Existence Of Solutions To Several Kinds Of Impulsive Differential Equations

As an important branch of the differential system,the impulsive differential equation is the main tool for studying impulsive phenomena.It can accurately describe the effect of the emergent phenomena on the system,and integrate the characteristics of continuous and discrete systems.Therefore,it is widely used in engineering,science,medicine and other fields.The research on the existence of its solution has more important theoretical significance and value.For example,in engineering control problems,by studying the existence of the solution,it can explain the controllability of the system,and reduce control costs.This thesis mainly studies the existence of solutions of several types of differential equations with impulses,and some sufficient conditions for the existence of solutions of the equations are obtained.The content consists of the following five chapters.The first chapter summarizes the research background,the main work of this thesis,and the preliminary knowledges such as definitions and lemmas used in this thesis.The second chapter considers the lack of positivity of the solution of a class of nonlinear second-order ordinary differential equations with instantanous impulses.Discuss that when the parameters in the boundary value conditions of the equations increase,the associated integral equation has a kernel that changes sign,so that the positivity of the solution of the equation is missing.The existence of nonzero solutions of the equations is established using the fixed point theorem of cone compression and cone extension under the condition that the nonlinear terms satisfy the superlinear and sublinear respectively,and the main results are explained with an example.The third chapter mainly studies the existence of solutions for a class of noninstantanous impuisive evolution systems with memory.The content of this chapter can be divided into two problems to discuss.The first problem discusses the existence of mild solutions and strong solutions for an evolution system with a finite number of noninstantanous impulses,and the mainly results are provided by the principle of Banach compression mapping and Krasnoselskii?s fixed point theorem;the second problem changes the nonlinear term on the basis of the first problem,and discusses the existence of bounded mild solutions for an evolution system with infinite noninstantaneous impulses on infinite intervals,the existing results are proved by the measure of non-compactness and Darbo?s theorem.The fourth chapter focuses on the existence of solutions for a class of noninstantaneous impulsive integro-differential equations with resolvent operator.The sufficient conditions for the existence of solutions are established by resolvent operator theory and Scheafer?s fixed point theorem,and an application of the abstract results is given.The fifth part summarizes the work of this thesis and prospects the future research direction of this kind of problems.