17
$\begingroup$

It there any known way of obtaining the Banach fixed-point theorem from the Tarski fixed-point theorem or vice-versa?

$\endgroup$
4
  • $\begingroup$ I'm intrigued - have you any indication that they might be? $\endgroup$ Aug 4, 2010 at 15:33
  • 4
    $\begingroup$ I was thinking that the metric (in the Banach version) induces a foliation of the space, which could be seen as a poset. If things 'line up' just right, contraction could preserve this foliation just right, so that the Tarski LFP exists and is the same as the Banach one. $\endgroup$ Aug 4, 2010 at 15:54
  • $\begingroup$ Okay, you've sold me. I'll follow this question ... $\endgroup$ Aug 4, 2010 at 16:24
  • $\begingroup$ @Michal: you should make that an answer. It isn't exactly right, but close enough. $\endgroup$ May 10, 2012 at 3:07

3 Answers 3

17
$\begingroup$

Hello,

I just found the question, so the answer might come a bit too lat, but.. Have a look at:

Paweł Waszkiewicz, "Common patterns for metric and ordered fixed point theorems.", In Proceedings of the 7th Workshop on Fixed Points in Computer Science (Luigi Santocanale ed.), 2010, pp. 83-87.

I attended this talk last summer, and it addresses exactly your question.

$\endgroup$
2
4
$\begingroup$

As suggested by Jacques, I turn my comment into an answer.

This is not exactly what you ask for, but it is related. Efe Ok in Section 3.4 in Chapter 6 of his yet-to-be-written book on ordered sets gives a proof of the Banach fixed point theorem using the Kantorovitch-Tarski fixed point theorem: files.nyu.edu/eo1/public/Book-PDF/CHAPTER%205.pdf https://sites.google.com/a/nyu.edu/efeok/books/CHAPTER%205.pdf

$\endgroup$
1
$\begingroup$

Look the article : M.Jawahiri, D. Misane, M. Pouzet. Retracts: graphs and ordrerd sets from the metric point of view. Contemporary Mathematics, 1986, vol. 57 pp. 175-226

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.