바로가기메뉴

본문 바로가기 주메뉴 바로가기

logo

ISSN : 1226-0657

Article Contents

Prev Next
e-Submission

Vol.17 No.1

Citation Share

ON THE FUZZY STABILITY OF QUADRATIC FUNCTIONAL EQUATIONS

Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics / Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics, (P)1226-0657; (E)2287-6081
2010, v.17 no.1, pp.65-80
Lee, Jung-Rye
Jang, Sun-Young
Shin, Dong-Yun
Lee,, J. , Jang,, S. , & Shin,, D. (2010). ON THE FUZZY STABILITY OF QUADRATIC FUNCTIONAL EQUATIONS. Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics, 17(1), 65-80.

Abstract

In [17, 18], the fuzzy stability problems for the Cauchy additive functional equation and the Jensen additive functional equation in fuzzy Banach spaces have been investigated. In this paper, we prove the generalized Hyers-Ulam stability of the following quadratic functional equations in fuzzy Banach spaces: (0.1) f(x + y) + f(x - y) = 2f(x) + 2f(y), (0.2) f(ax + by) + f(ax - by) = <TEX>$2a^2 f(x)\;+\;2b^2f(y)$</TEX> for nonzero real numbers a, b with <TEX>$a\;{\neq}\;{\pm}1$</TEX>.

keywords
fuzzy Banach space, quadratic functional equation, generalized Hyers-Ulam stability

Reference

1.

Park, C.-G.. (2003). Modified Trif's functional equations in Banach modules over a C<SUP>*</SUP>-algebra and approximate algebra homomorphisms. Journal of Mathematical Analysis and Applications, 278(1), 93-108. 10.1016/S0022-247X(02)00573-5.

2.

Mirmostafaee, A.K.;Moslehian, M.S.. (2008). Fuzzy versions of Hyers-Ulam-Rassias theorem. Fuzzy Sets and Systems, 159(6), 720-729. 10.1016/j.fss.2007.09.016.

3.

Park, C.-G.. (2002). On the stability of the linear mapping in Banach modules. Journal of Mathematical Analysis and Applications, 275(2), 711-720. 10.1016/S0022-247X(02)00386-4.

4.

Xiao, J.-z.;Zhu, X.-h.. (2003). Fuzzy normed space of operators and its completeness. Fuzzy Sets and Systems, 133(3), 389-399. 10.1016/S0165-0114(02)00274-9.

5.

Mirmostafaee, A.K.;Mirzavaziri, M.;Moslehian, M.S.. (2008). Fuzzy stability of the Jensen functional equation. Fuzzy Sets and Systems, 159(6), 730-738. 10.1016/j.fss.2007.07.011.

6.

Katsaras, A.K.. (1984). Fuzzy topological vector spaces II. Fuzzy Sets and Systems, 12(2), 143-154. 10.1016/0165-0114(84)90034-4.

7.

(1975). . Kybernetica, 11, 326-334.

8.

Krishna, S.V.;Sarma, K.K.M.. (1994). Separation of fuzzy normed linear spaces. Fuzzy Sets and Systems, 63(2), 207-217. 10.1016/0165-0114(94)90351-4.

9.

10.

11.

Skof. (1983). . Rendiconti del Seminario Matematico e Fisico di Milano, 53(1), 113-129. 10.1007/BF02924890.

12.

Gavruta, P.. (1994). A Generalization of the Hyers-Ulam-Rassias Stability of Approximately Additive Mappings. Journal of Mathematical Analysis and Applications, 184(3), 431-436. 10.1006/jmaa.1994.1211.

13.

Hyers, D H. (1941). On the Stability of the Linear Functional Equation.. Proceedings of the National Academy of Sciences, 27(4), 222-224. 10.1073/pnas.27.4.222.

14.

Rassias, Themistocles M.;Semrl, Peter. (1992). On the Behavior of Mappings which do not Satisfy Hyers-Ulam Stability. Proceedings of the American Mathematical Society, 114(4), 989-993. 10.1090/S0002-9939-1992-1059634-1.

15.

(1993). . J. Math. Anal. Appl., 193, 325-338.

16.

Rassias, T.M.;Shibata, K.. (1998). Variational Problem of Some Quadratic Functionals in Complex Analysis. Journal of Mathematical Analysis and Applications, 228(1), 234-253. 10.1006/jmaa.1998.6129.

17.

Rassias, Themistocles M.. (2000). On the Stability of Functional Equations and a Problem of Ulam. Acta Applicandae Mathematicae, 62(1), 23-130. 10.1023/A:1006499223572.

18.

Felbin, C.. (1992). Finite dimensional fuzzy normed linear space. Fuzzy Sets and Systems, 48(2), 239-248. 10.1016/0165-0114(92)90338-5.

19.

Gajda. (1991). . International Journal of Mathematics and Mathematical Sciences, 14(3), 431-434. 10.1155/S016117129100056X.

20.

21.

(1984). . Bull. Sci. Math., 108, 95-99.

22.

Cholewa. (1984). . Aequationes Mathematicae, 27(1), 76-86. 10.1007/BF02192660.

23.

Czerwik. (1992). . Abhandlungen aus dem Mathematischen Seminar der Universit채t Hamburg, 62(1), 59-64. 10.1007/BF02941618.

24.

25.

(1990). .

26.

Rassias, T.M.. (2000). The Problem of S. M. Ulam for Approximately Multiplicative Mappings. Journal of Mathematical Analysis and Applications, 246(2), 352-378. 10.1006/jmaa.2000.6788.

27.

(1998). . Studia Univ. Babes-Bolyai, XLIII, 89-124.

28.

AOKI. (1950). . Journal of the Mathematical Society of Japan, 2(1-2), 64-66. 10.2969/jmsj/00210064.

29.

Bag, T.;Samanta, S.K.. (2005). Fuzzy bounded linear operators. Fuzzy Sets and Systems, 151(3), 513-547. 10.1016/j.fss.2004.05.004.

30.

(1994). . Bull. Calcutta Math. Soc., 86, 429-436.

31.

Rassias, T.M.. (2000). On the Stability of Functional Equations in Banach Spaces. Journal of Mathematical Analysis and Applications, 251(1), 264-284. 10.1006/jmaa.2000.7046.

32.

Park*, Chun-Gil. (2005). Homomorphisms between Poisson JC*-Algebras. Bulletin of the Brazilian Mathematical Society, New Series, 36(1), 79-97. 10.1007/s00574-005-0029-z.

33.

(2003). . J. Fuzzy Math., 11, 687-705.

34.

Rassias, Themistocles M.. (1978). On the Stability of the Linear Mapping in Banach Spaces. Proceedings of the American Mathematical Society, 72(2), 297-300. 10.1090/S0002-9939-1978-0507327-1.

35.

Park, C.-G.. (2004). Lie *-homomorphisms between Lie C<SUP>*</SUP>-algebras and Lie *-derivations on Lie C<SUP>*</SUP>-algebras. Journal of Mathematical Analysis and Applications, 293(2), 419-434. 10.1016/j.jmaa.2003.10.051.

36.

Park, C.-G.. (2004). On an approximate automorphism on a C<SUP>*</SUP>-algebra. Proceedings of the American Mathematical Society, 132(6), 1739-1745. 10.1090/S0002-9939-03-07252-6.

Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics

e-Submission OA Policy