I am looking for a fixed point theorem for set valued maps that does not assume the set valued map should be convex valued.
Something like contractiblity or other properties can be replaced with convexity.
Although Vidit Nanda has addressed Lifchitz fixed point theorem that replace the convexity asumption with contractibility, however I wonder if we still can relax contractibility or acyclic set to a weaker condition?
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2 Answers
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In the absence of convex images, one typically relies on algebraic topology as you have guessed. If your set-valued map has a reasonably nice domain and contractible images, then you can easily string together two results:
Theorem: [Contractible Carrier] Let $K$ be a locally finite simplicial complex and $T$ a topological space. Assume the existence of a map $C$ from simplices of $K$ to subsets of $T$ so that for each simplex $\sigma \in K$ the image $C(\sigma) \subset T$ is contractible, and for each face relation $\sigma < \tau$ you have $C(\sigma) \subset C(\tau)$. Then, (a) there exists a continuous map $f:|K| \to T$ carried by $C$, i.e., $f(|\sigma|) \subset C(\sigma)$ for each $\sigma \in K$, and (b) any two maps carried by $C$ are homotopic.
This theorem is pretty much folklore at this point, see for instance Chapter II Theorem 9.2 in Lundell and Weingram's topology of CW complexes. There is an inductive process underlying (a): you can actually construct maps $f:|K| \to T$ carried by $C$ by inducting on skeleta of $K$. Things are easier in your case since $T = |K|$ as well.
Once you have any continuous $f:|K| \to |K|$ carried by $C$, compute its Lefschetz number, hope it is non-zero, and use the Lefschetz fixed point theorem.
answered Oct 1, 2013 at 16:33
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$\begingroup$ thank you very much, would you please insert the reference to that theorem too please. a reference that include the complete theorem with proof. $\endgroup$ Commented Oct 1, 2013 at 23:55
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$\begingroup$ @behradmahboobi Did you click on the wikipedia link? At the end you can find the original references, in particular see Solomon Lefschetz (1937). "On the fixed point formula". Ann. of Math. 38 (4): 819–822. doi:10.2307/1968838 $\endgroup$ Commented Oct 1, 2013 at 23:57
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$\begingroup$ would you please post the definition to these terms too ,locally finite simplicial complex, $\left|K\right|$ $\endgroup$ Commented Oct 1, 2013 at 23:59
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$\begingroup$ @behradmahboobi I'm afraid mathoverflow is not the right place for that. These are basic terms in algebraic topology that can be found in the Lundell-Weingram book and certainly do not count as research-level mathematics. I'd encourage you to read the introductory chapters of that book and post questions to math.stackexchange if you get stuck on something. $\endgroup$ Commented Oct 2, 2013 at 0:04
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$\begingroup$ seems that Lefschetz fixed point theorem is very powerfull fixed point theorem which is brought up in algebraic topology. do you remember any other relevant fixed point theorem that the set valued map can not be connected? actually contractible sets needs to be connected too ! ( if you prefer i can make a new) $\endgroup$ Commented Oct 2, 2013 at 11:34
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There is the Eilenberg-Montgomery fixed point theorem.
You can find it and many other results in Advanced Fixed Point Theory for Economics by Andy McLennan.
answered Oct 2, 2013 at 13:50
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$\begingroup$ thank you very much. however does that book needs background on algebraic topology or it is self contained for economists? $\endgroup$ Commented Oct 2, 2013 at 14:02
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$\begingroup$ It doesn't require a background in algebraic topology. I would say it is self-contained for economic theorists and advanced undergraduates in mathematics. $\endgroup$ Commented Oct 2, 2013 at 19:52
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$\begingroup$ I have seen this book before but since i wasn't expert in algebraic topology I wasn't confident to dig in to that. thanks to your recommendation I will start read that book. there is a beautiful fixed point theorem mentioned in the first chapter of that book that i have seen before for contractibility "Eilenberg and Montgomery fixed point theorem". This theorem was actually the source of this question that I have lost. $\endgroup$ Commented Oct 2, 2013 at 20:56