chapter
6 Pages

Introduction

In order to give an idea of the methods and to prepare for the subject of this

monograph, consider the following linear elliptic boundary value problem

u+mu = h in ; u = 0 on @ ; (1)

wherem is a nonnegative constant. The solutions of (1) are order preserving,

which means that if u

and u

are two solutions of (1) in some bounded

domain R

N

corresponding to the data h

and h

; respectively, then

the following property holds:

h

h

in implies u

u

in :

This property which immediately implies a uniqueness result is usually

called the inverse monotonicity of the operator L = + mI, and fol-

lows easily from the maximum principle. It is a key property to apply the

monotone iteration method to the nonlinear boundary value problem

u = f(u) in ; u = 0 on @ : (2)

Assuming the existence of an upper solution u and a lower solution u of

(2) such that u u; the existence of extremal solutions, i.e., least and

greatest solutions within the interval [u; u] can be proved by the monotone

iteration method, for instance, if the nonlinearity f : R ! R is continuous

and satises a one-sided Lipschitz condition

f(s

) f(s

) m(s

s

); s

s

: (3)

The monotone iterative technique combined with the upper and lower

solution method has widely been used as a powerful tool to get construc-

tive existence results for dierent kind of problems for both ordinary and

partial dierential equations, cf. [165, 189]. By means of a generalized it-

eration principle this technique has been extended in the monograph [141]

to deal with discontinuous dierential equations, such as problem (2) with

a nonlinearity f satisfying (3) but which need not be continuous. This

was possible, since due to the inverse monotonicity of the elliptic opera-

tor + mI problem (2) can be reformulated as a xed point equation

u = Gu with an increasing (not necessarily continuous) xed point opera-

tor G = (+mI)

(F +mI); where F denotes the Nemytskij operator

related with f: Many dierential operators occurring in models of real phe-

nomena are inverse monotone.