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
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.
Previous Page Next Page