The Existence Of Positive Solutions For Third Order Multipoint Boundary Value Problems With P-Laplacian Operator

Differential equations problems arise from many fields, such as geometry, physics and astronomy and so on. Boundary value problems are consisted of differential equation and a group of constraint conditions which called boundary conditions. For example, in physics, the distance from the both ends of the vibrating string to the equilibrium position and the heat of the endpoint in thermal thin rod are both boundary value problems. As an important part of the differential equations, boundary value problems also have a range of applications in real life, such as underground water flow, thermo-elasticity and chemical engineering and so on. So many researchers are very concerned about studying the boundary value problems for differential equations.In recent two decades, many researchers have done depth research for boundary value problems in recent two decades and have solved many problems by fixed point theorems. Boundary value problem with p-Laplacian operator is an important part of boundary value problems. And researchers have also made great progress in recent years. Aiming at these problems, we discuss the existence of positive solutions of the third order boundary value problems for differential equations with p-Laplacian operator, which includes three-point boundary value problem with two-order derivative p-Laplacian operator, m-point boundary value problem with one-order derivative p-Laplacian operator and m-point boundary value problem with two-order derivative p-Laplacian operator. Firstly, we solve the unique solution of the linear boundary value problem which corresponds to the original boundary value problem by using the konwledge of the uniqueness of solution in mathematical analysis, and then get the Green’s function. At last, combining with the related properties of Green’s function, we get the conclution that the boundary value problems have at least three positive solutions by using the Avery-Peterson fixed point theorem. The results improve the theory of seeking boundary value problems for differential equations with p-Laplacian operator.