Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 285807, 12 pages
http://dx.doi.org/10.1155/2012/285807
Research Article

Nearly Quadratic Mappings over -Adic Fields

1Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran
2Department of Mathematics, Kangnam University, Yongin, Gyeonggi 446-702, Republic of Korea

Received 30 October 2011; Revised 20 November 2011; Accepted 21 November 2011

Academic Editor: John Rassias

Copyright © 2012 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 establish some stability results over -adic fields for the generalized quadratic functional equation where and .

1. Introduction and Preliminaries

In 1899, Hensel [1] discovered the -adic numbers as a number of theoretical analogue of power series in complex analysis. Fix a prime number . For any nonzero rational number , there exists a unique integer such that , where and are integers not divisible by . Then, -adic absolute value defines a non-Archimedean norm on . The completion of with respect to the metric is denoted by , and it is called the -adic number field. In fact, is the set of all formal series , where are integers (see, e.g., [2, 3]). Note that if , then for each integer .

During the last three decades, -adic numbers have gained the interest of physicists for their research, in particular, in problems coming from quantum physics, -adic strings, and superstrings [4, 5]. A key property of -adic numbers is that they do not satisfy the Archimedean axiom: For , there exists such that .

Let denote a field and function (valuation absolute) from into . A non-Archimedean valuation is a function that satisfies the strong triangle inequality; namely, for all . The associated field is referred to as a non-Archimedean field. Clearly, and for all . A trivial example of a non-Archimedean valuation is the function taking everything except 0 into 1 and . We always assume in addition that is nontrivial, that is, there is a such that .

Let be a linear space over a field with a non-Archimedean nontrivial valuation . A function is said to be a non-Archimedean norm if it is a norm over with the strong triangle inequality (ultrametric); namely, for all . Then, is called a non-Archimedean space. In any such a space, a sequence 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.

The study of stability problems for functional equations is related to a question of Ulam [6] concerning the stability of group homomorphisms, which was affirmatively answered for Banach spaces by Hyers [7]. Subsequently, the result of Hyers was generalized by Aoki [8] for additive mappings and by Rassias [9] for linear mappings by considering an unbounded Cauchy difference. The paper by Rassias has provided a lot of influences in the development of what we now call the generalized Hyers-Ulam stability or Hyers-Ulam-Rassias stability of functional equations. Rassias [10] considered the Cauchy difference controlled by a product of different powers of norm. The above results have been generalized by Forti [11] and Găvruţa [12] who permitted the Cauchy difference to become arbitrary unbounded (see also [1322]). Arriola and Beyer [23] investigated stability of approximate additive functions . They showed that if is a continuous function for which there exists a fixed such that for all , then there exists a unique additive function such that for all . For more details about the results concerning such problems, the reader is referred to [2445].

Recently, Khodaei and Rassias [46] introduced the generalized additive functional equation and proved the generalized Hyers-Ulam stability of the above functional equation. The functional equation is related to symmetric biadditive function and is called a quadratic functional equation [47, 48]. Every solution of the quadratic equation (1.2) is said to be a quadratic function.

Now, we introduce the generalized quadratic functional equation in -variables as follows: where . Moreover, we investigate the generalized Hyers-Ulam stability of functional equation (1.3) over the -adic field .

As a special case, if in (1.3), then we have the functional equation (1.2). Also, if in (1.3), we obtain that is,

2. Stability of Quadratic Functional Equation (1.3) over -Adic Fields

We will use the following lemma.

Lemma 2.1. Let and be real vector spaces. A function satisfies the functional equation (1.3) if and only if the function is quadratic.

Proof. Let satisfy the functional equation (1.3). Setting () in (1.3), we have that is, or but , and also so .
Putting () in (1.3) and then using , we get that is, for all , this shows that satisfies the functional equation (1.2). So the function is quadratic.
Conversely, suppose that is quadratic, thus satisfies the functional equation (1.2). Hence, we have and is even.
We are going to prove our assumption by induction on . It holds on . Assume that it holds on the case where ; that is, we have for all . It follows from (1.2) that for all . Replacing by in (2.7), we obtain for all . Adding (2.7) to (2.8), we have for all . Replacing by in (2.9), we get for all . Adding (2.9) to (2.10), one gets for all . By using the above method, for until , we infer that for all . Now, by the case , we lead to for all , so (1.3) holds for . This completes the proof of the lemma.

Corollary 2.2. A function satisfies the functional equation (1.3) if and only if there exists a symmetric biadditive function such that for all .

Now, we investigate the stability of the functional equation (1.3) from a Banach space into -adic field . For convenience, we define the difference operator for a given function :

Theorem 2.3. Let be a Banach space and let be real numbers. Suppose that a function with satisfies the inequality for all . Then there exists a unique quadratic function such that for all nonzero .

Proof. Letting and () in (2.15), we obtain for all . Hence, for all nonnegative integers and with and for all . It follows from (2.18) that the sequence is a Cauchy sequence for all . Since is complete, the sequence converges. Therefore, one can define the function by for all . It follows from (2.15) and (2.19) that for all . So . By Lemma 2.1, the function is quadratic.
Taking the limit in (2.18) with , we find that the function is quadratic function satisfying the inequality (2.16) near the approximate function of (1.3).
To prove the aforementioned uniqueness, we assume now that there is another additive function which satisfies (1.3) and the inequality (2.16). So which tends to zero as for all nonzero . This proves the uniqueness of , completing the proof of uniqueness.

The following example shows that the above result is not valid over -adic fields.

Example 2.4. Let be a prime number and define by . Since , for all . Hence, the conditions of Theorem 2.3 for and hold. However for each , we have for all . Hence is not convergent for all nonzero .

In the next result, which can be compared with Theorem 2.3, we will show that the stability of the functional equation (1.3) in non-Archimedean spaces over -adic fields.

Theorem 2.5. Let be fixed. Let be a non-Archimedean space and be a complete non-Archimedean space over , where is a prime number. Suppose that a function satisfies the inequality for all , where and are nonnegative real numbers. Then, the limit exists for all and is a unique quadratic function satisfying for all .

Proof. By (2.24), for all , where . Putting () in (2.27) to obtain , setting () in (2.27), we obtain for all . So for all . Letting in (2.29), we have for all . By induction on , we will show that for each , for all . It holds on ; see (2.30). Let (2.31) hold for . Replacing and by and in (2.29), respectively, we get for all . It follows from (2.32) and our induction hypothesis that for all . This proves (2.31) for each . In particular, for all . So for all . Hence, for all . Since the right side of the above inequality tends to zero as , is a Cauchy sequence in complete non-Archimedean space , thus it converges to some function for all . Using (2.35) and induction, one can show that for any , we have for all . Letting in this inequality, we see that for all . Moreover, for all . So . By Lemma 2.1, the function is quadratic.
Now, let be another quadratic function satisfying (1.3) and (2.38). So which tends to zero as for all . This proves the uniqueness of .
The rest of the proof is similar to the above proof, hence it is omitted.

Acknowledgments

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 number: 2011-0005197).

References

  1. 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
  2. V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, p-Adic Analysis and Mathematical Physics, vol. 1 of Series on Soviet and East European Mathematics, World Scientific, River Edge, NJ, USA, 1994. View at Zentralblatt MATH
  3. F. Q. Gouvêa, p-Adic Numbers, Springer, Berlin, Germany, 2nd edition, 1997. View at Zentralblatt MATH
  4. A. Khrennikov, p-Adic Valued Distributions in Mathematical Physics, vol. 309 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1994. View at Zentralblatt MATH
  5. A. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models, vol. 427 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997. View at Zentralblatt MATH
  6. S. M. Ulam, Problems in Modern Mathematics, John Wiley & Sons, New York, NY, USA, 1964. View at Zentralblatt MATH
  7. 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
  8. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. 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
  10. 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
  11. G. L. Forti, “The stability of homomorphisms and amenability, with applications to functional equations,” Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 57, pp. 215–226, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. 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
  13. M. Eshaghi Gordji, A. Ebadian, and S. Zolfaghari, “Stability of a functional equation deriving from cubic and quartic functions,” Abstract and Applied Analysis, vol. 2008, Article ID 801904, 17 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. M. Eshaghi Gordji, M. B. Ghaemi, and H. Majani, “Generalized Hyers-Ulam-Rassias theorem in Menger probabilistic normed spaces,” Discrete Dynamics in Nature and Society, vol. 2010, Article ID 162371, 11 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. M. Eshaghi Gordji, S. Kaboli Gharetapeh, J. M. Rassias, and S. Zolfaghari, “Solution and stability of a mixed type additive, quadratic, and cubic functional equation,” Advances in Difference Equations, vol. 2009, Article ID 826130, 17 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. M. Eshaghi Gordji, H. Khodaei, and Th. M. Rassias, “Fixed points and stability for quadratic mappings in β-normed left Banach modules on Banach algebras,” Results in Mathematics. In press. View at Publisher · View at Google Scholar
  17. M. Eshaghi Gordji, S. Zolfaghari, J. M. Rassias, and M. B. Savadkouhi, “Solution and stability of a mixed type cubic and quartic functional equation in quasi-Banach spaces,” Abstract and Applied Analysis, vol. 2009, Article ID 417473, 14 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. 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
  19. K. Ravi, M. Arunkumar, and J. M. Rassias, “Ulam stability for the orthogonally general Euler-Lagrange type functional equation,” International Journal of Mathematics and Statistics, vol. 3, no. A08, pp. 36–46, 2008. View at Google Scholar
  20. H.-X. Cao, J.-R. Lv, and J. M. Rassias, “Superstability for generalized module left derivations and generalized module derivations on a Banach module. II,” Journal of Inequalities in Pure and Applied Mathematics, vol. 10, no. 3, pp. 1–8, 2009. View at Google Scholar · View at Zentralblatt MATH
  21. H.-X. Cao, J.-R. Lv, and J. M. Rassias, “Superstability for generalized module left derivations and generalized module derivations on a Banach module. I,” Journal of Inequalities and Applications, vol. 2009, Article ID 718020, 10 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. J. M. Rassias and H.-M. Kim, “Approximate homomorphisms and derivations between C*-ternary algebras,” Journal of Mathematical Physics, vol. 49, no. 6, article 063507, 10 pages, 2008. View at Publisher · View at Google Scholar
  23. 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, 2005/06. View at Google Scholar
  24. Y. J. Cho, C. Park, and R. Saadati, “Functional inequalities in non-Archimedean Banach spaces,” Applied Mathematics Letters, vol. 23, no. 10, pp. 1238–1242, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. M. B. Savadkouhi, M. E. Gordji, J. M. Rassias, and N. Ghobadipour, “Approximate ternary Jordan derivations on Banach ternary algebras,” Journal of Mathematical Physics, vol. 50, no. 4, article 042303, 9 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. A. Ebadian, N. Ghobadipour, and M. E. Gordji, “A fixed point method for perturbation of bimultipliers and Jordan bimultipliers in C*-ternary algebras,” Journal of Mathematical Physics, vol. 51, no. 1, 10 pages, 2010. View at Publisher · View at Google Scholar
  27. 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
  28. 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. View at Publisher · View at Google Scholar
  29. M. Eshaghi Gordji, M. B. Ghaemi, S. Kaboli Gharetapeh, S. Shams, and A. Ebadian, “On the stability of J*-derivations,” Journal of Geometry and Physics, vol. 60, no. 3, pp. 454–459, 2010. View at Publisher · View at Google Scholar
  30. M. Eshaghi Gordji and A. Najati, “Approximately J*-homomorphisms: a fixed point approach,” Journal of Geometry and Physics, vol. 60, no. 5, pp. 809–814, 2010. View at Publisher · View at Google Scholar
  31. M. E. Gordji and M. S. Moslehian, “A trick for investigation of approximate derivations,” Mathematical Communications, vol. 15, no. 1, pp. 99–105, 2010. View at Google Scholar · View at Zentralblatt MATH
  32. M. Eshaghi Gordji, J. M. Rassias, and N. Ghobadipour, “Generalized Hyers-Ulam stability of generalized (n, k)-derivations,” Abstract and Applied Analysis, vol. 2009, Article ID 437931, 8 pages, 2009. View at Publisher · View at Google Scholar
  33. M. Eshaghi Gordji, H. Khodaei, and R. Khodabakhsh, “General quartic-cubic-quadratic functional equation in non-Archimedean normed spaces,” “Politehnica” University of Bucharest Scientific Bulletin Series A, vol. 72, no. 3, pp. 69–84, 2010. View at Google Scholar
  34. S. 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
  35. 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
  36. M. Eshaghi Gordji and H. Khodaei, Stability of Functional Equations, Lap Lambert Academic Publishing, 2010.
  37. M. Eshaghi Gordji and H. Khodaei, “On the generalized Hyers-Ulam-Rassias stability of quadratic functional equations,” Abstract and Applied Analysis, vol. 2009, Article ID 923476, 11 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  38. M. Eshaghi Gordji and H. Khodaei, “Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces,” Nonlinear Analysis, Theory, Methods & Applications, vol. 71, no. 11, pp. 5629–5643, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  39. D. H. Hyers, G. Isac, and T. M. Rassias, Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and their Applications, Birkhäuser, Basel, Switzerland, 1998.
  40. S.-M. Jung, “On the Hyers-Ulam-Rassias stability of a quadratic functional equation,” Journal of Mathematical Analysis and Applications, vol. 232, no. 2, pp. 384–393, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  41. S.-M. Jung and P. K. Sahoo, “Stability of a functional equation for square root spirals,” Applied Mathematics Letters, vol. 15, no. 4, pp. 435–438, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  42. A. Najati and F. Moradlou, “Hyers-Ulam-Rassias stability of the Apollonius type quadratic mapping in non-Archimedean spaces,” Tamsui Oxford Journal of Mathematical Sciences, vol. 24, no. 4, pp. 367–380, 2008. View at Google Scholar · View at Zentralblatt MATH
  43. C.-G. Park, “On an approximate automorphism on a C*-algebra,” Proceedings of the American Mathematical Society, vol. 132, no. 6, pp. 1739–1745, 2004. View at Publisher · View at Google Scholar
  44. R. Saadati, Y. J. Cho, and J. Vahidi, “The stability of the quartic functional equation in various spaces,” Computers & Mathematics with Applications, vol. 60, no. 7, pp. 1994–2002, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  45. R. Saadati and C. Park, “Non-Archimedian -fuzzy normed spaces and stability of functional equations,” Computers & Mathematics with Applications, vol. 60, no. 8, pp. 2488–2496, 2010. View at Publisher · View at Google Scholar
  46. H. Khodaei and T. M. Rassias, “Approximately generalized additive functions in several variables,” International Journal of Nonlinear Analysis and Applications, vol. 1, pp. 22–41, 2010. View at Google Scholar
  47. 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.
  48. P. Kannappan, “Quadratic functional equation and inner product spaces,” Results in Mathematics, vol. 27, no. 3-4, pp. 368–372, 1995. View at Google Scholar · View at Zentralblatt MATH