An Extension and Refinement of the Brouwer-Schauder-Tychonoff Fixed Point Theorem
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.
Keywords
Cite
@article{arxiv.2602.09853,
title = {An Extension and Refinement of the Brouwer-Schauder-Tychonoff Fixed Point Theorem},
author = {Lixin Cheng and Chulei Liu and Wen Zhang},
journal= {arXiv preprint arXiv:2602.09853},
year = {2026}
}