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Nonoscillation and oscillation of first order neutral equations with variable coefficients. (English) Zbl 0809.34083
The neutral equations \[ \left[x(t)- \sum^ k_{i=1} c_ i(t) x(t- \nu_ i(t))\right]'+ Q(t) x(t- \sigma)= 0,\quad t\geq t_ 0, \] are considered, where \(\sigma>0\), \(Q(t)>0\), \(c_ i(t)> 0\), \(0< \nu_ 0< \nu_ i(t)\leq \nu\) when \(t\in [t_ 0,+\infty)\) \((i= 1,\dots, k)\). Sufficient conditions are given for the present equation to have nonoscillatory solutions (every of its regular solution to be oscillatory).

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
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