A Note on the Oscillatory Behavior of Impulsive Differential Equations with Piecewise Constant Arguments via Difference Equations

TL;DR

This paper investigates the oscillatory behavior of impulsive differential equations with piecewise constant arguments by reducing the problem to difference equations, extending classical criteria to impulsive systems.

Contribution

It introduces a reduction method to analyze oscillation in impulsive differential equations, extending classical difference equation criteria to impulsive cases.

Findings

Derived explicit oscillation criteria for impulsive differential equations.

Extended classical difference equation oscillation criteria to impulsive systems.

Provided an example demonstrating the applicability of the results.

Abstract

This paper studies the oscillatory behavior of solutions to linear nonautonomous impulsive differential equations with piecewise constant arguments, including both advanced and delayed cases \[ x'(t) = a(t)x(t) + b(t)x([t-k]), \quad k \in \mathbb{Z}. \] By exploiting the hybrid structure of these systems, we reduce the problem to an associated difference equation whose coefficients explicitly incorporate both the continuous dynamics and the impulsive effects. Classical oscillation criteria for difference equations do not account for impulsive phenomena. Through the proposed reduction, we extend these criteria to a class of impulsive and non-impulsive equations (IDEPCA and DEPCA), obtaining explicit sufficient conditions for oscillation in terms of the original system data. An example is provided to illustrate the applicability of the results.