The Singularly Perturbed Problems Of Some Classes Of Nonlinear Differential Equations

In this paper, there are two parts.In the frist part, main by the theory of differ-ential inequalities(upper and lower solutions method), we study the existence results of solution (or positive solution) of two classes of singular perturbation boundary value problems with a small parameter. Furthermore, 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. In the other part, based on the Method of Variation of Arbitrary Constant and the Principle of Compressed Map-ping, the quasi-diagonalizated technique of one class of singularly perturbed system with two parameters is considered. This paper is arranged as follows.In chapter 1, firstly, we introduce the background of singularly perturbation theory and some research of former scholars have studied. Secondly, the concepts of upper solution, lower solution, Dichotomy Method and Nagumo condition are showed, some fundamental results of second-order and third-order differential in-equality are introduced, and some lemmals are cited which will be utilized in the following chapters.In chapter 2, the existence of positive solution of singularly perturbed Robin boundary value problem for second-order ordinary differential equation is obtained. Moreover, the error estimate between the exact solution and the reduced solution is derived simultaneously under the suitable stability conditions.In chapter 3, the construction of higher order asymptotic solution of nonlinear three-point boundary value problem for third-order semilinear differential equations with singular perturbation is studied; Then using the relevant theory of differential inequalities,prove the existence of solution and get the error estimate of asymptotic solution and exact solution; at last, one example is given and the result of this paper is verified.In chapter 4, through twice linear-transform,consider one class of singularly perturbed system with two parameters. Based on the Method of Variation of Ar-bitrary Constant and the Principle of Compressed Mapping, proof the existence for the discussion above linear-transform, then the system is quasi-diagonalizated with some conditions.