Qualitative Theory Of Pulsed Two-parameter Periodic Systems

Periodic phenomena and impulsive phenomena are very important phenomena in nature,which usually occur in the evolution of systems and have many applications in the management of agricultural diseases and insect pests.In particular,(?,c)-periodic functions cover the classical periodic functions,antiperiodic functions,Bloch functions,etc.,which have generality.Compared with the instantaneous impulse phenomenon,the non-instantaneous impulse phenomenon can more accurately describe that the impulse interference depending on the current state and the duration is not negligible.It has many applications in pharmacy dynamics.In this paper,several kinds of impulsive double parametric periodic systems are analyzed qualitatively.It constructs appropriate Green's functions and adjoint systems by introducing the Cauchy operator of the linear system.It uses the comprehensive theory of constant variability and mathematical induction,nonlinear functional analysis,operator semigroup theory,fixed point theorem,etc.to study the problems of non-instantaneous impulse differential and development(?,c)-periodic system and instantaneous impulse(?,T)-periodic system.First,we study the(?,c)-periodic solutions existence and stability of linear homogeneous time-varying and time-invariant differential systems in finite and infinite dimensional spaces respectively.Firstly,introduce the corresponding Cauchy matrix/operator of the linear homogeneous non-instantaneous impulse differential system,and discuss the related properties of each Cauchy matrix/operator.Secondly,with the help of Cauchy matrices/operator,the specific expressions of the(?,c)-periodic solutions of linear homogeneous non-transient impulsive differential systems are given.Furthermore,the sufficient and necessary conditions for the existence of(?,c)-periodic solutions of linear homogeneous time-varying and time-invariant differential systems are given.Finally,some conditions of Cauchy matrix/operator norm estimation and exponential stability of non-instantaneous impulsive linear differential systems are established.Second,we study the existence and uniqueness of linear nonhomogeneous timevarying and time-invariant differential and evolutionary systems(?,c)-periodic solutions in finite and infinite dimensional spaces respectively.Firstly,according to the Cauchy matrix/operator of non-instantaneous impulse,the expression of the solution of the initial value non-homogeneous linear problem is derived by using the constant variation formula.Secondly,in the noncritical case,we study the concrete expression of the(?,c)-periodic solution of the non-homogeneous linear problem by constructing Green's function using spectral theory.Finally,in the critical case,the necessary and sufficient conditions for the existence and uniqueness of the non-homogeneous linear problem(?,c)-periodic solution and its existence are investigated with the help of the adjoint system.Third,we study the existence and uniqueness of non-linear time-varying and timeconstant differential and evolutionary systems(?,c)-periodic solutions in both finite and infinite dimensional spaces respectively.Firstly,the uniqueness of(?,c)-periodic solutions for nonlinear time-varying and time-invariant differential and evolutionary systems are obtained by using Banach compression fixed point theorem.Secondly,the appropriate complex two-parameter Poincaré operator is constructed.The equivalent theorem of the existence of fixed point of compound two-parameter Poincare operator and the existence of(?,c)-periodic solution is established.Sufficient conditions for the existence of the system(?,c)-periodic solutions are given.Thirdly,semigroup theory and LeraySchauder fixed point theorem are used to proof the existence of(?,c)-periodic solutions in systems.Finally,some sufficient conditions for exponential stability and asymptotic stability of(?,c)-periodic solutions of nonlinear time-invariant and time-varying differential systems are given.Fourth,extend the(?,c)-periodic function to the(?,T)-periodic function.The instantaneous impulse(?,T)-periodic evolution system and Sobolev type impulse evolution system are studied.Firstly,a judgment theorem for the existence of(?,T)-periodic solutions is established,and a sufficient condition for the existence of(?,T)-periodic solutions for instantaneous impulse linear homogeneous systems is given.Secondly,the necessary and sufficient conditions of the existence of(?,T)-periodic solutions for linear non-homogeneous systems are given in both noncritical and critical cases.Thirdly,the existence and uniqueness theorem of(?,T)-periodic solution for instantaneous semilinear impulse problem is established.Finally,the research content of instantaneous impulse(?,T)-period evolution system is extended to a class of Sobolev impulse development system.By using nonlinear functional analysis theory,the existence results of corresponding(?,T)-periodic solutions are obtained under suitable conditions.