Asymptotic behavior of positive solutions of fourth-order nonlinear difference equations
We consider a class of fourth-order nonlinear difference equations of the form $$ \Delta^2(p_n(\Delta^2y_n)^{\alpha})+q_n y^{\beta}_{n+3}=0, \quad n\in {\mathbb N} $$ where $\alpha, \beta$ are the ratios of odd positive integers, and $\{p_n\}, \{q_n\}$ are positive real sequences defined for all $n...
Saved in:
| Date: | 2008 |
|---|---|
| Main Authors: | , , , |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2008
|
| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3134 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| Summary: | We consider a class of fourth-order nonlinear difference equations of the form
$$ \Delta^2(p_n(\Delta^2y_n)^{\alpha})+q_n y^{\beta}_{n+3}=0, \quad n\in {\mathbb N} $$
where $\alpha, \beta$ are the ratios of odd positive integers, and $\{p_n\}, \{q_n\}$ are positive real sequences defined for all $n\in {\mathbb N} $.
We establish necessary and sufficient conditions for the existence of nonoscillatory solutions with specific asymptotic behavior under suitable combinations of the convergence or divergence conditions of the sums
$$ \sum\limits_{n=n_0}^{\infty}\frac n{p_n^{1/\alpha}}\quad \text{and}\quad \sum\limits_{n=n_0}^{\infty}\left(\frac n{p_n}\right)^{1/\alpha}.$$ |
|---|