Bvps Upper And Lower Solutions

In this paper, some boundary value problems of ordinary differential equation are discussed. By using upper and lower solution method, we get many significant conclusions.In the first chapter, we introduce some development of boundary value problems of ordinary differential equation briefly.For conveniently reading, we present some definitions, theorems in the second chapter.In the third chapter, we study second-order two-point boundary value problem and get the sufficient conditions of the existence and uniqueness of solutions. We get the iterative sequence for solving the minimal solution and maximal solution of the boundary value problem. When the conditions of uniqueness of solutions are satisfied, we give the iterative sequence for solving the solution and its formula in error estimate.Still using this method, in chapter four, we study second-order three-point boundary value problem and get the sufficient conditions of the existence and uniqueness of solutions, where 0 <η< 1,δ> 0.In the fifth chapter we study the second-order two-point boundary value problem and get the existence of three solutions. By using the corresponding Green function, the problem is translated into the problem of fixed point of the corresponding operator.Then we get the existence of three fixed points of the operator by using Leray Schauder degree theorem.In the sixth chapter, we study the second-order multi-point boundary value problem and get the existence of three solutions. Where f satisfies Carathéodory condition,and ,m. By using the corresponding Green function, the problem is translated into the problem of fixed point of the corresponding operator. If f satisfies Carathéodory condition,and the condition of at least growth, the completely continuity of the operator can be showed by Lebesgue dominated convergence theorem..And we get the existence of three fixed points of the operator by using Leray-Schauder degree theorem.In the last chapter, still using the same method as the fifth chapter, we study the third-order three-point boundary value problem and get the exietence of three solutions,where 0 <δ< 1,0 <η< b.We give some examples, apply the results we obtained to some specific boundary value problems, and give the judging method for the existence of solution of this kind of boundary value problems. It had the stronger actual application background.