TY - JOUR

T1 - Browder’s Theorem through Brouwer’s Fixed Point Theorem

AU - Solan, Eilon

AU - Solan, Omri N.

N1 - Publisher Copyright:
© 2023 The Mathematical Association of America.

PY - 2023

Y1 - 2023

N2 - A parametric version of Brouwer’s fixed point theorem, called Browder’s theorem, states that for every continuous mapping (Formula presented.), where X is a nonempty, compact, and convex set in a Euclidean space, the set of fixed points of f, namely, the set (Formula presented.), has a connected component whose projection onto the first coordinate is (Formula presented.). Browder’s original proof relies on the theory of the fixed point index. We provide an alternative proof that uses Brouwer’s fixed point theorem and is valid whenever X is a nonempty, compact, and convex subset of a Hausdorff topological vector space.

AB - A parametric version of Brouwer’s fixed point theorem, called Browder’s theorem, states that for every continuous mapping (Formula presented.), where X is a nonempty, compact, and convex set in a Euclidean space, the set of fixed points of f, namely, the set (Formula presented.), has a connected component whose projection onto the first coordinate is (Formula presented.). Browder’s original proof relies on the theory of the fixed point index. We provide an alternative proof that uses Brouwer’s fixed point theorem and is valid whenever X is a nonempty, compact, and convex subset of a Hausdorff topological vector space.

UR - http://www.scopus.com/inward/record.url?scp=85146972363&partnerID=8YFLogxK

U2 - 10.1080/00029890.2022.2160170

DO - 10.1080/00029890.2022.2160170

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AN - SCOPUS:85146972363

SN - 0002-9890

VL - 130

SP - 370

EP - 374

JO - American Mathematical Monthly

JF - American Mathematical Monthly

IS - 4

ER -