Oscillation of certain two-dimensional nonlinear difference systems. (English) Zbl 1056.39009

The paper deals with the nonlinear two-dimensional difference system \[ \Delta x_n = b_n g(y_n),\;\Delta y_n = -a_n f(x_n) + r_n, n=n_0,\;n_0+1,\dots, \tag{1} \] where \(\Delta x_n =x_{n+1}-x_n,\) \(n_0\) is a positive integer, \(b_n \geq 0, \sum_{s=1}^{\infty}| r_s| < \infty\), and the continuous real valued functions \(f\) and \(g\) satisfy inequalities \(uf(u) > 0\) and \(ug(u) > 0\) for all \(u \not= 0.\) A solution \(\{(x_n,y_n)\}\) of (1) is called oscillatory if both components are oscillatory. The authors give conditions for all solutions of (1) to be oscillatory or \(\liminf_{n \to \infty}| x_n| = 0.\)


39A11 Stability of difference equations (MSC2000)
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