6

FREDERICK A. HOWES

Theorem 1.1. Assume that F = F(t,x,x') satisfies a Nagumo condition with respect to the

pair a,3 which are, respectively, lower and upper solutions of equation (1.1)_on [a,b].

Then if a (a) A 3(a) and a(b) _ B _ 6(b), the boundary value problem (1.1), (1.2) has

a solution x = x(t) € C ^ [a,b] with a(t) _ x(t) _ 3(t) on [a,b].

Theorem 1.1 and the definitions which precede it are taken from Jackson [13]. The

theorem was originally proved in a slightly different form by M. Nagumo in his 1937 paper

[14]. The most interesting feature of the theorem, as far as perturbation theory is

concerned, is the restriction on F imposed by the Nagumo condition. This condition is an

assumption on the growth of the function F with respect to x'. It is clear that in order

to satisfy a Nagumo condition, essentially F cannot grow faster than^the square of x'.

This follows, on one hand, from the observation that, for X 0 , —^

=

°° while for

r

c\

{

s

any 6 0 , ~~?+T °° • On tne other hand, the simple equation of Nagumo [14]

X s

(1.3) x" =

2(x»)3

,

has solutions which always possess points of noncontinuability, i.e., two-point boundary

value problems associated with (1.3) have, in general, no solution.

If we restrict attention to the second-order singularly perturbed problem

(1.4) ey" = f(t,y,y',e) , 0 t 1 ,

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

Definitions 1.1 , 1.2 and 1.3 and Theorem 1.1 will certainly apply, provided f, A

and B are smooth functions of e. We will first of all be interested in the existence of

a solution of (1.4), (1.5) for each fixed e 0, e sufficiently small. If for this fixed

e, the function f satisfies a Nagumo condition for appropriate bounding solutions

a = a(t,e) and 3 = 3(t,e), we can deduce, via Theorem 1.1, the existence of a solution

y = y(t,e) of (1.4), (1.5) which satisfies a(t,e) _y(t,e) _3(t,e). Thus, in addition

to obtaining the existence of a solution, we simultaneously discover how this solution

behaves as a function of t and e.

In their 1952 paper Coddington and Levinson [5] gave the following example of a

problem of the form (1.4), (1.5) which fails to have a solution for small enough e 0:

ey" = -y» -

(y')3

, o t l ,

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