Brouwer fixed-point theorem

Brouwer's fixed-point theorem is a <a href="/facts/Fixed-point_theorem/Jvkz4SUu">fixed-point theorem</a> in <a href="/facts/Topology/2EqW47cU">topology</a>, named after <a href="/facts/Luitzen_Egbertus_Jan_Brouwer/6q6ULf3b">L. E. J. (Bertus) Brouwer</a>. It states that for any <a href="/facts/Continuous_function/sKbl02pB">continuous function</a> 
 
 
 
 f
 
 
 {\displaystyle f}
 
 mapping a nonempty <a href="/facts/Compactness/d0cgXJH7">compact</a> <a href="/facts/Convex_set/vdAuJRJl">convex set</a> to itself, there is a point 
 
 
 
 
 x
 
 0
 
 
 
 
 {\displaystyle x_{0}}
 
 such that 
 
 
 
 f
 (
 
 x
 
 0
 
 
 )
 =
 
 x
 
 0
 
 
 
 
 {\displaystyle f(x_{0})=x_{0}}
 
. The simplest forms of Brouwer's theorem are for continuous functions 
 
 
 
 f
 
 
 {\displaystyle f}
 
 from a closed interval 
 
 
 
 I
 
 
 {\displaystyle I}
 
 in the real numbers to itself or from a closed <a href="/facts/Disk_(mathematics)/DgMVr97H">disk</a> 
 
 
 
 D
 
 
 {\displaystyle D}
 
 to itself. A more general form than the latter is for continuous functions from a nonempty convex compact subset 
 
 
 
 K
 
 
 {\displaystyle K}
 
 of <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a> to itself.
Among hundreds of <a href="/facts/Fixed-point_theorem/Jvkz4SUu">fixed-point theorems</a>, Brouwer's is particularly well known, due in part to its use across numerous fields of mathematics. In its original field, this result is one of the key theorems characterizing the topology of Euclidean spaces, along with the <a href="/facts/Jordan_curve_theorem/zKpty4QT">Jordan curve theorem</a>, the <a href="/facts/Hairy_ball_theorem/pSX6zDBz">hairy ball theorem</a>, the <a href="/facts/Invariance_of_dimension/n1uFhWRY">invariance of dimension</a> and the <a href="/facts/Borsuk%25E2%2580%2593Ulam_theorem/3mzguHSE">Borsuk–Ulam theorem</a>. This gives it a place among the fundamental theorems of topology. The theorem is also used for proving deep results about <a href="/facts/Differential_equation/9MGbE3Ca">differential equations</a> and is covered in most introductory courses on <a href="/facts/Differential_geometry/c04XmyLu">differential geometry</a>. It appears in unlikely fields such as <a href="/facts/Game_theory/zkEIj1ya">game theory</a>. In economics, Brouwer's fixed-point theorem and its extension, the <a href="/facts/Kakutani_fixed-point_theorem/qMQNEjzU">Kakutani fixed-point theorem</a>, play a central role in the <a href="/facts/Arrow%25E2%2580%2593Debreu_model/MnxUyNFr">proof of existence</a> of <a href="/facts/General_equilibrium/WY0zX5UD">general equilibrium</a> in market economies as developed in the 1950s by economics Nobel prize winners <a href="/facts/Kenneth_Arrow/7KdWn4b2">Kenneth Arrow</a> and <a href="/facts/G%25C3%25A9rard_Debreu/h1lKfo20">Gérard Debreu</a>.
The theorem was first studied in view of work on differential equations by the French mathematicians around <a href="/facts/Henri_Poincar%25C3%25A9/AazL2j4X">Henri Poincaré</a> and <a href="/facts/Charles_%25C3%2589mile_Picard/fq6m32JE">Charles Émile Picard</a>. Proving results such as the <a href="/facts/Poincar%25C3%25A9%25E2%2580%2593Bendixson_theorem/lu3IWIUl">Poincaré–Bendixson theorem</a> requires the use of topological methods. This work at the end of the 19th century opened into several successive versions of the theorem. The case of differentiable mappings of the n-dimensional closed ball was first proved in 1910 by <a href="/facts/Jacques_Hadamard/TgCkP3AE">Jacques Hadamard</a> and the general case for continuous mappings by Brouwer in 1911.

Brouwer fixed-point theorem open-in-new

Brouwer fixed-point theorem