The canonical projection associated with certain possibly infinite generalized iterated function systems as a fixed point

Abstract

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