Quasilinear Impulsive Systems with Generalized Piecewise Constant Delay

This paper delves into the analysis of quasilinear systems of impulsive differential equations with generalized piecewise constant delay (IDEGPCDs), where the argument is characterized as a general step function. These systems merge the features of both continuous and discrete equations, with the discrete component playing a pivotal role. We derive explicit solutions for both homogeneous and non-homogeneous linear IDEGPCD systems. The paper thoroughly explores the existence, uniqueness, and stability of solutions for quasilinear IDEGPCDs, providing enhancements to previously established results. The study highlights the importance of delayed intervals and derives the corresponding Cauchy and Green matrices. Furthermore, the integral representation and Gronwall-type inequality developed herein offer powerful tools for investigating stability, periodicity, oscillations, and other related phenomena in IDEGPCDs. As a direct application, the paper addresses the stability of certain impulsive neural network models with generalized piecewise constant delay, and numerical simulations are presented to substantiate and improve upon the theoretical findings.