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Oscillations of neutral delay differential equations. (English) Zbl 0566.34054
Consider the neutral delay differential equation \((*)\quad (d/dt)[x(t)- px(t-\tau)]+Q(t)x(t-\sigma)=0,\) \(t\geq t_ 0\) where p, \(\tau\), and \(\sigma\) are positive constants and \(Q\in ([t_ 0,\infty),{\mathbb{R}}^+)\). We prove the following results: Theorem A. Assume that \(p<1\) and \(\int^{\infty}_{t_ 0}Q(s)ds=\infty\). Then every nonoscillatory solution of (*) tends to zero as \(t\to \infty\). Theorem B. Assume \(p=1\) and \(\int^{\infty}_{t_ 0}Q(s)ds=\infty\). Then every solution of (*) oscillates. Theorem C. Assume \(p<1\) and \(\lim \inf_{t\to \infty}\int^{t}_{t-\sigma}Q(s)ds>1/e\). Then every solution of (*) oscillates. Theorem D. Assume \(p<1\), \(\sigma >\tau\), Q is \(\tau\)- periodic, and \(\lim \inf_{t\to \infty}\int^{t}_{t-(\sigma - \tau)}Q(s)ds>(1-p)/e\). Then every solution of (*) 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
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