An Extension and Refinement of the Brouwer-Schauder-Tychonoff Fixed Point Theorem
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| Format: | Preprint |
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2026
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| _version_ | 1866911437958414336 |
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| author | Cheng, Lixin Liu, Chulei Zhang, Wen |
| author_facet | Cheng, Lixin Liu, Chulei Zhang, Wen |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_09853 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An Extension and Refinement of the Brouwer-Schauder-Tychonoff Fixed Point Theorem Cheng, Lixin Liu, Chulei Zhang, Wen Functional Analysis 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. |
| title | An Extension and Refinement of the Brouwer-Schauder-Tychonoff Fixed Point Theorem |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2602.09853 |