Iterated function systems consisting of F-contractions
© Secelean; licensee Springer. 2013
Received: 13 June 2013
Accepted: 5 September 2013
Published: 8 November 2013
In this paper we consider a particular case of a contractive self-mapping on a complete metric space, namely the F-contraction introduced by Wardowski (Fixed Point Theory Appl. 87, 2012, doi:10.1186/1687-1812-2012-94), and provide some new properties of it. As an application, we investigate the iterated function systems (IFS) composed of F-contractions extending some fixed point results from the classical Hutchinson-Barnsley theory of IFS consisting of Banach contractions. Some illustrative examples are given.
MSC:28A80, 47H10, 54E50.
One of the basic concepts of fractals theory is undubitably the iterated function system (IFS) introduced in 1981 by Hutchinson  and popularized by Barnsley , IFS being the main generator of fractals. This consists of a finite set of contractions on a complete metric space X into itself. For such an IFS, there is anyway a unique nonempty compact set such that . A is, generally, a fractal set and it is called the attractor of the respective IFS. During the last decades, many authors have been concerned with the extension of this framework to more general spaces, generalized contractions and infinite IFSs or, more generally, to multifunction systems.
Miculescu and Mihail introduced in [3, 4] the generalized iterated function system (GIFS) consisting of a collection of contractions (respectively, contractive functions) on endowed with the maximum metric into X, when X is a complete (respectively compact) metric space and I is finite. Dumitru in  and, respectively, Strobin and Swaczyna in  improved the work of Miculescu and Mihail by considering GIFSs composed of Meir-Keeler type mappings and, respectively, φ-contractions. Secelean  investigated the iterated function systems composed of a countable family of contractive, respectively φ-contractions, Meir-Keeler type maps. Some remarkable results concerning extensions of the metric spaces, on which the IFSs are defined, and generalizations of their contractions can be found, for example, in [8, 9] and others.
there exists a unique such that . Furthermore, for every , the sequence converges to e, where (p time composition of ω), . The map ω which satisfies the previous condition is called r-contraction or, more, Banach contraction. Also, in the above setting, we say that e is the fixed point of ω.
Niemytzki-Edelstein’s theorem (see, e.g., [, Th. 2.2, p.34]) states that each contractive mapping on a compact metric space into itself has a unique fixed point which is successively approximated by the iterates of the respective map at every point of the space. Some other particular cases in which the considered metric space is complete are investigated in fixed point theory. In this regard, starting from a function satisfying some suitable properties, Wardowski provided in  a new type of such a contractive mapping, namely F-contraction, and proved a fixed point theorem concerning F-contractions on a complete metric space. Further, an example of a map which is an F-contraction while it is not a Banach contraction was given.
In this paper, we simplify one of the required conditions for F (Remark 3.1) and describe a large class of such functions F (Proposition 3.1, Remark 3.2 and Remark 3.3). Next, we consider an iterated function system (IFS) composed of F-contractions and prove that it always has a unique attractor (Theorem 4.1). We further provide some conditions in which the limit of a sequence of IFSs is also an IFS and its attractor is the limit of the sequence of the corresponding attractors (Theorem 4.2). A significant example in this respect is given. Notice that all the mappings which compose the IFSs from that example are not Banach contractions.
Throughout the whole paper, by ℝ we understand the set of real numbers and by we denote the set of all positive real numbers . The set of all positive integers will be denoted by ℕ.
2 Preliminary facts: Hausdorff-Pompeiu metric
Let be a metric space and be the class of all nonempty compact subsets of X. The function , , where for all , is a metric called the Hausdorff-Pompeiu metric.
The metric space is complete provided that is complete.
Some simple standard facts, which will be used in the sequel, are described in the following lemmas (for details, see, e.g., [, §1.4.]).
Lemma 2.1 [, Prop. 1.1]
Let . Then
Lemma 2.2 [, Th. 1.13]
Lemma 2.3 [, Th. 1.14]
If for all , and the set is relatively compact, then .
If for every , then ,
the limit being taken in the Hausdorff-Pompeiu metric and the bar means the closure.
Throughout this paper, by we understand the metric space .
We describe here a new type of contractive mappings, namely F-contractions, defined by Wardowski in  and add some results about them.
Throughout this section, denotes a metric space.
Definition 3.1 
F is strictly increasing, i.e., for all , , one has ,
for each sequence of positive numbers, if and only if ,
there exists such that .
We denote by ℱ the family of all which satisfy conditions (F1), (F2) and (F3).
We will show that condition (F2) can be replaced by an equivalent but a more simple one. First, we need to prove the following two elementary lemmas.
Lemma 3.1 If is a bounded sequence of real numbers such that all its convergent subsequences have the same limit l, then is convergent and .
Therefore, by hypothesis, we infer that , that is, . □
if , then ;
if and , then .
Proof (a) First of all, we observe that is bounded. Indeed, if the sequence is unbounded above, one can find a subsequence such that . Then, for every , there is such that for any . So , that is, , which is a contradiction.
Thereby is bounded, hence it has a convergent subsequence. Let be such a subsequence and . Clearly .
Suppose that and choose , . Then there exists such that for all . As F is increasing, we deduce that which contradicts .
Accordingly . Next we apply Lemma 3.1.
(b) Assume that and . Choose . There is such that . Next, there exists such that for all . So, , for . Thus . □
Remark 3.1 According to Lemma 3.2, condition (F2) from Definition 3.1 may be replaced by
or, also, by
(F2″) there is such that .
By means of the previous remark, one can obtain new examples of functions of ℱ. In this respect, for , we can consider , and (to check (F3) we take ).
Some properties of the set ℱ are emphasized in the next proposition.
Proposition 3.1 Let consider , and . We define by , and . Then .
Proof If is such that for some , then and, also, . Next, since for all , we get .
In order to verify (F3), we set , where , . Then, for every , one has hence and for all and . So, and . Finally, as for every n, we deduce that . □
In the following remarks we provide a large subclass of ℱ.
f is strictly increasing, and there are such that for all ;
g is strictly increasing and there exists such that .
Proof Choose . Clearly, F is strictly increasing. If is such that , then and so , this happens since the sequence is bounded from above. Indeed, as , there exists such that , hence , so for each . According to Remark 3.1, it follows that F satisfies (F2).
Thereby (F3) is also verified.
Consequently, . □
f is strictly increasing and ;
g is strictly increasing and there exists such that ;
there exists such that . In particular, this condition holds if f is differentiable and there are such that for every .
Then the function , belongs to ℱ.
Proof Conditions (F1) and (F2) can be checked as in Remark 3.2 taking into account that , where , and (F3) follows from the hypothesis (c).
for every , . Since , we deduce . □
The next theorem states a similar result as the Banach contraction principle for F-contractions.
Theorem 3.1 [, Th. 2.1]
Assume that is a complete metric space, and is an F-contraction. Then ω has a unique fixed point e and for every , the sequence converges to e.
The Banach contractions are particular cases of F-contractions, where . In the following example, we show that there are F-contractions on a complete metric space which are not Banach contractions.
Example 3.1 Let be the sequence of real numbers given by and endowed with the Euclidian metric . Then is a complete metric space. Consider the maps , defined by , if and for , where , , . Then and ω is an F-contraction which is not a Banach contraction, being its fixed point.
Proof We first observe that , hence for all .
Choose such that and suppose, for instance, that . Three cases can occur.
for every .
for every .
for every .
In conclusion, in any case, inequality (1) is fulfilled for every , . More simple, since , we can take .
the sequence converges pointwise to a map ,
, where is a constant associated with from (1).
Then ω is an F-contraction. If we further have
is convergent and ,
where at the last inequality we used (4).
so ω is an F-contraction.
A property concerning the compositions of F-contractions which will be used in the next section is given in the following.
Proposition 3.2 If is a collection of F-contractions on X to itself, then the map is a F-contraction.
for all , . □
4 Application: iterated function systems
We assume that is a complete metric space.
The classical iterated function system (IFS) introduced by Hutchinson  consists of a finite family of Banach contractions on X to itself. There is a unique nonempty compact subset of X invariant with respect to these contractions. In what follows, we extend this IFS by considering a family of F-contractions.
Definition 4.1 For each , let and be an -contraction. The family is called an iterated function system, abbreviated IFS. The set function defined by is called the associated Hutchinson operator. A set is said to be an attractor of the IFS whenever .
We will prove that in a certain condition relating to the mappings , such an IFS has a unique attractor. We first need the following lemma.
Lemma 4.1 Let be an F-contraction, where . Then the mapping is an F-contraction too from into itself.
By hypothesis, there is such that for every , .
Consequently, , as required. □
Theorem 4.1 We consider , and define . Assume that the map is nondecreasing for all . For each , let be an -contraction. Then is an F-contraction and the IFS has a unique attractor which is successively approximated in the Hausdorff-Pompeiu metric by for every .
Proof First of all, notice that according to Proposition 3.1.
which assures that is an F-contraction.
The rest of assertions from the statement now follow by applying Theorem 3.1, the metric space being complete. □
Remark 4.1 If there exists a map such that is an F-contraction for all , then the IFS has a unique attractor which is successively approximated by for every .
Lemma 4.2 Let be a sequence of contractive self-mappings on X pointwise convergent to . Then for every , the converging process being taken with respect to the Hausdorff-Pompeiu metric.
Proof Choose . We have to show that . For this purpose, we suppose by contradiction that there is such that for infinitely many integers . Two cases can occur.
which is a contradiction.
Case II: The case when there is such that for every , can be treated analogously.
Accordingly, . □
for each , is continuous and the map , , where , is nondecreasing;
for each and , is an -contraction and , where is a constant associated with from (1);
the sequence converges pointwise to a map for every ;
if, for every , is the attractor of IFS , then the sequence converges in the space to a set A.
Then is an IFS which can be regarded as the limit of the sequence of IFSs . Moreover, A is the attractor of the respective IFS.
Proof Using (C2) and (C3), we deduce from Theorem 3.2 that is an IFS. According to Theorem 4.1, the Hutchinson operator associated with IFS is an F-contraction in the complete metric space , so it has a unique set-‘fixed point’ for all .
Next, condition (a2) follows from (C2), via the proof of Theorem 4.1, which assures that one of the constants associated with from (1) is . Condition (C4) implies (a3).
The conclusion of Theorem 3.2 says that , i.e., A is the attractor of the IFS . □
Example 4.1 We are in the settings from Example 3.1. Let , . Consider and, for every , , and , let define , , where , if and for . Then the hypotheses of Theorem 4.2 are fulfilled.
Proof From Example 3.1, it follows that and is an -contraction for each and . Hence is an IFS and, according to Theorem 4.1, the function being nondecreasing, we deduce that it has a unique attractor for every . Notice that all the mappings (, ) are F-contractions.
Clearly, , , are continuous hence (C1) is satisfied.
So, for every and , one can find such that . Therefore (C2) holds.
The pointwise convergence of the sequence to , for every , is obvious, hence (C3) is verified.
Hence so, applying again Lemma 2.1, .
Thus condition (P4) is verified. □
Remark 4.2 In the previous example, we can also take:
1 o ,
2 o , , if for every , . In this event, the sequence of attractors is descending and, according to Lemma 2.3, .
The author thanks the referees for their valuable comments and useful suggestions in improving the paper.
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