Study On The Existence And Multiplicity Of Solutions For Several Classes Of Boundary Value Problem Of Nonlinear Differential Equations

Due to the close relation with many problems in the physical, chemical, biological, economic and other fields, the existence and multiplicity of solutions for the boundary value problem of differential equations have become an important research topic. In this Ph.D thesis, by using the methods of nonlinear functional analysis, several nonlinear boundary value problems are studied and some new results about the existence and multiplicity of solutions are obtained, which improve or extend some known results in literatures. This Ph.D thesis is divided into five chapters. Its main contents are as follows:In chapter 1, the background and significance, status and present progress are introduced. Our works are also stated briefly and some preliminaries are given.In chapter 2, by using the critical point theory and the property of projective operator, we first study a class of nonlinear Hammerstein integral equations with quasi-positive definite kernel, and obtain results of the existence of one solution, two nontrivial solutions and finite pairs of solutions under some property conditions, respectively. Then, we apply these results on studying a class of two point boundary value problems of 2nth ordinary differential equations, and obtain the existence and multiplicity of solutions.In chapter 3, we study two classes of differential equation systems with impulse. In the first section, by using the variational method, we study the period boundary value problem of a class of second order Hamilton systems with impulse and obtain the results about the existence and multiplicity of solutions generated by the impulse. In the second one, by using the mountain pass lemma and three critical points theorem, we study a class of p-Laplacian systems with impulse and obtain the results about the existence and multiplicity of solutions generated by the impulse. The results in this chapter show that impulse can influence the existence and multiplicity of solutions.In chapter 4, we study several classes of boundary value problems of fractional differential equations. In the first section, we study the case of the order being in (3,4]. By using the u0 concave operator theory and fixed point theorem combined with iterative method, we obtain the uniqueness and multiplicity of solutions and get the iterative sequence whose initial value is a function easy to compute, or even a constant function. In the second one, by using the u0 concave operator theory, we study the uniqueness of positive solution for a class of boundary value problem of differential equation whose order is in (n-1,n]. In the third one and the fourth one, by using the fixed point index theory and fixed point theorem, we study a class of boundary value problem of fractional delay differential equations and a class of integral boundary value problem of fractional differential equations, respectively, and obtain the existences of solutions for them.In chapter 5, we study the boundary value problem of the fractional equations with impulse. In the first section, we study the nonlocal boundary value problem whose order is in (1,2] and nonlinear term include the first-order derivation of the unknown function. By using the fixed point theorem, we obtain the uniqueness and existence of the solution. In the second one, we study the anti-periodic boundary value problem whose order is in (0,1]. By using the fixed point theorems, we obtain the uniqueness and existence of the solution.