Browse > Article
http://dx.doi.org/10.7468/jksmeb.2018.25.2.135

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)
Publication Information
The Pure and Applied Mathematics / v.25, no.2, 2018 , pp. 135-147 More about this Journal
Abstract
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.
Keywords
fuzzy Banach algebra; additive functional equation; Hyers-Ulam stability; fixed point alternative; fuzzy Lie Banach algebra;
Citations & Related Records
연도 인용수 순위
  • Reference
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.   DOI
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.   DOI
5 V. Balopoulos, A.G. Hatzimichailidis & B.K. Papadopoulos: Distance and similarity measures for fuzzy operators. Inform. Sci. 177 (2007), 2336-2348.   DOI
6 R. Biswas: Fuzzy inner product spaces and fuzzy norm functions. Inform. Sci. 53 (1991), 185-190.   DOI
7 J. Brzdek: Hyperstability of the Cauchy equation on restricted domains. Acta Math. Hungarica 141 (2013), 58-67.   DOI
8 J. Brzdek, J. Chudziak & Zs. Pales: A fixed point approach to stability of functional equations. Nonlinear Anal.-TMA 74 (2011), 6728-6732.   DOI
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   DOI
13 I. Chang, M. Eshaghi Gordji, H. Khodaei & H. Kim: Nearly quartic mappings in fihomogeneous F-spaces. Results Math. 63 (2013), 529-541.   DOI
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.   DOI
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.   DOI
19 P. Gavruta: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184 (1994), 431-436.   DOI
20 D.H. Hyers: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 27 (1941), 222-224.   DOI
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.   DOI
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 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.   DOI
25 I. Kramosil & J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica 11 (1975), 326-334.
26 S.V. Krishna & K.K.M. Sarma: Separation of fuzzy normed linear spaces. Fuzzy Sets Syst. 63 (1994), 207-217.   DOI
27 G. Lu & C. Park: Hyers-Ulam stability of additive set-valued functional equations. Appl. Math. Lett. 24 (2011), 1312-1316.   DOI
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.   DOI
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.   DOI
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.   DOI
34 R. Saadati & S. M.Vaezpour: Some results on fuzzy Banach spaces. J. Appl. Math. Comput. 17 (2005), no. 1-2, 475-484.   DOI
35 B. Shieh: Infinite fuzzy relation equations with continuous t-norms. Inform. Sci. 178 (2008), 1961-1967.   DOI
36 J.Z. Xiao & X.-H. Zhu: Fuzzy normed spaces of operators and its completeness. Fuzzy Sets Syst. 133 (2003), 389-399.   DOI
37 S.M. Ulam: Problems in Modern Mathematics. Chapter VI, Science ed., Wiley, New York, 1940.
38 C. Wu & J. Fang: Fuzzy generalization of Klomogoroff's theorem. J. Harbin Inst. Technol. 1 (1984), 1-7.