Knowlpedia

Fixed point

A point x satisfying T(x)=x for a self-map T

Let XX be a set and let T:X→XT:X\to X be a function.

A point x∗∈Xx^\ast\in X is a fixed point of TT if

T(x∗)=x∗.T(x^\ast)=x^\ast.

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

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