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Zbl 0760.34031
Škerlík, Anton
Oscillation theorems for third order nonlinear differential equations. (English)
[J] Math. Slovaca 42, No.4, 471-484 (1992). ISSN 0139-9918; ISSN 1337-2211/e

The paper deals with oscillation and asymptotic behaviour of the nonlinear third order equation $(*)\ (r\sb 2(t)(r\sb 1(t)y')')'+p(t)y'+q(t)f(y)=0$, where $p,q$ are nonnegative functions and $r\sb 1(t)$, $r\sb 2(t)>0$ for $t\in[a,\infty)$. Conditions on the functions $r\sb 1,r\sb 2,p,q$ are given which guarantee that no nonoscillatory solution of $(*)$ has property $V\sb 2$ provided certain (nonlinear) second order equation associated with $(*)$ is oscillatory (a solution $y$ of $(*)$ is said to possess property $V\sb 2$ if $y(t)L\sb ky(t)>0$, $k=1,2$, $y(t)L\sb 3y(t)\le 0$ for large $t$, where $L\sb 0y=y$, $L\sb iy=r\sb i(L\sb{i-1}y)'$, $i=1,2$, $L\sb 3y=(L\sb 2y)')$. As a corollary of these results the following oscillation criterion is proved.\par Theorem. Let any condition which implies that no nonoscillatory solution of $(*)$ has property $V\sb 2$ be satisfied. A solution $y$ of $(*)$ which exists on an interval $[T,\infty)$ is oscillatory if and only if there exists $t\sb 0\in[T,\infty)$ such that $2yL\sb 2y-{r\sb 2\over r\sb 1}(L\sb 1y)\sp 2+py\sp 2\le 0$ for $t=t\sb 0$.
[O.Došlý (Brno)]

MSC 2000:: *34C10 Qualitative theory of oscillations of ODE: Zeros, etc.
34A34 Nonlinear ODE and systems, general

Keywords: property $V\sb 2$; oscillation; asymptotic behaviour; nonlinear third order equation

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