Common Attractors of Generalized Hutchinson–Wardowski Contractive Operators

Abstract

The aim of this paper is to obtain a fractal set of ℑ-iterated function systems comprising generalized ℑ-contractions. For a variety of Hutchinson–Wardowski contractive operators, we prove that this kind of system admits a unique common attractor. Consequently, diverse outcomes are obtained for generalized iterated function systems satisfying various generalized contractive conditions. An illustrative example is also provided. Finally, the existence results of common solutions to fractional boundary value problems are obtained.

1. Introduction and Preliminaries

Hutchinson [1] introduced an important and basic concept of fractal theory called an iterated function system (in short, IFS) in 1981. The Hutchinson operator, created by a finite system of contraction mappings on a Euclidean space, has a closed and bounded fixed point known as the attractor of the IFS. This concept is further developed by Barnsley [2]. IFSs are useful in a variety of fields, including engineering, medicine, forestry, economics, human anatomy, physics, and fractal picture compression. The IFS is a versatile tool that can handle complex structures and patterns, making it useful across a variety of disciplines. Its modeling, compression, and representation properties provide strong reasons to employ it in both theoretical and applied contexts.

Further, in Refs. [3,4], Miculescu and Mihail introduced the generalized iterated function system (in short, GIFS), which is composed of a finite number of Banach contractions, each defined on the Cartesian product ${\mathbb{U}}^m$ and taking values in $\mathbb{U}$. Dumitru [5] and Strobin and Swaczyna [6] built upon the work of Miculescu and Mihail by exploring generalized iterated function systems (GIFSs) consisting of Meir–Keeler-type mappings and ℑ-contractions, respectively. Additionally, Secelean [7] investigated IFSs comprising a countable family of contractive mappings, ℑ-contractions, and Meir–Keeler-type mappings, further expanding on this area of research. In a recent study [8], Khumalo et al. identified common attractors by utilizing a finite collection of generalized contractive mappings within a particular class of mappings in a partial metric space. The noteworthy findings about IFSs and generalizations of their contractions in different metric spaces can be found, for example, in Refs. [9,10,11,12,13] and others.

The Banach Fixed Point Theorem (BFPT), also known as the Contraction Mapping Principle, is a cornerstone of classical functional analysis, holding a prominent place among the most crucial results in the discipline, which was developed and demonstrated in Banach’s 1920 doctoral dissertation and published in 1922 [14]. A remarkable and important generalization of BFPT is stated by Wardowski [15]. He explained the concept of ℑ-contraction in the following manner:

Definition 1.

Consider a metric space $(\mathbb{U},d)$. A mapping $g:\mathbb{U}\to \mathbb{U}$ is classified as an ℑ-contraction if it happens that there is $\mathfrak{I}\in \mathcal{F}$ and $\lambda >0$ such that $\forall \kappa ,\xi \in \mathbb{U}$ with $d(\kappa ,\xi )>0$

$$\begin{array}{c}\hfill \lambda +\mathfrak{I}\left(d\right(g\kappa ,g\xi \left)\right)\le \mathfrak{I}\left(d\right(\kappa ,\xi \left)\right),\end{array}$$

(1)

where $\mathcal{F}$ is the group of all mappings $\mathfrak{I}:(0,\infty )\to (-\infty ,\infty )$ that meet the requirements listed below:

($\mathfrak{I}1$)

$\mathfrak{I}\left(\kappa \right)<\mathfrak{I}\left(\xi \right)\forall \kappa <\xi$;

($\mathfrak{I}2$)

${lim}_{\epsilon \to +\infty }{\mu }_{\epsilon }=0$, if and only if ${lim}_{\epsilon \to +\infty }\mathfrak{I}\left({\mu }_{\epsilon }\right)=-\infty$, for all sequences $\left\{{\mu }_{\epsilon }\right\}\subseteq (0,\infty )$;

($\mathfrak{I}3$)

it happens that there is $0<\mathit{𝚥}<1$ such that ${lim}_{\mu \to 0^{+}}{\mu }^{\mathit{𝚥}}\mathfrak{I}\left(\mu \right)=0.$

In Ref. [16], Secelean demonstrated that criterion ($\mathfrak{I}2$) can be substituted with an equivalent and more convenient one:

($\mathfrak{I}{2}^{\prime }$)

$inf\mathfrak{I}=-\infty$.

For convenience, we will denote the collection of all mappings $\mathfrak{I}:(0,\infty )\to (-\infty ,\infty )$ that satisfy $\left(\mathfrak{I}1\right)$, $\left(\mathfrak{I}{2}^{\prime }\right)$, and $\left(\mathfrak{I}3\right)$ by $\nabla \left(\mathfrak{I}\right)$.

Proposition 1

([16]). Let $F,G,H:(0,\infty )\to (-\infty ,\infty )$ be functions defined by $F:=min\{{\mathfrak{I}}_1,{\mathfrak{I}}_2,\cdots ,{\mathfrak{I}}_N\}$, $G:=max\{{\mathfrak{I}}_1,{\mathfrak{I}}_2,\cdots ,{\mathfrak{I}}_N\}$, and $H:={\rho }_1{\mathfrak{I}}_1+{\rho }_2{\mathfrak{I}}_2+\cdots +{\rho }_n{\mathfrak{I}}_N$, where ${\mathfrak{I}}_1,{\mathfrak{I}}_2,\cdots ,{\mathfrak{I}}_N\in \nabla \left(\mathfrak{I}\right)$ and ${\rho }_1,{\rho }_2,\cdots ,{\rho }_N\in (0,\infty )$ for some $N\in \mathbb{N}$. Then, $F,G,H\in \nabla \left(\mathfrak{I}\right)$.

Proposition 2

([16]). Let $\phi ,\psi :(0,\infty )\to (0,\infty )$ be two mappings satisfying the following conditions:

(a)

φ is strictly increasing and $inf\phi =0$;

(b)

ψ is strictly increasing and there is $\eta \in (0,1)$ such that ${lim}_{t\searrow 0}{t}^{\eta }\psi \left(t\right)=0$;

(c)

there exists $\lambda \in (0,1)$ such that ${lim}_{t\searrow 0}{t}^{\lambda }\phi \left(t\right)=0$. In particular, this condition holds if f is differentiable and there are $\pi ,\varrho \in (0,\infty )$ such that $t\phi \prime \left(t\right)\le \pi \phi \left(t\right)$ for every $t\in (0,\varrho )$.

Then, the function $F:(0,\infty )\to (-\infty ,\infty )$ defined by $ln\phi \left(t\right)+\psi \left(t\right)$ belongs to $\nabla \left(\mathfrak{I}\right)$.

Secelean also explored the IFSs composed of ℑ-contractions extending some fixed point results from the traditional Hutchinson–Barnsley theory of IFS consisting of Banach contractions. Cosentino and Vetro [17] introduced the notion of an ℑ-contraction of Hardy–Rogers type and obtained a fixed point theorem.

Definition 2

([17]). Let $(\mathbb{U},d)$ be a metric space. A self-mapping $g$ on $\mathbb{U}$ is called an ℑ-contraction of Hardy–Rogers type If it happens that there are $\mathfrak{I}\in \mathcal{F}$ and $\lambda >0$ such that

$$\begin{array}{c}\hfill \lambda +\mathfrak{I}\left(d\right(g\left(\kappa \right),g\left(\xi \right)\left)\right)\le \mathfrak{I}\left(\alpha d\right(\kappa ,\xi )+\beta d(\kappa ,g\kappa )+\gamma d(\xi ,g\xi )+\delta d(\kappa ,g\xi )+Ld(\xi ,g\kappa \left)\right),\end{array}$$

where $d\left(g\right(\kappa ),g(\xi \left)\right)>0$, $\alpha ,\beta ,\gamma ,\delta ,L\ge 0$, $\alpha +\beta +\gamma +2\delta =1$ and $\gamma \ne 1$.

To extend the theory of fractal sets, in this paper, we construct a fractal set of an ℑ-iterated function system, a certain finite collection of generalized ℑ-contractions. We prove that Hutchinson–Wardowski contractive operators defined with the help of a finite family of generalized ℑ-contractions on a complete metric space themselves represent a generalized ℑ-contraction mapping on a family of compact subsets. We obtain a final fractal via the successive application of a Hutchinson–Wardowski contractive operator in a metric space.

2. Fundamental Results

Let $(\mathbb{U},d)$ be a metric space and ${\mathcal{C}}^d\left(\mathbb{U}\right)$ be the collection of all nonempty compact subsets of $\mathbb{U}$. The function $H_d:{\mathcal{C}}^d\left(\mathbb{U}\right)\to {\mathcal{C}}^d\left(\mathbb{U}\right)$ defined by

$$\begin{array}{c}\hfill H_d(\mathcal{N},\mathcal{M})=max\left\{\underset{a\in \mathcal{N}}{sup}D(a,\mathcal{M}),\underset{b\in \mathcal{M}}{sup}D(b,\mathcal{N})\right\},\mathrm{for}\mathrm{all}\mathcal{N},\mathcal{M}\in {\mathcal{C}}^d\left(\mathbb{U}\right),\end{array}$$

where $D(a,\mathcal{M})=inf\left\{d(a,b):b\in \mathcal{M}\right\}$ is called the Hausdorff–Pompeiu metric. The metric space $({\mathcal{C}}^d\left(X\right),H_d)$ is complete provided that $(\mathbb{U},d)$ is complete.

Lemma 1

([18]). Let $\mathcal{N},\mathcal{M}\in {\mathcal{C}}^d\left(\mathbb{U}\right)$, and then

(1)

$\mathcal{N}\subset \mathcal{M}$ if and only if $D(\mathcal{N},\mathcal{M})=0$;

(2)

$D(\mathcal{N},S)\le D(\mathcal{N},\mathcal{M})+D(\mathcal{M},S)$.

Lemma 2

([18]). Let ${\left({\mathcal{N}}_i\right)}_{i\in \mathcal{I}}$,${\left({\mathcal{M}}_i\right)}_{i\in \mathcal{I}}$ be two finite collections of sets in $({\mathcal{C}}^d\left(X\right),H_d)$, and then

$$\begin{array}{c}\hfill H_d\left(\bigcup _{i\in \mathcal{I}}{\mathcal{N}}_i,\bigcup _{i\in \mathcal{I}}{\mathcal{M}}_i\right)\le \underset{i\in \mathcal{I}}{sup}{H}_d({\mathcal{N}}_i,{\mathcal{M}}_i).\end{array}$$

We begin by defining generalized ℑ-contraction as

Definition 3.

Let $(\mathbb{U},d)$ be a metric space and $g,\mathcal{\ell }:\mathbb{U}\to \mathbb{U}$ be two mappings. A pair $(g,\mathcal{\ell })$ is called a generalized ℑ-contraction if it happens that there are $\mathfrak{I}\in \nabla \left(\mathfrak{I}\right)$ and $\lambda >0$ such that, for all $\kappa ,\xi \in \mathbb{U}$,

$$\begin{array}{c}\hfill d\left(g\right(\kappa ),\mathcal{\ell }(\xi \left)\right)>0,implying\lambda +\mathfrak{I}\left(d\right(g\left(\kappa \right),\mathcal{\ell }\left(\xi \right)\left)\right)\le \mathfrak{I}\left(d\right(\kappa ,\xi \left)\right).\end{array}$$

(2)

For two mappings $g,\mathcal{\ell }:\mathbb{U}\to \mathbb{U}$, we define for any $\mathcal{N}\in {\mathcal{C}}^d\left(\mathbb{U}\right)$,

$$\begin{array}{ccc}\hfill g\left(\mathcal{N}\right)& =\left\{g\right(a):a\in \mathcal{N}\}\mathrm{and}\mathcal{\ell }\left(\mathcal{N}\right)& =\left\{\mathcal{\ell }\right(a):a\in \mathcal{N}\}.\hfill \end{array}$$

Definition 4.

Let $(\mathbb{U},d)$ be a metric space. If for each $\kappa =1,2,3,\cdots ,\epsilon$, $g_{\kappa },{\mathcal{\ell }}_{\kappa }:\mathbb{U}\to \mathbb{U}$ are continuous mappings and the pair $(g_{\kappa },{\mathcal{\ell }}_{\kappa })$ is generalized ${\mathfrak{I}}_{\kappa }$-contraction for ${\mathfrak{I}}_{\kappa }\in \nabla \left(\mathfrak{I}\right)$ and ${\lambda }_{\kappa }>0$, then $\{Y;(g_{\kappa },{\mathcal{\ell }}_{\kappa }),\kappa =1,2,3,\cdots ,\epsilon \}$ is called an ℑ-iterated function system (in short, ℑ-IFS). The functions $\Theta ,\Omega :{\mathcal{C}}^d\left(\mathbb{U}\right)\to {\mathcal{C}}^d\left(\mathbb{U}\right)$ defined by

$$\begin{array}{c}\hfill \Theta \left(\mathcal{N}\right)=\bigcup _{\kappa =1}^{\epsilon }{g}_{\kappa }\left(\mathcal{N}\right)and\Omega \left(\mathcal{M}\right)=\bigcup _{\kappa =1}^{\epsilon }{\mathcal{\ell }}_{\kappa }\left(\mathcal{M}\right)forall\mathcal{N},\mathcal{M}\in {\mathcal{C}}^d\left(\mathbb{U}\right)\end{array}$$

(3)

are called associated Hutchinson operators.

Definition 5.

Let $\mathcal{N}\in {\mathcal{C}}^d\left(\mathbb{U}\right)$, and then $\mathcal{N}$ is called a common attractor of ℑ-IFS if

(i)

$\Theta \left(\mathcal{N}\right)=\Omega \left(\mathcal{N}\right)=\mathcal{N}$;

(ii)

there exists an open set $V\subseteq \mathbb{U}$ such that $\mathcal{N}\subseteq V$ and ${lim}_{\kappa \to +\infty }{\Theta }^{\kappa }\left(\mathcal{M}\right)={lim}_{\kappa \to +\infty }{\Omega }^{\kappa }\left(\mathcal{M}\right)$ for any compact set $\mathcal{M}\subseteq V$,

where Θ and Ω are provided in (3). The maximal open set V such that (ii) is satisfied is known as a basin of common attraction.

Next, we prove two basic results that play a key role in converting a pair of Hutchinson operators into generalized ℑ-contraction on ${\mathcal{C}}^d\left(\mathbb{U}\right)$ and to ensure the existence of a common attractor for these operators.

Lemma 3.

Let $(\mathbb{U},d)$ be a metric space and $g,\mathcal{\ell }:\mathbb{U}\to \mathbb{U}$ be two continuous mappings. If the pair $(g,\mathcal{\ell })$ is a generalized ℑ-contraction for $\mathfrak{I}\in \nabla \left(\mathfrak{I}\right)$ and $\lambda >0$, then

(1)

$\mathcal{N}\in {\mathcal{C}}^d\left(\mathbb{U}\right)$ implies $g\left(\mathcal{N}\right)\in {\mathcal{C}}^d\left(\mathbb{U}\right)$ and $\mathcal{\ell }\left(\mathcal{N}\right)\in {\mathcal{C}}^d\left(\mathbb{U}\right)$ for any $\mathcal{N}\in {\mathcal{C}}^d\left(\mathbb{U}\right)$;

(2)

the pair $(g,\mathcal{\ell })$ is a generalized ℑ-type contraction on $({\mathcal{C}}^d\left(\mathbb{U}\right),H_d)$.

Proof. 

(1)

Since an image of a compact subset under a continuous mapping is compact, continuity of $g$ and ℓ thus signifies that $\mathcal{N}\in {\mathcal{C}}^d\left(\mathbb{U}\right)$ implies $g\left(\mathcal{N}\right)\in {\mathcal{C}}^d\left(\mathbb{U}\right)$ and $\mathcal{\ell }\left(\mathcal{N}\right)\in {\mathcal{C}}^d\left(\mathbb{U}\right)$ for any $\mathcal{N}\in {\mathcal{C}}^d\left(\mathbb{U}\right)$.

(2)

Let $\mathcal{N},\mathcal{M}\in {\mathcal{C}}^d\left(\mathbb{U}\right)$ such that $H_d(g\mathcal{N},\mathcal{\ell }\mathcal{M})>0$. Assume that

$$\begin{array}{c}\hfill H_d(g\mathcal{N},\mathcal{\ell }\mathcal{M})=\underset{\kappa \in \mathcal{N}}{sup}\underset{\xi \in \mathcal{M}}{inf}d(g\kappa ,\mathcal{\ell }\xi ),\end{array}$$

(4)

which further implies that $d(g\kappa ,\mathcal{\ell }\xi )>0$. So, there exists $\lambda >0$ such that

$$\begin{array}{c}\hfill \lambda +\mathfrak{I}\left(d\right(g\left(\kappa \right),\mathcal{\ell }\left(\xi \right)\left)\right)\le \mathfrak{I}\left(d\right(\kappa ,\xi \left)\right)\mathrm{for}\mathrm{all}\kappa ,\xi \in \mathbb{U}.\end{array}$$

(5)

Due to compactness of $\mathcal{N}$ and continuity of $g$ & $\mathcal{\ell }$, we have $u\in \mathcal{N}$ such that ${inf}_{\xi \in \mathcal{M}}d(gu,\mathcal{\ell }\xi )>0$ for all $\xi \in \mathcal{M}$. Therefore,

$$\begin{array}{c}\hfill \lambda +\mathfrak{I}\left(\underset{\xi \in \mathcal{M}}{inf}d(g\left(u\right),\mathcal{\ell }\left(\xi \right))\right)\le \lambda +\mathfrak{I}\left(d(g\left(u\right),\mathcal{\ell }\left(\xi \right))\right)\le \mathfrak{I}\left(d(u,\xi )\right)\mathrm{for}\mathrm{all}\xi \in \mathcal{M}.\end{array}$$

(6)

Hence, by using (4) and (6), we obtain

$$\begin{array}{c}\hfill \lambda +\mathfrak{I}\left(H_d(g\mathcal{N},\mathcal{\ell }\mathcal{M})\right)\le \mathfrak{I}\left(d(u,\xi )\right),\mathrm{for}\mathrm{all}\xi \in \mathcal{M}.\end{array}$$

(7)

Now, let $v\in \mathcal{M}$ such that $d(u,v)={inf}_{\xi \in \mathcal{M}}d(u,\xi )$, and then (7) yields

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \lambda +\mathfrak{I}\left(H_d(g\mathcal{N},\mathcal{\ell }\mathcal{M})\right)\le \mathfrak{I}\left(d(u,\xi )\right)& =\mathfrak{I}\left(\underset{\xi \in \mathcal{M}}{inf}d(u,\xi )\right)\hfill \\ & \le \mathfrak{I}\left(\underset{\kappa \in \mathcal{N}}{sup}\underset{\xi \in \mathcal{M}}{inf}d(u,\xi )\right)\hfill \\ & \le \mathfrak{I}\left(H_d(\mathcal{N},\mathcal{M})\right).\hfill \end{array}\end{array}$$

(8)

If we assume that

$$H_d(g\mathcal{N},\mathcal{\ell }\mathcal{M})=\underset{\xi \in \mathcal{M}}{sup}\underset{\kappa \in \mathcal{N}}{inf}d(g\kappa ,\mathcal{\ell }\xi ),$$

then, by similar arguments as above, we obtain

$$\lambda +\mathfrak{I}\left(H_d(g\mathcal{N},\mathcal{\ell }\mathcal{M})\right)\le \mathfrak{I}\left(H_d(\mathcal{N},\mathcal{M})\right).$$

Consequently, we have the pair $(g,\mathcal{\ell })$, which is a generalized ℑ-contraction on $({\mathcal{C}}^d\left(\mathbb{U}\right),H_d)$. □

Remark 1.

By considering $g=\mathcal{\ell }$ in Lemma 3, we return to Lemma 4.1 of [16] and Theorem 1.10 of [10].

Lemma 4.

Let $(\mathbb{U},d)$ be a metric space and $g_{\kappa },{\mathcal{\ell }}_{\kappa }:\mathbb{U}\to \mathbb{U}$ be continuous mappings for $\kappa =1,2,3,\cdots ,\epsilon$. If for each $\kappa =1,2,3,\cdots ,\epsilon$ there exist ${\mathfrak{I}}_1,{\mathfrak{I}}_2,\cdots ,{\mathfrak{I}}_{\epsilon }\in \nabla \left(\mathfrak{I}\right)$ and ${\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_{\epsilon }>0$ such that the pair $(g_{\kappa },{\mathcal{\ell }}_{\kappa })$ satisfy

$$\begin{array}{c}\hfill d(g_{\kappa }\left(\kappa \right),{\mathcal{\ell }}_{\kappa }\left(\xi \right))>0,implying\lambda +{\mathfrak{I}}_{\kappa }\left(d(g_{\kappa }\left(\kappa \right),{\mathcal{\ell }}_{\kappa }\left(\xi \right))\right)\le {\mathfrak{I}}_{\kappa }\left(d(\kappa ,\xi )\right)\end{array}$$

(9)

for all $\kappa ,\xi \in \mathbb{U}$, and the mapping $G_{\kappa }:=\mathfrak{I}-{\mathfrak{I}}_{\kappa }$ is nondecreasing. Then, the pair $(\Theta ,\Omega )$ is a generalized ℑ-contraction on ${\mathcal{C}}^d\left(\mathbb{U}\right)$ for $\mathfrak{I}={max}_{1\le \kappa \le \epsilon }{\mathfrak{I}}_{\kappa }$ and $\lambda ={min}_{1\le \kappa \le \epsilon }{\lambda }_{\kappa }$, where Θ and Ω are defined in (3).

Proof. 

By hypothesis, there exist ${\mathfrak{I}}_1,{\mathfrak{I}}_2,\cdots ,{\mathfrak{I}}_{\epsilon }\in \nabla \left(\mathfrak{I}\right)$ and ${\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_{\epsilon }>0$ such that the pair $(g_{\kappa },{\mathcal{\ell }}_{\kappa })$ satisfy (9) for each $\kappa =1,2,3,\cdots ,\epsilon$ and $\kappa ,\xi \in \mathbb{U}$. Let $\mathfrak{I}=max\{{\mathfrak{I}}_1,{\mathfrak{I}}_2,\cdots ,{\mathfrak{I}}_{\epsilon }\}$ and $\lambda =min\{{\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_{\epsilon }\}$, and then $\lambda >0$, and, by using Proposition 1, we have $\mathfrak{I}\in \nabla \left(\mathfrak{I}\right)$.

Now, let $\mathcal{N},\mathcal{M}\in {\mathcal{C}}^d\left(\mathbb{U}\right)$ such that $H_d(\Theta \mathcal{N},\Omega \mathcal{M})>0$. Then, due to Lemma 2, for some ${\kappa }_0\in \{1,2,\cdots ,\epsilon \}$, we obtain

$$\begin{array}{c}\hfill 0<H_d(\Theta \mathcal{N},\Omega \mathcal{M})\le \underset{1\le \kappa \le \epsilon }{sup}{H}_d(g_{\kappa }\left(\mathcal{N}\right),{\mathcal{\ell }}_{\kappa }\left(\mathcal{M}\right))=H_d(g_{{\kappa }_0}\left(\mathcal{N}\right),{\mathcal{\ell }}_{{\kappa }_0}\left(\mathcal{M}\right)).\end{array}$$

(10)

With the aid of Lemma 3, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \lambda +\mathfrak{I}\left(H_d(\Theta \mathcal{N},\Omega \mathcal{M})\right)& \le \lambda +\mathfrak{I}\left(H_d(g_{{\kappa }_0}\mathcal{N},{\mathcal{\ell }}_{{\kappa }_0}\mathcal{M})\right)\hfill \\ & \le {\lambda }_{{\kappa }_0}+{\mathfrak{I}}_{{\kappa }_0}\left(H_d(g_{{\kappa }_0}\mathcal{N},{\mathcal{\ell }}_{{\kappa }_0}\mathcal{M})\right)+G_{{\kappa }_0}\left(H_d(g_{{\kappa }_0}\mathcal{N},{\mathcal{\ell }}_{{\kappa }_0}\mathcal{M})\right)\hfill \\ & \le {\mathfrak{I}}_{{\kappa }_0}\left(H_d(\mathcal{N},\mathcal{M})\right)+G_{{\kappa }_0}\left(H_d(\mathcal{N},\mathcal{M})\right)\hfill \\ & =\mathfrak{I}\left(H_d(\mathcal{N},\mathcal{M})\right);\hfill \end{array}\end{array}$$

(11)

that is, the pair $(\Theta ,\Omega )$ is a generalized ℑ-contraction on ${\mathcal{C}}^d\left(\mathbb{U}\right)$. □

3. Main Results

This section is devoted to proving the existence results of common attractors of Hutchinson–Wardowski contractive operators. We start with the following definition.

Definition 6.

Let $(\mathbb{U},d)$ be a metric space and $\Theta ,\Omega :{\mathcal{C}}^d\left(\mathbb{U}\right)\to {\mathcal{C}}^d\left(\mathbb{U}\right)$ be mappings as defined in (3). A pair $(\Theta ,\Omega )$ is called Hardy–Rogers-type Hutchinson–Wardowski contractive operator if there exist $\mathfrak{I}\in \nabla \left(\mathfrak{I}\right)$ and $\lambda >0$ such that, for all $\mathcal{N},\mathcal{M}\in {\mathcal{C}}^d\left(\mathbb{U}\right)$, the following holds

$$\begin{array}{c}\hfill H_d(\Theta \left(\mathcal{N}\right),\Omega \left(\mathcal{M}\right))>0,implying\lambda +\mathfrak{I}\left(H_d(\Theta \left(\mathcal{N}\right),\Omega \left(\mathcal{M}\right))\right)\le \mathfrak{I}\left({\mathbb{Q}}_{\Theta ,\Omega }^{H_d}(\mathcal{N},\mathcal{M})\right),\end{array}$$

(12)

where

$$\begin{array}{c}\hfill {\mathbb{Q}}_{\Theta ,\Omega }^{H_d}(\mathcal{N},\mathcal{M})=\alpha H_d(\mathcal{N},\mathcal{M})+\beta H_d(\mathcal{N},\Theta \mathcal{N})+\gamma H_d(\mathcal{M},\Omega \mathcal{M})+\delta H_d(\mathcal{N},\Omega \mathcal{M})+L{H}_d(\mathcal{M},\Theta \mathcal{N})\end{array}$$

with $\alpha ,\beta ,\gamma ,\delta ,L\ge 0$, $\alpha +\beta +\gamma +2\delta =1$ and $\gamma \ne 1$.

Definition 7.

Let $(\mathbb{U},d)$ be a metric space and $\Theta ,\Omega :{\mathcal{C}}^d\left(\mathbb{U}\right)\to {\mathcal{C}}^d\left(\mathbb{U}\right)$ be mappings as defined in (3). A pair $(\Theta ,\Omega )$ is called weak Hutchinson–Wardowski contractive operator if there exist $\mathfrak{I}\in \nabla \left(\mathfrak{I}\right)$ and $\lambda >0$ such that, for all $\mathcal{N},\mathcal{M}\in {\mathcal{C}}^d\left(\mathbb{U}\right)$, the following holds

$$\begin{array}{c}\hfill H_d(\Theta \left(\mathcal{N}\right),\Omega \left(\mathcal{M}\right))>0,implying\lambda +\mathfrak{I}\left(H_d(\Theta \left(\mathcal{N}\right),\Omega \left(\mathcal{M}\right))\right)\le \mathfrak{I}\left({\mathbb{P}}_{\Theta ,\Omega }^{H_d}(\mathcal{N},\mathcal{M})\right),\end{array}$$

(13)

where

$$\begin{array}{c}\hfill {\mathbb{P}}_{\Theta ,\Omega }^{H_d}(\mathcal{N},\mathcal{M})=max\left\{H_d(\mathcal{N},\mathcal{M}),H_d(\mathcal{N},\Theta \mathcal{N}),H_d(\mathcal{M},\Omega \mathcal{M}),{\displaystyle \frac{H_d(\mathcal{N},\Omega \mathcal{M})+H_d(\mathcal{M},\Theta \mathcal{N})}{2}}\right\}.\end{array}$$

Remark 2.

From $\left(\mathfrak{I}1\right)$, (12), and (13), we deduce that every Hardy–Rogers-type Hutchinson–Wardowski contractive operator and weak Hutchinson–Wardowski contractive operator satisfy the following conditions, respectively:

$$\begin{array}{c}\hfill H_d(\Theta \left(\mathcal{N}\right),\Omega \left(\mathcal{M}\right))<{\mathbb{Q}}_{\Theta ,\Omega }^{H_d}(\mathcal{N},\mathcal{M})\end{array}$$

(14)

and

$$\begin{array}{c}\hfill H_d(\Theta \left(\mathcal{N}\right),\Omega \left(\mathcal{M}\right))<{\mathbb{P}}_{\Theta ,\Omega }^{H_d}(\mathcal{N},\mathcal{M}).\end{array}$$

(15)

Theorem 1.

Let $(\mathbb{U},d)$ be a metric space, $\{\mathbb{U};(g_{\kappa },{\mathcal{\ell }}_{\kappa }),\kappa =1,2,3,\cdots ,\epsilon \}$ be an ℑ-IFS, and $\Theta ,\Omega :{\mathcal{C}}^d\left(\mathbb{U}\right)\to {\mathcal{C}}^d\left(\mathbb{U}\right)$ be mappings as defined in (3). If the pair $(\Theta ,\Omega )$ is a Hardy–Rogers-type Hutchinson–Wardowski contractive operator, then Θ and Ω share a common attractor $\mathcal{A}\in {\mathcal{C}}^d\left(\mathbb{U}\right)$. If $\alpha +\delta +L\le 1$, then this common attractor is unique.

Moreover, for any initial compact set ${\mathcal{N}}_0$, the sequence

$$\begin{array}{c}\hfill \{{\mathcal{N}}_0,\Theta \left({\mathcal{N}}_0\right),\Omega \Theta \left({\mathcal{N}}_0\right),\Theta \Omega \Theta \left({\mathcal{N}}_0\right),\cdots \}\end{array}$$

of compact sets generated by Θ and Ω will converge to the common attractor $\mathcal{A}$.

Proof. 

Let ${\mathcal{N}}_0\in {\mathcal{C}}^d\left(\mathbb{U}\right)$ be an arbitrary point. Define the following sequence for initial point ${\mathcal{N}}_0$ as

$$\begin{array}{c}\hfill \left.\begin{array}{c}{\mathcal{N}}_1=\Theta \left({\mathcal{N}}_0\right),{\mathcal{N}}_3=\Theta \left({\mathcal{N}}_2\right),\cdots ,{\mathcal{N}}_{2r+1}=\Theta \left({\mathcal{N}}_{2r}\right),\hfill \\ {\mathcal{N}}_2=\Omega \left({\mathcal{N}}_1\right),{\mathcal{N}}_4=\Omega \left({\mathcal{N}}_3\right),\cdots ,{\mathcal{N}}_{2r+2}=\Omega \left({\mathcal{N}}_{2r+1}\right),\hfill \end{array}\right\}\mathrm{for}r\in {\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}.\end{array}$$

(16)

If ${\mathcal{N}}_{2r+1}={\mathcal{N}}_{2r}$ for some $r\in {\mathbb{N}}_0$, then ${\mathcal{N}}_{2r}$ is the common attractor of $\Theta$ and $\Omega$. Assume that ${\mathcal{N}}_{2r+1}\ne {\mathcal{N}}_{2r}$ for all $r\in {\mathbb{N}}_0$. So, by using $\mathcal{N}={\mathcal{N}}_{2r}$ and $\mathcal{M}={\mathcal{N}}_{2r+1}$ in inequality (12), we have

$$\begin{array}{c}\hfill \lambda +\mathfrak{I}\left(H_d({\mathcal{N}}_{2r+1},{\mathcal{N}}_{2r+2})\right)=\lambda +\mathfrak{I}\left(H_d(\Theta {\mathcal{N}}_{2r},\Omega {\mathcal{N}}_{2r+1})\right)\le \mathfrak{I}\left({\mathbb{Q}}_{\Theta ,\Omega }^{H_d}({\mathcal{N}}_{2r},{\mathcal{N}}_{2r+1})\right),\end{array}$$

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathbb{Q}}_{\Theta ,\Omega }^{H_d}({\mathcal{N}}_{2r},{\mathcal{N}}_{2r+1}))=& \alpha H_d({\mathcal{N}}_{2r},{\mathcal{N}}_{2r+1})+\beta H_d({\mathcal{N}}_{2r},\Theta {\mathcal{N}}_{2r})+\gamma H_d({\mathcal{N}}_{2r+1},\Omega {\mathcal{N}}_{2r+1})\hfill \\ & +\delta H_d({\mathcal{N}}_{2r},\Omega {\mathcal{N}}_{2r+1})+L{H}_d({\mathcal{N}}_{2r+1},\Theta {\mathcal{N}}_{2r})\hfill \\ \hfill \le & (\alpha +\beta )H_d({\mathcal{N}}_{2r},{\mathcal{N}}_{2r+1})+\gamma H_d({\mathcal{N}}_{2r+1},{\mathcal{N}}_{2r+2})+\delta H_d({\mathcal{N}}_{2r},{\mathcal{N}}_{2r+2})\hfill \\ \hfill \le & (\alpha +\beta +\delta )H_d({\mathcal{N}}_{2r},{\mathcal{N}}_{2r+1})+(\gamma +\delta )H_d({\mathcal{N}}_{2r+1},{\mathcal{N}}_{2r+2}).\hfill \end{array}\end{array}$$

Thus,

$$\begin{array}{c}\hfill \begin{array}{cc}& \lambda +\mathfrak{I}\left(H_d({\mathcal{N}}_{2r+1},{\mathcal{N}}_{2r+2})\right)\hfill \\ \hfill \le & \mathfrak{I}((\alpha +\beta +\delta )H_d({\mathcal{N}}_{2r},{\mathcal{N}}_{2r+1})+(\gamma +\delta )H_d({\mathcal{N}}_{2r+1},{\mathcal{N}}_{2r+2})).\hfill \end{array}\end{array}$$

Similarly, by using $\mathcal{N}={\mathcal{N}}_{2r+1}$ and $\mathcal{M}={\mathcal{N}}_{2r+2}$ in inequality (12), we have

$$\begin{array}{c}\hfill \lambda +\mathfrak{I}\left(H_d({\mathcal{N}}_{2r+2},{\mathcal{N}}_{2r+3})\right)=\lambda +\mathfrak{I}\left(H_d(\Theta {\mathcal{N}}_{2r+1},\Omega {\mathcal{N}}_{2r+2})\right)\le \mathfrak{I}\left({\mathbb{Q}}_{\Theta ,\Omega }^{H_d}({\mathcal{N}}_{2r+1},{\mathcal{N}}_{2r+2})\right),\end{array}$$

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathbb{Q}}_{\Theta ,\Omega }^{H_d}({\mathcal{N}}_{2r+1},{\mathcal{N}}_{2r+2}))=& \alpha H_d({\mathcal{N}}_{2r+1},{\mathcal{N}}_{2r+2})+\beta H_d({\mathcal{N}}_{2r+1},\Theta {\mathcal{N}}_{2r+1})+\gamma H_d({\mathcal{N}}_{2r+2},\Omega {\mathcal{N}}_{2r+2})\hfill \\ & +\delta H_d({\mathcal{N}}_{2r+1},\Omega {\mathcal{N}}_{2r+2})+L{H}_d({\mathcal{N}}_{2r+2},\Theta {\mathcal{N}}_{2r+1})\hfill \\ \hfill \le & (\alpha +\beta )H_d({\mathcal{N}}_{2r+1},{\mathcal{N}}_{2r+2})+\gamma H_d({\mathcal{N}}_{2r+2},{\mathcal{N}}_{2r+3})+\delta H_d({\mathcal{N}}_{2r+1},{\mathcal{N}}_{2r+3})\hfill \\ \hfill \le & (\alpha +\beta +\delta )H_d({\mathcal{N}}_{2r+1},{\mathcal{N}}_{2r+2})+(\gamma +\delta )H_d({\mathcal{N}}_{2r+2},{\mathcal{N}}_{2r+3}).\hfill \end{array}\end{array}$$

Thus,

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \lambda +\mathfrak{I}\left(H_d({\mathcal{N}}_{2r+2},{\mathcal{N}}_{2r+3})\right)\le \mathfrak{I}((\alpha +\beta +\delta )H_d({\mathcal{N}}_{2r+1},{\mathcal{N}}_{2r+2})+(\gamma +\delta )H_d({\mathcal{N}}_{2r+2},{\mathcal{N}}_{2r+3})).\end{array}\end{array}$$

In general, for all $r\in {\mathbb{N}}_0$, we have

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \lambda +\mathfrak{I}\left(H_d({\mathcal{N}}_{r+1},{\mathcal{N}}_{r+2})\right)\le \mathfrak{I}((\alpha +\beta +\delta )H_d({\mathcal{N}}_r,{\mathcal{N}}_{r+1})+(\gamma +\delta )H_d({\mathcal{N}}_{r+1},{\mathcal{N}}_{r+2})).\end{array}\end{array}$$

(17)

Since ℑ is strictly increasing, we deduce that

$$\begin{array}{c}\hfill H_d({\mathcal{N}}_{r+1},{\mathcal{N}}_{r+2})<(\alpha +\beta +\delta )H_d({\mathcal{N}}_r,{\mathcal{N}}_{r+1})+(\gamma +\delta )H_d({\mathcal{N}}_{r+1},{\mathcal{N}}_{r+2}),\end{array}$$

which further implies that

$$\begin{array}{c}\hfill (1-\gamma -\delta )H_d({\mathcal{N}}_{r+1},{\mathcal{N}}_{r+2})<(\alpha +\beta +\delta )H_d({\mathcal{N}}_r,{\mathcal{N}}_{r+1}),\mathrm{for}\mathrm{all}r\in {\mathbb{N}}_0.\end{array}$$

Since $\alpha +\beta +\gamma +2\delta =1$ and $\gamma \ne 1$, $1-\gamma -\delta >0$ and thus

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill H_d({\mathcal{N}}_{r+1},{\mathcal{N}}_{r+2})& <{\displaystyle \frac{(\alpha +\beta +\delta )}{(1-\gamma -\delta )}}{H}_d({\mathcal{N}}_r,{\mathcal{N}}_{r+1})\hfill \\ & =H_d({\mathcal{N}}_r,{\mathcal{N}}_{r+1}).\hfill \end{array}\end{array}$$

Consequently,

$$\begin{array}{c}\hfill \lambda +\mathfrak{I}\left(H_d({\mathcal{N}}_{r+1},{\mathcal{N}}_{r+2})\right)\le \mathfrak{I}\left(H_d({\mathcal{N}}_r,{\mathcal{N}}_{r+1})\right),\mathrm{for}\mathrm{all}r\in {\mathbb{N}}_0.\end{array}$$

(18)

Inequality (18) implies that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathfrak{I}\left(H_d({\mathcal{N}}_r,{\mathcal{N}}_{r+1})\right)& \le \mathfrak{I}\left(H_d({\mathcal{N}}_{r-1},{\mathcal{N}}_r)\right)-\lambda \hfill \\ & \le \mathfrak{I}\left(H_d({\mathcal{N}}_{r-2},{\mathcal{N}}_{r-1})\right)-2\lambda \hfill \\ & ⋮\hfill \\ & \le \mathfrak{I}\left(H_d({\mathcal{N}}_0,{\mathcal{N}}_1)\right)-r\lambda \hfill \end{array}\end{array}$$

(19)

for all $r\in \mathbb{N}$; thus,

$$\begin{array}{c}\hfill \underset{r\to \infty }{lim}\mathfrak{I}\left(H_d({\mathcal{N}}_r,{\mathcal{N}}_{r+1})\right)=-\infty .\end{array}$$

(20)

By using $\left(\mathfrak{I}{2}^{\prime }\right)$ and (20), we obtain that $H_d({\mathcal{N}}_r,{\mathcal{N}}_{r+1})\to 0$ as $r\to \infty$. Now, from $\left(\mathfrak{I}3\right)$, there exists $j\in (0,1)$ such that

$$\begin{array}{c}\hfill \underset{r\to \infty }{lim}{\left[H_d({\mathcal{N}}_r,{\mathcal{N}}_{r+1})\right]}^j\mathfrak{I}\left(H_d({\mathcal{N}}_r,{\mathcal{N}}_{r+1})\right).\end{array}$$

Thus, by using (19), the following holds for all $r\in \mathbb{N}$,

$$\begin{array}{c}\hfill \begin{array}{cc}& {\left[H_d({\mathcal{N}}_r,{\mathcal{N}}_{r+1})\right]}^j\mathfrak{I}\left(H_d({\mathcal{N}}_r,{\mathcal{N}}_{r+1})\right)-{\left[H_d({\mathcal{N}}_r,{\mathcal{N}}_{r+1})\right]}^j\mathfrak{I}\left(H_d({\mathcal{N}}_0,{\mathcal{N}}_1)\right)\hfill \\ \hfill \le & {\left[H_d({\mathcal{N}}_r,{\mathcal{N}}_{r+1})\right]}^j(\mathfrak{I}\left(H_d({\mathcal{N}}_0,{\mathcal{N}}_1)\right)-r\lambda )-{\left[H_d({\mathcal{N}}_r,{\mathcal{N}}_{r+1})\right]}^j\mathfrak{I}\left(H_d({\mathcal{N}}_0,{\mathcal{N}}_1)\right)\hfill \\ \hfill =& -r\lambda {\left[H_d({\mathcal{N}}_r,{\mathcal{N}}_{r+1})\right]}^j\le 0\hfill \end{array}\end{array}$$

(21)

On letting limit as $r\to \infty$ in (21), we obtain

$$\begin{array}{c}\hfill \underset{r\to \infty }{lim}r{\left[H_d({\mathcal{N}}_r,{\mathcal{N}}_{r+1})\right]}^j=0\end{array}$$

and hence

$$\begin{array}{c}\hfill \underset{r\to \infty }{lim}{\left(r\right)}^{{\displaystyle \frac{1}{j}}}{H}_d({\mathcal{N}}_r,{\mathcal{N}}_{r+1})=0.\end{array}$$

(22)

Equation (22) guarantees that the series ${\displaystyle \sum _{r=1}^{\infty }{H}_d({\mathcal{N}}_r,{\mathcal{N}}_{r+1})}$ is convergent. This implies that $\left\{{\mathcal{N}}_r\right\}$ is a Cauchy sequence in ${\mathcal{C}}^d\left(\mathbb{U}\right)$. Completeness of $({\mathcal{C}}^d\left(\mathbb{U}\right),H_d)$ ensures the existence of $\mathcal{A}\in {\mathcal{C}}^d\left(\mathbb{U}\right)$ such that

$$\begin{array}{c}\hfill \underset{r\to \infty }{lim}{\mathcal{N}}_r=\mathcal{A},\end{array}$$

(23)

which implies

$$\begin{array}{c}\hfill \underset{r\to \infty }{lim}{H}_d({\mathcal{N}}_r,\mathcal{A})=\underset{r\to \infty }{lim}{H}_d({\mathcal{N}}_r,{\mathcal{N}}_{r+1})=H_d(\mathcal{A},\mathcal{A}),\end{array}$$

so we have

$$\begin{array}{c}\hfill \underset{r\to \infty }{lim}{H}_d({\mathcal{N}}_r,\mathcal{A})=0.\end{array}$$

(24)

Next, assume that $\mathcal{A}\ne \Theta \mathcal{A}$, and then we can assume that $\Theta {\mathcal{N}}_r\ne \Theta \mathcal{A}$ for all $r\in {\mathbb{N}}_0$. Now, from (14), we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill H_d(\Theta \mathcal{A},\mathcal{A})& \le H_d(\Theta \mathcal{A},{\mathcal{N}}_{2r+2})+H_d({\mathcal{N}}_{2r+2},\mathcal{A})\hfill \\ & =H_d(\Theta \mathcal{A},\Omega {\mathcal{N}}_{2r+1})+H_d({\mathcal{N}}_{2r+2},\mathcal{A})\hfill \\ & <{\mathbb{Q}}_{\Theta ,\Omega }^{H_d}(\mathcal{A},{\mathcal{N}}_{2r+1})+H_d({\mathcal{N}}_{2r+2},\mathcal{A}),\hfill \end{array}\end{array}$$

(25)

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathbb{Q}}_{\Theta ,\Omega }^{H_d}(\mathcal{A},{\mathcal{N}}_{2r+1})=& \alpha H_d(\mathcal{A},{\mathcal{N}}_{2r+1})+\beta H_d(\mathcal{A},\Theta \mathcal{A})+\gamma H_d({\mathcal{N}}_{2r+1},\Omega {\mathcal{N}}_{2r+1})\hfill \\ & +\delta H_d(\mathcal{A},\Omega {\mathcal{N}}_{2r+1})+L{H}_d({\mathcal{N}}_{2r+1},\Theta \mathcal{A})\hfill \\ \hfill =& \alpha H_d(\mathcal{A},{\mathcal{N}}_{2r+1})+\beta H_d(\mathcal{A},\Theta \mathcal{A})+\gamma H_d({\mathcal{N}}_{2r+1},{\mathcal{N}}_{2r+2})\hfill \\ & +\delta H_d(\mathcal{A},{\mathcal{N}}_{2r+2})+L{H}_d({\mathcal{N}}_{2r+1},\Theta \mathcal{A}).\hfill \end{array}\end{array}$$

(26)

By letting limit as $r\to \infty$ in (26) and combining with (25), we obtain

$$\begin{array}{c}\hfill H_d(\Theta \mathcal{A},\mathcal{A})<(\beta +L)H_d(\Theta \mathcal{A},\mathcal{A})<H_d(\Theta \mathcal{A},\mathcal{A}),\end{array}$$

a contradiction; hence, $\mathcal{A}=\Theta \mathcal{A}$.

Similarly, assume that $\mathcal{A}\ne \Omega \mathcal{A}$, and then we can assume that $\Omega {\mathcal{N}}_r\ne \Omega \mathcal{A}$ for all $r\in {\mathbb{N}}_0$. Now, from (14), we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill H_d(\mathcal{A},\Omega \mathcal{A})& \le H_d(\mathcal{A},{\mathcal{N}}_{2r+1})+H_d({\mathcal{N}}_{2r+1},\Omega \mathcal{A})\hfill \\ & =H_d(\mathcal{A},{\mathcal{N}}_{2r+1})+H_d(\Theta {\mathcal{N}}_{2r},\Omega \mathcal{A})\hfill \\ & <H_d(\mathcal{A},{\mathcal{N}}_{2r+1})+{\mathbb{Q}}_{\Theta ,\Omega }^{H_d}({\mathcal{N}}_{2r},\mathcal{A}),\hfill \end{array}\end{array}$$

(27)

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathbb{Q}}_{\Theta ,\Omega }^{H_d}({\mathcal{N}}_{2r},\mathcal{A})=& \alpha H_d({\mathcal{N}}_{2r},\mathcal{A})+\beta H_d({\mathcal{N}}_{2r},\Theta {\mathcal{N}}_{2r})+\gamma H_d(\mathcal{A},\Omega \mathcal{A})\hfill \\ & +\delta H_d({\mathcal{N}}_{2r},\Omega \mathcal{A})+L{H}_d(\mathcal{A},\Theta {\mathcal{N}}_{2r})\hfill \\ \hfill =& \alpha H_d({\mathcal{N}}_{2r},\mathcal{A})+\beta H_d({\mathcal{N}}_{2r},{\mathcal{N}}_{2r+1})+\gamma H_d(\mathcal{A},\Omega \mathcal{A})\hfill \\ & +\delta H_d({\mathcal{N}}_{2r},\Omega \mathcal{A})+L{H}_d(\mathcal{A},{\mathcal{N}}_{2r+1}).\hfill \end{array}\end{array}$$

(28)

By letting limit as $r\to \infty$ in (28) and combining with (27), we obtain

$$\begin{array}{c}\hfill H_d(\mathcal{A},\Omega \mathcal{A})<(\gamma +\delta )H_d(\mathcal{A},\Omega \mathcal{A})<H_d(\mathcal{A},\Omega \mathcal{A}),\end{array}$$

a contradiction; hence, $\mathcal{A}=\Omega \mathcal{A}$. Thus, $\mathcal{A}$ is a common attractor of $\Theta$ and $\Omega$.

Next, we prove the uniqueness of the common attractor. Let $\mathcal{B}$ be another common attractor of $\Theta$ and $\Omega$ such that $\mathcal{A}\ne \mathcal{B}$. Then, $H_d(\mathcal{A},\mathcal{B})>0$, so (12) yields

$$\begin{array}{c}\hfill \lambda +\mathfrak{I}(H_d(\mathcal{A},\mathcal{B})=\lambda +\mathfrak{I}(H_d(\Theta \mathcal{A},\Omega \mathcal{B})\le \mathfrak{I}\left({\mathbb{Q}}_{\Theta ,\Omega }^{H_d}(\mathcal{A},\mathcal{B})\right),\end{array}$$

(29)

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathbb{Q}}_{\Theta ,\Omega }^{H_d}(\mathcal{A},\mathcal{B})=& \alpha H_d(\mathcal{A},\mathcal{B})+\beta H_d(\mathcal{A},\Theta \mathcal{A})+\gamma H_d(\mathcal{B},\Omega \mathcal{B})+\delta H_d(\mathcal{A},\Omega \mathcal{B})+L{H}_d(\mathcal{B},\Theta \mathcal{A})\hfill \\ \hfill =& (\alpha +\delta +L)H_d(\mathcal{A},\mathcal{B}).\hfill \end{array}\end{array}$$

If $\alpha +\delta +L\le 1$, inequality (29) yields a contradiction to the fact that $\lambda >0$. Hence, $\mathcal{A}=\mathcal{B}$. □

Theorem 2.

Let $(\mathbb{U},d)$ be a metric space, $\{\mathbb{U};(g_{\kappa },{\mathcal{\ell }}_{\kappa }),\kappa =1,2,3,\cdots ,\epsilon \}$ be an ℑ-IFS, and $\Theta ,\Omega :{\mathcal{C}}^d\left(\mathbb{U}\right)\to {\mathcal{C}}^d\left(\mathbb{U}\right)$ be mappings as defined in (3). If the pair $(\Theta ,\Omega )$ is a weak Hutchinson–Wardowski contractive operator, then Θ and Ω share at most one common attractor $\mathcal{A}\in {\mathcal{C}}^d\left(\mathbb{U}\right)$. Moreover, for any initial compact set ${\mathcal{N}}_0$, the sequence

$$\begin{array}{c}\hfill \{{\mathcal{N}}_0,\Theta \left({\mathcal{N}}_0\right),\Omega \Theta \left({\mathcal{N}}_0\right),\Theta \Omega \Theta \left({\mathcal{N}}_0\right),\cdots \}\end{array}$$

of compact sets generated by Θ and Ω will converge to the common attractor ${\mathcal{A}}_0$.

Proof. 

Let ${\mathcal{N}}_0\in {\mathcal{C}}^d\left(\mathbb{U}\right)$ be an arbitrary point. Define the sequence as in (16) for initial point ${\mathcal{N}}_0$. If ${\mathcal{N}}_{2r+1}={\mathcal{N}}_{2r}$ for some $r\in {\mathbb{N}}_0$, then ${\mathcal{N}}_{2r}$ is the common attractor of $\Theta$ and $\Omega$. Assume that ${\mathcal{N}}_{2r+1}\ne {\mathcal{N}}_{2r}$ for all $r\in {\mathbb{N}}_0$; thus, by using $\mathcal{N}={\mathcal{N}}_{2r}$ and $\mathcal{M}={\mathcal{N}}_{2r+1}$ in inequality (13), we have

$$\begin{array}{c}\hfill \lambda +\mathfrak{I}\left(H_d({\mathcal{N}}_{2r+1},{\mathcal{N}}_{2r+2})\right)=\lambda +\mathfrak{I}\left(H_d(\Theta {\mathcal{N}}_{2r},\Omega {\mathcal{N}}_{2r+1})\right)\le \mathfrak{I}\left({\mathbb{P}}_{\Theta ,\Omega }^{H_d}({\mathcal{N}}_{2r},{\mathcal{N}}_{2r+1})\right),\end{array}$$

(30)

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathbb{P}}_{\Theta ,\Omega }^{H_d}({\mathcal{N}}_{2r},{\mathcal{N}}_{2r+1}))=& max\left\{H_d({\mathcal{N}}_{2r},{\mathcal{N}}_{2r+1}),H_d({\mathcal{N}}_{2r},\Theta {\mathcal{N}}_{2r}),H_d({\mathcal{N}}_{2r+1},\Omega {\mathcal{N}}_{2r+1}),\right.\hfill \\ & \left.{\displaystyle \frac{H_d({\mathcal{N}}_{2r},\Omega {\mathcal{N}}_{2r+1})+H_d({\mathcal{N}}_{2r+1},\Theta {\mathcal{N}}_{2r})}{2}}\right\}\hfill \\ \hfill =& max\left\{H_d({\mathcal{N}}_{2r},{\mathcal{N}}_{2r+1}),H_d({\mathcal{N}}_{2r+1},{\mathcal{N}}_{2r+2}),{\displaystyle \frac{H_d({\mathcal{N}}_{2r},{\mathcal{N}}_{2r+2})}{2}}\right\}\hfill \\ \hfill \le & max\left\{H_d({\mathcal{N}}_{2r},{\mathcal{N}}_{2r+1}),H_d({\mathcal{N}}_{2r+1},{\mathcal{N}}_{2r+2}),\right.\hfill \\ & \left.{\displaystyle \frac{H_d({\mathcal{N}}_{2r},{\mathcal{N}}_{2r+1})+H_d({\mathcal{N}}_{2r+1},{\mathcal{N}}_{2r+2})}{2}}\right\}\hfill \\ \hfill =& max\left\{H_d({\mathcal{N}}_{2r},{\mathcal{N}}_{2r+1}),H_d({\mathcal{N}}_{2r+1},{\mathcal{N}}_{2r+2})\right\}.\hfill \end{array}\end{array}$$

Now, if ${\mathbb{P}}_{\Theta ,\Omega }^{H_d}({\mathcal{N}}_{2r},{\mathcal{N}}_{2r+1}))=H_d({\mathcal{N}}_{2r+1},{\mathcal{N}}_{2r+2})$, then (30) reduces to

$$\begin{array}{c}\hfill \lambda +\mathfrak{I}\left(H_d({\mathcal{N}}_{2r+1},{\mathcal{N}}_{2r+2})\right)\le \mathfrak{I}\left(H_d({\mathcal{N}}_{2r+1},{\mathcal{N}}_{2r+2})\right),\end{array}$$

(31)

which leads to contradiction because $\lambda >0$. Hence, ${\mathbb{P}}_{\Theta ,\Omega }^{H_d}({\mathcal{N}}_{2r},{\mathcal{N}}_{2r+1}))=H_d({\mathcal{N}}_{2r},{\mathcal{N}}_{2r+1})$ and

$$\begin{array}{c}\hfill \lambda +\mathfrak{I}\left(H_d({\mathcal{N}}_{2r+1},{\mathcal{N}}_{2r+2})\right)\le \mathfrak{I}\left(H_d({\mathcal{N}}_{2r},{\mathcal{N}}_{2r+1})\right).\end{array}$$

(32)

Similarly, by using $\mathcal{N}={\mathcal{N}}_{2r+1}$ and $\mathcal{M}={\mathcal{N}}_{2r+2}$ in inequality (13), we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \lambda +\mathfrak{I}\left(H_d({\mathcal{N}}_{2r+2},{\mathcal{N}}_{2r+3})\right)& =\lambda +\mathfrak{I}\left(H_d(\Theta {\mathcal{N}}_{2r+1},\Omega {\mathcal{N}}_{2r+2})\right)\hfill \\ & \le \mathfrak{I}\left({\mathbb{P}}_{\Theta ,\Omega }^{H_d}({\mathcal{N}}_{2r+1},{\mathcal{N}}_{2r+2})\right),\hfill \end{array}\end{array}$$

(33)

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathbb{P}}_{\Theta ,\Omega }^{H_d}({\mathcal{N}}_{2r+1},{\mathcal{N}}_{2r+2})=& max\left\{H_d({\mathcal{N}}_{2r+1},{\mathcal{N}}_{2r+2}),H_d({\mathcal{N}}_{2r+1},\Theta {\mathcal{N}}_{2r+1}),H_d({\mathcal{N}}_{2r+2},\Omega {\mathcal{N}}_{2r+2}),\right.\hfill \\ & \left.{\displaystyle \frac{H_d({\mathcal{N}}_{2r+1},\Omega {\mathcal{N}}_{2r+2})+H_d({\mathcal{N}}_{2r+2},\Theta {\mathcal{N}}_{2r+1})}{2}}\right\}\hfill \\ \hfill =& max\left\{H_d({\mathcal{N}}_{2r+1},{\mathcal{N}}_{2r+2}),H_d({\mathcal{N}}_{2r+2},{\mathcal{N}}_{2r+3}),{\displaystyle \frac{H_d({\mathcal{N}}_{2r+1},{\mathcal{N}}_{2r+3})}{2}}\right\}\hfill \\ \hfill \le & max\left\{H_d({\mathcal{N}}_{2r+1},{\mathcal{N}}_{2r+2}),H_d({\mathcal{N}}_{2r+2},{\mathcal{N}}_{2r+3}),\right.\hfill \\ & \left.{\displaystyle \frac{H_d({\mathcal{N}}_{2r+1},{\mathcal{N}}_{2r+2})+H_d({\mathcal{N}}_{2r+2},{\mathcal{N}}_{2r+3})}{2}}\right\}\hfill \\ \hfill =& max\left\{H_d({\mathcal{N}}_{2r+1},{\mathcal{N}}_{2r+2}),H_d({\mathcal{N}}_{2r+2},{\mathcal{N}}_{2r+3})\right\}.\hfill \end{array}\end{array}$$

Thus,

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \lambda +\mathfrak{I}\left(H_d({\mathcal{N}}_{2r+2},{\mathcal{N}}_{2r+3})\right)\le \mathfrak{I}\left(H_d({\mathcal{N}}_{2r+1},{\mathcal{N}}_{2r+2})\right).\end{array}\end{array}$$

In general, for all $r\in {\mathbb{N}}_0$, we have

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \lambda +\mathfrak{I}\left(H_d({\mathcal{N}}_{r+1},{\mathcal{N}}_{r+2})\right)\le \mathfrak{I}\left(H_d({\mathcal{N}}_r,{\mathcal{N}}_{r+1})\right).\end{array}\end{array}$$

(34)

Inequality (34) implies that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathfrak{I}\left(H_d({\mathcal{N}}_r,{\mathcal{N}}_{r+1})\right)& \le \mathfrak{I}\left(H_d({\mathcal{N}}_{r-1},{\mathcal{N}}_r)\right)-\lambda \hfill \\ & \le \mathfrak{I}\left(H_d({\mathcal{N}}_{r-2},{\mathcal{N}}_{r-1})\right)-2\lambda \hfill \\ & ⋮\hfill \\ & \le \mathfrak{I}\left(H_d({\mathcal{N}}_0,{\mathcal{N}}_1)\right)-r\lambda ,\hfill \end{array}\end{array}$$

(35)

for all $r\in \mathbb{N}$; thus,

$$\begin{array}{c}\hfill \underset{r\to \infty }{lim}\mathfrak{I}\left(H_d({\mathcal{N}}_r,{\mathcal{N}}_{r+1})\right)=-\infty .\end{array}$$

(36)

By using $\left(\mathfrak{I}{2}^{\prime }\right)$ and (36), we obtain that $H_d({\mathcal{N}}_r,{\mathcal{N}}_{r+1})\to 0$ as $r\to \infty$. Now, from $\left(\mathfrak{I}3\right)$, there exists $j\in (0,1)$ such that

$$\begin{array}{c}\hfill \underset{r\to \infty }{lim}{\left[H_d({\mathcal{N}}_r,{\mathcal{N}}_{r+1})\right]}^j\mathfrak{I}\left(H_d({\mathcal{N}}_r,{\mathcal{N}}_{r+1})\right).\end{array}$$

Thus, by using (35), the following holds for all $r\in \mathbb{N}$,

$$\begin{array}{c}\hfill \begin{array}{cc}& {\left[H_d({\mathcal{N}}_r,{\mathcal{N}}_{r+1})\right]}^j\mathfrak{I}\left(H_d({\mathcal{N}}_r,{\mathcal{N}}_{r+1})\right)-{\left[H_d({\mathcal{N}}_r,{\mathcal{N}}_{r+1})\right]}^j\mathfrak{I}\left(H_d({\mathcal{N}}_0,{\mathcal{N}}_1)\right)\hfill \\ \hfill \le & {\left[H_d({\mathcal{N}}_r,{\mathcal{N}}_{r+1})\right]}^j(\mathfrak{I}\left(H_d({\mathcal{N}}_0,{\mathcal{N}}_1)\right)-r\lambda )-{\left[H_d({\mathcal{N}}_r,{\mathcal{N}}_{r+1})\right]}^j\mathfrak{I}\left(H_d({\mathcal{N}}_0,{\mathcal{N}}_1)\right)\hfill \\ \hfill =& -r\lambda {\left[H_d({\mathcal{N}}_r,{\mathcal{N}}_{r+1})\right]}^j\le 0\hfill \end{array}\end{array}$$

(37)

On setting limit as $r\to \infty$ in (37), we obtain

$$\begin{array}{c}\hfill \underset{r\to \infty }{lim}r{\left[H_d({\mathcal{N}}_r,{\mathcal{N}}_{r+1})\right]}^j=0\end{array}$$

and hence

$$\begin{array}{c}\hfill \underset{r\to \infty }{lim}{\left(r\right)}^{{\displaystyle \frac{1}{j}}}{H}_d({\mathcal{N}}_r,{\mathcal{N}}_{r+1})=0.\end{array}$$

(38)

Equation (38) guarantees that the series ${\displaystyle \sum _{r=1}^{\infty }{H}_d({\mathcal{N}}_r,{\mathcal{N}}_{r+1})}$ is convergent. This implies that $\left\{{\mathcal{N}}_r\right\}$ is a Cauchy sequence in ${\mathcal{C}}^d\left(\mathbb{U}\right)$. Completeness of $({\mathcal{C}}^d\left(\mathbb{U}\right),H_d)$ ensures the existence of $\mathcal{A}\in {\mathcal{C}}^d\left(\mathbb{U}\right)$ such that

$$\begin{array}{c}\hfill \underset{r\to \infty }{lim}{\mathcal{N}}_r=\mathcal{A},\end{array}$$

(39)

which implies

$$\begin{array}{c}\hfill \underset{r\to \infty }{lim}{H}_d({\mathcal{N}}_r,\mathcal{A})=\underset{r\to \infty }{lim}{H}_d({\mathcal{N}}_r,{\mathcal{N}}_{r+1})=H_d(\mathcal{A},\mathcal{A}),\end{array}$$

so we have

$$\begin{array}{c}\hfill \underset{r\to \infty }{lim}{H}_d({\mathcal{N}}_r,\mathcal{A})=0.\end{array}$$

(40)

Next, assume that $\mathcal{A}\ne \Theta \mathcal{A}$, and then we can assume that $\Theta {\mathcal{N}}_r\ne \Theta \mathcal{A}$ for all $r\in {\mathbb{N}}_0$. Now, from (15), we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill H_d(\Theta \mathcal{A},\mathcal{A})& \le H_d(\Theta \mathcal{A},{\mathcal{N}}_{2r+2})+H_d({\mathcal{N}}_{2r+2},\mathcal{A})\hfill \\ & =H_d(\Theta \mathcal{A},\Omega {\mathcal{N}}_{2r+1})+H_d({\mathcal{N}}_{2r+2},\mathcal{A})\hfill \\ & <{\mathbb{P}}_{\Theta ,\Omega }^{H_d}(\mathcal{A},{\mathcal{N}}_{2r+1})+H_d({\mathcal{N}}_{2r+2},\mathcal{A}),\hfill \end{array}\end{array}$$

(41)

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathbb{P}}_{\Theta ,\Omega }^{H_d}(\mathcal{A},{\mathcal{N}}_{2r+1})=& max\{H_d(\mathcal{A},{\mathcal{N}}_{2r+1}),H_d(\mathcal{A},\Theta \mathcal{A}),H_d({\mathcal{N}}_{2r+1},\Omega {\mathcal{N}}_{2r+1})\hfill \\ & {\displaystyle \frac{H_d(\mathcal{A},\Omega {\mathcal{N}}_{2r+1})+H_d({\mathcal{N}}_{2r+1},\Theta \mathcal{A})}{2}}\}\hfill \\ \hfill =& max\{H_d(\mathcal{A},{\mathcal{N}}_{2r+1}),H_d(\mathcal{A},\Theta \mathcal{A}),H_d({\mathcal{N}}_{2r+1},{\mathcal{N}}_{2r+2})\hfill \\ & {\displaystyle \frac{H_d(\mathcal{A},{\mathcal{N}}_{2r+2})+H_d({\mathcal{N}}_{2r+1},\Theta \mathcal{A})}{2}}\}.\hfill \end{array}\end{array}$$

(42)

By setting limit as $r\to \infty$ in (41) and combining with (42), we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill H_d(\Theta \mathcal{A},\mathcal{A})& <max\left\{H_d(\mathcal{A},\Theta \mathcal{A}),{\displaystyle \frac{H_d(\mathcal{A},\Theta \mathcal{A})}{2}}\right\}\hfill \\ & =H_d(\mathcal{A},\Theta \mathcal{A}),\hfill \end{array}\end{array}$$

(43)

a contradiction; hence, $\mathcal{A}=\Theta \mathcal{A}$.

Similarly, assume that $\mathcal{A}\ne \Omega \mathcal{A}$, and then we can assume that $\Omega {\mathcal{N}}_r\ne \Omega \mathcal{A}$ for all $r\in {\mathbb{N}}_0$. Now, from (15), we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill H_d(\mathcal{A},\Omega \mathcal{A})& \le H_d(\mathcal{A},{\mathcal{N}}_{2r+1})+H_d({\mathcal{N}}_{2r+1},\Omega \mathcal{A})\hfill \\ & =H_d(\mathcal{A},{\mathcal{N}}_{2r+1})+H_d(\Theta {\mathcal{N}}_{2r},\Omega \mathcal{A})\hfill \\ & <H_d(\mathcal{A},{\mathcal{N}}_{2r+1})+{\mathbb{P}}_{\Theta ,\Omega }^{H_d}({\mathcal{N}}_{2r},\mathcal{A}),\hfill \end{array}\end{array}$$

(44)

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathbb{P}}_{\Theta ,\Omega }^{H_d}({\mathcal{N}}_{2r},\mathcal{A})=& max\{H_d({\mathcal{N}}_{2r},\mathcal{A}),H_d({\mathcal{N}}_{2r},\Theta {\mathcal{N}}_{2r}),H_d(\mathcal{A},\Omega \mathcal{A})\hfill \\ & {\displaystyle \frac{H_d({\mathcal{N}}_{2r},\Omega \mathcal{A})+H_d(\mathcal{A},\Theta {\mathcal{N}}_{2r})}{2}}\}\hfill \\ \hfill =& max\{H_d({\mathcal{N}}_{2r},\mathcal{A}),H_d({\mathcal{N}}_{2r},{\mathcal{N}}_{2r+1}),H_d(\mathcal{A},\Omega \mathcal{A})\hfill \\ & {\displaystyle \frac{H_d({\mathcal{N}}_{2r},\Omega \mathcal{A})+H_d(\mathcal{A},{\mathcal{N}}_{2r+1})}{2}}\}.\hfill \end{array}\end{array}$$

(45)

By setting limit as $r\to \infty$ in (44) and combining with (45), we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill H_d(\mathcal{A},\Omega \mathcal{A})& <max\left\{H_d(\mathcal{A},\Omega \mathcal{A}),{\displaystyle \frac{H_d(\mathcal{A},\Omega \mathcal{A})}{2}}\right\}\hfill \\ & =H_d(\mathcal{A},\Omega \mathcal{A}),\hfill \end{array}\end{array}$$

(46)

a contradiction; hence, $\mathcal{A}=\Omega \mathcal{A}$. Thus, $\mathcal{A}$ is a common attractor of $\Theta$ and $\Omega$.

Next, we prove the uniqueness of the common attractor. Let $\mathcal{B}$ be another common attractor of $\Theta$ and $\Omega$ such that $\mathcal{A}\ne \mathcal{B}$. Then, $H_d(\mathcal{A},\mathcal{B})>0$, so (13) yields

$$\begin{array}{c}\hfill \lambda +\mathfrak{I}(H_d(\mathcal{A},\mathcal{B})=\lambda +\mathfrak{I}(H_d(\Theta \mathcal{A},\Omega \mathcal{B})\le \mathfrak{I}\left({\mathbb{P}}_{\Theta ,\Omega }^{H_d}(\mathcal{A},\mathcal{B})\right),\end{array}$$

(47)

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathbb{P}}_{\Theta ,\Omega }^{H_d}(\mathcal{A},\mathcal{B})=& max\left\{H_d(\mathcal{A},\mathcal{B}),H_d(\mathcal{A},\Theta \mathcal{A}),H_d(\mathcal{B},\Omega B),{\displaystyle \frac{H_d(\mathcal{A},\Omega \mathcal{B})+H_d(\mathcal{B},\Theta \mathcal{A})}{2}}\right\}\hfill \\ \hfill =& H_d(\mathcal{A},\mathcal{B}).\hfill \end{array}\end{array}$$

Inequality (47) yields a contradiction because $\lambda >0$. Hence, $\mathcal{A}=B$. □

4. Consequences

By considering $\alpha =1$, $\beta =\gamma =\delta =L=0$ in (12) and then by using Theorem 1, we obtain the following existence result for a pair of Hutchinson operators.

Corollary 1.

Let $(\mathbb{U},d)$ be a metric space, $\{\mathbb{U};(g_{\kappa },{\mathcal{\ell }}_{\kappa }),\kappa =1,2,3,\cdots ,\epsilon \}$ be an ℑ-IFS, and $\Theta ,\Omega :{\mathcal{C}}^d\left(\mathbb{U}\right)\to {\mathcal{C}}^d\left(\mathbb{U}\right)$ be mappings as defined in (3). If there exist $\mathfrak{I}\in \nabla \left(\mathfrak{I}\right)$ and $\lambda >0$ such that, for all $\mathcal{N},\mathcal{M}\in {\mathcal{C}}^d\left(\mathbb{U}\right)$, the pair $(\Theta ,\Omega )$ satisfies

$$\begin{array}{c}\hfill H_d(\Theta \left(\mathcal{N}\right),\Omega \left(\mathcal{M}\right))>0,implying\lambda +\mathfrak{I}\left(H_d(\Theta \left(\mathcal{N}\right),\Omega \left(\mathcal{M}\right))\right)\le \mathfrak{I}\left(H_d(\mathcal{N},\mathcal{M})\right),\end{array}$$

(48)

then Θ and Ω have a unique common attractor $\mathcal{A}\in {\mathcal{C}}^d\left(\mathbb{U}\right)$. Moreover, for an arbitrarily chosen initial set ${\mathcal{N}}_0\in {\mathcal{C}}^d\left(\mathbb{U}\right)$, the sequence $\{{\mathcal{N}}_0,\Theta \left({\mathcal{N}}_0\right),\Omega \Theta \left({\mathcal{N}}_0\right),\Theta \Omega \Theta \left({\mathcal{N}}_0\right),\cdots \}$ of compact sets converges to the common attractor $\mathcal{A}$ of Θ and Ω.

Further, setting $\alpha =\delta =L=0$, $\beta +\gamma =1$ and $\beta \ne 0$ in Theorem 1, we obtain the following existence result of common attractors of Kannan-type Hutchinson–Wardowski contractive operator.

Corollary 2.

Let $(\mathbb{U},d)$ be a metric space, $\{Y;(g_{\kappa },{\mathcal{\ell }}_{\kappa }),\kappa =1,2,3,\cdots ,\epsilon \}$ be an ℑ-IFS, and $\Theta ,\Omega :{\mathcal{C}}^d\left(\mathbb{U}\right)\to {\mathcal{C}}^d\left(\mathbb{U}\right)$ be mappings as defined in (3). If there exist $\mathfrak{I}\in \nabla \left(\mathfrak{I}\right)$ and $\lambda >0$, for all $\mathcal{N},\mathcal{M}\in {\mathcal{C}}^d\left(\mathbb{U}\right)$, the pair $(\Theta ,\Omega )$ satisfies

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill H_d(\Theta \left(\mathcal{N}\right),\Omega \left(\mathcal{M}\right))& >0,implying\hfill \\ & \lambda +\mathfrak{I}\left(H_d(\Theta \left(\mathcal{N}\right),\Omega \left(\mathcal{M}\right))\right)\le \mathfrak{I}(\beta H_d(\mathcal{N},\Theta \mathcal{N})+\gamma H_d(\mathcal{M},\Omega \mathcal{M})),\hfill \end{array}\end{array}$$

(49)

where $\beta +\gamma =1$ and $\beta \ne 0$. Then, Θ and Ω have a unique common attractor $\mathcal{A}$ in ${\mathcal{C}}^d\left(\mathbb{U}\right)$. Moreover, for an arbitrarily chosen initial set ${\mathcal{N}}_0\in {\mathcal{C}}^d\left(\mathbb{U}\right)$, the sequence $\{{\mathcal{N}}_0,\Theta \left({\mathcal{N}}_0\right),\Omega \Theta \left({\mathcal{N}}_0\right),\Theta \Omega \Theta \left({\mathcal{N}}_0\right),\cdots \}$ of compact sets converges to the common attractor $\mathcal{A}$ of Θ and Ω.

Next, if we choose $\alpha =\beta =\gamma =0$ and $\delta ={\displaystyle \frac{1}{2}}$ in Theorem 1, we obtain the following result for Chatterjea-type Hutchinson–Wardowski contractive operator.

Corollary 3.

Let $(\mathbb{U},d)$ be a metric space, $\{Y;(g_{\kappa },{\mathcal{\ell }}_{\kappa }),\kappa =1,2,3,\cdots ,\epsilon \}$ be an ℑ-IFS, and $\Theta ,\Omega :{\mathcal{C}}^d\left(\mathbb{U}\right)\to {\mathcal{C}}^d\left(\mathbb{U}\right)$ be mappings as defined in (3). If there exist $\mathfrak{I}\in \nabla \left(\mathfrak{I}\right)$ and $\lambda >0$ such that, for all $\mathcal{N},\mathcal{M}\in {\mathcal{C}}^d\left(\mathbb{U}\right)$, the pair $(\Theta ,\Omega )$ satisfies

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill H_d(\Theta \left(\mathcal{N}\right),& \Omega \left(\mathcal{M}\right))>0,implying\hfill \\ & \lambda +\mathfrak{I}\left(H_d(\Theta \left(\mathcal{N}\right),\Omega \left(\mathcal{M}\right))\right)\le \mathfrak{I}\left({\displaystyle \frac{1}{2}}{H}_d(\mathcal{N},\Omega \mathcal{M})+L{H}_d(\mathcal{M},\Theta \mathcal{N})\right).\hfill \end{array}\end{array}$$

(50)

Then, Θ and Ω have a common attractor $\mathcal{A}\in {\mathcal{C}}^d\left(\mathbb{U}\right)$. If $L\le {\displaystyle \frac{1}{2}}$, then common attractor of Θ and Ω is unique. Moreover, for an arbitrarily chosen initial set ${\mathcal{N}}_0\in {\mathcal{C}}^d\left(\mathbb{U}\right)$, the sequence $\{{\mathcal{N}}_0,\Theta \left({\mathcal{N}}_0\right),\Omega \Theta \left({\mathcal{N}}_0\right),\Theta \Omega \Theta \left({\mathcal{N}}_0\right),\cdots \}$ of compact sets converges to the common attractor $\mathcal{A}$ of Θ and Ω.

Finally, if we set $\delta =L=0$ in Theorem 1, we obtain the following existence of common attractors of Reich-type Hutchinson–Wardowski contractive operator.

Corollary 4.

Let $(\mathbb{U},d)$ be a metric space, $\{\mathbb{U};(g_{\kappa },{\mathcal{\ell }}_{\kappa }),\kappa =1,2,3,\cdots ,\epsilon \}$ be an ℑ-IFS, and $\Theta ,\Omega :{\mathcal{C}}^d\left(\mathbb{U}\right)\to {\mathcal{C}}^d\left(\mathbb{U}\right)$ be mappings as defined in (3). If there exist $\mathfrak{I}\in \nabla \left(\mathfrak{I}\right)$ and $\lambda >0$, for all $\mathcal{N},\mathcal{M}\in {\mathcal{C}}^d\left(\mathbb{U}\right)$, the pair $(\Theta ,\Omega )$ satisfies

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill H_d(& \Theta \left(\mathcal{N}\right),\Omega \left(\mathcal{M}\right))>0,implying\hfill \\ & \lambda +\mathfrak{I}\left(H_d(\Theta \left(\mathcal{N}\right),\Omega \left(\mathcal{M}\right))\right)\le \mathfrak{I}(\alpha H_d(\mathcal{N},\mathcal{M})+\beta H_d(\mathcal{N},\Theta \mathcal{N})+\gamma H_d(\mathcal{M},\Omega \mathcal{M})),\hfill \end{array}\end{array}$$

(51)

where $\alpha \beta +\gamma =1$ and $\gamma \ne 1$. Then, Θ and Ω have a unique common attractor $\mathcal{A}\in {\mathcal{C}}^d\left(\mathbb{U}\right)$. Moreover, for an arbitrarily chosen initial set ${\mathcal{N}}_0\in {\mathcal{C}}^d\left(\mathbb{U}\right)$, the sequence

$\{{\mathcal{N}}_0,\Theta \left({\mathcal{N}}_0\right),\Omega \Theta \left({\mathcal{N}}_0\right),\Theta \Omega \Theta \left({\mathcal{N}}_0\right),\cdots \}$ of compact sets converges to the common attractor $\mathcal{A}$ of Θ and Ω.

Now, if we consider in Theorems 1 and 2 ${\mathcal{S}}^d\left(\mathbb{U}\right)$, the collection of all singleton subsets of the space $\mathbb{U}$, then ${\mathcal{S}}^d\left(\mathbb{U}\right)\subseteq {\mathcal{C}}^d\left(\mathbb{U}\right)$. Furthermore, if we take a pair of mappings $(g_{\kappa },{\mathcal{\ell }}_{\kappa })=(g,\mathcal{\ell })$ for each $\kappa$, where $g=g_1$ and $\mathcal{\ell }={\mathcal{\ell }}_1$, then the pair of operators $(\Theta ,\Omega )$ becomes $(\Theta \left(a_1\right),\Omega \left(a_2\right))=(g\left(a_1\right),\mathcal{\ell }\left(a_2\right))$. As a result, the subsequent common fixed point results are attained, respectively.

Corollary 5.

Let $(\mathbb{U},d)$ be a metric space, $\{\mathbb{U};(g_{\kappa },{\mathcal{\ell }}_{\kappa }),\kappa =1,2,3,\cdots ,\epsilon \}$ be an ℑ-IFS, and $g,\mathcal{\ell }:\mathbb{U}\to \mathbb{U}$ be mappings defined as $(g_{\kappa },{\mathcal{\ell }}_{\kappa })=(g,\mathcal{\ell })$ for each κ, where $g=g_1$ and $\mathcal{\ell }={\mathcal{\ell }}_1$. If there exist $\mathfrak{I}\in \nabla \left(\mathfrak{I}\right)$ and $\lambda >0$, for all $a_1,a_1\in \mathbb{U}$, the pair $(g,\mathcal{\ell })$ satisfies

$$\begin{array}{c}\hfill d(g\left(a_1\right),\mathcal{\ell }\left(a_2\right))>0,implying\lambda +\mathfrak{I}\left(d(g\left(a_1\right),\mathcal{\ell }\left(a_2\right))\right)\le \mathfrak{I}\left({\mathbb{Q}}_{g,\mathcal{\ell }}^d(a_1,a_2)\right),\end{array}$$

(52)

where

$$\begin{array}{c}\hfill {\mathbb{Q}}_{g,\mathcal{\ell }}^d(a_1,a_2)=\alpha d(a_1,a_2)+\beta d(a_1,g{a}_1)+\gamma d(a_2,\mathcal{\ell }{a}_2)+\delta d(a_1,\mathcal{\ell }{a}_2)+Ld(a_2,g{a}_1)\end{array}$$

with $\alpha ,\beta ,\gamma ,\delta ,L\ge 0$, $\alpha +\beta +\gamma +2\delta =1$ and $\gamma \ne 1$. Then, $g$ and $\mathcal{\ell }$ have a common fixed point $u\in \mathbb{U}$. Moreover, if $\alpha +\delta +L\le 1$, then the common fixed point is unique. Furthermore, for an arbitrarily chosen initial set $u_0\in \mathbb{U}$, the sequence $\{u_0,g\left(u_0\right),\mathcal{\ell }g\left(u_0\right),g\mathcal{\ell }g\left(u_0\right),\cdots \}$ converges to the common fixed point u of $g$ and $\mathcal{\ell }$.

Corollary 6.

Let $(\mathbb{U},d)$ be a metric space, $\{\mathbb{U};(g_{\kappa },{\mathcal{\ell }}_{\kappa }),\kappa =1,2,3,\cdots ,\epsilon \}$ be an ℑ-IFS, and $g,\mathcal{\ell }:\mathbb{U}\to \mathbb{U}$ be mappings defined as $(g_{\kappa },{\mathcal{\ell }}_{\kappa })=(g,\mathcal{\ell })$ for each κ, where $g=g_1$ and $\mathcal{\ell }={\mathcal{\ell }}_1$. If there exist $\mathfrak{I}\in \nabla \left(\mathfrak{I}\right)$ and $\lambda >0$, for all $a_1,a_1\in \mathbb{U}$, the pair $(g,\mathcal{\ell })$ satisfies

$$\begin{array}{c}\hfill d(g\left(a_1\right),\mathcal{\ell }\left(a_2\right))>0,implying\lambda +\mathfrak{I}\left(d(g\left(a_1\right),\mathcal{\ell }\left(a_2\right))\right)\le \mathfrak{I}\left({\mathbb{P}}_{g,\mathcal{\ell }}^d(a_1,a_2)\right),\end{array}$$

(53)

where

$$\begin{array}{c}\hfill {\mathbb{P}}_{g,\mathcal{\ell }}^d(a_1,a_2)=max\left\{d(a_1,a_2),d(a_1,g{a}_1),d(a_2,\mathcal{\ell }{a}_2),{\displaystyle \frac{d(a_1,\mathcal{\ell }{a}_2)+H_d(a_2,g{a}_1)}{2}}\right\}.\end{array}$$

Then, $g$ and $\mathcal{\ell }$ have a unique common fixed point $u\in \mathbb{U}$. Moreover, for an arbitrarily chosen initial set $u_0\in \mathbb{U}$, the sequence $\{u_0,g\left(u_0\right),\mathcal{\ell }g\left(u_0\right),g\mathcal{\ell }g\left(u_0\right),\cdots \}$ converges to the common fixed point u of $g$ and $\mathcal{\ell }$.

By considering $g=\mathcal{\ell }$ and $\Theta =\Omega$ in Theorem 2, we return to Theorem 2.1 of [10].

Corollary 7

([10]). Let $(\mathbb{U},d)$ be a metric space, $\{\mathbb{U};g_{\kappa }),\kappa =1,2,3,\cdots ,\epsilon \}$ be an ℑ-IFS, and $\Theta :{\mathcal{C}}^d\left(\mathbb{U}\right)\to {\mathcal{C}}^d\left(\mathbb{U}\right)$ be mappings as defined in (3). If there exist $\mathfrak{I}\in \nabla \left(\mathfrak{I}\right)$ and $\lambda >0$, for all $\mathcal{N},\mathcal{M}\in {\mathcal{C}}^d\left(\mathbb{U}\right)$ with $H_d(\Theta \left(\mathcal{N}\right),\Theta \left(\mathcal{M}\right))>0$, the following holds

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \lambda +\mathfrak{I}\left(H_d(\Theta \left(\mathcal{N}\right),\Theta \left(\mathcal{M}\right))\right)\le & \mathfrak{I}(max\{H_d(\mathcal{N},\mathcal{M}),H_d(\mathcal{N},\Theta \mathcal{N}),H_d(\mathcal{M},\Theta \mathcal{M}),\hfill \\ & {\displaystyle \frac{H_d(\mathcal{N},\Theta \mathcal{M})+H_d(\mathcal{M},\Theta \mathcal{N})}{2}}\}).\hfill \end{array}\end{array}$$

(54)

Then, Θ has a unique attractor $\mathcal{A}\in {\mathcal{C}}^d\left(\mathbb{U}\right)$. Furthermore, for an arbitrarily chosen initial set ${\mathcal{N}}_0\in {\mathcal{C}}^d\left(\mathbb{U}\right)$, the sequence $\{{\mathcal{N}}_0,\Theta \left({\mathcal{N}}_0\right),{\Theta }^2\left({\mathcal{N}}_0\right),{\Theta }^3\left({\mathcal{N}}_0\right),\cdots \}$ of compact sets converges to the attractor $\mathcal{A}$ of Θ.

By defining $\mathfrak{I}\left(t\right)=ln\left(t\right)$ for all $t\in (0,\infty )$ in Theorem 2, we obtain the following:

Corollary 8.

Let $(\mathbb{U},d)$ be a metric space, $\{\mathbb{U};(g_{\kappa },{\mathcal{\ell }}_{\kappa }),\kappa =1,2,3,\cdots ,\epsilon \}$ be an ℑ-IFS, and $\Theta ,\Omega :{\mathcal{C}}^d\left(\mathbb{U}\right)\to {\mathcal{C}}^d\left(\mathbb{U}\right)$ be mappings as defined in (3). If there exist $\kappa \in (0,1)$, for all $\mathcal{N},\mathcal{M}\in {\mathcal{C}}^d\left(\mathbb{U}\right)$ with $H_d(\Theta \left(\mathcal{N}\right),\Theta \left(\mathcal{M}\right))>0$, the following holds

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathfrak{I}\left(H_d(\Theta \left(\mathcal{N}\right),\Omega \left(\mathcal{M}\right))\right)\le & \kappa (max\{H_d(\mathcal{N},\mathcal{M}),H_d(\mathcal{N},\Theta \mathcal{N}),H_d(\mathcal{M},\Omega \mathcal{M}),\hfill \\ & {\displaystyle \frac{H_d(\mathcal{N},\Omega \mathcal{M})+H_d(\mathcal{M},\Theta \mathcal{N})}{2}}\}).\hfill \end{array}\end{array}$$

(55)

Then, Θ and Ω share at most one common attractor $\mathcal{A}\in {\mathcal{C}}^d\left(\mathbb{U}\right)$. Moreover, for any initial compact set ${\mathcal{N}}_0$, the sequence $\{{\mathcal{N}}_0,\Theta \left({\mathcal{N}}_0\right),\Omega \Theta \left({\mathcal{N}}_0\right),\Theta \Omega \Theta \left({\mathcal{N}}_0\right),\cdots \}$ of compact sets generated by Θ and Ω will converge to the common attractor $\mathcal{A}$.

With the aid of Lemma 4, Theorems 1 and 2 provide the following corollary:

Corollary 9.

Let $(\mathbb{U},d)$ be a metric space and $\{\mathbb{U};(g_{\kappa },{\mathcal{\ell }}_{\kappa }),\kappa =1,2,3,\cdots ,\epsilon \}$ be an ℑ-IFS. If there exist ${\mathfrak{I}}_1,{\mathfrak{I}}_2,\cdots ,{\mathfrak{I}}_{\epsilon }\in \nabla \left(\mathfrak{I}\right)$ and ${\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_{\epsilon }>0$ such that the pair $(g_{\kappa },{\mathcal{\ell }}_{\kappa })$ satisfy (9) and the mapping $G_{\kappa }:=\mathfrak{I}-{\mathfrak{I}}_{\kappa }$ is nondecreasing for each $\kappa =1,2,3,\cdots ,\epsilon$, then the mappings $\Theta ,\Omega :{\mathcal{C}}^d\left(\mathbb{U}\right)\to {\mathcal{C}}^d\left(\mathbb{U}\right)$ defined in (3) have a unique common attractor $\mathcal{A}\in {\mathcal{C}}^d\left(\mathbb{U}\right)$. Moreover, for an arbitrarily chosen initial set ${\mathcal{N}}_0\in {\mathcal{C}}^d\left(\mathbb{U}\right)$, the sequence $\{{\mathcal{N}}_0,\Theta \left({\mathcal{N}}_0\right),\Omega \Theta \left({\mathcal{N}}_0\right),\Theta \Omega \Theta \left({\mathcal{N}}_0\right),\cdots \}$ of compact sets converges to the common attractor $\mathcal{A}$ of Θ and Ω.

Next, we provide a supporting example of Corollary 9.

Example 1.

Let $\mathbb{U}=[0,\infty )$ be endowed with the Euclidian metric $d(\kappa ,\xi )=|\kappa -\xi |$. Define $g_{\kappa },{\mathcal{\ell }}_{\kappa }:\mathbb{U}\to \mathbb{U}$, $\mathfrak{I}:(0,\infty )\to \mathbb{R}$ and ${\mathfrak{I}}_{\kappa }:(0,\infty )\to \mathbb{R}$, $\kappa =1,2$ as

$$\begin{array}{c}\hfill g_1\left(\kappa \right)={\displaystyle \frac{\kappa }{2}},g_2\left(\kappa \right)={\displaystyle \frac{\kappa }{4}}forall\kappa \in \mathbb{U},\end{array}$$

$$\begin{array}{c}\hfill {\mathcal{\ell }}_1\left(\kappa \right)={\displaystyle \frac{\kappa }{2}}+{\displaystyle \frac{1}{4}},{\mathcal{\ell }}_2\left(\kappa \right)={\displaystyle \frac{\kappa +1}{4}}forall\kappa \in \mathbb{U},\end{array}$$

and

$$\begin{array}{c}\hfill \mathfrak{I}\left(a\right)=lna+\eta a{\mathfrak{I}}_{\kappa }\left(a\right)=lna+{\eta }_{\kappa }aforalla\in (0,\infty ),\end{array}$$

where $\eta ,{\eta }_{\kappa }\in (0,\infty )$ for all $\kappa =1,2$. Then, by Proposition 2, $\mathfrak{I},{\mathfrak{I}}_{\kappa }\in \nabla \left(\mathfrak{I}\right)$ and $G_{\kappa }:=\mathfrak{I}-{\mathfrak{I}}_{\kappa }$ are nondecreasing for each κ. Now, we will prove that there exist ${\lambda }_1,{\lambda }_2>0$ such that the pair $(g_{\kappa },{\mathcal{\ell }}_{\kappa });\kappa =1,2$ satisfy (9), which is equivalent to

$$\begin{array}{c}\hfill {\displaystyle \frac{|g_{\kappa }\kappa -{\mathcal{\ell }}_{\kappa }\xi |}{|\kappa -\xi |}}{e}^{{\eta }_{\kappa }\left(\right|g_{\kappa }\kappa -{\mathcal{\ell }}_{\kappa }\xi |-|\kappa -\xi \left|\right)}\le e^{-\lambda };\kappa =1,2.\end{array}$$

(56)

Let $\kappa ,\xi \in X$ such that $d(g_1,{\mathcal{\ell }}_1)>0$ and $\kappa \ne \xi$. Suppose that $\xi <\kappa$; then,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\displaystyle \frac{|g_1\kappa -{\mathcal{\ell }}_1\xi |}{|\kappa -\xi |}}{e}^{{\eta }_1\left(\right|g_1\kappa -{\mathcal{\ell }}_1\xi |-|\kappa -\xi \left|\right)}& ={\displaystyle \frac{\left|\frac{1}{2}(\kappa -\xi )-\frac{1}{4}\right|}{|\kappa -\xi |}}{e}^{{\eta }_1\left(\left|{\displaystyle \frac{1}{2}}(\kappa -\xi )-{\displaystyle \frac{1}{4}}\right|-|\kappa -\xi |\right)}\hfill \\ & \le {\displaystyle \frac{\left|\frac{1}{2}(\kappa -\xi )\right|}{|\kappa -\xi |}}{e}^{{\eta }_1\left(\left|{\displaystyle \frac{1}{2}}(\kappa -\xi )-{\displaystyle \frac{1}{4}}\right|-|\kappa -\xi |\right)}\hfill \\ & ={\displaystyle \frac{1}{2}}{e}^{{\eta }_1(\left|{\displaystyle \frac{1}{2}}(\kappa -\xi )-{\displaystyle \frac{1}{4}}\right|-|\kappa -\xi |)}\hfill \\ & <{\displaystyle \frac{1}{2}}\hfill \\ & \le e^{-0.2}=e^{-{\lambda }_1}.\hfill \end{array}\end{array}$$

Also, for $\kappa ,\xi \in X$ such that $d(g_2,{\mathcal{\ell }}_2)>0$ and $\kappa \ne \xi$, suppose that $\xi <\kappa$; then,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\displaystyle \frac{|g_2\kappa -{\mathcal{\ell }}_2\xi |}{|\kappa -\xi |}}{e}^{{\eta }_2\left(\right|g_2\kappa -{\mathcal{\ell }}_2\xi |-|\kappa -\xi \left|\right)}& ={\displaystyle \frac{\left|\frac{1}{4}(\kappa -\xi )-\frac{1}{4}\right|}{|\kappa -\xi |}}{e}^{{\eta }_2\left(\left|{\displaystyle \frac{1}{4}}(\kappa -\xi )-{\displaystyle \frac{1}{4}}\right|-|\kappa -\xi |\right)}\hfill \\ & \le {\displaystyle \frac{\left|\frac{1}{4}(\kappa -\xi )\right|}{|\kappa -\xi |}}{e}^{{\eta }_2\left(\left|{\displaystyle \frac{1}{4}}(\kappa -\xi )-{\displaystyle \frac{1}{4}}\right|-|\kappa -\xi |\right)}\hfill \\ & ={\displaystyle \frac{1}{4}}{e}^{{\eta }_2(\left|{\displaystyle \frac{1}{4}}(\kappa -\xi )-{\displaystyle \frac{1}{4}}\right|-|\kappa -\xi |)}\hfill \\ & <{\displaystyle \frac{1}{4}}\hfill \\ & \le e^{-0.2}=e^{-{\lambda }_2}.\hfill \end{array}\end{array}$$

Consider the ℑ-IFS $\{\mathbb{U};(g_{\kappa },{\mathcal{\ell }}_{\kappa }),\kappa =1,2\}$ with the mappings $\Theta ,\Omega :{\mathcal{C}}^d\left(\mathbb{U}\right)\to {\mathcal{C}}^d\left(\mathbb{U}\right)$ defined as

$$\begin{array}{c}\hfill \Theta \left(\mathcal{N}\right)=g_1\left(\mathcal{N}\right)\bigcup g_2\left(\mathcal{N}\right)\mathrm{and}\Omega \left(\mathcal{M}\right)={\mathcal{\ell }}_1\left(\mathcal{M}\right)\bigcup {\mathcal{\ell }}_1\left(\mathcal{M}\right)forall\mathcal{N},\mathcal{M}\in {\mathcal{C}}^d\left(\mathbb{U}\right).\end{array}$$

From Lemma 4, for all $\mathcal{N},\mathcal{M}\in {\mathcal{C}}^d\left(\mathbb{U}\right)$ such that $H_d(\Theta \mathcal{N},\Omega \mathcal{M})>0$, we have

$$\begin{array}{c}\hfill \lambda +\mathfrak{I}\left(H_d(\Theta \mathcal{N},\Omega \mathcal{M})\right)\le \mathfrak{I}\left(H_d(\mathcal{N},\mathcal{M})\right)\end{array}$$

for $\mathfrak{I}=max\{{\mathfrak{I}}_1,{\mathfrak{I}}_2\}$ and $\lambda =min\{{\lambda }_1,{\lambda }_2\}=0.2$. Thus, all conditions of Corollary 9 are satisfied. Moreover, for an arbitrarily chosen initial set ${\mathcal{N}}_0\in {\mathcal{C}}^d\left(\mathbb{U}\right)$, the sequence $\{{\mathcal{N}}_0,\Theta \left({\mathcal{N}}_0\right),\Omega \Theta \left({\mathcal{N}}_0\right),\Theta \Omega \Theta \left({\mathcal{N}}_0\right),\cdots \}$ of compact sets is convergent and has a limit point that is the common attractor of Θ and Ω.

5. Application to Fractional Differential Equations

Let $C_J$ be the space of all continuous real-valued functions on J, where $J=[0,1]$. Then, $C_J$ is a complete metric space with respect to metric $d:C_J\times C_J\to [0,\infty )$ defined by

$$\begin{array}{c}\hfill d(\omega ,\upsilon )={\parallel \omega -\upsilon \parallel }_{\infty }=\underset{\mu \in J}{max}|\omega \left(\mu \right)-\upsilon \left(\mu \right)|,\mathrm{for}\mathrm{all}\omega ,\upsilon \in C_J.\end{array}$$

For a continuous function $q:[0,\infty )\to \mathbb{R}$, the Caputo–Fabrizio derivative of order $\sigma$, ${}^{c}D^{\sigma }$, is defined as

$$\begin{array}{c}\hfill {}^{c}D^{\sigma }q\left(\mu \right)={\displaystyle \frac{1}{\mathrm{\Gamma }(n-\sigma )}}{\int }_0^{\mu }{(\mu -s)}^{n-\sigma -1}{q}^{\left(n\right)}\left(s\right)ds,n-1<\sigma <n,n=\left[\sigma \right]+1,\end{array}$$

(57)

where $\left[\sigma \right]$ denotes the integer part of the real number $\sigma$ and the Riemann–Liouville fractional integral of order $\sigma$ is defined as

$$\begin{array}{c}\hfill I^{\sigma }q\left(\mu \right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\sigma \right)}}{\int }_0^{\mu }{(\mathcal{\ell }-s)}^{\sigma -1}q\left(s\right)ds,\sigma >0,\end{array}$$

(58)

provided the integral exists.

In this part, we apply our findings to demonstrate the existence of common solutions to the following Caputo–Fabrizio fractional differential equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^{c}D^{\sigma }q\left(\mu \right)={\mathbb{K}}_1(\mu ,q\left(\mu \right))\hfill \\ q\left(0\right)=0,Iq\left(1\right)=q^{\prime }\left(0\right),\hfill \end{array}\right.\end{array}$$

(59)

and

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^{c}D^{\sigma }\wp \left(\mu \right)={\mathbb{K}}_2(\mu ,\wp \left(\mu \right))\hfill \\ \wp \left(0\right)=0,I\wp \left(1\right)={\wp }^{\prime }\left(0\right),\hfill \end{array}\right.\end{array}$$

(60)

where $\mu \in [0,1]$ and ${\mathbb{K}}_1,{\mathbb{K}}_2:[0,1]\times \mathbb{R}\to \mathbb{R}$.

Lemma 5

([19]). Given $\mu \in [0,1]$, problem (59) is equivalent to the following integral equation:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill q\left(\mu \right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\sigma \right)}}{\int }_0^{\mu }{(\mu -s)}^{\sigma -1}{\mathbb{K}}_1(s,q\left(s\right))ds+{\displaystyle \frac{2\mu }{\mathrm{\Gamma }\left(\sigma \right)}}{\int }_0^1{\int }_0^s{(s-w)}^{\sigma -1}{\mathbb{K}}_1(w,q\left(w\right))dwds.\end{array}\end{array}$$

(61)

Now, define the operators ${\mathcal{L}}_1,{\mathcal{L}}_2:C_J\to C_J$ as follows:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathcal{L}}_1\left(q\left(\mu \right)\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\sigma \right)}}{\int }_0^{\mu }{(\mu -s)}^{\sigma -1}{\mathbb{K}}_1(s,q\left(s\right))ds+{\displaystyle \frac{2\mu }{\mathrm{\Gamma }\left(\sigma \right)}}{\int }_0^1{\int }_0^s{(s-w)}^{\sigma -1}{\mathbb{K}}_1(w,q\left(w\right))dwds\end{array}\end{array}$$

(62)

and

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathcal{L}}_2(\wp \left(\mu \right))={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\sigma \right)}}{\int }_0^{\mu }{(\mu -s)}^{\sigma -1}{\mathbb{K}}_2(s,\wp \left(s\right))ds+{\displaystyle \frac{2\mu }{\mathrm{\Gamma }\left(\sigma \right)}}{\int }_0^1{\int }_0^s{(s-w)}^{\sigma -1}{\mathbb{K}}_2(w,\wp \left(w\right))dwds.\end{array}\end{array}$$

(63)

Note that a common fixed point of operators (62) and (63) is the common solution of (59) and (60).

Theorem 3.

Boundary value problems (59) and (60) have a common solution in $C_J$ given that

(H1)

there exists $\lambda >0$ such that, for all $q,\wp \in C_J$, we have

$$\left|{\mathbb{K}}_1(\mu ,q\left(\mu \right))-{\mathbb{K}}_2(\mu ,\wp \left(\mu \right))\right|\le e^{-\lambda }\mathbb{Q}(q\left(\mu \right),\wp \left(\mu \right)),$$

where $\mathbb{Q}(q\left(\mu \right),\wp \left(\mu \right))=\alpha |q\left(\mu \right)-\wp \left(\mu \right)|+\beta |q\left(\mu \right)-{\mathcal{L}}_{1}q\left(\mu \right))+\gamma |\wp \left(\mu \right)-{\mathcal{L}}_2\wp \left(\mu \right)|+\delta |q\left(\mu \right)-{\mathcal{L}}_2\wp \left(\mu \right)|+L|\wp \left(\mu \right)-{\mathcal{L}}_{1}q\left(\mu \right)|$ with $\alpha ,\beta ,\gamma ,\delta ,L\ge 0$, $\alpha +\beta +\gamma +2\delta =1$ and $\gamma \ne 1$;

(H2)

${\rm Y}<\mathrm{\Gamma }\left(\sigma \right)$, where

$$\begin{array}{c}\hfill {\rm Y}={\int }_0^{\mu }{(\mu -s)}^{\sigma -1}ds+2\mu {\int }_0^1{\int }_0^s{(s-w)}^{\sigma -1}\left(e^{-\lambda }\mathbb{Q}(q\left(w\right),\wp \left(w\right))\right)dwds.\end{array}$$

Proof. 

Let $q,\wp \in C_J$; then, for all $\mu \in [0,1]$, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |{\mathcal{L}}_{1}q\left(\mu \right)-{\mathcal{L}}_2\wp \left(\mu \right)|=& \left|{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\sigma \right)}}{\int }_0^{\mu }{(\mu -s)}^{\sigma -1}[{\mathbb{K}}_1(s,q\left(s\right))-{\mathbb{K}}_2(s,\wp \left(s\right))]ds\right.\hfill \\ & \left.+{\displaystyle \frac{2\mu }{\mathrm{\Gamma }\left(\sigma \right)}}{\int }_0^1{\int }_0^s{(s-w)}^{\sigma -1}[{\mathbb{K}}_1(w,q\left(w\right))-{\mathbb{K}}_2(w,\wp \left(w\right))]dwds\right|\hfill \\ \hfill \le & {\displaystyle \frac{1}{\mathrm{\Gamma }\left(\sigma \right)}}{\int }_0^{\mu }{(\mu -s)}^{\sigma -1}\left|{\mathbb{K}}_1(s,q\left(s\right))-{\mathbb{K}}_2(s,\wp \left(s\right))\right|ds\hfill \\ & +{\displaystyle \frac{2\mu }{\mathrm{\Gamma }\left(\sigma \right)}}{\int }_0^1{\int }_0^s{(s-w)}^{\sigma -1}\left|{\mathbb{K}}_1(w,q\left(w\right))-{\mathbb{K}}_2(w,\wp \left(w\right))\right|dwds\hfill \\ \hfill \le & {\displaystyle \frac{1}{\mathrm{\Gamma }\left(\sigma \right)}}{\int }_0^{\mu }{(\mu -s)}^{\sigma -1}\left(e^{-\lambda }\mathbb{Q}(q\left(s\right),\wp \left(s\right))\right)ds\hfill \\ & +{\displaystyle \frac{2\mu }{\mathrm{\Gamma }\left(\sigma \right)}}{\int }_0^1{\int }_0^s{(s-w)}^{\sigma -1}\left(e^{-\lambda }\mathbb{Q}(q\left(w\right),\wp \left(w\right))\right)dwds\hfill \\ \hfill \le & {\displaystyle \frac{1}{\mathrm{\Gamma }\left(\sigma \right)}}\left(e^{-\lambda }\mathbb{Q}(q\left(s\right),\wp \left(s\right))\right){\int }_0^{\mu }{(\mu -s)}^{\sigma -1}ds\hfill \\ & +{\displaystyle \frac{2\mu }{\mathrm{\Gamma }\left(\sigma \right)}}\left(e^{-\lambda }\mathbb{Q}(q\left(w\right),\wp \left(w\right))\right){\int }_0^1{\int }_0^s{(s-w)}^{\sigma -1}\left(e^{-\lambda }\mathbb{Q}(q\left(w\right),\wp \left(w\right))\right)dwds\hfill \\ \hfill \le & {\displaystyle \frac{{\rm Y}}{\mathrm{\Gamma }\left(\sigma \right)}}\left(e^{-\lambda }\mathbb{Q}(q\left(s\right),\wp \left(s\right))\right)\hfill \\ \hfill \le & e^{-\lambda }\mathbb{Q}(q\left(s\right),\wp \left(s\right)).\hfill \end{array}\end{array}$$

Hence, (52) is satisfied for $\mathfrak{I}\left(t\right)=ln\left(t\right)$ for all $t\in (0,\infty )$. Thus, with the aid of Corollary (5), operators ${\mathcal{L}}_1$ and ${\mathcal{L}}_2$ admit a common fixed point, and therefore boundary value problems (59) and (60) have a common solution in J. □

6. Conclusions

This paper effectively creates a fractal set for an ℑ-iterated function system consisting of generalized ℑ-contractions, establishing the existence of a unique common attractor for a range of Hutchinson–Wardowski contractive operators. Our findings produce a wide range of results for generalized iterated function systems that meet a variety of generalized contractive requirements, contributing to the advancement of the field. The provided illustrative example further validates our results, offering a comprehensive understanding of the subject matter. Finally, the existence results of common solutions to fractional boundary value problems are obtained, further extending the applicability of our work to a broader range of mathematical problems.