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Extensions of Brouwer’s Theorem

Chapter
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Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 124)

Abstract

Chapter II makes clear the need for more general fixed-point theorems and algorithms to approximate the resulting fixed points. Two extensions are necessary: one to relax the continuity of the function involved, and one to relax the requirements on the domain and range of the function. Section I discusses upper semi-continuity of point-to-set mappings. Section 2 introduces the fundamental concept of piecewise-linear approximation and with this tool proves Kakutani’s theorem on the existence of fixed points for upper semi-continuous mappings. Section 3 extends Kakutani’s theorem in ways that will be useful in applications.

Keywords

Convex Subset Compact Convex Compact Convex Subset Nonempty Convex Piecewise Linear Approximation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1976

Authors and Affiliations

  1. 1.School of Operations Research and Industrial EngineeringCornell UniversityIthacaUSA

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