Researches On The Existence Of Positive Solutions Of Nonlinear Sturm-Liouville Boundary Value Problems And The Numerical Solutions

The boundary value problem(BVP) of ordinary differential equations has become an important component part of the discipline of differential equations. Its original research was begun by Sturm and Liouville in 1830s, for solving the BVP of the second order linear differential equations. The BVP has important applications in engineering, finance and other fields, and it is a very powerful tool for analyzing and solving problems in the modern science and technology. But the existence of solution of the BVP is an important research topic, so that we first clarify the existence of solutions and the number of solutions of differential equation, then find its numerical solutions and apply them to actual problems. This thesis mainly studies the second order nonlinear Sturm-Liouville boundary value problems. First using the Green’s function we construct an equivalent integral operator equation with the differential equation. Next the existence of solutions for the differential equations can be converted to study existence problems of fixed points for the corresponding integral operator on a cone of in a Banach space. Via the fixed point theorem on a cone and with the related results of topological degree, we prove the existence of positive solutions of the second order nonlinear Sturm-Liouville boundary value problems.This thesis is structured as follows:In chapter one, the introduction presents the basic knowledge and research status of Sturm-Liouville boundary value problems and some important definitions and lemmas used in this thesis. In addition, this part introduces a method for finding the Green’s function of a class of boundary value problems of second order ordinary differential equations.In chapter two, we study the existence of positive solutions of nonlinear Sturm-Liouville boundary value problems. First of all, by restricting the values of the upper and lower limit of the nonlinear term, the existence of positive solutions of nonlinear Sturm-Liouville boundary value problem is proved under the condition of nonnegative nonlinear term. Then, by improving the existence conditions for the second order nonlinear boundary value problems, we break through the limitation of nonnegative nonlinear term, and obtain the existence theorem of solutions for the BVP with sign-changing nonlinearities and the case of parameters. Finally, we prove the existence of multiple solutions of nonlinear Sturm-Liouville boundary value problems.In chapter three, we transfer on the basis of the given theory, finding the numerical solution of differential equations into discussing the numerical solution of the integral equation of Hammerstein’s type. Firstly, we introduce the common solving method for numerical solutions of nonlinear ordinary differential equation BVP and the classification of nonlinear integral equations. Then, on the basis of the original iterative method, we modify the iterative item such that the speed of new method of iteration is faster than the old one. Finally we give a concrete example, and calculate the numerical solution of the equation by using Matlab. The new acceleration method is effective indeed, compared with the original iterative method.In chapter four, we give the summary of this thesis and some prospects.