Oscillation and nonoscillation of two-dimensional difference systems. (English) Zbl 1117.39004

This paper is concerned with the oscillation and nonoscillation behavior of solution of the nonlinear two-dimensional difference system
\[ \Delta x_n =a_n g (y_n),\quad \Delta y_{n-1}=-f (n ,x_n),\quad n \in N (n_0) \]
where \(\Delta\) is defined by \(\Delta y_n=y_{n+1}-y_n\), \(N (n_0) =\{n_0,n_0+1,\dots\}\), \(n_0\in N=\{1,2,\dots\}\), \(\{a_n\}\) is the nonnegative real sequence, \(g(u):\mathbb R \to\mathbb R\) is continuous function with properties, \(ug(u)>0\) for \(u\neq 0\), \(f(n,u):N(n_0)\times \mathbb R\to \mathbb R\) is continuous as a function of \(u\in \mathbb R\), \(u f (n,u)>0\) for \(n \in N (n_0)\) and \(u\neq 0\), where \(N_0\geq n_0\). Some necessary and sufficient conditions are given for the system to admit the existence of oscillatory and nonoscillatary solution with special asymptotic properties. Some important results in the literatures are generalized. Finally, examples to illustrate the result are included.


39A11 Stability of difference equations (MSC2000)
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