Existence And Stability Of Solutions For Non-instantaneous Impulsive Periodic Systems

In this paper,we mainly study the existence and stability of solutions for noninstantaneous impulsive periodic systems.Firstly,the concept of non-instantaneous impulsive Cauchy operator is introduced and some basic properties such as periodicity and exponential estimation are proved.The influence of non-instantaneous impulsive process on the exponential stability of the system is fully considered.The expression of the solution of linear non-homogeneous non-instantaneous impulsive system is derived by the method of constant variation.In the non-critical case,the concrete expressions of periodic solutions for linear nonhomogeneous non-instantaneous impulsive systems are given.Under critical conditions,the adjoint system of linear homogeneous non-instantaneous impulsive periodic system is constructed.The relationship between the original system and its adjoint system is studied,and the necessary and sufficient conditions for the existence of periodic solutions of linear non-homogeneous non-instantaneous impulsive system are given.By constructing compound Poincar?e operator,using Brouwer fixed point theorem,sufficient conditions for the existence of periodic solutions of semi-linear non-instantaneous impulsive systems are given.Secondly,for non-instantaneous impulsive periodic development systems,the definition of non-instantaneous impulsive Cauchy operator as development family is given by using operator semigroup theory.The existence of periodic mild solutions for linear non-homogeneous non-instantaneous impulsive periodic development systems is proved by the convergence method of weak* and Banach-Alaoglu's theorem,and the upper bound estimates of periodic mild solutions and sufficient conditions for the uniqueness and global asymptotic stability of periodic mild solutions are given.The existence and uniqueness of periodic mild solutions for semi-linear non-instantaneous impulsive periodic development systems are further proved by the contraction mapping principle.In the case of the development family with exponential dichotomy,a Green function on a half line is constructed,and a new form of mild solution is given.Then,the existence and uniqueness of periodic mild solution for linear non-homogeneous and semi-linear periodic development systems are proved,and the norm estimates between mild solution and periodic mild solution for semi-linear non-instantaneous impulsive periodic development systems are given.Finally,some examples are given to verify the correctness of the theoretical results.