Several Nonlinear Singularly Perturbed Problems With Infinite Boundary Value Conditions

In this thesis, by the theory of differential inequalities (upper and lower solutions method), we are mainly concerned with the existence, the asymptotic behaviors and the asymptotic expansion of solutions of several nonlinear singularly perturbed problems with infinite boundary values. The paper is divided into four chapters.Chapter1is the introduction, in which, the research background concerning singularly per-turbed problems with infinity boundary values and several related works which had been done in this direction are first introduced, and then, some fundamental lemmas needed in the following chapters are given.In chapter2, the existence and the asymptotic behaviors of solutions in a second-order non-linear singularly perturbed Neumann problem with infinite boundary value is studied. We are concerned with the impacts of the singularity degree in the boundary value on the asymptotic behaviors of solutions. The comparisons between the results of Chang and Howes and the present situation are performed. Also, an example is given to verified the obtained results.In Chapter3, the double boundary layer phenomenon in a second-order nonlinear singularly perturbed Robin problem with infinite boundary value is studied. We construct the boundary layer correction functions and then define a couple of upper and lower solutions. Hence the existence and the asymptotic behaviors of solutions can be shown.In Chapter4, under suitable conditions, the higher-order asymptotic expansion of solution to a singularly perturbed third-order nonlinear differential equation with infinite two-point boundary conditions is studied. We prove the existence of solution and obtain the asymptotic estimates.