Fixed Point Theorem

Theorem

A Fixed Point Theorem is a theorem that gives conditions under which a continuous mapping $f: S \to S$ is guaranteed to have at least one fixed point, that is:

$\exists x \in S: \map f x = x$

The following are examples:

Let $\struct {M, d}$ be a complete metric space.

Let $f: M \to M$ be a contraction.

That is, there exists $q \in \hointr 0 1$ such that for all $x, y \in M$:

$\map d {\map f x, \map f y} \le q \, \map d {x, y}$

Then there exists a unique fixed point of $f$.

Let $\struct {X, \le}$ be a non-empty chain complete ordered set (that is, an ordered set in which every chain has a supremum).

Let $f: X \to X$ be an inflationary mapping, that is, so that $\map f x \ge x$.

Then for every $x \in X$ there exists $y \in X$ where $y \ge x$ such that $\map f y = y$.

Let $f: \closedint a b \to \closedint a b$ be a real function which is continuous on the closed interval $\closedint a b$.

Then:

$\exists \xi \in \closedint a b: \map f \xi = \xi$