Dynamics of the fuzzy difference equation \(z_n =\max\{\frac{ 1}{ z_{n-m}} , \frac{\alpha_n }{z_{n-r} }\}\)
Volume 11, Issue 4, pp 477--485
http://dx.doi.org/10.22436/jnsa.011.04.04
Publication Date: March 16, 2018
Submission Date: February 04, 2017
Revision Date: November 25, 2017
Accteptance Date: January 11, 0018
-
2299
Downloads
-
4792
Views
Authors
Taixiang Sun
- Guangxi Key Laboratory Cultivation Base of Cross-border E-commerce Intelligent Information Processing, Guangxi Univresity of Finance and Economics, Nanning, 530003, China.
Hongjian Xi
- Guangxi Key Laboratory Cultivation Base of Cross-border E-commerce Intelligent Information Processing, Guangxi Univresity of Finance and Economics, Nanning, 530003, China.
Guangwang Su
- Guangxi Key Laboratory Cultivation Base of Cross-border E-commerce Intelligent Information Processing, Guangxi Univresity of Finance and Economics, Nanning, 530003, China.
Bin Qin
- Guangxi Key Laboratory Cultivation Base of Cross-border E-commerce Intelligent Information Processing, Guangxi Univresity of Finance and Economics, Nanning, 530003, China.
Abstract
In this paper, we study the eventual periodicity of the following
fuzzy max-type difference equation
\[z_n=\max\{\frac{1}{z_{n-m}},\frac{\alpha_n}{z_{n-r}}\},\ \
n=0,1,\ldots,\] where \(\{\alpha_n\}_{n\geq 0}\) is a periodic
sequence of positive fuzzy numbers and the initial values
\(z_{-d},z_{-d+1},\ldots,z_{-1}\)
are positive fuzzy numbers with
\(d=\max\{m,r\}\). We show that if
\(\max(\mbox{supp}\ \alpha_n)<1\), then every positive solution of
this equation is eventually periodic with period \(2m\).
Share and Cite
ISRP Style
Taixiang Sun, Hongjian Xi, Guangwang Su, Bin Qin, Dynamics of the fuzzy difference equation \(z_n =\max\{\frac{ 1}{ z_{n-m}} , \frac{\alpha_n }{z_{n-r} }\}\), Journal of Nonlinear Sciences and Applications, 11 (2018), no. 4, 477--485
AMA Style
Sun Taixiang, Xi Hongjian, Su Guangwang, Qin Bin, Dynamics of the fuzzy difference equation \(z_n =\max\{\frac{ 1}{ z_{n-m}} , \frac{\alpha_n }{z_{n-r} }\}\). J. Nonlinear Sci. Appl. (2018); 11(4):477--485
Chicago/Turabian Style
Sun, Taixiang, Xi, Hongjian, Su, Guangwang, Qin, Bin. "Dynamics of the fuzzy difference equation \(z_n =\max\{\frac{ 1}{ z_{n-m}} , \frac{\alpha_n }{z_{n-r} }\}\)." Journal of Nonlinear Sciences and Applications, 11, no. 4 (2018): 477--485
Keywords
- Fuzzy max-type difference equation
- positive solution
- eventual periodicity
MSC
References
-
[1]
K. A. Chrysafis, B. K. Papadopoulos, G. Papaschinopoulos , On the fuzzy difference equations of finance , Fuzzy Sets and Systems, 159 (2008), 3259–3270
-
[2]
E. Hatir, T. Mansour, I. Yalçinkaya, On a fuzzy difference equation, Util. Math., 93 (2014), 135–151.
-
[3]
Q. He, C. Tao, T. Sun, X. Liu, D. Su, Periodicity of the positive solutions of a fuzzy max-difference equation, Abstr. Appl. Anal., 2014 (2014 ), 4 pages.
-
[4]
R. HorĨík, Solution of a system of linear equations with fuzzy numbers, Fuzzy Sets and Systems, 159 (2008), 1788–1810.
-
[5]
R. Kargar, T. Allahviranloo, M. Rostami-Malkhalifeh, G. R. Jahanshaloo, A proposed method for solving fuzzy system of linear equations, Sci. World J., 2014 (2014 ), 6 pages.
-
[6]
G. J. Klir, B. Yuan , Fuzzy sets and fuzzy logic, Prentice-Hall PTR, New Jersey (1995)
-
[7]
V. Lakshmikantham, A. S. Vatsala, Basic theory of fuzzy difference equations, J. Difference Equ. Appl., 8 (2002), 957–968.
-
[8]
H. T. Nguyen, E. A. Walker, A first course in fuzzy logic, CRC Press, Florida (1997)
-
[9]
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.
-
[10]
G. Papaschinopoulos, B. K. Papadopoulos , On the fuzzy difference equation \(x_{n+1} = A + B/x_n\), Soft Comput., 6 (2002), 456–461.
-
[11]
G. Papaschinopoulos, G. Stefanidou, Boundedness and asymptotic behavior of the solutions of a fuzzy difference equation, Fuzzy Sets and Systems, 140 (2003), 523–539.
-
[12]
G. Stefanidou, G. Papaschinopoulos , A fuzzy difference equation of a rational form, J. Nonlinear Math. Phys., 12 (2005), 300–315.
-
[13]
G. Stefanidou, G. Papaschinopoulos , Behavior of the positive solutions of fuzzy max- difference equations, Adv. Difference Equ., 2005 (2005), 153–172.
-
[14]
G. Stefanidou, G. Papaschinopoulos , The periodic nature of the positive solutions of a nonlinear fuzzy max-difference equation, Inform. Sci., 176 (2006), 3694–3710.
-
[15]
G. Stefanidou, G. Papaschinopoulos, C. J. Schinas, On an exponential-type fuzzy difference equation, Adv. Difference Equ., 2010 (2010), 19 pages.
-
[16]
C. Wu, B. Zhang, Embedding problem of noncompact fuzzy number space E~(I), Fuzzy Sets and Systems, 105 (1999), 165–169.
-
[17]
Q. H. Zhang, J. Liu , The first order fuzzy difference equation \(x_{n+1} = Ax_n + B\) , (Chinese), Mohu Xitong yu Shuxue, 23 (2009), 74–79.
-
[18]
Q. H. Zhang, J. Liu, Z. Luo, Dynamical behavior of a third-order rational fuzzy difference equation, Adv. Difference Equ., 2015 (2015), 18 pages.
-
[19]
Q. H. Zhang, L. Yang, D. Liao , On the fuzzy difference equation \(x_{n+1} = A + \sum^k_{i =0} B/x_{n-i }\), International J. Math. Comput. Phys. Elect. Comput. Eng., 5 (2011), 490–495.
-
[20]
Q. H. Zhang, L. Yang, D. Liao, Behavior of solutions to a fuzzy nonlinear difference equation, Iran J. Fuzzy Sys., 9 (2012), 1–12.
-
[21]
Q. H. Zhang, L. Yang, D. Liao , On first order fuzzy Ricatti difference equation, Inform. Sci., 270 (2014), 226–236.