  Miculescu and Mihail in 2008 introduced a concept of a generalized iterated
function system (GIFS in short), a particular extension of classical IFS.
Instead of families of selfmaps of a metric space $X$, they considered families
of mappings defined on finite Cartesian product $X^m$. It turned out that a
great part of the classical Hutchinson--Barnsley theory has natural counterpart
in this GIFSs' case. Recently, Secelean extended these considerations to
mappings defined on the space $\ell_\infty(X)$ of all bounded sequences of
elements of $X$ and obtained versions of the Hutchinson--Barnsley theorem for
appropriate families of such functions. In the paper we study some further
aspects of Secelean's setting. In particular, we introduce and investigate a
bit more restrictive framework and we show that some problems of the theory
have more natural solutions within such a case. Finally, we present an example
which shows that this extended theory of GIFSs gives us fractal sets that
cannot be obtained by any IFSs or even by any GIFSs.%In a recent paper of the
authors 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 this
approach to study a version of Hutchinson--Barnsley theory of fractals for
function systems consisting of such mappings. In particular, we generalize the
earlier concept of generalized generated function systems introduced by Mihail
and Miculescu in 2008, in which mappings defined on finite Cartesian product
are considered, and compare our approach with the one introduced recently by
Secelean.
