Two Singularly Perturbed Boundary Value Problems Of Nonlinear Differential Equations

Singularly perturbed theory is one of the powerful tools to deal with nonlinear problems,which have important applications in celestial mechanics,fluids mechanics,optical,chemical,biomathematics and cybernetics.In recent years,singular perturba-tion method is used to study singularly perturbed system and boundary value problem,which get a lot of attention.In this paper,by nonlinear analysis technique and the theory of differential in-equalities,we study the existence results of two classes of boundary value problems without small parameter for nonlinear differential systems.Furthermore,by construct-ing appropriate a lower solution-upper solution pair,as well as analysis technique,the existence and asymptotic estimates of solutions for the singularly perturbed differential systems are established.This paper includes three chapters as follows:In chapter 1,we introduce the backgrounds and significance of our studies briefly.Next,we introduce some important results of former scholars have studied.Further-more,we introduce the main work of this paper.In chapter 2,we consider a class of third-order nonlinear differential equation with singular perturbation subject to three-point boundary value conditions.By using the Schauder fixed point theorem,Green's function and the method of upper-lower so-lutions,we first establish an existence result of corresponding boundary value problem without perturbation.Furthermore,by constructing an appropriate lower solution-upper solution pair and boundary layer,the existence and asymptotic estimates of the solu-tions for the singularly perturbed differential equation are obtained.In chapter 3,we deal with a third-order singularly perturbed nonlinear multi-point boundary value problem for nonlinear vector differential systems.By using the topo-logical degree theory,Nagumo condition and the upper and lower solutions method,we present the existence of solutions for the nonlinear differential systems.By the eigen-values of the comparison equation to construct an appropriate lower solution-upper solution pair,the existence,uniqueness and asymptotic estimates of solutions for the singularly perturbed differential systems are established.