Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Hyers-Ulam Stability of Cubic Mappings in Non-Archimedean Normed Spaces

  • Mirmostafaee, Alireza Kamel (Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad)
  • Received : 2009.10.29
  • Accepted : 2010.04.23
  • Published : 2010.06.30
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Abstract

We give a xed point approach to the generalized Hyers-Ulam stability of the cubic equation f(2x + y) + f(2x - y) = 2f(x + y) + 2f(x - y) + 12f(x) in non-Archimedean normed spaces. We will give an example to show that some known results in the stability of cubic functional equations in real normed spaces fail in non-Archimedean normed spaces. Finally, some applications of our results in non-Archimedean normed spaces over p-adic numbers will be exhibited.

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References

  1. T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2(1950) 64-66. https://doi.org/10.2969/jmsj/00210064
  2. L. M. Arriola and W. A. Beyer, Stability of the Cauchy functional equation over p-adic fields, Real Analysis Exchange, 31(2005/2006), 125-132 https://doi.org/10.14321/realanalexch.31.1.0125
  3. D. G. Bourgin, Classes of transformations and bordering transformations, Bull. AMS, 57(1951), 223-237. https://doi.org/10.1090/S0002-9904-1951-09511-7
  4. L. Cadariu and V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Grazer Math. Ber., 346(2004), 43-52.
  5. J. B. Diaz and B. Margolis, A fixed point theorem of the alternative for the con- tractions on a generalized complete metric space, Bull. Amer. Math. Soc., 74(1968), 305-309. https://doi.org/10.1090/S0002-9904-1968-11933-0
  6. P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184(1994), 431-436. https://doi.org/10.1006/jmaa.1994.1211
  7. K. Hensel, Uber eine news Begrundung der Theorie der algebraischen Zahlen, Jahresber. Deutsch. Math. Verein, 6(1897), 83-88.
  8. D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci.U.S.A., 27(1941), 222-224. https://doi.org/10.1073/pnas.27.4.222
  9. D. H. Hyers, G. Isac and Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser, Basel, 1998.
  10. S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, 2001.
  11. Y.-S Jung and I.-S. Chang, The stability of a cubic type functional equation with the fixed point alternative, J. Math. Anal. Appl., 306b(2005) 752-760.
  12. Z. Kaiser, On stability of the monomial functional equation in normed spaces over fields with valuation, J. Math. Anal. Appl., 322(2006), No. 2, 1188-1198. https://doi.org/10.1016/j.jmaa.2005.04.087
  13. K.W. Jun and H.M. Kim, The generalized Hyers-Ulam-Rassias stability of a cubic functional equation, J. Math. Anal. Appl., 274(2002), 867-878. https://doi.org/10.1016/S0022-247X(02)00415-8
  14. S.-M. Jung and T.-S. Kim, A fixed point approach to the stability of the cubic func- tional equation, Bol. Soc. Mat. Mexicana, (3) 12(2006), no. 1, 51-57.
  15. A. K. Mirmostafaee, Approximately additive mappings in non-Archimedean normed spaces, Bull. Korean Math. Soc., 46(2009) no.2, 378-400. https://doi.org/10.4134/BKMS.2009.46.2.387
  16. A.K. Mirmostafaee, Stability of quartic mappings in non-Archimedean normed spaces, Kyungpook Math. J., 49(2009), 289-297. https://doi.org/10.5666/KMJ.2009.49.2.289
  17. A. K. Mirmostafaee and M. S. Moslehian, Fuzzy approximately cubic mappings, Inform. Sci., 178(2008), no 19, 3791-3798. https://doi.org/10.1016/j.ins.2008.05.032
  18. M. Mirzavaziri and M. S. Moslehian, A fixed point approach to stability of a quadratic equation, Bull. Braz. Math. Soc., (N. S.), 37(2006), no. 3, 361-376. https://doi.org/10.1007/s00574-006-0016-z
  19. M.S. Moslehian, The Jensen functional equation in non-Archimedean normed spaces, J. Funct. Spaces Appl., 7(2009), no. 1, 13-24. https://doi.org/10.1155/2009/802032
  20. M. S. Moslehian and T. M. Rassias, Stability of functional equations in non Archimedean spaces, Appl. Anal. Discrete Math., 1(2007), 325-334. https://doi.org/10.2298/AADM0702325M
  21. M. S. Moslehian and G. Sadeghi, Stability of two types of cubic functional equations in non-Archimedean spaces, Real Anal. Exchange, 33(2008), no. 2, 375-383. https://doi.org/10.14321/realanalexch.33.2.0375
  22. A. Najati, The generalized Hyers-Ulam-Rassias stability of a cubic functional equa- tion, Turk J. Math., 31(2007), 395-408.
  23. L. Narici and E. Beckenstein, Strange terrain|non-Archimedean spaces, Amer. Math. Monthly, 88(1981), no. 9, 667-676. https://doi.org/10.2307/2320670
  24. J. M. Rassias, Alternative contraction principle and Ulam stability problem, Math. Sci. Res. J., 9(7)(2005), 190-199.
  25. V. Radu, The fixed point alternative and stability of functional equations, Fixed Point Theory, 4(2003), no. 1. 91-96.
  26. J. M. Rassias, Solution of the stability problem for cubic mappings, Glasnik Math., Vol. 36, 56(2001), 63-72.
  27. Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72(1978), 297-300. https://doi.org/10.1090/S0002-9939-1978-0507327-1
  28. S. M. Ulam, Problems in Modern Mathematics (Chapter VI, Some Questions in Analysis: 1, Stability), Science Editions, John Wiley & Sons, New York, 1964.

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