Non-Newtonian Fuzzy numbers and \related applications

Document Type : Research Paper


Department of Mathematics, Bozok University, Yozgat, Turkey


Although there are many excellent ways presenting the principle of the classical calculus, the novel presentations probably leads most naturally to the development of the non-Newtonian calculus. The important point to note is that the non-Newtonian calculus is a self-contained system independent of any other system of calculus. Since this self-contained work is intended for a wide audience, including engineers, scientists and mathematicians. The main purpose of the present paper is to construct of fuzzy numbers with respect to the non-Newtonian calculus and is to give the necessary and sufficient conditions according to the generalization of the notion of fuzzy numbers by using the generating functions. Also we introduce the concept of non-Newtonian fuzzy distance and give some properties regarding convergence of sequences and series of fuzzy numbers with some illustrative examples.


[1] A. E. Bashirov, E. Kurpnar and A. Ozyapc, Multiplicative calculus and its applications, J.
Math. Anal. Appl., 337(1) (2008), 36-48.
[2] A. F. Cakmak and F. Basar, Some new results on sequence spaces with respect to non-
Newtonian calculus, J. Inequal. Appl., 2012(1) (2012), 228.
[3] M. Grossman and R. Katz, Non-Newtonian Calculus, Lee Press, Pigeon Cove, MA, 1972.
[4] M. Grossman, Bigeometric Calculus, Archimedes Foundation, Rockport, Mass, USA, 1983.
[5] M. Grossman, The First Nonlinear System of Di erential and Integral Calculus, Mathco,
[6] U. Kadak and H. Efe, Matrix transformations between certain sequence spaces over the non-
Newtonian complex eld, Sci. World J., 2014 (2014).
[7] U. Kadak and M. Ozluk, Generalized Runge-Kutta method with respect to the non-Newtonian
calculus, Abstr. Appl. Anal., 2014 (2014).
[8] U. Kadak and F. Basar, On Fourier series of fuzzy-valued function, Sci. World J., 2014
[9] M. Matloka, Sequences of fuzzy numbers, BUSEFAL, 28 (1986), 28-37.
[10] E. Msrl, and Y. Gurefe, Multiplicative Adams-Bashforth-Moulton methods, Numer. Algorithms,
57(4) (2011), 425-439.
[11] M. Stojakovic and Z. Stojakovic, Series of fuzzy sets, Fuzzy Sets Syst., 160(21) (2009),
[12] M. Stojakovic and Z. Stojakovic, Addition and series of fuzzy sets, Fuzzy Sets Syst., 83(3)
(1996), 341{346.
[13] O. Talo and F. Basar, Determination of the duals of classical sets of sequences of fuzzy
numbers and related matrix transformations, Comput. Math. Appl., 58(4) (2009), 717-733.
[14] O. Talo and F. Basar, Quasilinearity of the classical sets of sequences of the fuzzy numbers
and some applications, Taiwanese J. Math., 14(5) (2010), 1799-1819.
[15] S. Tekin and F. Basar, Certain sequence spaces over the non-Newtonian complex eld, Abstr.
Appl. Anal., 2012 (2013).
[16] L. A. Zadeh, Fuzzy sets, Inform. and Control, 8 (1965), 338{353.