The Upper And Lower Solution Method For Several Classes Of Multi-Point Boundary Value Problems Of Higher-Order Differential Equations

Multi-point boundary value problems (BVPs for short) of differential equa-tions are an important branch in the theory of nonlinear analysis, which arise in different fields of applied mathematics and physics and have wide applications on elasticity and stability theory. Especially, motivated by the wide application, multi-point BVPs of higher-order differential equations have received much attention. At present, it is one of the most active fields that is studied in analytic mathematics. Recently, the existence of solutions to multi-point BVPs of higher-order differential equations has received much attention. The tools used are Leray-Schauder con-tinuation principle, upper and lower solution method, monotone iterative method, coincidence degree theory and so on. Among them, the upper and lower solution method is a powerful tool in obtaining the existence of solutions.This thesis carries out research work in the following three aspects:first, aiming at a class of nonlinear third-order three-point BVP, we establish a new maximum principle and obtain some existence criteria for the nonlinear third-order three-point BVP by using the upper and lower solution method. Second, we discuss a class of nonlinear third-order m-point BVP. By imposing some conditions on the nonlinear term, we construct a lower solution and an upper solution and prove the existence of solution to the nonlinear third-order m-point BVP. Our main tools are upper and lower solution method and Schauder fixed point theorem. Finally, we establish some existence criteria for the nonlinear fourth-order m-point BVP by using the upper and lower solution method and the Leray-Schauder continuation principle.