Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 584215, 15 pages
doi:10.1155/2010/584215

Generalized IFSs on noncompact spaces

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 f1,…,fn:Xm→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) and m,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.