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Oscillation results for second order neutral differential equations. (English) Zbl 0787.34057

The authors consider the oscillatory behavior of the neutral functional differential equation

${\left[y\left(t\right)-cy\left(t-\tau \right)\right]}^{\text{'}\text{'}}+p\left(t\right)f\left(y\left(t-\sigma \left(t\right)\right)\right)=0$

under the assumption

(H) $c$ and $\tau$ are positive numbers; $p$ and $\sigma \in C\left({R}_{+},{R}_{+}\right)$, $p\left(t\right)¬\equiv 0$, $t-\sigma \left(t\right)$ is increasing and tends to $\infty$ as $t\to \infty$, $\sigma \left(t\right)>\tau$; $f\in C\left(R,R\right)$ is increasing, $f\left(-x\right)=-f\left(x\right)$, $f\left(xy\right)\ge f\left(x\right)f\left(y\right)$, $xy>0$, $f\left(\infty \right)=\infty$, and $f\left(y\right)/y\to \infty$ or 1 as $y\to \infty$.

The main result is the following one:

Suppose that assumption (H) holds and that the equation

${z}^{\text{'}\text{'}}+p\left(t\right)f\left(\frac{\lambda \left(t-\sigma \left(t\right)\right)}{t}z\left(t\right)\right)=0$

is oscillatory for some $0<\lambda <1$. Let in addition

$\underset{t\to \infty }{lim}{\int }_{t-\sigma \left(t\right)+\tau }^{t}\left(u-\left(t-\sigma \left(t\right)+\tau \right)\right)p\left(u\right)du>\left\{\begin{array}{cc}c\phantom{\rule{1.em}{0ex}}\text{if}\phantom{\rule{4.pt}{0ex}}f\left(y\right)/y\to 1,\hfill & y\to \infty \hfill \\ 0\phantom{\rule{1.em}{0ex}}\text{if}\phantom{\rule{4.pt}{0ex}}f\left(y\right)/y\to \infty ,\hfill & y\to 0·\hfill \end{array}\right\$

Then the considered equation is oscillatory.

##### MSC:
 34K99 Functional-differential equations 34K40 Neutral functional-differential equations 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory