Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
권호 표지

ON THE STABILITY OF AN AQCQ-FUNCTIONAL EQUATION

  • Received : 2009.09.02
  • Accepted : 2009.11.06
  • Published : 2009.12.30
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Abstract

In this paper, we prove the generalized Hyers-Ulam stability of the following additive-quadratic-cubic-quartic functional equation (0.1) f(x + 2y) + f(x - 2y) = 4f(x + y) + 4f(x - y) - 6f(x) + f(2y) + f(-2y) - 4f(y) - 4f(-y) in Banach spaces.

Keywords

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. P.W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76-86. https://doi.org/10.1007/BF02192660
  3. S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59-64. https://doi.org/10.1007/BF02941618
  4. P. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific Publishing Company, New Jersey, Hong Kong, Singapore and London, 2002.
  5. M. Eshaghi-Gordji, S. Abbaszadeh and C. Park, On the stability of a generalized quadratic and quartic type functional equation in quasi-Banach spaces, preprint.
  6. M. Eshaghi-Gordji, S. Kaboli-Gharetapeh, C. Park and S. Zolfaghri, Stability of an additive-cubic-quartic functional equation, preprint.
  7. Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), 431-434. https://doi.org/10.1155/S016117129100056X
  8. P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436.
  9. D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222–224.
  10. D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser, Basel, 1998.
  11. G. Isac and Th.M. Rassias, Stability of $\psi$-additive mappings: Appications to nonlinear analysis, Internat. J. Math. Math. Sci. 19 (1996), 219–228.
  12. K. Jun and H. Kim, The generalized Hyers-Ulam-Rassias stability of a cubic functional equation, J. Math. Anal. Appl. 274 (2002), 867–878.
  13. S. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press lnc., Palm Harbor, Florida, 2001.
  14. S. Lee, S. Im and I. Hwang, Quartic functional equations, J. Math. Anal. Appl. 307 (2005), 387–394.
  15. D. Mihet and V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343 (2008), 567–572.
  16. M. Mirzavaziri and M.S. Moslehian, A fixed point approach to stability of a quadratic equation, Bull. Braz. Math. Soc. 37 (2006), 361–376.
  17. C. Park, Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras, Fixed Point Theory and Applications 2007, Art. ID 50175 (2007).
  18. C. Park, Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach, Fixed Point Theory and Applications 2008, Art. ID 493751 (2008).
  19. C. Park, Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras, Bull. Sci. Math. 132 (2008), 87–96.
  20. C. Park, Hyers-Ulam-Rassias stability of a generalized Apollonius-Jensen type additive mapping and isomorphisms between C-algebras, Math. Nachr. 281 (2008), 402–411.
  21. C. Park and J. Cui, Generalized stability of C-ternary quadratic mappings, Abstract Appl. Anal. 2007, Art. ID 23282 (2007).
  22. C. Park and A. Najati, Homomorphisms and derivations in C-algebras, Abstract Appl. Anal. 2007, Art. ID 80630 (2007).
  23. J.M. Rassias, On approximation of approximately linear mappings by linear mappings, Bull. Sci. Math. 108 (1984), 445–446.
  24. J.M. Rassias, Refined Hyers-Ulam approximation of approximately Jensen type mappings, Bull. Sci. Math. 131 (2007), 89–98.
  25. J.M. Rassias and M.J. Rassias, Asymptotic behavior of alternative Jensen and Jensen type functional equations, Bull. Sci. Math. 129 (2005), 545–558.
  26. Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300.
  27. Th.M. Rassias, Problem 16; 2, Report of the 27th International Symp. on Functional Equations, Aequationes Math. 39 (1990), 292–293; 309.
  28. Th.M. Rassias, On the stability of the quadratic functional equation and its applications, Studia Univ. Babes-Bolyai XLIII (1998), 89–24.
  29. Th.M. Rassias, The problem of S.M. Ulam for approximately multiplicative mappings, J. Math. Anal. Appl. 246 (2000), 352–378.
  30. Th.M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000), 264–284.
  31. Th.M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62 (2000), 23–130.
  32. Th.M. Rassias and P. Semrl, On the behaviour of mappings which do not satisfy Hyers-Ulam stability, Proc. Amer. Math. Soc. 114 (1992), 989–993.
  33. Th.M. Rassias and P. Semrl, On the Hyers-Ulam stability of linear mappings, J. Math. Anal. Appl. 173 (1993), 325–338.
  34. Th.M. Rassias and K. Shibata, Variational problem of some quadratic functionals in complex analysis, J. Math. Anal. Appl. 228 (1998), 234–253.
  35. F. Skof, Propriet`a locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129.
  36. S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960.