zbMATH — the first resource for mathematics

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).

39A11 Stability of difference equations (MSC2000)
39A20 Multiplicative and other generalized difference equations, e.g., of Lyness type
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.