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.
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© 1976 Springer-Verlag Berlin Heidelberg
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Todd, M.J. (1976). Extensions of Brouwer’s Theorem. In: The Computation of Fixed Points and Applications. Lecture Notes in Economics and Mathematical Systems, vol 124. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-50327-6_5
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DOI: https://doi.org/10.1007/978-3-642-50327-6_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-07685-8
Online ISBN: 978-3-642-50327-6
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