Brouwer’s fixed point theorem

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.