Statement
''Definition.'' Let be aBanach Fixed Point Theorem. Let''Remark 1.'' The following inequalities are equivalent and describe the speed of convergence: :(X, d) be a Empty set">non-emptycomplete metric space In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou ...with a contraction mappingT : X \to X. Then ''T'' admits a unique Fixed point (mathematics)">fixed-pointx^* in ''X'' (i.e.T(x^*) = x^*) . Furthermore,x^* can be found as follows: start with an arbitrary elementx_0 \in X and define a sequence(x_n)_ byx_n = T(x_) forn \geq 1. Then\lim_ x_n = x^* .
Proof
LetApplications
*A standard application is the proof of the Picard–Lindelöf theorem about the existence and uniqueness of solutions to certain ordinary differential equations. The sought solution of the differential equation is expressed as a fixed point of a suitable integral operator which transforms 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 bi-lipschitz homeomorphisms. Let Ω be an open set of a Banach space ''E''; let ''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 ''B''(''x'', ''r'') ⊂ Ω one has ''B''((''I''+''g'')(''x''), ''r''(1−''k'')) ⊂ Ω′; #''I''+''g'' : Ω → Ω′ is a bi-lipschitz homeomorphism; :precisely, (''I''+''g'')−1 is still of the form ''I'' + ''h'' : Ω → Ω′ with ''h'' a Lipschitz map of constant ''k''/(1−''k''). A direct consequence of this result yields the proof of theConverses
Several converses of the Banach contraction principle exist. The following is due to Czesław Bessaga, from 1959: Let ''f'' : ''X'' → ''X'' be a map of an abstract set such that eachGeneralizations
There are a number of generalizations (some of which are immediateSee also
*Notes
References
* * * * See chapter 7. * {{DEFAULTSORT:Banach Fixed-Point Theorem Articles containing proofs Fixed-point theorems Metric geometry Topology