Generalized IFSs on Noncompact Spaces

Abstract

Abstract The aim of this paper is to continue the research work that we have done in a previous paper published in this journal (see Mihail and Miculescu, 2008). We introduce the notion of GIFS, which is a family of functions "Equation missing", where "Equation missing" is a complete metric space (in the above mentioned paper the case when "Equation missing" is a compact metric space was studied) and "Equation missing". In case that the functions "Equation missing" are Lipschitz contractions, we prove the existence of the attractor of such a GIFS and explore its properties (among them we give an upper bound for the Hausdorff-Pompeiu distance between the attractors of two such GIFSs, an upper bound for the Hausdorff-Pompeiu distance between the attractor of such a GIFS, and an arbitrary compact set of "Equation missing" and we prove its continuous dependence in the "Equation missing"'s). Finally we present some examples of attractors of GIFSs. The last example shows that the notion of GIFS is a natural generalization of the notion of IFS.