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.


The Organization of the Paper
The paper is organized as follows. Section 2 contains a short presentation of the notion of an iterated function system IFS , one of the most common and most general ways to generate fractals. This will serve as a framework for our generalization of an iterated function system. Then, we introduce the notion of a GIFS, which is a finite family of Lipschitz contractions f k : X m → X, where X, d is a complete metric space and m ∈ N.
In Section 3 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 . Section 4, the last one, contains some examples and remarks. The last example shows that the notion of GIFS is a natural generalization of the notion of IFS.

Some Generalizations of the Notion of IFS
IFSs were introduced in their present form by John Hutchinson and popularized by Barnsley see 1 . There is a current effort to extend Hutchinson's classical framework for fractals to more general spaces and infinite IFSs. Some papers containing results on this direction are 2-7 .

Some Physical Applications of IFSs
In the last period IFSs have attracted much attention being used by researchers who work on autoregressive time series, engineer sciences, physics, and so forth. For applications of IFSs in image processing theory, in the theory of stochastic growth models, and in the theory of random dynamical systems one can consult 8-10 . Concerning the physical applications of iterated function systems we should mention the seminal paper 11 of El Naschie which draws attention to an informal but instructive analogy between iterated function systems and the two-slit experiment which is quite valuable in illuminating the role played by the possibly DNA-like Cantorian nature of microspacetime and clarifies the way in which probability enters into this subject. We also mention the paper 12 of Słomczyński where a new definition of quantum entropy is introduced and one method using the theory of iterated function systems of calculating coherent states entropy is presented. The coherent states entropy is computed as the integral of the Boltzmann-Shannon entropy over a fractal set.
In 13 , Bahar described bifurcation from a fixed-point generated by an iterated function system IFS as well as the generation of "chaotic" orbits by an IFS, and in 14 unusual and quite interesting patterns of bifurcation from a fixed-point in an IFS system, as well as the routes to chaos taken by IFS-generated orbits, are discussed. Moreover, in 15 it is shown that random selection of transformation in the IFS is essential for the generation of a chaotic attractor. In 16, Section 6.4 , one can find a lengthy but elementary explanation which features of randomness play the main role.

Preliminaries
Notations. Let X, d X and Y, d Y be two metric spaces.
As usual, C X, Y denotes the set of continuous functions from X to Y , and d : For a sequence f n n of elements of C X, Y and f ∈ C X, Y , f n s → f denotes the punctual convergence, f n u.c → f denotes the uniform convergence on compact sets, and f n u → f denotes the uniform convergence, that is, the convergence in the generalized metric d. For a function f : X m × m k 1 X → X, the number We will use the notation LCon m X for the set {f : X m → X : Lip f < 1}.
Notations. P X denotes the subsets of a given set X and P * X denotes the set P X − {∅}.
Given a metric space X, d , K X denotes the set of compact subsets of X and B X denotes the set of closed bounded subsets of X.
For a metric space X, d , we consider on P * X the generalized Hausdorff-Pompeiu pseudometric h : Remark 2.4. The Hausdorff-Pompeiu pseudometric is a metric on B * X and, in particular, on K * X .
Remark 2.5. The metric spaces B * X , h and K * X , h are complete, provided that X, d is a complete metric space see 1, 7, 17 .

Fixed Point Theory and Applications
ii if H i i∈I and K i i∈I are two families of nonempty subsets of X, then iii if H and K are two nonempty subsets of X and f : X → X is a Lipschitz function, then Definition 2.7. An iterated function system on X consists of a finite family of Lipschitz contractions f k k 1,n on X and is denoted S X, f k k 1,n .
The set A S is called the attractor of the IFS S X, f k k 1,n .
Given a metric space X, d , the idea of our generalization of the notion of an IFS is to consider contractions from X m × m k 1 X to X, rather than contractions from X to itself.
Definition 2.9. Let X, d be a complete metric space and m ∈ N. A generalized iterated function system on X of order m for short a GIFS or a GmIFS , denoted S X, f k k 1,n , consists of a finite family of functions f k k 1,n , f k : X m → X such that f 1 , . . . , f n ∈ LCon m X .
Earlier several authors tried to coin the name generalized IFS. One should note the paper 19 in which notion tightly corresponds to contractive multivalued IFS from 2 .

The Existence of the Attractor of a GIFS for Lipschitz Contractions
In this section m is a fixed natural number, X, d will be a fixed complete metric space, and all the GIFSs are of order m and have the form S X, f k k 1,n , where n is a natural number. We prove the existence of the attractor of S Theorem 3.9 and study its properties among them we give an upper bound for the Hausdorff-Pompeiu distance between the attractors of two such GIFSs Theorem 3.12 , an upper bound for the Hausdorff-Pompeiu distance between the attractor of such a GIFS, and an arbitrary compact set of X Theorem 3.17 and we prove its continuous dependence in the f k 's Theorem 3.15 .
for all K 1 , K 2 , . . . , K m ∈ K * X , is called the set function associated to the function f.

5
The function F S : K * X m → K * X defined by for all K 1 , K 2 , . . . , K m ∈ K * X , is called the set function associated to the GIFS S.

Lemma 3.2.
For a sequence f n n of elements of C X m , X and f ∈ C X m , X such that f n u → f, one has

3.4
Proof. In this proof, by M we mean sup n≥1 Lip f n . Let us consider A {x ∈ X | f n x → f x }, which is a dense set in X, let K be a compact set in X, and let ε > 0.
Since f is uniformly continuous on K, there exists δ ∈ 0, ε/3 M 1 such that if x, y ∈ K and d X x, y < δ, then Since K is compact, there exist x 1 , x 2 , . . . , x p ∈ K such that Taking into account the fact that A is dense in X, we can choose y 1 , y 2 , . . . , y p ∈ A such that Since, for all i ∈ {1, . . . , p}, lim n → ∞ f n y i f y i , there exists n ε ∈ N such that for every n ∈ N, n ≥ n ε , we have for every i ∈ {1, . . . , p}.

Fixed Point Theory and Applications
For x ∈ K, there exists i ∈ {1, . . . , p}, such that x ∈ B x i , δ/2 and therefore and so Hence, for n ≥ n ε , we have

3.11
Consequently, as x was arbitrary chosen in K, we infer that f n u → f on K, and so The inequality For every x 0 , x 1 , . . . , x m−1 ∈ X, the sequence x n n≥1 , defined by x k m f x k m−1 , x k m−2 , . . . , x k , for all k ∈ N, has the property that 3.17 Concerning the speed of the convergence, one has the following estimation: Proof. See 20, Remark 5.1 .

3.26
From Theorem 3.11 and Lemma 3.6, we have the following.
Fixed Point Theory and Applications 9 Theorem 3.12. In the framework of this section, if S X, f k k 1,n and S X, g k k 1,n are two m dimensional GIFSs, then g 1 , . . . , d f n , g n , 3.27 where μ min max{Lip f 1 , . . . , Lip f n }, max{Lip g 1 , . . . , Lip g n } .

Examples
In this section we present some examples of attractors of GIFSs. Example 4.3 shows that the notion of GIFS is a natural generalization of the notion of IFS.

Fixed Point Theory and Applications
Example 4.1. Let A 1 , A 2 , . . . , A m ∈ B X and α ∈ X, where X is a Banach space and B X is the set of linear and continuous operators from X to X.
Let us consider the function f : X m → X, given by for every x 1 , x 2 , . . . , x m ∈ X. Then