Zhang, Yanlei

In this thesis, we study modelling with non-linear ordinary differential equations, and the existence of positive solutions for Boundary Value Problems (BVPs). These problems have wide applications in many areas. The focus is on the extensions of previous work done on non-linear second-order differential equations with boundary conditions involving first-order derivative. The contribution of this thesis has four folds. First, using a fixed point theorem on order intervals, the existence of a positive solution on an interval for a non-local boundary value problem is obtained. Second, considering a different boundary value problem that consists of the first-order derivative in the non-linear term, an increasing solution is obtained by applying the Krasnoselskii-Guo fixed point theorem. Third, the existence of two solutions, one solution and no solution for a BVP is proved by using fixed point index and iteration methods. Last, the results of Green's function unify some methods in studying the existence of positive solutions for BVPs of nonlinear differential equations. Examples are presented to illustrate the applications of our results.

Author Keywords: Banach Space, Boundary Value Problems, Differential Equations, Fixed Point, Norm, Positive Solutions