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) .
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