SINGULAR PERTURBATIONS

3

discussed most completely by Coddington and Levinson [5],is considered next in Chapter 5.

We obtain explicit boundary layer estimates for the solution y under the original smooth-

ness assumptions in [5], These considerations lead to a study of related differential

inequality theorems for more general problems. Such theorems have been recently treated

by means of the maximum principle by Dorr, Parter and Shampine [7]. Our results amplify

some of theirs by providing explicit boundary layer data and by relaxing some of their

monotonicity conditions.

In the following two chapters we discuss approximation theorems for problems of the

form (0.1), (0.2). We succeed in answering a question of D. Willett [27] and A. Erdelyi

[9] regarding the admissibility of the order of the approximation. Briefly, Willett and

Erdelyi proved that if there exists an approximate solution u = u(t,e) in the sense that

eu" = f(t,u,u',e) + 0(n) + 0Le^e"11^^ , 0 t 1 ,

I )

u(0,e) = A(e) + 0(n) , u(l,e) = B(e) + 0(n) ,

with n = 0(e), then the exact problem

eyM = f(t,y,y',e) , 0 t 1 ,

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

has a solution y = y(t,e) satisfying

y(t,e) - u(t,e) = 0(e) , 0 _t _1 .

The question naturally arises as to the significance of the order 0(e)-restriction to the

error estimate, i.e., if u is an approximate solution of order 0(n), n = o(l), is it true

that an actual solution y exists and satisfies y - u = 0(n)? We show that n = o(l) is

sufficient to guarantee the existence of an actual solution y, and that y - u = 0(n). We

are further able to weaken some of Erdelyi's boundedness restrictions on the function f,

i.e., 82f/9yf8y' = 0(e) is replaced by 82f/3yt8y' = 0(1).

In the last chapter we indicate several applications of the techniques developed in

the course of the paper to some problems not specifically considered. These problems in-

clude turning point problems, a certain quasilinear equation and equations whose right-

hand sides are independent of y'. Finally, the Appendix contains several references to

the other possible approaches to the study of singularly perturbed boundary value

problems of the form (0.1), (0.2).