# zbMATH — the first resource for mathematics

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
##### Keywords:
neutral equations; nonoscillatory solutions; oscillatory
Full Text: