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Global asymptotic stability of a system of two nonlinear difference equations. (English) Zbl 1207.39024
The authors consider the system of rational difference equations $$ z_{n+1}=\frac{t_n + z_{n-1}}{t_n z_{n-1} + a}, \quad t_{n+1}=\frac{z_n + t_{n-1}}{z_n t_{n-1} + a}, \quad n=0,1,2, \dots,$$ where the parameter $a \in (0, \infty)$ and the initial values are positive, i.e., $z_k,t_k\in(0, \infty)$ for $k=-1,0$. The change of variables $(z_n,t_n)=(\sqrt{a}x_n,\sqrt{a}y_n)$ reduces this system to $$x_{n+1}=\frac{y_n + x_{n-1}}{y_n x_{n-1} + 1}, \quad y_{n+1}=\frac{x_n + y_{n-1}}{x_n y_{n-1} + 1}, \quad n=0,1,2,\dots,$$ with initial values $x_k,y_k\in(0, \infty)$, $k=-1,0$. As their main result the authors show that the positive equilibrium point $(\overline{x},\overline{y})=(1,1)$ of the reduced system is globally asymptotically stable.

39A30Stability theory (difference equations)
39A20Generalized difference equations
39A22Growth, boundedness, comparison of solutions (difference equations)