Necessary and Sufficient Conditions for oscillations to a Second-Order Neutral Differential Equations with Impulses

Arun Kumar Tripathy, Shyam Sundar Santra

In this work, we obtain 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)' +um_{i=1}^m q_i(t)x^{lpha_i}(igma_i(t))=0, \quad t\geq t_0,\, teq ambda_k, \Delta \Big(r(ambda_k)\big(z'(ambda_k)\big)^\gamma\Big) + um_{i=1}^m h_i(ambda_k)x^{lpha_i} (igma_i(ambda_k))=0,\quad k=1,2,3,…, \end{cases} \end{align*} where $z(t)=x(t)+p(t)x(\tau(t))$. Under the assumption $\int_{0}^{\infty}\big(r(\eta)\big)^{-1/\gamma} d\eta=\infty$, we consider two cases when $\gamma>\alpha_i$ and $\gamma<\alpha_i$. Our main tool is Lebesgue's Dominated Convergence theorem. Examples are given to illustrate our main results and we state an open problem.