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In [[mathematics]], the '''Banach–Caccioppoli [[fixed-point theorem]]''' (also known as the '''contraction mapping theorem''' or '''contractive mapping theorem''') is an important 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 |first=David |last=Kinderlehrer |
==Statement==
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*It can be used to give sufficient conditions under which Newton's method of successive approximations is guaranteed to work, and similarly for Chebyshev's third order method.
*It can be used to prove existence and uniqueness of solutions to integral equations.
*It can be used to give a proof to the [[Nash embedding theorem]].<ref>{{cite journal |first=Matthias|last=Günther|title=Zum Einbettungssatz von J. Nash | trans-title=On the embedding theorem of J. Nash | language=
*It can be used to prove existence and uniqueness of solutions to value iteration, policy iteration, and policy evaluation of [[reinforcement learning]].<ref>{{cite book |first=Frank L. |last=Lewis |first2=Draguna |last2=Vrabie |first3=Vassilis L. |last3=Syrmos |title=Optimal Control |chapter=Reinforcement Learning and Optimal Adaptive Control |___location=New York |publisher=John Wiley & Sons |year=2012 |isbn=978-1-118-12272-3 |pages=461–517 [p. 474] |
*It can be used to prove existence and uniqueness of an equilibrium in [[Cournot competition]],<ref>{{cite journal |first=Ngo Van |last=Long |first2=Antoine |last2=Soubeyran |title=Existence and Uniqueness of Cournot Equilibrium: A Contraction Mapping Approach |journal=[[Economics Letters]] |volume=67 |issue=3 |year=2000 |pages=345–348 |doi=10.1016/S0165-1765(00)00211-1 |url=https://www.cirano.qc.ca/pdf/publication/99s-22.pdf }}</ref> and other dynamic economic models.<ref>{{cite book |first=Nancy L. |last=Stokey |first2=Robert E. Jr. |last2=Lucas |
==Converses==
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Let ''f'' : ''X'' → ''X'' be a map of an abstract [[set (mathematics)|set]] such that each [[iterated function|iterate]] ''f<sup>n</sup>'' has a unique fixed point. Let ''q'' ∈ (0, 1), then there exists a complete metric on ''X'' such that ''f'' is contractive, and ''q'' is the contraction constant.
Indeed, very weak assumptions suffice to obtain such a kind of converse. For example if ''f'' : ''X'' → ''X'' is a map on a [[T1 space|''T''<sub>1</sub> topological space]] with a unique [[fixed point (mathematics)|fixed point]] ''a'', such that for each ''x'' in ''X'' we have ''f<sup>n</sup>''(''x'') → ''a'', then there already exists a metric on ''X'' with respect to which ''f'' satisfies the conditions of the Banach contraction principle with contraction constant 1/2.<ref>{{cite journal |first=Pascal |last=Hitzler |
==Generalizations==
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In applications, the existence and unicity of a fixed point often can be shown directly with the standard Banach fixed point theorem, by a suitable choice of the metric that makes the map ''T'' a contraction. Indeed, the above result by Bessaga strongly suggests to look for such a metric. See also the article on [[fixed point theorems in infinite-dimensional spaces]] for generalizations.
A different class of generalizations arise from suitable generalizations of the notion of [[metric space]], e.g. by weakening the defining axioms for the notion of metric.<ref>{{cite book |first=Pascal |last=Hitzler |
==See also==
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==References==
*{{cite book |first=Praveen |last=Agarwal |first2=Mohamed |last2=Jleli |first3=Bessem |last3=Samet |chapter=Banach Contraction Principle and Applications |title=Fixed Point Theory in Metric Spaces |publisher=Springer |___location=Singapore |year=2018 |isbn=978-981-13-2912-8 |pages=1–23 |doi=10.1007/978-981-13-2913-5_1 }}
*{{cite book |first=Carmen |last=Chicone |title=Ordinary Differential Equations with Applications |___location=New York |publisher=Springer |edition=2nd |year=2006 |isbn=0-387-30769-9 |chapter=Contraction |
*{{cite book |first=Andrzej |last=Granas |first2=James |last2=Dugundji |
*{{cite book |first=Vasile I. |last=Istrăţescu |title=Fixed Point Theory: An Introduction |publisher=D. Reidel |___location=The Netherlands |year=1981 |isbn=90-277-1224-7 }} See chapter 7.
*{{cite book |last1=Kirk |first1=William A. |last2=Khamsi |first2=Mohamed A. |title=An Introduction to Metric Spaces and Fixed Point Theory |year=2001 |publisher=John Wiley |___location=New York |isbn=0-471-41825-0 }}
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