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Zbl 0787.34057
Erbe, L.H.; Kong, Q.
Oscillation results for second order neutral differential equations.
(English)
[J] Funkc. Ekvacioj, Ser. Int. 35, No.3, 545-555 (1992). ISSN 0532-8721

The authors consider the oscillatory behavior of the neutral functional differential equation $$[y(t)-cy(t-\tau)]''+p(t) f(y(t-\sigma(t)))=0$$ under the assumption\par (H) $c$ and $\tau$ are positive numbers; $p$ and $\sigma \in C(R\sb +,R\sb +)$, $p(t) \not\equiv 0$, $t-\sigma(t)$ is increasing and tends to $\infty$ as $t \to \infty$, $\sigma (t)>\tau$; $f \in C(R,R)$ is increasing, $f(-x)=-f(x)$, $f(xy) \ge f(x)f(y)$, $xy>0$, $f(\infty)= \infty$, and $f(y)/y \to \infty$ or 1 as $y \to \infty$.\par The main result is the following one:\par Suppose that assumption (H) holds and that the equation $$z''+p(t)f\left( {\lambda(t-\sigma(t)) \over t} z(t) \right)=0$$ is oscillatory for some $0<\lambda<1$. Let in addition $$\lim\sb{t \to \infty} \int\sb{t- \sigma(t)+\tau}\sp t (u-(t-\sigma(t)+\tau)) p(u) du> \cases c \quad \text {if } f(y)/y \to 1, & y \to \infty \\ 0 \quad \text {if } f(y)/y \to \infty, & y \to 0. \endcases$$ Then the considered equation is oscillatory.
[I.Ginchev (Varna)]
MSC 2000:
*34K99 Functional-differential equations
34K40 Neutral equations
34C10 Qualitative theory of oscillations of ODE: Zeros, etc.

Keywords: oscillatory behavior; neutral functional differential equation

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