• Journal title : The Pure and Applied Mathematics
• Volume 23, Issue 2,  2016, pp.163-179
• Publisher : Korea Society of Mathematical Education
• DOI : 10.7468/jksmeb.2016.23.2.163
Title & Authors
LEE, SUNG JIN; SEO, JEONG PIL;

Abstract
Let $\small{M_1f(x,y):=\frac{3}{4}f(x+y)-\frac{1}{4}f(-x-y)+\frac{1}{4}(x-y)+\frac{1}{4}f(y-x)-f(x)-f(y)}$, $\small{M_2f(x,y):=2f(\frac{x+y}{2})+f(\frac{x-y}{2})+f(\frac{y-x}{2})-f(x)-f(y)}$ Using the direct method, we prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequalities (0.1) $\small{N(M_1f(x,y)-{\rho}M_2f(x,y),t){\geq}\frac{t}{t+{\varphi}(x,y)}}$ and (0.2) $\small{N(M_2f(x,y)-{\rho}M_1f(x,y),t){\geq}\frac{t}{t+{\varphi}(x,y)}}$ in fuzzy Banach spaces, where ρ is a fixed real number with ρ ≠ 1.
Keywords
Language
English
Cited by
References
1.
M. Adam: On the stability of some quadratic functional equation. J. Nonlinear Sci. Appl. 4 (2011), 50-59.

2.
T. Aoki: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Japan 2 (1950), 64-66.

3.
T. Bag & S.K. Samanta: Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math. 11 (2003), 687-705.

4.
_______: Fuzzy bounded linear operators. Fuzzy Sets and Systems 151 (2005), 513-547.

5.
L. Cădariu, L. Găvruta & P. Găvruta: On the stability of an affine functional equation. J. Nonlinear Sci. Appl. 6 (2013), 60-67.

6.
A. Chahbi & N. Bounader: On the generalized stability of d’Alembert functional equation J. Nonlinear Sci. Appl. 6 (2013), 198-204.

7.
I. Chang & Y. Lee: Additive and quadratic type functional equation and its fuzzy stability. Results Math. 63 (2013), 717-730.

8.
S.C. Cheng & J.M. Mordeson: Fuzzy linear operators and fuzzy normed linear spaces. Bull. Calcutta Math. Soc. 86 (1994), 429-436.

9.
P.W. Cholewa: Remarks on the stability of functional equations. Aequationes Math. 27 (1984), 76-86.

10.
G.Z. Eskandani & P. Găvruta: Hyers-Ulam-Rassias stability of pexiderized Cauchy functional equation in 2-Banach spaces. J. Nonlinear Sci. Appl. 5 (2012), 459-465.

11.
C. Felbin: Finite dimensional fuzzy normed linear spaces. Fuzzy Sets and Systems 48 (1992), 239-248.

12.
P. Găvruta: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184 (1994), 431-436.

13.
D.H. Hyers: On the stability of the linear functional equation. Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222-224.

14.
A.K. Katsaras: Fuzzy topological vector spaces II. Fuzzy Sets and Systems 12 (1984), 143-154.

15.
I. Kramosil & J. Michalek: Fuzzy metric and statistical metric spaces. Kybernetica 11 (1975), 326-334.

16.
S.V. Krishna & K.K.M. Sarma: Separation of fuzzy normed linear spaces. Fuzzy Sets and Systems 63 (1994), 207-217.

17.
G. Lu, Y. Wang & P. Ye: n-Jordan *-derivations on induced fuzzy C*-algebras. J. Comput. Anal. Appl. 20 (2016), 266-276.

18.
D. Mihet & R. Saadati: On the stability of the additive Cauchy functional equation in random normed spaces. Appl. Math. Lett. 24 (2011), 2005-2009.

19.
A.K. Mirmostafaee, M. Mirzavaziri & M.S. Moslehian: Fuzzy stability of the Jensen functional equation. Fuzzy Sets and Systems 159 (2008), 730-738.

20.
A.K. Mirmostafaee & M.S. Moslehian: Fuzzy versions of Hyers-Ulam-Rassias theorem. Fuzzy Sets and Systems 159 (2008), 720-729.

21.
_______: Fuzzy approximately cubic mappings, Inform. Sci. 178 (2008), 3791-3798.

22.
E. Movahednia, S.M. Mosadegh, C. Park & D. Shin: Stability of a lattice preserving functional equation on Riesz space: fixed point alternative. J. Comput. Anal. Appl. 21 (2016), 83-89.

23.
C. Park: Additive ρ-functional inequalities and equations. J. Math. Inequal. 9 (2015), 17-26.

24.
_______: Additive ρ-functional inequalities in non-Archimedean normed spaces. J. Math. Inequal. 9 (2015), 397-407.

25.
_______: Stability of ternary quadratic derivation on ternary Banach algebras: revisited. J. Comput. Anal. Appl. 20 (2016), 21-23.

26.
C. Park, K. Ghasemi, S.G. Ghaleh & S. Jang: Approximate n-Jordan *-homomorphisms in C*-algebras. J. Comput. Anal. Appl. 15 (2013), 365-368.

27.
C. Park, A. Najati & S. Jang: Fixed points and fuzzy stability of an additive-quadratic functional equation. J. Comput. Anal. Appl. 15 (2013), 452-462.

28.
Th. M. Rassias: On the stability of the linear mapping in Banach spaces. Proc. Amer. Math. Soc. 72 (1978), 297-300.

29.
K. Ravi, E. Thandapani & B.V. Senthil Kumar: Solution and stability of a reciprocal type functional equation in several variables. J. Nonlinear Sci. Appl. 7 (2014), 18-27.

30.
S. Schin, D. Ki, J. Chang & M. Kim: Random stability of quadratic functional equations: a fixed point approach. J. Nonlinear Sci. Appl. 4 (2011), 37-49.

31.
S. Shagholi, M. Bavand Savadkouhi & M. Eshaghi Gordji: Nearly ternary cubic homomorphism in ternary Fréchet algebras. J. Comput. Anal. Appl. 13 (2011), 1106-1114.

32.
S. Shagholi, M. Eshaghi Gordji & M. Bavand Savadkouhi: Stability of ternary quadratic derivation on ternary Banach algebras. J. Comput. Anal. Appl. 13 (2011), 1097-1105.

33.
D. Shin, C. Park & Sh. Farhadabadi: On the superstability of ternary Jordan C*-homomorphisms. J. Comput. Anal. Appl. 16 (2014), 964-973.

34.
_______: Stability and superstability of J*-homomorphisms and J*-derivations for a generalized Cauchy-Jensen equation. J. Comput. Anal. Appl. 17 (2014), 125-134.

35.
F. Skof: Propriet locali e approssimazione di operatori. Rend. Sem. Mat. Fis. Milano 53 (1983), 113-129.

36.
S.M. Ulam: A Collection of the Mathematical Problems. Interscience Publ. New York, 1960.

37.
J.Z. Xiao & X.H. Zhu: Fuzzy normed spaces of operators and its completeness. Fuzzy Sets and Systems 133 (2003), 389-399.

38.
C. Zaharia: On the probabilistic stability of the monomial functional equation. J. Nonlinear Sci. Appl. 6 (2013), 51-59.

39.
S. Zolfaghari: Approximation of mixed type functional equations in p-Banach spaces. J. Nonlinear Sci. Appl. 3 (2010), 110-122.