LAUSR

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