Sufficient and necessary condition for the global uniqueness of fix-points

Question

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 or Schauder fix points.

For the case of Brouwer fix point, we are looking for the conditions that we need to put on $f:[0,1]\to[0,1]$, such that the equation $f(x)=x$ has a unique solution.

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For another example, in Banach fix point theorem, the point is unique because the map $T$ is assumed to be contracting. "Contracting" is a sufficient condition for the uniqueness, but it is not necessary.

Answer 1

Well, here's a necessary and sufficient condition for uniqueness of the fixed point of a continuous function from $[0,1]$ to itself:

For every $x \in [0,1]$, if $f(x) \ge x$ then $f(t) > t$ for all $t < x$, and if $f(x) \le x$ then $f(t) < t$ for all $t > x$.

Of course it's not likely to be a useful necessary and sufficient condition. It's hard to imagine what a useful necessary and sufficient condition could possibly look like.