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FUZZY STABILITY OF THE CAUCHY ADDITIVE AND QUADRATIC TYPE FUNCTIONAL EQUATION

  • Received : 2011.03.14
  • Published : 2012.07.31

Abstract

In this paper, we investigate a fuzzy version of stability for the functional equation $$2f(x+y)+f(x-y)+f(y-x)-3f(x)-f(-x)-3f(y)-f(-y)=0$$ in the sense of M. Mirmostafaee and M. S. Moslehian.

Keywords

fuzzy normed space;fuzzy almost quadratic-additive mapping;Cauchy additive and quadratic type functional equation

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. S. C. Cheng and J. N. Mordeson, Fuzzy linear operator and fuzzy normed linear spaces, Bull. Calcutta Math. Soc. 86 (1994), no. 5, 429-436.
  4. 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
  5. 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
  6. 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
  7. D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941), 222-224. https://doi.org/10.1073/pnas.27.4.222
  8. K.-W. Jun and Y.-H. Lee, A generalization of the Hyers-Ulam-Rassias stability of the Pexiderized quadratic equations. II, Kyungpook Math. J. 47 (2007), no. 1, 91-103.
  9. G.-H. Kim, On the stability of functional equations with square-symmetric operation, Math. Inequal. Appl. 4 (2001), no. 2, 257-266.
  10. H.-M. Kim, On the stability problem for a mixed type of quartic and quadratic functional equation, J. Math. Anal. Appl. 324 (2006), no. 1, 358-372. https://doi.org/10.1016/j.jmaa.2005.11.053
  11. I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetika (Prague) 11 (1975), no. 5, 326-334.
  12. Y.-H. Lee, On the Hyers-Ulam-Rassias stability of the generalized polynomial function of degree 2, J. Chuncheong Math. Soc. 22, (2009) 201-209.
  13. Y.-H. Lee, On the stability of the monomial functional equation, Bull. Korean Math. Soc. 45 (2008), no. 2, 397-403. https://doi.org/10.4134/BKMS.2008.45.2.397
  14. Y. H. Lee and K. W. Jun, A generalization of the Hyers-Ulam-Rassias stability of Jensen's equation, J. Math. Anal. Appl. 238 (1999), no. 1, 305-315. https://doi.org/10.1006/jmaa.1999.6546
  15. Y. H. Lee and K. W. Jun, A generalization of the Hyers-Ulam-Rassias stability of the Pexider equation, J. Math. Anal. Appl. 246 (2000), no. 2, 627-638. https://doi.org/10.1006/jmaa.2000.6832
  16. Y. H. Lee and K. W. Jun, On the stability of approximately additive mappings, Proc. Amer. Math. Soc. 128 (2000), no. 5, 1361-1369. https://doi.org/10.1090/S0002-9939-99-05156-4
  17. A. K. Mirmostafaee and M. S. Moslehian, Fuzzy almost quadratic functions, Results Math. 52 (2008), no. 1-2, 161-177. https://doi.org/10.1007/s00025-007-0278-9
  18. 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
  19. C.-G. Park, On the stability of the Cauchy additive and quadratic type functional equation, to appear.
  20. Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), no. 2, 297-300. https://doi.org/10.1090/S0002-9939-1978-0507327-1
  21. F. Skof, Local properties and approximations of operators, Rend. Sem. Mat. Fis. Milano 53 (1983), 113-129. https://doi.org/10.1007/BF02924890
  22. S. M. Ulam, A Collection of Mathematical Problems, Interscience, New York, 1968.

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  2. A General Uniqueness Theorem concerning the Stability of Additive and Quadratic Functional Equations vol.2015, 2015, https://doi.org/10.1155/2015/643969