Document Type : Research Paper


Department of Mathematics, Institute for Advanced Studies in, Basic Sciences (IASBS), Zanjan 45137-66731, Iran


In this study, we consider two different inequivalent formulations of the logistic
difference equation $x_{n+1}= \beta x_n(1- x_n),\ \ n=0,1,...,
$ where $x_n$ is a sequence of fuzzy numbers and $\beta$ is a positive fuzzy number. The major contribution of this paper is to study the existence, uniqueness and global behavior of the solutions for two corresponding equations, using the concept of Hukuhara difference for fuzzy numbers. Finally, some examples are given to illustrate our results.


[1] N. Bacaer, A Short History of Mathematical Population Dynamics, Springer, Bondy, 2011.
[2] B. Bede, Mathematics of fuzzy sets and fuzzy logic, Springer, Berlin, 2013.
[3] T. G. Bhaskar, V. Lakshmikantham and V. Devi, Revisiting fuzzy differential equations,
Nonlinear Anal., 58 (2004) 351-358.
[4] E. Deeba, A. Korvin and E. L. Koh, A fuzzy difference equation with and Application, Journal
of Difference Equations and applications, 2 (1996), 365-374.
[5] P. Diamond and P. Kloeden, Metric Spaces of Fuzzy Sets, World Scientific, Singapore, 1994.
[6] S. N. Elaydi, An introduction to difference equations, Springer, Texas, 1995.
[7] A. Khastan, New solutions for first order linear fuzzy difference equations, Journal of Computational
and Applied Mathematics, 312 (2017), 156-166.
[8] V. Lakshmikantham and D. Trigiante, Theory of Difference Equations Numerical Methods
and Applications, Marcel Dekker, New York, 2002.
[9] J. E. Macias-Diaz and S. Tomasiello, A differential quadrature-based approach a la Picard
for systems of partial differential equations associated with fuzzy differential equations ,
http://dx.doi.org/10.1016/j.cam.2015.08.009, (2015).
[10] R. May, Simple mathematical models with very complicated dynamics, Nature, 261 (1976),
[11] H. E. Metwally, E. A. Grove, G. Ladas, R. Levins and M. Radin, On the difference equation
xn+1 = + xn􀀀1e􀀀xn, Nonlinear Anal., 47 (2001), 4623-4634.
[12] J. J. Nieto and R. Rodriguez-Lopez, Analysis of a logistic differential model with uncertainty,
International Journal of Dynamical Systems and Differential Equations, DOI:
10.1504/IJDSDE.2008.019678, 1(3) (2008).
[13] J. J. Nieto, M. V. Otero-Espinar and R. Rodriguez-Lopez, Dynamics of the Fuzzy Logistic
Family, Discrete and Continuous Dynamical Systems: Series B, 14 (2010), 699-717.
[14] G. Papaschinopoulos and G. Stefanidou, Boundedness and asymptotic behaviour of the solution
of a fuzzy difference equation, Fuzzy Sets and Systems, 140 (2003), 523-539.
[15] C. G. Philos, I. K. Purnaras and Y. G. Sficas, Global attractivity in a nonlinear difference
equation, Appl. Math. Comput, 62 (1994), 249-258.
[16] Q. Zhang, W.Zhang, J. Liu and Y. Shao, On a fuzzy logistic difference equation, Wseas
transaction on mathematics, 13 (2014), 282-285.