Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

FUZZY STABILITY OF A CUBIC-QUARTIC FUNCTIONAL EQUATION: A FIXED POINT APPROACH

  • Received : 2009.09.02
  • Published : 2011.05.31
Download PDF
⟨ Previous Next ⟩

Abstract

Using the fixed point method, we prove the generalized Hyers-Ulam stability of the following cubic-quartic functional equation (0.1) f(2x + y) + f(2x - y) = 3f(x + y) + f(-x - y) + 3f(x - y) + f(y - x) + 18f(x) + 6f(-x) - 3f(y) - 3f(-y) in fuzzy 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. T. Bag and S. K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math. 11 (2003), no. 3, 687-705.
  3. T. Bag and S. K. Samanta,Fuzzy bounded linear operators, Fuzzy Sets and Systems 151 (2005), no. 3, 513-547. https://doi.org/10.1016/j.fss.2004.05.004
  4. L. Cadariu and V. Radu, Fixed points and the stability of Jensen's functional equation, J. Inequal. Pure Appl. Math. 4 (2003), no. 1, Art. 4, 7 pp.
  5. L. Cadariu and V. Radu,On the stability of the Cauchy functional equation: a fixed point approach, Iteration theory (ECIT '02), 43-52, Grazer Math. Ber., 346, Karl-Franzens-Univ. Graz, Graz, 2004.
  6. L. Cadariu and V. Radu, Fixed point methods for the generalized stability of functional equations in a single variable, Fixed Point Theory and Applications 2008 (2008), Art. ID 749392.
  7. S. C. Cheng and J. M. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces, Bull. Calcutta Math. Soc. 86 (1994), no. 5, 429-436.
  8. P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), no. 1-2, 76-86. https://doi.org/10.1007/BF02192660
  9. 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
  10. J. Diaz and B. Margolis, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968), 305-309. https://doi.org/10.1090/S0002-9904-1968-11933-0
  11. C. Felbin, Finite dimensional fuzzy normed linear spaces, Fuzzy Sets and Systems 48 (1992), no. 2, 239-248. https://doi.org/10.1016/0165-0114(92)90338-5
  12. P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), no. 3, 431-436. https://doi.org/10.1006/jmaa.1994.1211
  13. 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
  14. D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser, Basel, 1998.
  15. G. Isac and Th. M. Rassias, Stability of $\psi$-additive mappings: Appications to nonlinear analysis, Internat. J. Math. Math. Sci. 19 (1996), no. 2, 219-228. https://doi.org/10.1155/S0161171296000324
  16. K. Jun and H. Kim, The generalized Hyers-Ulam-Rassias stability of a cubic functional equation, J. Math. Anal. Appl. 274 (2002), no. 2, 267-278.
  17. S. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press lnc., Palm Harbor, Florida, 2001.
  18. A. K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets and Systems 12 (1984), no. 2, 143-154. https://doi.org/10.1016/0165-0114(84)90034-4
  19. I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica 11 (1975), no. 5, 336-344.
  20. S. V. Krishna and K. K. M. Sarma, Separation of fuzzy normed linear spaces, Fuzzy Sets and Systems 63 (1994), no. 2, 207-217. https://doi.org/10.1016/0165-0114(94)90351-4
  21. S. Lee, S. Im, and I. Hwang, Quartic functional equations, J. Math. Anal. Appl. 307 (2005), no. 2, 387-394. https://doi.org/10.1016/j.jmaa.2004.12.062
  22. D. Mihet and V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343 (2008), no. 1, 567-572. https://doi.org/10.1016/j.jmaa.2008.01.100
  23. M. Mirzavaziri and M. S. Moslehian, A fixed point approach to stability of a quadratic equation, Bull. Braz. Math. Soc. 37 (2006), no. 3, 361-376. https://doi.org/10.1007/s00574-006-0016-z
  24. A. K. Mirmostafaee, M. Mirzavaziri, and M. S. Moslehian, Fuzzy stability of the Jensen functional equation, Fuzzy Sets and Systems 159 (2008), no. 6, 730-738. https://doi.org/10.1016/j.fss.2007.07.011
  25. A. K. Mirmostafaee and M. S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets and Systems 159 (2008), no. 6, 720-729. https://doi.org/10.1016/j.fss.2007.09.016
  26. 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
  27. C. Park, Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras, Fixed Point Theory Appl. 2007 (2007), Art. ID 50175.
  28. C. Park, Generalized Hyers-Ulam stability of quadratic functional equations: a fixed point approach, Fixed Point Theory Appl. 2008 (2008), Art. ID 493751.
  29. C. Park, Y. Cho, and M. Han, Functional inequalities associated with Jordan-von Neumann type additive functional equations, J. Inequal. Appl. 2007 (2007), Art. ID 41820.
  30. C. Park and J. Cui, Generalized stability of $C^{*}$- ternary quadratic mappings, Abstr. Appl. Anal. 2007 (2007), Art. ID 23282.
  31. C. Park and A. Najati, Homomorphisms and derivations in $C^{*}$- algebras, Abstr. Appl. Anal. 2007 (2007), Art. ID 80630.
  32. V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), no. 1, 91-96.
  33. Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), no. 2, 297-300.
  34. Th. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62 (2000), no. 1, 23-130. https://doi.org/10.1023/A:1006499223572
  35. F. Skof, Local properties and approximation of operators, Rend. Sem. Mat. Fis. Milano 53 (1983), 113-129. https://doi.org/10.1007/BF02924890
  36. S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960.
  37. J. Z. Xiao and X. H. Zhu, Fuzzy normed spaces of operators and its completeness, Fuzzy Sets and Systems 133 (2003), no. 3, 389-399. https://doi.org/10.1016/S0165-0114(02)00274-9