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In this paper, necessary and sufficient conditions for asymptotic behavior are established of the solutions to second-order neutral delay differential equations of the form \begin{equation} \frac{d}{d{}t}\Biggl(r(t)\biggl(\frac{d}{d{}t}[x(t)-p(t)x(\tau(t))]\biggr)^{\gamma}\Biggr)+\sum_{i=1}^{m}q_{i}(t)f_{i}\bigl(x(\sigma_{i}(t))\bigr)=0 \quad\text{for}\ t\geq{}t_{0}.\nonumber \end{equation} We consider two cases when $f_{i}(u)/u^{\beta}$ is nonincreasing for $\gamma>\beta$, and nondecreasing for $\beta>\gamma$, where $\beta$ and $\gamma$ are quotients of two positive odd integers. Our main tool is Lebesgue's dominated convergence theorem. Examples illustrating the applicability of the results are also given, and state an open problem.
Alan : Fen Bilimleri ve Matematik
Dergi Türü : Uluslararası
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