Existence And Asymptotic Behaviors Of Solutions In Several Boundary Value Problems For Differential Equations

In this thesis, we study the existence and asymptotic behaviors of solutions in several boundary value problems for differential equations, including the asymp-totic behavior of solutions in second-order semi-linear singularly perturbed boundary value problem, in which, the autonomous and non-autonomous cases are discussed respectively, and the two-point boundary value problem and periodic boundary value problem for four-order differential equation. This thesis is divided into four chapters.Chapter1is the introduction, in which, some basic concepts including singular and regular perturbations, normally hyperbolicity and stability conditions and so on. Also, the recent advance on the study of singular perturbation is reviewed and the main works in this thesis are outlined.In chapters2and3, the asymptotic behavior of solutions in second-order semi-linear singularly perturbed boundary value problem without normal hyperbolic con-dition is studied. We consider the autonomous and non-autonomous cases in chap-ters2and3respectively. Based on an appropriate rescaling, we give the framework to construct the higher order asymptotic expansion by using the method of boundary layer functions. So we obtain the algebraic boundary layers and hence the uniformly valid asymptotic solution. Then with the method of the upper and lower solution, the existence of solutions and the error estimate between the asymptotic and exact solutions are verified. Based on the above results, the information of the asymptotic behavior is available.In chapter4, we study the existence of solutions of a two-point boundary vaule problem and a periodic boundary value problem four-order differential equation under contain conditions. The main technique in the proof is the introduction a transition boundary value problem, which benefits our proof largely.