On the oscillation of a nonlinear two-dimensional difference system (Q2783578)

Consider the system of nonlinear difference equations NEWLINE\[NEWLINE\begin{aligned} & \Delta x_n=b_ng(y_n),\\ & \Delta y_n=-f(n,x_{n+1})\;n\in\{n_0, n_{0+1}, \dots\}. \end{aligned} \tag{S}NEWLINE\]NEWLINE 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.NEWLINENEWLINENEWLINELet (S) be either superlinear or sublinear. If \(ug(v) \leq g(u,v)\) for all sufficiently small \(u\) and every \(v>0\) and NEWLINE\[NEWLINE\sum^\infty_{n =n_0} B_n\bigl|f(n,k)\bigr|<\infty\quad\Bigl(\text{resp. }\sum^\infty_{n=n_0} \biggl|f\bigl(n,g(k)\bigr) B_{n+1} \biggr|<\infty\Bigr)\text{ for some }k\neq 0,NEWLINE\]NEWLINE where \(B_n=\sum^\infty_{s=n_0}b_s\), then (S) has a nonoscillatory solution \(((x_n),(y_n))\) such that NEWLINE\[NEWLINE\lim_{n\to\infty} x_n=k\text{ and } \lim_{n\to\infty} B_ny_n=0\text{ (resp. }\lim_{n\to\infty} x_n/B_n=k\text{ and } \lim_{n\to\infty} y_n=-k).NEWLINE\]