discussed most completely by Coddington and Levinson ,is considered next in Chapter 5.
We obtain explicit boundary layer estimates for the solution y under the original smooth-
ness assumptions in , 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 . Our results amplify
some of theirs by providing explicit boundary layer data and by relaxing some of their
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  and A. Erdelyi
 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 ,
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).