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.


39A30 Stability theory for difference equations
39A20 Multiplicative and other generalized difference equations
39A22 Growth, boundedness, comparison of solutions to difference equations