한국수학교육학회지시리즈B:순수및응용수학 (The Pure and Applied Mathematics)

  • 제25권2호
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  • Pages.135-147
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  • 2018
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  • 1226-0657(pISSN)
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  • 2287-6081(eISSN)
한국수학교육학회 (Korean Society of Mathematical Education)
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HYERS-ULAM STABILITY OF DERIVATIONS IN FUZZY BANACH SPACE: REVISITED

  • Lu, Gang (Department of Mathematics, Zhejiang University) ;
  • Jin, Yuanfeng (Department of Mathematics, Yanbian University) ;
  • Wu, Gang (Department of Mathematics and Applied Mathematics, Harbin University of Commerce) ;
  • Yun, Sungsik (Department of Financial Mathematics, Hanshin University)
  • 투고 : 2017.11.28
  • 심사 : 2018.05.04
  • 발행 : 2018.05.31
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초록

Lu et al. [27] defined derivations on fuzzy Banach spaces and fuzzy Lie Banach spaces and proved the Hyers-Ulam stability of derivations on fuzzy Banach spaces and fuzzy Lie Banach spaces. It is easy to show that the definitions of derivations on fuzzy Banach spaces and fuzzy Lie Banach spaces are wrong and so the results of [27] are wrong. Moreover, there are a lot of seroius problems in the statements and the proofs of the results in Sections 2 and 3. In this paper, we correct the definitions of biderivations on fuzzy Banach algebras and fuzzy Lie Banach algebras and the statements of the results in [27], and prove the corrected theorems.

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참고문헌

  1. S. Alizadeh & F. Moradlou: Approximate a quadratic mapping in multi-Banach spaces, a fixed point approach. Int. J. Nonlinear Anal. Appl. 7 (2016), no. 1, 63-75.
  2. 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
  3. T. Bag & S.K. Samanta: Finite dimensional fuzzy normed linear spaces. J. Fuzzy Math. 11 (2003), 687-705.
  4. T. Bag & S.K. Samanta: Fuzzy bounded linear operators. Fuzzy Sets Syst. 151 (2005), 513-547. https://doi.org/10.1016/j.fss.2004.05.004
  5. V. Balopoulos, A.G. Hatzimichailidis & B.K. Papadopoulos: Distance and similarity measures for fuzzy operators. Inform. Sci. 177 (2007), 2336-2348. https://doi.org/10.1016/j.ins.2007.01.005
  6. R. Biswas: Fuzzy inner product spaces and fuzzy norm functions. Inform. Sci. 53 (1991), 185-190. https://doi.org/10.1016/0020-0255(91)90063-Z
  7. J. Brzdek: Hyperstability of the Cauchy equation on restricted domains. Acta Math. Hungarica 141 (2013), 58-67. https://doi.org/10.1007/s10474-013-0302-3
  8. J. Brzdek, J. Chudziak & Zs. Pales: A fixed point approach to stability of functional equations. Nonlinear Anal.-TMA 74 (2011), 6728-6732. https://doi.org/10.1016/j.na.2011.06.052
  9. L. Cadariu, L. Gavruta & P. Gavruta: Fixed points and generalized Hyers-Ulam stability. Abs. Appl. Anal. 2012, Article ID 712743 (2012).
  10. L. Cadariu & V. Radu: Fixed points and the stability of Jensen's functional equation. J. Inequal. Pure Appl. Math. 4 (2003), No. 1, Article ID 4.
  11. L.S. Chadli, S. Melliani, A. Moujahid & M. Elomari: Generalized solution of sine Gordon equation. Int. J. Nonlinear Anal. Appl. 7 (2016), no. 1, 87-92.
  12. A. Chahbi & N. Bounader: On the generalized stability of d'Alembert functional equation. J. Nonlinear Sci. Appl. 6 (2013), 198-204 https://doi.org/10.22436/jnsa.006.03.05
  13. I. Chang, M. Eshaghi Gordji, H. Khodaei & H. Kim: Nearly quartic mappings in fihomogeneous F-spaces. Results Math. 63 (2013), 529-541. https://doi.org/10.1007/s00025-011-0215-9
  14. S.C. Cheng & J.N. Mordeson: Fuzzy linear operator and fuzzy normed linear spaces. Bull. Calcutta Math. Soc. 86 (1994), 429-436.
  15. K. Cieplinski: Applications of fixed point theorems to the Hyers-Ulam stability of functional equations-a survey. Ann. Funct. Anal. 3 (2012), 151-164. https://doi.org/10.15352/afa/1399900032
  16. J.B. Diaz & B. Margolis: A fixed point theorem of the alternative for contractions on a generalized complete metric space. Bull. Amer. Math. Soc. 44 (1968), 305-309.
  17. M. Eshaghi Gordji, H. Khodaei, Th.M. Rassias & R. Khodabakhsh: $J^{\ast}$-homo-morphisms and $J^{\ast}$-derivations on $J^{\ast}$-algebras for a generalized Jensen type functional equation. Fixed Point Theory 13 (2012), 481-494.
  18. C. Felbin: Finite dimensional fuzzy normed linear space. Fuzzy Sets Syst. 48 (1992), 239-248. https://doi.org/10.1016/0165-0114(92)90338-5
  19. 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
  20. 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
  21. D.H. Hyers, G. Isac & Th.M. Rassias: Stability of Functional Equations in Several Variables. Birkhauser, Basel, 1998.
  22. 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
  23. H. Khodaei, R. Khodabakhsh & M. Eshaghi Gordji: Fixed points, Lie*-homo-morphisms and Lie *-derivations on Lie $C^{\ast}$-algebras. Fixed Point Theory 14 (2013), 387-400.
  24. I. Kramosil & J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica 11 (1975), 326-334.
  25. S.V. Krishna & K.K.M. Sarma: Separation of fuzzy normed linear spaces. Fuzzy Sets Syst. 63 (1994), 207-217. https://doi.org/10.1016/0165-0114(94)90351-4
  26. G. Lu & C. Park: Hyers-Ulam stability of additive set-valued functional equations. Appl. Math. Lett. 24 (2011), 1312-1316. https://doi.org/10.1016/j.aml.2011.02.024
  27. G. Lu, J. Xie, Q. Liu & Y. Jin: Hyers-Ulam stability of derivations in fuzzy Banach space. J. Nonlinear Sci. Appl. 9 (2016), 5970-5979. https://doi.org/10.22436/jnsa.009.12.05
  28. D. Mihet & 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. F. Moradlou & M. Eshaghi Gordji: Approximate Jordan derivations on Hilbert $C^{\ast}$-modules. Fixed Point Theory 14 (2013), 413-425.
  30. C. Park: Homomorphisms between Poisson $JC^{\ast}$-algebras. Bull. Braz. Math. Soc. 36 (2005), 79-97. https://doi.org/10.1007/s00574-005-0029-z
  31. C. Park, K. Ghasemi & S. Ghaleh: Fuzzy n-Jordan *-derivations on induced fuzzy $C^{\ast}$-algebras. J. Comput. Anal. Appl. 16 (2014), 494-502.
  32. C. Park & A. Najati: Generalized additive functional inequalities in Banach algebras. Int. J. Nonlinear Anal. Appl. 1 (2010), no. 2, 54-62.
  33. Th.M. Rassias: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72 (1978), 297-300. https://doi.org/10.1090/S0002-9939-1978-0507327-1
  34. R. Saadati & S. M.Vaezpour: Some results on fuzzy Banach spaces. J. Appl. Math. Comput. 17 (2005), no. 1-2, 475-484. https://doi.org/10.1007/BF02936069
  35. 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
  36. S.M. Ulam: Problems in Modern Mathematics. Chapter VI, Science ed., Wiley, New York, 1940.
  37. C. Wu & J. Fang: Fuzzy generalization of Klomogoroff's theorem. J. Harbin Inst. Technol. 1 (1984), 1-7.
  38. J.Z. Xiao & X.-H. Zhu: Fuzzy normed spaces of operators and its completeness. Fuzzy Sets Syst. 133 (2003), 389-399. https://doi.org/10.1016/S0165-0114(02)00274-9