Composition of piecewise pseudo almost periodic functions and applications to abstract impulsive differential equations

Junwei Liu,Chuanyi Zhang

doi:10.1186/1687-1847-2013-11

Abstract

In this paper, we introduce the concept of piecewise pseudo almost periodic functions on a Banach space and establish some composition theorems of piecewise pseudo almost periodic functions. We apply these composition theorems to investigate the existence of piecewise pseudo almost periodic (mild) solutions to abstract impulsive differential equations. In addition, the stability of piecewise pseudo almost periodic solutions is considered.

Highlights

The notion of pseudo almost periodic functions was introduced by Zhang as a natural generalization of the classical concept of almost periodic functions in [, ]
The study of impulsive differential equations is important [ – ] because many evolution processes, optimal control models in economics, stimulated neural networks, population models, artificial intelligence, and robotics are characterized by the fact that at certain moments of time they undergo abrupt changes of state
The existence of almost periodic solutions of abstract impulsive differential equations has been considered by many authors; see, e.g., [ – ]

Summary

Introduction

The notion of pseudo almost periodic functions was introduced by Zhang as a natural generalization of the classical concept of almost periodic functions in [ , ]. We give some results about the existence and stability of piecewise pseudo almost periodic solutions to the following abstract impulsive differential equation: A function f ∈ PC(R, X) is said to be piecewise pseudo almost periodic if it can be decomposed as f = g + h, where g ∈ APT (R, X) and h ∈ PAP T (R, X).

Results

Conclusion