Discontinuous compartmental periodic Poisson stable functions

Abstract

Among recurrent functions the most sophisticated are Poisson stable functions. For discontinuous functions, there are very few results, for the stability. Discontinuous compartmental Poisson stable functions are in the focus of this research. As the discontinuity points of the functions, a special time sequences, Poisson sequences, are considered. It the first time, the discontinuous functions of two compartments, periodic and Poisson stable, are investigated.  To combine periodicity and Poisson stability, in the case of continuous functions, a convergence sequence with a special kappa property was used [1,2]. For discontinuous functions, this property is not enough, because we also should consider the discontinuity points of the function. For this reason, we need a new concept known as Poisson couple, that is, a couple of a sequence of discontinuity points and convergence sequence that has the kappa property.  Moreover, we meet the challenges for the stability by considering functions on diagonals in the space of arguments. Examples of Poisson stable functions are given to illustrate the theoretical results. The method and results can be effectively used in the study of different types of functional differential equations, impulsive differential equations and differential equations generalized piecewise constant argument, as well as their application.