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

TL;DR

This paper extends and refines the Brouwer-Schauder-Tychonoff fixed point theorem for locally convex spaces, providing a more general and concise result for compact star-shaped sets with new fixed point and eigenvalue properties.

Contribution

It introduces a generalized fixed point theorem for locally convex spaces, extending the classical results to compact star-shaped sets with a new eigenvalue characterization.

Findings

Fixed point existence under new conditions

Uncountably many eigenvalues and eigenvectors for certain mappings

Simplification of fixed point theorem in locally convex spaces

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.