Abstract
In this paper we consider a particular case of a contractive selfmapping on a complete metric space, namely the Fcontraction introduced by Wardowski (Fixed Point Theory Appl. 87, 2012, doi:10.1186/16871812201294), and provide some new properties of it. As an application, we investigate the iterated function systems (IFS) composed of Fcontractions extending some fixed point results from the classical HutchinsonBarnsley theory of IFS consisting of Banach contractions. Some illustrative examples are given.
MSC:28A80, 47H10, 54E50.
Similar content being viewed by others
1 Introduction
One of the basic concepts of fractals theory is undubitably the iterated function system (IFS) introduced in 1981 by Hutchinson [1] and popularized by Barnsley [2], IFS being the main generator of fractals. This consists of a finite set of contractions {({\omega}_{k})}_{k=1}^{K} on a complete metric space X into itself. For such an IFS, there is anyway a unique nonempty compact set A\subset X such that A={\bigcup}_{k=1}^{K}{\omega}_{k}(A). 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 {X}^{I}=\{{({x}_{i})}_{i\in I};{x}_{i}\in X\} endowed with the maximum metric into X, when X is a complete (respectively compact) metric space and I is finite. Dumitru in [5] and, respectively, Strobin and Swaczyna in [6] improved the work of Miculescu and Mihail by considering GIFSs composed of MeirKeeler type mappings and, respectively, φcontractions. Secelean [7] investigated the iterated function systems composed of a countable family of contractive, respectively φcontractions, MeirKeeler 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.
The famous BanachPicardCaccioppoli theorem (also called the contraction principle) states that, given a complete metric space (X,\mathrm{d}) and a function \omega :X\to X for which there is r\in (0,1) such that
there exists a unique e\in X such that \omega (e)=e. Furthermore, for every x\in X, the sequence {({\omega}^{p}(x))}_{p} converges to e, where {\omega}^{p}=\omega \circ {\omega}^{p1} (p time composition of ω), p\ge 1. The map ω which satisfies the previous condition is called rcontraction or, more, Banach contraction. Also, in the above setting, we say that e is the fixed point of ω.
Many authors have provided several extensions of this result. In this respect, they considered the contractive mappings, namely those functions \omega :X\to X that satisfy the inequality
NiemytzkiEdelstein’s theorem (see, e.g., [[10], 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 F:(0,\mathrm{\infty})\to \mathbb{R} satisfying some suitable properties, Wardowski provided in [11] a new type of such a contractive mapping, namely Fcontraction, and proved a fixed point theorem concerning Fcontractions on a complete metric space. Further, an example of a map which is an Fcontraction 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 Fcontractions 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 {\mathbb{R}}_{+} we denote the set of all positive real numbers (0,\mathrm{\infty}). The set of all positive integers will be denoted by ℕ.
2 Preliminary facts: HausdorffPompeiu metric
Let (X,\mathrm{d}) be a metric space and \mathcal{K}(X) be the class of all nonempty compact subsets of X. The function h:\mathcal{K}(X)\times \mathcal{K}(X)\to [0,\mathrm{\infty}), h(A,B)=max\{\mathrm{D}(A,B),\mathrm{D}(B,A)\}, where \mathrm{D}(A,B)={sup}_{x\in A}{inf}_{y\in B}\mathrm{d}(x,y) for all A,B\in \mathcal{K}(X), is a metric called the HausdorffPompeiu metric.
The metric space (\mathcal{K}(X),h) is complete provided that (X,\mathrm{d}) is complete.
Some simple standard facts, which will be used in the sequel, are described in the following lemmas (for details, see, e.g., [[12], §1.4.]).
Lemma 2.1 [[12], Prop. 1.1]
Let A,B,C\in \mathcal{K}(X). Then

(α) A\subset B\iff \mathrm{D}(A,B)=0,

(β) \mathrm{D}(A,C)\le \mathrm{D}(A,B)+\mathrm{D}(B,C).
Lemma 2.2 [[12], Th. 1.13]
If {({E}_{i})}_{i\in \mathrm{\Im}}, {({F}_{i})}_{i\in \mathrm{\Im}} are two finite collections of sets in (\mathcal{K}(X),h), then
Lemma 2.3 [[12], Th. 1.14]
Let {({A}_{n})}_{n} be a sequence of nonempty compact subsets of X.

(a)
If {A}_{n}\subset {A}_{n+1} for all n\ge 1, and the set A:={\bigcup}_{n\ge 1}{A}_{n} is relatively compact, then {lim}_{n}{A}_{n}=\overline{A}.

(b)
If {A}_{n+1}\subset {A}_{n} for every n\ge 1, then {lim}_{n}{A}_{n}={\bigcap}_{n\ge 1}{A}_{n},
the limit being taken in the HausdorffPompeiu metric and the bar means the closure.
Throughout this paper, by \mathcal{K}(X) we understand the metric space (\mathcal{K}(X),h).
3 Fcontractions
We describe here a new type of contractive mappings, namely Fcontractions, defined by Wardowski in [11] and add some results about them.
Throughout this section, (X,\mathrm{d}) denotes a metric space.
Definition 3.1 [11]
Let F:{\mathbb{R}}_{+}\to \mathbb{R} be a mapping satisfying:

(F1)
F is strictly increasing, i.e., for all t,s\in {\mathbb{R}}_{+}, t<s, one has F(t)<F(s),

(F2)
for each sequence {({t}_{k})}_{k} of positive numbers, {lim}_{k}{t}_{k}=0 if and only if {lim}_{k}F({t}_{k})=\mathrm{\infty},

(F3)
there exists \lambda \in (0,1) such that {lim}_{t\searrow 0}{t}^{\lambda}F(t)=0.
A mapping \omega :X\to X is said to be an Fcontraction if there is \tau >0 such that
We denote by ℱ the family of all F:{\mathbb{R}}_{+}\to \mathbb{R} 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 {({t}_{k})}_{k} is a bounded sequence of real numbers such that all its convergent subsequences have the same limit l, then {({t}_{k})}_{k} is convergent and {lim}_{k}{t}_{k}=l.
Proof There are two subsequences {({t}_{{k}_{n}})}_{n}, {({t}_{{k}_{p}})}_{p} of {({t}_{k})}_{k} such that
Therefore, by hypothesis, we infer that lim\hspace{0.17em}inf{t}_{k}=lim\hspace{0.17em}sup{t}_{k}=l, that is, {lim}_{k}{t}_{k}=l. □
Lemma 3.2 Let F:{\mathbb{R}}_{+}\to \mathbb{R} be an increasing map and {({t}_{k})}_{k} be a sequence of positive real numbers. Then the following assertions hold:

(a)
if F({t}_{k})\underset{k}{\u27f6}\mathrm{\infty}, then {t}_{k}\underset{k}{\u27f6}0;

(b)
if infF=\mathrm{\infty} and {t}_{k}\underset{k}{\u27f6}0, then F({t}_{k})\underset{k}{\u27f6}\mathrm{\infty}.
Proof (a) First of all, we observe that {({t}_{k})}_{k} is bounded. Indeed, if the sequence is unbounded above, one can find a subsequence {({t}_{{k}_{p}})}_{p} such that {t}_{{k}_{p}}\underset{p}{\u27f6}\mathrm{\infty}. Then, for every \epsilon >0, there is {p}_{\epsilon}\in \mathbb{N} such that {t}_{{k}_{p}}\ge \epsilon for any p\ge {p}_{\epsilon}. So F(\epsilon )\le F({t}_{{k}_{p}}), that is, F(\epsilon )\le {lim}_{p}F({t}_{{k}_{p}})=\mathrm{\infty}, which is a contradiction.
Thereby {({t}_{k})}_{k} is bounded, hence it has a convergent subsequence. Let {({t}_{{k}_{n}})}_{n} be such a subsequence and \alpha ={lim}_{n}{t}_{{k}_{n}}. Clearly \alpha \ge 0.
Suppose that \alpha >0 and choose \epsilon >0, \epsilon <\alpha. Then there exists {n}_{\epsilon}\in \mathbb{N} such that {t}_{{k}_{n}}\in (\alpha \epsilon ,\alpha +\epsilon ) for all n\ge {n}_{\epsilon}. As F is increasing, we deduce that F(\alpha \epsilon )\le {lim}_{n}F({t}_{{k}_{n}})=\mathrm{\infty} which contradicts F(\alpha \epsilon )\in \mathbb{R}.
Accordingly \alpha =0. Next we apply Lemma 3.1.
(b) Assume that infF=\mathrm{\infty} and {t}_{k}\underset{k}{\u27f6}0. Choose \epsilon >0. There is \alpha >0 such that F(\alpha )<\epsilon. Next, there exists {k}_{\alpha}\in \mathbb{N} such that {t}_{k}<\alpha for all k\ge {k}_{\alpha}. So, F({t}_{k})\le F(\alpha )<\epsilon, for k\ge {k}_{\alpha}. Thus F({t}_{k})\underset{k}{\u27f6}\mathrm{\infty}. □
Remark 3.1 According to Lemma 3.2, condition (F2) from Definition 3.1 may be replaced by
(F2′) infF=\mathrm{\infty}
or, also, by
(F2″) there is {({t}_{k})}_{k}\subset {\mathbb{R}}_{+} such that {lim}_{k}F({t}_{k})=\mathrm{\infty}.
By means of the previous remark, one can obtain new examples of functions of ℱ. In this respect, for \alpha \in (0,1), we can consider {F}_{1}(t)={t}^{\alpha}, {F}_{2}(t)=\frac{{t}^{\alpha}}{1{e}^{t}} and {F}_{3}(t)=\frac{1}{arctan{t}^{\alpha}} (to check (F3) we take \lambda \in (\alpha ,1)).
Some properties of the set ℱ are emphasized in the next proposition.
Proposition 3.1 Let consider N\in \mathbb{N}, {F}_{1},\dots ,{F}_{N}\in \mathcal{F} and {\alpha}_{1},\dots ,{\alpha}_{N}\in {\mathbb{R}}_{+}. We define F,G,H:{\mathbb{R}}_{+}\to \mathbb{R} by F:=min\{{F}_{1},\dots ,{F}_{N}\}, G:=max\{{F}_{1},\dots ,{F}_{N}\} and H:={\alpha}_{1}{F}_{1}+\cdots +{\alpha}_{N}{F}_{N}. Then F,G,H\in \mathcal{F}.
Proof If {({t}_{k})}_{k}\subset {\mathbb{R}}_{+} is such that {lim}_{k}{F}_{n}({t}_{k})=\mathrm{\infty} for some n\in \{1,\dots ,N\}, then {lim}_{k}F({t}_{k})=\mathrm{\infty} and, also, {lim}_{k}H({t}_{k})=\mathrm{\infty}. Next, since inf{F}_{n}=\mathrm{\infty} for all 1\le n\le N, we get infG=\mathrm{\infty}.
In order to verify (F3), we set \lambda :=max\{{\lambda}_{1},\dots ,{\lambda}_{N}\}, where {lim}_{t\searrow 0}{t}^{{\lambda}_{n}}{F}_{n}(t)=0, 1\le n\le N. Then, for every t\in (0,1), one has {t}^{\lambda}\le {t}^{{\lambda}_{n}} hence {t}^{\lambda}F(t)\le {t}^{{\lambda}_{n}}{F}_{n}(t) and {t}^{\lambda}H(t)\le {\alpha}_{1}{t}^{\lambda}{F}_{1}(t)+\cdots +{\alpha}_{N}{t}^{\lambda}{F}_{N}(t) for all t\in {\mathbb{R}}_{+} and n\in \{1,\dots ,N\}. So, {lim}_{t\searrow 0}{t}^{\lambda}F(t)=0 and {lim}_{t\searrow 0}{t}^{\lambda}H(t)=0. Finally, as {lim}_{t\searrow 0}{t}^{\lambda}{F}_{n}(t)\le {lim}_{t\searrow 0}{t}^{{\lambda}_{n}}{F}_{n}(t)=0 for every n, we deduce that {lim}_{t\searrow 0}{t}^{\lambda}G(t)=0. □
In the following remarks we provide a large subclass of ℱ.
Remark 3.2 Let {\mathcal{F}}_{1} be the family of mappings F:{\mathbb{R}}_{+}\to \mathbb{R} given by F=G\circ f+g, where G\in \mathcal{F}, and the functions f:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+} and g:{\mathbb{R}}_{+}\to \mathbb{R} satisfy the following conditions:

(a)
f is strictly increasing, inff=0 and there are M,\delta \in {\mathbb{R}}_{+} such that t\le Mf(t) for all t\in (0,\delta );

(b)
g is strictly increasing and there exists \eta \in (0,1) such that {lim}_{t\searrow 0}{t}^{\eta}g(t)=0.
Then {\mathcal{F}}_{1}\subset \mathcal{F}.
Proof Choose F\in {\mathcal{F}}_{1}. Clearly, F is strictly increasing. If {({t}_{k})}_{k}\subset {\mathbb{R}}_{+} is such that {lim}_{k}f({t}_{k})=0, then {lim}_{k}G(f({t}_{k}))=\mathrm{\infty} and so {lim}_{k}F({t}_{k})=\mathrm{\infty}, this happens since the sequence {(g({t}_{k}))}_{k} is bounded from above. Indeed, as {lim}_{k}f({t}_{k})=0, there exists {k}_{0} such that \frac{{t}_{k}}{M}\le f({t}_{k})\le 1, hence {t}_{k}\le M, so g({t}_{k})\le g(M) for each k>{k}_{0}. According to Remark 3.1, it follows that F satisfies (F2).
Next, let {\lambda}_{1}\in (0,1) be such that {lim}_{s\searrow 0}{s}^{{\lambda}_{1}}G(s)=0. Then, for every \lambda \ge max\{\eta ,{\lambda}_{1}\}, \lambda <1, one has
Thereby (F3) is also verified.
Consequently, F\in \mathcal{F}. □
Remark 3.3 Let f:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+} and g:{\mathbb{R}}_{+}\to \mathbb{R} be two maps satisfying the following conditions:

(a)
f is strictly increasing and inff=0;

(b)
g is strictly increasing and there exists \eta \in (0,1) such that {lim}_{t\searrow 0}{t}^{\eta}g(t)=0;

(c)
there exists {\lambda}_{1}\in (0,1) such that {lim}_{t\searrow 0}{t}^{{\lambda}_{1}}lnf(t)=0. In particular, this condition holds if f is differentiable and there are M,\delta \in {\mathbb{R}}_{+} such that t{f}^{\prime}(t)\le Mf(t) for every t\in (0,\delta ).
Then the function F:{\mathbb{R}}_{+}\to \mathbb{R}, F(t):=lnf(t)+g(t) belongs to ℱ.
Proof Conditions (F1) and (F2) can be checked as in Remark 3.2 taking into account that G\in \mathcal{F}, where G(t)=lnt, and (F3) follows from the hypothesis (c).
In the particular case, when f is differentiable, we use L’Hospital’s rule and obtain
for every \lambda \ge max\{\eta ,{\lambda}_{1}\}, \lambda <1. Since {lim}_{t\searrow 0}{t}^{\lambda}g(t)=0, we deduce {lim}_{t\searrow 0}{t}^{\lambda}F(t)=0. □
Some examples of functions f which satisfy the requirement (a) from Remark 3.2 and the requirements (a) and (c) from Remark 3.3 are the following:
for every t\in {\mathbb{R}}_{+}, where a>0, 0<\alpha \le 1, b>1. For Remark 3.3, we can further consider {f}_{5}(t)={a}_{n}{t}^{{\alpha}_{n}}+{a}_{n1}{t}^{{\alpha}_{n1}}+\cdots +{a}_{1}{t}^{{\alpha}_{1}}, where {\alpha}_{1},\dots ,{\alpha}_{n}\in {\mathbb{R}}_{+} and {a}_{1},\dots ,{a}_{n} are nonnegative numbers, {a}_{n}\ne 0, n\in \mathbb{N}. Indeed, as example for {f}_{5}, assuming that {\alpha}_{1}\le {\alpha}_{2}\le \cdots \le {\alpha}_{n}, one has
The next theorem states a similar result as the Banach contraction principle for Fcontractions.
Theorem 3.1 [[11], Th. 2.1]
Assume that (X,\mathrm{d}) is a complete metric space, F\in \mathcal{F} and \omega :X\to X is an Fcontraction. Then ω has a unique fixed point e and for every x\in X, the sequence {({\omega}^{p}(x))}_{p} converges to e.
The Banach contractions are particular cases of Fcontractions, where F(t)=lnt. In the following example, we show that there are Fcontractions on a complete metric space which are not Banach contractions.
Example 3.1 Let {({\alpha}_{m})}_{m\ge 1} be the sequence of real numbers given by {\alpha}_{m}={m}^{2}lnm and X=[0,1]\cup \{{\alpha}_{m};m=1,2,\dots \} endowed with the Euclidian metric \mathrm{d}(x,y)=xy. Then (X,\mathrm{d}) is a complete metric space. Consider the maps F:{\mathbb{R}}_{+}\to \mathbb{R}, \omega :X\to X defined by F(t)=ln{t}^{\delta}+\beta t, \omega (x)=\eta x+\lambda if x\in [0,1] and \omega ({\alpha}_{m})={\alpha}_{m1} for m\ge 2, where \delta ,\beta ,\eta \in {\mathbb{R}}_{+}, \eta <1, \lambda \in [0,1\eta ]. Then F\in \mathcal{F} and ω is an Fcontraction which is not a Banach contraction, e=\frac{\lambda}{1\eta} being its fixed point.
Proof We first observe that {\alpha}_{m+1}{\alpha}_{m}=2m+1ln(1+{m}^{1})>0, hence 1\le {\alpha}_{m}<{\alpha}_{m+1} for all m\ge 1.
According to Remark 3.3, we deduce that F\in \mathcal{F}. We will prove that there is \tau >0 such that ω satisfies condition (1), that is,
Choose x,y\in X such that \omega (x)\ne \omega (y) and suppose, for instance, that x<y. Three cases can occur.
Case I: x,y\in [0,1]. Then
for every \tau \in (0,\delta ln\eta ).
Case II: x={\alpha}_{m}, y={\alpha}_{m+p}, m\ge 2, p\ge 1. Then
for every \tau \in (0,2\beta ].
Case III: x\in [0,1], y={\alpha}_{m}, m\ge 2. Since
we get
for every \tau \in (0,\beta (2ln2+\eta )].
In conclusion, in any case, inequality (1) is fulfilled for every \tau >0, \tau \le min\{\delta ln\eta ,2\beta ,\beta (2ln2+\eta )\}. More simple, since 1ln2>0, we can take \tau \le min\{\delta ln\eta ,\beta (\eta +1)\}.
The mapping ω is not a Banach contraction because
□
Theorem 3.2 Let F\in \mathcal{F} be continuous and suppose that (X,\mathrm{d}) is complete. For every n\ge 1, let {\omega}_{n}:X\to X be an Fcontraction and {e}_{n} be its fixed point. Assume that:

(a1)
the sequence {({\omega}_{n})}_{n} converges pointwise to a map \omega :X\to X,

(a2)
{inf}_{n}{\tau}_{n}>0, where {\tau}_{n} is a constant associated with {\omega}_{n} from (1).
Then ω is an Fcontraction. If we further have

(a3)
{({e}_{n})}_{n} is convergent and e={lim}_{n}{e}_{n},
then \omega (e)=e.
Proof Set \tau ={inf}_{n}{\tau}_{n}. Then
Let x,y\in X be such that \omega (x)\ne \omega (y). Since {lim}_{n}\mathrm{d}({\omega}_{n}(x),{\omega}_{n}(y))=\mathrm{d}(\omega (x),\omega (y))>0 for every \delta >0, \delta <\mathrm{d}(\omega (x),\omega (y)), one can find {N}_{1}\in \mathbb{N} such that
Choose \epsilon >0. By the pointwise convergence of {({\omega}_{n})}_{n}, there exists {N}_{2}\in \mathbb{N} such that
Let N=max\{{N}_{1},{N}_{2}\}. We have
for any n\ge N. Hence, for each n\ge N, we get by (3)
where at the last inequality we used (4).
Letting \epsilon \searrow 0 and \delta \searrow 0 it follows, using the continuity of F,
so ω is an Fcontraction.
The last assertion of statement comes taking into consideration that each {\omega}_{n} is contractive as follows:
□
A property concerning the compositions of Fcontractions which will be used in the next section is given in the following.
Proposition 3.2 If {({\omega}_{k})}_{k=1}^{K} is a collection of Fcontractions on X to itself, then the map \omega :={\omega}_{1}\circ \cdots \circ {\omega}_{K} is a Fcontraction.
Proof Set \tau =min\{{\tau}_{1},\dots ,{\tau}_{K}\}, where {\tau}_{k} is the constant from (1) associated to {\omega}_{k}, k=1,\dots ,K. We consider K=2. For the general case, one can proceed inductively.
for all x,y\in X, {\omega}_{1}({\omega}_{2}(x))\ne {\omega}_{1}({\omega}_{2}(y)). □
4 Application: iterated function systems
We assume that (X,\mathrm{d}) is a complete metric space.
The classical iterated function system (IFS) introduced by Hutchinson [1] 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 Fcontractions.
Definition 4.1 For each k=1,\dots ,K, let {F}_{k}\in \mathcal{F} and {\omega}_{k}:X\to X be an {F}_{k}contraction. The family {({\omega}_{k})}_{k=1}^{K} is called an iterated function system, abbreviated IFS. The set function \mathcal{S}:\mathcal{K}(X)\to \mathcal{K}(X) defined by \mathcal{S}(B)={\bigcup}_{k=1}^{K}{\omega}_{k}(B) is called the associated Hutchinson operator. A set A\in \mathcal{K}(X) is said to be an attractor of the IFS whenever \mathcal{S}(A)=A.
We will prove that in a certain condition relating to the mappings {F}_{1},\dots ,{F}_{K}, such an IFS has a unique attractor. We first need the following lemma.
Lemma 4.1 Let \omega :X\to X be an Fcontraction, where F\in \mathcal{F}. Then the mapping A\mapsto \omega (A) is an Fcontraction too from \mathcal{K}(X) into itself.
Proof Choose A,B\in \mathcal{K}(X) such that h(\omega (A),\omega (B))>0. Assume that
By hypothesis, there is \tau >0 such that \tau +F(\mathrm{d}(\omega (x),\omega (y)))\le F(\mathrm{d}(x,y)) for every x,y\in X, \omega (x)\ne \omega (y).
Using (5), the compactness of A and the continuity of ω, one can find a\in A such that \mathrm{D}(\omega (A),\omega (B))={inf}_{y\in B}\mathrm{d}(\omega (a),\omega (y))>0, so \mathrm{d}(\omega (a),\omega (y))>0 for all y\in B. Therefore
that is,
Let b\in B be such that \mathrm{d}(a,b)={inf}_{y\in B}\mathrm{d}(a,y). Then, by (6), we get
Consequently, \tau +F(h(\omega (A),\omega (B)))\le F(h(A,B)), as required. □
Theorem 4.1 We consider K\in \mathbb{N}, {F}_{1},\dots ,{F}_{K}\in \mathcal{F} and define F={max}_{1\le k\le K}{F}_{k}. Assume that the map {g}_{k}:=F{F}_{k} is nondecreasing for all 1\le k\le K. For each k=1,\dots ,K, let {\omega}_{k}:X\to X be an {F}_{k}contraction. Then \mathcal{S} is an Fcontraction and the IFS {({\omega}_{k})}_{k=1}^{K} has a unique attractor which is successively approximated in the HausdorffPompeiu metric by {({\mathcal{S}}^{p}(B))}_{p} for every B\in \mathcal{K}(X).
Proof First of all, notice that F\in \mathcal{F} according to Proposition 3.1.
By hypothesis, there are {\tau}_{1},\dots ,{\tau}_{K}\in {\mathbb{R}}_{+} such that
Set \tau =min\{{\tau}_{1},\dots ,{\tau}_{K}\}>0.
Let A,B\in \mathcal{K}(X) be such that h(\mathcal{S}(A),\mathcal{S}(B))>0. By Lemma 2.2, we get
for some {k}_{0}\in \{1,\dots ,K\}. Using now Lemma 4.1 and the hypotheses, one obtains
which assures that \mathcal{S} is an Fcontraction.
The rest of assertions from the statement now follow by applying Theorem 3.1, the metric space (\mathcal{K}(X),h) being complete. □
Remark 4.1 If there exists a map F\in \mathcal{F} such that {\omega}_{k} is an Fcontraction for all 1\le k\le K, then the IFS {({\omega}_{k})}_{k=1}^{K} has a unique attractor which is successively approximated by {({\mathcal{S}}^{p}(B))}_{p} for every B\in \mathcal{K}(X).
Lemma 4.2 Let {({\omega}_{n})}_{n} be a sequence of contractive selfmappings on X pointwise convergent to \omega :X\to X. Then {\omega}_{n}(B)\underset{n}{\u27f6}\omega (B) for every B\in \mathcal{K}(X), the converging process being taken with respect to the HausdorffPompeiu metric.
Proof Choose B\in \mathcal{K}(X). We have to show that h({\omega}_{n}(B),\omega (B))\underset{n}{\u27f6}0. For this purpose, we suppose by contradiction that there is {\epsilon}_{0}>0 such that h({\omega}_{n}(B),\omega (B))\ge {\epsilon}_{0} for infinitely many integers n\ge 1. Two cases can occur.
Case I: There exists a sequence of positive integers {({n}_{p})}_{p} such that
So, for every p\in \mathbb{N}, one can find {x}_{p}\in B such that {inf}_{y\in B}\mathrm{d}({\omega}_{{n}_{p}}({x}_{p}),\omega (y))\ge {\epsilon}_{0}, that is,
Since B is compact, the sequence {({x}_{p})}_{p} admits a convergent subsequence which, for simplicity, will be denoted in the same way. Thus {x}_{p}\to y\in B. Now, using the hypotheses and (8), we get
which is a contradiction.
Case II: The case when there is {({n}_{p})}_{p}\subset \mathbb{N} such that \mathrm{D}(\omega (B),{\omega}_{{n}_{p}}(B))\ge {\epsilon}_{0} for every p\ge 1, can be treated analogously.
Accordingly, \omega (B)={lim}_{n}{\omega}_{n}(B). □
Theorem 4.2 Let K\in \mathbb{N} be given. For every n\ge 1 and k\in \{1,\dots ,K\}, we consider the mappings {F}_{k}\in \mathcal{F} and {\omega}_{k}^{n}:X\to X. Assume that the following conditions are fulfilled:

(C1)
for each k=1,\dots ,K, {F}_{k} is continuous and the map {g}_{k}:{\mathbb{R}}_{+}\to [0,\mathrm{\infty}), {g}_{k}=F{F}_{k}, where F=max\{{F}_{1},\dots ,{F}_{K}\}, is nondecreasing;

(C2)
for each n\in \mathbb{N} and 1\le k\le K, {\omega}_{k}^{n} is an {F}_{k}contraction and {inf}_{n}{\tau}_{k}^{n}>0, where {\tau}_{k}^{n} is a constant associated with {\omega}_{k}^{n} from (1);

(C3)
the sequence {({\omega}_{k}^{n})}_{n} converges pointwise to a map {\omega}_{k}:X\to X for every k=1,\dots ,K;

(C4)
if, for every n\ge 1, {A}_{n} is the attractor of IFS {({\omega}_{k}^{n})}_{k=1}^{K}, then the sequence {({A}_{n})}_{n} converges in the space (\mathcal{K}(X),h) to a set A.
Then {({\omega}_{k})}_{k=1}^{K} is an IFS which can be regarded as the limit of the sequence of IFSs {({({\omega}_{k}^{n})}_{k=1}^{K})}_{n}. Moreover, A is the attractor of the respective IFS.
Proof Using (C2) and (C3), we deduce from Theorem 3.2 that {({\omega}_{k})}_{k=1}^{K} is an IFS. According to Theorem 4.1, the Hutchinson operator {\mathcal{S}}_{n} associated with IFS {({\omega}_{k}^{n})}_{k=1}^{K} is an Fcontraction in the complete metric space (\mathcal{K}(X),h), so it has a unique set‘fixed point’ {A}_{n} for all n=1,2,\dots .
In order to apply again Theorem 3.2, we notice that {\mathcal{S}}_{n}(B)\underset{n}{\u27f6}\mathcal{S}(B) with respect to the HausdorffPompeiu metric for every B\in \mathcal{K}(X). Indeed, by means of Lemmas 2.2 and 4.2, one has
Next, condition (a2) follows from (C2), via the proof of Theorem 4.1, which assures that one of the constants associated with {\mathcal{S}}_{n} from (1) is min\{{\tau}_{1}^{n},{\tau}_{2}^{n},\dots ,{\tau}_{K}^{n}\}. Condition (C4) implies (a3).
The conclusion of Theorem 3.2 says that \mathcal{S}(A)=A, i.e., A is the attractor of the IFS {({\omega}_{k})}_{k=1}^{K}. □
Example 4.1 We are in the settings from Example 3.1. Let K\in \mathbb{N}, K\ge 2. Consider \delta ,{\beta}_{1},\dots ,{\beta}_{K}\in {\mathbb{R}}_{+} and, for every k=1,\dots ,K, n\in \mathbb{N}, t\in {\mathbb{R}}_{+} and x\in X, let define {F}_{k}(t)=ln{t}^{\delta}+{\beta}_{k}t, {\omega}_{k}^{n}(x)={\eta}_{n}x+{\lambda}_{k}, where {\eta}_{n}=\frac{n}{(n+1)(K+1)}, {\lambda}_{k}=\frac{K(k1)}{{K}^{2}1} if x\in [0,1] and {\omega}_{k}^{n}({\alpha}_{m})={\alpha}_{m1} for m\ge 2. Then the hypotheses of Theorem 4.2 are fulfilled.
Proof From Example 3.1, it follows that {F}_{k}\in \mathcal{F} and {\omega}_{k}^{n} is an {F}_{k}contraction for each k=1,\dots ,K and n\ge 1. Hence {({\omega}_{k}^{n})}_{k=1}^{K} is an IFS and, according to Theorem 4.1, the function F{F}_{k} being nondecreasing, we deduce that it has a unique attractor {A}_{n} for every n\ge 1. Notice that all the mappings {\omega}_{k}^{n} (n\ge 1, 1\le k\le K) are Fcontractions.
Clearly, {F}_{k}, 1\le k\le K, are continuous hence (C1) is satisfied.
Next, as {\eta}_{n}\in [\frac{1}{2(K+1)},\frac{1}{K+1}), we get
So, for every k=1,\dots ,K and n\ge 1, one can find {\tau}_{k}^{n}\in (0,min\{\delta ln{\eta}_{n},{\beta}_{k}({\eta}_{n}+1)\}] such that {inf}_{n}{\tau}_{k}^{n}>0. Therefore (C2) holds.
The pointwise convergence of the sequence {({\omega}_{k}^{n})}_{n} to {\omega}_{k}:=\frac{1}{K+1}x+{\lambda}_{k}, for every k=1,\dots ,K, is obvious, hence (C3) is verified.
It remains to show that also condition (C4) is satisfied. We claim that, for every n\in \mathbb{N}, {A}_{n}\subset [0,1]. Indeed, using Proposition 3.2, it follows that for each p\ge 1, the map {\omega}_{{i}_{1}{i}_{2}\dots {i}_{p}}^{n} is an Fcontraction, where we denoted {\omega}_{{i}_{1}\dots {i}_{p}}^{n}:={\omega}_{{i}_{1}}^{n}\circ \cdots \circ {\omega}_{{i}_{p}}^{n} for {i}_{1},\dots ,{i}_{p}\in \{1,\dots ,K\}. It is easy to check that {A}_{n} is also an attractor for the IFS {({\omega}_{{i}_{1}\dots {i}_{p}}^{n})}_{{i}_{1},\dots ,{i}_{p}=1}^{K}. Next, since {A}_{n} is compact, there is M\in \mathbb{N} such that {A}_{n}\subset [0,1]\cup \{{\alpha}_{1},\dots ,{\alpha}_{M}\}. Obviously, if p\ge M, then {\omega}_{{i}_{1}\dots {i}_{p}}^{n}({\alpha}_{m})\le 1 for every m=1,\dots ,M, hence
Now, we intend to prove that {A}_{n}\subset {A}_{n+1} for each n\ge 1. Set B=[0,1]\in \mathcal{K}(X) and choose n\in \mathbb{N}. Since {\eta}_{n}<{\eta}_{n+1}, it follows
Then, inductively, we obtain {\omega}_{{i}_{1}\dots {i}_{p}}^{n}(B)\subset {\omega}_{{i}_{1}\dots {i}_{p}}^{n+1}(B) for every p\ge 1, 1\le {i}_{1},\dots ,{i}_{p}\le K. Therefore
By Theorem 4.1, one has {\mathcal{S}}_{n}^{p}(B)\underset{p}{\u27f6}{A}_{n} and {\mathcal{S}}_{n+1}^{p}(B)\underset{p}{\u27f6}{A}_{n+1}. Then, using Lemma 2.1 and (10), we get
Hence \mathrm{D}({A}_{n},{A}_{n+1})=0 so, applying again Lemma 2.1, {A}_{n}\subset {A}_{n+1}.
The convergence of {({A}_{n})}_{n} now follows from Lemma 2.3 taking into consideration that, by (9), we have
Thus condition (P4) is verified. □
Remark 4.2 In the previous example, we can also take:
1^{o} {F}_{k}(t)=ln{t}^{{\delta}_{k}}+\beta t, {\delta}_{1},\dots ,{\delta}_{K},\beta \in {\mathbb{R}}_{+}
and/or
2^{o} {\omega}_{k}^{n}(x)={\eta}_{n}x+{\lambda}_{k}, {\eta}_{n}=\frac{n+1}{2n(K+1)}, {\lambda}_{k}=\frac{K(k1)}{{K}^{2}1} if x\in [0,1] for every n\ge 1, 1\le k\le K. In this event, the sequence of attractors {({A}_{n})}_{n} is descending and, according to Lemma 2.3, {lim}_{n}{A}_{n}={\bigcap}_{n}{A}_{n}.
References
Hutchinson J: Fractals and selfsimilarity. Indiana Univ. Math. J. 1981, 30: 713–747. 10.1512/iumj.1981.30.30055
Barnsley MF: Fractals Everywhere. 2nd edition. Academic Press, Boston; 1993.
Mihail A, Miculescu R: Applications of fixed point theorems in the theory of generalized IFS. Fixed Point Theory Appl. 2008., 2008: Article ID 312876 10.1155/2008/312876
Mihail A, Miculescu R: Generalized IFSs on noncompact spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 584215 10.1155/2010/584215
Dumitru D: Generalized iterated function systems containing MeirKeeler functions. An. Univ. Bucureşti, Math. 2009, LVIII: 3–15.
Strobin F, Swaczyna J: On a certain generalization of the iterated function system. Bull. Aust. Math. Soc. 2013, 87(1):37–54. 10.1017/S0004972712000500
Secelean NA: The existence of the attractor of countable iterated function systems. Mediterr. J. Math. 2012, 9(1):61–79. 10.1007/s000090110116x
LlorensFuster E, Petruşel A, Yao JC: Iterated function systems and wellpossedness. Chaos Solitons Fractals 2009, 41: 1561–1568. 10.1016/j.chaos.2008.06.019
Petruşel A: Dynamical systems, fixed points and fractals. Pure Math. Appl. 2002, 13(1–2):275–281.
Berinde V Lecture Notes in Mathematics. In Iterative Approximation of Fixed Points. Springer, Berlin; 2006.
Wardowski D: Fixed points of a new type of contractive mappings in complete metric spaces. Fixed Point Theory Appl. 2012., 87: Article ID 2012/1/94 10.1186/16871812201294
Secelean NA: Countable Iterated Function Systems. Lambert Academic Publishing, Colne; 2013.
Acknowledgements
The author thanks the referees for their valuable comments and useful suggestions in improving the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The author declares that they have no competing interests.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Secelean, NA. Iterated function systems consisting of Fcontractions. Fixed Point Theory Appl 2013, 277 (2013). https://doi.org/10.1186/168718122013277
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/168718122013277