A Generalization of the "Brouwer-Schauder-Tychonoff" Fixed-Point Theorem

TL;DR

This paper presents a generalized fixed-point theorem extending classical results like Brouwer, Schauder, and Tychonoff, applicable to continuous functions from compact convex subsets of locally convex spaces.

Contribution

It introduces a new fixed-point result for functions into product spaces, broadening the scope of classical fixed-point theorems.

Findings

Generalizes Brouwer-Schauder-Tychonoff fixed point theorem

Applicable to continuous functions into product spaces with specific conditions

Includes classical fixed-point theorems as special cases

Abstract

We prove a new fixed - point result for the image Im(j) of any continuous function j from K to (K x K), where K is a compact convex subset of a Hausdorff locally convex space, provided that the projection of Im(j) to the first factor is onto, and a condition on the convex hull of Im(j) holds. A special case of our result is the Brouwer - Schauder - Tychonoff fixed point theorem for continuous functions from K to K.