SINGULAR PERTURBATIONS 7

We remark that the function f(t,y,y',e) = -y1 - (y') does not satisfy aNaguno condition;

thus, this condition is quite sensitive in the setting of problems of the form (1.4),

(1.5).

We shall also consider boundary value problems whose boundary conditions are more

general than (1.5); specifically, problems of the form

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

a1(e)y(0,e) + a2(e)y'(0,e) = A(e) ,

b1Ce)y(l,e) + b2(e)y»(l,e) = B(e) .

Bris [3] attacked this problem with the aid of the following theorem of Nagumo [15].

Theorem 1.2. Suppose g = g(t,y,z) and h = h(t,y,z) are two functions continuous' in the

domain D: 0 £ t _ i, |y-Y(t) | a(t), |y!-Y!(t)l pe" c t + b(t), where &,p and c are

positive constants, Y = Y(t) is a twice continuously differentiable function on [0,£] and

a,b are positive functions continuous on [0,£]. In addition, assume that

[g(t,y,z)-h(t,y,z)| X ,

|h(t,y

2

,z)-h(t,y

1

,z)| _ K|y2-y1| and

h(t,y,z

?

)-h(t,y,z

1

)

-T ; — L 0 ,

Z2 " Zl

where X,K and L are positive constants. Further suppose that Y is a solution of the dif-

ferential equation

h(t,Y,Y') = 0 , Y(0) = A ,

satisfying |Y"| _ M. Then the initial value problem

ey" + g(t,y,y') = 0 , 0 t £ ,

y(0) = A , y'(0) = A' ,

has a solution y = y(t,e) which satisfies the inequalities

|y(t,e)-Y(t) I _

{XK_1+ (pL^+MK"1)^*1 t

, 0 _ t _ I ,

T _1

t

where |A'-Y'(0) | £ p, and_ |y' (t,e)-Y' (t) | £ pe + c^ + c2A, 0 £ t _ £, for_e and X

sufficiently small.