Discrete Dynamics in Nature and Society
Volume 2012, Issue 1 961642
Research Article
Open Access

Nearly Quadratic n-Derivations on Non-Archimedean Banach Algebras

Madjid Eshaghi Gordji, 

Madjid Eshaghi Gordji

Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran semnan.ac.ir

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Badrkhan Alizadeh, 

Badrkhan Alizadeh

Technical and Vocational University of Iran, Technical and Vocational Faculty of Tabriz, P.O. Box 51745-135, Tabriz, Iran

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Young Whan Lee, 

Corresponding Author

Young Whan Lee

Department of Computer Hacking and Information Security, Daejeon University, Dong-gu, Daejeon 300-716, Republic of Korea dju.ac.kr

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Gwang Hui Kim, 

Gwang Hui Kim

Department of Mathematics, Kangnam University, Yongin, Gyeonggi 446-702, Republic of Korea kangnam.ac.kr

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First published: 20 May 2012
https://doi.org/10.1155/2012/961642
Academic Editor: John Rassias

Abstract

Let n > 1 be an integer, let A be an algebra, and X be an A-module. A quadratic function D : A → X is called a quadratic n-derivation if for all a1,…,an ∈ A. We investigate the Hyers-Ulam stability of quadratic n-derivations from non-Archimedean Banach algebras into non-Archimedean Banach modules by using the Banach fixed point theorem.

1. Introduction

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

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 ∥f(x + y) − f(x) − f(y)∥≤ϵ(∥x∥p + ∥y∥p),   (ϵ > 0, p ∈ [0,1)). In 1994, a generalization of Th. M. Rassias theorem was obtained by Gvruţa [4], who replaced the bound ϵ(∥x∥p + ∥y∥p) by a general control function φ(x, y) (see also [5–7]).

Every solution of the following functional equation
(1.1)

is said to be a quadratic function [8]. It is well known that a mapping f between real vector spaces is quadratic mapping if and only if there exists a unique symmetric biadditive mapping B1 such that f(x) = B1(x, x) for all x. The biadditive mapping B1 is given by B1(x, y) = (1/4)(f(x + y) − f(x − y)).

The stability problem of the quadratic functional equation was proved by Skof [9] for mappings f : A → B, where A is a normed space and B is a Banach space (see also [10, 11]). Let A be an algebra and let X be a A-bimodule. A quadratic function D : A → X is called a quadratic n-derivation if
(1.2)

for all a1, …, an ∈ A. 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 n-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 𝕂 equipped with a function (valuation) |·| from 𝕂 into [0, ∞) such that |r | = 0 if and only if r = 0,      | rs | = |r| | s|, and |r + s | ≤ max {|r | , |s|} for all r, s ∈ 𝕂. An example of a non-Archimedean valuation is the mapping |·| taking everything but 0 into 1 and |0| = 0. This valuation is called trivial (see [34]).

Definition 2.1. Let X be a vector space over a scalar field 𝕂 with a non-Archimedean non-trivial valuation |·|. A function ∥·∥   : X → ℝ 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 r ∈ 𝕂 and x ∈ X;

  • (NA3)

    ∥x + y∥ ≤ max {∥x∥, ∥y∥} for all x, y ∈ X (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 p-adic numbers. Let p be a prime number. For any nonzero rational number such that a and b are integers not divisible by p, define the p-adic absolute value . Then |·|p is a non-Archimedean norm on ℚ. The completion of ℚ with respect to |·|p is denoted by ℚp which is called the p-adic number field.

Definition 2.2. Let X be a nonempty set and let d : X × 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 inequality),

for all x, y, z ∈ X. Then (X, d) is called a non-Archimedean metric space. (X, d) is called a non-Archimedean complete metric space if every d-Cauchy sequence in X is d-convergent.

Theorem 2.3 (Non-Archimedean Banach Contraction Principle). Let (X, d) be a non-Archimedean complete metric space and let T : X → X be a contraction; that is, there exists α ∈ [0,1) such that

(2.1)
Then there exists a unique element a ∈ X such that Ta = a. Moreover, a = limn→∞Tnx, and
(2.2)

Proof. A similar argument as Archimedean case can be applied to show that T has a unique element a ∈ X such that Ta = a and a = limn→∞Tnx. It follows from strong triangle inequality that for all x ∈ X and for each n ∈ ℕ, we have

(2.3)

3. Main Results

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

Theorem 3.1. Let φ : A × A → [0, ∞),   ψ : A × ⋯×A → [0, ∞) be functions. Let f : A → X be a given mapping such that f(0) = 0,

(3.1)
and that
(3.2)
for all x1, …, xn, x, y ∈ A. Suppose that there exist a natural number k ∈ 𝕂 and L, K ∈ (0,1), such that
(3.3)
for all x1, …, xn, x, y ∈ A. Then there exists a unique quadratic n-derivation h from A into X such that
(3.4)
for all x ∈ A, where
(3.5)

Proof. By induction on i, one can show that for all x ∈ A and i ≥ 2,

(3.6)
Let x = y in (3.1). Then
(3.7)
This proves (3.6) for i = 2. Let (3.6) hold for i = 1,2, …, j. Replacing x by jx and y by x in (3.1) for all x ∈ A, we get
(3.8)
for all x ∈ A. Since
(3.9)
for all x ∈ A, it follows from induction hypothesis and (3.8) that for all x ∈ A,
(3.10)
This proves (3.6) for all i ≥ 2. In particular
(3.11)
Replacing x by k−1x in (3.11), we get
(3.12)
for all x ∈ A. Let Ω be the set of all functions u : A → X. We define the metric d on Ω as follows:
(3.13)
where D(x) = (∥u(x) − v(x)∥)/Φ(x) if Φ(x) ≠ 0 and D(x) = ∥u(x) − v(x)∥ if Φ(x) = 0. One has the operator J : Ω → Ω by J(u)(x) = k2u(k−1x). Then J is strictly contractive on Ω; in fact, if
(3.14)
then by (3.3),
(3.15)
It follows that
(3.16)
Hence J is a contractive with Lipschitz constant L. By Theorem 2.3, J has a unique fixed point h : A → X and
(3.17)
for all x ∈ A.

Therefore

(3.18)
for all x, y ∈ A. This shows that h is quadratic. It follows from Theorem 2.3 that
(3.19)
that is,
(3.20)
Replacing xi by k−mxi, i = 1, …, n in (3.2), we get
(3.21)
and so
(3.22)
for all x1, …, xn ∈ A and each m ∈ ℕ. By taking m → ∞, we have
(3.23)
for all x1, …, xn ∈ A.

In the following corollaries we will assume that A is a non-Archimedean Banach algebra over 𝕂 = ℚp the field of p-adic numbers, where p > 2 is a prime number.

Corollary 3.2. Let r < 1 and let ε be δ be positive real numbers. Suppose that f : A → X is a mapping such that

(3.24)
for all x1, …, xn, x, y ∈ A. Then there exists a unique quadratic n-derivation h from A into X such that
(3.25)
for all x ∈ A.

Proof. By (3.24), f(0) = 0. Let φ(x, y) = ε∥x∥r∥y∥r and for all x1, …, xn, x, y ∈ A. Then

(3.26)
for all x1, …, xn, x, y ∈ A.

Moreover,

(3.27)
Put L = p2r−2 and K = pr−2 in Theorem 3.1. Then there exists a unique quadratic n-derivation h from A into X such that
(3.28)
for all x ∈ A.

Similarly, we can prove the following result.

Corollary 3.3. Let r < 2 and let ε be δ be positive real numbers. Suppose that f : A → X is a mapping such that

(3.29)
for all x1, …, xn, x, y ∈ A. Then there exists a unique quadratic n-derivation h from A into X such that
(3.30)
for all x ∈ A.

Remark 3.4. We can use similar arguments to obtain corollaries like Corollaries 3.2 and 3.3, when r > 1 and r > 2.

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

Remark 3.5. Let φ : A × A → [0, ∞), ψ : A × ⋯×A → [0, ∞) be functions. Let f : A → X be a given mapping such that f(0) = 0,

(3.31)
and that
(3.32)
for all x1, …, xn, x, y ∈ A. Suppose that there exist a natural number k ∈ 𝕂 and L, K ∈ (0,1), such that
(3.33)
for all x1, …, xn, x, y ∈ A. Then there exists a unique quadratic n-derivation d from A into X such that
(3.34)
for all x ∈ A, where
(3.35)

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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