Sharp oscillation criteria for fourth order sub-half-linear and super-half-linear differential equations
This paper is concerned with the oscillatory behavior of the fourth-order nonlinear differential equation $$ \bigl(p(t)|x^{\prime\prime}|^{\alpha-1}\,x^{\prime\prime}\bigr)^{\prime\prime} +q(t)|x|^{\beta-1}x=0\,,\tag{E} $$ where $\alpha>0$, $\beta>0$ are constants and $p,q:[a,\infty)\to(0,\inf...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2008-11-01
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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=346 |
| Summary: | This paper is concerned with the oscillatory behavior of the fourth-order nonlinear differential equation
$$
\bigl(p(t)|x^{\prime\prime}|^{\alpha-1}\,x^{\prime\prime}\bigr)^{\prime\prime}
+q(t)|x|^{\beta-1}x=0\,,\tag{E}
$$
where $\alpha>0$, $\beta>0$ are constants and $p,q:[a,\infty)\to(0,\infty)$ are continuous functions satisfying conditions
$$
\int_a^{\infty}\left( \frac{t}{p(t)}\right)^{\frac{1}{\alpha}}\,dt<\infty,
\int_a^{\infty}\frac{t}{\left(p(t)\right)^{\frac{1}{\alpha}}}\,dt<\infty .
$$
We will establish necessary and sufficient condition for oscillation of all solutions of the sub-half-linear equation (E) (for $\beta<\alpha$) as well as of the super-half-linear equation (E) (for $\beta>\alpha$). |
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| ISSN: | 1417-3875 |