An Extension and Refinement of the Brouwer-Schauder-Tychonoff Fixed Point Theorem
Lixin Cheng, Chulei Liu, Wen Zhang
TL;DR
Problem: extend fixed-point results like $\text{Brouwer}$, $\text{Schauder}$, and $\text{Tychonoff}$ to compact star-shaped sets in locally convex spaces, beyond convex domains. Approach: define a convexity index via Minkowski gauges $\rho_{S,p}$ and reduce fixed-point questions to scaled cones $\lambda\alpha_p(C-p)$ where $C=\overline{\mathrm{co}}(S)$; use the $\text{Tychonoff}$ fixed-point theorem to construct a family of eigenpoints. Key contributions: a concise extension of the fixed-point property to locally convex spaces for compact star-shaped sets with $\alpha_p>0$, a precise eigenvalue-eigenvector structure $f(x_\lambda)=p+\frac{1}{\lambda\alpha_p}(x_\lambda-p)$ with $\lambda\in(0,1]$, and density and geometric analysis of the convexity index. Significance: broadens fixed-point theory to non-convex compact domains in locally convex spaces and yields an explicit uncountable eigenstructure for nontrivial mappings, with potential applications to nonexpansive mappings and nonlinear functional analysis.
Abstract
In this paper, we present the Brouwer-Schauder-Tychonoff fixed point theorem on locally convex spaces as the following extension and improvement: Suppose that S is a compact star-shaped subset with respect to p in S with its convexity index alpha(p)>0. Then every continuous self-mapping f has one of the following two properties: (a) The point p is a fixed point of f; (b) f has uncountably many different eigenvalues and eigenvectors. Note that a closed bounded star-shaped set in a locally convex space is convex if and only if alpha=1, and we extend a Brouwer's type fixed-point theorem on compact star-shaped sets in Banach spaces in a more concise manner to locally convex spaces, thereby this is a simplification and an improvement of the Tychonoff fixed-point theorem to compact star-shaped sets.