Periodic solutions of a system with impulsive action at nonfixed times

Abstract

We establish conditions for the existence of periodic solutions of a system with impulsive action at nonfixed times. The idea of application of the method of averaging to the investigation of equations with impulsive action was proposed by Krylov and Bogolyubov [1] and later developed by Samoilenko, Mytropol’skyi, Perestyuk, and other researchers. It was extended to a broad class of systems with impulsive action. Thus, in particular, Samoilenko and Perestyuk [2‐4] studied the problem of existence of limiting discontinuous cycles for impulsive systems with one point of discontinuity, established conditions for the existence of periodic solutions of the analyzed impulsive systems, and substantiated the possibility of application of the Krylov‐Bogolyubov averaging scheme to these impulsive systems. The main aim of the present paper is to establish conditions for the coefficients and functions of an impulsive system with two points of discontinuity at nonfixed times under which this system possesses 2 -periodic solutions. In our investigation, we use the Krylov‐Bogolyubov method of averaging and the results of the theory of numericalanalytic methods for periodic solutions presented in [5]. Consider a system _