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

Abstract

Abstract 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 ) x α ( σ ( t ) ) = 0 , t ≥ t 0 ,       t ≠ λ k , Δ ( r ( λ k ) ( z ′ ( λ k ) ) γ ) + h ( λ k ) x α ( σ ( λ k ) ) = 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 / α d η = ∞ \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.