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Miculescu and Mihail in 2008 introduced a concept of a generalized iteratedfunction system (GIFS in short), a particular extension of classical IFS.Instead of families of selfmaps of a metric space $X$, they considered familiesof mappings defined on finite Cartesian product $X^m$. It turned out that agreat part of the classical Hutchinson--Barnsley theory has natural counterpartin this GIFSs' case. Recently, Secelean extended these considerations tomappings defined on the space $\ell_\infty(X)$ of all bounded sequences ofelements of $X$ and obtained versions of the Hutchinson--Barnsley theorem forappropriate families of such functions. In the paper we study some furtheraspects of Secelean's setting. In particular, we introduce and investigate abit more restrictive framework and we show that some problems of the theoryhave more natural solutions within such a case. Finally, we present an examplewhich shows that this extended theory of GIFSs gives us fractal sets thatcannot be obtained by any IFSs or even by any GIFSs.%In a recent paper of theauthors and Jachymski we itroduced fixed point theory for mappings defined on$\ell_\infty(X)$ product of a metric space $X$. In this note we use thisapproach to study a version of Hutchinson--Barnsley theory of fractals forfunction systems consisting of such mappings. In particular, we generalize theearlier concept of generalized generated function systems introduced by Mihailand Miculescu in 2008, in which mappings defined on finite Cartesian productare considered, and compare our approach with the one introduced recently bySecelean.

On generalized iterated function systems defined on $\ell_\infty$-sum of a metric space