In this paper, influenced by the ideas from Mihail (Fixed Point Theory Appl 2015:15, 2015), we associate to every generalized iterated function system F\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {F}$$\end{document} (of order m) an operator HF:Cm→C\documentclass[12pt]{minimal}
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\begin{document}$$H_{\mathcal {F}}:\mathcal {C} ^{m}\rightarrow \mathcal {C}$$\end{document}, where C\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {C}$$\end{document} 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\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {F}$$\end{document}) for the operator HF\documentclass[12pt]{minimal}
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\begin{document}$$H_{\mathcal {F}}$$\end{document} to be continuous, contraction, φ\documentclass[12pt]{minimal}
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\begin{document}$$\varphi $$\end{document}-contraction, Meir–Keeler or contractive. We also give sufficient condition under which HF\documentclass[12pt]{minimal}
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\begin{document}$$H_{\mathcal {F}}$$\end{document} has a unique fixed point π0\documentclass[12pt]{minimal}
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\begin{document}$$\pi _{0}$$\end{document}. Moreover, we prove that, under these circumstances, the closure of the imagine of π0\documentclass[12pt]{minimal}
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\begin{document}$$\pi _{0}$$\end{document} is the attractor of F\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {F}$$\end{document} and that π0\documentclass[12pt]{minimal}
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\begin{document}$$\pi _{0}$$\end{document} is the canonical projection associated with F\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {F}$$\end{document}. In this way we give a partial answer to the open problem raised on the last paragraph of the above-mentioned Mihail’s paper.
