3 problems

Austin's conjecture. If $\{f_1,\ldots,f_r\}$ is a pairwise commuting family of $\lambda$-contractions on $X$, then the maps have a common fixed point; that is, there exists…

Stein's fixed-point conjecture. Some composition of members of $\mathcal{F}$ has a fixed point.

Let $X$ be a compact metric space, and let $f:X\to X$ be a weak local contraction, meaning that for every $x\in X$ there exists an $r_x>0$ such that $d(x,y)<r_x$ implies…