Study On Several Nonlinear Singularly Perturbation Phenomenon

In this paper,by the theory of differential inequalities,we study the existence (or uniqueness) results of some classes of BVP(without small parameter). Furthermore, we apply some of the existence results(or uniqueness) to singular perturbed BVP which involve small parameters. By the method of boundary layer function,we construct the higher order asymptotic solution and get the error estimate of asymptotic solution and exact solution.This paper is divided into four chapters.In chapter 1, firstly,we introduce the background of singularly perturbation theory and some important results of former scholars have studied. Secondly,the concept of lower solution,upper solution and Nagumo condition are showed , two principal theorems of second-order differential inequality are introduced ,and some principal lemmas are cited which will be utilized in the follower chapters.In chapter 2, by the technical of differential inequality ,we prove the existence and uniqueness of the solution to the three-point boundary value problem for third order differential equations. Purthermore,based on former results, we study singular perturbation of three-point boundary value problem to third order quasilinear differential equation and semilinear differential equation, construct the higher order asymptotic solution and get the error estimate of asymptotic solution and perturbed solution.In chapter 3, we: study singularly perturbation of second order quasilinear BVP with turning point. Under the weakness condition, we prove the existence of solution to BVP, and get the error estimate of solution.And the result of this section can be extended to BVP with multi-turning points.In chapter 4, by the theory of differential inequalities and the method of " shooting target ", we prove the existence of solution to the nonlinear boundary value problem for second order nonlinear differential equation. Furthermore, we study singular perturbation of nonlinear boundary value problem for second order nonlinear differential equation, construct the high order asymptotic solution, and get the error estimate of asymptotic solution and exact solution.