Browder's Theorem through Brouwer's Fixed Point Theorem. Issue 4 (21st April 2023) - Solan, Eilon; Solan, Omri N.
Abstract: A parametric version of Brouwer's fixed point theorem, called Browder's theorem, states that for every continuous mapping f : [ 0, 1 ] × X → X, where X is a nonempty, compact, and convex set in a Euclidean space, the set of fixed points of f, namely, the set { ( t, x ) ∈ [ 0, 1 ] × X : f ( t, x ) = x }, has a connected component whose projection onto the first coordinate is [ 0, 1 ] . 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.