A fixed point theorem for mappings on the $\ell_\infty$-sum of a metric space and its application

TL;DR

This paper extends the Banach fixed point theorem to mappings from the space of bounded sequences in a metric space to the space itself, broadening fixed point theory applications.

Contribution

It generalizes previous fixed point results to the setting of $ ext{ell}_ ext{infty}$-sums of metric spaces, providing a new fixed point theorem and applications.

Findings

Established a fixed point theorem for mappings on $ ext{ell}_ ext{infty}(X)$

Compared new results with previous theorems by Miculescu, Mihail, and Secelean

Provided examples and an application demonstrating the theorem's utility

Abstract

The aim of this paper is to prove a counterpart of the Banach fixed point principle for mappings $f: \ell_\infty(X) \to X$, where $X$ is a metric space and $\ell_\infty(X)$ is the space of all bounded sequences of elements from~$X$. Our result generalizes the theorem obtained by Miculescu and Mihail in 2008, who proved a~counterpart of the Banach principle for mappings $f:X^m\to X$, where $X^m$ is the Cartesian product of $m$ copies of $X$. We also compare our result with a recent one due to Secelean, who obtained a weaker assertion under less restrictive assumptions. We illustrate our result with several examples and give an application.