  In this paper, influenced by the ideas from A. Mihail, The canonical
projection between the shift space of an IIFS and its attractor as a fixed
point, Fixed Point Theory Appl., 2015, Paper No. 75, 15 p., we associate to
every generalized iterated function system F (of order m) an operator H defined
on C^m and taking values on C, where C stands for the space of continuous
functions from the shift space on the metric space corresponding to the system.
We provide sufficient conditions (on the constitutive functions of F) for the
operator H to be continuous, contraction, phi-contraction, Meir-Keeler or
contractive. We also give sufficient condition under which H has a unique fixed
point. Moreover, we prove that, under these circumstances, the closer of the
imagine of the fixed point is the attractor of F and that the fixed point is
the canonical projection associated to F. In this way we give a partial answer
to the open problem raised on the last paragraph of the above mentioned
Mihail's paper.
