In this paper we indicate a way to generalize a series of fixed point results in the framework of b-metric spaces and we exemplify it by extending Nadler's contraction principle for set-valued functions (see Multi-valued contraction mappings, Pac. J. Math., 30 (1969), 475-488) and a fixed point theorem for set-valued quasi-contractions functions due to H. Aydi, M.F. Bota, E. Karapinar and S. Mitrović (see A fixed point theorem for set-valued quasi-contractions in b-metric spaces, Fixed

We introduce the notion of a generalized iterated function system GIFS , which is a finite family of functions f k : X m → X, where X, d is a metric space and m ∈ N. In case that X, d is a compact metric space and the functions f k are contractions, using some fixed point theorems for contractions from X m to X, we prove the existence of the attractor of such a GIFS and its continuous dependence in the f k 's.

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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 f 1 , . . . , f n : X m → 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 f k 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 f k '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.

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