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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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##### References:
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