Browse > Article
http://dx.doi.org/10.7468/jksmeb.2020.27.1.25

QUADRATIC (ρ1, ρ2)-FUNCTIONAL EQUATION IN FUZZY BANACH SPACES  

Paokant, Siriluk (Department of Mathematics, Research Institute for Natural Sciences, Hanyang University)
Shin, Dong Yun (Department of Mathematics, University of Seoul)
Publication Information
The Pure and Applied Mathematics / v.27, no.1, 2020 , pp. 25-33 More about this Journal
Abstract
In this paper, we consider the following quadratic (ρ1, ρ2)-functional equation (0, 1) $$N(2f({\frac{x+y}{2}})+2f({\frac{x-y}{2}})-f(x)-f(y)-{\rho}_1(f(x+y)+f(x-y)-2f(x)-2f(y))-{\rho}_2(4f({\frac{x+y}{2}})+f(x-y)-f(x)-f(y)),t){\geq}{\frac{t}{t+{\varphi}(x,y)}}$$, where ρ2 are fixed nonzero real numbers with ρ2 ≠ 1 and 2ρ1 + 2ρ2≠ 1, in fuzzy normed spaces. Using the fixed point method, we prove the Hyers-Ulam stability of the quadratic (ρ1, ρ2)-functional equation (0.1) in fuzzy Banach spaces.
Keywords
fuzzy Banach space; quadratic (${\rho}_1$, ${\rho}_2$)-functional equation; fixed point method; Hyers-Ulam stability;
Citations & Related Records
연도 인용수 순위
  • Reference
1 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
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 L. Cadariu & V. Radu: Fixed points and the stability of Jensen's functional equation. J. Inequal. Pure Appl. Math. 4, no. 1, Art. ID 4 (2003).
6 L. Cadariu & V. Radu: On the stability of the Cauchy functional equation: a fixed point approach. Grazer Math. Ber. 346 (2004), 43-52.
7 L. Cadariu & V. Radu: Fixed point methods for the generalized stability of functional equations in a single variable. Fixed Point Theory Appl. 2008, Art. ID 749392 (2008).
8 I. Chang & Y. Lee: Additive and quadratic type functional equation and its fuzzy stability, Results Math. 63 (2013), 717-730.   DOI
9 S.C. Cheng & J.M. Mordeson: Fuzzy linear operators and fuzzy normed linear spaces. Bull. Calcutta Math. Soc. 86 (1994), 429-436.
10 J. Diaz & B. Margolis: A fixed point theorem of the alternative for contractions on a generalized complete metric space. Bull. Am. Math. Soc. 74 (1968), 305-309.   DOI
11 C. Felbin: Finite dimensional fuzzy normed linear spaces. Fuzzy Sets Syst. 48 (1992), 239-248.   DOI
12 P. Gavruta: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184 (1994), 431-436.   DOI
13 D.H. Hyers: On the stability of the linear functional equation. Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222-224.   DOI
14 D.H. Hyers, G. Isac & Th.M. Rassias: Stability of Functional Equations in Several Variables. Birkhauser, Basel, 1998.
15 S.V. Krishna & K.K.M. Sarma: Separation of fuzzy normed linear spaces. Fuzzy Sets Syst. 63 (1994), 207-217.   DOI
16 G. Isac & Th.M. Rassias: Stability of $\psi$-additive mappings: Applications to nonlinear analysis. Internat. J. Math. Math. Sci. 19 (1996), 219-228.   DOI
17 A.K. Katsaras: Fuzzy topological vector spaces II. Fuzzy Sets Syst. 12 (1984), 143-154.   DOI
18 I. Kramosil & J. Michalek: Fuzzy metric and statistical metric spaces. Kybernetica 11 (1975), 326-334.
19 A.K. Mirmostafaee, M. Mirzavaziri & M.S. Moslehian: Fuzzy stability of the Jensen functional equation. Fuzzy Sets Syst. 159 (2008), 730-738.   DOI
20 A.K. Mirmostafaee & M.S. Moslehian: Fuzzy versions of Hyers-Ulam-Rassias theorem. Fuzzy Sets Syst. 159 (2008), 720-729.   DOI
21 A.K. Mirmostafaee & M.S. Moslehian: Fuzzy approximately cubic mappings. Inform. Sci. 178 (2008), 3791-3798.   DOI
22 V. Radu: The fixed point alternative and the stability of functional equations. Fixed Point Theory 4 (2003), 91-96.
23 Th.M. Rassias: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72 (1978), 297-300.   DOI
24 S.M. Ulam: A Collection of the Mathematical Problems. Interscience Publ. New York, 1960.
25 J.Z. Xiao & X.H. Zhu: Fuzzy normed spaces of operators and its completeness. Fuzzy Sets Syst. 133 (2003), 389-399.   DOI