Oscillation of two-dimensional difference systems. (English) Zbl 0964.39012

The paper presents some oscillation results for the two-dimensional difference system \[ \begin{aligned} \triangle x_n&=b_ng(y_n),\\ \triangle y_{n-1}&=-a_nf(x_n), \end{aligned} \qquad n\in\mathbb N (n_0)=\{n_0,n_0+1,\dots \}, \tag{1} \] where \(n_0\in\mathbb N\), \(\{a_n\}, \{b_n\}\), \(n\in\mathbb N (n_0)\) are real sequences and \(f, g\) are real continuous functions with the sign property \(uf(u)>0\) and \(ug(u)>0\) for all \(u\neq 0\). Assuming both sequences \(\{a_n\}, \{b_n\}\) are nonnegative and not identically zero for infinitely many values of \(n\) the authors establish the sufficient conditions for all solutions to the system (1) to be oscillatory. Moreover, some of their results do not require the sign condition on \(\{a_n\}\), i.e., \(\{a_n\}\) can assume both positive and negative values. The difference system (1) can be reduced via a special choice of \(f, g\) and \(b_n\) to the second order superlinear or sublinear equation. It is shown that the derived oscillation results generalize the known oscillation criteria for this type of difference equations.


39A11 Stability of difference equations (MSC2000)
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