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https://www.ams.org/journals/proc/1976-060-01/S0002-9939-1976-0423137-6/S0002-9939-1976-0423137-6.pdf

This paper gives a sufficient condition for the uniqueness of SCHAUDER fix point. I wonder if there are any sufficient and necessary results, whether published or not, for the uniqueness of Brouwer, Banach, or Schauder fix point.

Many thanks!

Here is a related question.

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    $\begingroup$ Regarding the Banach fixed point theorem: a fixed point is always unique. Another (fairly) simple result: if $(K,d)$ is a compact metric space and $f\colon K\to K$ has the property that $d(f(x),f(y))<d(x,y)$ for any $x,y\in K$ such that $x\ne y$ then $f$ has a unique fixed point in $K$. $\endgroup$ – user539887 Apr 4 at 12:00
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    $\begingroup$ @user539887 you are right! Banach assumes the map to be contracting, and of course, the fix-point is unique. The condition of "contracting" is sufficient but not necessary to have a unique fixed point. $\endgroup$ – High GPA Apr 5 at 0:14

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