Positive Solution To Nonlinear Multi-point Boundary Value Problems And Its Application

Boundary value problems of ordinary differential equations, as one of the important parts of the subject of ordinary differential equation, exist in classical mechanics and electrics pervasively. Ordinary differential equations two-point boundary value problems (e.g. Dirichlet Boundary value problems, Neumann Boundary value problems, Robin Boundary value problems, Strum-Liouville Boundary value problems ) have been widely studied. In fact, since Picard made researches on the existence and uniqueness of second order two-point boundary value problems of nonlinear ordinary differential equations by employing iterative methods in 1893, the study on two-point boundary value problems of ordinary differential has been thriving.Since 20th century, functional analysis has become the important theoretical basis of boundary value problems of ordinary differential equations. In the mid-thirties of the last century, French mathematicians J. Leray and J. Schauder set up the Leray-Schauder theory. By employing the theory, researches made great success in linear ordinary equations, integral equations and functional differential equations. In particular, the application of the theory to boundary value problems of ordinary differential equations forms the topological degree methods and functional methods [55]-[60]. The core of the theory is the foundation and application of fixed point theorem.By employing cone theory, the method of lower and upper solutions, monotone iterative technique, the fixed point theorem of cone expansion and compression and the theory of fixed point index, this dissertation made researches on the necessary and sufficient condition of nonlinear multi-point boundary value problem for the first time and obtained new results, most of which have been published in J. Math. Anal. Appl. (SCI), Appl. Math. Comput. (SCI), Dynamic of Continuous, Discrete and Impulsive Systems (SCI) and Nonlinear Funct. Anal. Appl.This dissertation includes five parts. In chapter 1. a necessary and sufficient condition for the existence of C[0,1] as well as C~1[0, 1] positive solutions is given by constructing lower and upper solutions and with the maximal theorem. Also, the uniqueness, iterative methods and convergence rate of the C~1[0,1] positive solutions are studied.In chapter 2, we continue to discuss the problems in chapter 1 under the weaker condition and obtain similar results.In chapter 3, we investigate a class of singular three-point boundary value problems under the condtion of the nonlinear terms mixed monotone and obtain a necessary and sufficient condition fot the existence of C[0,1] positive solutions. The results extend the main results in chapter 1 and 2.In chapter 4, we investigate a class of singular three-point boundary value problems under the assumptions that the nonlinearity mixed monotone and obtain a necessary and sufficient condition fot the existence of smooth positive solutions. We also pointing out that the smooth positive solution is exact the unique C[0,1] positive solution. The results in this chapter complement the main results in chapter 3.In chapter 5, we investigate the existence and uniqueness of positive solutions to a class of singular m-point boundary value problems of second order ordinary differential equations. A sufficient condition for the existence and uniqueness of C[0,1] positive solutions as well as C~1[0,1] positive solutions is given by constructing lower and upper solutions and with the maximal theorem. What's more, we do not require the existence of upper and lower solutions.