Oscillatory and asymptotic behavior of fourth order quasilinear difference equations
The authors consider the fourth order quasilinear difference equation $$\Delta^{2}\left(p_{n}|\Delta^{2}x_n|^{\alpha-1}\Delta^{2}x_n\right)+q_{n}|x_{n+3}|^{\beta -1}x_{n+3}=0,$$ where $\alpha$ and $\beta$ are positive constants, and ${\{p_{n}\}}$ and ${\{q_{n}\}}$ are positive real sequences. They o...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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University of Szeged
2009-11-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=451 |
| Summary: | The authors consider the fourth order quasilinear difference equation $$\Delta^{2}\left(p_{n}|\Delta^{2}x_n|^{\alpha-1}\Delta^{2}x_n\right)+q_{n}|x_{n+3}|^{\beta -1}x_{n+3}=0,$$ where $\alpha$ and $\beta$ are positive constants, and ${\{p_{n}\}}$ and ${\{q_{n}\}}$ are positive real sequences. They obtain sufficient conditions for oscillation of all solutions when $\sum\limits_{n=n_{0}}^{\infty}\left(\frac{n}{p_{n}}\right)^\frac{1}{\alpha}<\infty $ and $\sum\limits_{n=n_{0}}^{\infty}\left(\frac{n}{{p_{n}}^{\frac{1}{\alpha}}}\right)<\infty.$ The results are illustrated with examples. |
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| ISSN: | 1417-3875 1417-3875 |