Banach and Knaster-Tarski fixed point theorems -- are they related? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T15:49:38Z http://mathoverflow.net/feeds/question/34511 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/34511/banach-and-knaster-tarski-fixed-point-theorems-are-they-related Banach and Knaster-Tarski fixed point theorems -- are they related? Jacques Carette 2010-08-04T15:04:59Z 2012-05-15T12:11:57Z <p>It there any known way of obtaining the <a href="http://en.wikipedia.org/wiki/Banach_fixed_point_theorem" rel="nofollow">Banach fixed-point theorem</a> from the <a href="http://en.wikipedia.org/wiki/Knaster-Tarski_theorem" rel="nofollow">Tarski fixed-point theorem</a> or vice-versa?</p> http://mathoverflow.net/questions/34511/banach-and-knaster-tarski-fixed-point-theorems-are-they-related/53035#53035 Answer by Matteo Mio for Banach and Knaster-Tarski fixed point theorems -- are they related? Matteo Mio 2011-01-24T10:16:19Z 2011-01-24T10:16:19Z <p>Hello,</p> <p>I just found the question, so the answer might come a bit too lat, but.. Have a look at:</p> <p><em>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.</em></p> <p>I attended this talk last summer, and it addresses exactly your question.</p> http://mathoverflow.net/questions/34511/banach-and-knaster-tarski-fixed-point-theorems-are-they-related/89000#89000 Answer by Driss Misane for Banach and Knaster-Tarski fixed point theorems -- are they related? Driss Misane 2012-02-20T08:31:13Z 2012-02-20T08:31:13Z <p>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</p> http://mathoverflow.net/questions/34511/banach-and-knaster-tarski-fixed-point-theorems-are-they-related/96993#96993 Answer by Michał Kukieła for Banach and Knaster-Tarski fixed point theorems -- are they related? Michał Kukieła 2012-05-15T12:11:57Z 2012-05-15T12:11:57Z <p>As suggested by Jacques, I turn my comment into an answer.</p> <p>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: <a href="http://files.nyu.edu/eo1/public/Book-PDF/CHAPTER%25205.pdf" rel="nofollow">files.nyu.edu/eo1/public/Book-PDF/CHAPTER%205.pdf</a> </p>