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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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