Generalized IFSs on Noncompact Spaces

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 andMiculescu, 2008 . We introduce the notion of GIFS, which is a family of functions f1, . . . , fn : X → X, where X, d is a complete metric space in the above mentioned paper the case when X, d is a compact metric space was studied andm,n ∈ N. In case that the functions fk 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 X and we prove its continuous dependence in the fk’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.