2 FREDERICK A. HOWES

solution of the terminal value problem

u1

= 0 , u(l) = 1 ,

obtained by setting e = 0 in (0.3) and cancelling the lefthand boundary condition in

(0.4). For more general nonlinear problems (0.1), (0.2) we will proceed in basically the

same manner, aided by a powerful theorem from the theory of second-order boundary value

problems. That is, we will use the solution of an appropriate reduced or approximate

problem to deduce both the existence and the asymptotic behavior as E - 0 of a solution

of the full problem (0.1), (0.2).

The method we use in the work to follow is not a new one. Indeed, it was introduced

more than twenty years ago by N. I. Bris [3]. The method consists of using the reduced

or approximate solution to construct lower and upper solutions (to be defined in the next

chapter) of the original equation and then applying a Nagumo-type existence and comparison

theorem. Brio's techniques have not received much attention; perhaps this is due to the

fact that his paper is written in an abbreviated style which leaves most of the proofs to

the reader. However, his method of attack seems sound and deserving of wider application,

and we hope that the material offered here will support this judgment.

We attempt to give a unified presentation of some of the classical problems and

questions involving singularly perturbed nonlinear second-order boundary value problems.

In the process of doing this, we make some useful extensions of known results, useful, in

particular, to the applied mathematician who is faced with the difficulty of solving

actual problems. This is especially evident in the chapters on approximation theory.

The first chapter contains,the mathematical tools and concepts which are employed in

the later chapters. In the next chapter we prove completely a fundamental result of

Bris [3] in a more general setting. This result and the method used to establish it form

the basis for the rest of the presentation.

In Chapter 3 we apply this theorem of Bris to extend a result of A. Erdelyi [8],

Erdelyifs

approach is discussed as well as the effects of our weakening of his hypotheses.

Chapter 4 treats boundary value problems with coupled boundary conditions from the

point of view of Bris. The theorem of Erdelyi [8] is then extended in this more general

setting.

The classical quasilinear problem

ey" + g(t,y)y' + h(t,y) = 0 , 0 t 1 ,

y(0) = A , y(l) = B ,