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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
##### Keywords:
neutral delay differential equation
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