequality),for all . Then is called a non-Archimedean metric space. is called a non-Archimedean complete metric space if every -Cauchy sequence in is -convergent.

Theorem 2.3 (Non-Archimedean Banach Contraction Principle). Let be a non-Archimedean complete metric space and let be a contraction; that is, there exists such that Then there exists a unique element such that . Moreover, , and

Proof. A similar argument as Archimedean case can be applied to show that has a unique element such that and . It follows from strong triangle inequality that for all and for each , we have

3. Main Results

In this section denotes a non-Archimedean Banach algebra over a non-Archimedean field and is a non-Archimedean Banach -module.

Theorem 3.1. Let be functions. Let be a given mapping such that , and that for all . Suppose that there exist a natural number and , such that for all . Then there exists a unique quadratic -derivation from into such that for all , where

Proof. By induction on , one can show that for all and , Let in (3.1). Then This proves (3.6) for . Let (3.6) hold for . Replacing by and by in (3.1) for all , we get for all . Since for all , it follows from induction hypothesis and (3.8) that for all , This proves (3.6) for all . In particular Replacing by in (3.11), we get for all . Let be the set of all functions . We define the metric on as follows: where if and if . One has the operator by . Then is strictly contractive on ; in fact, if then by (3.3), It follows that Hence is a contractive with Lipschitz constant . By Theorem 2.3, has a unique fixed point and for all .Therefore for all . This shows that is quadratic. It follows from Theorem 2.3 that that is, Replacing by in (3.2), we get and so for all and each . By taking , we have for all .

In the following corollaries we will assume that is a non-Archimedean Banach algebra over the field of -adic numbers, where is a prime number.

Corollary 3.2. Let and let be be positive real numbers. Suppose that is a mapping such that for all . Then there exists a unique quadratic -derivation from into such that for all .

Proof. By (3.24), . Let and for all . Then for all .Moreover, Put and in Theorem 3.1. Then there exists a unique quadratic -derivation from into such that for all .

Similarly, we can prove the following result.

Corollary 3.3. Let and let be be positive real numbers. Suppose that is a mapping such that for all . Then there exists a unique quadratic -derivation from into such that for all .

Remark 3.4. We can use similar arguments to obtain corollaries like Corollaries 3.2 and 3.3, when and .

By using the same technique of proving Theorem 3.1, we can prove the following result.

Remark 3.5. Let be functions. Let be a given mapping such that , and that for all . Suppose that there exist a natural number and , such that for all . Then there exists a unique quadratic -derivation from into such that for all , where

Acknowledgment

The third author of this work was partially supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant no. 2011-0021253).

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