Document Type : Research Paper


Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran


In this paper, we define the notion of (dual) multi-fuzzy normed
spaces and describe some properties of them.
 We then investigate Ulam-Hyers stability of Jensen's functional equation for mappings from linear spaces into
  multi-fuzzy normed spaces. We establish an asymptotic behavior of the Jensen equation in the framework of multi-fuzzy normed spaces.


[1] A. Alotaibi, M. Mursaleen, H. Dutta and S. A. Mohiuddine, On the Ulam stability of Cauchy
functional equation in IFN-spaces, Appl. Math. Inf. Sci. 8(3) (2014), 1135{1143.
[2] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan,
2 (1950), 64{66.
[3] T. Bag and S. K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math.
11(3) (2003), 687{705.
[4] T. Bag and S. K. Samanta, Fuzzy bounded linear operators, Fuzzy Sets and Syst. 151 (2005),
[5] S. C. Cheng and J. N. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces,
Bull. Calcutta Math. Soc., 86(5) (1994), 429{436.
[6] H. G. Dales and M. E. Polyakov, Multi-normed spaces and multi-Banach algebras, University
of Leeds, preprint, (2008), 1{156.
[7] C. Felbin, Finite-dimensional fuzzy normed linear space, Fuzzy Sets and Syst., 48(2) (1992),
[8] A. Gha ari and A. Alinejad, Stabilities of cubic mappings in fuzzy normed spaces, Adv.
Di erence Equ., 2010 (2010), 1{15.
[9] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A.,
27 (1941), 222{224.
[10] S. M. Jung, Hyers-Ulam-Rassias stability of Jensens equation and its application, Proc.
Amer. Math. Soc., 126 (1998), 3137{3143.
[11] A. K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets and Syst., 12(2) (1984), 143{
[12] I. Kramosil and J. Michalek, Fuzzy metrics and statistical metric spaces, Kybernetika, 11(5)
(1975), 336-344.
[13] S. V. Krishna and K. K. M. Sarma, Separation of fuzzy normed linear spaces, Fuzzy Sets and
Syst., 63(2) (1994), 207{217.
[14] L. Li, J. Chung and D. Kim, Stability of Jensen equations in the space of generalized func-
tions, J. Math. Anal. Appl., 299 (2004), 578{586.
[15] T. Li, A. Zada and S. Faisal, Hyers-Ulam stability of n-th order linear di erential equations,
J. Nonlinear Sci. Appl., 9(5) (2016), 2070{2075.
[16] A. K. Mirmostafaee, Perturbation of generalized derivations in fuzzy Menger normed alge-
bras, Fuzzy Sets and Syst., 195 (2012), 109-117.
[17] A. K. Mirmostafaee, M. Mirzavaziri and M. S. Moslehian, Fuzzy stability of the Jensen
functional equation, Fuzzy Sets and Syst., 159(6) (2008), 730{738.
[18] M. S. Moslehian and H. M. Sirvastava, Jensen's functional equation in multi-normed spaces,
Taiwanese J. Math., 14(2) (2010), 453{462.
[19] E. Movahednia, S. Eshtehar and Y. Son, Stability of quadratic functional equations in fuzzy
normed spaces, Int. J. Math. Anal., 6(48) (2012), 2405{2412.
[20] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math.
Soc., 72 (1978), 297{300.
[21] Y. Shen, An integrating factor approach to the Hyers-Ulam stability of a class of exact
di erential equations of second order, J. Nonlinear Sci. Appl., 9(5) (2016), 2520{2526.
[22] S. M. Ulam, A collection of mathematical problems, Interscience Publ., New York, 1960.
[23] S. M. Vaezpour and F. Karimi, t-Best approximation in fuzzy normed spaces, Iranian Journal
of Fuzzy Systems, 5(2) (2008), 93{99.
[24] J. Z. Xiao and X. H. Zhu, Fuzzy normed space of operators and its completeness, Fuzzy Sets
and Syst., 133(3) (2003), 389{399.
[25] B. Yood, On the non-existence of norms for some algebras of functions, Stud. Math., 111(1)
(1994), 97-101.