Necessary and sufficient conditions for oscillatory and asymptotic behaviour of solutions to second-order nonlinear neutral differential equations with several delays

Abstract

In this paper, necessary and sufficient conditions are obtained for oscillatory and asymptotic behaviour of solutions to second-order nonlinear neutral delay differential equations of the form

\begin{equation}

\frac{d}{dt}\left[r(t)\frac{d}{dt}[x(t)+p(t)x(t-\tau)]^\alpha\right]+\sum_{i=1}^{m}q_i(t)H\bigl(x(t-\sigma_i)\bigr)=0

\quad\text{for}\ t\geq{}t_0,\notag

\end{equation}

under the assumption $\int^{\infty}\big(r(\eta)\big)^{-1/\alpha}d\eta=\infty$. Our main tool is Lebesque's dominated convergence theorem. Further, some illustrative examples showing the applicability of the new results are included.