The Pure and Applied Mathematics (한국수학교육학회지시리즈B:순수및응용수학)

Korean Society of Mathematical Education (한국수학교육학회)
권호 표지

ON THE FUZZY STABILITY OF QUADRATIC FUNCTIONAL EQUATIONS

  • Published : 2010.02.28
Download PDF
⟨ Previous Next ⟩

Abstract

In [17, 18], the fuzzy stability problems for the Cauchy additive functional equation and the Jensen additive functional equation in fuzzy Banach spaces have been investigated. In this paper, we prove the generalized Hyers-Ulam stability of the following quadratic functional equations in fuzzy Banach spaces: (0.1) f(x + y) + f(x - y) = 2f(x) + 2f(y), (0.2) f(ax + by) + f(ax - by) = $2a^2 f(x)\;+\;2b^2f(y)$ for nonzero real numbers a, b with $a\;{\neq}\;{\pm}1$.

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 & S.K. Samanta: Finite dimensional fuzzy normed linear spaces. J. Fuzzy Math. 11 (2003), 687-705.
  3. T. Bag & S.K. Samanta: Fuzzy bounded linear operators. Fuzzy Sets and Systems 151 (2005), 513-547. https://doi.org/10.1016/j.fss.2004.05.004
  4. S.C. Cheng & J.M. Mordeson: Fuzzy linear operators and fuzzy normed linear spaces. Bull. Calcutta Math. Soc. 86 (1994), 429-436.
  5. P.W. Cholewa: Remarks on the stability of functional equations. Aequationes Math. 27 (1984), 76-86. https://doi.org/10.1007/BF02192660
  6. 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
  7. S. Czerwik: Functional Equations and Inequalities in Several Variables. World Scientific Publishing Company, New Jersey, Hong Kong, Singapore and London, 2002.
  8. C. Felbin: Finite dimensional fuzzy normed linear spaces. Fuzzy Sets and Systems 48 (1992), 239-248. https://doi.org/10.1016/0165-0114(92)90338-5
  9. Z. Gajda: On stability of additive mappings. Internat. J. Math. Math. Sci. 14 (1991), 431-434. https://doi.org/10.1155/S016117129100056X
  10. 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
  11. 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
  12. D.H. Hyers, G. Isac & Th.M. Rassias: Stability of Functional Equations in Several Variables. Birkhauser, Basel, 1998.
  13. S. Jung: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press lnc., Palm Harbor, Florida, 2001.
  14. A.K. Katsaras: Fuzzy topological vector spaces II . Fuzzy Sets and Systems 12 (1984), 143-154. https://doi.org/10.1016/0165-0114(84)90034-4
  15. I. Kramosil & J. Michalek: Fuzzy metric and statistical metric spaces. Kybernetica 11 (1975), 326-334.
  16. S.V. Krishna & K.K.M. Sarma: Separation of fuzzy normed linear spaces. Fuzzy Sets and Systems 63 (1994), 207-217. https://doi.org/10.1016/0165-0114(94)90351-4
  17. A.K. Mirmostafaee, M. Mirzavaziri & M.S. Moslehian: Fuzzy stability of the Jensen functional equation. Fussy Sets and Systems 159 (2008), 730-738. https://doi.org/10.1016/j.fss.2007.07.011
  18. A.K. Mirmostafaee & M.S. Moslehian: Fuzzy versions of Hyers-Ulam-Rassias theorem. Fuzzy Sets and Systems 159 (2008), 720-729. https://doi.org/10.1016/j.fss.2007.09.016
  19. C. Park: On the stability of the linear mapping in Banach modules. J. Math. Anal. Appl. 275 (2002), 711-720. https://doi.org/10.1016/S0022-247X(02)00386-4
  20. C. Park: Modified Trif's functional equations in Banach modules over a C*-algebra and approximate algebra homomorphisms. J. Math. Anal. Appl. 278 (2003), 93-108. https://doi.org/10.1016/S0022-247X(02)00573-5
  21. C. Park: On an approximate automorphism on a C*-algebra. Proc. Amer. Math. Soc. 132 (2004), 1739-1745. https://doi.org/10.1090/S0002-9939-03-07252-6
  22. C. Park: Lie *-homomorphisms between Lie C*-algebras and Lie *-derivations on Lie C*-algebras. J. Math. Anal. Appl. 293 (2004), 419-434. https://doi.org/10.1016/j.jmaa.2003.10.051
  23. C. Park: Homomorphisms between Poisson C*-algebras. Bull. Braz. Math. Soc. 36 (2005), 79-97. https://doi.org/10.1007/s00574-005-0029-z
  24. 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
  25. Th.M. Rassias: New characterizations of inner product spaces. Bull. Sci. Math. 108 (1984), 95-99.
  26. Th.M. Rassias: On the stability of the quadratic functional equation and its applications. Studia Univ. Babes-Bolyai XLIII (1998), 89-124.
  27. Th.M. Rassias: The problem of S.M. Ulam for approximately multiplicative mappings. J. Math. Anal. Appl. 246 (2000), 352-378. https://doi.org/10.1006/jmaa.2000.6788
  28. Th.M. Rassias: On the stability of functional equations in Banach spaces. J. Math. Anal. Appl. 251 (2000), 264-284. https://doi.org/10.1006/jmaa.2000.7046
  29. Th.M. Rassias: On the stability of functional equations and a problem of Ulam. Acta Appl. Math. 62 (2000), 23-130. https://doi.org/10.1023/A:1006499223572
  30. Th.M. Rassias, Problem 16; 2, Report of the 27th International Symp. on Functional Equations, Aequationes Math. 39 (1990), 292-293; 309.
  31. Th.M. Rassias & P. Semrl: On the behaviour of mappings which do not satisfy Hyers-Ulam stability. Proc. Amer. Math. Soc. 114 (1992), 989-993. https://doi.org/10.1090/S0002-9939-1992-1059634-1
  32. Th.M. Rassias & P. Semrl: On the Hyers-Ulam stability of linear mappings. J. Math. Anal. Appl. 173 (1993), 325-338.
  33. Th.M. Rassias & K. Shibata: Variational problem of some quadratic functionals in complex analysis. J. Math. Anal. Appl. 228 (1998), 234-253. https://doi.org/10.1006/jmaa.1998.6129
  34. F. Skof: Proprieta locali e approssimazione di operatori. Rend. Sem. Mat. Fis. Milano 53 (1983), 113-129. https://doi.org/10.1007/BF02924890
  35. S.M. Ulam: A Collection of the Mathematical Problems. Interscience Publ. New York, 1960
  36. J.Z. Xiao & X.H. Zhu: Fuzzy normed spaces of operators and its completeness. Fuzzy Sets and Systems 133 (2003), 389-399. https://doi.org/10.1016/S0165-0114(02)00274-9