Study On The Positive Solution Of Two Classes Of Multi-point Boundary Value Problems For Impulsive Differential Equations

Differential equation is a very powerful tool in problem-analyzing and -solving in the modem science and technology, it originates from the production practice and science and technology, and applied to them in return. It has made great contributions to the development of modern science and technology. Along with the development of differential equation, impulsive differential equation is gradually becoming a very important branch of the differential equation, it not only reflects the instantaneous mutations phenomenon of things in the development process, i.e. pulse phenomenon, but also considers the influence of the phenomenon to the entire state. It has very good applications in many fields of science. These instantaneous mutations phenomena tend to affect the researching of the rule in the practical problem, and in recent years obtained the wide attention of scholars, and access to the further development. Its theories are richer than the non-impulsive differential equations, and can more truly reflect the phenomenon of the objective world, so it's with more research significance and value.Multi-point boundary value problems are from various physical and applied-mathematics fields; its studies include the ordinary differential equation, impulsive differential equation, functional differential equation with Laplace operator and differential equation. With the development of the impulsive differential equation and the multi-point boundary value problems theory, scholars begin to pay close attention to the research of the multi-point boundary value problem of impulsive differential equation and have had some achievements.This paper is about the existence of positive solutions to the positive solutions to three boundary value problems for second order impulsive functional differential equations with two parameters and positive solutions of m-Point boundary value problems for second order impulsive equations. I hope that the result is a beneficial supplement to this field. This paper is divided into four parts:First, the purpose and meaning of the research of impulsive differential equations boundary value problem, and the research status and the main content at home and abroad.Second, the related concepts and theorem in the research of the boundary value problems of impulsive differential equation.Third, we employ a well-known fxed-point index theorem (Krasnoselskii fixed point theorem) to study the existence multiplicity and non-existence of positive solutions to three boundary value problems for second order impulsive functional differential equations with two parameters. Several existence non-existence results are established.Finally, based on the Leggett-Williams fixed point theorem, the research of the existence of positive solutions to the second order differential equations with impulses point boundary value problem, namely in some condition, there are at least three positive solutions u₁ , u₂, u₃,to‖u₁‖2)and‖u₁‖3)≤a.