Brouwer’s Fixed-Point Theorem
It is more than a century since Brouwer  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 , and later to locally convex linear topological spaces by Tychonoff . Brouwer’s theorem was generalized to multifunctions first by Kakutani , and later to locally convex linear topological spaces by Glicksberg  and Ky Fan . Brouwer’s theorem admits of several proofs. Notable among them are those based on Sperner’s lemma  or concepts of homotopy/homology from algebraic topology (see Dugundji  or Munkres ) or concepts and results from real analysis (see Milnor , Seki , Rogers , Kannai , Traynor ). 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.
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