Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
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STABILITY OF THE JENSEN FUNCTIONAL EQUATION IN FUZZY BANACH ALGEBRAS

  • Received : 2012.01.09
  • Accepted : 2012.02.25
  • Published : 2012.03.30
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Abstract

In this paper, we prove the Hyers-Ulam stability of the Jensen functional equation in fuzzy Banach algebras by using fixed point method and by using direct method.

Keywords

References

  1. R. P. Agarwal, Y. J. Cho and R. Saadati, On Random Topological Structures, Appl. Anal. Vol. 2011, Article ID 762361, 41 pages doi:10.1155/2011/762361.
  2. T. Aoki, On the stability of the linear transformation in Banach spaces, 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
  3. T. Bag and S.K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math. 11 (2003), 687-705.
  4. T. Bag and 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
  5. Y. J. Cho, M. Eshaghi Gordji and S. Zolfaghari, Solutions and Stability of Generalized Mixed Type QC Functional Equations in Random Normed Spaces, Inequal. Appl. Vol. 2010, Article ID 403101, pp. 16.
  6. Y. J. Cho, C. Park and R. Saadati, Fuzzy Functional Inequalities, J. Comput. Anal. Appl. 13(2011), 305-320.
  7. Y. J. Cho and R. Saadati, Lattice Non-Archimedean Random Stability of ACQ Functional Equation, Advan. in Diff. Equat. 2011, 2011:31. https://doi.org/10.1186/1687-1847-2011-31
  8. L. Cadariu and V. Radu, Fixed points and the stability of Jensen's functional equation, J. Inequal. Pure Appl. Math. 4, no. 1, Art. ID 4 (2003).
  9. L. Cadariu and V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Grazer Math. Ber. 346 (2004), 43-52.
  10. 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, Art. ID 749392 (2008).
  11. S.C. Cheng and J.M. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces, Bull. Calcutta Math. Soc. 86 (1994), 429-436.
  12. Y. Cho, C. Park and R. Saadati, Functional inequalities in non-Archimedean Banach spaces, Appl. Math. Letters 23 (2010), 1238-1242. https://doi.org/10.1016/j.aml.2010.06.005
  13. P.W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76-86. https://doi.org/10.1007/BF02192660
  14. 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
  15. S. Czerwik, The stability of the quadratic functional equation. in: Stability of mappings of Hyers-Ulam type, (ed. Th.M. Rassias and J.Tabor), Hadronic Press, Palm Harbor, Florida, 1994, 81-91.
  16. P. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific Publishing Company, New Jersey, Hong Kong, Singapore and London, 2002.
  17. 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
  18. S. S. Dragomir, Y. J. Cho and J. K. Kim, Subadditivity of some Functionals associated to Jensen's Inequality with Applications, Taiwan. J. Math. 15(2011), 1815-1828.
  19. 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
  20. 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
  21. 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
  22. D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser, Basel, 1998.
  23. 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
  24. S. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press lnc., Palm Harbor, Florida, 2001.
  25. 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
  26. I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica 11 (1975), 326-334.
  27. S.V. Krishna and 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
  28. 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
  29. M. Mirzavaziri and M.S. Moslehian, A fixed point approach to stability of a quadratic equation, Bull. Braz. Math. Soc. 37 (2006), 361-376. https://doi.org/10.1007/s00574-006-0016-z
  30. A.K. Mirmostafaee, M. Mirzavaziri and M.S. Moslehian, Fuzzy stability of the Jensen functional equation, Fuzzy Sets and Systems 159 (2008), 730-738. https://doi.org/10.1016/j.fss.2007.07.011
  31. A.K. Mirmostafaee and 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
  32. A.K. Mirmostafaee and M.S. Moslehian, Fuzzy approximately cubic mappings, Inform. Sci. 178 (2008), 3791-3798. https://doi.org/10.1016/j.ins.2008.05.032
  33. M. Mohammadi, Y. J. Cho, C. Park, P. Vetro and R. Saadati, Random Stability of an Additive-quadratic-quartic Functional Equation, J. Inequal. Appl. Vol. 2010, Article ID 754210, pp. 18 pages.
  34. A. Najati and Y. J. Cho, Generalized Hyers-Ulam Stability of the Pexiderized Cauchy Functional Equation in Non-Archimedean Spaces, Fixed Point Theory Appl. Vol. 2011, Article ID 309026, 11 pages, doi:10.1155/2011/309026.
  35. A. Najati, J. I. Kang and Y. J. Cho, Local Stability of the Pexiderized Cauchy and Jensen's equations in Fuzzy Spaces, J. Inequal. Appl. 2011, 2011:78 doi:10.1186/1029-242X-2011-78.
  36. C. Park, Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras, Fixed Point Theory and Applications 2007, Art. ID 50175 (2007).
  37. C. Park, Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach, Fixed Point Theory and Applications 2008, Art. ID 493751 (2008).
  38. V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), 91-96.
  39. 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
  40. Th.M. Rassias, Problem 16; 2, Report of the 27th International Symp. on Functional Equations, Aequationes Math. 39 (1990), 292-293; 309.
  41. Th.M. Rassias, On the stability of the quadratic functional equation and its applications, Studia Univ. Babes-Bolyai XLIII (1998), 89-124.
  42. 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
  43. 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
  44. 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
  45. Th.M. Rassias and 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
  46. Th.M. Rassias and P. Semrl, On the Hyers-Ulam stability of linear mappings, J. Math. Anal. Appl. 173 (1993), 325-338. https://doi.org/10.1006/jmaa.1993.1070
  47. Th.M. Rassias and 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
  48. R. Saadati and C. Park, Non-Archimedean L-fuzzy normed spaces and stability of functional equations, Computers Math. Appl. 60 (2010), 2488-2496. https://doi.org/10.1016/j.camwa.2010.08.055
  49. F. Skof, Proprieta locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113-129. https://doi.org/10.1007/BF02924890
  50. S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960.
  51. J.Z. Xiao and 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