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Oscillation theorems for third order nonlinear differential equations. (English) Zbl 0760.34031

The paper deals with oscillation and asymptotic behaviour of the nonlinear third order equation $\left(*\right)\phantom{\rule{4pt}{0ex}}{\left({r}_{2}\left(t\right){\left({r}_{1}\left(t\right){y}^{\text{'}}\right)}^{\text{'}}\right)}^{\text{'}}+p\left(t\right){y}^{\text{'}}+q\left(t\right)f\left(y\right)=0$, where $p,q$ are nonnegative functions and ${r}_{1}\left(t\right)$, ${r}_{2}\left(t\right)>0$ for $t\in \left[a,\infty \right)$. Conditions on the functions ${r}_{1},{r}_{2},p,q$ are given which guarantee that no nonoscillatory solution of $\left(*\right)$ has property ${V}_{2}$ provided certain (nonlinear) second order equation associated with $\left(*\right)$ is oscillatory (a solution $y$ of $\left(*\right)$ is said to possess property ${V}_{2}$ if $y\left(t\right){L}_{k}y\left(t\right)>0$, $k=1,2$, $y\left(t\right){L}_{3}y\left(t\right)\le 0$ for large $t$, where ${L}_{0}y=y$, ${L}_{i}y={r}_{i}{\left({L}_{i-1}y\right)}^{\text{'}}$, $i=1,2$, ${L}_{3}y={\left({L}_{2}y\right)}^{\text{'}}\right)$. As a corollary of these results the following oscillation criterion is proved.

Theorem. Let any condition which implies that no nonoscillatory solution of $\left(*\right)$ has property ${V}_{2}$ be satisfied. A solution $y$ of $\left(*\right)$ which exists on an interval $\left[T,\infty \right)$ is oscillatory if and only if there exists ${t}_{0}\in \left[T,\infty \right)$ such that $2y{L}_{2}y-\frac{{r}_{2}}{{r}_{1}}{\left({L}_{1}y\right)}^{2}+p{y}^{2}\le 0$ for $t={t}_{0}$.

Reviewer: O.Došlý (Brno)
##### MSC:
 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory 34A34 Nonlinear ODE and systems, general