Erbe, L. H.; Kong, Q. Oscillation results for second order neutral differential equations. (English) Zbl 0787.34057 Funkc. Ekvacioj, Ser. Int. 35, No. 3, 545-555 (1992). 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) Cited in 5 Documents 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 Keywords:oscillatory behavior; neutral functional differential equation PDFBibTeX XMLCite \textit{L. H. Erbe} and \textit{Q. Kong}, Funkc. Ekvacioj, Ser. Int. 35, No. 3, 545--555 (1992; Zbl 0787.34057)