TITLE: Two commuting mappings in the disk
QUESTION [183 upvotes]: Suppose that $f$ and $g$ are two commuting continuous mappings from the closed unit disk (or, if you prefer, the closed unit ball in $R^n$) to itself. Does there always exist a point $x$ such that $f(x)=g(x)$? 
If one of the mappings is invertible, then it is just a restatement of Brouwer's fixed point theorem but I do not know the answer in the general case and would not even dare to guess what it must be. Also, the answer is well-known to be "Yes" in dimension $1$.

REPLY [11 votes]: There is also a result due to Shields which implies a common fixed point (and hence a coincidence point) assuming that $f$ and $g$ are both analytic on the disk. 
A. L. Shields, On fixed points of commuting analytic functions, Proc. Amer. Math. Soc. 15(1964), 703-706. MR 29 #2790.