On the Behavior of Solutions of the System of Rational Difference Equations x<sub>n+1</sub>=x<sub>n-1</sub>/y<sub>n</sub>x<sub>n-1</sub>-1, y<sub>n+1</sub>=y<sub>n-1</sub>/x<sub>n</sub>y<sub>n-1</sub>-1, z<sub>n+1</sub>=x<sub>n</sub>/y<sub>n</sub>z<sub>n-1</sub> — Oak Academic Publishing
Research ArticleOpen AccessGoogle Scholar indexed
On the Behavior of Solutions of the System of Rational Difference Equations x<sub>n+1</sub>=x<sub>n-1</sub>/y<sub>n</sub>x<sub>n-1</sub>-1, y<sub>n+1</sub>=y<sub>n-1</sub>/x<sub>n</sub>y<sub>n-1</sub>-1, z<sub>n+1</sub>=x<sub>n</sub>/y<sub>n</sub>z<sub>n-1</sub>
In this paper, we investigate the solutions of the system of difference equations x<sub>n+1</sub>=x<sub>n-1</sub>/y<sub>n</sub>x<sub>n-1</sub>-1,y<sub>n+1</sub>=y<sub>n-1</sub>/x<sub>n</sub>y<sub>n-1</sub>-1,z<sub>n+1</sub>=x<sub>n</sub>/y<sub>n</sub>z<sub>n-1</sub> where x<sub>0</sub>,x<sub>-1</sub>,y<sub>0</sub>,y<sub>-1</sub>,z<sub>0</sub>,z<sub>-1</sub>∈R.
A. S. Kurbanli, C. Cinar and I. Yalcinkaya, “On the Behavaior of Positive Solutions of the System of Rational Difference Equations x n+1 =x n-1 /y n x n-1 +1 y n+1 =y n-1 /x n y n-1 +1 ,” Mathematical and Computer Modelling, Vol. 53, No. 5-6, 2011, pp. 1261-1267. doi:10.1016/j.mcm.2010.12.009
A. S. Kurbanli, “On the Behavior of Solutions of the System of Rational Difference Equations x n+1 =x n-1 /y n x n-1 -1,y n+1 =y n-1 /x n y n-1 -1 ,” World Applied Sciences Journal, 2010, in Press.
A. S. Kurbanli, “On the Behavior of Solutions of the System of Rational Difference Equations x n+1 =x n-1 /y n x n-1 -1 , y n+1 =y n-1 /x n y n-1 -1 , z n+1 =z n-1 /y n z n-1 -1 ,” Discrete Dynamics in Nature and Society, 2011, in Press. doi:10.1155/2011/932362
C. Cinar, “On the Solutions of the Difference Equation x n+1 =x n-1 /-1+ax n x n-1 ,” Applied Mathematics and Com- pulation, Vol. 158, 2004, pp. 793-797.
C. ?inar, “On the Positive Solutions of the Difference Equation Systemx n+1 =1/y n ,y n+1 =y n /x n-1 y n-1 ,” Applied Mathematics and Computation, Vol. 158, 2004, pp. 303-305. doi:10.1016/j.amc.2003.08.073
G. Papaschinopoulos and C. J. Schinas, “On a System of Two Nonlinear Difference Equations,” Journal of Mathematical Analysis and Applications, Vol. 219, No. 1998, pp. 415-426. doi:10.1006/jmaa.1997.5829
G. Papaschinopoulos and C. J. Schinas, “On the System of Two Difference Equations,” Journal of Mathematical Analysis and Applications, Vol. 273, No. 2, 2002, pp. 294-309. doi:10.1016/S0022-247X(02)00223-8
A. Y. ?zban, “On the System of Rational Difference Equations x n =a/y n-3 ,y n+1 =by n-3 /x n-q y n-q ,” Applied Mathe- matics and Computation, Vol. 188, No. 1, 2007, pp. 833-837. doi:10.1016/j.amc.2006.10.034
D. Clark and M. R. S. Kulenovi?, “A Coupled System of Rational Difference Equations,” Computers & Mathe- matics with Applications, Vol. 43, 2002, pp. 49-867.
D. Clark, M. R. S. Kulenovi? and J. F. Selgrade, “Global Asymptotic Behavior of a Two-Dimensional Diserence Equation Modelling Competition,” Nonlinear Analysis, Vol. 52, No. 7, 2003, pp. 1765-1776. doi:10.1016/S0362-546X(02)00294-8
E. Camouzis and G. Papaschinopoulos, “Global Asymptotic Behavior of Positive Solutions on the System of Rational Difference Equations x n+1 =1+x n /y n-m ,y n+1 =1+y n /x n-m ,” Applied Mathematics Letters, Vol. 17, No. 6, 2004, pp. 733-737. doi:10.1016/S0893-9659(04)90113-9
X. Yang, Y. Liu and S. Bai, “On the System of High Order Rational Difference Equations x n =a/y n-p ,y n =by n-p /x n-q y n-q ,” Applied Mathematics and Computation, Vol. 171, No. 2, 2005, pp. 853-856. doi:10.1016/j.amc.2005.01.092
X. Yang, “On the System of Rational Difference Equations x n =A+y n-1 /x n-p y n-p ,y n =A+x n-1 /x n-r y n-s ,” Journal of Mathe- matical Analysis and Applications, Vol. 307, No. 1, 2005, pp. 305-311. doi:10.1016/j.jmaa.2004.10.045
M. R. S. Kulenovi? and Z. Nurkanovi?, “Global Behavior of a Three-Dimensional Linear Fractional System of Difference Equations,” Journal of Mathematical Analysis and Applications, Vol. 310, No. 2, 2005, pp. 673-689.
A. Y. ?zban, “On the Positive Solutions of the System of Rational Difference Equations x n+1 =1/y n-k ,y n+1 =y n /x n-m y n-m-k ,” Journal of Mathematical Analysis and Applications, Vol. 323, No. 1, 2006, pp. 26-32. doi:10.1016/j.jmaa.2005.10.031
Y. Zhang, X. Yang, G. M. Megson and D. J. Evans, “On the System of Rational Difference Equations x n =A+1/y n-p ,y n =A+y n-1 /x n-r y n-s ,” Applied Mathematics and Computation, Vol. 176, No. 2, 2006, pp. 403-408. doi:10.1016/j.amc.2005.09.039
Y. Zhang, X. Yang, D. J. Evans and C. Zhu, “On the Nonlinear Difference Equation System x n+1 =A+y n-m /x n ,y n+1 =A+x n-m /y n ,” Computers & Mathematics with Appli- cations, Vol. 53, No. 10, 2007, pp. 1561-1566.
I. Yalcinkaya and C. Cinar, “Global Asymptotic Stability of two nonlinear Difference Equations z n+1 =t n z n-1 +a/t n +z n-1 ,t n+1 =z n t n-1 +a/z n +t n-1 ,” Fasciculi Mathematici, Vol. 43, 2010, pp. 171-180.
E. M. Elsayed, “On the Solutions of Higher Order Rational System of Recursive Sequences,” Mathematica Balkanica, Vol. 21, No. 3-4, 2008, pp. 287-296.
I. Yalcinkaya, “On the Global Asymptotic Stability of a Second-Order System of Difference Equations,” Discrete Dynamics in Nature and Society, 2008, Article ID 860152, 12 Pages.
E. M. Elabbasy, H. EI-Metwally and E. M. Elsayed, “On the Solutions of a Class of Difference Equations Systems,” Demonstratio Mathematica, Vol. 41, No. 1, 2008, pp. 109-122.
E. M. Elsayed, “Dynamics of a Recursive Sequence of Higher Order,” Communications on Applied Nonlinear Analysis, Vol. 16, No. 2, 2009, pp. 37-50.
R. Abu-Saris, C. Cinar and I. Yalcinkaya, “On the Asym- ptotic Stability of x n+1 =a+x n x n-k /x n +x n-k ,” Computers & Mathematics with Applications, Vol. 56, No. 5, 2008, pp. 1172-1175. doi:10.1016/j.camwa.2008.02.028
R. P. Agarwal, W. T. Li and P. Y. H. Pang, “Asymptotic Behavior of a Class of Nonlinear Delay Difference Equa- tions,” Journal of Difference Equations and Applications, Vol. 8, 2002, pp. 719-728. doi:10.1080/1023619021000000735
R. P. Agarwal, “Difference Equations and Inequalites,” 2nd Editions, Marcel Dekker, New York, 2000.
I. Yalcinkaya, C. Cinar and D. Simsek, “Global Asymp- totic Stability of a System of Difference Equations,” Applicable Analysis, Vol. 87, No. 6, June 2008, pp. 689- 699. doi:10.1080/00036810802140657
E. M. Elsayed, “On the Solutions of a Rational System of Difference Equations,” Fasciculi Mathematici, Vol. 45 2010, pp. 25-36.
B. Iri?anin and S. Stevi?, “Some Systems of Nonlinear Difference Equations of Higher Order with Periodic Solutions,” Dynamics of Continuous, Discrete and Impulsive Systems. Series A Mathematical Analysis, Vol. 13, No. 3-4, 2006, pp. 499-507.