An Extension of Tychonoff Fixed Point Theorem

TL;DR

This paper extends the Tychonoff fixed point theorem to a broader class of quasi-locally convex topological vector spaces by introducing new concepts of weaknorm and quasi-weaknorm, supported by a fixed point proof.

Contribution

It introduces the concepts of weaknorm and quasi-weaknorm, and extends the Tychonoff fixed point theorem to quasi-locally convex spaces, broadening its applicability.

Findings

Established a fixed point theorem in quasi-locally convex spaces.

Showed that the extension properly generalizes the classical Tychonoff theorem.

Provided an example demonstrating the extension's novelty.

Abstract

In this paper, we introduce the concepts of weaknorm, quasi-weaknorm on real vector spaces. By these concepts, we introduce the concept of quasi-locally convex topological vector spaces, which include locally convex topological vector spaces as special cases. By the Fan-KKM theorem, we prove a fixed point theorem in quasi-locally convex topological vector spaces, that is a natural extension of Tychonoff fixed point theorem in locally convex topological vector spaces. Then we provide an example to show that this extension is a proper extension.