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Abstract and Applied Analysis
Volume 2010, Article ID 741942, 22 pages
http://dx.doi.org/10.1155/2010/741942
Research Article

AQCQ-Functional Equation in Non-Archimedean Normed Spaces

1Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran
2Research Group of Nonlinear Analysis and Applications (RGNAA), Semnan, Iran
3Center of Excellence in Nonlinear Analysis and Applications (CENAA), Semnan University, Semnan, Iran
4Mathematics Section, College of Science and Technology, Hongik University, 339-701 Jochiwon, Republic of Korea

Received 30 June 2010; Revised 11 September 2010; Accepted 11 October 2010

Academic Editor: John M. Rassias

Copyright © 2010 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.

Linked References

  1. K. Hensel, “Über eine neue Begründung der Theorie der algebraischen Zahlen,” Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 6, pp. 83–88, 1897. View at Google Scholar
  2. 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.
  3. 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, 2005/2006. View at Google Scholar
  4. 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
  5. M. Eshaghi Gordji and M. B. Savadkouhi, “Stability of a mixed type cubicquartic functional equation in non-Archimedean spaces,” Applied Mathematics Letters, vol. 23, no. 10, pp. 1198–1202, 2010. View at Publisher · View at Google Scholar
  6. M. Eshaghi Gordji, H. Khodaei, and R. Khodabakhsh, “General quartic-cubic-quadratic functional equation in non-archimedean normed spaces,” UPB Scientific Bulletin, Series A, vol. 72, no. 3, pp. 69–84, 2010. View at Google Scholar
  7. M. S. Moslehian and Th. 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
  8. L. Narici and E. Beckenstein, “Strange terrain—non-Archimedean spaces,” American Mathematical Monthly, vol. 88, no. 9, pp. 667–676, 1981. View at Publisher · View at Google Scholar
  9. C. Park, D. H. Boo, and Th. M. Rassias, “Approximately addtive mappings over p-adic fields,” Journal of Chungcheong Mathematical Society, vol. 21, pp. 1–14, 2008. View at Google Scholar
  10. 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.
  11. S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience, New York, NY, USA, 1960.
  12. 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
  13. 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
  14. D. G. Bourgin, “Classes of transformations and bordering transformations,” Bulletin of the American Mathematical Society, vol. 57, pp. 223–237, 1951. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. Th. 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
  16. P. Găvruta, “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
  17. S.-M. Jung, “On the Hyers-Ulam-Rassias stability of approximately additive mappings,” Journal of Mathematical Analysis and Applications, vol. 204, no. 1, pp. 221–226, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. 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
  19. 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
  20. 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
  21. J. M. Rassias, “On the stability of the Euler-Lagrange functional equation,” Chinese Journal of Mathematics, vol. 20, no. 2, pp. 185–190, 1992. View at Google Scholar · View at Zentralblatt MATH
  22. J. M. Rassias, “Solution of a stability problem of Ulam,” Discussiones Mathematicae, vol. 12, pp. 95–103, 1992. View at Google Scholar · View at Zentralblatt MATH
  23. J. M. Rassias, “Complete solution of the multi-dimensional problem of Ulam,” Discussiones Mathematicae, vol. 14, pp. 101–107, 1994. View at Google Scholar · View at Zentralblatt MATH
  24. P. Găvruta, “An answer to a question of John. M. Rassias concerning the stability of Cauchy equation,” in Advances in Equations and Inequalities, Hardronic Mathematics Series, pp. 67–71, 1999. View at Google Scholar
  25. Z. Gajda, “On stability of additive mappings,” International Journal of Mathematics and Mathematical Sciences, vol. 14, no. 3, pp. 431–434, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. 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
  27. B. Bouikhalene, E. Elqorachi, and J. M. Rassias, “The superstability of d'Alembert's functional equation on the Heisenberg group,” Applied Mathematics Letters, vol. 23, no. 1, pp. 105–109, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. 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 Google Scholar · View at Zentralblatt MATH
  29. 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, article 85, 8 pages, 2009. View at Google Scholar
  30. 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
  31. M. Eshaghi Gordji, T. Karimi, and S. Kaboli Gharetapeh, “Approximately n-Jordan homomorphisms on Banach algebras,” Journal of Inequalities and Applications, vol. 2009, Article ID 870843, 8 pages, 2009. View at Google Scholar
  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 Google Scholar
  33. 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 Google Scholar · View at Zentralblatt MATH
  34. 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
  35. 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 Google Scholar · View at Zentralblatt MATH
  36. P. Găvruta and L. Găvruta, “A new method for the generalized Hyers-Ulam-Rassias stability,” International Journal of Nonlinear Analysis and Applications, vol. 1, no. 2, pp. 11–18, 2010. View at Google Scholar
  37. M. S. Moslehian and J. M. Rassias, “Power and Euler-Lagrange norms,” Australian Journal of Mathematical Analysis and Applications, vol. 4, no. 1, article 17, 4 pages, 2007. View at Google Scholar · View at Zentralblatt MATH
  38. M. S. Moslehian and J. M. Rassias, “A characterization of inner product spaces concerning an Euler-Lagrange identity,” Communications in Mathematical Analysis, vol. 8, no. 2, pp. 16–21, 2010. View at Google Scholar · View at Zentralblatt MATH
  39. A. Pietrzyk, “Stability of the Euler-Lagrange-Rassias functional equation,” Demonstratio Mathematica, vol. 39, no. 3, pp. 523–530, 2006. View at Google Scholar · View at Zentralblatt MATH
  40. J. M. Rassias, “Two new criteria on characterizations of inner products,” Discussiones Mathematicae, vol. 9, pp. 255–267, 1989. View at Google Scholar
  41. J. M. Rassias, “Four new criteria on characterizations of inner products,” Discussiones Mathematicae, vol. 10, pp. 139–146, 1991. View at Google Scholar · View at Zentralblatt MATH
  42. Gh. A. Tabadkan and A. Rahmani, “Hyers-Ulam-Rassias and Ulam-Gavruta-Rassias stability of generalized quadratic functional equations,” Advances in Applied Mathematical Analysis, vol. 4, no. 1, pp. 31–38, 2009. View at Google Scholar
  43. 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
  44. Pl. 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
  45. F. Skof, “Proprieta’ locali e approssimazione di operatori,” 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
  46. 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
  47. S.-M. Jung, “Stability of the quadratic equation of Pexider type,” Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 70, pp. 175–190, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  48. K.-W. Jun and H.-M. Kim, “The generalized Hyers-Ulam-Rassias stability of a cubic functional equation,” Journal of Mathematical Analysis and Applications, vol. 274, no. 2, pp. 267–278, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  49. 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
  50. S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, Fla, USA, 2001. View at Zentralblatt MATH
  51. S.-M. Jung and T.-S. Kim, “A fixed point approach to the stability of the cubic functional equation,” Boletin Sociedad Matemática Mexicana, vol. 12, no. 1, pp. 51–57, 2006. View at Google Scholar · View at Zentralblatt MATH
  52. H. Khodaei and Th. 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
  53. A. Najati, “Hyers-Ulam-Rassias stability of a cubic functional equation,” Bulletin of the Korean Mathematical Society, vol. 44, no. 4, pp. 825–840, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  54. A. Najati and C. Park, “On the stability of a cubic functional equation,” Acta Mathematica Sinica, vol. 24, no. 12, pp. 1953–1964, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  55. P. K. Sahoo, “A generalized cubic functional equation,” Acta Mathematica Sinica, vol. 21, no. 5, pp. 1159–1166, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  56. S. H. Lee, S. M. Im, and I. S. Hwang, “Quartic functional equations,” Journal of Mathematical Analysis and Applications, vol. 307, no. 2, pp. 387–394, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  57. A. Najati and M. B. Moghimi, “Stability of a functional equation deriving from quadratic and additive functions in quasi-Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 337, no. 1, pp. 399–415, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  58. A. Najati and Th. M. Rassias, “Stability of a mixed functional equation in several variables on Banach modules,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 3-4, pp. 1755–1767, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  59. A. Najati and G. Z. Eskandani, “Stability of a mixed additive and cubic functional equation in quasi-Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 342, no. 2, pp. 1318–1331, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  60. K.-W. Jun and H.-M. Kim, “Ulam stability problem for a mixed type of cubic and additive functional equation,” Bulletin of the Belgian Mathematical Society. Simon Stevin, vol. 13, no. 2, pp. 271–285, 2006. View at Google Scholar · View at Zentralblatt MATH
  61. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  62. C. Park, “Fuzzy stability of a functional equation associated with inner product spaces,” Fuzzy Sets and Systems, vol. 160, no. 11, pp. 1632–1642, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  63. M. Eshaghi Gordji, H. Khodaei, and Th. M. Rassias, “On the Hyers-Ulam-Rassias stability of generalized mixed type of quartic, cubic, quadratic and additive functional equation in quasi-Banach spaces,” to appear in International Journal of Nonlinear Science.