Necessary and Sufficient Conditions for oscillation to Second-order Neutral Differential Equations with Impulses

Abstract

In this work, necessary and sufficient conditions

for oscillation of solutions of second-order neutral impulsive differential system

\begin{align*}

\begin{cases}

\Big(r(t)\big(z'(t)\big)^\gamma\Big)' + q(t)x^{\alpha}(\sigma(t))=0, \quad t\geq t_0,\; t\neq \lambda_k,\\

\Delta \Big(r(\lambda_k)\big(z'(\lambda_k)\big)^\gamma\Big) + h(\lambda_k)x^{\alpha} (\sigma(\lambda_k))=0,\,k=1,2,3,\dots

\end{cases}

\end{align*}

are established, where $z(t)=x(t)+p(t)x(\tau(t))$. Under the assumption $\int^{\infty}\big(r(\eta)\big)^{-1/\alpha} d\eta=\infty$,

two cases when $\gamma>\alpha$ and $\gamma<\alpha$ are considered. The main tool is Lebesgue's Dominated Convergence theorem. Examples are given to illustrate the main results, and state an open problem.