Banach fixed-point theorem: Difference between revisions

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==Applications==

* A standard application is the proof of the [[Picard–Lindelöf theorem]] about the existence and uniqueness of solutions to certain [[ordinary differential equation]]s. The sought solution of the differential equation is expressed as a fixed point of a suitable integral operator which changes continuous functions into continuous functions. The Banach fixed-point theorem is then used to show that this integral operator has a unique fixed point.

* One consequence of the Banach fixed-point theorem is that small Lipschitz perturbations of the identity are [[Lipschitz continuity#Definitions|bi-lipschitz]] homeomorphisms. Let Ω be an open set of a Banach space ''E''; let {{nobr|''I'' : Ω → ''E''}} denote the identity (inclusion) map and let ''g'' : Ω → ''E'' be a Lipschitz map of constant ''k'' < 1. Then

# Ω′ := (''I'' + ''g'')(Ω) is an open subset of ''E'': precisely, for any ''x'' in Ω such that {{nobr|''B''(''x'', ''r'') ⊂ Ω}} one has {{nobr|''B''((''I'' + ''g'')(''x''), ''r''(1−1 − ''k'')) ⊂ Ω′; }}

# ''I'' + ''g'' : Ω → Ω′ is a bi-lipschitz homeomorphism;

: precisely, (''I'' + ''g'')⁻¹ is still of the form {{nobr|''I'' + ''h'' : Ω → Ω′}} with ''h'' a Lipschitz map of constant ''k''/(1−1&nbsp;−&nbsp;''k''). A direct consequence of this result yields the proof of the [[inverse function theorem]].

* 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=de | journal=[[Mathematische Nachrichten]]|volume= 144 |year=1989|pages= 165–187|doi=10.1002/mana.19891440113 | mr=1037168}}</ref>

* 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 |first1=Frank L. |last1=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] |chapter-url=https://books.google.com/books?id=U3Gtlot_hYEC&pg=PA474 }}</ref>

* It can be used to prove existence and uniqueness of an equilibrium in [[Cournot competition]],<ref>{{cite journal |first1=Ngo Van |last1=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 |archive-url=https://web.archive.org/web/20041230225125/http://www.cirano.qc.ca/pdf/publication/99s-22.pdf |archive-date=2004-12-30 |url-status=live }}</ref> and other dynamic economic models.<ref>{{cite book |first1=Nancy L. |last1=Stokey|author1-link=Nancy Stokey |first2=Robert E. Jr. |last2=Lucas |author-link2=Robert Lucas Jr. |title=Recursive Methods in Economic Dynamics |___location=Cambridge |publisher=Harvard University Press |year=1989 |isbn=0-674-75096-9 |pages=508–516 |url=https://books.google.com/books?id=BgQ3AwAAQBAJ&pg=PA508 }}</ref>

==Converses==