Abstract and Applied Analysis
Volume 2015 (2015), Article ID 345856, 16 pages
http://dx.doi.org/10.1155/2015/345856

Research Article

Applications of Multivalued Contractions on Graphs to Graph-Directed Iterated Function Systems

T. Dinevari and M. Frigon

Département de Mathématiques et de Statistique, Université de Montréal, C.P. 6128, Succursale Centre-Ville, Montréal, QC, Canada H3C 3J7

Received 12 August 2014; Accepted 18 December 2014

Academic Editor: Naseer Shahzad

Copyright © 2015 T. Dinevari and M. Frigon. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We apply a fixed point result for multivalued contractions on complete metric spaces endowed with a graph to graph-directed iterated function systems. More precisely, we construct a suitable metric space endowed with a graph and a suitable -contraction such that its fixed points permit us to obtain more information on the attractor of a graph-directed iterated function system.

1. Introduction

Based on the work of Hutchinson [1] and being popularized by Barnsley [2], the method of iterated function systems (IFS) permits us to generate fractals by iterating a collection of transformations . If each is a contraction on a complete metric space , it was shown in [1] that there exists a unique nonempty compact set which is invariant with respect to ; that is,

This attractor is such that, for every compact ,

where

The existence of can be deduced from the Banach fixed point theorem.

A fixed point result which is, in some sense, a combination of the Banach contraction principle and the Knaster-Tarski fixed point theorem in a partially ordered set was obtained by Ran and Reurings [3] in 2004. They considered a monotone, order preserving single-valued map defined on a complete metric space endowed with a partial ordering. They assumed that satisfies a contraction condition not necessarily for all and , but for those such that . Subsequently, their result was generalized by many authors, in particular by Nieto, Rodríguez-López, Pouso, Petruşel, and Rus [4–7]. In 2008, Jachymski [8] presented a nice unification of most of the previous results by considering complete metric spaces endowed with a graph . He introduced the notion of single-valued -contraction for which he obtained fixed point results.

Using those fixed point results, Gwóźdź-Łukawska and Jachymski [9] developed the Hutchinson-Barnsley theory on complete metric space endowed with a graph for iterated function systems of single-valued -contractions.

Different extensions of the concept of single-valued contractions on complete metric spaces endowed with a graph to multivalued maps were presented by Dinevari and Frigon [10] and by Nicolae et al. [11]. Those extensions led to generalizations of Jachymski’s fixed point results and of the Nadler fixed point theorem for multivalued contractions.

In 1988, Mauldin and Williams [12] introduced the notion of geometric graph-directed construction.

Definition 1. A geometric graph-directed construction in consists of

(i)a collection of nonoverlapping, compact, nonempty subsets of , with nonempty interior;(ii)a directed-graph such that is the set of its vertices, and, for each , there exists some edge ;(iii)for each , there is a similarity map with similarity ratios such that

(iv)for each , is a nonoverlapping family of sets;(v)if is a cycle in , then

They showed that a geometric graph-directed construction has an attractor.

Theorem 2 (Mauldin and Williams [12]). For a geometric graph-directed construction as above, there exists a unique collection of nonempty compact sets such that

The set

is called the attractor of this geometric graph-directed construction.

Geometric graph-directed constructions have been studied and generalized by many authors; see [13–16]. In particular, it was shown in [13] that with an appropriate rescaling, condition (v) can be replaced by

(v)′for each , .

Also, in some of those generalizations, similarities on were replaced by contractions on complete metric spaces and the terminology of graph-directed iterated function system was used. Again, the existence of an attractor was established.

In this paper, we take into account the graph to obtain more information on the attractor of a graph-directed iterated function system. To do so, we apply a fixed point result obtained by the authors [10] for multivalued contractions on complete metric spaces endowed with a graph.

The paper is organized as follows. In Section 2, we present some notations and we recall some results. In Section 3, we consider a space such that and on which we define a suitable graph and a suitable metric. In Section 4, we define an appropriate multivalued -contraction . In the last three sections, taking into account the maximal connected component of the graph , we obtain more information on the attractor from some fixed points of .

2. -Iterated Function System

First of all, we introduce the notion of MW-directed graph and we consider iterated function systems which takes into account the structure of an -directed graph.

Definition 3. A directed-graph is called an MW-directed graph if , has no parallel edges, and for every , there exists such that .

Definition 4. Let be an MW-directed graph. A graph-directed iterated function system over the graph (-IFS) is a collection of nonempty, bounded, complete metric spaces, , and, for each , a contraction with constant of contraction . An -IFS is denoted .

Definition 5. Let be an -IFS. An attractor of the -IFS is a collection of nonempty compact sets such that and

The Banach contraction principle insures the existence of an attractor of an -IFS. We present the proof for sake of completeness. For more information on graph-directed iterated function systems, the reader is referred to [12, 15].

Theorem 6. An -IFS, , has a unique attractor .

Proof. Consider

endowed with the metric

where is the Hausdorff metric on ; that is,

where

Let us define by

Using the fact that every is a contraction, one verifies that is a contraction with constant of contraction

The Banach contraction principle insures the existence of a unique fixed point of . Thus, is the unique attractor of .

More information on will be obtained by applying a fixed point result for multivalued contractions on complete metric spaces endowed with a graph. We recall the notion of -contraction introduced in [10].

For a complete metric space, we consider a directed graph such that , the diagonal in is contained in , and has no parallel edges.

Definition 7. Let be a multivalued map with nonempty values. We say that is a -contraction if there exists such that

for all and all , there exists such that and .

We consider suitable trajectories in .

Definition 8. Let be a multivalued mapping and . We say that a sequence is a -Picard trajectory from if and for all . The set of all such -Picard trajectories from is denoted by .

The reader is referred to [10] for the proof of the following fixed point result for multivalued -contractions.

Theorem 9. Let be a multivalued -contraction such that there exists such that . In addition, assume that one of the following conditions holds.

(i) is -Picard continuous from ; that is, the limit of any convergent sequence is a fixed point of .(ii) has closed values and, for every in converging to some , there exists a subsequence such that for all .

Then, there exists a -Picard trajectory from , , converging to a fixed point of . Moreover, every converging -Picard trajectory from converges to a fixed point of .

In what follows, we consider an MW-directed graph. We will use the following definitions and notations.

A path from to in is denoted by , where , , and for every .

We say that a subgraph of is connected if for every there exists a path from to in . A connected component of is a maximal connected subgraph of . We denote

It follows from the definition of MW-directed graph that

We can define a partial order on as follows:

We write to mean and . We say that and are incomparable if and .

We denote the set of vertices from which there is a path in reaching by

Similarly, for , we denote the set of vertices from which there is a path in reaching by

3. A Suitable Metric Space Endowed with a Directed Graph

Let be an MW-directed graph with . For , let be a bounded complete metric space.

In this section, using and the spaces , we define a complete metric space endowed with a suitable directed graph. Let us recall that

We consider the space of -tuples satisfying the following properties:

(Xi) is compact for every ;(Xii)if for some and , then for all ;(Xiii)there exists and such that .

It is important to point out that, for , some can be empty.

We endow with the metric

where

where is the Hausdorff metric in and is a constant which will be fixed later, with

It is clear that is a complete metric space.

Taking into account the graph , we want to endow with a directed graph. To do so, we distinguish vertices of which are in a connected component from the others. We set

We define the graph as follows: , and for , if and only if

(G)for every , one of the following properties holds:

(i), or and ;(ii), , and one of the following statements is true:

(a) and there exists such that and ;(b) for some and there exist and such that and ;

(iii), , , and one of the following properties is satisfied:

(a)there is no such that ;(b)for every such that , one has .

Example 10. Let be the MW-graph of Figure 1. We consider the associated metric space satisfying (Xi)–(Xiii) endowed with the graph satisfying the condition (G). Let be nonempty compact subsets of for all and . We consider the following elements of :

Here is the list of all edges of between them:

Figure 1: An MW-graph .

Now, we want to fix in (22) in such a way that we will be able to define a suitable multivalued -contraction on in the next section. To this aim, we decompose in appropriate subsets with a totally ordered set.

Lemma 11. Let be an MW-directed graph. Then there exist a totally ordered set and a family of nonempty disjoint subsets, and, for every , there exists such that

(1);(2)if for some and some , then ;(3)if in , for all , and , then ;(4)for every , one has for every ;(5)for every , one has for every , .

Proof. We want to separate vertices of in suitable subsets. Let us recall that some vertices are in a connected component, and some others are not:

where and are defined in (24) and (25), respectively.
First of all, we examine vertices in . Let

We denote

We define

Observe that

Now, we separate vertices in in suitable subsets. We first separate them in two sets: those which can be reached by a path starting from a vertex in a connected component, and those which cannot. This last set is denoted:

If , let

We define

Observe that

If , it follows from Definition 3 that, for every , there exist such that

In other words, is on a path from to . Hence, , where is defined in (29).
If , we first examine vertices on a path from some to some . Let

If , we define

If , we define

We define inductively .
We denote the set of vertices on a path from to by

If , we examine vertices on a path from some to some . Let

If , we define

If , we define

Similarly, we define for .
So, inductively, we define the following subsets of :

Each vertex in one of those sets is on a path from one connected component to another.
We have decomposed in a collection of disjoint sets:

We denote

We endow with the order

By construction,

Also, for every , there exists such that . Moreover, for such that , one has for every , and .
Finally, we choose a strictly increasing map. We define

By construction, statements (4) and (5) are satisfied.

Remark 12. Let . From the definition of the graph and Lemma 11, we can make the following observations.

(1)If for some , (G)(ii) holds with some such that ; let be such that and . Then, .(2)If for some , (G)(iii)(b) holds, let be such that . Then, for all such that , there is such that and one has .

Example 13. We consider the MW-graph of Figure 2 for which we describe the collection of subsets constructed as in the proof of Lemma 11. In this graph,

Since , one has , and

By considering the paths from to , one sees that , and

By considering the paths from to , one sees that , and

Similarly, one has , and

So, the vertices which are not in one of the previous sets are in . Similarly, , and

So, is the totally ordered set:

Figure 2: An MW-graph with .

4. A -Contraction

In this section, we consider a graph-directed iterated function system over the graph , . We will define an appropriate multivalued -contraction on , where and are, respectively, the graph and the metric space endowed with this graph and defined in the previous section. This -contraction will be used to get more information on the attractor of this -IFS.

Let . For each such that , for all such that . So, it is important to distinguish all those edges. To this aim, we introduce the following notations.

Let be the subset of vertices in which are not in connected components of and defined in (25). So, for , we denote

For , we define

Let be the subset of vertices in which are in connected components of and defined in (24). So, for , there exists such that . We consider the set of edges from a vertex of to a vertex outside of for which the component of is nonempty:

For , we denote

and we define by

We define

We also define

where .

We have all the ingredients to define the multivalued map . For ,

where is defined as follows.

For ,

For for some ,

Observe that is well defined. Indeed, if is such that for in some , then for all . Also, there exists such that for all . Moreover, the values of are finite and hence closed.

We show that is a multivalued -contraction.

Proposition 14. Let be the multivalued map defined above. Then is a -contraction.

Proof. We want to show that is a -contraction with constant of contraction:

where , , and for are given in Lemma 11. For for some , we denote

where is given in (61). Observe that .
Let be such that and . We look for such that and .
Step  1 ( ). Let be such that .
Case  1 ( and for every ). In this case, and by (66). Choose some . Therefore, , , and for such that , one has .
By condition (G)(ii)(a), if , there exists such that and . So, .
On the other hand, if for some , by condition (G)(ii)(b), there exist and such that and . So, and .
So, for the case and the case , we obtain by (66) and (67),

Moreover, by (21), (22), and (68),

Case 2 ( and for every ). In this case, and by (66). Choose some . Therefore, , , and for such that , one has . By conditions (G)(i) and (G)(iii), one has and . By (66), (67) and since , one has

Also, by (21), (22), and (68),

Case 3 ( and for every ). In this case, for some , and by (66).
If , one has by (21), (59), and (68),

If , choose some . So, , , and, for such that , one has . Thus, by (21), (22), and (68),

Combining (74) and (75), for for some , we choose such that

and we get

Step 2 ( for some ). Let be such that .
Case 4 ( and for every ). In this case, and by (67).
If , by condition (G)(ii)(b), there exist and such that and . So, . This contradicts the fact that .
If , by (60), there exist and such that and and, for such that , one has . Since , . If , by condition (G)(ii)(a), there exists such that and . So, , and by (66). On the other hand, if for some , by condition (G)(ii)(b), there exist , such that and . So, and by (67). Thus, for the case and the case , we obtain

Moreover, by (21), (22), and (68),

Case 5 ( and for every ). In this case, and by (67). From condition (G)(iii), we deduce that . Let . One has and since . By condition (G)(iii), and since . This implies that by condition (Xii) since . This is a contradiction. Thus,

Case 6 ( and for every ). In this case, and by (67).
If , by condition (G)(iii), . So , , and, by (21), (64), and (68),

If , for such that , one has by (21), (62), (63), (68), and (69),

If and , there exists such that , and, for such that , one has . Hence, by (21), (22), and (68),

Combining (67), (81), (82), and (83), we choose such that

and we get

Step  3 (choice of an appropriate ). Finally, we choose as follows:

It follows from (70), (72), (78), and (80) that

Finally, from (71), (73), (77),   (79), and (85), we deduce that

Therefore, is a -contraction.

Here is another property satisfied by the multivalued map .

Lemma 15. Let be the multivalued map defined above. Then, for every and every -Picard trajectory from converging to some , there exists such that for all .

Proof. Let and a -Picard trajectory from such that . Thus, there exists such that for all . So, by (21) and (22), and are such that, for all and all , if and only if . Thus, (G)(i) is satisfied and for all .

5. Attractor of an -IFS and Elements of

For an MW-directed graph, and a graph-directed iterated function system over the graph , we consider the attractor of this -IFS insured by Theorem 6. We want to get more information on by taking into account the connected components of .

Theorem 16. Let be an MW-directed graph. Let be an -IFS and its attractor. Then the following statements hold.

(1)For every , there exists such that

(a) for every ;(b) for every , where is defined in (19).(c) for every .

(2)If are such that , then .(3)If are incomparable, then

(4)There exists such that and

(a)for every , for every and for every ;(b)if are such that , then

(c)if are incomparable, then,

Proof. Let be the multivalued map defined in (65), (66), and (67). We know that is a -contraction by Proposition 14. Also, it follows from Lemma 15 that satisfies condition (ii) of Theorem 9.
Theorem 6 and the definition of imply that fixed points of are included in .
Let . We want to show that there exists a fixed point of satisfying the required properties. Fix

For , we choose inductively

That is, by (66) and (67), is chosen as follows.
For ,

where and are defined in (58) and (59), respectively.
For for some ,

where , , and are defined in (60), (63), and (64), respectively.
Arguing as in the proof of Proposition 14, one has that and

By Theorem 9, is a -Picard trajectory converging to some a fixed point of .
Observe that, for every and every , . Therefore,

Similarly, observe that, by construction, for every . Indeed, for such , if , and if for some . Thus,

On the other hand, let

Again by construction, for all , for all . So,

Finally, observe that is independent of chosen as in (92). Indeed, for

we define inductively as in (93). Observe that for all . Arguing as in Proposition 14, one has

This inequality combined with the fact that implies that .
Let be such that . One has

Let be such that

By and (G)(i), one has and . Let be the biggest element in ; that is, is chosen similarly to (94) and (95). Observe that , since , , and by the definitions of and . Arguing as in the proof of Proposition 14, one has and

Repeating this argument, we obtain a -Picard trajectory from such that

Therefore, and

If are incomparable, it follows directly from (1)(c) that

(4) For every , is an MW-directed graph and

is a graph-directed iterated function system over the graph . Let

be the attractor of this graph-directed iterated system insured by Theorem 6.
We define by

Let and the -Picard trajectory from defined in (92) and (93). By (95), for all , and for all . So, using (64) and (67) and the fact that for every , we deduce that

By definition of ,

The uniqueness of the fixed point of this operator implies that

On the other hand, if , one has . If for some , then . It follows from (114) and (2) that .
The properties (b) and (c) follow directly from and (a).

Example 17. Let be the MW-graph of Figure 3.
We consider the -IFS, , with the metric spaces:

and the contractions:

where

Figures 4 and 5 present and , respectively.

Figure 3: An MW-graph with .

Figure 4: The set .

Figure 5: The set .

6. Attractor of an -IFS and Subsets of

We obtain other pieces of information on the attractor of the graph-directed iterated function system by considering subsets of .

Theorem 18. Let be an MW-directed graph. Let be an -IFS and its attractor. Then the following statements hold:

(1)for every , there exists such that

(a) for every ;(b) for every and every maximal element ;(c) if and only if

(2)if are such that, for every , there exists such that , then ,(3)for , one has

(4)the attractor is such that .

Proof. By Proposition 14 and Lemma 15, the map defined in (65), (66), and (67) is a -contraction satisfying condition (ii) of Theorem 9. Also, from the proof of Theorem 16, we know that fixed points of are included in .
Let . We want to show that there exists a fixed point of satisfying the required properties. Fix

For , we choose inductively

Arguing as in the proof of Theorem 16, one deduces that is a -Picard trajectory converging to some a fixed point of . Also, is independent of chosen as in (119).
For , observe that for all , where is defined in (92) and (93). Since and , we deduce that

It follows from this inclusion and Theorem 16(1)(b) that

On the other hand,

Thus, (1)(c) holds.
In the particular case where is maximal, one has

where is defined in (93). Since

one has

((2) and (3)) The proofs are, respectively, analogous to those of (2) and (3) in Theorem 16.
(4) Let . Since is independent of the choice of in (119), we can fix

where and are defined in (24) and (25), respectively. Let be defined as in (120). We know that . On the other hand, since is the unique attractor of this -IFS obtained in Theorem 6, we deduce that .

In the following result, we see that the maximal elements of play a key role.

Corollary 19. Let be an MW-directed graph and an -IFS. Then, for every such that

one has

Proof. Let and let

To conclude, it is sufficient to show that

It follows from Theorem 18(2) that

7. Other Fixed Points of Our -Contraction

In the proofs of Theorems 16 and 18, and were obtained as fixed points of the multivalued -contraction . In fact, much more fixed points of can be obtained in order to get more information on the attractor .

Let . For a vertex , we consider the set of edges from on a path to some vertex in

Similarly, for , we consider

Finally, we consider suitable subsets of edges on paths in reaching , that is, subsets of and ,

Using , we can obtain more information on .

Theorem 20. Let be an MW-directed graph and an -IFS. Then, the following statements hold.

(1)For every and every , there exists such that

(a);(b) if and only if ;(c) for every and every maximal element in .

(2)For every , if are such that , then .(3)Let be such that . If , then .(4)Let be such that, for every , there exists such that . If and are such that , then .(5)For every and every , for every and every such that .

Proof. Let . From Proposition 14 and Lemma 15, the multivalued map defined in (65), (66), and (67) is a -contraction satisfying condition (ii) of Theorem 9. We want to show that there exists a fixed point of satisfying the required properties.
Fix

where is defined in (119) and (120). From the definition of , we can observe that

is such that

Moreover, for every ,

where is defined in (58). Similarly, for every ,

where is defined in (60).
For , we choose inductively

with

where , , and are defined in (59), (63), and (64), respectively.
Arguing as in the proof of Theorem 16, one deduces that is a -Picard trajectory converging to some a fixed point of . So, satisfies (1)(b). Again, it can be shown that is independent of chosen as in (136).
Observe that

where is defined in (120) and . Moreover, for every maximal element in , and

Therefore, satisfies (1)(a),(c).
(2) Let be such that . From (141) and (142), one sees that

Since and , one has that

(3) Let be such that and let . From (141) and (142), one sees that

Since and , one has that

(4) Let be such that, for every , there exists such that . One has

Let and be such that . Fix

to be such that

One has and . For , we define

by

Since , , and using the definitions of and , we deduce that . Also, . Arguing as in the proof of Proposition 14, one has

Repeating this argument, we obtain for every , such that . Therefore,

(5) Let and be such that . Let

We define

by

Clearly, and . It follows from (2), (4), and Theorem 16(4) that

Example 21. Let be the -IFS considered in Example 17. One has , , , and . For let be given by

Figures 6 and 7 present and , respectively. Observe that

where is presented in Figure 4.

Figure 6: The set .

Figure 7: The set .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was partially supported by CRSNG Canada.

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