Differential Research And Improvement For Singularly Perturbated Boundary Value Problems Under Nagumo Condition

In this paper, taking advantage of differential inequalities(upper and lower so-lutions method),we prove the existence results of some classes of boundary value problems for nonlinear differential equations without small parameter. Further-more, we apply the results to singularly perturbed boundary value problems which involve small parameter by constructing the higher order asymptotic solution and getting the error estimate between asymptotic solution and exact solution.We also pose a condition which substitutes for Nagumo condition, and study the differential inequality of some kinds of two order nonlinear two-point and n-point boundary value problems which don't satisfy the Nagumo condition. This paper is divided into four chapters.In chapter 1, firstly, we introduce simplely the background of singularly pertur-bation theory and some important results which former scholars have get. Secondly, the concept of upper solution, lower solution and Nagumo condition are introduced, then some principal results of second-order and third-order differential inequality are showed, and some principal lemmals are introduced which will be used in the following chaptersIn chapter 2, we study a class of two order nonlinear n-point boundary value problems with small parameter, prove the existence of solution and obtain the error estimate of the solution under the conditions of strong stability and weak stability respectively.In chapter 3, we consider a class of two point boundary value problem with singular perturbation for third-order nonlinear differential equation. By using the boundary layer function method at first,the higher order asymptotic solution is constructed,then we prove the existence of solution and obtain the error estimate of the solution by using Banach contracion mapping principle.In chapter 4, we pose a condition which substitutes for Nagumo condition, study the differential inequalities of some two order nonlinear boundary value problems whose right functions don't satisfy the Nagumo condition and get the existence of solutions.