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Weak θcontractions and some fixed point results with applications to fractal theory
 Mohammad Imdad^{1},
 Waleed M. Alfaqih^{1, 2}Email authorView ORCID ID profile and
 Idrees A. Khan^{3}
https://doi.org/10.1186/s1366201819008
© The Author(s) 2018
 Received: 27 March 2018
 Accepted: 21 November 2018
 Published: 28 November 2018
Abstract
In this paper, we define weak θcontractions on a metric space into itself by extending θcontractions introduced by Jleli and Samet (J. Inequal. Appl. 2014:38, 2014) and utilize the same to prove some fixed point results besides proving some relationtheoretic fixed point results in generalized metric spaces. Moreover, we give some applications to fractal theory improving the classical Hutchinson–Barnsley′s theory of iterated function systems. We also give illustrative examples to exhibit the utility of our results.
Keywords
 Fixed point
 θcontraction
 Weak θcontraction
 Iterated function system
 Countable iterated function system
 Attractor
 Binary relation
1 Introduction
The Banach contraction principle is one of the pivotal results of nonlinear analysis, which asserts that every contraction mapping defined on a complete metric space \((M,d)\) to itself admits a unique fixed point. This principle is a very effective and popular tool for guaranteeing the existence and uniqueness of solutions of certain problems arising within and beyond mathematics. The Banach contraction principle has been extended and generalized in many directions (see [1–6] and references therein). With a similar quest, beginning from a function \(\theta:(0,\infty)\to(1,\infty)\) satisfying suitable properties (see Definition 3.1 to be given later), Jleli and Samet [5] proposed a new type of contractive mappings known as θcontraction (or JScontraction) and proved a fixed point result in generalized metric spaces wherein the authors showed that the Banach contraction principle remains a particular case of θcontraction.
In this paper, we observe that the first condition in Definition 3.1 is unnecessarily stringent; its omission enlarges the class of functions \(\theta:(0,\infty)\to(1,\infty)\). In fact, we consider the families \(\varTheta_{2,3}\) and \(\varTheta_{2,4}\) and utilize the same to define a weak θcontraction (see Definition 3.2 to be introduced shortly) such that every weak θcontraction on a complete metric space is a Picard operator. Also, we provide an example of a weak θcontraction that is not a Banach contraction.
The basic concept of fractal theory is the iterated function system (IFS) introduced by Hutchinson [7] and generalized by Barnsley [8], IFS being the main generator of fractals. This consists of a finite set of contractions \(\{f_{i}\}_{i=1}^{N}\) on a complete metric space \((M,d)\) into itself. For such an IFS, there is always a unique nonempty compact set \(A\subset M\) such that \(A=\bigcup_{i=1}^{N}f_{i}(A)\), wherein generally A is a fractal set called the attractor of the respective IFS.
In Sect. 4, we apply Theorem 3.2 to obtain the existence and uniqueness of the attractor of some iterated function system on a complete metric space and, also, to provide an example to demonstrate our results.
The fixed point theory on metric spaces endowed with a binary relation is a relatively new area initiated by Turinici [9]. This area becomes very active after the appearance of the very interesting results of Ran and Reurings [1] and Nieto and RodriguezLopez [2, 10] with their nice applications. Recently, this branch of fixed point theory has been developed by many researchers. To mention a few, we recall Bhaskar and Lakshmikantham [11], BenElMechaiekh [12], Samet and Turinici [13], Alam and Imdad [14], Imdad et al. [15–17], and several others.
In Sect. 5, we provide some fixed point results on a generalized metric space equipped with a binary relation under weak θcontractions without completeness requirement. Also, we adopt some examples to exhibit the utility of our results.
Finally, in Sect. 6, we apply Theorems 5.2 and 5.3 to obtain the existence and uniqueness of an attractor for a countable iterated function system, which is also composed by contractions on a complete metric space besides furnishing an example to exhibit the validity of our results.
2 Preliminaries
In this section, we recall some notions, notations, and basic results.
Throughout this presentation, \(\mathbb{N}\) is the set of natural numbers, and \(\mathbb{N}_{0}=\mathbb{N}\cup\{0\}\). We write \(\{u_{n}\} \to u\) whenever \(\{u_{n}\}\) converges to u. If M is a nonempty set, \(u\in M\), and \(f:M\to M\), then we write fu instead of \(f(u)\). The sequence \(\{u_{n}\}\) defined by \(u_{n}=f^{n}u_{0}\) is called a Picard sequence based at the point \(u_{0}\in M\).
Definition 2.1
([18])
A selfmapping f on a metric space \((M,d)\) is said to be a Picard operator if it has a unique fixed point \(z\in M\) and \(z=\lim_{n\to\infty}f^{n}u\) for all \(u\in M\).
Lemma 2.1
([19])
Let \(\{u_{n}\}\) be a sequence in a metric space \((M,d)\). If \(\{u_{n}\}\) is not a Cauchy sequence, then there exist \(\epsilon>0\) and two subsequences \(\{u_{n(k)}\}\) and \(\{u_{m(k)}\}\) of \(\{u_{n}\}\) such that \(k\leq m(k)< n(k), d(u_{m(k)},u_{n(k)})\geq\epsilon \) and \(d(u_{m(k)},u_{n(k)1})<\epsilon\ \forall k\in\mathbb {N}\).
Furthermore, \(\lim_{k\to\infty}\,d(u_{m(k)},u_{n(k)})=\epsilon\), provided that \(\lim_{n\to\infty}\,d(u_{n},u_{n+1})=0\).
Definition 2.2
Let \((M,d)\) be a metric space, and let \(\mathcal{K}(M)\) the class of all nonempty compact subsets of M. The function \(\eta:\mathcal {K}(M)\times\mathcal{K}(M)\to[0,\infty)\) defined by \(\eta (A,B)=\max\{D(A,B),D(B,A)\}\), where \(D(A,B)=\sup_{u\in A}\inf_{v\in B}\,d(u,v)\), for all \(A,B\in\mathcal{K}(M)\), is a metric known as Hausdorff–Pompeiu metric. It is well known that if \((M,d)\) is complete, then \((\mathcal{K}(M),\eta)\) is also complete.
Lemma 2.2
([20])
 (i)
\(A\subset B\) if and only if \(D(A,B)=0\);
 (ii)
\(D(A,B)\leq D(A,C)+D(C,B)\).
Lemma 2.3
([21])
Definition 2.3
([22])
 (G1)
\(d(u,v)=0\) if and only if \(u=v\);
 (G2)
\(d(u,v)=d(v,u)\);
 (G3)
\(d(u,v)\leq d(u,z)+d(z,v)\).
Wardowski [3] introduced a new class of auxiliary functions and utilized the same to define Fcontractions as follows:
Definition 2.4
([3])
 F1::

F is strictly increasing;
 F2::

for every sequence \(\{\alpha_{n}\}\subset(0,\infty)\),$$\lim_{n\to\infty}F(\alpha_{n})=\infty\quad\Leftrightarrow\quad\lim _{n\to \infty}\beta_{n}=0; $$
 F3::

there exists \(k \in(0,1)\) such that \(\lim_{\alpha\to 0^{+}}\alpha^{k}F(\alpha)=0\).
Wardowski [3] proved that every Fcontraction mapping on a complete metric space is a Picard operator. Thereafter, Piri and Kumam [23] replaced condition F3 by
F4: F is a continuous mapping.
3 Weak θcontractions
Definition 3.1
 Θ1::

θ is nondecreasing;
 Θ2::

for each sequence \(\{\alpha_{n}\}\) in \((0,\infty)\),$$\lim_{n\to\infty}\theta(\alpha_{n})=1\quad \Leftrightarrow\quad \lim _{n\to \infty}\alpha_{n}=0^{+}; $$
 Θ3::

there exist \(r\in(0,1)\) and \(l\in(0,\infty]\) such that \(\lim_{\alpha\to0^{+}}\frac{\theta(\alpha)1}{\alpha^{r}}=l\);
 Θ4::

θ is continuous.

\(\varTheta_{1,2,3}\), the family of all functions θ that satisfy Θ1–Θ3;

\(\varTheta_{1,2,4}\), the family of all functions θ that satisfy \(\varTheta1,\varTheta2\), and Θ4;

\(\varTheta_{2,3}\), the family of all functions θ that satisfy Θ2 and Θ3;

\(\varTheta_{2,4}\), the family of all functions θ that satisfy Θ2 and Θ4;

\(\varTheta_{2}\), the family of all functions θ that satisfy Θ2.
Remark 3.1
 (i)
\(\theta:(0,\infty)\to(1,\infty)\) satisfies, respectively, Θ1, Θ2, or Θ4 if and only if \(\ln \ln\theta:(0,\infty)\to(\infty,\infty)\) satisfies F1, F2, or F4;
 (ii)
\(F:(0,\infty)\to(\infty,\infty)\) satisfies, respectively, F1, F2, or F4 if and only if \(e^{e^{F}}:(0,\infty )\to(1,\infty)\) satisfies Θ1, Θ2, or Θ4.
Example 3.1
([5])
Define \(\theta:(0,\infty)\to(1,\infty)\) by \(\theta(\alpha )=e^{\sqrt{\alpha}}\). Then \(\theta\in\varTheta_{1,2,3,4}\).
Example 3.2
([5])
Define \(\theta:(0,\infty)\to(1,\infty)\) by \(\theta(\alpha )=2\frac{2}{\pi}\arctan (\frac{1}{\alpha^{r}} )\), \(0< r<1\). Then \(\theta\in\varTheta_{1,2,3,4}\).
Example 3.3
([24])
Define \(\theta:(0,\infty)\to(1,\infty)\) by \(\theta(\alpha )=e^{\alpha}\). Then \(\theta\in\varTheta_{1,2,4}\).
Now, we add some more examples to this effect.
Example 3.4
Define \(\theta:(0,\infty)\to(1,\infty)\) by \(\theta(\alpha )=e^{\alpha e^{\frac{1}{\alpha}}}\). Then \(\theta\in\varTheta_{1,2,4}\).
Example 3.5
Define \(\theta:(0,\infty)\to(1,\infty)\) by \(\theta(\alpha )=e^{\sqrt{\frac{\alpha}{2}+\sin\alpha}}\). Then \(\theta\in \varTheta_{2,3}\).
Example 3.6
 1
\(\theta(\alpha)=e^{\frac{\alpha}{2}+\sin\alpha}\);
 2
\(\theta(\alpha)=\alpha^{r}+1\), \(r\in(0,\infty)\).
For more examples, see [5, 24].
Jleli and Samet [5] proved the following theorem.
Theorem 3.1
(see [5])
The following proposition shows that f in Theorem 3.1 is continuous due to Θ2.
Proposition 3.1
Let \((M,d)\) be a metric space, and let \(f:M\to M\). If f satisfies (3.1) for some \(\theta\in\varTheta_{2}\) and \(h\in (0,1)\), then f is continuous.
Proof
Let \(u,v\in M\) be such that \(d(u,v)\to0\). Then we must have \(\theta (d(u,v))\to1\) (due to Θ2). This, together with (3.1), implies that \(\theta(d(fu,fv))\to1\). Therefore \(d(fu,fv)\to0\) (due to Θ2). Hence f is continuous. □
Remark 3.2
Observe that condition Θ1 can be withdrawn, and still Theorem 3.1 (also, most of the existence results in the literature (e.g., results of [25–29])) survives (in view of Proposition 3.1).
Now, Proposition 3.1 and Remark 3.2 led us to define a weaker contraction under the name of weak θcontraction as follows.
Definition 3.2
Let \((M,d)\) be a metric space, and let \(f:M\to M\). We say that f is a weak θcontraction if there exist \(\theta\in \varTheta_{2,3}\) (or \(\theta\in\varTheta_{2,4}\)) and \(h\in(0,1)\) such that (3.1) holds for all \(u,v\in M\).
Remark 3.3
It is easy to verify that every Banach contraction is a weak θcontraction w.r.t. \(\theta(\alpha)=e^{\alpha}\) (or \(\theta (\alpha)=e^{\sqrt{\alpha}}\)) for all \(\alpha>0\).
The following result shows that the completeness assumption of a metric space is a sufficient condition to show that a weak θcontraction is a Picard operator.
Theorem 3.2
Every weak θcontraction on a complete metric space is a Picard operator.
Proof
Remark 3.4
In view of Remark 3.1, Theorem 3.2 with \(\theta\in\varTheta_{2,4}\) remains a weaker version of the main result of Piri and Kumam [23].
In the following example (inspired by [30]), we show that weak θcontractions are a proper generalization of Banach contractions.
Example 3.7
 (a)
f is not a Banach contraction;
 (b)
f is a weak θcontraction;
 (c)
f is a Picard operator.
Proof
(c) Follows immediately from Theorem 3.2 as \((M,d)\) is a complete metric space and in view of (b). The fixed point of f is \(z=\frac{\beta i+\sqrt{\beta^{2}i^{2}+4\beta i}}{2i}\). □
Theorem 3.2 can be improved as follows.
Theorem 3.3
Let \((M,d)\) be a complete metric space, and let \(f:M\to M\). If there exists \(n\in\mathbb{N}\) such that \(f^{n}\) is a weak θcontraction, then f is a Picard operator.
Proof
Theorem 3.2 ensures that \(f^{n}\) is a Picard operator, so that there exists a unique \(z\in M\) such that \(f^{n}z=z\) and \(\lim_{m\to\infty}(f^{n}u)^{m}=z\) for all \(u\in M\). Observe that \(f^{n+1}z=fz\). Thus fz is also a fixed point of \(f^{n}\). Therefore \(fz=z\). Moreover, if \(z^{*}\) is another fixed point of f, then it is also a fixed point of \(f^{n}\). Hence \(z=z^{*}\). Thus f has a unique fixed point.
4 Application: weak θiterated function systems
In this section, we apply our results to obtain the existence and uniqueness of the attractors of some iterated function systems composed by weak θcontractions on a complete metric space. In the following, \((M,d)\) is a complete metric space, \(N\in\mathbb{N}\), and \(\theta\in\varTheta_{1,2,4}\).
Definition 4.1
Let \(\{f_{i}\}_{i=1}^{N}\) be a finite family of selfmappings on M. If (for each i) \(f_{i}:M\to M\) is a weak θcontraction, then the family \(\{f_{i}\}_{i=1}^{N}\) is called a weak θiterated function system (weak θIFS).
The set function \(\mathcal{G}:\mathcal{K}(M)\to\mathcal{K}(M)\) defined by \(\mathcal{G}(B)=\bigcup_{i=1}^{N}f_{i}(B)\) (for all \(B\in \mathcal{K}(M)\)) is called the associated Hutchinson operator. A set \(A\in\mathcal{K}(M)\) is called an attractor of the weak θIFS if \(\mathcal{G}(A)=A\).
Now, we prove that the weak θIFS has a unique attractor. To do so, we begin with the following:
Lemma 4.1
Let \(f:M\to M\) be a weak θcontraction. Then the mapping \(A\longmapsto f(A)\) is also a weak θcontraction from \(\mathcal {K}(M)\) into itself.
Proof
Now, we can state and prove our main result in this section.
Theorem 4.1
If \(\{f_{i}\}_{i=1}^{N}\) is a weak θIFS, then it has a unique attractor A. Moreover, \(A=\lim_{n\to\infty}\mathcal{G}^{n}(B)\) for all \(B\in\mathcal{K}(M)\), the limit being taken w.r.t. the Hausdorff–Pompeiu metric.
Proof
In support of Theorem 4.1, we provide the following:
Example 4.1
Let \(M=[0,\infty)\) be endowed with the usual metric, \(i\in\{ 1,2,\ldots,N\}\), and \(\beta_{i}\in M\) (for all i). Define \(f_{i}:M\to M\) by: \(f_{i}(u)=\frac{u}{iu+1}+\beta_{i}\). Consider the function \(\theta :(0,\infty)\to(1,\infty)\) given in Example 3.4. Then \(\{f_{i}\}_{i=1}^{N}\) is a weak θIFS and has a unique attractor, which is approximated (w.r.t. the Hausdorff–Pompeiu metric) by the sequence \(\{\mathcal{G}^{n}(B)\}\) for all \(B\in\mathcal{K}(M)\). Furthermore \(\{f_{i}\}_{i=1}^{N}\) is not a classical Hutchinson IFS.
5 Relationtheoretic fixed point results
Let M be a nonempty set, and let \(u,v\in M\). A subset \(\mathcal{R}\) of \(M\times M\) is called a binary relation on M. If \((u,v)\in \mathcal{R}\), then we write \(u\mathcal{R}v\). Two elements \(u,v\in M\) are said to be comparable under \(\mathcal{R}\) if either \(u\mathcal{R}v\) or \(v\mathcal{R}u\), which is often denoted by \([u,v]\in\mathcal{R}\). A binary relation \(\mathcal{R}\) is said to be: reflexive if \(u\mathcal{R}u\) for any \(u\in M\); transitive if for any \(u,v,z\in M\), \(u\mathcal{R}v\) and \(v\mathcal {R}z\) imply \(u\mathcal{R}z\); antisymmetric if for any \(u,v\in M\), \(u\mathcal{R}v\) and \(v\mathcal{R}u\) imply \(u=v\); partial order if it is reflexive, transitive, and antisymmetric. If \(f:M\to M\), then \(\mathcal{R}\) is said to be ftransitive if it is transitive on \(f(M)\).
In the following results, \(\theta:(0,\infty]\to(1,\infty]\) is such that \(\theta(\alpha)=\infty\) if and only if \(\alpha=\infty\). Now, we can state and prove our results in this section.
Theorem 5.1
 (a)
\(u_{0}\mathcal{R}fu_{0}\), and \(\mathcal{R}\) is ftransitive;
 (b)
for any \(u,v\in M\), \(u\mathcal{R}v\) implies \(fu\mathcal{R}fv\);
 (c)
every Cauchy sequence \(\{u_{n}\}\subseteq M\) with \(u_{n}\mathcal{R}u_{n+1}\) converges to some \(u\in M\);
 (d)
there exist \(\theta\in\varTheta_{2,3}\) and \(h\in (0,1)\) such that (3.1) holds for all \(u,v\in M\) with \(u\mathcal{R}v\) and \(d(u,v)<\infty\);
 (e)
f is continuous.
 (I)
\(d(f^{n}u_{0},f^{n+1}u_{0})=\infty\) for all \(n\in\mathbb {N}\); or
 (II)
f has a fixed point that is approximated by \(\{f^{n}u_{0}\}\).
Proof
 (i)
for every \(n\in\mathbb{N}\), \(d(u_{n},u_{n+1})=\infty\), which is precisely the alternative (I) of the conclusion of the theorem; or
 (ii)
there exists \(n\in\mathbb{N}\) such that \(d(u_{n},u_{n+1})<\infty\); in such a case, we will show that conclusion (II) of the theorem is fulfilled.
Now, we give the following example, which exhibits the utility of Theorem 5.1.
Example 5.1
Remark 5.1
 1
Theorems 3.1 and 3.2 are not applicable in the context of Example 5.1 as condition (3.1) does not hold on \((0,3]\) and also \((M,d)\) is not a complete space.
 2
Theorems of [14, 22, 31] do not work in the context of Example 5.1 as \(\lim_{n\to\infty}\frac{d(f\pi_{n},f1)}{d(\pi_{n},1)}=1\), so that their contraction conditions do not hold.
Next, we present an analogue of Theorem 5.1 avoiding the continuity assumption of f.
Theorem 5.2
 (e′):

for every convergent sequence \(\{u_{n}\}\subseteq M\) and all \(u\in M\),$$\bigl[u_{n}\mathcal{R}u_{n+1}\textit{ and }\{u_{n}\} \to u \bigr]\quad\Rightarrow\quad[ u_{n}\mathcal{R}u \textit{ for all } n\in \mathbb{N}]. $$
Proof
The following example exhibits the utility of Theorem 5.2.
Example 5.2

\(d(u,v)<\infty\) for all \(u,v\in M\), so that alternative (I) of Theorem 5.2 is excluded;

\(0\in M\) and \(0\mathcal{R}f0\) (as \((0,0)\in\mathcal{R}\));

\(\mathcal{R}\) is ftransitive (as \(\mathcal{R}\) is transitive on \(\{0,1\}\));

for any \(u,v\in M\), \(u\mathcal{R}v\) implies \(fu\mathcal{R}fv\);

if \(\{u_{n}\}\) is a Cauchy sequence in M with \(u_{n}\mathcal {R}u_{n+1}\), then there exists \(N\in\mathbb{N}\) such that either \(u_{n}=0\) for all \(n\geq N\) or \(u_{n}=1\) for all \(n\geq N\) so that \(\{u_{n}\} \) converges to either 0 or 1, which are in M;

f satisfies (3.1) for all \(u,v\in M\) with \(u\mathcal{R}v\) and \(fu\neq fv\) (namely, for \(u,v\in\{0,3\}\)) with θ given in Example 3.1 and any \(h\in[\frac{1}{2},1)\);

if \(\{u_{n}\}\) is a sequence in M such that \(u_{n}\mathcal {R}u_{n+1}\), then we may observe that \((u_{n},u_{n+1})\notin\{(0,3)\}\), so that \((u_{n},u_{n+1})\in\{(0,0),(0,1),(1,0),(1,1)\}\), and hence \(\{ u_{n}\}\subset\{0,1\}\), which is closed, so that \(u_{n}\mathcal{R}u\) for all n.
Now, we present a corresponding uniqueness result as follows.
Theorem 5.3
If in addition to the hypotheses of Theorem 5.1 (or Theorem 5.2), we assume that, for each \(u,v\in \operatorname{Fix}(f)\), there exists \(z_{0}\in M\) comparable to both u and v, \(d(u,z_{0})<\infty\), and \(d(v,z_{0})<\infty\), then f is a Picard operator.
Proof
In view of Theorem 5.1 (or Theorem 5.2), the set \(\operatorname{Fix}(f)\) is nonempty. Let \(u,v \in \operatorname{Fix}(f)\). By our assumption there exists \(z_{0} \in M\) such that \([u,z_{0}]\in\mathcal{R}\) and \([v,z_{0}]\in\mathcal{R}\). Let \(\{z_{n}\}\) be the Picard sequence under f based on \(z_{0}\), that is, \(z_{n}=f^{n}z_{0}\) for all \(n\geq0\). Now, we show that \(u=v\) by proving \(\{ z_{n}\}\to u\) and \(\{z_{n}\}\to v\).
The following example exhibits the utility of Theorem 5.3.
Example 5.3
Setting \(\mathcal{R}=M\times M\) in Theorem 5.3, we deduce the following corollary.
Corollary 5.1
Let \((M,d)\) be a generalized complete metric space, and let \(f:M\to M\). If there exist \(\theta\in\varTheta_{2,3}\) and \(h\in(0,1)\) such that (3.1) holds for all \(u,v\in M\) with \(d(u,v)<\infty\), then f is a Picard operator.
6 Application: weak θcountable iterated function systems
In this section, inspired by [30], we apply our results to obtain the existence and uniqueness of the attractors of some countable iterated function systems composed by weak θcontractions on a generalized complete metric space. In the following, \((M,d)\) is a complete metric space, \(\mathit{CL}(M)\) is the class of all nonempty closed subsets of M, and \(\theta\in\varTheta_{1,2,3}\).
Definition 6.1
Let \(\{f_{i}\}_{i\geq1}\) be a countable family of selfmappings on M. If (for each i) \(f_{i}:M\to M\) is a weak θcontraction, then the countable family \(\{f_{i}\}_{i\geq1}\) is called a weak θcountable iterated function system (weak θCIFS).
The set function \(\mathcal{G}:\mathit{CL}(M)\to \mathit{CL}(M)\) defined by \(\mathcal {G}(B)=\overline{\bigcup_{i\geq1}f_{i}(B)}\) (for all \(B\in \mathit{CL}(M)\)) is called the associated Hutchinson operator. A set \(A\in \mathit{CL}(M)\) is called an attractor of the weak θCIFS if \(\mathcal {G}(A)=A\).
Before giving our main result in this section, we prove the following lemma.
Lemma 6.1
Proof
Now, we can present our main result of this section.
Theorem 6.1
 (i)
\(\eta(\mathcal{G}^{n}(C_{K}),\mathcal {G}^{n+1}(C_{K}))=\infty\) for all \(n\in\mathbb{N}\), or
 (ii)
there exists an attractor \(A\in \mathit{CL}(M)\) of the considered weak θCIFS and \(A=\lim_{n\to\infty}\mathcal{G}^{n}(C_{K})\), the limit being taken w.r.t. the generalized Hausdorff–Pompeiu metric.
Proof

In view of Theorem 3.2, for each \(i\geq1\), there is a unique fixed point \(z_{i}\) of \(f_{i}\). Therefore, for every nonempty \(K\subset\mathbb{N}\), \(\{z_{i}:i\in K\}=\bigcup_{i\in K}\{f_{i}(z_{i})\}\subset\bigcup_{i\geq1}f_{i}(C_{K})\), and hence \(C_{K}\subset\mathcal{G}(C_{K})\), so that assumption (a) is satisfied.

for any \(B,C\in \mathit{CL}(M)\), \(B\subset C\) clearly implies that \(\mathcal{G}(B)\subset\mathcal{G}(C)\), and therefore assumption (b) holds.

According to [30, 32], the completeness of \((M,d)\) implies the completeness of \((\mathit{CL}(M),\eta)\), and hence assumption (c) clearly holds.

Let \(B,C\in \mathit{CL}(M)\) be such that \(C\subset B\), \(\eta(B,C)<\infty \), and \(\mathcal{G}(B)\neq\mathcal{G}(C)\). As \(\mathcal{G}(B)\neq \mathcal{G}(C)\), we can find \(i\in\mathbb{N}\) such that \(f_{i}(B)\neq f_{i}(C)\), so that \(D(f_{i}(B),f_{i}(C))>0\). Hence, using Lemma 2.2, Lemma 6.1 and the continuity of θ, we havewhere in the last inequality we used (6.3). As \(C\subset B\) and \(\mathcal{G}(C)\subset\mathcal{G}(B)\), we have \(D(C,B)=D(\mathcal{G}(C),\mathcal{G}(B))=0\), so that$$\begin{aligned} \theta \bigl(D \bigl(\mathcal{G}(B),\mathcal{G}(C) \bigr) \bigr)&\leq \theta \Bigl(\sup_{i\geq1}D \bigl(f_{i}(B),f_{i}(C) \bigr) \Bigr) \\ &\leq \sup_{i\geq1}\theta \bigl(D \bigl(f_{i}(B),f_{i}(C) \bigr) \bigr) \\ &\leq \sup_{i\geq1} \bigl[\theta \bigl(D(B,C) \bigr) \bigr]^{h_{i}}= \bigl[\theta \bigl(D(B,C) \bigr) \bigr]^{h}, \end{aligned}$$Therefore, assumption (d) is fulfilled.$$\theta \bigl(\eta \bigl(\mathcal{G}(B),\mathcal{G}(C) \bigr) \bigr)\leq \bigl[ \theta \bigl(\eta(B,C) \bigr) \bigr]^{h}. $$

Next, we show that assumption (e′) is satisfied. To this end, let \(\{B_{n}\}\) be a sequence of closed subsets of M such that \(B_{n}\subset B_{n+1}\) (for all \(n\in\mathbb{N}\)) and \(\lim_{n\to \infty}B_{n}=B\) for some \(B\in \mathit{CL}(M)\). We will show that \(B_{n}\subset B\) for all \(n\in\mathbb{N}\). Let \(n_{0}\in\mathbb{N}\) and \(b\in B_{n_{0}}\) be fixed. Observe that, for each \(n\geq n_{0}\), we have \(b\in B_{n}\) (as \(B_{n_{0}}\subset B_{n}\)). Therefore, for all \(n\geq n_{0}\), we haveHence \(b\in\overline{B}=B\), and therefore \(B_{n_{0}}\subset B\).$$\inf_{u\in B}d(b,u)\leq\sup_{v\in B_{n}}\inf _{u\in B}d(v,u)=D(B_{n},B)\to0 \quad\text{as } n\to\infty \Bigl(\because\lim_{n\to\infty}B_{n}=B \Bigr). $$
In support of Theorem 6.1, we provide the following example.
Example 6.1
 (i)
for each \(i\in\mathbb{N}\), \(f_{i}\) is not a Banach contraction;
 (ii)\(\{f_{i}\}_{i\geq1}\) is a weak θCIFS, and for each \(i\in\mathbb{N}\), \(h_{i}=e^{1}\in(0,1)\), \(h_{i}\) being the constant associated with \(f_{i}\) from (3.1). Furthermore, if \(C_{K}=\overline{\{\frac{i\beta_{i}+\sqrt{i^{2}\beta _{i}^{2}+4i\beta_{i}}}{2i}; i\in K\}}\) for every nonempty \(K\subset \mathbb{N}\) and \(\mathcal{G}:\mathit{CL}(M)\to \mathit{CL}(M)\) be defined as in Definition 6.1, then one of the following cases occurs:
 (I)
\(\eta(\mathcal{G}^{n}(C_{K}),\mathcal {G}^{n+1}(C_{K}))=\infty\) for all \(n\in\mathbb{N}\); or
 (II)
there exists an attractor \(A\in \mathit{CL}(M)\) of the considered weak θCIFS and \(A=\lim_{n\to\infty}\mathcal{G}^{n}(C_{K})\), the limit being taken w.r.t. the generalized Hausdorff–Pompeiu metric.
 (I)
 (iii)
if \(\sup_{i\geq1}\beta_{i}<\infty\), then the attractor A is unique, and \(A=\lim_{n\to\infty}\mathcal{G}^{n}(B)\) for all \(B\in \mathit{CL}(M)\).
Proof
In view of Example 3.7, for each \(i\in\mathbb {N}\), \(f_{i}\) is a weak θcontraction (with \(h_{i}=e^{1}\)), not a Banach contraction, and its fixed point is \(z_{i}=\frac{i\beta_{i}+\sqrt {i^{2}\beta_{i}^{2}+4i\beta_{i}}}{2i}\). Observe that \(\sup_{i\geq 1}h_{i}=e^{1}<1\) and, for each \(i\in\mathbb{N}\), \(f_{i}(M)\subset[\beta _{i},\beta_{i}+\frac{1}{i}]\), so that \(\mathcal{G}(M)\subset[0,\sup_{i\geq1}\beta_{i}+1]\), and hence \(\mathcal{G}(M)\) is bounded. Therefore, the conclusion of this example follows from Theorem 6.1. □
Declarations
Acknowledgements
All the authors are grateful to the anonymous referees for their excellent suggestions, which greatly improved the presentation of the paper.
Funding
This work was supported by Hajjah University (Yemen) and Integral University (India).
Authors’ contributions
All the authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Competing interests
All the authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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