Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
권호 표지
DOI QR코드

DOI QR Code

GENERALIZED CUBIC FUNCTIONS ON A QUASI-FUZZY NORMED SPACE

  • Kang, Dongseung (Mathematics Education Dankook University) ;
  • Kim, Hoewoon B. (Department of Mathematics Oregon State University)
  • Received : 2018.07.25
  • Accepted : 2018.11.29
  • Published : 2019.02.15
Download PDF
⟨ Previous Next ⟩

Abstract

We introduce a generalized cubic functional equation and investigate the Hyers-Ulam stability of the cubic functions as solutions to the generalized cubic functional equation on a quasi-fuzzy anti-${\beta}$-Banach space by both the direct method and the fixed point method.

Keywords

References

  1. T. Aoki, On the stability of the linear transformation in Banach spaces, Journal of the Mathematical Society of Japan, 2 (1950), 64-66. https://doi.org/10.2969/jmsj/00210064
  2. R. Biswas, Fuzzy inner product spaces and fuzzy normed functions, Inform. Sci. 53 (1991), 185-190. https://doi.org/10.1016/0020-0255(91)90063-Z
  3. B. Belaid, E. Elhoucien, and R. Ahmed, Hyers-Ulam Stability of the Generalized Quadratic Functional Equation in Amenable Semigroups, Journal of inequalities in pure and applied mathematics, 8 (2007), no. 2, Article 56, 18 pages.
  4. Y. Benyamini and J. Lindenstrauss, Geometric nonlinear functional analysis, vol. 1, Colloq Publ, 48, Amer Math Soc, Providence, 2000.
  5. J. H. Bae and W. G. Park, On the generalized Hyers-Ulam-Rassias stability in Banach modules over a C*-algebra, Journal of Mathematical Analysis and Applications, 294 (2004), no. 1, 196-205. https://doi.org/10.1016/j.jmaa.2004.02.009
  6. T. Bag and S. K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math. 11 (2003), no. 3, 687-705.
  7. T. Bag and S.K. Samanta, A comparative study of fuzzy norms on a linear space, Fuzzy Sets and Sys. 159 (2008), 670-684. https://doi.org/10.1016/j.fss.2007.09.011
  8. S. C. Cheng and J. N. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces, Bull. Calcutta Math. Soc. 86 (1994), 429-436.
  9. I. S. Chang, K. W. Jun, and Y. S. Jung The modified Hyers-Ulam-Rassias stability of a cubic type functional equation, Math. Inequal. Appl. 8 (2005), no. 4, 675-683.
  10. PW Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76-86. https://doi.org/10.1007/BF02192660
  11. St. Czerwik, On the stability of the quadratic mapping in normed spaces, Abhandlungen aus dem Mathematischen Seminar der Universit'at Hamburg, 62 (1992), 59-64. https://doi.org/10.1007/BF02941618
  12. J. B. 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
  13. 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
  14. Z. Gajda, On stability of additive mappings, International Journal of Mathematics and Mathematical Sciences, 14 (1991), no. 3, 431-434. https://doi.org/10.1155/S016117129100056X
  15. S. Romaguera and V.Gregori, On completion of fuzzy metric spaces, Fuzzy Sets and Systems, 130, no. 3, 399-404. https://doi.org/10.1016/S0165-0114(02)00115-X
  16. D. H. Hyers, On the stability of the linear functional equation, Proceedings of the National Academy of Sciences of the United States of America, 27 (1941), 222-224. https://doi.org/10.1073/pnas.27.4.222
  17. 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
  18. K. W. Jun and H. M. Kim, On the Hyers-Ulam-Rassias stability of a general cubic functional equation, Math Inequal Appl. 6 (2003), no. 2, 289-302.
  19. I. H. Jebril and T. K. Samanta, Fuzzy anti-normed linear space, J. Math. Tech. (2010), 66-77.
  20. A.K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets Sys. 12 (1984), 143-154. https://doi.org/10.1016/0165-0114(84)90034-4
  21. I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica, 11 (1975), 326-334.
  22. S. V. Krishna and K. K. M. Sarma, Seperation of fuzzy normed linear spaces, Fuzzy Sets and Systems, 63 (1994), 207-217. https://doi.org/10.1016/0165-0114(94)90351-4
  23. D. Mihett 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
  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 Sys 159 (2008), 720-729. https://doi.org/10.1016/j.fss.2007.09.016
  26. A. Najati, The generalized Hyers-Ulam-Rassias stability of a cubic functional equation, Turk J Math. 31 (2007), 395-408.
  27. C. Park, Generalized Hyers-Ulam Stability of Quadratic Functional Equations: A Fixed Point Approach, Fixed Point Theory and Applications, 2008, Article ID 493751, 9 pages.
  28. K. Peeva and Y. Kyosev, Fuzzy Relational Calculus-Theory, Applications and Software. World Scientific, New Jersey, 2004.
  29. Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proceedings of the American Mathematical Society, 72 (1978), no. 2, 297-300. https://doi.org/10.1090/S0002-9939-1978-0507327-1
  30. Th. M. Rassias, On the stability of functional equations in Banach spaces, Journal of Mathematical Analysis and Applications, 251 (2000), no. 1, 264-284. https://doi.org/10.1006/jmaa.2000.7046
  31. J. M. Rassias and H. M. Kim, Generalized, Hyers-Ulam stability for general additive functional equations in quasi-${\beta}$-normed spaces, J. Math. Anal. Appl. 356 (2009), 302-309. https://doi.org/10.1016/j.jmaa.2009.03.005
  32. S. Rolewicz, Metric Linear Spaces, Reidel/PWN-Polish Sci. Publ, Dordrecht, 1984.
  33. Th. M. Rassias and P. Semrl, On the Hyers-Ulam stability of linear mappings, Journal of Mathematical Analysis and Applications, 173 (1993), no. 2, 325-338. https://doi.org/10.1006/jmaa.1993.1070
  34. Th. M. Rassias and K. Shibata, Variational problem of some quadratic functionals in complex analysis, Journal of Mathematical Analysis and Applications, 228 (1998), no. 1, 234-253. https://doi.org/10.1006/jmaa.1998.6129
  35. P. K. Sahoo, A generalized cubic functional equation, Acta Math Sinica 21 (2005), no. 5, 1159-1166. https://doi.org/10.1007/s10114-005-0551-3
  36. B. Shieh, Infinite fuzzy relation equations with continuous t-norms, Inform. Sci., 178 (2008), 1961-1967. https://doi.org/10.1016/j.ins.2007.12.006
  37. F. Skof Proprieta locali e approssimazione di operatori, Rend Sem Mat Fis Milano. 53 (1983), 113-129. https://doi.org/10.1007/BF02924890
  38. S. M. Ulam, Problems in Morden Mathematics, Wiley, New York, 1960.
  39. C. Wu and J. Fang, Fuzzy generalization of Klomogoroffs theorem, J Harbin Inst Technol, 1 (1984), 1-7.
  40. T. Z. Xu, J.M. Rassias, and W. X. Xu, A generalized mixed quadratic-quartic functional equation, Bull. Malays. Math. Sci. Soc. 35 (2012), no. 3, 633-649.
  41. J. Z. Xiao and X. H. Zhu, Fuzzy normed space of operators and its completeness, Fuzzy Sets and Systems 133 (2003), 389-399. https://doi.org/10.1016/S0165-0114(02)00274-9
  42. Dilian Yang, The stability of the quadratic functional equation on amenable groups, Journal of Math. Anal. Appl. 291 (2004), 666-672. https://doi.org/10.1016/j.jmaa.2003.11.021
  43. L. A. Zadeh, Fuzzy sets, Inform Control, 8 (1965), 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X
  44. L. A. Zadeh, editor. Fuzzy sets and their applications to cognitive and decision processes, Academic Press, Inc. New York, 1975.