Differential Inequalities And Singularly Perturbated Phenomenon

In this paper,by the theory of differential inequalities(upper and lower solutions method),we study the existence(or uniqueness) results of some classes of nonlinear differential equations(without small parameter).Furthermore,we apply the results to singularly perturbed BVPs(Boundary Value Problem for short) which involve small parameters.By the method of boundary layer function,we construct the higher order asymptotic solution and get the error estimate between asymptotic solution and exact solution.We also consider the differential inequality of the boundary value problem of a differential system that doesn't accord with the Nagumo condition.This paper is arranged following.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 upper solution,lower solution and Nagumo condition are showed,two principal theorems of second-order differential inequality are introduced,and some principal lemmals are cited,which will be utilized in the following chapters.In chapter 2,by using the upper and lower solutions method,the existence and uniqueness of a class of boundary value problems for third-order differential equation with a turning point have been studied.Then we construct the higher order asymptotic solution and get the error estimate of asymptotic solution and perturbed by using the differential inequalities theory.In chapter 3,we study the theory of differential inequalities of solutions of a class of two-point boundary value problem for second order nonlinear differential system by the upper and lower solutions method,and get the existence and uniqueness of solutions.Then using the Lypapunov function,we study the differential theory of the differential system with another boundary condition.In chapter 4,we study the existence and uniqueness of a class of two-pointed boudary value problem for second order differential equation by the method of differential inequalities.Then under some conditions,we extend the solution to be a periodic solution of the differential equation.Finally,we consider the perturbed phenomenon.