Let $(X, d)$ be a nonempty complete metric space, and let $T: X \rightarrow X$ be a continuous map such that $T^{2}$ is a contraction, i.e., there exists $a<1$ such that for all distinct $x, y \in X$, we have $d\left(T^{2}(x), T^{2}(y)\right)<$ $a d(x, y)$. Then there exists a unique element $x \in X$ such that $T(x)=$ $x$.
