Existence And Stability Of Solutions For Initial Value Problems Of A Class Of Non-transient Impulsive Differential Equations

Non-instantaneous impulsive refers to the process depends on the state and effects for a period of time.The intravenous injection is a typical non-instantaneous impulsive therapy behavior in clinical medicine.Considering medicine is transforming in absorption,distribution and metabolism,and it indeed exists resistant virus,thus,it is more suitable for describing the dynamics of clinical medical treatment process by using noninstantaneous impulsive differential equations.This paper researches a class of linear non-instaneous impulsive differential equations.At first,we introduce the notation of non-instantaneous impulsive Cauchy matrix(2(·,·),and give the solution expressions of linear homogeneous and inhomogeneous initial value problems.Secondly,in terms of eigenvalues of matrix via the distance between impulsive points and junction points,we derive some exponential estimation about(2(·,·),even obtain many sufficient conditions about linear equations and perturbation equation zero solutions which are asymptotically stable.Then on this basis,introducing an appropriate weighting function space and using the classical contracting mapping principle,we discuss existence and uniqueness of solution about nonlinear problem.Thirdly,introducing the notation of non-instantaneous impulsive differential equations Ulam Hyers-Rassias stability,many sufficient conditions Ulam–Hyers–Rassias stability of nonlinear non-instantaneous impulsive equations are also derived.