Browse > Article
http://dx.doi.org/10.4134/BKMS.2011.48.3.491

FUZZY STABILITY OF A CUBIC-QUARTIC FUNCTIONAL EQUATION: A FIXED POINT APPROACH  

Jang, Sun-Young (Department of Mathematics University of Ulsan)
Park, Choon-Kil (Department of Mathematics Hanyang University)
Shin, Dong-Yun (Department of Mathematics University of Seoul)
Publication Information
Bulletin of the Korean Mathematical Society / v.48, no.3, 2011 , pp. 491-503 More about this Journal
Abstract
Using the fixed point method, we prove the generalized Hyers-Ulam stability of the following cubic-quartic functional equation (0.1) f(2x + y) + f(2x - y) = 3f(x + y) + f(-x - y) + 3f(x - y) + f(y - x) + 18f(x) + 6f(-x) - 3f(y) - 3f(-y) in fuzzy Banach spaces.
Keywords
fuzzy Banach space; fixed point; generalized Hyers-Ulam stability; quartic mapping; cubic mapping;
Citations & Related Records

Times Cited By Web Of Science : 0  (Related Records In Web of Science)
Times Cited By SCOPUS : 0
연도 인용수 순위
  • Reference
1 C. Park, Generalized Hyers-Ulam stability of quadratic functional equations: a fixed point approach, Fixed Point Theory Appl. 2008 (2008), Art. ID 493751.
2 C. Park, Y. Cho, and M. Han, Functional inequalities associated with Jordan-von Neumann type additive functional equations, J. Inequal. Appl. 2007 (2007), Art. ID 41820.
3 C. Park and J. Cui, Generalized stability of $C^{*}$- ternary quadratic mappings, Abstr. Appl. Anal. 2007 (2007), Art. ID 23282.
4 C. Park and A. Najati, Homomorphisms and derivations in $C^{*}$- algebras, Abstr. Appl. Anal. 2007 (2007), Art. ID 80630.
5 V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), no. 1, 91-96.
6 Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), no. 2, 297-300.
7 Th. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62 (2000), no. 1, 23-130.   DOI
8 F. Skof, Local properties and approximation of operators, Rend. Sem. Mat. Fis. Milano 53 (1983), 113-129.   DOI
9 S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960.
10 J. Z. Xiao and X. H. Zhu, Fuzzy normed spaces of operators and its completeness, Fuzzy Sets and Systems 133 (2003), no. 3, 389-399.   DOI   ScienceOn
11 K. Jun and H. Kim, The generalized Hyers-Ulam-Rassias stability of a cubic functional equation, J. Math. Anal. Appl. 274 (2002), no. 2, 267-278.
12 S. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press lnc., Palm Harbor, Florida, 2001.
13 A. K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets and Systems 12 (1984), no. 2, 143-154.   DOI   ScienceOn
14 D. Mihet and V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343 (2008), no. 1, 567-572.   DOI   ScienceOn
15 I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica 11 (1975), no. 5, 336-344.
16 S. V. Krishna and K. K. M. Sarma, Separation of fuzzy normed linear spaces, Fuzzy Sets and Systems 63 (1994), no. 2, 207-217.   DOI   ScienceOn
17 S. Lee, S. Im, and I. Hwang, Quartic functional equations, J. Math. Anal. Appl. 307 (2005), no. 2, 387-394.   DOI   ScienceOn
18 M. Mirzavaziri and M. S. Moslehian, A fixed point approach to stability of a quadratic equation, Bull. Braz. Math. Soc. 37 (2006), no. 3, 361-376.   DOI
19 A. K. Mirmostafaee, M. Mirzavaziri, and M. S. Moslehian, Fuzzy stability of the Jensen functional equation, Fuzzy Sets and Systems 159 (2008), no. 6, 730-738.   DOI   ScienceOn
20 A. K. Mirmostafaee and M. S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets and Systems 159 (2008), no. 6, 720-729.   DOI   ScienceOn
21 A. K. Mirmostafaee and M. S. Moslehian, Fuzzy approximately cubic mappings, Inform. Sci. 178 (2008), no. 19, 3791-3798.   DOI   ScienceOn
22 C. Park, Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras, Fixed Point Theory Appl. 2007 (2007), Art. ID 50175.
23 L. Cadariu and V. Radu,On the stability of the Cauchy functional equation: a fixed point approach, Iteration theory (ECIT '02), 43-52, Grazer Math. Ber., 346, Karl-Franzens-Univ. Graz, Graz, 2004.
24 S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59-64.   DOI
25 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 (2008), Art. ID 749392.
26 S. C. Cheng and J. M. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces, Bull. Calcutta Math. Soc. 86 (1994), no. 5, 429-436.
27 P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), no. 1-2, 76-86.   DOI
28 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.   DOI
29 C. Felbin, Finite dimensional fuzzy normed linear spaces, Fuzzy Sets and Systems 48 (1992), no. 2, 239-248.   DOI   ScienceOn
30 P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), no. 3, 431-436.   DOI   ScienceOn
31 D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222-224.   DOI   ScienceOn
32 D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser, Basel, 1998.
33 G. Isac and Th. M. Rassias, Stability of $\psi$-additive mappings: Appications to nonlinear analysis, Internat. J. Math. Math. Sci. 19 (1996), no. 2, 219-228.   DOI   ScienceOn
34 L. Cadariu and V. Radu, Fixed points and the stability of Jensen's functional equation, J. Inequal. Pure Appl. Math. 4 (2003), no. 1, Art. 4, 7 pp.
35 T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64-66.   DOI
36 T. Bag and S. K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math. 11 (2003), no. 3, 687-705.
37 T. Bag and S. K. Samanta,Fuzzy bounded linear operators, Fuzzy Sets and Systems 151 (2005), no. 3, 513-547.   DOI   ScienceOn