^{}Department of Mathematics, Sahand university of technology, Tabriz- Iran

Abstract

n this paper we study the Hyers-Ulam-Rassias stability of Cauchy equation in Felbin's type fuzzy normed linear spaces. As a result we give an example of a fuzzy normed linear space such that the fuzzy version of the stability problem remains true, while it fails to be correct in classical analysis. This shows how the category of fuzzy normed linear spaces differs from the classical normed linear spaces in general.

[1] T. Bag and S. K. Samanta, A comparative study of fuzzy norms on a linear space, Fuzzy Sets and Systems, 159(6) (2008), 670{684. [2] T. Bag and S. K. Samanta, Fixed point theorems in Felbin type fuzzy normed linear spaces, The Journal of Fuzzy Mathematics, to appear. [3] C. Borelli and G. L. Forti, On a general Hyers{Ulam stability result, Internat. J. Math. Math. Sci., 18 (1995), 229{236. [4] S. Czerwik, The stability of the quadratic functional equation, in: Th. M. Rassias, J. Tabor, eds., Stability of Mappings of Hyers{Ulam Type, Hadronic Press, Florida, (1994), 81{91. [5] V. A. Faiziev, T. M. Rassias and P. K. Sahoo, The space of ( ; )- additive mappings on semigroup, Trans. Amer. Math. Soc., 354(11) (2002), 4455{4472. [6] C. Felbin, Finite dimensional fuzzy normed linear spaces, Fuzzy Sets and Systems, 48 (1992), 239-248. [7] T. Gantner, R. Steinlage and R. Warren, Compactness in fuzzy topological spaces, J. Math. Anal. Appl., 62 (1978) 547562. [8] P. Gavruta, A generalization of the Hyers{Ulam{Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431{436. [9] U. Hoehle, Fuzzy real numbers as Dedekind cuts with respect to a multiple-valued logic, Fuzzy Sets and Systems, 24 (1987) 263-278. [10] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A., 27 (1941), 222{224. [11] K. Jun, H. Kim and J. M. Rassias, Extended Hyers{Ulam stability for Cauchy{Jensen map- pings, J. Dierence Equ. Appl., 13 (2007), 1139{1153. [12] K. Jun and Y. Lee, On the Hyers{Ulam{Rassias stability of a Pexiderized quadratic inequal- ity, Math. Inequal. Appl., 4 (2001), 93{118. [13] S. M. Jung, Hyers{Ulam{Rassias stability of functional equations in nonlinear analysis, Springer Science, New York, 2011. [14] O. Kaleva and S. Seikkala, On fuzzy metric spaces, Fuzzy Sets and Systems, 12 (1984), 215-229. [15] O. Kaleva, The completion of fuzzy metric spaces, J. Math. Anal. Appl., 109 (1985), 194-198. [16] O. Kaleva, A comment on the completion of fuzzy metric spaces, Fuzzy Sets and Systems, 159(16) (2008), 2190-2192. [17] P. Kannappan, Functional equations and inequalities with applications, Springer Science, New York, 2009. [18] R. Lowen, Fuzzy set theory, Ch. 5 : Fuzzy Real Numbers, Kluwer, Dordrecht, 1996. [19] A. Maturo, On some structures of fuzzy numbers, Iranian Journal of Fuzzy Systems, 6 (2009), 49{59. [20] A. K. Mirmostafaee and M. S. Moslehian, Fuzzy version of Hyers-Ulam-Rassias theorem, Fuzzy Sets and Systems, 159(6) (2008), 720-729.

[21] F. Moradlou, H. Vaezi and C. Park, Fixed points and stability of an additive functional equation of n-Apollonius type in C-algebras, Abstract and Applied Analysis, 2008, Article ID 672618, 13 pages, 2008. doi:10.1155/2008/672618. [22] F. Moradlou, A. Najati and H. Vaezi, Stability of homomorphisms and derivations on C- ternary rings associated to an Euler{Lagrange type additive mapping, Result. Math., 55 (2009), 469-486. [23] M. S. Moslehian, On the orthogonal stability of the Pexiderized quadratic equation, J. Dier- ence Equ. Appl., 11 (2005), 999{1004. [24] C. Park, Modied Trif 's functional equations in Banach modules over a C-algebra and approximate algebra homomorphisms, J. Math. Anal. Appl., 278 (2003), 93{108. [25] C. Park, On an approximate automorphism on a C-algebra, Proc. Amer. Math. Soc., 132 (2004), 1739{1745. [26] C. Park and T. M. Rassias, Hyers{Ulam stability of a generalized Apollonius type quadratic mapping, J. Math. Anal. Appl., 322 (2006), 371{381. [27] J. M. Rassias, On approximation of approximately linear mappings by linear mappings, J. Funct. Anal., 46 (1982), 126{130. [28] J. M. Rassias, On approximation of approximately linear mappings by linear mappings, Bull. Sci. Math., 108 (1984), 445{446. [29] J. M. Rassias, Solution of a problem of Ulam, J. Approx. Theory, 57 (1989), 268{273. [30] T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297{300. [31] T. M. Rassias, Problem 16; 2, Report of the 27th International Symp. on Functional Equa- tions, Aequationes Math., 39 (1990), 292{293. [32] T. M. Rassias, The problem of S.M. Ulam for approximately multiplicative mappings, J. Math. Anal. Appl., 246 (2000), 352{378. [33] T. M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl., 251 (2000), 264{284. [34] T. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math., 62 (2000), 23{130. [35] S. E. Rodabaugh, Fuzzy addition in the L-fuzzy real line, Fuzzy Sets and Systems, 8 (1982) 3951. [36] I. Sadeqi and M. Salehi, Fuzzy compacts operators and topological degree theory , Fuzzy Sets and Systems, 160(9) (2009), 1277-1285. [37] F. Skof, Local properties and approximations of operators, Rend. Sem. Mat. Fis. Milano, 53 (1983), 113{129. [38] S. M. Ulam, A collection of the mathematical problems, Interscience Publ. New York, 1960. [39] J. Xiao and X. Zhu, On linearly topological structure and property of fuzzy normed linear space, Fuzzy Sets and Systems, 125 (2002), 153-161. [40] J. Xiao and X. Zhu, Topological degree theory and xed point theorems in fuzzy normed space, Fuzzy Sets and Systems, 147 (2004), 437-452.