Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 896032, 10 pages
http://dx.doi.org/10.1155/2012/896032
Research Article

Nearly Radical Quadratic Functional Equations in p-2-Normed Spaces

1Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran
2Center of Excellence in Nonlinear Analysis and Applications (CENAA), Semnan University, Semnan, Iran
3Department of Mathematics, Payame Noor University, P.O. Box 19395-3697, Tehran, Iran
4Department of Mathematics, Kangnam University, Yongin, Gyeonggi 446-702, Republic of Korea

Received 16 November 2011; Accepted 13 February 2012

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.

Linked References

  1. G. Póolya and G. Szeg, Aufgaben und Lehrsätze aus der Analysis, vol. 1, Springer, Berlin, Germany, 1925.
  2. S. M. Ulam, A Collection of Mathematical Problems, Interscience, New York, NY, USA, 1960. View at Zentralblatt MATH
  3. S. M. Ulam, Problems in Modern Mathematics, Wiley, New York, NY, USA, 1964. View at Zentralblatt MATH
  4. 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
  5. 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
  6. 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
  7. 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
  8. 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
  9. P. Gavruta, “An answer to a question of John M. Rassias concerning the stability of Cauchy equation,” in Advances in Equations and Inequalities, Hadronic Mathematics, pp. 67–71, 1999. View at Google Scholar
  10. J. M. Rassias and H.-M. Kim, “Generalized Hyers-Ulam stability for general additive functional equations in quasi-β-normed spaces,” Journal of Mathematical Analysis and Applications, vol. 356, no. 1, pp. 302–309, 2009. View at Publisher · View at Google Scholar
  11. 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
  12. J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,” Bulletin des Sciences Mathematiques, vol. 108, no. 4, pp. 445–446, 1984. View at Google Scholar · View at Zentralblatt MATH
  13. 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
  14. G. Z. Eskandani, P. Gavruta, J. M. Rassias, and R. Zarghami, “Generalized Hyers-Ulam stability for a general mixed functional equation in quasi-β-normed spaces,” Mediterranean Journal of Mathematics, vol. 8, no. 3, pp. 331–348, 2011. View at Publisher · View at Google Scholar
  15. 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
  16. 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
  17. 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
  18. M. Eshaghi Gordji, M. B. Ghaemi, Y. J. Cho, and H. Majani, “A general system of Euler-Lagrange-type quadratic functional equations in Menger probabilistic non-Archimedean 2-normed spaces,” Abstract and Applied Analysis, vol. 2011, Article ID 208163, 21 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. 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 Google Scholar · View at Zentralblatt MATH
  20. 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 Google Scholar · View at Zentralblatt MATH
  21. 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 ID 042303, 9 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. C. Park and M. E. Gordji, “Approximate ternary Jordan derivations on Banach ternary algebras,” Journal of Mathematical Physics, vol. 51, no. 4, Article ID 044102, 7 pages, 2010. View at Publisher · View at Google Scholar
  23. M. Eshaghi Gordji, S. Abbaszadeh, and C. Park, “On the stability of a generalized quadratic and quartic type functional equation in quasi-Banach spaces,” Journal of Inequalities and Applications, vol. 2009, Article ID 153084, 26 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. 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 Google Scholar · View at Zentralblatt MATH
  25. F. Rahbarnia, T. M. Rassias, R. Saadati, and G. Sadeghi, “Forti's approach in fixed point theory and the stability of a functional equation on metric and ultra metric spaces,” Journal of Computational Analysis and Applications, vol. 13, no. 3, pp. 458–462, 2011. View at Google Scholar · View at Zentralblatt MATH
  26. R. Saadati, S. M. Vaezpour, and C. Park, “The stability of the cubic functional equation in various spaces,” Mathematical Communications, vol. 16, no. 1, pp. 131–145, 2011. View at Google Scholar · View at Zentralblatt MATH
  27. R. Saadati, M. M. Zohdi, and S. M. Vaezpour, “Nonlinear -random stability of an ACQ functional equation,” Journal of Inequalities and Applications, vol. 2011, Article ID 194394, 23 pages, 2011. View at Publisher · View at Google Scholar
  28. 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
  29. M. M. Pourpasha, J. M. Rassias, R. Saadati, and S. M. Vaezpour, “A fixed point approach to the stability of Pexider quadratic functional equation with involution,” Journal of Inequalities and Applications, vol. 2010, Article ID 839639, 18 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  30. L. Gavruta and P. Gavruta, “On a problem of John M.Rassias concerning the stability in Ulam sense of Euler-Lagrange equation,” in Functional Equations, Difference Inequalities, pp. 47–53, Nova Sciences, 2010. View at Google Scholar
  31. Z. Gajda and R. Ger, “Subadditive multifunctions and Hyers-Ulam stability,” in General Inequalities, vol. 5 of International Series of Numerical Mathematics, Birkhäuser, Basel, Switzerland, 1987. View at Google Scholar · View at Zentralblatt MATH
  32. 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
  33. M. Eshaghi Gordji and H. Khodaei, Stability of Functional Equations, Lap Lambert Academic, 2010.
  34. D. H. Hyers, G. Isac, and T. M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser, Basel, Switzerland, 1998.
  35. 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
  36. M. Eshaghi Gordji, “Stability of a functional equation deriving from quartic and additive functions,” Bulletin of the Korean Mathematical Society, vol. 47, no. 3, pp. 491–502, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  37. M. Eshaghi Gordji, “Stability of an additive-quadratic functional equation of two variables in F-spaces,” Journal of Nonlinear Science and its Applications, vol. 2, no. 4, pp. 251–259, 2009. View at Google Scholar
  38. 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 Google Scholar · View at Zentralblatt MATH
  39. 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
  40. 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
  41. M. E. 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
  42. C. Park and A. Najati, “Homomorphisms and derivations in C-algebras,” Abstract and Applied Analysis, vol. 2007, Article ID 80630, 12 pages, 2007. View at Google Scholar
  43. J. Aczél and J. Dhombres, Functional Equations in Several Variables, Cambridge University Press, Cambridge, UK, 1989.
  44. 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
  45. Y. J. Cho, P. C. S. Lin, S. S. Kim, and A. Misiak, Theory of 2-Inner Product Spaces, Nova Science, New York, NY, USA, 2001.
  46. S. Gähler, “Lineare 2-normierte Räume,” Mathematische Nachrichten, vol. 28, pp. 1–43, 1964. View at Publisher · View at Google Scholar
  47. S. Gähler, “Über 2-banach-Räume,” Mathematische Nachrichten, vol. 42, pp. 335–347, 1969. View at Publisher · View at Google Scholar
  48. W.-G. Park, “Approximate additive mappings in 2-banach spaces and related topics,” Journal of Mathematical Analysis and Applications, vol. 376, no. 1, pp. 193–202, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  49. A. G. White,, “2-banach spaces,” Mathematische Nachrichten, vol. 42, pp. 43–60, 1969. View at Publisher · View at Google Scholar · View at Zentralblatt MATH