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,
<math>d(f(x),f(y))\leq k\,d(x,y).</math>
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
Kief, North Dakota

... with 23.1% under the age of 18, 0.0% from 18 to 24, 38.5% from 25 to 44, 7.7% from 45 to 64, and 30.8% who are 65 years of age or older. The median age is 42 years. For ...