On the Behavior of the Positive Solutions of the System of Two Higher-Order Rational Difference Equations

Abstract

We study the convergence of the positive solutions of the system of the following two difference equations:

where K is a positive integer, the parameters A,B, α, β  and the initial conditions are positive real numbers. Our results generalize well known results in [1,2].

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Q. Wang, G. Zhang and L. Fu, "On the Behavior of the Positive Solutions of the System of Two Higher-Order Rational Difference Equations," Applied Mathematics, Vol. 4 No. 8, 2013, pp. 1220-1225. doi: 10.4236/am.2013.48164.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] A. S. Kurbanli, C. Ginar and ì. Yalcinkaya, “On the Be havior of Positive Solutions of the System of Rational Difference Equation 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] S. Stevic, “On a System of Difference Equations,” Appli ed Mathematics and Computation, Vol. 218, No. 7, 2011, pp. 3372-3378. doi:10.1016/j.amc.2011.08.079
[3] E. Camouzis and G. Papaschinopoulos, “Global Asymp totic Behavior of Positive Solutions on the System of Ra tional 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
[4] X. Yang, “On the System of Rational Difference Equa tions xn+1=A+yn-1+a/xn-pyn-p,yn+1=A+xn-1+a/yn-rxn-s ,” Journal of Mathematical Analysis and Applications, Vol. 307, No. 1, 2005, pp. 305-311. doi:10.1016/j.jmaa.2004.10.045
[5] Q. Wang, F. P. Zeng, G. R. Zhang and X. H. Liu, “Dy namics of the Difference Equation xn+1=α+B1xn-1+B3xn-3+...+B2k+1xn-2k-1/A+B0xn+B2xn-2+...+B2kxn-2k ,” Journal of Difference Equations and Applications, Vol. 12, No. 5, 2006, pp. 399-417. doi:10.1080/10236190500453695
[6] 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 & Mathema tics with Applications, Vol. 53, No. 10, 2007, pp. 1561 1566. doi:10.1016/j.camwa.2006.04.030
[7] K. S. Berenhaut, J. D. Foley and S. Stevic, “The Global Attractivity of the Rational Difference Equation yn=yn-k+yn-m/1+yn-k+yn-m ,” Applied Mathematics Letters, Vol. 20, No. 1, 2007, pp. 54-58. doi:10.1016/j.aml.2006.02.022
[8] K. S. Berenhaut, J. D. Foley and S. Stevic, “The Global Attractivity of the Rational Difference Equation yn=1+yn-k/yn-m,” Proceedings of the American Mathema tical Society, Vol. 135, No. 1, 2007, pp. 1133-1140. doi:10.1090/S0002-9939-06-08580-7
[9] K. S. Berenhaut, J. D. Foley and S. Stevic, “The Global Attractivity of the Rational Difference Equation yn=A+[yn-k/yn-m],” Proceedings of the American Mathe matical Society, Vol. 136, No. 1, 2008, pp. 103-110. doi:10.1090/S0002-9939-07-08860-0
[10] B. Iricanin and S. Stevic, “Eventually Constant Solutions of a Rational Difference Equation,” Applied Mathematics and Computation, Vol. 215, No. 2, 2009, pp. 854-856. doi:10.1016/j.amc.2009.05.044
[11] I. Yalcinkaya and C. Ginar, “Global Asymptotic Stability of Two Nonlinear Difference Equations zn+1=tn+zn-1/tnzn-1+a,tn+1=zn+tn-1/zntn-1+a,” Fasciculi Mathematici, Vol. 43, 2010, pp. 171-180.

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