Research On The Existence And Multiplicity Of Solutions For Some Classes Of Bounday Value Problems Of Nonlinear Differential Equations

This thesis investigates the existence of solutions and positive solutions for some classes of boundary value problems of nonlinear differential equations in abstract spaces, the existence of mulitple positive solutions for a class of n-point nonhomogeneous boundary value problem of nonlinear differential equation in abstract spaces, the existence of solution and multiple positive solutions for two kinds of multi-point boundary value problems of nonlinear differential equations on half-line in abstract spaces, the existence of at least one and three positive solutions for a class of n-points boundary value problems for impulsive dynamic equations with p-Laplacian operator on time scales, the existence of multiple positive solutions for two kinds of boundary value problems of impulsive differential equations with p-Laplacian operator and integral boundary conditions. The thesis consists of six chapters.Chapter 1 is introduction, 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 development on the existence of solutions for boundary value problems of nonlinear differential equations in this thesis is given. We also introduce the background and main results of our works. At last, some preliminary knowledge needed in this thesis is summarized.In Chapter 2, using the Sadovskii fixed point theorem, the Monch fixed point theorem, the theory of Kuratovski noncompactness measure and some techniques of analysis, we discuss the existence of solutions and positive solutions for three classes of three point, multi-point boundary value problems with the nonlinear term f including the first-order derivative for second-order nonlinear differential equations in abstract spaces, and obtain some results which generalize and improve some known results in literatures. In Chapter 3, we study the existence of multiple positive solutions of a class of n-point nonhomogeneous boundary value problem for second-order nonlinear differential equation in abstract spaces by using the theory of strict-set-contraction operator, Schauder fixed point theorem and some techniques of analysis. We obtain some new results.In Chapter 4, firstly, by the Monch fixed point theorem, the Kuratovski noncompactness measure theory, we discuss the existence of solution for a kind of multi-point boundary value problem for nonlinear differential equation on half-line in abstract spaces. Secondly, the existence of multiple positive solutions for a class of second order multi-point boundary value problem on half-line in abstract spaces is considered through applying the fixed point index theory. We obtain some results which extend and improve the related known works in literatures.In Chapter 5, by using the Leray-Schauder fixed point theorem, the nonlinear alternative of Leray-Schauder type and a new functional fixed-point theorem due to Professor Weigao Ge and Jingli Ren, we study the existence of multiple positive solutions for a class of n-point boundary value problems for impulsive dynamic equations with p-Laplacian operator on time scales. We obtain some results which extend the related known works in literatures.In Chapter 6, first, we investigate the existence of multiple positive solutions of a class of boundary value problem for second order impulsive differential equations with p-Laplacian operator and integral boundary conditions. We get some results which generalize some known results. Second, the existence of at least one and triple positive solutions of a class of nth-order boundary value problems for impulsive differential equations with p-Laplacian operator and integral boundary conditions is considered. Some new results are obtained. The main tools are the nonlinear transformation, the Leray-Schauder fixed point theorem, a functional fixed-point theorem in a cone due to Avery and Peterson.