Banach fixed-point theorem: Difference between revisions

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The '''Banach fixed point theorem''' is an important tool in the theory of [[metric space]]s; it guarantees the existence and uniqueness of fixed points of certain self maps of metric spaces, and provides a constructive method to find those fixed points.

Let F be a non-empty closed subset of a complete metric space (X, d). Let T : X -> X be a contraction operator on F, , i.e: There is a constant q ε (0, 1) such that

Let F(''X'', d) be a non-empty closed subset of a [[completeness|complete]] metric space (X, d). Let ''T'' : ''X'' <tt>-></tt> ''X'' be a ''contraction operator'' on F, ''X'', i.e: Therethere is a constant[[real number]] ''q'' &epsilon; (0,< 1) such that

<center>d(Tx, Ty) <u>&lt;</u> q &bull; d(x, y)</center>

: d(''Tx'', ''Ty'') &le; ''q'' &middot; d(''x'', ''y'')

for all ''x'', ''y'' in ''X''. Then the map ''T'' admits one and only one fixed point ''x''^* in ''X'' (this means ''Tx''^* = ''x''^*). Furthermore, this fixed point can be found as follows: start with an arbitrary element ''x''₀ in ''X'' and define a sequence by ''x''_''n'' = ''Tx''_''n''-1 for ''n'' = 1, 2, 3, ... This sequence [[mathematical limit|converges]], and its limit is ''x''^*. The following inequality describes the speed of convergence:

for all x, y &epsilon; X. Then:

1. For any x₀ &epsilon; F rhe sequence defined by x_n+1 = Tx_n, n &epsilon; {1, 2, 3, ...} remains within F

2. There exists an x* &epsilon; F such that:

lim_{n -> INFTY} x_n = x*

3. x* is a fixed point of T i.e. Tx* = x*

4. x* is the only fixed point of T in F

5.

qⁿ

d(x*, x_n) <u>&ltle;</u> ----- d(x₁,x₀)

1-q

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An earlier version of this article was posted on [http://planetmath.org/encyclopedia/BanachFixedPointTheorem.html Planet Math]. This article is [[open content]]