On the Behavior of Solutions of the System of Rational Difference Equations xn+1=xn-1/ynxn-1-1, yn+1=yn-1/xnyn-1-1, zn+1=xn/ynzn-1
Abdullah Selçuk Kurbanlı, Cengiz Çinar, Mehmet Emre Erdoğan
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DOI: 10.4236/am.2011.28143   PDF    HTML     4,603 Downloads   9,849 Views   Citations

Abstract

In this paper, we investigate the solutions of the system of difference equations xn+1=xn-1/ynxn-1-1,yn+1=yn-1/xnyn-1-1,zn+1=xn/ynzn-1 where x0,x-1,y0,y-1,z0,z-1∈R.

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A. Kurbanlı, C. Çinar and M. Erdoğan, "On the Behavior of Solutions of the System of Rational Difference Equations xn+1=xn-1/ynxn-1-1, yn+1=yn-1/xnyn-1-1, zn+1=xn/ynzn-1," Applied Mathematics, Vol. 2 No. 8, 2011, pp. 1031-1038. doi: 10.4236/am.2011.28143.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] A. S. Kurbanli, C. Cinar and I. Yalcinkaya, “On the Behavaior of Positive Solutions of the System of Rational Difference Equations xn+1=xn-1/ynxn-1+1 yn+1=yn-1/xnyn-1+1 ,” Mathematical and Computer Modelling, Vol. 53, No. 5-6, 2011, pp. 1261-1267. doi:10.1016/j.mcm.2010.12.009
[2] A. S. Kurbanli, “On the Behavior of Solutions of the System of Rational Difference Equations xn+1=xn-1/ynxn-1-1,yn+1=yn-1/xnyn-1-1 ,” World Applied Sciences Journal, 2010, in Press.
[3] A. S. Kurbanli, “On the Behavior of Solutions of the System of Rational Difference Equations xn+1=xn-1/ynxn-1-1 , yn+1=yn-1/xnyn-1-1 , zn+1=zn-1/ynzn-1-1 ,” Discrete Dynamics in Nature and Society, 2011, in Press. doi:10.1155/2011/932362
[4] C. Cinar, “On the Solutions of the Difference Equation xn+1=xn-1/-1+axnxn-1 ,” Applied Mathematics and Com- pulation, Vol. 158, 2004, pp. 793-797.
[5] C. ?inar, “On the Positive Solutions of the Difference Equation Systemxn+1=1/yn,yn+1=yn/xn-1yn-1 ,” Applied Mathematics and Computation, Vol. 158, 2004, pp. 303-305. doi:10.1016/j.amc.2003.08.073
[6] 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
[7] 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
[8] A. Y. ?zban, “On the System of Rational Difference Equations xn=a/yn-3,yn+1=byn-3/xn-qyn-q,” Applied Mathe- matics and Computation, Vol. 188, No. 1, 2007, pp. 833-837. doi:10.1016/j.amc.2006.10.034
[9] 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.
[10] 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
[11] E. Camouzis and G. Papaschinopoulos, “Global Asymptotic Behavior of Positive Solutions on the System of Rational Difference Equations xn+1=1+xn/yn-m ,yn+1=1+yn/xn-m,” Applied Mathematics Letters, Vol. 17, No. 6, 2004, pp. 733-737. doi:10.1016/S0893-9659(04)90113-9
[12] X. Yang, Y. Liu and S. Bai, “On the System of High Order Rational Difference Equations xn=a/yn-p,yn=byn-p/xn-qyn-q ,” Applied Mathematics and Computation, Vol. 171, No. 2, 2005, pp. 853-856. doi:10.1016/j.amc.2005.01.092
[13] X. Yang, “On the System of Rational Difference Equations xn=A+yn-1/xn-pyn-p ,yn=A+xn-1/xn-ryn-s ,” Journal of Mathe- matical Analysis and Applications, Vol. 307, No. 1, 2005, pp. 305-311. doi:10.1016/j.jmaa.2004.10.045
[14] 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.
[15] A. Y. ?zban, “On the Positive Solutions of the System of Rational Difference Equations xn+1=1/yn-k ,yn+1=yn/xn-myn-m-k ,” Journal of Mathematical Analysis and Applications, Vol. 323, No. 1, 2006, pp. 26-32. doi:10.1016/j.jmaa.2005.10.031
[16] Y. Zhang, X. Yang, G. M. Megson and D. J. Evans, “On the System of Rational Difference Equations xn=A+1/yn-p ,yn=A+yn-1/xn-ryn-s ,” Applied Mathematics and Computation, Vol. 176, No. 2, 2006, pp. 403-408. doi:10.1016/j.amc.2005.09.039
[17] Y. Zhang, X. Yang, D. J. Evans and C. Zhu, “On the Nonlinear Difference Equation System xn+1=A+yn-m/xn ,yn+1=A+xn-m/yn ,” Computers & Mathematics with Appli- cations, Vol. 53, No. 10, 2007, pp. 1561-1566.
[18] I. Yalcinkaya and C. Cinar, “Global Asymptotic Stability of two nonlinear Difference Equations zn+1=tnzn-1+a/tn+zn-1 ,tn+1=zntn-1+a/zn+tn-1 ,” Fasciculi Mathematici, Vol. 43, 2010, pp. 171-180.
[19] 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.
[20] 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.
[21] 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.
[22] E. M. Elsayed, “Dynamics of a Recursive Sequence of Higher Order,” Communications on Applied Nonlinear Analysis, Vol. 16, No. 2, 2009, pp. 37-50.
[23] R. Abu-Saris, C. Cinar and I. Yalcinkaya, “On the Asym- ptotic Stability of xn+1=a+xnxn-k/xn+xn-k ,” Computers & Mathematics with Applications, Vol. 56, No. 5, 2008, pp. 1172-1175. doi:10.1016/j.camwa.2008.02.028
[24] 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
[25] R. P. Agarwal, “Difference Equations and Inequalites,” 2nd Editions, Marcel Dekker, New York, 2000.
[26] 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
[27] E. M. Elsayed, “On the Solutions of a Rational System of Difference Equations,” Fasciculi Mathematici, Vol. 45 2010, pp. 25-36.
[28] 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.

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