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

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"<!-- image only, no MathML or LaTex -->, where "Equation missing"<!-- image only, no MathML or LaTex --> is a complete metric space (in the above mentioned paper the case when "Equation missing"<!-- image only, no MathML or LaTex --> is a compact metric space was studied) and "Equation missing"<!-- image only, no MathML or LaTex -->. In case that the functions "Equation missing"<!-- image only, no MathML or LaTex --> 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"<!-- image only, no MathML or LaTex --> and we prove its continuous dependence in the "Equation missing"<!-- image only, no MathML or LaTex -->'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.

Georganopoulos (see [1] ) has shown that a continuous function f : X → B, where X is a compact metric space and B a convex subset of a real normed space Y, is the uniform limit of a sequence of Lipschitz maps from X to B. In this note we obtain a similar result, namely we show that a continuous function f : X → ℝ, where X is a metric space, is a uniform limit of a sequence of locally Lipschitz maps from X to ℝ. When X is compact and B = Y = ℝ, we get the Georganopoulos' result.