ThmDex – An index of mathematical definitions, results, and conjectures.
Result R4174 on D49: Proper contraction
Proper contraction has at most a single fixed point
Formulation 0
Let $M = (X, d)$ be a D1107: Metric space such that
(i) $f : X \to X$ is a D49: Proper contraction on $M$
Then \begin{equation} \# \{ x \in X : f(x) = x \} \leq 1 \end{equation}
Proofs
Proof 0
Let $M = (X, d)$ be a D1107: Metric space such that
(i) $f : X \to X$ is a D49: Proper contraction on $M$
If $f$ has no fixed points, then the claim clearly holds. Assume thus that $f$ has at least one fixed point and let $x, y \in X$ be fixed points of $f$. Since $f$ is a proper contraction, there is a constant $0 \leq C < 1$ such that \begin{equation} d(x, y) = d(f(x), f(y)) \leq C d(x, y) \end{equation} Since $d(x, y) \geq 0$ is an unsigned quantity, this can only happen when either $C$ or $d(x, y)$ is zero. If $d(x, y) = 0$, then $x = y$ by definition of a D58: Metric. Since $x$ and $y$ were arbitrary fixed points of $f$, result R2833: Nonempty set is singleton iff all elements equal each other guarantees that the set $\{ x \in X : f(x) = x \}$ is a singleton.

If $C = 0$, then $0 \leq d(x, y) \leq 0$ and thus $d(x, y) = 0$. Metric axioms imply $x = y$ and the claim again becomes a consequence of R2833: Nonempty set is singleton iff all elements equal each other. $\square$