On the fractal operator of a mixed possibly infinite iterated function system

Abstract

In this paper we introduce a new class of iterated function systems. More precisely, we study the fractal operator associated with a mixed possibly infinite iterated function system (briefly mIIFS). Such a system is a possibly infinite iterated function system (i.e. a possibly infinite family of Banach contractions on a complete metric space, satisfying some extra conditions) enriched with an orbital possibly infinite iterated function system (i.e. a possible infinite family of nonexpansive functions which need not be Banach contractions on the entire previously mentioned complete metric space, but just on the orbits of the space’s elements). Our main result states that the fractal operator associated with an mIIFS is a weakly Picard operator. Its fixed points are called attractors of the system. We present concrete examples of mIIFSs and graphical representations of certain attractors’ approximates.