# Banach fixed point theorem

Let $(X,d)$ be a complete metric space. A function $T:X\to X$ is said to be a *contraction mapping* if there is a constant $q$ with $$ such that

$$d(Tx,Ty)\le q\cdot d(x,y)$$ |

for all $x,y\in X$. Contractions have an important property.

###### Theorem 1 (Banach Theorem).

Every contraction has a unique http://planetmath.org/node/2777fixed point^{}.

There is an estimate to this fixed point that can be useful in applications. Let $T$ be a contraction mapping on $(X,d)$ with constant $q$ and unique fixed point ${x}^{*}\in X$. For any ${x}_{0}\in X$, define recursively the following sequence

${x}_{1}$ | $:=$ | $T{x}_{0}$ | ||

${x}_{2}$ | $:=$ | $T{x}_{1}$ | ||

$\mathrm{\vdots}$ | ||||

${x}_{n+1}$ | $:=$ | $T{x}_{n}.$ |

The following inequality^{} then holds:

$$d({x}^{*},{x}_{n})\le \frac{{q}^{n}}{1-q}d({x}_{1},{x}_{0}).$$ |

So the sequence $({x}_{n})$ converges to ${x}^{*}$. This estimate is occasionally responsible for this result being known as *the method of successive approximations*.

Title | Banach fixed point theorem |

Canonical name | BanachFixedPointTheorem |

Date of creation | 2013-03-22 12:31:10 |

Last modified on | 2013-03-22 12:31:10 |

Owner | mathwizard (128) |

Last modified by | mathwizard (128) |

Numerical id | 21 |

Author | mathwizard (128) |

Entry type | Theorem |

Classification | msc 54A20 |

Classification | msc 47H10 |

Classification | msc 54H25 |

Synonym | contraction principle |

Synonym | contraction mapping theorem |

Synonym | method of successive approximations |

Synonym | Banach-Caccioppoli fixed point theorem |

Related topic | FixedPoint |

Defines | contraction mapping |

Defines | contraction operator |