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 \[ \lim_{n\to\infty} x_n=k\text{ and } \lim_{n\to\infty} B_ny_n=0\text{ (resp. }\lim_{n\to\infty} x_n/B_n=k\text{ and } \lim_{n\to\infty} y_n=-k). \]


39A11 Stability of difference equations (MSC2000)