Necessary and Sufficient Conditions for Oscillation of Solutions to Second-Order Neutral Differential Equations with Impulses

In this work, necessary and sufficient conditions for oscillation of solutions of second-order neutral impulsive differential system

{(r(t)(z′(t))γ)′+q(t)xα(σ(t))=0,t≥t0, t≠λk,Δ(r(λk)(z′(λk))γ)+h(λk)xα(σ(λk))=0,k∈𝕅\left\{ {\matrix{{{{\left( {r\left( t \right){{\left( {z'\left( t \right)} \right)}^\gamma }} \right)}^\prime } + q\left( t \right){x^\alpha }\left( {\sigma \left( t \right)} \right) = 0,} \hfill & {t \ge {t_0},\,\,\,t \ne {\lambda _k},} \hfill \cr {\Delta \left( {r\left( {{\lambda _k}} \right){{\left( {z'\left( {{\lambda _k}} \right)} \right)}^\gamma }} \right) + h\left( {{\lambda _k}} \right){x^\alpha }\left( {\sigma \left( {{\lambda _k}} \right)} \right) = 0,} \hfill & {k \in \mathbb{N}} \hfill \cr } } \right. are established, where z(t)=x(t)+p(t)x(τ(t))z\left( t \right) = x\left( t \right) + p\left( t \right)x\left( {\tau \left( t \right)} \right)

Under the assumption ∫∞(r(η))-1/αdη=∞\int {^\infty {{\left( {r\left( \eta \right)} \right)}^{ - 1/\alpha }}d\eta = \infty } two cases when γ>α and γ<α are considered. The main tool is Lebesgue’s Dominated Convergence theorem. Examples are given to illustrate the main results, and state an open problem.