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Existence for nonoscillatory solutions of second-order nonlinear differential equations. (English) Zbl 1111.34049
Summary: The existence of nonoscillatory solutions of the second-order nonlinear neutral differential equation $$[r(t)(x(t)+P(t)x(t-\tau))']'+ \sum_{i=1}^m Q_i(t)f_i(x(t-\sigma_i))=0, \quad t\ge t_0,$$ where $m\ge1$ is an integer,$\tau>0$, $\sigma_i\ge 0$, $r,P,Q_i\in C([t_0,\infty),\Bbb R)$, $f_i\in C(\Bbb R,\Bbb R)$, $i=1,2,\dots,m$, is studied. Some new sufficient conditions for the existence of a nonoscillatory solution are obtained for general $P(t)$ and $Q_i(t)$, $i=1,2,\dots,m$, which means that we allow oscillatory $P(t)$ and $Q_i(t)$, $i=1,2,\dots,m$. In particular, our results improve essentially and extend some known results in the recent references.

34K11Oscillation theory of functional-differential equations
34K40Neutral functional-differential equations
Full Text: DOI
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