Oscillation of a Linear Delay Impulsive Differential Equation

TL;DR

This paper establishes that the oscillation behavior of a linear delay impulsive differential equation is equivalent to that of a related non-impulsive equation, providing explicit criteria for oscillation.

Contribution

It introduces a novel equivalence between impulsive and non-impulsive delay differential equations regarding oscillation properties.

Findings

Oscillation of impulsive equations is equivalent to non-impulsive counterparts.

Explicit oscillation criteria are derived.

Results apply to a broad class of linear delay impulsive differential equations.

Abstract

The main result of the paper is that the oscillation (non-oscillation) of the impulsive delay differential equation $\dot {x}(t)+\sum_{k=1}^m A_k(t)x[h_k(t)]=0,~~t\geq 0$, $x(\tau_j)=B_jx(\tau_j-0), \lim \tau_j = \infty$ is equivalent to the oscillation (non-oscillation) of the equation without impulses $\dot {x}(t)=\sum_{k=1}^m A_k(t) \prod_{h_k(t)<\tau_j\leq t} B_j^{-1}x[h_k(t)]=0, t \geq 0$. Explicit oscillation results are presented.