Brouwer’s Fixed-Point Theorem

Abstract

It is more than a century since Brouwer [4] proved a fixed- point theorem of great consequence, in the setting of finite-dimensional Euclidean spaces. It was subsequently extended to normed linear spaces by Schauder [25], and later to locally convex linear topological spaces by Tychonoff [31]. Brouwer’s theorem was generalized to multifunctions first by Kakutani [12], and later to locally convex linear topological spaces by Glicksberg [8] and Ky Fan [6]. Brouwer’s theorem admits of several proofs. Notable among them are those based on Sperner’s lemma [28] or concepts of homotopy/homology from algebraic topology (see Dugundji [5] or Munkres [17]) or concepts and results from real analysis (see Milnor [16], Seki [26], Rogers [23], Kannai [13], Traynor [30]). However, we provide here only the analytic proof of Brouwer’s theorem and a proof based on Sperner’s lemma. Needless to state that Brouwer’s theorem and its generalizations/variants find a wide range of applications in the solution of nonlinear equations, differential and integral equations, mathematical biology and mathematical economics.