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

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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!

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