(Note: I have a very little knowledge in the area related to this question. (My research is more related to Combinatorics.) So, I apologize in advance if I say something wrong or trivial.)
One version of the Brouwer fixed point theorem states that any continuous function from the closed unit ball in n-dimensional Euclidean space to itself must have a fixed point. Moreover, it is possible that many fixed points exist.
One specific question I would like to ask is whether there exists a unique fixed point if we restrict the function to be convex. Any pointers to something related or counter examples would be appreciated.
More generally, what are the restrictions on the functions that are sufficient to guarantee the uniqueness. The only result I'm aware of is Banach fixed point theorem which says that if the function is a contraction mapping (thus continuous) then there is a unique fixed point. Are there other restrictions like this?