Existence Of Solutions And Oscillation Analysis For Some Boundary Value Problems To Nonlinear Differential Equations

This dissertation is mainly concerned with multi-point boundary value problems of nonlinear differential equations with the resonance case and non-resonance case, discrete boundary value problems for difference equations, even-order distributed delay differential equations with damping and singularly perturbed boundary value problems for nonlinear differential systems. We study the existence of solutions and the uniqueness of solution, positive solutions and the multiplicity of positive solutions and the oscillation. The proofs of the results are based Mawhin's coincidence degree theorem, differential inequality technique, topology degree theory, fixed point theorems in cones and the methods of singular perturbation respectively.The whole thesis contains five chapters.The first chapter introduces concisely the historical situation, the present development and the research background of dynamical systems and boundary value problems for nonlinear differential equations and nonlinear oscillation, as well as the main work done in this thesis.Chapter 2 devotes to the study of the solvability of third order multi point boundary value problems at resonance with the case dim Ker L=1 and dim Ker L=2, respectively. We also discuss the solvability of multi point and nonlocal boundary value problems at resonance for higher order differential equations. The method is based upon the theory of Mawhin's coincident degree. The approaches are different from those used in the past and the results improve and extend what are given in the relevant literatures.In chapter 3, we first study a kind of third order boundary value problem with nonlinear multi-point boundary conditions. The existence and uniqueness of solutions are given by use of upon lower and supper solutions methods and the theory of Leray-Schauder degree. We also investigate the multiplicity of solutions for a n order three-point boundary value problem with nonlinear terms depending explicity on the higher order derivative. Compared with the work done by others, we mainly deal with more generalized nonlinear boundary conditions. Our results improve and extend some work in the past.In chapter 4, by applying Leggett-Williams fixed point theorem, the Green's function and exploring its properties, the existence of the multiplicity of positive solutions for second order difference equations with three-point boundary value problems is given. Similarly, the multiplicity of second order boundary value problems on infinite intervals is discussed. Our theorems include some available results as special cases.In chapter 5, by using the generalized Riccati technique and an inequality due to Hardy, Littlewood and Polya, several new oscillation criteria are established to even-order distributed delay differential equations with damping. We also apply differential inequality theory, fixed point theorem and singular perturbation theory to study the existence of solution and the asymptotic estimate of solution for a singularly perturbed three-point boundary value problem for nonlinear differential systems. The results obtained extend and improve earlier results in existing literature.