Fixed point

Let $X$ be a set and let $T:X\to X$ be a function.

A point $x^\ast\in X$ is a fixed point of $T$ if $T(x^\ast)=x^\ast.$

Fixed points are solutions of the equation $T(x)=x$. Many existence and uniqueness problems can be reformulated as fixed point problems.

Examples:

• On $\mathbb{R}$, the map $T(x)=\cos x$ has a fixed point (indeed exactly one).
• On $\mathbb{R}$, $T(x)=x+1$ has no fixed point.
• On any set $X$, the identity map $T(x)=x$ has every point as a fixed point.