Homoclinic Solutions And Boundary Value Problems Of Impulsive Differential Equations

Variational methods and critical point theory are not only the main re-search objects of non-linear functional analysis, but also powerful tools for s-tudying differential equation solutions. This dissertation mainly discusses the bound value problem of impulsive differential equations, periodic solutions, and homoclimic orbits in the theoretical framework of variational methods and crit-ical point theory.This dissertation consists of four chapters and the arrangement is as follows.In Chapter one, an overview of the research status quo of impulsive differen-tial equations and some theories of non-linear functional analysis are provided, including some important theorems of critical point.In Chapter two, applying variational methods and critical point theorems, firstly, we discuss second-order impulsive differential equations, coming up with the sufficient conditions for the unique solution and infinitely many solutions; secondly, we discuss the parameter-controlled Sturm-Liouville impulsive differential equation in the right part of the following equation, and come up with the sufficient conditions for the infinitely many solution-s;thirdly, we discuss the parameter-controlled forth-order impulsive differential equations, come up with the sufficient conditions for the infinitely many solutions; finally, we discuss ODE with p-Laplacian, and come up with the new sufficient conditions for the existence of three solu-tions. All the results mentioned above are newly found.In Chapter three, guided by variational methods and critical point theo-rems, we first investigate the parameter-controlled second-order impulsive dif-ferential equations in the right part of the following equation, and come up with the sufficient conditions for the existence of periodic solution; then we investigate the second-order impulsive Hamiltonian systems, among which is the F(t,u)=W(u)+H(t,u), and come up with the sufficient conditions for the existence of periodic solution, so the results in the relative researches have been both improved and extended; lastly, we investigate the second-order Hamiltonian systems, and come up with the sufficient conditions for the existence of periodic solution under the new conditions. The findings in this chapter have made the results in the relative researches both improved and extended. In Chapter four, with the inspiration of the mountain pass theorem, we study the second-order impulsive Hamiltonian systems, among which is the V(t, x)=-K(t, q)+W(t, q), and come up with the newly found sufficient conditions for the existence of homoclimic orbits under more common conditions, and the results in the relative researches have been both improved and extended; secondly, we discuss the impulsive Hamiltonian systems with p-Laplacian, and come up with the newly found sufficient conditions for the existence of homoclimic orbits resulted from impulses under new conditions. The findings in this chapter have made the results in the relative researches both improved and extended.