Discrete Dynamics in Nature and Society

Volume 2012 (2012), Article ID 961642, 10 pages

http://dx.doi.org/10.1155/2012/961642

## Nearly Quadratic *n*-Derivations on Non-Archimedean Banach Algebras

^{1}Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran^{2}Technical and Vocational University of Iran, Technical and Vocational Faculty of Tabriz, P.O. Box 51745-135, Tabriz, Iran^{3}Department of Computer Hacking and Information Security, Daejeon University, Dong-gu, Daejeon 300-716, Republic of Korea^{4}Department 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*: *(NA_{1}) if and only if ;(NA_{2}) for all and ;(NA_{3}) 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:(D_{1}) if and only if ,(D_{2}) (symmetry),(D_{3}) (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).

#### References

- S. M. Ulam,
*Problems in Modern Mathematics*, chapter 6, John Wiley & Sonsc, New York, NY, USA, 2nd edition, 1964. - D. H. Hyers, “On the stability of the linear functional equation,”
*Proceedings of the National Academy of Sciences of the United States of America*, vol. 27, pp. 222–224, 1941. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - T. M. Rassias, “On the stability of the linear mapping in Banach spaces,”
*Proceedings of the American Mathematical Society*, vol. 72, no. 2, pp. 297–300, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - P. Găvruţa, “A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings,”
*Journal of Mathematical Analysis and Applications*, vol. 184, no. 3, pp. 431–436, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. Badora, “Report of Meeting: The Thirty-fourth International Symposium on Functional Equations, June 10 to 19, 1996, Wisła-Jawornik, Poland,”
*Aequationes Mathematicae*, vol. 53, no. 1-2, pp. 162–205, 1997. View at Publisher · View at Google Scholar - H. Khodaei and T. M. Rassias, “Approximately generalized additive functions in several variables,”
*International Journal of Nonlinear Analysis and Applications*, vol. 1, no. 1, pp. 22–41, 2010. View at Google Scholar - J. Tabor, “Remark 20, In Report on the 34th ISFE,”
*Aequationes Mathematicae*, vol. 53, pp. 194–196, 1997. View at Google Scholar - J. Aczél and J. Dhombres,
*Functional Equations in Several Variables*, vol. 31 of*Encyclopedia of Mathematics and its Applications*, Cambridge University Press, Cambridge, UK, 1989. View at Zentralblatt MATH - F. Skof, “Local properties and approximation of operators,”
*Rendiconti del Seminario Matematico e Fisico di Milano*, vol. 53, pp. 113–129, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - P. W. Cholewa, “Remarks on the stability of functional equations,”
*Aequationes Mathematicae*, vol. 27, no. 1-2, pp. 76–86, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - St. Czerwik, “On the stability of the quadratic mapping in normed spaces,”
*Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg*, vol. 62, pp. 59–64, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Eshaghi Gordji and N. Ghobadipour, “Hyers–Ulam–Aoki–Rassias stability and Ulam–Gavruta–Rassias stability of quadratic homomorphisms and quadratic derivations on Banach Algebras,” in
*Functional Equations, Difference Inequalities*, vol. and Ulam Stability Notions (F.U.N.) of*Mathematics Research Developments*, NOVA Publishers, 2010. View at Google Scholar - M. Eshaghi Gordji, M. B. Ghaemi, and B. Alizadeh, “A fixed point method for perturbation of higher ring derivations in non–Archimedean Banach algebras,”
*International Journal of Geometric Methods in Modern Physics*, vol. 8, no. 7, pp. 1611–1625, 2011. View at Google Scholar - E. H. Lee, I.-S. Chang, and Y.-S. Jung, “On stability of the functional equations having relation with a multiplicative derivation,”
*Bulletin of the Korean Mathematical Society*, vol. 44, no. 1, pp. 185–194, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H.-M. Kim and I.-S. Chang, “Stability of the functional equations related to a multiplicative derivation,”
*Journal of Applied Mathematics & Computing A*, vol. 11, no. 1-2, pp. 413–421, 2003. View at Google Scholar - M. Eshaghi Gordji, “Nearly ring homomorphisms and nearly ring derivations on non-Archimedean Banach algebras,”
*Abstract and Applied Analysis*, vol. 2010, Article ID 393247, 12 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Eshaghi Gordji and Z. Alizadeh, “Stability and superstability of ring homomorphisms on non-Archimedean Banach algebras,”
*Abstract and Applied Analysis*, vol. 2011, Article ID 123656, 10 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Eshaghi Gordji, M. B. Ghaemi, and B. Alizadeh, “A fixed point approach to superstability of generalized derivations on non-Archimedean Banach algebras,”
*Abstract and Applied Analysis*, vol. 2011, Article ID 587097, 9 pages, 2011. View at Google Scholar · View at Zentralblatt MATH - M. E. Gordji and H. Khodaei,
*Stability of Functional Equations*, LAP LAMBERT Academic Publishing, 2010. - M. Eshaghi Gordji, H. Khodaei, and R. Khodabakhsh, “General quartic-cubic-quadratic functional equation in non-Archimedean normed spaces,”
*University of Bucharest: Scientific Bulletin A*, vol. 72, no. 3, pp. 69–84, 2010. View at Google Scholar - M. Eshaghi Gordji and M. B. Savadkouhi, “Stability of cubic and quartic functional equations in non-Archimedean spaces,”
*Acta Applicandae Mathematicae*, vol. 110, no. 3, pp. 1321–1329, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Eshaghi Gordji and M. B. Savadkouhi, “Stability of a mixed type cubic-quartic functional equation in non-Archimedean spaces,”
*Applied Mathematics Letters*, vol. 23, no. 10, pp. 1198–1202, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Eshaghi Gordji, M. B. Savadkouhi, and M. Bidkham, “Stability of a mixed type additive and quadratic functional equation in non-Archimedean spaces,”
*Journal of Computational Analysis and Applications*, vol. 12, no. 2, pp. 454–462, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,”
*Journal of Functional Analysis*, vol. 46, no. 1, pp. 126–130, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,”
*Bulletin des Sciences Mathématiques*, vol. 108, no. 4, pp. 445–446, 1984. View at Google Scholar · View at Zentralblatt MATH - J. M. Rassias, “Solution of a problem of Ulam,”
*Journal of Approximation Theory*, vol. 57, no. 3, pp. 268–273, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - G. Z. Eskandani, H. Vaezi, and Y. N. Dehghan, “Stability of a mixed additive and quadratic functional equation in non-Archimedean Banach modules,”
*Taiwanese Journal of Mathematics*, vol. 14, no. 4, pp. 1309–1324, 2010. View at Google Scholar · View at Zentralblatt MATH - T. Z. Xu, J. M. Rassias, and W. X. Xu, “Stability of a general mixed additive-cubic functional equation in non-Archimedean fuzzy normed spaces,”
*Journal of Mathematical Physics*, vol. 51, no. 9, Article ID 093508, 19 pages, 2010. View at Publisher · View at Google Scholar - T. Z. Xu, J. M. Rassias, and W. X. Xu, “Intuitionistic fuzzy stability of a general mixed additive-cubic equation,”
*Journal of Mathematical Physics*, vol. 51, no. 6, Article ID 063519, 21 pages, 2010. View at Publisher · View at Google Scholar - T. Z. Xu, J. M. Rassias, and W. X. Xu, “On the stability of a general mixed additive-cubic functional equation in random normed spaces,”
*Journal of Inequalities and Applications*, vol. 2010, Article ID 328473, 16 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - T. Z. Xu, J. M. Rassias, and W. X. Xu, “A fixed point approach to the stability of a general mixed AQCQ-functional equation in non-Archimedean normed spaces,”
*Discrete Dynamics in Nature and Society*, vol. 2010, Article ID 812545, 24 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - T. Z. Xu, “Stability of multi-Jensen mappings in non-Archimedean normed spaces,”
*Journal of Mathematical Physics*, vol. 53, Article ID 10.1063/1.3684746, 9 pages, 2012. View at Publisher · View at Google Scholar - P. Găvruta and L. Găvruta, “A new method for the generalized Hyer-Ulam-Rassias stability,”
*International Journal of Nonlinear Analysis and Applications*, vol. 1, no. 2, pp. 11–18, 2010. View at Google Scholar - L. M. Arriola and W. A. Beyer, “Stability of the Cauchy functional equation over $p$-adic fields,”
*Real Analysis Exchange*, vol. 31, no. 1, pp. 125–132, 2005. View at Google Scholar - K. Hensel, “Uber eine neue Begrundung der Theorie der algebraischen Zahlen,”
*Jahresbericht der Deutschen Mathematiker-Vereinigung*, vol. 6, pp. 83–88, 1897. View at Google Scholar