Browder's Theorem with General Parameter Space

TL;DR

This paper extends Browder's fixed point theorem from the unit interval to more general connected and compact Hausdorff spaces, broadening its applicability in fixed point theory.

Contribution

It generalizes Browder's theorem to arbitrary connected, compact Hausdorff spaces as the parameter space, beyond the original interval case.

Findings

Connected component of fixed points projects onto the entire space

Extension from interval to Hausdorff spaces

Maintains connectedness of fixed point set

Abstract

Browder (1960) proved that for every continuous function $F : X \times Y \to Y$, where $X$ is the unit interval and $Y$ is a nonempty, convex, and compact subset of $\dR^n$, the set of fixed points of $F$, defined by $C_F := \{ (x,y) \in X \times Y \colon F(x,y)=y\}$ has a connected component whose projection to the first coordinate is $X$. We extend this result to the case where $X$ is a connected and compact Hausdorff space.