The Existence Of Solutions Of Higher Order Differential Equation With Impulses

As a powerful tool, variational method is applied more and more on the s-tudy of boundary-value problems with impulsive effects, especially of the prob-lems with Neumann or Dirichlet boundary conditions. The increasing research-es, generating more critical points theorems, enrich the theory itself in return.By using the variational method with the latest critical points theorems, the existence of multiple solutions and positive solutions are studied, multi solution problem based on four order Sturm-Liouville differential boundary value, also containing the p-Laplace operator of order four Sturm-Liouville pulse differ-ential solution of the boundary value problem of existence and multiplicity of. The full text is divided into six chapters:The first chapter is the introduction, briefly introducing fourth-order Sturm-Liouville boundary-value problems with impulsive effects and some common methods to study them, the variational method with its historical background· and research status is also introduced, followed by the main research contents of this thesis.The second chapter introduces some basic concepts and theorems, prepar-ing for the rest study.The third chapter studies the application of variational method in the ex-istence of multiple solutions of the fourth-order Sturm-Liouville differential boundary-value problem with impulsive effects. Given some different assump-tion, we get results by using several critical point theorems.The fourth chapter, with transforming the original problem to a new boundary-value problem, applies a three-critical-point theorem to the study of positive solutions of the fourth-order Sturm-Liouville differential boundary-value problem with impulsive effects.The fifth chapter studies he fourth-order Sturm-Liouville differential boundary value problem with impulsive effects when a p-Laplace operator is added to the problem.Finally in the sixth chapter, the research contents of this paper are summa-rized, and the possible follow-up research is prospected.