A study of the attractor of a $$\varvec{\varphi }$$ φ - $$\varvec{\max }$$ max -IFS via a relatively new method

Abstract

In this paper, inspired by the ideas from Mihail (Fixed Point Theory Appl 75:15, 2015) we associate to every iterated function system \(\mathcal {S}\) (i.e., a finite system of continuous selfmaps of a metric space) a continuous operator \(G_{\mathcal {S}}:\mathcal {C}\rightarrow \mathcal {C}\), where \(\mathcal {C}\) stands for the space of continuous functions from the shift space to the metric space corresponding to the system, and we prove that if \(G_{\mathcal {S}}\) is a Picard operator, then the fractal operator associated to \(\mathcal {S}\) is a Picard operator and \(\mathcal {S}\) admits canonical projection. Moreover we introduce the new notion of iterated function system consisting of \(\varphi \)-\(\max \)-contractions and prove that the map \(G_{\mathcal {S}}\) associated to such a system \(\mathcal {S}\) is a Picard operator. Iterated function systems consisting of Matkowski (in particular, Banach) contractions and iterated function systems consisting of convex contractions are particular cases of iterated function systems consisting of \(\varphi \)-\(\max \)-contractions.