Existence And Multiplicity Of Solutions For Impulsive Boundary Value Problem Via Critical Point Theory

In this dissertation, the existence and multiplicity of solutions for impulsive boundary value problems are considered by using critical point theory. A class of impulsive differential equation with Dirich-let boundary conditions, a class of Sturm-Liouville boundary value problem with impulsive effects and a class of impulsive damped prob-lem are included. In short, we first reduce the solutions of impulsive boundary value problem to the critical points of corresponding func-tional on suitable space, then consider the existence and multiplicity of the critical points for the functional, thus the existence and mul-tiplicity of solutions for the impulsive boundary value problem are obtained. The organization of this dissertation is as follows.In the first chapter, impulsive differential equation and critical point theory are briefly introduced. After that, some basic facts in critical point theory, which will be used in the proof of our main re-sults in this dissertation, are recalled. Then we present the recent development of impulsive differential equations related to our prob-lems. At the same time, the main contents of this dissertation are also outlined.In Chapter2, a class of impulsive differential equation with Dirich-let boundary conditions is considered. Multiplicity results are ob-tained via critical point theory. It is worth stressing that the exis-tence of at least three solutions has not been considered before and the existence of infinitely many solutions in this chapter is different from the result in the literature. Examples are presented to illustrate the feasibility and effectiveness of the results.In Chapter3, a class of p-Laplacian Sturm-Liouville boundary value problem with impulsive effects is considered. By using critical point theory, some criteria are obtained to guarantee that the impul-sive boundary value problem has at least one solution, two solutions and infinitely many solutions when the parameter lies in different in-tervals. Moreover, multiplicity of solutions is also obtained by using three critical points theorem. Examples are presented to illustrate main results.In Chapter4, solvability of a class of impulsive damped problem, which can be reduced to impulsive Hamiltonian system and Hamilto-nian system, is considered via critical point theory. The existence of solution for Hamiltonian system has been studied extensively. Inspired by the previous results, the aim of Chapter4is to study solvability of the impulsive damped problem under some more general conditions. It is a remarkable fact that the obtained results are also valid and new even if the impulsive damped problem is reduced to impulsive Hamiltonian system and Hamiltonian system.