ABSTRACT
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.
