Necessary and Sufficient Condition for Oscillatory and Asymptotic Behaviour of Second-Order Functional Differential Equations

Shyam Sundar Santra

In this paper, necessary and sufficient conditions are obtained for oscillatory and asymptotic behaviour of solutions of second-order neutral delay differential equations of the form \begin{equation} \frac{d}{dt}eft[r(t)\frac{d}{dt}[x(t)+p(t)x(au(t))]\right]+q(t)Geft(x(igma(t))\right)=0, \quadext{for} t\geq{}t_{0},otag \end{equation} under the assumption $\int^{\infty}\frac{1}{r(\eta)}d\eta=\infty$ for various ranges of the bounded neutral coefficient $p$. Our main tools are Lebesgue's dominated convergence theorem and Banach's contraction mapping principle. Further, an illustrative example showing the applicability of the new results is included.