Share This Article:

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

Abstract Full-Text HTML Download Download as PDF (Size:227KB) PP. 1220-1225
DOI: 10.4236/am.2013.48164    2,777 Downloads   4,185 Views  

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].

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

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.

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.

  
comments powered by Disqus

Copyright © 2018 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.