Let $(X, \rho)$ be a compact metric space.
Suppose that $T: X \rightarrow X$ and for all $u \neq v \in X$,
$$ \rho(T(u),T(v)) \leq \rho(u,v) $$
Must $T$ have a fixed point?
My intuition is telling me that the answer is no, since the contraction mapping principle requires
$$ \rho(T(u),T(v)) \leq c\rho(u,v) $$
with $c \in (0,1)$ for there to be a unique fixed point. However, I am struggling to come up with a counterexample.
Thank you in advance for the help!