Banach fixed-point theorem

Banach fixed-point theorem In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of 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 Picard's method of successive approximations.[1] The theorem is named after Stefan Banach (1892–1945) who first stated it in 1922.[2][3] Contents 1 Statement 2 Proof 3 Applications 4 Converses 5 Generalizations 6 See also 7 Notes 8 References Statement Definition. Let {displaystyle (X,d)} be a complete metric space. Then a map {displaystyle T:Xto X} is called a contraction mapping on X if there exists {displaystyle qin [0,1)} such that {displaystyle d(T(x),T(y))leq qd(x,y)} for all {displaystyle x,yin X.} Banach Fixed Point Theorem. Let {displaystyle (X,d)} be a non-empty complete metric space with a contraction mapping {displaystyle T:Xto X.} Then T admits a unique fixed-point {displaystyle x^{*}} in X (i.e. {displaystyle T(x^{*})=x^{*})} . Furthermore, {displaystyle x^{*}} can be found as follows: start with an arbitrary element {displaystyle x_{0}in X} and define a sequence {displaystyle (x_{n})_{nin mathbb {N} }} by {displaystyle x_{n}=T(x_{n-1})} for {displaystyle ngeq 1.} Then {displaystyle lim _{nto infty }x_{n}=x^{*}} .

Remark 1. The following inequalities are equivalent and describe the speed of convergence: {displaystyle {begin{aligned}d(x^{*},x_{n})&leq {frac {q^{n}}{1-q}}d(x_{1},x_{0}),\d(x^{*},x_{n+1})&leq {frac {q}{1-q}}d(x_{n+1},x_{n}),\d(x^{*},x_{n+1})&leq qd(x^{*},x_{n}).end{aligned}}} Any such value of q is called a Lipschitz constant for {displaystyle T} , and the smallest one is sometimes called "the best Lipschitz constant" of {displaystyle T} .

Remark 2. {displaystyle d(T(x),T(y)) n: {displaystyle {begin{aligned}d(x_{m},x_{n})&leq d(x_{m},x_{m-1})+d(x_{m-1},x_{m-2})+cdots +d(x_{n+1},x_{n})\&leq q^{m-1}d(x_{1},x_{0})+q^{m-2}d(x_{1},x_{0})+cdots +q^{n}d(x_{1},x_{0})\&=q^{n}d(x_{1},x_{0})sum _{k=0}^{m-n-1}q^{k}\&leq q^{n}d(x_{1},x_{0})sum _{k=0}^{infty }q^{k}\&=q^{n}d(x_{1},x_{0})left({frac {1}{1-q}}right).end{aligned}}} Let ε > 0 be arbitrary. Since q ∈ [0, 1), we can find a large {displaystyle Nin mathbb {N} } so that {displaystyle q^{N}<{frac {varepsilon (1-q)}{d(x_{1},x_{0})}}.} Therefore, by choosing m and n greater than N we may write: {displaystyle d(x_{m},x_{n})leq q^{n}d(x_{1},x_{0})left({frac {1}{1-q}}right)qd(p_{1},p_{2}).} Applications 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 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.[4] It can be used to prove existence and uniqueness of solutions to value iteration, policy iteration, and policy evaluation of reinforcement learning.[5] It can be used to prove existence and uniqueness of an equilibrium in Cournot competition,[6] and other dynamic economic models.[7] Converses 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 each iterate fn has a unique fixed point. Let {displaystyle qin (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 {displaystyle f:Xto X} is a map on a T1 topological space with a unique fixed point a, such that for each {displaystyle xin X} we have fn(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.[8] In this case the metric is in fact an ultrametric. Generalizations There are a number of generalizations (some of which are immediate corollaries).[9] Let T : X → X be a map on a complete non-empty metric space. Then, for example, some generalizations of the Banach fixed-point theorem are: Assume that some iterate Tn of T is a contraction. Then T has a unique fixed point. Assume that for each n, there exist cn such that d(Tn(x), Tn(y)) ≤ cnd(x, y) for all x and y, and that {displaystyle sum nolimits _{n}c_{n}

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