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On the oscillation of a nonlinear two-dimensional difference system. (English) Zbl 1009.39009
Consider the system of nonlinear difference equations \begin{aligned} & \Delta x_n=b_ng(y_n),\\ & \Delta y_n=-f(n,x_{n+1})\;n\in\{n_0, n_{0+1}, \dots\}. \end{aligned} \tag{S} If for each fixed $$n$$ $$f(n,u)/u$$ is nondecreasing in $$u$$ for $$u>0$$ and nonincreasing in $$u$$ for $$u<0$$ then the system (S) is called to be superlinear. Likewise the sublinearity of the system is defined.
Let (S) be either superlinear or sublinear. If $$ug(v) \leq g(u,v)$$ for all sufficiently small $$u$$ and every $$v>0$$ and $\sum^\infty_{n =n_0} B_n\bigl|f(n,k)\bigr|<\infty\quad\Bigl(\text{resp. }\sum^\infty_{n=n_0} \biggl|f\bigl(n,g(k)\bigr) B_{n+1} \biggr|<\infty\Bigr)\text{ for some }k\neq 0,$ where $$B_n=\sum^\infty_{s=n_0}b_s$$, then (S) has a nonoscillatory solution $$((x_n),(y_n))$$ such that $\lim_{n\to\infty} x_n=k\text{ and } \lim_{n\to\infty} B_ny_n=0\text{ (resp. }\lim_{n\to\infty} x_n/B_n=k\text{ and } \lim_{n\to\infty} y_n=-k).$
##### MSC:
 39A11 Stability of difference equations (MSC2000)