Banach fixed-point theorem: Difference between revisions

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Let (''X'', d) be a non-empty [[complete metric space]]. Let ''T'' : ''X'' <tt>-></tt> ''X'' be a ''[[contraction mapping]]'' on ''X'', i.e: there is a [[real number]] ''q''&nbsp;<&nbsp;1 such that

:<math>d(Tx,Ty) \le q\cdot d(x,y)</math>

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 an [[iteratediterative functionmethod|iterative]] sequence by ''x''_''n'' = ''Tx''_''n''-1 for ''n'' = 1, 2, 3, ... This sequence [[limit (mathematics)|converges]], and its limit is ''x''^*. The following inequality describes the speed of convergence:

:<math>d(x^*, x_n) \leq \frac{q^n}{1-q} d(x_1,x_0)</math>.