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Nonoscillatory solutions for discrete equations. (English) Zbl 1052.39003
The authors consider the discrete equation $$ \Delta(a(k)\Delta(y(k)+ py(k-\tau)))+F(k+1,y(k+1-\sigma))=0 \quad (k\in{\Bbb N}), $$ here $\Delta$ is the difference operator, $F$ is a continuous map from ${\Bbb N}\times (0,\infty)$ into $[0,\infty), \tau,\sigma\in{\Bbb N}\cup\{0\}, a:{\Bbb N}\to(0,\infty)$, and $p\in{\Bbb R}$. A nontrivial solution $y$ of this equation is called oscillatory if it is neither eventually positive nor eventually negative. Otherwise, it is called nonoscillatory. The main result of the paper under review says that if $\vert p\vert \ne 1$ and there exist $K>0$ and $k_0\in{\Bbb N}$ such that $$ \sum_{k=k_0}^\infty {1 \over a(k)} \sum_{i=k}^\infty\sup_{w\in[K/2,K]} F(i+1,w)<\infty, $$ then the above discrete equation has a bounded nonoscillatory solution. The authors notice that the latter condition can be replaced by the less restrictive condition $$ \sum_{k=k_0}^\infty {1 \over a(k)} \sum_{i=k_0}^{k-1}\sup_{w\in[K/2,K]} F(i+1,w)<\infty, $$ and the theorem remains valid. The results extend and correct the results of {\it R. P. Agarwal} and {\it P. J. Y. Wong} [Advanced topics in difference equations (1997; Zbl 0878.39001)].

39A11Stability of difference equations (MSC2000)
39A10Additive difference equations
Full Text: DOI
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