| Differential equations have long history, they arise from practical problems, such as gas dynamics, nuclear physics, fluid mechanics, material mechanics, calculation of trajectory, the stability study of aircraft and missiles flight, the stability research of chemical reaction process and so on. Because differential equations have close contact with actual problems, it can be to solve the actual problems by establishing a proper differential equation(s), it provides very proper mathematical model, so many mechanics, engineering practical problems boiled down to solving differential equations boundary value problems. So, it has important theoretical significance and practical applications to research differential equations boundary value problem. Due to differential equation plays a very important role in the actual problems, so the differential equation has become a new study direction in mathematics field.This thesis mainly research resonance boundary value problems for the existence of solutions, the paper is divided into three parts, main contents are as follows: Firstly, we introduce the historical background of the boundary value problems and research current situation analysis about boundary value problems of resonance at home and abroad and the main content of this paper.Secondly, we study a second-order multi-point boundary value problems at resonance for equations by applying cone tensile - fixed point theorem on a cone, we change the original resonant boundary value problem into the equivalent non-resonant problem, then we transform the differential equations into integral equations, we obtained the condition for the existence of fixed points of integral operator, namely, the existence of the solution of sufficient conditions.Thirdly, using Mawhin coincidence degree theory to study the existence of the solutions for a third-order multi-point boundary value problem at resonance on condition that corresponding operator nuclear space dimension is 3.
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