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Reference.org
Banach fixed-point theorem
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem or Banach–Caccioppoli theorem) is an important <a href="/facts/Convergence_proof_techniques/3SJPmrxI">tool</a> in the theory of <a href="/facts/Metric_space/gnBBRXtc">metric spaces</a>; it guarantees the existence and uniqueness of <a href="/facts/Fixed_point_(mathematics)/vVVddTKc">fixed points</a> of certain self-maps of metric spaces and provides a constructive method to find those fixed points. It can be understood as an abstract formulation of <a href="/facts/Fixed-point_iteration/05dQGazd">Picard's method of successive approximations</a>. The theorem is named after <a href="/facts/Stefan_Banach/b1qHSNg2">Stefan Banach</a> (1892–1945) who first stated it in 1922. </p>