The cornerstone of applications of order-theoretical
fixed point methods to analysis so far
appears to be the Abian-Brown-Pelczar theorem:
In a chain-complete ordered set P every order-preserving map for which
there is a with has a fixed point.
Abian and Brown prove it in [2] as a result on ordered sets.
Pelczar in the introduction of
[84] mentions the possible application to
integral inequalities, but gives no specific examples.
Recently these methods have been used sucessfully by Heikkilä,
Lakhshmikantham, Carl and Sun to prove existence results for
solutions for various differential and integral equations
(cf. [16], [17], [49]-[57]).
Similar iterations are used by Heikkilä and Salonen
to derive results in game theory (cf. [58]).
Due to the often very specific nature of results in nonlinear
analysis we will only present one example and the order-theoretical
iteration method that is used. The presentation is only intended
to give a general idea of the arguments
(the order-theoretical results can be derived fairly
quickly and for a complete introduction to nonlinear
analysis there certainly is not enough room in this paper).
For precise proofs the
reader is referred to [49]. For a larger set of results and
references cf. the text [56].
The underlying order-theoretical iteration used is similar in spirit to
Abian's f-chains (cf. [1]) respectively
Cousot and Cousot's iteration in [20] and [21].
Note however that it is neither assumed here that the function
is order-preserving, nor that the underlying set is
chain-complete.
Clearly this sequence would be the maximal sequence generated by a transfinite iteration scheme (cf. [49], Lemma 1.1) such that
where the iteration stops when one of the steps 2 or 3 cannot be carried out. As noted in Theorem 1.1 in [49], the increasing sequence of G-iterations always exists and is unique. Moreover the proof of this result does not depend on the Axiom of Choice (a similar point was made in [1] for the Abian-Brown-Pelczar theorem). Unfortunately due to the extremely weak hypotheses the sequence might have only one element, which need not be a fixed point. However ([49], Lemma 2.1) if the increasing sequence of G-iterations has a last element such that , then is a fixed point of G. (Otherwise we could extend the sequence by adding contradicting the maximality of the sequence.)
Condition 2 insures that nothing goes wrong at successor
ordinals and condition 3 insures the existence of all
needed suprema.
Condition 3 looks a little strange to the order-theorist,
especially considering that ([49], Lemma 2.2) it is equivalent to
the simpler condition 3 with n=0. However when applying
the result in analysis it is sometimes easier to prove condition 3
for an than it would be to prove it for n=0.
The included example of a Hammerstein integral equation is
an example.
This explains the format of condition 3.
It would be interesting to see how far this idea can
be taken under mild hypotheses on the underlying ordered set.
It is easy to see that every space , where
and
is a measure space, is conditionally complete
(i.e., every set with an upper bound has a lowest upper bound) with
the pointwise almost everywhere order.
Thus conditional completeness might be a good hypothesis.
Then one could try to devise an iteration as described above
also for the case for which is not comparable
to : As long as all is bounded above, take the supremum
as the next element of the sequence.
This is similar to the approach in [21], yet the author would
hope that milder conditions than assuming the underlying set is a complete
lattice would lead to success.
As an example let us consider the
following Hammerstein integral
equation
which is considered in [49], section 3. is a closed and bounded subset of and all functions except k assume values in an ordered Banach space with a closed and regular positive cone K. (One could envision with its natural order here to assimilate the general flavor of the following.) Now assume:
The fact that k maps into and f maps into K insures that . Condition 2b insures that G is order-preserving. Condition 3 together with condition 2c insures (via an analytical argument) that there is function such that . Thus G maps to itself. Unfortunately this interval is not chain-complete, so we are not trivially done. As done in section 3 of [49] a short analytical argument shows that condition 3 of Theorem 5.9 is satisfied with n=1. This proves that G has a fixed point and thus that the equation in question has a solution.
It can be proved that the set of all functions on [0,1] that are constant on intervals bounded by dyadic rationals and take dyadic rational numbers as values forms a pseudo cone as defined in [6]. This construction is such that the restriction of the pseudo cone structure to natural subsets, such as functions constant on intervals bounded by dyadic rationals of order n (denominator at most ), taking dyadic rational numbers of order n as values, also is a pseudo cone. The domain [0,1] is chosen for convenience only. Thus there is a possibility that Baclawski's algorithm for pseudo cones can have further impact on the use of order in analysis.