Possibly infinite generalized iterated function systems comprising ϕ-max contractions
DOI:
https://doi.org/10.24193/subbmath.2019.2.01Keywords:
Possibly infinite generalized iterated function system, ϕ-max contraction, attractor, canonical projection.Abstract
One way to generalize the concept of iterated function system was proposed by R. Miculescu and A. Mihail under the name of generalized iterated function system (for short GIFS). More precisely, given m ∈ N∗ and a metric space (X, d), a generalized iterated function system of order m is a finite family of functions f1, . . . , fn : Xm → X satisfying certain contractive conditions. Another generalization of the notion of iterated function system, due to F. Georgescu, R. Miculescu and A. Mihail, is given by those systems consisting of ϕ-max contractions. Combining these two lines of research, we prove that the fractal operator associated to a possibly infinite generalized iterated function system comprising ϕ-max contractions is a Picard operator (whose fixed point is called the attractor of the system). We associate to each possibly infinite generalized iterated function m system comprising ϕ-max contractions F (of order m) an operator HF : C → C, where C stands for the space of continuous and bounded functions from the shift space on the metric space corresponding to the system. We prove that HF is a Picard operator whose fixed point is the canonical projection associated to F.
Mathematics Subject Classification (2010): 28A80, 37C70, 41A65, 54H25.
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