Research On The Existence Of Solutions For Some Class Of Boundary Value Problems Of Differential Equations

This thesis investigates the existence of solutions for some class of boundary value problems of differential equations. The existence of positive solutions for several class of nonlinear differential equations (systems) boundary value problems are obtained by using the fixed point theorems (including functional fixed point theorems on cone); We investigate the existence of solutions for two kinds of boundary value problems of nonlinear differential equations by applying the monotone iterative technique,the method of upper and lower solutions and the quasilinearization method, respectively; We study the existence of solution for a class of boundary value problem at resonance on the half-line by the degree theory. The thesis consists of six chapters.Chapter 1 is preface, the historical background of the problems and the significance of this thesis are introduced. Boundary value problems occur in various fields of natural science. The existence of solutions for boundary value problems attracts much attention of many famous mathematicians all around the world. The recent developments on the existence of solutions for boundary value problems of nonlinear differential equations in this thesis are given. And some knowledge needed in this thesis are also summarized.In Chapter 2, using the Krasnosel'skii fixed point theorem,the fixed-point index theory on cone and some techniques of analysis, we discuss the existence of positive solutions for four kinds of multi-point boundary value problems of second-order nonlinear differential equations, and obtain a set of new results which generalize and improve some known results in literatures.In Chapter 3, firstly, applying the quasilinearization method, we study the existence of solutions to a class of boundary value problem of differential equation with nonlinear boundary conditions. Secondly, we discuss the existence of solutions to a class of boundary value problem of second-order nonlinear impulsive functional differential equation by using the monotone iterative technique and the method of general upper and lower solutions. We obtain some results which extend and improve the related known works in literatures.Chapter 4 deals with boundary value problems on time scales. By using functional fixed point theorems, we study the existence of multiple positive solutions for a class of boundary value problem of nonlinear impulsive dynamic equation with p -Laplacian operator on time scales and the existence of multiple positive solutions for a type of boundary value problem of nonlinear impulsive functional dynamic equation with p -Laplacian operator on time scales. We obtain some news results.In Chapter 5, we study the existence of positive solutions for boundary value problems of nonlinear differential system. Using the Krasnosel'skii fixed point theorem, we investigate the existence of multiple positive solutions for a nonlinear differential system of second-order multi-point boundary value problem with variable parameters. By the Leggett-Williams functional fixed point theorem, we study the existence of multiple positive solutions for a class of second-order differential system of Sturm-Liouville nonlocal boundary value problem. We obtain some news results.In Chapter 6, we investigate the existence of solution to a kind of multi-point boundary value problem with p -Laplacian operator at resonance on the half line. The main tool is the Ge theorem, which is more general than Mawhin continuous theorem. New results are obtained.