ChapterI

Basic Existence and Comparison Results

In this first chapter we discuss the existence and comparison theorems which are

used inthe rest ofthe work. Our principal reference isthe paper of L. Jackson [13]

which treats general second-order nonlinear boundary value problems with a varietyof

differential inequality techniques. The theorems presented here will not be proved;

ample references are given. We will, however, discuss the results inthe contextof

singularly perturbed second-order boundary value problems.

We begin with the definitions ofthe concepts that are fundamental toour approach.

Consider the boundary value problem

(1.1) x" = F(t,x,x') , a t b ,

(1.2) x(a) = A , x(b) = B ,

where the function F is assumed to be continuous on[a,b]xR .

Definition 1.1. A function a = a(t) iscalled a lower solution ofequation (1.1) on

[a,b] if a € C(2)[a,b] and a"(t) _ F(t,a(t),a'(t)) on (a,b).

Definition 1.2. A function 3 = 3(t) iscalled an upper solution of equation (1.1) on

[a,b] ifB€ C ® [a,b] and 3"(t) _ F(t,3(t) ,3' (t)) on (a,b).

Definition 1.3. The function F i s said to satisf y a Nagumo condition on [a,b] with

respect to the pai r a,3 € C[a,b] in case a(t ) _ 3(t) on [a,b] and there exist s a positiv e

continuous function (f on [0,°°) such tha t | F ( t , x , x ' ) | £ K|x' |) for a l l a _ t _ b ,

a(t) _ x _ 3 ( t ) , |x' | oo and

00

I ITIY ma x 3(t) - min a(t) ,

[ a ^ atb

where X(b-a) = max{|a(a)-3(b) | , |a(b)-3(a) |} .

(Such a function f will be called a Nagumo function.) With these definitions wecan now

state the principal existence and comparison theorem.

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