East Asian mathematical journal

The Youngnam Mathematical Society (영남수학회)
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FUZZY STABILITY OF QUADRATIC-CUBIC FUNCTIONAL EQUATIONS

  • Kim, Chang Il (Department of Mathematics Education, Dankook University) ;
  • Yun, Yong Sik (Department of Mathematics and research institute for basic sciences, Jeju National University)
  • Received : 2016.04.19
  • Accepted : 2016.05.24
  • Published : 2016.05.31
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Abstract

In this paper, we consider the functional equation f(x + 2y) - 3f(x + y) + 3f(x) - f(x - y) - 3f(y) + 3f(-y) = 0 and prove the generalized Hyers-Ulam stability for it when the target space is a fuzzy Banach space. The usual method to obtain the stability for mixed type functional equation is to split the cases according to whether the involving mappings are odd or even. In this paper, we show that the stability of a quadratic-cubic mapping can be obtained without distinguishing the two cases.

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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. 3(2003), 687-705.
  3. S. C. Cheng and J. N. Mordeson, Fuzzy linear operator and fuzzy normed linear spaces, Bull. Calcutta Math. Soc. 86(1994), 429-436.
  4. P.W.Cholewa, Remarkes on the stability of functional equations, Aequationes Math., 27(1984), 76-86. https://doi.org/10.1007/BF02192660
  5. K Cieplinski, Applications of fixed point theorems to the hyers-ulam stability of functional equation-A survey, Ann. Funct. Anal. 3 (2012), no. 1, 151-164. https://doi.org/10.15352/afa/1399900032
  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. 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
  8. P. Gavruta, A generalization of the Hyer-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184(1994), 431-436. https://doi.org/10.1006/jmaa.1994.1211
  9. D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. 27(1941), 222-224. https://doi.org/10.1073/pnas.27.4.222
  10. G. Isac and Th. M. Rassias, Stability of $\psi$-additive mappings, Appications to nonlinear analysis, Internat. J. Math. Math. Sci. 19(1996), 219-228. https://doi.org/10.1155/S0161171296000324
  11. I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica 11(1975), 326-334.
  12. A. K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets Syst, 12(1984), 143-154. https://doi.org/10.1016/0165-0114(84)90034-4
  13. 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. https://doi.org/10.1016/j.jmaa.2008.01.100
  14. A. K. Mirmostafaee, M. Mirzavaziri, and M. S. Moslehian, Fuzzy stability of the Jensen functional equation, Fuzzy Sets Syst. 159(2008), 730-738. https://doi.org/10.1016/j.fss.2007.07.011
  15. A. K. Mirmostafaee and M. S. Moslehian, Fuzzy almost quadratic functions, Results Math. 52(2008), 161-177. https://doi.org/10.1007/s00025-007-0278-9
  16. A. K. Mirmostafaee and M. S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets Syst. 159(2008), 720-729. https://doi.org/10.1016/j.fss.2007.09.016
  17. M. Mirzavaziri and M. S. Moslehian, A fixed point approach to stability of a quadratic equation, Bulletin of the Brazilian Mathematical Society,37(2006), no. 3, 361-376. https://doi.org/10.1007/s00574-006-0016-z
  18. S. A. Mohiuddine and A. Alotaibi, Fuzzy stability of a cubic functional equation via fixed point technique, Advances in Difference Equations, 48(2012).
  19. C. Park, S. H. Lee and S. H. Lee, Fuzzy stability of a cubic-cubic functional equation: A fixed point approach, Korean J. Math. 17(2009), no.3, 315-330.
  20. J. M. Rassias, Solution of the Ulam stability problem for cubic mappings, Glasnik Matematicki, 36(2001), 63-72.
  21. Th. M. Rassias, On the stability of the linear mapping in Banach sapces, Proc. Amer. Math. Sco. 72(1978), 297-300. https://doi.org/10.1090/S0002-9939-1978-0507327-1
  22. F. Skof, Proprieta locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53(1983), 113-129. https://doi.org/10.1007/BF02924890
  23. S. M. Ulam, A collection of mathematical problems, Interscience Publisher, New York, 1964.