Solvability And Stability Of Initial Value Problems For Ordinary Differential Equations With Noninstantaneous Impulses In Banach Space

The utility and application of instantaneous impulsive differential equations in simulating the process and phenomenon of disturbance in a short time is studied,and the disturbance process is discrete,the total duration of the phenomenon is negligible.In other words,the basic characteristic of pulse is suddenly in an instant.The instantaneous pulse interference refers to the process depends on the status and effect for a period of time.The instantaneous impulsive differential equations is an extension of the classical impulsive differential equations.The characteriz of pulse function of time relative to the whole development process is not to be ignored.In recent years,the instantaneous pulse received extensive attention of scholars.As we all know,most of the existence results of fractional order differential equations are obtained in real space R,but are still fresh in the abstract Banach space.Moreover,we propose that the conditions of nonlinear term f are harsh and have not reached the optimal conditions similar to those in integer order differential equations.In view of this,using new tools,methods and techniques,this paper discusses the ex-istence,uniqueness and UHR stability of solutions to nonlinear ordinary differential equations with non-instantaneous impulses and we obtain Lipschitz stability of zero solution to nonlinear ordinary differential equations with non-instantaneous impulsesThe main results of this paper are as follows1.This paper describes the ordinary differential equations with noninstanta-neous impulses and the research status of this topic,gives the structure arrangement of this paper,and finally introduces some basic knowledge and lemma to be involved in the discussion2.Using the fixed point theorem of k-set compression mapping;The monotone iterative method combined with the new non-compactness estimation technique s-tudies the existence of the(1)solution and the uniqueness of the(1)solution by using the compression mapping theorem3.Using the definition of Ulam-Hyers-Rassias stability,the Ulam-Hyers-Rassias stability of(1)solution is studied4.Using the definition of Lyapunov function and Lipschitz stability,the Lips-chitz stability of(2)zero solution is studied.