Advertisement

Acta Applicandae Mathematicae

, Volume 110, Issue 3, pp 1321–1329 | Cite as

Stability of Cubic and Quartic Functional Equations in Non-Archimedean Spaces

  • M. Eshaghi Gordji
  • M. B. Savadkouhi
Article

Abstract

We prove generalized Hyers-Ulam–Rassias stability of the cubic functional equation f(kx+y)+f(kxy)=k[f(x+y)+f(xy)]+2(k 3k)f(x) for all \(k\in \Bbb{N}\) and the quartic functional equation f(kx+y)+f(kxy)=k 2[f(x+y)+f(xy)]+2k 2(k 2−1)f(x)−2(k 2−1)f(y) for all \(k\in \Bbb{N}\) in non-Archimedean normed spaces.

Keywords

Generalized Hyers–Ulam–Rassias stability Cubic functional equation Quartic functional equation Non-Archimedean space p-adic 

Mathematics Subject Classification (2000)

39B22 39B82 46S10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arriola, L.M., Beyer, W.A.: Stability of the Cauchy functional equation over p-adic fields. Real Anal. Exch. 31, 125–132 (2005/2006) MathSciNetGoogle Scholar
  2. 2.
    Borelli, C., Forti, G.L.: On a general Hyers–Ulam stability result. Int. J. Math. Math. Sci. 18, 229–236 (1995) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Cholewa, P.W.: Remarks on the stability of functional equations. Aequ. Math. 27, 76–86 (1984) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Czerwik, S.: On the stability of the quadratic mapping in normed spaces. Abh. Math. Sem. Univ. Hambg. 62, 59–64 (1992) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Eshaghi Gordji, M.: Stability of a functional equation deriving from quartic and additive functions. Bull. Korean Math. Soc. (to appear) Google Scholar
  6. 6.
    Eshaghi Gordji, M., Park, C., Savadkouhi, M.B.: Stability of a quartic type functional equation. Fixed Point Theory (to appear) Google Scholar
  7. 7.
    Eshaghi Gordji, M., Ebadian, A., Zolfaghari, S.: Stability of a functional equation deriving from cubic and quartic functions. Abstr. Appl. Anal. 2008 (2008), Article ID 801904, 17 pages Google Scholar
  8. 8.
    Gajda, Z.: On stability of additive mappings. Int. J. Math. Math. Sci. 14, 431–434 (1991) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Gǎvruta, P.: A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184, 431–436 (1994) CrossRefMathSciNetGoogle Scholar
  10. 10.
    Gouvêa, F.Q.: p-adic Numbers. Springer, Berlin (1997) zbMATHGoogle Scholar
  11. 11.
    Grabiec, A.: The generalized Hyers–Ulam stability of a class of functional equations. Publ. Math. Debr. 48, 217–235 (1996) MathSciNetGoogle Scholar
  12. 12.
    Hensel, K.: Über eine neue Begründung der Theorie der algebraischen Zahlen. Jahresber. Dtsch. Math. Ver. 6, 83–88 (1897) Google Scholar
  13. 13.
    Hyers, D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 27, 222–224 (1941) CrossRefMathSciNetGoogle Scholar
  14. 14.
    Hyers, D.H., Isac, G., Rassias, Th.M.: Stability of Functional Equations in Several Variables. Birkhäuser, Basel (1998) zbMATHGoogle Scholar
  15. 15.
    Jung, S.M.: Hyers–Ulam–Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press Inc., Palm Harbor (2001) zbMATHGoogle Scholar
  16. 16.
    Jung, K.W., Kim, H.M.: The generalized Hyers–Ulam–Rassias stability of a cubic functional equation. J. Math. Anal. Appl. 274(2), 267–278 (2002) CrossRefMathSciNetGoogle Scholar
  17. 17.
    Khrennikov, A.: Non-Archimedean Analysis, Quantum Paradoxes, Dynamical Systems and Biological Models. Kluwer Academic Publishers, Dordrecht (1997) zbMATHGoogle Scholar
  18. 18.
    Moslehian, M.S., Rassias, Th.M.: Stability of functional equations in non-Archimedean spaces. Appl. Anal. Discrete Math. 1, 325–334 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Najati, A.: On the stability of a quartic functional equation. J. Math. Anal. Appl. 340(1), 569–574 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Park, C.: On the stability of the quadratic mapping in Banach modules. J. Math. Anal. Appl. 276, 135–144 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Park, C.: Generalized quadratic mappings in several variables. Nonlinear Anal. 57, 713–722 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Park, W.G., Bae, J.H.: On the stability a bi-quartic functional equation. Nonlinear Anal. 62(4), 643–654 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Rassias, Th.M.: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72, 297–300 (1978) zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Rassias, Th.M.: On the stability of the quadratic functional equation and its applications. Stud. Univ. Babes-Bolyai 43, 89–124 (1998) zbMATHMathSciNetGoogle Scholar
  25. 25.
    Rassias, T.M. (ed.): Functional Equations and Inequalities. Kluwer Academic Publishers, Dordrecht (2000) zbMATHGoogle Scholar
  26. 26.
    Robert, A.M.: A Course in p-adic Analysis. Springer, New York (2000) zbMATHGoogle Scholar
  27. 27.
    Skof, F.: Proprietá localie approssimazione dioperatori. Rend. Sem. Mat. Fis. Milano 53, 113–129 (1983) zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Ulam, S.M.: A Collection of Mathematical Problems. Interscience Tracts in Pure and Applied Mathematics. Interscience, New York (1960) zbMATHGoogle Scholar
  29. 29.
    Vladimirov, V.S., Volovich, I.V., Zelenov, E.I.: p-adic Analysis and Mathematical Physics. World Scientific, Singapore (1994) Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Department of MathematicsSemnan UniversitySemnanIran

Personalised recommendations