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Zbl 1009.39009
Thandapani, E.; Ponnammal, B.
On the oscillation of a nonlinear two-dimensional difference system. (English)
[J] Tamkang J. Math. 32, No.3, 201-209 (2001). ISSN 0049-2930; ISSN 2073-9826/e

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

MSC 2000:: *39A11 Stability of difference equations

Keywords: oscillatory solution; system of nonlinear difference equations; superlinear; sublinear; nonoscillatory solution

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