A fixed point theorem for mappings on the ${\mathrm{\ell }}_{\mathrm{\infty }}$-sum of a metric space and its application

The aim of this paper is to prove a counterpart of the Banach fixed point principle for mappings $f:{\mathrm{\ell }}_{\mathrm{\infty }}(X)\to X$, where $X$ is a metric space and ${\mathrm{\ell }}_{\mathrm{\infty }}(X)$ 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.