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On the system of rational difference equations \(x_n=A+y_{n-1}/x_{n-p}y_{n-q}\), \(y_n=A+x_{n-1}/x_{n-r}y_{n-s}\). (English) Zbl 1072.39011
The paper is mainly concerned with the long-term behavior of the positive solutions of the system of rational difference equations: \[ \begin{aligned} x_n &= A+ \frac{y_{n-1}}{x_{n-p}y_{n-q}}, \\ y_n &= A+ \frac{x_{n-1}}{x_{n-r}y_{n-s}}, \end{aligned} \] where \(A\) is a positive real number, \(p,q,r,s\) are positive integers greater than \(2\), and the initial conditions \(x_{1-\max(p,r)},x_{2-\max(p,r)},\dots ,x_0\), \(y_{1-\max(q,s)},y_{2-\max(q,s)},\dots ,y_0\) are positive real numbers.
The authors establish the following results:
(1) If \(A>1\), then every positive solution is bounded. Explicit upper bounds can be found in Theorem 2.1.
(2) If \(A>2/\sqrt{3}\), then the equilibrium solution \[ (\overline{x},\bar{y})= \left(\frac{A+\sqrt{A^2+4}} {2},\frac{A+\sqrt{A^2+4}}{2}\right) \] is locally asymptotically stable (Theorem 3.1). Furthermore, if \(A>\sqrt{2}\), then the equilibrium solution is globally attractive and, hence, globally asymptotically stable (Theorems 3.3–3.4).

MSC:
39A11 Stability of difference equations (MSC2000)
39A20 Multiplicative and other generalized difference equations, e.g., of Lyness type
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[1] Cinas, C.: On the positive solutions of the difference equation system xn+1=1/yn, yn+1=yn/xn - 1yn - 1. Appl. math. Comput. 158, 303-305 (2004)
[2] Clark, D.; Kulenovic, M. R. S.: A coupled system of rational difference equations. Comput. math. Appl. 43, 849-867 (2002) · Zbl 1001.39017
[3] Clark, D.; Kulenovic, M. R. S.; Selgrade, J. F.: Global asymptotic behavior of a two-dimensional difference equation modeling competition. Nonlinear anal. 52, 1765-1776 (2003) · Zbl 1019.39006
[4] Grove, E. A.; Ladas, G.; Mcgrath, L. C.; Teixeira, C. T.: Existence and behavior of solutions of a rational system. Commun. appl. Nonlinear anal. 8, 1-25 (2001) · Zbl 1035.39013
[5] Papaschinopoulos, G.; Schinas, C. J.: On a system of two nonlinear difference equations. J. math. Anal. appl. 219, 415-426 (1998) · Zbl 0908.39003
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