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arXiv · 2602.09853

An Extension and Refinement of the Brouwer-Schauder-Tychonoff Fixed Point Theorem

Lixin Cheng·Chulei Liu·Wen Zhang
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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.

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Lixin Cheng, Chulei Liu, Wen Zhang. 2026-02-10. An Extension and Refinement of the Brouwer-Schauder-Tychonoff Fixed Point Theorem. https://arxiv.org/abs/2602.09853

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