The Existence Of Positive Solutions For Nonlinear Third-order Three-point Boundary Value Problems

This paper studies the two classes of nonlinear third-order boundary value problem, it gives sufficient conditions for existence of positive solutions by studying the three-point boundary value problems, and it uses the fixed point theorem to proved the solutions existence. It also discuss if the boundary value problems has positive solutions when the parameter changes.In the first chapter, it describes the research background of the nonlinear third-order differential equations boundary value problem and the research status at home and abroad. It introduces the main work in this paper at the same time.In the second chapter, it studies the third-order three-point boundary value problem in the following conditionsThe third-order three-point boundary value problem is converted into the second-order boundary value problem and the equivalent integral equation was derived by order reduction method and appropriate transformation. It constructs Green function and gives estimate of upper and lower bound. It also proved the existence of positive solution of boundary value problem by using cone fixed point theorem.In the third chapter, it studies nonlinear third-order three-point boundary value problem that contains the following parameterWe still use the order reduction method to convent third-order boundary value problem into the second order boundary value problem and the equivalent integral equation is derived. By using the Krasnoselskii fixed point theorem, it discusses if the problem has the positive solutions when the parameter ? changes. It proved the existence of*? ?(0,??).It also proved when parameters*? ?(0,?], boundary value problem has at least one positive solution; When*? ?(?,??), it has no positive solution.