Boundary Value Problems And Periodic Solutions Of Impulsive Differential Equations

This thesis mainly studies the existence of solutions for impulsive differen-tial equations singular boundary value problem and boundary value problem, existence of periodic solutions of two kinds of functional differential equations and a neutral impulsive delayed stage-structured predator-prey model, and the minimal period problems for a class of second order impulsive differential equa-tions.This thesis consists of five chapters. It is arranged as follows.In Chapter one, we mainly introduce the background of impulsive differ-ential equations and basic knowledges which will be used later. Meanwhile, we state our main results and the arrangement of the whole thesis.In Chapter two, we consider Multiple positive solutions of the singular boundary value problem for second-order impulsive differential equations on the half-line and Multiple positive solutions of a boundary value problem for nth-order impulsive singular integro-differential equations in a Banach space. Using fixed point theorem in cones, the conditions for the existence of multiple positive solutions are established. The results are new.In Chapter three, using variational methods, we get the sufficient condi-tions for the existence of singer or multi-solutions for the impulsive differential equations with Dirichlet,Sturm-Liouville and integral boundary value condi-tions. In order to get the solutions of impulsive differential system, we find the proper variational structure for the corresponding systems, then come down the problems to find the existence of critical point of some functional.In Chapter four, by variational methods, we investigate the existence of two and infinitely many periodic solutions for tow kinds of functional differen-tial equations. We also discuss the Existence of positive periodic solutions of a neutral impulsive delayed stage-structured predator-prey model using coinci-dence degree thoery. In Chapter five, we mainly consider the subharmonic solutions with pre-scribed minimal period for a class of second order impulsive differential equa-tions. The sufficient conditions for the existence of subharmonic solutions are established. Our results rectify some known results in the literature and will allow in the future to deal with the subharmonic solutions for more extensive impulsive problems.