Encyclopedia > Contraction mapping

Article Content

Contraction mapping

In mathematics, a contraction mapping, or contraction, on a metric space M is a function f from M to itself, with the property that there is some real number k < 1 such that, for all x and y in M,
$d(f(x),f(y))\leq k\,d(x,y).$
Every contraction mapping is continuous, and has at most one fixed point.

An important property of contraction mappings is given by the Banach fixed point theorem. This states that every contraction mapping on a nonempty complete metric space has a unique fixed point, and that, for any x in M, the sequence x, f (x), f (f (x)), f (f (f (x))), ... converges to the fixed point.

All Wikipedia text is available under the terms of the GNU Free Documentation License

Search Encyclopedia
 Search over one million articles, find something about almost anything!

Featured Article
 Residential community association ... There are at least 400 private-street associations within St. Louis County, Missouri. These organizations limit street access by closing and barricading streets that ...