Extensions of Brouwer’s Theorem

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.