Brouwer-Schauder-Tychonoff不动点定理的扩展与完善
An Extension and Refinement of the Brouwer-Schauder-Tychonoff Fixed Point Theorem
AI总结：
本文在局部凸空间中扩展并改进了Brouwer-Schauder-Tychonoff不动点定理，提出了关于紧星形集的更简洁条件和更广泛的应用。
摘要：
本文提出了一种在局部凸空间中Brouwer-Schauder-Tychonoff不动点定理的扩展和改进：假设S是关于p∈S的紧星形集，其凸性指数alpha(p)>0。则每个连续自映射f具有以下两种性质之一：(a)点p是f的不动点；(b)f有不可数多个不同的特征值和特征向量。注意，在局部凸空间中，闭有界星形集是凸的当且仅当alpha=1，并且我们将Banach空间中紧星形集上的Brouwer型不动点定理以更简洁的方式扩展到局部凸空间，从而是对Tychonoff不动点定理在紧星形集上的改进。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
