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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 \[ [y(t)-cy(t-\tau)]''+p(t) f(y(t-\sigma(t)))=0 \] under the assumption
(H) \(c\) and \(\tau\) are positive numbers; \(p\) and \(\sigma \in C(R_ +,R_ +)\), \(p(t) \not\equiv 0\), \(t-\sigma(t)\) is increasing and tends to \(\infty\) as \(t \to \infty\), \(\sigma (t)>\tau\); \(f \in C(R,R)\) is increasing, \(f(-x)=-f(x)\), \(f(xy) \geq f(x)f(y)\), \(xy>0\), \(f(\infty)= \infty\), and \(f(y)/y \to \infty\) or 1 as \(y \to \infty\).
The main result is the following one:
Suppose that assumption (H) holds and that the equation \[ z''+p(t)f\left( {\lambda(t-\sigma(t)) \over t} z(t) \right)=0 \] is oscillatory for some \(0<\lambda<1\). Let in addition \[ \lim_{t \to \infty} \int_{t- \sigma(t)+\tau}^ t (u-(t-\sigma(t)+\tau)) p(u) du> \begin{cases} c \quad \text{if } f(y)/y \to 1, & y \to \infty \\ 0 \quad \text{if } f(y)/y \to \infty, & y \to 0. \end{cases} \] Then the considered equation is oscillatory.
Reviewer: I.Ginchev (Varna)

MSC:

34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
34K40 Neutral functional-differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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