## 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.

### MSC:

 39A11 Stability of difference equations (MSC2000)

### Keywords:

nonlinear difference system; oscillation
Full Text:

### References:

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