Banach fixed-point theorem: Difference between revisions

Revision as of 02:20, 29 November 2022

Revision as of 13:53, 2 January 2023

Line 7:

for all <math>x, y \in X.</math>

<blockquote>'''Banach Fixed Point Theorem.''' Let <math>(X, d)</math> be a [[Empty set|non-empty]] [[complete metric space]] with a contraction mapping <math>T : X \to X.</math> Then ''T'' admits a unique [[Fixed point (mathematics)|fixed-point]] <math>x^*</math> in ''X'' (i.e. <math>T(x^*) = x^*)</math>). Furthermore, <math>x^*</math> can be found as follows: start with an arbitrary element <math>x_0 \in X</math> and define a [[sequence]] <math>(x_n)_{n\in\mathbb N}</math> by <math>x_n = T(x_{n-1})</math> for <math>n \geq 1.</math> Then <math>\lim_{n \to \infty} x_n = x^*</math>.</blockquote>

''Remark 1.'' The following inequalities are equivalent and describe the [[Rate of convergence|speed of convergence]]: