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Some further results on oscillation of neutral differential equations. (English) Zbl 0729.34051
Consider the first order neutral delay differential equation $(1)\quad (d/dt)(y(t)-R(t)y(t-r))+P(t)y(t-\tau)-Q(t)y(t-\sigma)=0$ where $$P,Q,R\in C([t_ 0,\infty),{\mathbb{R}}^+)$$, $$r\in (0,\infty)$$ and $$\tau$$,$$\sigma\in [0,\infty)$$. Set $A=\liminf_{t\to \infty}\int^{t}_{t-\tau}(P(s)-Q(s+\sigma -\tau))(1+R(s- \tau)+\int^{s}_{s-\tau}Q(u-\tau)du) ds,$
$M=\limsup_{t\to \infty}\int^{t}_{t-\tau}(P(s)-Q(s+\sigma -\tau))(1+R(s- \tau)+\int^{s}_{s-\tau}Q(u-\tau)du) ds.$ The main result is the following: Theorem. Assume that $$\tau\geq \sigma$$, $$P(t)-Q(t+\sigma - \tau)\geq 0$$ and not identically zero, $$1-R(t)-\int^{t}_{t-(\tau - \sigma)}Q(s)ds\geq 0$$ for all sufficiently large t, and that either $$A>1/e$$ or $$A\leq 1/e$$ and $$M>1-(1/2)(1-A-\sqrt{1-2A-A^ 2}).$$ Then every solution of (1) oscillates.

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