In [[mathematics]], the '''Banach [[fixed-point theorem]]''' (also known as the '''contraction mapping theorem''' or '''contractive mapping theorem''' or '''Banach-Caccioppoli theorem''') is an important [[Convergence proof techniques#contraction mapping|tool]] in the theory of [[metric space]]s; it guarantees the existence and uniqueness of [[fixed point (mathematics)|fixed points]] 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 [[Fixed-point iteration|Picard's method of successive approximations]].<ref>{{cite book |first1=David |last1=Kinderlehrer |author-link=David Kinderlehrer |first2=Guido |last2=Stampacchia |author-link2=Guido Stampacchia |chapter=Variational Inequalities in '''R'''<sup>N</sup> |title=An Introduction to Variational Inequalities and Their Applications |___location=New York |publisher=Academic Press |year=1980 |isbn=0-12-407350-6 |pages=7–22 |chapter-url=https://books.google.com/books?id=eCDnoB3Np5oC&pg=PA7 }}</ref> The theorem is named after [[Stefan Banach]] (1892–1945) who first stated it in 1922.<ref>{{cite journal |last=Banach|first= Stefan|author-link=Stefan Banach| title=Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales|journal=[[Fundamenta Mathematicae]]|volume= 3|year=1922|pages= 133–181 |url=http://matwbn.icm.edu.pl/ksiazki/or/or2/or215.pdf |archive-url=https://web.archive.org/web/20110607002842/http://matwbn.icm.edu.pl/ksiazki/or/or2/or215.pdf |archive-date=2011-06-07 |url-status=live |doi=10.4064/fm-3-1-133-181}}</ref><ref>{{cite journal |first=Krzysztof |last=Ciesielski |title=On Stefan Banach and some of his results |journal=Banach J. Math. Anal. |volume=1 |year=2007 |issue=1 |pages=1–10 |url=http://www.emis.de/journals/BJMA/tex_v1_n1_a1.pdf |archive-url=https://web.archive.org/web/20090530012258/http://www.emis.de/journals/BJMA/tex_v1_n1_a1.pdf |archive-date=2009-05-30 |url-status=live |doi=10.15352/bjma/1240321550 |doi-access=free }}</ref>
