In this section is a fixed natural number, will be a fixed complete metric space, and all the GIFSs are of order and have the form , where is a natural number.

We prove the existence of the attractor of (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 (Theorem 3.17) and we prove its continuous dependence in the 's (Theorem 3.15)).

Definition 3.1.

Let be a function. The function defined by

for all , is called the set function associated to the function .

The function defined by

for all , is called the set function associated to the GIFS .

Lemma 3.2.

For a sequence of elements of and such that , one has

in, for all.

Proposition 3.3.

Let and be two complete metric spaces and let be such that and on a dense set in . Then

Proof.

In this proof, by we mean .

Let us consider , which is a dense set in , let be a compact set in , and let .

Since is uniformly continuous on , there exists such that if and , then

Since is compact, there exist such that

Taking into account the fact that is dense in , we can choose such that

Since, for all , , there exists such that for every , , we have

for every .

For , there exists , such that and therefore

and so

Hence, for , we have

Consequently, as was arbitrary chosen in , we infer that on , and so

The inequality

is obvious.

Lemma 3.4.

Let be subsets of .

Then

(1);

(2).

Lemma 3.5.

If is a Lipschitz function, then

Lemma 3.6.

In the framework of this section, one has

The proofs of the above lemmas are almost obvious.

Theorem 3.7 (Banach contraction principle for ).

For every , there exists a unique , such that

For every , the sequence , defined by , for all , has the property that

Concerning the speed of the convergence, one has the following estimation:

for every .

Proof.

See [20, Remark ].

Remark 3.8.

The point from the above theorem is called the fixed point of .

From Theorem 3.7 and Lemma 3.6 we have the following.

Theorem 3.9.

In the framework of this section, there exists a unique such that

Moreover, for any , the sequence defined by , for all , has the property that

Concerning the speed of the convergence, one has the following estimation:

for all .

Definition 3.10.

The unique set given by the previous theorem is called the attractor of the GIFS .

Theorem 3.11.

If have the fixed points and , then

Proof.

We have

so

and in a similar manner we get

Therefore

From Theorem 3.11 and Lemma 3.6, we have the following.

Theorem 3.12.

In the framework of this section, if andare two dimensional GIFSs, then

where .

Theorem 3.13.

Let with fixed points and , respectively, such that

on a dense set in .

Then

Proof.

From the fact that and on a dense set in , it follows, using Proposition 3.3, that

on and

From Theorem 3.11, we have

and hence

for all .

Since on , we obtain that

and consequently, using the above inequality, we obtain that

Proposition 3.14.

Let , where , and let be -dimensional generalized iterated function systems such that

on a dense set in , for every .

Then

Proof.

Using Proposition 3.3, we obtain that

on and

Then, using Lemma 3.2 and Proposition 2.6(ii), we get

Since, according to Lemma 3.6, we have

for all , we obtain, using again the arguments from Proposition 3.3, that

From Theorem 3.13, Proposition 3.14, and Lemma 3.6, we have the following.

Theorem 3.15.

Let , where , and let be -dimensional generalized iterated function systems having the property that

on a dense set in , for every .

Then

Theorem 3.16.

For having the unique fixed point and for every , one has

Proof.

We can use the Banach contraction principle for , where

for all .

Theorem 3.17.

For a generalized iterated function system and , the following inequality is valid:

Proof.

The function , defined by

for all , is a contraction and