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All of the fixed point theorem I have seen (like Kakutani and Brower, Browder) required the set valued map to be hemi-continuous (lower). Is any fixed point theorem that can assure the existence of fixed point without the continuity of the set valued map (or even singleton set valued map).

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  • $\begingroup$ For what purpose? There are order-theoretic fixed point theorems that do not rely on any topological assumptions. $\endgroup$ Oct 2, 2013 at 13:41
  • $\begingroup$ would you please post your answer about order theoretic fixed point theorem in a more descriptive way? my set valued map in game theoretic discipline is not continuous and i need a proper theorem for proving the existence of the Nash equilibrium for that game. $\endgroup$ Oct 2, 2013 at 13:54
  • $\begingroup$ Does the following apply to your setting: en.wikipedia.org/wiki/Knaster%E2%80%93Tarski_theorem $\endgroup$
    – Suvrit
    Oct 2, 2013 at 15:41
  • $\begingroup$ There is a rather extensive literature on existence of Nash equilibria in discontinuous games. Guilherme Carmona has a nice book on the topic. Order theoretic approaches have been used too for proving the existence of Nash equilibria. A classic book on the approach by its inventor is Supermodularity and Complementarity by Donald Topkis. $\endgroup$ Oct 2, 2013 at 19:58
  • $\begingroup$ thank you Michael and Sux, Actually modular game and fixed point theorem on lattices are usefull however I am looking for non-lattice set that has fixed point. $\endgroup$ Oct 2, 2013 at 21:00

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Here is a very simple FPT that requires no continuity. Suppose that $g$ is a self mapping of a set $X$ that has a unique fixed point. Then every mapping which commutes with $g$ has a fixed point. Application: If $X$ is a non-empty, complete metric space, then a mapping $f$ thereon, which is such that some power is a contraction, has a fixed point. (This property does not imply continuity).

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  • $\begingroup$ Is nt this the consequence of kakutani FPT $\endgroup$ Mar 28, 2017 at 9:58
  • $\begingroup$ Might be--I added it since the proof is less than half a line. I first saw it in print in the 60's. Can't remeber by whom but will try to trace it. $\endgroup$
    – traun
    Mar 28, 2017 at 10:15
  • $\begingroup$ The above result is contained in an appendix to a longer paper on pde's by J. B. Diaz published in "Theory of Distributions" (Lisbon, 1964)--- the whole volume is available online. It requires no structure on the underlying set---in contrast to the Kakutani FPT, at least in the versions known to me. $\endgroup$
    – traun
    Mar 28, 2017 at 10:29
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There are some results on fixed points for isotone mappigs of POSets. These results do not use the concepts of topology and continuity. Some of them can be found in the book "Fixed point theory" by A. Granas and J. Dugundji. For set-valued mappings of POSets there is a result by R. Smithson (see http://projecteuclid.org/euclid.pjm/1102947733).

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  • $\begingroup$ Can you bring at least one of those and draw the application? $\endgroup$ Mar 28, 2017 at 17:45

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