Linearized Stability Analysis of Nonlinear Delay Differential Equations with Impulses

This paper explores the linearized stability of nonlinear delay differential equations (DDEs) with impulses. The classical results on the existence of periodic solutions are extended from ordinary differential equations (ODEs) to DDEs with impulses. Furthermore, the classical results of linearized stability for nonlinear semigroups are generalized to periodic DDEs with impulses. A significant challenge arises from the need for a discontinuous initial function to obtain periodic solutions. To address this, first-kind discontinuous spaces R(a,b,Rn) are introduced for defining DDEs with impulses, providing key existence and uniqueness results. This study also establishes linear stability results by linearizing the Poincaré operator for DDEs with impulses. Additionally, the stability properties of equilibrium solutions for these equations are analyzed, highlighting their importance due to the wide range of applications in various scientific fields.