Abstract and Applied AnalysisVolume 2012 (2012), Article ID 401762, 9 pagesdoi:10.1155/2012/401762

Research Article

Stability of the -Dimensional Mixed-Type Additive and Quadratic Functional Equation in Non-Archimedean Normed Spaces

Yang-Hi Lee,¹ Soon-Mo Jung,² and Themistocles M. Rassias³

¹Department of Mathematics Education, Gongju National University of Education, Gongju 314-711, Republic of Korea²Mathematics Section, College of Science and Technology, Hongik University, 339-701 Jochiwon, Republic of Korea³Department of Mathematics, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece

Received 19 November 2011; Accepted 23 January 2012

Academic Editor: Muhammad Aslam Noor

Copyright © 2012 Yang-Hi Lee 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 will prove the stability of the functional equation in non-Archimedean normed spaces.

1. Introduction

A classical question in the theory of functional equations is “when is it true that a function, which approximately satisfies a functional equation, must be somehow close to an exact solution of the equation?’’ Such a problem, called a stability problem of the functional equation, was formulated by Ulam in 1940 (see [1]). In the following year, Hyers [2] gave a partial solution of Ulam’s problem for the case of approximate additive functions. Subsequently, his result was generalized by Aoki [3] for additive functions and by Rassias [4] for linear functions. Indeed, they considered the stability problem for unbounded Cauchy differences. During the last decades, the stability problems of functional equations have been extensively investigated by a number of mathematicians (see [5–23]).

A non-Archimedean field is a field equipped with a function (valuation) such that() if and only if ;();() for all .

Clearly, it holds that and for all .

Let be a vector space over a scalar field with a non-Archimedean and nontrivial valuation . A function is a non-Archimedean norm (valuation) if it satisfies the following conditions:() if and only if ;() for all and ;() for all .

Then is called a non-Archimedean space. Due to the fact that a sequence is Cauchy if and only if converges to zero in a non-Archimedean space. A complete non-Archimedean space is a non-Archimedean space in which every Cauchy sequence is convergent.

Recently, Moslehian and Rassias [24] proved the Hyers-Ulam stability of the Cauchy functional equation and the quadratic functional equation in non-Archimedean normed spaces.

We now consider the -dimensional mixed-type quadratic and additive functional equation whose solution is called a quadratic-additive function.

In 2009, Towanlong and Nakmahachalasint [25] obtained a stability result for the functional equation (1.4), in which they constructed a quadratic-additive function by composing an additive function and a quadratic function , where and approximate the odd part and the even part of the given function , respectively.

In this paper, we investigate a general stability problem for the -dimensional mixed-type quadratic and additive functional equation (1.4) in non-Archimedean normed spaces.

2. Solutions of (1.4)

In this section, we prove the generalized Hyers-Ulam stability of the -dimensional mixed-type quadratic and additive functional equation (1.4). Assume that is an additive group and is a complete non-Archimedean space.

For a given function , we use the abbreviations for all and for an arbitrarily fixed .

Theorem 2.1. Assume that is an integer. Let and be an additive group and a complete non-Archimedean space, respectively. A function is a solution of (1.4) if and only if is quadratic, is additive, and .

Proof. If a function is a solution of (1.4), then we have , for all , that is, is quadratic and is additive.Conversely, assume that is quadratic, is additive, and . We apply an induction on to prove for all . For , we have If and for some integer   () and for all , then a routine calculation yields for all . Hence, we conclude that for all .Since is additive, a long calculation yields Hence, it follows from (2.5) and (2.6) that for all ; that is, is a solution of (1.4).

3. Generalized Hyers-Ulam Stability of (1.4)

In the following theorem, we will investigate the stability problem of the functional equation (1.4).

Theorem 3.1. Assume that is an integer. Let and be an additive group and a complete non-Archimedean space, respectively. Assume that is a function such that for all . Moreover, assume that the limit exists for each . If a function satisfies the inequality for any , then there exists a unique quadratic-additive function such that for each . In particular, is given by for all .

Proof. If we replace in (3.1) with 0 for each , then we have Since , it holds that and Hence, we conclude that .Let be a function defined by for all and . A tedious calculation, together with (), (), and (3.3), yields for all and . It follows from (3.1) and (3.9) that the sequence is Cauchy. Since is complete, we conclude that is convergent.Let us define for any . It follows from and (3.9) that for all and . In view of (3.2), if we let in (3.11), then we obtain the inequality (3.4).Replacing in (3.3) with for and considering () and (), we get for all and . If we let in the last inequality, then it follows from the condition (3.1) that for all ; that is, is a quadratic-additive function.Assume that is another quadratic-additive function satisfying (3.4). By the definition of , a routine calculation yields for each and . Hence, it follows from (3.8) that for any and . Since is a solution of (1.4), it follows from the last equality that for any and . Obviously, this equality also holds for .Consequently, by considering that , it follows from (), (3.1), (3.4), and (3.8) that for all . Therefore, , which proves the uniqueness of .

Corollary 3.2. Let and be non-Archimedean normed spaces over with . If is complete and satisfies the inequality for all and for some , then there exists a unique quadratic-additive function such that for all .

Proof. Let . Since and , we get for all . Therefore, the conditions of Theorem 3.1 are satisfied. Indeed, it is easy to see that . By Theorem 3.1, there exists a unique quadratic-additive function such that (3.18) holds.

Acknowledgments

The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (no. 2011-0004919).

References

S. M. Ulam, A Collection of Mathematical Problems, Interscience Publishers, London, UK, 1960.

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.

T. Aoki, “On the stability of the linear transformation in Banach spaces,” Journal of the Mathematical Society of Japan, vol. 2, pp. 64–66, 1950.

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.

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, 2006.

I.-S. Chang, E. H. Lee, and H.-M. Kim, “On Hyers-Ulam-Rassias stability of a quadratic functional equation,” Mathematical Inequalities & Applications, vol. 6, no. 1, pp. 87–95, 2003.

S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific Publishing, River Edge, NJ, USA, 2002.

A. Ebadian, N. Ghobadipour, Th. M. Rassias, and M. Eshaghi Gordji, “Functional inequalities associated with Cauchy additive functional equation in non-Archimedean spaces,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 929824, 14 pages, 2011.

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.

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.

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.

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.

D. H. Hyers, G. Isac, and T. M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser Boston Inc., Boston, Mass, USA, 1998.

K.-W. Jun and Y.-H. Lee, “A generalization of the Hyers-Ulam-Rassias stability of the Pexiderized quadratic equations—II,” Kyungpook Mathematical Journal, vol. 47, no. 1, pp. 91–103, 2007.

S.-M. Jung, “Hyers-Ulam-Rassias stability of Jensen's equation and its application,” Proceedings of the American Mathematical Society, vol. 126, no. 11, pp. 3137–3143, 1998.

S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, vol. 48, Springer, New York, NY, USA, 2011.

G. H. Kim, “On the stability of functional equations with square-symmetric operation,” Mathematical Inequalities & Applications, vol. 4, no. 2, pp. 257–266, 2001.

H.-M. Kim, “On the stability problem for a mixed type of quartic and quadratic functional equation,” Journal of Mathematical Analysis and Applications, vol. 324, no. 1, pp. 358–372, 2006.

Y.-H. Lee, “On the stability of the monomial functional equation,” Bulletin of the Korean Mathematical Society, vol. 45, no. 2, pp. 397–403, 2008.

Y.-H. Lee and K.-W. Jun, “A generalization of the Hyers-Ulam-Rassias stability of Jensen's equation,” Journal of Mathematical Analysis and Applications, vol. 238, no. 1, pp. 305–315, 1999.

Y.-H. Lee and K.-W. Jun, “A generalization of the Hyers-Ulam-Rassias stability of the Pexider equation,” Journal of Mathematical Analysis and Applications, vol. 246, no. 2, pp. 627–638, 2000.

A. K. Mirmostafaee, “Approximately additive mappings in non-Archimedean normed spaces,” Bulletin of the Korean Mathematical Society, vol. 46, no. 2, pp. 387–400, 2009.

F. Skof, “Local properties and approximation of operators,” Rendiconti del Seminario Matematico e Fisico di Milano, vol. 53, pp. 113–129, 1983.

M. S. Moslehian and T. M. Rassias, “Stability of functional equations in non-Archimedean spaces,” Applicable Analysis and Discrete Mathematics, vol. 1, no. 2, pp. 325–334, 2007.

W. Towanlong and P. Nakmahachalasint, “An n-dimensional mixed-type additive and quadratic functional equation and its stability,” Science Asia, vol. 35, pp. 381–385, 2009.