Banach fixed-point theorem

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.

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 real number q < 1 such that

${\displaystyle d(Tx,Ty)\leq q\cdot d(x,y)}$

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 iterated function 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:

${\displaystyle d(x^{*},x_{n})\leq {\frac {q^{n}}{1-q}}d(x_{1},x_{0})}$.

Equivalently,

${\displaystyle d(x^{*},x_{n+1})\leq {\frac {q}{1-q}}d(x_{n+1},x_{n})}$

and

${\displaystyle d(x^{*},x_{n+1})\leq qd(x_{n},x^{*})}$.

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.

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 Bessaga, from 1959:

Let ${\displaystyle f:X\rightarrow X}$  be a map of an abstract set such that each iterate fⁿ 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, Holland (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