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Order vs. Analysis

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 tex2html_wrap_inline7890 with tex2html_wrap_inline7816 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

  1. tex2html_wrap_inline12258 ,
  2.   tex2html_wrap_inline12260 iff tex2html_wrap_inline12262 ,
  3.   tex2html_wrap_inline12264 if tex2html_wrap_inline9282 is a limit ordinal,
where the iteration stops when one of the steps 2 or 3 cannot be carried out.gif 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 tex2html_wrap_inline12272 such that tex2html_wrap_inline12274 , then tex2html_wrap_inline12272 is a fixed point of G. (Otherwise we could extend the sequence by adding tex2html_wrap_inline12272 contradicting the maximality of the sequence.)
Unfortunately this result still is too general to be of much apparent use, as the iteration could stop at any successor or limit ordinal and the stopping need not guarantee a fixed point. Introducing the hypothesis of being order-preserving however one obtains the following variant of the Abian-Brown-Pelczar theorem:


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 tex2html_wrap_inline12306 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 tex2html_wrap_inline12310 , where tex2html_wrap_inline12312 and tex2html_wrap_inline12314 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 tex2html_wrap_inline12316 is not comparable to tex2html_wrap_inline12318 : 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. tex2html_wrap_inline12314 is a closed and bounded subset of tex2html_wrap_inline12324 and all functions except k assume values in an ordered Banach space tex2html_wrap_inline12328 with a closed and regular positive cone K. (One could envision tex2html_wrap_inline12332 with its natural order here to assimilate the general flavor of the following.) Now assume:

  1. tex2html_wrap_inline12334 (i.e., k does not take negative values) is continuous,
  2. tex2html_wrap_inline12338 (i.e., f takes values tex2html_wrap_inline12342 in E) is such that
    1. tex2html_wrap_inline12346 is strongly measurable for each tex2html_wrap_inline12348 ,
    2.   tex2html_wrap_inline12350 is increasing for almost every tex2html_wrap_inline12352 ,
    3.   There are tex2html_wrap_inline12354 and a Bochner-integrable tex2html_wrap_inline12356 such that for all tex2html_wrap_inline12358 and almost every tex2html_wrap_inline12352 we have


  3.   r>0 is such that tex2html_wrap_inline12366 for each tex2html_wrap_inline12352 .
Then for each tex2html_wrap_inline12370 the above integral equation has a solution.
Clearly the operator to work with is


The fact that k maps into tex2html_wrap_inline12376 and f maps into K insures that tex2html_wrap_inline12382 . Condition 2b insures that G is order-preserving. Condition 3 together with condition 2c insures (via an analytical argument) that there is function tex2html_wrap_inline12386 such that tex2html_wrap_inline12388 . Thus G maps tex2html_wrap_inline12392 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 tex2html_wrap_inline12402 ), 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.

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