Brouwer’s fixed point theorem

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Written by Stephan C. Carlson

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Key People:: Luitzen Egbertus Jan Brouwer

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Brouwer’s fixed point theorem, in mathematics, a theorem of algebraic topology that was stated and proved in 1912 by the Dutch mathematician L.E.J. Brouwer. Inspired by earlier work of the French mathematician Henri Poincaré, Brouwer investigated the behaviour of continuous functions (see continuity) mapping the ball of unit radius in n-dimensional Euclidean space into itself. In this context, a function is continuous if it maps close points to close points. Brouwer’s fixed point theorem asserts that for any such function f there is at least one point x such that f(x) = x; in other words, such that the function f maps x to itself. Such a point is called a fixed point of the function.

When restricted to the one-dimensional case, Brouwer’s theorem can be shown to be equivalent to the intermediate value theorem, which is a familiar result in calculus and states that if a continuous real-valued function f defined on the closed interval [−1, 1] satisfies f(−1) < 0 and f(1) > 0, then f(x) = 0 for at least one number x between −1 and 1; less formally, an unbroken curve passes through every value between its endpoints. An n-dimensional version of the intermediate value theorem was shown to be equivalent to Brouwer’s fixed point theorem in 1940.

There are many other fixed point theorems, including one for the sphere, which is the surface of a solid ball in three-dimensional space and to which Brouwer’s theorem does not apply. The fixed point theorem for the sphere asserts that any continuous function mapping the sphere into itself either has a fixed point or maps some point to its antipodal point.

Fixed point theorems are examples of existence theorems, in the sense that they assert the existence of objects, such as solutions to functional equations, but not necessarily methods for finding such solutions. However, some of these theorems are coupled with algorithms that produce solutions, especially for problems in modern applied mathematics.

Stephan C. Carlson

fixed-point theorem

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fixed-point theorem, any of various theorems in mathematics dealing with a transformation of the points of a set into points of the same set where it can be proved that at least one point remains fixed. For example, if each real number is squared, the numbers zero and one remain fixed; whereas the transformation whereby each number is increased by one leaves no number fixed. The first example, the transformation consisting of squaring each number, when applied to the open interval of numbers greater than zero and less than one (0,1), also has no fixed points. However, the situation changes for the closed interval [0,1], with the endpoints included. A continuous transformation is one in which neighbouring points are transformed into other neighbouring points. (See continuity.) Brouwer’s fixed-point theorem states that any continuous transformation of a closed disk (including the boundary) into itself leaves at least one point fixed. The theorem is also true for continuous transformations of the points on a closed interval, in a closed ball, or in abstract higher dimensional sets analogous to the ball.

Fixed-point theorems are very useful for finding out if an equation has a solution. For example, in differential equations, a transformation called a differential operator transforms one function into another. Finding a solution of a differential equation can then be interpreted as finding a function unchanged by a related transformation. By considering these functions as points and defining a collection of functions analogous to the above collection of points comprising a disk, theorems analogous to Brouwer’s fixed-point theorem can be proved for differential equations. The most famous theorem of this type is the Leray-Schauder theorem, published in 1934 by the Frenchman Jean Leray and the Pole Julius Schauder. Whether or not this method yields a solution (i.e.,whether or not a fixed-point can be found) depends on the exact nature of the differential operator and the collection of functions from which a solution is sought.

This article was most recently revised and updated by William L. Hosch.