Boundary Layer Phenomena For Several Classes Of Singular Perturbed Problems Of Second-order Nonlinear Equations

In this thesis, we mainly study boundary layer phenomena for several classes of singularly perturbed problems of second-order nonlinear equations. Under the principal assumption that the reduced solutions are locally weak stable, the existence of solutions which exhibit boundary layer behavior are proved and the asymptotic estimation of solutions is given using the method of bounding functions and the theory of differential inequality.This thesis consists of four chapters:In the first chapter, we mainly introduce the research significance and situation of the singular perturbed problem, introduce the origin and the research significance of the boundary layer phenomenon, give the main lemmas which we use, the main work and the innovation of this thesis.In the second chapter, we mainly discuss the singular perturbed Dirichlet prob-lems of second-order nonlinear equations with locally weak stable reduced solutions εy"=F(t,y,y’), a<t<b, y(α,ε)=A, y(b,ε)=B by constructing auxiliary problem, selecting bounding function and using the esti-mate of quantity order. Under the six different conditions, we prove the existence of the solutions and give asymptotic estimate of solutions by the theory of differential inequalities. Finally, two examples are given to illustrate the applying value of the study achievements. In the third chapter, we consider the singular perturbed Robin problems1of second-order nonlinear equations with locally weak stable reduced solutions εy"=F(t,y,y’),α<t<b, y(a,ε)-p1y’(α,ε)=A, y(b,ε)+p2y’(b,ε)=B by comparing the equation, constructing bounding function and using the techniques of inequality amplifying. Under the three different conditions, we prove the existence and asymptotic behavior of solutions by the theory of differential inequalities. Finally, we give an example to illustrate the applying value of the study achievements.In the fourth chapter, we mainly talk about the singular perturbed Robin prob-lems2of second-order nonlinear equations with locally weak stable reduced solutions εy"=F(t,y,y’),α<t<b, y(α,ε)-p1y’(α,ε)=A,y(b,ε)=B by comparing the equation and constructing bounding function. Under the six dif-ferent conditions, we prove the existence and asymptotic behavior of solutions by the theory of differential inequalities. We give two examples to illustrate the applying value.of the study achievements.