A note on a fixed point theorem on topological cylinders

TL;DR

This paper introduces a new fixed point theorem for topological cylinders in normed spaces, generalizing previous results and providing alternative proofs for classical fixed point theorems like Brouwer, Schauder, and Krasnosel'skii.

Contribution

It presents a generalized fixed point theorem on topological cylinders and offers alternative proofs for well-known fixed point theorems.

Findings

Established a fixed point theorem for maps on topological cylinders.

Extended the theorem to various domain types.

Provided alternative proofs for classical fixed point theorems.

Abstract

We present a fixed point theorem on topological cylinders in normed linear spaces for maps satisfying a property of stretching a space along paths. This result is a generalization of a similar theorem obtained by D. Papini and F. Zanolin. In view of the main result we discuss the existence of fixed points for maps defined on different types of domains and we propose alternative proofs for classical fixed point theorems, as Brouwer, Schauder and Krasnosel'ski\u{\i} ones.