On the dynamics of a five-order fuzzy difference equation

Volume 10, Issue 6, pp 3303--3319 http://dx.doi.org/10.22436/jnsa.010.06.40

Publication Date: June 25, 2017 Submission Date: March 23, 2017

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Authors

Changyou Wang - College of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, P. R. China. - School of Applied Mathematics, Chengdu University of Information Technology, Chengdu, Sichuan 610225, P. R. China. Xiaolin Su - College of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, P. R. China. Ping Liu - College of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, P. R. China. Xiaohong Hu - College of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, P. R. China. Rui Li - College of Automation, Chongqing University of Posts and Telecommunications, Chongqing 400065, P. R. China.

Abstract

Our aim in this paper is to investigate the existence and uniqueness of the positive solutions and the asymptotic behavior of the equilibrium points of the fuzzy difference equation \[x_{n+1}=\frac{Ax_{n-1}x_{n-2}}{D+Bx_{n-3}+Cx_{n-4}}, n=0,1,2,...,\] where \(x_n\) is a sequence of positive fuzzy numbers, the parameters \(A, B, C, D\) and the initial conditions \(x_{-4}, x_{-3}, x_{-2}, x_{-1}, x_0\) are positive fuzzy numbers. Moreover, some numerical examples to the difference system are given to verify our theoretical results.

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Keywords

• Fuzzy difference equation
• existence
• uniqueness
• equilibrium point
• asymptotic behavior.

MSC

•  39A11
•  03E72

References

• [1] T. Allahviranloo, S. Salahshour, M. Khezerloo, Maximal- and minimal symmetric solutions of fully fuzzy linear systems, J. Comput. Appl. Math., 235 (2011), 4652–4662.
  • View Article
  • Google Scholar
  • MATH


• [2] B. Bede, Mathematics of fuzzy sets and fuzzy logic, Studies in Fuzziness and Soft Computing, Springer, Heidelberg (2013)
  • View Article
  • MathSciNet


• [3] E. Camouzis, G. Ladas, Dynamics of third-order rational difference equations with open problems and conjectures, Advances in Discrete Mathematics and Applications, Chapman & Hall/CRC, Boca Raton, FL (2008)
  • MathSciNet
  • Google Scholar


• [4] K. A. Chrysafis, B. K. Papadopoulos, G. Papaschinopoulos, On the fuzzy difference equations of finance, Fuzzy Sets and Systems, 159 (2008), 3259–3270.
  • View Article
  • MathSciNet
  • Google Scholar


• [5] E. Y. Deeba, A. De Korvin, Analysis by fuzzy difference equations of a model of \(CO_2\) level in the blood, Appl. Math. Lett., 12 (1999), 33–40.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [6] E. Y. Deeba, A. De Korvin, E. L. Koh, A fuzzy difference equation with an application, J. Differ. Equations Appl., 2 (1996), 365–374.
  • View Article
  • MathSciNet
  • Google Scholar


• [7] E. Y. Deeba, A. De Korvin, E. L. Koh, On a fuzzy logistic difference equation, Differential Equations Dynam. Systems, 4 (1996), 149–156.
  • MathSciNet
  • Google Scholar


• [8] P. Diamond, P. Kloeden, Metric spaces of fuzzy sets, Theory and applications, World Scientific Publishing Co., Inc., River Edge, NJ (1994)
  • Google Scholar


• [9] E. M. Elsayed, Solutions of rational difference systems of order two, Math. Comput. Modelling, 55 (2012), 378–384.
  • View Article
  • MathSciNet
  • Google Scholar


• [10] E. M. Elsayed, On the solutions and periodic nature of some systems of difference equations, Int. J. Biomath., 7 (2014), 26 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [11] A. Ghomashi, S. Salahshour, A. Hakimzadeh, Approximating solutions of fully fuzzy linear systems: a financial case study, J. Intell. Fuzzy Systems, 26 (2014), 367–378.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [12] E. Hatir, T. Mansour, I. Yalçinkaya, On a fuzzy difference equation, Util. Math., 93 (2014), 135–151.
  • MathSciNet


• [13] A. Khastan, New solutions for first order linear fuzzy difference equations, J. Comput. Appl. Math., 312 (2017), 156–166.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [14] V. L. Kocić, G. Ladas, Global behavior of nonlinear difference equations of higher order with applications, Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht (1993)
  • View Article
  • MathSciNet
  • Google Scholar


• [15] V. Lakshmikantham, R. N. Mohapatra, Theory of fuzzy differential equations and inclusions, Series in Mathematical Analysis and Applications, Taylor & Francis, Ltd., London (2003)
  • View Article
  • MathSciNet
  • Google Scholar


• [16] V. Lakshmikantham, A. S. Vatsala, Basic theory of fuzzy difference equations, Special issue in honour of Professor Allan Peterson on the occasion of his 60th birthday, Part I, J. Difference Equ. Appl., 8 (2002), 957–968.
  • View Article
  • MathSciNet
  • Google Scholar


• [17] J. N. Mordeson, P. S. Nair, Fuzzy mathematics, An introduction for engineers and scientists, Second edition, With a foreword by Azriel Rosenfeld, Studies in Fuzziness and Soft Computing, Physica-Verlag, Heidelberg (2001)


• [18] M. Pakdaman, S. Effati, On fuzzy linear projection equation and applications, Fuzzy. Optim. Decis. Mak., 15 (2016), 219–236.
  • View Article
  • MathSciNet
  • Google Scholar


• [19] G. Papaschinopoulos, B. K. Papadopoulos, On the fuzzy difference equation \(x_{n+1} = A + x_n/x_{n-m}\), Fuzzy Sets and Systems, 129 (2002), 73–81.
  • View Article
  • MathSciNet
  • MATH


• [20] G. Papaschinopoulos, G. Stefanidou, Boundedness and asymptotic behavior of the solutions of a fuzzy difference equation, Theme: Topology, Fuzzy Sets and Systems, 140 (2003), 523–539.
  • View Article
  • MathSciNet
  • Google Scholar


• [21] E. C. Pielou, Population and community ecology: principles and methods, CRC Press, London (1974)
  • Google Scholar
  • MATH


• [22] E. P. Popov, Automatic regulation and control, (Russian) Nauka, Moscow (1966)
  • Google Scholar


• [23] H. Sedaghat, Nonlinear difference equations, Theory with applications to social science models, Mathematical Modelling: Theory and Applications, Kluwer Academic Publishers, Dordrecht (2003)
  • Google Scholar


• [24] G. Stefanidou, G. Papaschinopoulos, The periodic nature of the positive solutions of a nonlinear fuzzy max-difference equation, Inform. Sci., 176 (2006), 3694–3710.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [25] S. Stević, Behavior of the positive solutions of the generalized Beddington-Holt equation, PanAmer. Math. J., 10 (2000), 77–85.
  • MathSciNet
  • Google Scholar
  • MATH


• [26] S. Stević, Global stability and asymptotics of some classes of rational difference equations, J. Math. Anal. Appl., 316 (2006), 60–68.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [27] C.-Y. Wang, X.-J. Fang, R. Li, On the solution for a system of two rational difference equations, J. Comput. Anal. Appl., 20 (2016), 175–186.
  • MathSciNet
  • Google Scholar


• [28] C.-Y. Wang, X.-J. Fang, R. Li, On the dynamics of a certain four-order fractional difference equations, J. Comput. Anal. Appl., 22 (2017), 968–976.
  • MathSciNet
  • Google Scholar


• [29] C.-Y.Wang, S.Wang,W.Wang, Global asymptotic stability of equilibrium point for a family of rational difference equations, Appl. Math. Lett., 24 (2011), 714–718.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [30] C.-Y.Wang, S.Wang, Z.-W.Wang, F. Gong, R.-F.Wang, Asymptotic stability for a class of nonlinear difference equations, Discrete Dyn. Nat. Soc., 2010 (2010), 10 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [31] C.-X.Wu, B.-K. Zhang, Embedding problem of noncompact fuzzy number space E, Fuzzy Sets and Systems, 105 (1999), 165–169.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [32] Q.-H. Zhang, J.-Z. Liu, Z.-G. Luo, Dynamical behavior of a third-order rational fuzzy difference equation, Adv. Difference Equ., 2015 (2015), 18 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [33] Q.-H. Zhang, L.-H. Yang, D.-X. Liao, Behavior of solutions to a fuzzy nonlinear difference equation, Iran. J. Fuzzy Syst., 9 (2012), 1–12.
  • MathSciNet
  • Google Scholar
  • MATH


• [34] Q.-H. Zhang, L.-H. Yang, D.-X. Liao, On first order fuzzy Ricatti difference equation, Inform. Sci., 270 (2014), 226–236.
  • View Article
  • MathSciNet
  • Google Scholar