Banach fixed point theorem

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The Banach fixed point 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. The theorem is named after Stefan Banach (1892-1945), and was first stated by Banach in 1922.

The theorem

Let (X, d) be a non-empty complete metric space. Let T : X → X be a contraction mapping on X, i.e: there is a nonnegative real number q < 1 such that

[d(Tx,Ty) \le q\cdot d(x,y)]

for all x, y in X. Then the map T admits one and only one fixed point x^* in X (this means Tx^* = x^*). Furthermore, this fixed point can be found as follows: start with an arbitrary element x₀ in X and define an iterative sequence by x_n = Tx_n-1 for n = 1, 2, 3, ... This sequence converges, and its limit is x^*. The following inequality describes the speed of convergence:

[d(x^*, x_n) \leq \frac d(x_1,x_0)].

Equivalently,

[d(x^*, x_) \leq \frac d(x_,x_n)]

and

[d(x^*, x_) \leq q d(x_n,x^*)].

The smallest such value of q is sometimes called the Lipschitz constant.

Note that the requirement d(Tx, Ty) < d(x, y) for all unequal x and y is in general not enough to ensure the existence of a fixed point, as is shown by the map T : [1,∞) → [1,∞) with T(x) = x + 1/x, which lacks a fixed point. However, if the space X is compact, then this weaker assumption does imply all the statements of the theorem.

When using the theorem in practice, the most difficult part is typically to define X properly so that T actually maps elements from X to X, i.e. that Tx is always an element of X.

Proof

Choose any [x_0 \in (X, d)]. For each [n \in \], define [x_n = Tx_\,\!]. We claim that for all [n \in \], the following is true:

:[d(x_, x_n) \leq q^n d(x_1, x_0)].

To show this, we will proceed using induction. The above statement is true for the case [n = 1\,\!], for

:[d(x_, x_1) = d(x_2, x_1) = d(Tx_1, Tx_0) \leq qd(x_1, x_0)].

Suppose the above statement holds for some [k \in \]. Then we have

:

[d(x_, x_)\,\!]	[= d(x_, x_)\,\!]
	[= d(Tx_, Tx_k)\,\!]
	[\leq q d(x_, x_k)]
	[\leq q \cdot q^kd(x_1, x_0)]
	[= q^d(x_1, x_0)\,\!].

The inductive assumption is used going from line three to line four. By the principle of mathematical induction, for all [n \in \], the above claim is true.

Let [\epsilon > 0\,\!]. Since [0 \leq q < 1], we can find a large [N \in \] so that

:[q^N < \frac].

Using the claim above, we have that for any [m\,\!], [n \in \] with [m > n \geq N],

:

[d\left(x_m, x_n\right)]	[\leq d(x_m, x_) + d(x_, x_) + \cdots + d(x_, x_n)]
	[\leq q^d(x_1, x_0) + q^d(x_1, x_0) + \cdots + q^nd(x_1, x_0)]
	[= d(x_1, x_0)q^n \cdot \sum_^ q^k]
	[< d(x_1, x_0)q^n \cdot \sum_^\infty q^k]
	[= d(x_1, x_0)q^n \frac]
	[= q^n \frac]
	[< \frac\cdot\frac]
	[= \epsilon\,\!].

The inequality in line one follows from repeated applications of the triangle inequality; the series in line four is a geometric series with [0 \leq q < 1] and hence it converges. The above shows that [\_] is a Cauchy sequence in [(X, d)\,\!] and hence convergent by completeness. So let [x^* = \lim_ x_n]. We make two claims: (1) [x^*\,\!] is a fixed point of [T\,\!]. That is, [Tx^* = x^*\,\!]; (2) [x^*\,\!] is the only fixed point of [T\,\!] in [(X, d)\,\!].

To see (1), we note that for any [n \in \],

:[0 \leq d(x_, Tx^*) = d(Tx_n, Tx^*) \leq q d(x_n, x^*)].

Since [qd(x_n, x^*) \to 0] as [n \to \infty], the squeeze theorem shows that [\lim_ d(x_, Tx^*) = 0]. This shows that [x_n \to Tx^*] as [n \to \infty]. But [x_n \to x^*] as [n \to \infty], and limits are unique; hence it must be the case that [x^* = Tx^*\,\!].

To show (2), we suppose that [y\,\!] also satisfies [Ty = y\,\!]. Then

:[0 \leq d(x^*, y) = d(Tx^*, Ty) \leq q d(x^*, y)].

Remembering that [0 \leq q < 1], the above implies that [0 \leq (1-q) d(x^*, y) \leq 0], which shows that [d(x^*, y) = 0\,\!], whence by positive definiteness, [x^* = y\,\!] and the proof is complete.

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.

Converses

Several converses of the Banach contraction principle exist. The following is due to Czeslaw Bessaga, from 1959:

Let [f:X\rightarrow X] be a map of an abstract set such that each iterate f ⁿ has a unique fixed point. Let q be a real number, 0 < q < 1. Then there exists a complete metric on X such that f is contractive, and q is the contraction constant.

Generalizations

See the article on fixed point theorems in infinite-dimensional spaces for generalizations.

References

• Vasile I. Istratescu, Fixed Point Theory, An Introduction, D.Reidel, the Netherlands (1981). ISBN 90-277-1224-7 See chapter 7.
• Andrzej Granas and James Dugundji, Fixed Point Theory (2003) Springer-Verlag, New York, ISBN 0-387-00173-5.
• William A. Kirk and Brailey Sims, Handbook of Metric Fixed Point Theory (2001), Kluwer Academic, London ISBN 0-7923-7073-2.

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An earlier version of this article was posted on [Planet Math]. This article is open content.

 

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