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. 5
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