Enriched $\mathcal{Z}$-Contractions and Fixed-Point Results with Applications to IFS

Abstract

In this manuscript, we initiate a large class of enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-contractions defined on Banach spaces and prove the existence and uniqueness of the fixed point of these contractions. We also provide an example to support our results and give an existence condition for the uniqueness of the solution to the integral equation. The results provided in the manuscript extend, generalize, and modify the existence results. Our research introduces novel fixed-point results under various contractive conditions. Furthermore, we discuss the iterated function system associated with enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-contractions in Banach spaces and define the enriched $\mathcal{Z}$-Hutchinson operator. A result regarding the convergence of Krasnoselskii’s iteration method and the uniqueness of the attractor via enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-contractions is also established. Our discoveries not only confirm but also significantly build upon and broaden several established findings in the current body of literature.

1. Introduction

The theory of the fixed point (FP) is an essential part of fractals and the iterated function system (IFS). Basically, the simplest forms of fractals are the compact subsets in Hausdorff spaces that remain unchanged under Hutchinson–Barnsley operators. The concept of an iterated function system (IFS) was initiated to study fractals by Hutchinson [1] and Barnsley [2]. IFSs are, in fact, the natural extension of the classical contraction principle given by Banach [3] in 1922. Fractals, as an IFS, are very important due to their applications in many fields. For example, IFSs have applications in image compression, quantum physics, graphics, wavelet analysis, and many others areas. That is why many computer experts and mathematicians have shown their interest in this active research area. For example, see the work of Andres and Fišer [4], Duvall et al. [5], Kieninger [6], Barnsley and Demko [7], Zhou et al. [8], and the references therein for deep understanding.

Over time, the concept of an IFS has been generalized in many directions. In the past decades, many tools have been created to analyze the unique attractor (or the unique FP of Hutchinson–Barnsley operators) of the fractals. This theory of IFS has been expanded via generalized contractions, multifunctions, countable IFSs, and more. In particular, Kashyap et al. [9] generalized the fractal given by Mandelbrot [10] by using the Krasnoselskii theorem. Maślanka and Strobin [11] explored the generalized IFSs given by the $l_{\infty }$-sum of a metric space (MS), and Klimek and Kosek [12] discussed the multifunctions, generalized IFSs, and Cantor sets. Torre and others [13,14,15] studied the more generalized multifunctions. Khumalo et al. [16] studied generalized IFSs for shared attractors in partial MSs.

Very recently, Rizwan et al. [17] generalized the work of Ahmad et al. [18] on fractals with the generalized $\mathrm{\Theta }$-Hutchinson operator by using the enriched contraction given by Berinde and Păcurar [19]. Prithvi and colleagues in [20,21,22] discussed the IFS over generalized Kannan mappings and also gave remarks on countable IFSs via the partial Hausdorff metric and non-conventional IFSs. Sahu et al. [23], Thangaraj et al. [24], and Chandra and Verma [25] constructed fractals via an IFS on Kannan contractions with different conditions. Amit et al. [26] presented the idea of IFS via non-stationary $\mathrm{\Phi }$-contractions, while Verma and Priyadarshi [27] worked on general datasets and generated a new type of fractal function. Khojasteh et al. [28] presented FP results via the notion of a simulation function (SF) and $\mathcal{Z}$-contraction. Rhoades [29] studied the continuity and nondecreasing behavior of $\mathrm{\Phi }$-contractions.

Throughout the article, we represent the MS by $(\widetilde{\mathrm{II}},\mathfrak{u})$, linear normed space by $(\widetilde{\mathrm{II}},||·|\left|\right)$, and collection of all non-empty and compact subsets of $\widetilde{\mathrm{II}}$ by $\mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right)$. The distance between an element ${\iota }^o$ of $\widetilde{\mathrm{II}}$ and a subset ${\mathrm{{\rm Y}}}^o$ of $\widetilde{\mathrm{II}}$ is given by:

$$\mathfrak{u}({\iota }^o,{\mathrm{{\rm Y}}}^o)=\underset{{\varrho }^o\in {\mathrm{{\rm Y}}}^o}{min}\mathfrak{u}({\iota }^o,{\varrho }^o),$$

while the distance between two subsets ${\mathrm{{\rm Y}}}^o$ and ℧ of $\widetilde{\mathrm{II}}$ is given by:

$$\mathfrak{D}({\mathrm{{\rm Y}}}^o,\mho )=\underset{{\iota }^o\in {\mathrm{{\rm Y}}}^o}{max}\mathfrak{u}({\iota }^o,\mho ).$$

Using the above notions, the Hausdorff metric is given by:

$$\mathring{\partial }({\mathrm{{\rm Y}}}^o,\mho )=max\{\mathfrak{D}({\mathrm{{\rm Y}}}^o,\mho ),\mathfrak{D}(\mho ,{\mathrm{{\rm Y}}}^o)\}.$$

It is noted that, if $\widetilde{\mathrm{II}}$ is complete, then the Hausdorff space $\mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right)$ is also complete. We used the following technical lemma and notions in our main findings.

Lemma 1

([2]). Consider $\widetilde{\mathrm{II}}$ be a MS and ${\mathrm{{\rm Y}}}^o,\mho ,\mathfrak{C}\in \mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right)$. Then, the following holds:

(i)

$\mathfrak{C}\subseteq \mho \Rightarrow {sup}_{{\iota }^o\in {\mathrm{{\rm Y}}}^o}\mathfrak{u}({\iota }^o,\mathfrak{C})\le {sup}_{{\iota }^o\in {\mathrm{{\rm Y}}}^o}\mathfrak{u}({\iota }^o,\mho )$;

(ii)

${sup}_{{\iota }^o\in {\mathrm{{\rm Y}}}^o\cup \mathfrak{C}}\mathfrak{u}({\iota }^o,\mho )=max\{{sup}_{{\iota }^o\in {\mathrm{{\rm Y}}}^o}\mathfrak{u}({\iota }^o,\mho ),{sup}_{{\iota }^o\in \mathfrak{C}}\mathfrak{u}({\iota }^o,\mho )\}$;

(iii)

If $\{{{\mathrm{{\rm Y}}}^o}_{\mathrm{i}}:i=1,2,\cdots ,\mathsf{ȷ}\}$ and $\{{\mathfrak{C}}_{\mathrm{i}}:i=1,2,\cdots ,\mathsf{ȷ}\}$ are two finite collections of subsets of $\widetilde{\mathrm{II}}$, then

$$\mathring{\partial }(\bigcup _{i=1}^{\mathsf{ȷ}}{{\mathrm{{\rm Y}}}^o}_{\mathrm{i}},\bigcup _{i=1}^{\mathsf{ȷ}}{\mathfrak{C}}_{\mathrm{i}})\le \underset{i=1}{\overset{\mathsf{ȷ}}{max}}\left\{\mathring{\partial }({{\mathrm{{\rm Y}}}^o}_{\mathrm{i}},{\mathfrak{C}}_{\mathrm{i}})\right\}.$$

Definition 1

([2]). Consider $(\widetilde{\mathrm{II}},\mathfrak{u})$ to be a complete MS and $\{{\mathrm{\Theta }}^{\left(i\right)}:\widetilde{\mathrm{II}}\to \widetilde{\mathrm{II}},\mathrm{for}\mathrm{all}i=1,2,\cdots ,\mathsf{ȷ}\}$ to be a family of all continuous contraction mappings with contraction factors ${\theta }_i,\forall i=1,2,\cdots ,\mathsf{ȷ}$. Then, $(\widetilde{\mathrm{II}},{\mathrm{\Theta }}^{\left(1\right)},{\mathrm{\Theta }}^{\left(2\right)},\cdots ,{\mathrm{\Theta }}^{\left(\mathsf{ȷ}\right)})$ is named as IFS.

Definition 2

([2]). Any set ${\mathrm{{\rm Y}}}^o$ from $\mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right)$ is known as the attractor of the IFS if:

1:

$\mathrm{\Theta }\left({\mathrm{{\rm Y}}}^o\right)={\mathrm{{\rm Y}}}^o$;

2:

∃ an open set $\mho \subseteq \widetilde{\mathrm{II}}$ for which $\mho \subseteq {\mathrm{{\rm Y}}}^o$ and ${lim}_{q\to \infty }{\mathrm{\Theta }}^q\left(\mathfrak{C}\right)={\mathrm{{\rm Y}}}^o$, for any $\mathfrak{C}\in \mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right)$ with $\mathfrak{C}\subseteq \mho$, where the limit is taken with respect to the Hausdorff metric $\mathring{\partial }$.

The primary outcome in this area was presented by Barnsley in [2], which is expressed as follows:

Theorem 1

([2]). Let $(\widetilde{\mathrm{II}},{\mathrm{\Theta }}^{\left(1\right)},{\mathrm{\Theta }}^{\left(2\right)},\cdots ,{\mathrm{\Theta }}^{\left(\mathsf{ȷ}\right)})$ be an IFS with contraction factors ${\theta }_i$, for all $i=1,2,\cdots ,\mathsf{ȷ}$. Then, the operator ${\mathrm{\Theta }}^{*}:\mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right)\to \mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right)$, defined by

$${\mathrm{\Theta }}^{*}\left(\mathcal{Z}\right)=\bigcup _{i=1}^{\mathsf{ȷ}}{\mathrm{\Theta }}^{\left(i\right)}\left({\mathrm{{\rm Y}}}^o\right),\forall {\mathrm{{\rm Y}}}^o\in \mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right),$$

is also a contraction on $(\mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right),\mathring{\partial })$, with contraction factor $\theta ={max}_{i=1}^{\mathsf{ȷ}}{\theta }_i$. Further, ${\mathrm{\Theta }}^{*}$ has a unique attractor, that is, $\exists {{\mathrm{{\rm Y}}}^o}^{*}\in \mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right)$, such that

$${{\mathrm{{\rm Y}}}^o}^{*}={\mathrm{\Theta }}^{*}\left({{\mathrm{{\rm Y}}}^o}^{*}\right)=\bigcup _{i=1}^{\mathsf{ȷ}}{\mathrm{\Theta }}^{\left(i\right)}\left({{\mathrm{{\rm Y}}}^o}^{*}\right),$$

and is obtained by ${{\mathrm{{\rm Y}}}^o}^{*}={lim}_{q\to \infty }{\mathrm{\Theta }}^{*q}(\mho )$ for any initial choice $\mho \in \mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right)$. Here, ${\mathrm{\Theta }}^{*q}(\mho )$ is given by ${\mathrm{\Theta }}^{*q}(\mho )={\mathrm{\Theta }}^{*}\left({\mathrm{\Theta }}^{*q-1}(\mho )\right)$.

Remark 1

([17]). For a given normed space $(\widetilde{\mathrm{II}},||·|\left|\right)$, we have

• for all ${\mathrm{{\rm Y}}}^o$ and ℧ of $\widetilde{\mathrm{II}}$,

  $${\mathrm{{\rm Y}}}^o+\mho :=\{{\iota }^o+{\varrho }^o:{\iota }^o\in {\mathrm{{\rm Y}}}^o,{\varrho }^o\in \mho \};$$
• for any ${\mathrm{{\rm Y}}}^o\subseteq \widetilde{\mathrm{II}}$, and a real number ν,

  $$\nu {\mathrm{{\rm Y}}}^o:=\{\nu {\iota }^o:{\iota }^o\in {\mathrm{{\rm Y}}}^o\}.$$

Recently, Berinde and Păcurar [19] established a wide and novel class of operators called “enriched contractions”. This class comprises Banach contractions as well as various other nonexpansive contractions that have been introduced in the literature. They discovered that each enriched contraction has a distinct FP, which can be determined using a Krasnoselskii iteration sequence in the context of Banach spaces. Enriched contraction operators are important because they can include both Banach contractions and non-expansive mappings. In particular, the non-expansive mappings do not always ensure the FPs, but the enriched contraction mappings consistently demonstrated the unique FP.

Definition 3

([19]). Let $(\widetilde{\mathrm{II}},\parallel ·\parallel )$ be a linear normed space. An operator $\mathrm{\Theta }:\widetilde{\mathrm{II}}\to \widetilde{\mathrm{II}}$ is known as an enriched contraction if $\exists \mathrm{d}\ge 0$ and $\delta \in [0,\mathrm{d}+1)$ for which, for all ${\iota }^o,{\varrho }^o\in \widetilde{\mathrm{II}}$:

$$\parallel \mathrm{d}({\iota }^o-{\varrho }^o)+\mathrm{\Theta }{\iota }^o-\mathrm{\Theta }{\varrho }^o\parallel \le \delta \parallel {\iota }^o-{\varrho }^o\parallel .$$

(1)

Lemma 2.

For any mapping Θ and its average operator ${\mathrm{\Theta }}_{\delta }\left({\iota }^o\right)=(1-\delta ){\iota }^o+\delta \mathrm{\Theta }{\iota }^o$ for some $\delta \in [0,1)$, the set of FPs for both mappings Θ and ${\mathrm{\Theta }}_{\delta }$ is the same.

Recently, Khojasteh et al. [28] presented the notion of a well-known SF as well as the $\mathcal{Z}$-contraction and its FP results, which generalize the several classical FP theorems in the documented literature. A SF with some important examples is given by:

Definition 4

([28]). A function $\mathfrak{Z}:{[0,\infty )}^2\to \mathbb{R}$ is said to be a SF if it fulfills the conditions listed below:

(λ₁):

$\mathfrak{Z}(0,0)=0$;

(λ₂):

$\mathfrak{Z}({\iota }^o,{\varrho }^o)<{\varrho }^o-{\iota }^o,\forall {\iota }^o,{\varrho }^o>0$;

(λ₃):

for sequences $\left\{{{\iota }^o}_l\right\},\left\{{{\varrho }^o}_l\right\}\subseteq (0,\infty )$ satisfying ${lim}_{l\to \infty }{{\iota }^o}_l={lim}_{l\to \infty }{{\varrho }^o}_l>0$ implies

$$\underset{l\to \infty }{lim\; sup}\mathfrak{Z}\left({{\iota }^o}_l,{{\varrho }^o}_l\right)<0.$$

The notation $\mathcal{Z}$ is used for the family of all the simulation functions $\mathfrak{Z}$ contained in $\mathcal{Z}$.

Example 1

([28]). Consider the mappings ${\mathfrak{Z}}_i:{[0,\infty )}^2\to \mathbb{R}$ for $i=1,2,3$ given by:

1. 

${\mathfrak{Z}}_1({\iota }^o,{\varrho }^o)=\nu \left({\varrho }^o\right)-\pi \left({\iota }^o\right)$ for all ${\iota }^o,{\varrho }^o\in [0,\infty )$, where $\pi ,\nu :[0,\infty )\to [0,\infty )$ are the two continuous functions for which $\nu \left({\iota }^o\right)=\pi \left({\iota }^o\right)=0$ if and only if ${\iota }^o=0$ and $\nu \left({\iota }^o\right)<{\iota }^o\le \pi \left({\iota }^o\right)$ for all ${\iota }^o>0$.

2. 

${\mathfrak{Z}}_2({\iota }^o,{\varrho }^o)={\varrho }^o-{\displaystyle \frac{\pi ({\iota }^o,{\varrho }^o)}{\nu ({\iota }^o,{\varrho }^o)}}$ for all ${\iota }^o,{\varrho }^o\in [0,\infty )$, where $\pi ,\nu :[0,\infty )\to (0,\infty )$ are the two continuous functions for which $\pi ({\iota }^o,{\varrho }^o)>\nu ({\iota }^o,{\varrho }^o)$ for all ${\iota }^o,{\varrho }^o>0$.

3. 

${\mathfrak{Z}}_3({\iota }^o,{\varrho }^o)={\varrho }^o-\pi \left({\varrho }^o\right)-{\iota }^o$ for all ${\iota }^o,{\varrho }^o\in [0,\infty )$, where $\pi :[0,\infty )\to [0,\infty )$ is the continuous function for which $\pi \left({\iota }^o\right)=0$ if and only if ${\iota }^o=0$.

Then, ${\mathfrak{Z}}_i$ for $i=1,2,3$ satisfies all conditions $({\lambda }_1-{\lambda }_3)$, so these are SFs.

Khojasteh et al. [28], by using the notion of SFs, established the following definition of $\mathcal{Z}$-contraction as follows.

Definition 5

([28]). Let $(\widetilde{\mathrm{II}},\mathfrak{u})$ be a MS, $\mathrm{\Theta }:\widetilde{\mathrm{II}}\to \widetilde{\mathrm{II}}$ a mapping, and $\mathfrak{Z}\in \mathcal{Z}$. Then, Θ is called a $\mathcal{Z}$-contraction via SF $\mathfrak{Z}$ if:

$$\mathfrak{Z}(\mathfrak{u}(\mathrm{\Theta }{\iota }^o,\mathrm{\Theta }{\varrho }^o),\mathfrak{u}({\iota }^o,{\varrho }^o))\ge 0\mathit{for}\mathit{all}{\iota }^o,{\varrho }^o\in \widetilde{\mathrm{II}}.$$

(2)

By using the notion of enriched contraction given by Berinde and Păcurar [19] and the $\mathcal{Z}$-contraction via SF by Khojasteh et al. [28], we found a large class of enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-contractions and proved the existence and uniqueness of the fixed point of these contractions in the setting of Banach spaces. We also present an example to support our results and give an existence condition for the uniqueness of the solution of the integral equation. Our research introduces novel FP results under various contractive conditions. Moreover, we also discuss the IFS associated with enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-contractions and define the enriched $\mathcal{Z}$-Hutchinson operator in Banach spaces. The convergence of Krasnoselskii’s iteration scheme and uniqueness of the attractor via enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-contractions is also established.

2. Main Results

In the following section, we present the concept of $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-contraction operators and derive results regarding their existence and approximation of the fixed point.

Definition 6.

Let $(\widetilde{\mathrm{II}},||·|\left|\right)$ be a normed space, $\mathrm{\Theta }:\widetilde{\mathrm{II}}\to \widetilde{\mathrm{II}}$ a mapping, and $\mathfrak{Z}\in \mathcal{Z}$. Then, Θ is called an enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-contraction via some $\mathfrak{Z}$ if $\exists \mathrm{d}\in [0,\infty )$ such that

$$\mathfrak{Z}({\displaystyle \frac{1}{\mathrm{d}+1}}\left|\right|\mathrm{d}({\iota }^o-{\varrho }^o)+\mathrm{\Theta }{\iota }^o-\mathrm{\Theta }{\varrho }^o\left|\right|,\left|\right|{\iota }^o-{\varrho }^o\left|\right|)\ge 0,\forall {\iota }^o,{\varrho }^o\in \widetilde{\mathrm{II}}.$$

(3)

To highlight the constant $\mathrm{d}$ and SF $\mathfrak{Z}$ involved in Definition (6), we call it an enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-contraction on $\widetilde{\mathrm{II}}$. We will now establish some properties of $\mathcal{Z}$-contractions defined in the setting of normed spaces.

Remark 2.

It is noted that, if we put $\mathrm{d}=0$ in Definition (6), then we obtain Definition (5) given by Khojasteh et al. [28]. Therefore, every $\mathcal{Z}$-contraction via any $\mathfrak{Z}\in \mathcal{Z}$ is an enriched $(0,\mathfrak{Z})$-$\mathcal{Z}$-contraction.

Remark 3.

Note that the definition of a SF implies $\mathfrak{Z}({\iota }^o,{\varrho }^o)<0$, $\forall {\iota }^o\ge {\varrho }^o>0$. Therefore, if Θ is a $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-contraction, then

$${\displaystyle \frac{1}{\mathrm{d}+1}}\left|\right|\mathrm{d}({\iota }^o-{\varrho }^o)+\mathrm{\Theta }{\iota }^o-\mathrm{\Theta }{\varrho }^o\left|\right|<\left|\right|{\iota }^o-{\varrho }^o\left|\right|,\forall {\iota }^o,{\varrho }^o\in \widetilde{\mathrm{II}},$$

or equivalently,

$$\left|\right|{\mathrm{\Theta }}_{\delta }{\iota }^o-{\mathrm{\Theta }}_{\delta }{\varrho }^o\left|\right|<\left|\right|{\iota }^o-{\varrho }^o\left|\right|,\forall {\iota }^o,{\varrho }^o\in \widetilde{\mathrm{II}},$$

(4)

where $\delta ={\displaystyle \frac{1}{\mathrm{d}+1}}$. This shows that the transformation of ${\mathrm{\Theta }}_{\delta }$ is continuous. Thus, Θ being the translation and scaling of a continuous function is also continuous. Therefore, every $\mathcal{Z}$-contraction mapping is continuous.

Initially, we present the following result, where we prove the uniqueness of the FP of an enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-contraction, provided it possesses a FP.

Lemma 3.

Let $(\widetilde{\mathrm{II}},||·|\left|\right)$ be a normed space and $\mathrm{\Theta }:\widetilde{\mathrm{II}}\to \widetilde{\mathrm{II}}$ be an enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-contraction on $\widetilde{\mathrm{II}}$. The FP of Θ is unique in $\widetilde{\mathrm{II}}$, provided it possesses a FP.

Proof. 

Consider ${\iota }^o\in \widetilde{\mathrm{II}}$ be any FP of $\mathrm{\Theta }$. On the contrary, we suppose that ${\varrho }^o\in \mathrm{\Theta }$ is any other FP of $\mathrm{\Theta }$, with ${\iota }^o\ne {\varrho }^o$. It is to be noted that the collection of FPs of both $\mathrm{\Theta }$ and ${\mathrm{\Theta }}_{\delta }$ is the same. Thus, using (3), we obtain the following:

$$0\le \mathfrak{Z}({\displaystyle \frac{1}{\mathrm{d}+1}}\left|\right|\mathrm{d}({\iota }^o-{\varrho }^o)+\mathrm{\Theta }{\iota }^o-\mathrm{\Theta }{\varrho }^o\left|\right|,\left|\right|{\iota }^o-{\varrho }^o\left|\right|)=\mathfrak{Z}\left(\right||{\mathrm{\Theta }}_{\delta }{\iota }^o-{\mathrm{\Theta }}_{\delta }{\varrho }^o\left|\right|,\left|\right|{\iota }^o-{\varrho }^o\left|\right|)=\mathfrak{Z}\left(\right||{\iota }^o-{\varrho }^o\left|\right|,\left|\right|{\iota }^o-{\varrho }^o).$$

(5)

In light of Remark (3), inequality (5) implies a contradiction, and this proves our result.  □

Next, we show that, for any enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-contraction, the corresponding transformation ${\mathrm{\Theta }}_{\delta }:=(1-\delta )I+\delta \mathrm{\Theta }$ is always asymptotically regular.

Lemma 4.

Let $(\widetilde{\mathrm{II}},||·|\left|\right)$ be a normed space and Θ be any enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-contraction on $\widetilde{\mathrm{II}}$. Then, the averaged operator ${\mathrm{\Theta }}_{\delta }$ is asymptotically regular.

Proof. 

Suppose any arbitrary element ${\iota }^o\in \widetilde{\mathrm{II}}$ and $p\in \mathbb{N}$. If ${\mathrm{\Theta }}_{\delta }^p{\iota }^o={\mathrm{\Theta }}^{p-1}{\iota }^o$, which further becomes ${\mathrm{\Theta }}_{\delta }{\varrho }^o={\varrho }^o$, then some ${\varrho }^o={\mathrm{\Theta }}_{\delta }^{p-1}{\iota }^o$. Then, ${\mathrm{\Theta }}_{\delta }^q{\varrho }^o={\mathrm{\Theta }}_{\delta }^{q-1}{\mathrm{\Theta }}_{\delta }{\varrho }^o={\mathrm{\Theta }}_{\delta }^{q-1}{\varrho }^o=\dots ={\mathrm{\Theta }}_{\delta }{\varrho }^o={\varrho }^o,$$\forall q\in \mathbb{N}$. So, for a sufficiently large $q\in \mathbb{N}$, we obtain

$$\begin{array}{cc}\hfill \left|\right|{\mathrm{\Theta }}_{\delta }^q{\iota }^o-{\mathrm{\Theta }}_{\delta }^{q+1}{\iota }^o\left|\right|& =\left|\right|{\mathrm{\Theta }}_{\delta }^{q-p+1}{\mathrm{\Theta }}_{\delta }^{p-1}{\iota }^o-{\mathrm{\Theta }}_{\delta }^{q-p+2}{\mathrm{\Theta }}_{\delta }^{p-1}{\iota }^o\left|\right|=\left|\right|{\mathrm{\Theta }}_{\delta }^{q-p+1}{\varrho }^o-{\mathrm{\Theta }}_{\delta }^{q-p+2}{\varrho }^o\left|\right|\hfill \\ & =\left|\right|{\varrho }^o-{\varrho }^o\left|\right|=0.\hfill \end{array}$$

Letting $q\to \infty$, we obtain ${lim}_{q\to \infty }\left|\right|{\mathrm{\Theta }}_{\delta }^q{\iota }^o-{\mathrm{\Theta }}_{\delta }^{q+1}{\iota }^o\left|\right|=0,\forall {\iota }^o\in \widetilde{\mathrm{II}}$. On the other hand, consider ${\mathrm{\Theta }}_{\delta }^q{\iota }^o\ne {\mathrm{\Theta }}_{\delta }^{q-1}{\iota }^o$, $\forall q\in \mathbb{N}$. So, by using inequality (3), we obtain

$$\begin{array}{cc}\hfill 0& \le \mathfrak{Z}\left(\right||{\mathrm{\Theta }}_{\delta }^{q+1}{\iota }^o-{\mathrm{\Theta }}_{\delta }^q{\iota }^o||,||{\mathrm{\Theta }}_{\delta }^q{\iota }^o-{\mathrm{\Theta }}_{\delta }^{q-1}{\iota }^o|\left|\right)\hfill \\ & =\mathfrak{Z}\left(\left|\right|{\mathrm{\Theta }}_{\delta }{\mathrm{\Theta }}_{\delta }^q{\iota }^o-{\mathrm{\Theta }}_{\delta }{\mathrm{\Theta }}_{\delta }^{q-1}{\iota }^o\left|\right|,\left|\right|{\mathrm{\Theta }}_{\delta }^q{\iota }^o-{\mathrm{\Theta }}_{\delta }^{q-1}{\iota }^o\left|\right|\right)\hfill \\ & <\left|\right|{\mathrm{\Theta }}_{\delta }^q{\iota }^o-{\mathrm{\Theta }}_{\delta }^{q-1}{\iota }^o\left|\right|-\left|\right|{\mathrm{\Theta }}_{\delta }^{q+1}{\iota }^o-{\mathrm{\Theta }}_{\delta }^q{\iota }^o\left|\right|\hfill \\ \hfill ⟹\left|\right|{\mathrm{\Theta }}_{\delta }^{q+1}{\iota }^o-{\mathrm{\Theta }}_{\delta }^q{\iota }^o\left|\right|& <\left|\right|{\mathrm{\Theta }}_{\delta }^q{\iota }^o-{\mathrm{\Theta }}_{\delta }^{q-1}{\iota }^o\left|\right|.\hfill \end{array}$$

This shows that $\left\{\left|\right|{\mathrm{\Theta }}_{\delta }^q{\iota }^o-{\mathrm{\Theta }}_{\delta }^{q-1}{\iota }^o\left|\right|\right\}$ is a monotonically decreasing sequence of positive real numbers. Therefore, it must be convergent. Let ${lim}_{q\to \infty }\left|\right|{\mathrm{\Theta }}_{\delta }^q{\iota }^o-{\mathrm{\Theta }}_{\delta }^{q+1}{\iota }^o\left|\right|=s\ge 0$. If $s>0$, and as $\mathrm{\Theta }$ is an enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-contraction, therefore by $\left({\mathfrak{Z}}_3\right)$, we obtain

$$0\le \underset{q\to \infty }{lim\; sup}\mathfrak{Z}\left(\left|\right|{\mathrm{\Theta }}_{\delta }^{q+1}{\iota }^o-{\mathrm{\Theta }}_{\delta }^q{\iota }^o\left|\right|,\left|\right|{\mathrm{\Theta }}_{\delta }^q{\iota }^o-{\mathrm{\Theta }}_{\delta }^{q-1}{\iota }^o\left|\right|\right)<0,$$

which is a contradiction. This implies that $s=0$. Equivalently, ${lim}_{q\to \infty }\left|\right|{\mathrm{\Theta }}_{\delta }^q{\iota }^o-{\mathrm{\Theta }}_{\delta }^{q+1}{\iota }^o\left|\right|=0$. Thus, ${\mathrm{\Theta }}_{\delta }$ is an asymptotically regular mapping on $\widetilde{\mathrm{II}}$.  □

The following result demonstrates that the Krasnoselskii sequence $\left\{{{\iota }^o}_q\right\}$ generated by an enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-contraction is always bounded.

Lemma 5.

Let $(\widetilde{\mathrm{II}},||·|\left|\right)$ be a normed space and $\mathrm{\Theta }:\widetilde{\mathrm{II}}\to \widetilde{\mathrm{II}}$ be an enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-contraction. Then, the Krasnoselskii sequence $\left\{{{\iota }^o}_q\right\}$ generated by Θ with initial value ${{\iota }^o}_0\in {\iota }^o$ is a bounded sequence, where ${{\iota }^o}_q=(1-\delta ){{\iota }^o}_{q-1}+\delta \mathrm{\Theta }{{\iota }^o}_{q-1},\forall q\in \mathbb{N}$ and $\delta ={\displaystyle \frac{1}{\mathrm{d}+1}}$.

Proof. 

For any arbitrary point ${{\iota }^o}_0\in \widetilde{\mathrm{II}}$, define the Krasnoselskii sequence $\left\{{{\iota }^o}_q\right\}$ given by ${{\iota }^o}_q=(1-\delta ){{\iota }^o}_{q-1}+\delta \mathrm{\Theta }{{\iota }^o}_{q-1}={\mathrm{\Theta }}_{\delta }{{\iota }^o}_{q-1},\forall q\in \mathbb{N}$. Assume that $\left\{{{\iota }^o}_q\right\}$ is not bounded. Then, without loss of generality, we can suppose that ${{\iota }^o}_{q+p}\ne {{\iota }^o}_q,\forall q,p\in \mathbb{N}$. As the sequence $\left\{{{\iota }^o}_q\right\}$ is not bounded, so we must find a sub-sequence $\left\{{{\iota }^o}_{q_k}\right\}$ such that $q_1=1$ and, for each $k\in \mathbb{N},q_{k+1}$ is the minimum integer such that $\left|\right|{{\iota }^o}_{q_{k+1}}-{{\iota }^o}_{q_k}\left|\right|>1$. Also, we obtain

$$\left|\right|{{\iota }^o}_m-{{\iota }^o}_{q_k}\left|\right|\le 1,q_k\le m\le q_{k+1}-1.$$

(6)

Therefore, by utilizing inequality (6) and the triangular inequality, we have

$$\begin{array}{cc}\hfill 1<\left|\right|{{\iota }^o}_{q_{k+1}}-{{\iota }^o}_{q_k}\left|\right|& \le \left|\right|{{\iota }^o}_{q_{k+1}}-{{\iota }^o}_{q_{k+1}-1}\left|\right|+\left|\right|{{\iota }^o}_{q_{k+1}-1}-{{\iota }^o}_{q_k}\left|\right|\hfill \\ & \le \left|\right|{{\iota }^o}_{q_{k+1}}-{{\iota }^o}_{q_{k+1}-1}\left|\right|+1.\hfill \end{array}$$

Taking $k\to \infty$ and using Lemma (4), we obtain

$$\underset{k\to \infty }{lim}\left|\right|{{\iota }^o}_{q_{k+1}}-{{\iota }^o}_{q_k}\left|\right|=1.$$

By inequality (3), we conclude that $\left|\right|{{\iota }^o}_{q_{k+1}}-{{\iota }^o}_{q_k}\left|\right|\le \left|\right|{{\iota }^o}_{q_{k+1}-1}-{{\iota }^o}_{q_k-1}\left|\right|$. Therefore, we obtain the following by the aid of triangular inequality as follows:

$$\begin{array}{cc}\hfill 1<\left|\right|{{\iota }^o}_{q_{k+1}}-{{\iota }^o}_{q_k}\left|\right|& \le \left|\right|{{\iota }^o}_{q_{k+1}-1}-{{\iota }^o}_{q_k-1}\left|\right|\hfill \\ & \le \left|\right|{{\iota }^o}_{q_{k+1}-1}-{{\iota }^o}_{q_k}\left|\right|+\left|\right|{{\iota }^o}_{q_k}-{{\iota }^o}_{q_k-1}\left|\right|\hfill \\ & \le 1+\left|\right|{{\iota }^o}_{q_k}-{{\iota }^o}_{q_k-1}\left|\right|.\hfill \end{array}$$

Taking $k\to \infty$ and using Lemma (4), we obtain

$$\underset{k\to \infty }{lim}\left|\right|{{\iota }^o}_{q_{k+1}-1}-{{\iota }^o}_{q_k-1}\left|\right|=1.$$

Now, since $\mathrm{\Theta }$ is an enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-contraction, from condition ${\mathfrak{Z}}_3$, we have

$$\begin{array}{cc}\hfill 0& \le \underset{k\to \infty }{lim\; sup}\mathfrak{Z}\left(\left|\right|{\mathrm{\Theta }}_{\delta }{{\iota }^o}_{q_{k+1}-1}-{\mathrm{\Theta }}_{\delta }{{\iota }^o}_{q_k-1}\left|\right|,\left|\right|{{\iota }^o}_{q_{k+1}-1}-{{\iota }^o}_{q_k-1}\left|\right|\right)\hfill \\ & =\underset{k\to \infty }{lim\; sup}\mathfrak{Z}\left(\left|\right|{{\iota }^o}_{q_{k+1}}-{{\iota }^o}_{q_k}\left|\right|,\left|\right|{{\iota }^o}_{q_{k+1}-1}-{{\iota }^o}_{q_k-1}\left|\right|\right)<0,\hfill \end{array}$$

which is a contradiction. This complete the proof.  □

In the next result, we prove the existence of the FP of an enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-contraction.

Theorem 2.

Let $(\widetilde{\mathrm{II}},||·|\left|\right)$ be a Banach space and $\mathrm{\Theta }:\widetilde{\mathrm{II}}\to \widetilde{\mathrm{II}}$ be an enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-contraction. Then, Θ has a unique FP in $\widetilde{\mathrm{II}}$ and, for every initial guess ${{\iota }^o}_0\in \widetilde{\mathrm{II}}$, the Krasnoselskii sequence $\left\{{{\iota }^o}_q\right\}$, defined by ${{\iota }^o}_q={\mathrm{\Theta }}_{\delta }{{\iota }^o}_{q-1},\forall q\in \mathbb{N}$, converges to the FP of Θ, where $\delta ={\displaystyle \frac{1}{\mathrm{d}+1}}$.

Proof. 

From the definition of an enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-contraction, we can write

$$\mathfrak{Z}\left(\delta \right||({\displaystyle \frac{1}{\delta }}-1)({\iota }^o-{\varrho }^o)+\mathrm{\Theta }{\iota }^o-\mathrm{\Theta }{\varrho }^o\left|\right|,\left|\right|{\iota }^o-{\varrho }^o\left|\right|)\ge 0,\forall {\iota }^o,{\varrho }^o\in \widetilde{\mathrm{II}},$$

or equivalently,

$$\mathfrak{Z}\left(\right||{\mathrm{\Theta }}_{\delta }{\iota }^o-{\mathrm{\Theta }}_{\delta }{\varrho }^o\left|\right|,\left|\right|{\iota }^o-{\varrho }^o\left|\right|)\ge 0,\forall {\iota }^o,{\varrho }^o\in \widetilde{\mathrm{II}}.$$

(7)

Let ${{\iota }^o}_0\in \widetilde{\mathrm{II}}$ be an arbitrary initial point, and let $\left\{{{\iota }^o}_q\right\}$ be the Krasnoselskii sequence defined by ${{\iota }^o}_q={\mathrm{\Theta }}_{\delta }{{\iota }^o}_{q-1}$ for all $q\in \mathbb{N}$. First, we will demonstrate that the sequence $\left\{{{\iota }^o}_q\right\}$ is Cauchy. To accomplish this, take

$${\mathfrak{C}}_q=sup\left\{\left|\right|{{\iota }^o}_i-{{\iota }^o}_j\left|\right|:i,j\ge q\right\}.$$

Observe that the sequence $\left\{{\mathfrak{C}}_q\right\}$ is a monotonically decreasing sequence of positive real numbers. According to Lemma (5), the sequence $\left\{{{\iota }^o}_q\right\}$ is bounded, which implies that ${\mathfrak{C}}_q<\infty$ for all $q\in \mathbb{N}$. Therefore, the sequence $\left\{{\mathfrak{C}}_q\right\}$ is monotonic and bounded, which implies it is convergent. This means there exists a non-negative real number $\mathfrak{C}\ge 0$ such that ${lim}_{q\to \infty }{\mathfrak{C}}_q=\mathfrak{C}$. We aim to prove that $\mathfrak{C}=0$. If $\mathfrak{C}>0$, then according to the definition of ${\mathfrak{C}}_q$, for every $k\in \mathbb{N}$, there exist indices $q_k$ and $m_k$ such that $m_k>q_k\ge k$ and

$${\mathfrak{C}}_k-{\displaystyle \frac{1}{k}}<\left|\right|{{\iota }^o}_{m_k}-{{\iota }^o}_{q_k}\left|\right|\le {\mathfrak{C}}_k.$$

Hence,

$$\underset{k\to \infty }{lim}\left|\right|{{\iota }^o}_{m_k}-{{\iota }^o}_{q_k}\left|\right|=\mathfrak{C}.$$

(8)

Using inequality (4) and the triangular inequality we have

$$\begin{array}{cc}\hfill \left|\right|{{\iota }^o}_{m_k}-{{\iota }^o}_{q_k}\left|\right|& =\left|\right|{\mathrm{\Theta }}_{\delta }{{\iota }^o}_{m_k-1}-{\mathrm{\Theta }}_{\delta }{{\iota }^o}_{q_k-1}\left|\right|\hfill \\ & <\left|\right|{{\iota }^o}_{m_k-1}-{{\iota }^o}_{q_k-1}\left|\right|\hfill \\ & <\left|\right|{{\iota }^o}_{m_k-1}-{{\iota }^o}_{m_k}\left|\right|+\left|\right|{{\iota }^o}_{m_k}-{{\iota }^o}_{q_k}\left|\right|+\left|\right|{{\iota }^o}_{q_k}-{{\iota }^o}_{q_k-1}\left|\right|.\hfill \end{array}$$

Using Lemma (4), inequality (8) and letting $k\to \infty$ in the above inequality, we obtain

$$\underset{k\to \infty }{lim}\left|\right|{{\iota }^o}_{m_k-1}-{{\iota }^o}_{q_k-1}\left|\right|=\mathfrak{C}.$$

(9)

Since $\mathrm{\Theta }$ is an enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-contraction, using inequalities (4), (8), (9), and (${\lambda }_3$), we have

$$0\le \underset{k\to \infty }{lim\; sup}\mathfrak{Z}\left(\left|\right|{{\iota }^o}_{m_k-1}-{{\iota }^o}_{q_k-1}\left|\right|,\left|\right|{{\iota }^o}_{m_k}-{{\iota }^o}_{q_k}\left|\right|\right)<0,$$

which is a contradiction, and it proves that $\mathfrak{C}=0$. So, $\left\{{{\iota }^o}_q\right\}$ is a Cauchy sequence. Since $\widetilde{\mathrm{II}}$ is a Banach space, there exists ${{\iota }^o}^{*}\in \widetilde{\mathrm{II}}$ such that ${lim}_{n\to \infty }{{\iota }^o}_q={{\iota }^o}^{*}$. Next, we prove that the point ${{\iota }^o}^{*}$ will remain fixed under ${\mathrm{\Theta }}_{\delta }$, and therefore it would also be the FP of $\mathrm{\Theta }$. Suppose ${\mathrm{\Theta }}_{\delta }{{\iota }^o}^{*}\ne {{\iota }^o}^{*}$; then,

$$\begin{array}{cc}\hfill 0& \le \underset{q\to \infty }{lim\; sup}\mathfrak{Z}\left(\left|\right|{\mathrm{\Theta }}_{\delta }{{\iota }^o}_q-{\mathrm{\Theta }}_{\delta }{{\iota }^o}^{*}\left|\right|,\left|\right|{{\iota }^o}_q-{{\iota }^o}^{*}\left|\right|\right)\hfill \\ & \le \underset{q\to \infty }{lim\; sup}\left[\left|\right|{{\iota }^o}_q-{{\iota }^o}^{*}\left|\right|-\left|\right|{{\iota }^o}_{q+1}-{\mathrm{\Theta }}_{\delta }{{\iota }^o}^{*}\left|\right|\right]\hfill \\ & =-\left|\right|{{\iota }^o}^{*}-{\mathrm{\Theta }}_{\delta }{{\iota }^o}^{*}\left|\right|.\hfill \end{array}$$

This leads to a contradiction. Thus, this shows that $\left|\right|{{\iota }^o}^{*}-{\mathrm{\Theta }}_{\delta }{{\iota }^o}^{*}\left|\right|=0$, that is, ${\mathrm{\Theta }}_{\delta }{{\iota }^o}^{*}={{\iota }^o}^{*}$. Thus, ${{\iota }^o}^{*}$ is a FP of ${\mathrm{\Theta }}_{\delta }$ and so is the FP of $\mathrm{\Theta }$ as well. Uniqueness of the FP follows from Lemma (3).  □

Example 2.

Let $\widetilde{\mathrm{II}}=[0,1]$ be a real normed space with the norm defined by $\left|\right|{\iota }^o-{\varrho }^o\left|\right|=|{\iota }^o-{\varrho }^o|,\forall {\iota }^o,{\varrho }^o\in \widetilde{\mathrm{II}}$. Then, $(\widetilde{\mathrm{II}},||·|\left|\right)$ is a Banach space. Define an operator $\mathrm{\Theta }:\widetilde{\mathrm{II}}\to \widetilde{\mathrm{II}}$ as $\mathrm{\Theta }\left({\iota }^o\right)={\displaystyle \frac{{\iota }^o(1-{\iota }^o)}{1+{\iota }^o}},\forall {\iota }^o\in [0,1]$. If $\mathrm{d}=1$, then $\delta ={\displaystyle \frac{1}{2}}$. Thus, the mapping ${\mathrm{\Theta }}_{\delta }$ becomes ${\mathrm{\Theta }}_{\delta }{\iota }^o={\displaystyle \frac{{\iota }^o}{{\iota }^o+1}},\forall {\iota }^o\in \widetilde{\mathrm{II}}$. Then, Θ is an enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-contraction, where $\mathrm{d}=1$ and $\mathfrak{Z}({\iota }^o,{\varrho }^o)={\displaystyle \frac{{\varrho }^o}{{\varrho }^o+1}}-{\iota }^o,\forall {\iota }^o,{\varrho }^o\in [0,\infty ).$ In particular, if ${\iota }^o,{\varrho }^o\in \widetilde{\mathrm{II}}$, then

$$\begin{array}{cc}\hfill \mathfrak{Z}(||{\mathrm{\Theta }}_{\frac{1}{2}}{\iota }^o-{\mathrm{\Theta }}_{\frac{1}{2}}{\varrho }^o||,||{\iota }^o-{\varrho }^o||)& ={\displaystyle \frac{\left|\right|{\iota }^o-{\varrho }^o\left|\right|}{1+\left|\right|{\iota }^o-{\varrho }^o\left|\right|}}-\left|\right|{\mathrm{\Theta }}_{\frac{1}{2}}{\iota }^o-{\mathrm{\Theta }}_{\frac{1}{2}}{\varrho }^o\left|\right|\hfill \\ & ={\displaystyle \frac{|{\iota }^o-{\varrho }^o|}{1+|{\iota }^o-{\varrho }^o|}}-\left|{\displaystyle \frac{{\iota }^o}{{\iota }^o+1}}-{\displaystyle \frac{{\varrho }^o}{{\varrho }^o+1}}\right|\hfill \\ & ={\displaystyle \frac{|{\iota }^o-{\varrho }^o|}{1+|{\iota }^o-{\varrho }^o|}}-\left|{\displaystyle \frac{|{\iota }^o-{\varrho }^o|}{({\iota }^o+1)({\varrho }^o+1)}}\right|\ge 0.\hfill \end{array}$$

Note that all the conditions of Theorem (2) are satisfied and hence both ${\mathrm{\Theta }}_{\frac{1}{2}}$ and Θ have a unique FP ${{\iota }^o}^{*}=0\in \widetilde{\mathrm{II}}$.

If we choose $\mathrm{d}=0$ in Theorem (2), then we obtain Theorem 2.8 in Khojasteh et al. [28] in the setting of Banach spaces as follows.

Corollary 1.

Consider a Banach space $(\widetilde{\mathrm{II}},||·|\left|\right)$ and an operator $\mathrm{\Theta }:\widetilde{\mathrm{II}}\to \widetilde{\mathrm{II}}$, which is an enriched $(0,\mathfrak{Z})$-$\mathcal{Z}$-contraction. That is,

$$\mathfrak{Z}\left(\right||\mathrm{\Theta }{\iota }^o-\mathrm{\Theta }{\varrho }^o\left|\right|,\left|\right|\mathfrak{x}-\mathfrak{y}\left|\right|)\ge 0,\forall \mathfrak{x},\mathfrak{y}\in \widetilde{\mathrm{II}}.$$

Then, Θ has a unique FP in $\widetilde{\mathrm{II}}$.

In the following, we obtain some well-known and novel results in the FP theory with an enriched-type contraction and the SFs. For example, the FP result of Berinde and Păcurar [19] is given in terms of the SF as follows.

Corollary 2.

Consider a Banach space $(\widetilde{\mathrm{II}},||·|\left|\right)$ with an operator $\mathrm{\Theta }:\widetilde{\mathrm{II}}\to \widetilde{\mathrm{II}}$ satisfying

$$\left|\right|\mathrm{d}({\iota }^o-{\varrho }^o)-\mathrm{\Theta }{\iota }^o+\mathrm{\Theta }{\varrho }^o\left|\right|\le \theta (\mathrm{d}+1)\left|\right|{\iota }^o-{\varrho }^o\left|\right|,\forall {\iota }^o,{\varrho }^o\in \widetilde{\mathrm{II}},$$

where $\theta \in [0,1)$ and $\mathrm{d}\in [0,\infty )$. Then, Θ has a unique FP in $\widetilde{\mathrm{II}}$.

Proof. 

Define ${\mathfrak{Z}}_E:{[0,\infty )}^2\to \mathbb{R}$ by

$${\mathfrak{Z}}_E({\iota }^o,{\varrho }^o)=\theta {\varrho }^o-{\iota }^o,\forall {\varrho }^o,{\iota }^o\in [0,\infty ).$$

It is clear that the mapping $\mathrm{\Theta }$ is an enriched $(\mathrm{d},{\mathfrak{Z}}_E)$-$\mathcal{Z}$-contraction with respect to ${\mathfrak{Z}}_E\in \mathcal{Z}$. Therefore, the result follows by taking $\mathfrak{Z}={\mathfrak{Z}}_E$ in Theorem (2).  □

Next, we have the Rhoades FP theorem [29] in terms of the enriched and SFs in the setting of normed spaces as follows.

Corollary 3.

Consider a Banach space $(\widetilde{\mathrm{II}},||·|\left|\right)$ with an operator $\mathrm{\Theta }:\widetilde{\mathrm{II}}\to \widetilde{\mathrm{II}}$ satisfying

$${\displaystyle \frac{1}{\mathrm{d}+1}}\left|\right|\mathrm{d}({\iota }^o-{\varrho }^o)-\mathrm{\Theta }{\iota }^o-\mathrm{\Theta }{\varrho }^o\left|\right|\le \left|\right|{\iota }^o-{\varrho }^o\left|\right|-\pi \left(\right||{\iota }^o-{\varrho }^o\left|\right|),\forall {\iota }^o,{\varrho }^o\in \widetilde{\mathrm{II}},$$

where $\pi :[0,\infty )\to [0,\infty )$ is a lower semi-continuous function, and ${\pi }^{-1}\left(0\right)=\left\{0\right\}$. Then, Θ has a unique FP in $\widetilde{\mathrm{II}}$.

Proof. 

Define ${\mathfrak{Z}}_r:{[0,\infty )}^2\to \mathbb{R}$ by

$${\mathfrak{Z}}_r({\iota }^o,{\varrho }^o)={\varrho }^o-\pi \left({\varrho }^o\right)-{\iota }^o,\forall {\varrho }^o,{\iota }^o\in [0,\infty ).$$

It is clear that the mapping $\mathrm{\Theta }$ is an enriched $(\mathrm{d},{\mathfrak{Z}}_r)$-$\mathcal{Z}$-contraction with respect to ${\mathfrak{Z}}_r\in \mathcal{Z}$. Therefore, the result follows by taking $\mathfrak{Z}={\mathfrak{Z}}_r$ in Theorem (2).  □

Rhoades [29] studied the continuity and nondecreasing behavior of the function $\mathrm{\Phi }$ with ${lim}_{t\to \infty }\psi \left({\iota }^o\right)=\infty$. In Corollary (3), we changed these assumptions by the lower semi-continuity of $\mathrm{\Phi }$. Hence, our result is a proper generalization of the results given by Rhoades [29] in the setting of Banach spaces via enriched techniques.

Corollary 4.

Consider a Banach space $(\widetilde{\mathrm{II}},||·|\left|\right)$ with an operator $\mathrm{\Theta }:\widetilde{\mathrm{II}}\to \widetilde{\mathrm{II}}$ satisfying

$${\displaystyle \frac{1}{\mathrm{d}+1}}\left|\right|\mathrm{d}({\iota }^o-{\varrho }^o)-\mathrm{\Theta }{\iota }^o-\mathrm{\Theta }{\varrho }^o\left|\right|\le \pi \left(\right||{\iota }^o-{\varrho }^o\left|\right|\left)\right||{\iota }^o-{\varrho }^o\left|\right|,\forall {\iota }^o,{\varrho }^o\in \widetilde{\mathrm{II}},$$

where $\pi :[0,+\infty )\to [0,1)$ is a function for which ${lim\; sup}_{{\iota }^o\to {\mathfrak{r}}^{+}}\pi \left({\iota }^o\right)<1$, for all $\mathfrak{r}>0$. Then, Θ has a unique FP.

Proof. 

Define ${\mathfrak{Z}}_w:{[0,\infty )}^2\to \mathbb{R}$ by

$${\mathfrak{Z}}_w({\iota }^o,{\varrho }^o)={\varrho }^o\pi \left({\varrho }^o\right)-{\iota }^o,\forall {\varrho }^o,{\iota }^o\in [0,\infty ),$$

and follow Theorem (2) to achieve the result.  □

Corollary 5.

Consider a Banach space $(\widetilde{\mathrm{II}},||·|\left|\right)$ with an operator $\mathrm{\Theta }:\widetilde{\mathrm{II}}\to \widetilde{\mathrm{II}}$ satisfying

$${\displaystyle \frac{1}{\mathrm{d}+1}}\left|\right|\mathrm{d}({\iota }^o-{\varrho }^o)-\mathrm{\Theta }{\iota }^o-\mathrm{\Theta }{\varrho }^o\left|\right|\le \pi \left(\right||{\iota }^o-{\varrho }^o\left|\right|),\forall {\iota }^o,{\varrho }^o\in \widetilde{\mathrm{II}},$$

where $\pi :[0,+\infty )\to [0,+\infty )$ is an upper semi-continuous function for which $\pi \left({\iota }^o\right)<{\iota }^o,\forall {\iota }^o>0$ and $\pi \left(0\right)=0$. Then, Θ has a unique FP.

Proof. 

Define the simulation operator ${\mathfrak{Z}}_q:{[0,\infty )}^2\to \mathbb{R}$ by

$${\mathfrak{Z}}_q({\iota }^o,{\varrho }^o)=\pi \left({\varrho }^o\right)-{\iota }^o,\forall {\varrho }^o,{\iota }^o\in [0,\infty ),$$

and apply Theorem (2) to complete the proof.  □

Corollary 6.

Consider a Banach space $(\widetilde{\mathrm{II}},||·|\left|\right)$ with an operator $\mathrm{\Theta }:\widetilde{\mathrm{II}}\to \widetilde{\mathrm{II}}$ satisfying

$${\int }_0^{{\displaystyle \frac{1}{\mathrm{d}+1}}\left|\right|\mathrm{d}({\iota }^o-{\varrho }^o)+\mathrm{\Theta }{\iota }^o-\mathrm{\Theta }{\varrho }^o\left|\right|}\pi \left(\theta \right)d\theta \le \left|\right|{\iota }^o-{\varrho }^o\left|\right|,\forall {\iota }^o,{\varrho }^o\in \widetilde{\mathrm{II}},$$

where $\pi :[0,\infty )\to [0,\infty )$ is a function such that ${\int }_0^{ϵ}\pi \left(\theta \right)d\theta$ exists and ${\int }_0^{ϵ}\pi \left(\theta \right)d\theta >ϵ$ for each $ϵ>0$. Then, Θ has a unique FP in $\widetilde{\mathrm{II}}$.

Proof. 

Define ${\mathfrak{Z}}_l:[0,\infty )\to \mathbb{R}$ by

$${\mathfrak{Z}}_l({\iota }^o,{\varrho }^o)={\varrho }^o-{\int }_0^{{\iota }^o}\pi \left(\theta \right)d\theta ,\forall {\varrho }^o,{\iota }^o\in [0,\infty ).$$

Then, apply Theorem (2) to obtain the conclusion.  □

Enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-contractions and their IFSs provide advanced methods for solving complex problems in optimization, image processing, and dynamical systems. They enable more robust algorithms, improved computational efficiency, and enhanced simulation accuracy, offering better modeling and new approaches for iterative processes and fixed points.

3. An Application

We suppose the following integral equation, $\forall \lambda \in \mathcal{I}=[a,b]$,

$${\iota }^o\left(\lambda \right)={\varrho }^o\left(\lambda \right)+{\int }_a^b\mathcal{J}(\lambda ,\sigma )\mathrm{z}(\sigma ,{\iota }^o\left(\sigma \right))d\sigma -(1-\delta ){\iota }^o\left(\lambda \right),$$

(10)

where ${\varrho }^o:\mathcal{I}\to \mathbb{R},\mathcal{J}:{\mathcal{I}}^2\to \mathbb{R},\mathrm{z}:\mathcal{I}\times \mathbb{R}\to \mathbb{R}$ are continuous functions, and $\delta ={\displaystyle \frac{1}{\mathrm{d}+1}}$ with $\mathrm{d}\in [0,\infty )$. In the following, we prove the existence of a unique solution to the integral Equation (10) in $\widetilde{\mathrm{II}}=\mathfrak{C}(\mathcal{I},\mathbb{R})$ as an application of our previous results. For this, define a self-mapping $\mathrm{\Theta }:\widetilde{\mathrm{II}}\to \widetilde{\mathrm{II}}$ by

$$\mathrm{\Theta }{\iota }^o\left(\lambda \right)={\varrho }^o\left(\lambda \right)+{\int }_a^b\mathcal{J}(\lambda ,\sigma )\mathrm{z}(\sigma ,{\iota }^o\left(\sigma \right))d\sigma -(1-\delta ){\iota }^o\left(\lambda \right),\forall \lambda \in \mathcal{I}.$$

(11)

For $\delta \in [0,1)$, we obtain

$${\mathrm{\Theta }}_{\delta }{\iota }^o\left(\lambda \right)=\delta {\varrho }^o\left(\lambda \right)+\delta {\int }_a^b\mathcal{J}(\lambda ,\sigma )\mathrm{z}(\sigma ,{\iota }^o\left(\sigma \right))d\sigma ,\forall \lambda \in \mathcal{I}.$$

(12)

Then, the existence of the FP of (11) and the existence of the solution to the integral Equation (10) are equivalent to each other. We use the FP technique to show the existence of the solution to (10).

We take the following norm $\widetilde{\mathrm{II}}$, which makes it the Banach space

$$\left|\right|{\iota }^o-{\varrho }^o\left|\right|=\underset{\lambda \in \mathcal{I}}{sup}|{\iota }^o\left(\lambda \right)-{\varrho }^o\left(\lambda \right)|.$$

Further, we assume the following conditions to analyze the existence of the solution of the integral Equation (10):

• ${sup}_{\lambda \in \mathcal{I}}{\int }_a^b\left|\mathcal{J}(\lambda ,\sigma )\right|d\sigma \le {\displaystyle \frac{1}{b-a}}$;
• $|\mathrm{z}(\sigma ,{\iota }^o\left(\sigma \right))-\mathrm{z}(\sigma ,{\varrho }^o\left(\sigma \right))|\le {\displaystyle \frac{1}{\delta }}\theta \left(\right||{\iota }^o-{\varrho }^o\left|\right|),\forall {\iota }^o,{\varrho }^o\in \widetilde{\mathrm{II}}$,

where $\theta :[0,\infty )\to [0,\infty )$ is a nondecreasing upper semi-continuous operator with $\theta \left({\iota }^o\right)<{\iota }^o,\forall {\iota }^o>0$ and $\theta \left(0\right)=0$.

Theorem 3.

The solution to the integral Equation (10) is unique in $\widetilde{\mathrm{II}}$ if assumptions 1 and 2 are satisfied.

Proof. 

Consider the following $\forall \lambda \in \mathcal{I}$ and $\mathrm{d}\in [0,\infty )$,

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{\mathrm{d}+1}}\left|\right|\mathrm{d}({\iota }^o-{\varrho }^o)+\mathrm{\Theta }{\iota }^o-\mathrm{\Theta }{\varrho }^o\left|\right|& =& \left|\right|{\mathrm{\Theta }}_{\delta }{\iota }^o-{\mathrm{\Theta }}_{\delta }{\varrho }^o\left|\right|\hfill \\ & =& \underset{\lambda \in \mathcal{I}}{sup}|{\mathrm{\Theta }}_{\delta }{\iota }^o-{\mathrm{\Theta }}_{\delta }{\varrho }^o|\hfill \\ & =& \underset{\lambda \in \mathcal{I}}{sup}|\delta {\varrho }^o\left(\lambda \right)+\delta {\int }_a^b\mathcal{J}(\lambda ,\sigma )\mathrm{z}(\sigma ,{\iota }^o\left(\sigma \right))d\sigma \hfill \\ & -& \delta {\varrho }^o\left(\lambda \right)-\delta {\int }_a^b\mathcal{J}(\lambda ,\sigma )\mathrm{z}(\sigma ,{\varrho }^o\left(\sigma \right))d\sigma |\hfill \\ & =& \delta \underset{\lambda \in \mathcal{I}}{sup}\left|{\int }_a^b\mathcal{J}(\lambda ,\sigma )[\mathrm{z}(\sigma ,{\iota }^o\left(\sigma \right))-\mathrm{z}(\sigma ,{\varrho }^o\left(\sigma \right))]d\sigma \right|\hfill \\ & \le & \delta \underset{\lambda \in \mathcal{I}}{sup}{\int }_a^b\left|\mathcal{J}(\lambda ,\sigma )[\mathrm{z}(\sigma ,{\iota }^o\left(\sigma \right))-\mathrm{z}(\sigma ,{\varrho }^o\left(\sigma \right))]\right|d\sigma \hfill \\ & \le & \delta \underset{\lambda \in \mathcal{I}}{sup}{\int }_a^b\left|\mathcal{J}(\lambda ,\sigma )\right|·|\mathrm{z}(\sigma ,{\iota }^o\left(\sigma \right))-\mathrm{z}(\sigma ,{\varrho }^o\left(\sigma \right))|d\sigma \hfill \\ & \le & \theta \left(\right||{\iota }^o-{\varrho }^o\left|\right|)·\underset{\lambda \in \mathcal{I}}{sup}{\int }_a^b\left|\mathcal{J}(\lambda ,\sigma )\right|d\sigma \hfill \\ & \le & {\displaystyle \frac{1}{b-a}}·\theta \left(\right||{\iota }^o-{\varrho }^o\left|\right|)\hfill \\ & \le & \theta \left(\right||{\iota }^o-{\varrho }^o|\left|\right).\hfill \end{array}$$

Hence, all the assumptions of Corollary (5) are satisfied, so $\mathrm{\Theta }$ has a unique FP. Equivalently, the solution to the integral Equation (10) is unique in $\widetilde{\mathrm{II}}$.  □

4. Application to the Iterated Function System

In this part, we list applications of our results to the iterated functions system via enrichment and SFs $\mathfrak{Z}\in \mathcal{Z}$. The first result in this direction is given below.

Theorem 4.

Let Θ be an enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-contraction on linear normed space $\widetilde{\mathrm{II}}$ and define the operator $\tilde{\mathrm{\Theta }}:\mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right)\to \mathcal{P}\left(\widetilde{\mathrm{II}}\right)$ by $\tilde{\mathrm{\Theta }}\left({\mathrm{{\rm Y}}}^o\right)=\{\mathrm{\Theta }\left({\iota }^o\right):{\iota }^o\in {\mathrm{{\rm Y}}}^o\},\forall {\mathrm{{\rm Y}}}^o\in \mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right)$. Then,

1. 

$\tilde{\mathrm{\Theta }}$ maps $\mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right)$ to $\mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right)$;

2. 

$\tilde{\mathrm{\Theta }}$ is also an enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-contraction on $\mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right)$,

where $\mathcal{P}\left(\widetilde{\mathrm{II}}\right)$ is the power set of $\widetilde{\mathrm{II}}$.

Proof. 

Initially, we demonstrate that $\tilde{\mathrm{\Theta }}$ maps elements from $\mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right)$ to $\mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right)$. Since $\mathrm{\Theta }$ is an enriched $(\mathrm{d},\mathfrak{Z})$-contraction, from ${\lambda }_2$ and inequality (4), we obtain

$$\begin{array}{c}\hfill \left|\right|{\displaystyle \frac{\mathrm{d}({\iota }^o-{\varrho }^o)+\mathrm{\Theta }{\iota }^o-\mathrm{\Theta }{\varrho }^o}{\mathrm{d}+1}}\left|\right|<\left|\right|{\iota }^o-{\varrho }^o\left|\right|\\ \hfill ⟹\left|\right|{\mathrm{\Theta }}_{\delta }{\iota }^o-{\mathrm{\Theta }}_{\delta }{\varrho }^o\left|\right|<\left|\right|{\iota }^o-{\varrho }^o\left|\right|.\end{array}$$

This implies that ${\mathrm{\Theta }}_{\delta }$ is a contractive mapping and, is therefore continuous. Thus,

$${\mathrm{{\rm Y}}}^o\in \mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right)\Rightarrow {\mathrm{\Theta }}_{\delta }\left({\mathrm{{\rm Y}}}^o\right)\in \mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right).$$

This means that ${\mathrm{\Theta }}_{\delta }$ sends elements from $\mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right)$ to $\mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right)$. Subsequently, the sum of any number of compact sets and the scalar multiplication of a compact set by any constant remain compact. Consequently, $\tilde{\mathrm{\Theta }}$ also maps elements from $\mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right)$ to $\mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right)$ as ${\mathrm{\Theta }}_{\delta }\left({\mathrm{{\rm Y}}}^o\right)=(1-\delta ){\mathrm{{\rm Y}}}^o+\delta \tilde{\mathrm{\Theta }}\left({\mathrm{{\rm Y}}}^o\right)$.

Next, take ${\mathrm{{\rm Y}}}^o,\mho \in \mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right)$. Then, from ${\lambda }_2$ and inequality (4), we obtain

$$\left|\right|{\mathrm{\Theta }}_{\delta }{\iota }^o-{\mathrm{\Theta }}_{\delta }{\varrho }^o\left|\right|<\left|\right|{\iota }^o-{\varrho }^o\left|\right|,\forall {\iota }^o,{\varrho }^o\in \widetilde{\mathrm{II}}.$$

Thus,

$$\mathfrak{D}({\mathrm{\Theta }}_{\delta }{\iota }^o,{\mathrm{\Theta }}_{\delta }\mho )=\underset{{\varrho }^o\in \mho }{inf}\left|\right|{\mathrm{\Theta }}_{\delta }{\iota }^o-{\mathrm{\Theta }}_{\delta }{\varrho }^o\left|\right|<\underset{{\varrho }^o\in \mho }{inf}\left|\right|{\iota }^o-{\varrho }^o\left|\right|=\mathfrak{D}({\iota }^o,\mho ).$$

(13)

Similarly,

$$\mathfrak{D}({\mathrm{\Theta }}_{\delta }{\varrho }^o,{\mathrm{\Theta }}_{\delta }{\mathrm{{\rm Y}}}^o)<\mathfrak{D}({\varrho }^o,{\mathrm{{\rm Y}}}^o).$$

(14)

Now, using the definition of Hausdorff metric $\mathring{\partial }$, (13), and (14), we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathring{\partial }(\mathrm{d}{\mathrm{{\rm Y}}}^o+\tilde{\mathrm{\Theta }}{\mathrm{{\rm Y}}}^o,\mathrm{d}\mho +\tilde{\mathrm{\Theta }}\mho )}{\mathrm{d}+1}}& =\mathring{\partial }({\mathrm{\Theta }}_{\delta }{\mathrm{{\rm Y}}}^o,{\mathrm{\Theta }}_{\delta }\mho )\hfill \\ \hfill & =max\{\underset{{\iota }^o\in {\mathrm{{\rm Y}}}^o}{sup}D({\mathrm{\Theta }}_{\delta }{\iota }^o,{\mathrm{\Theta }}_{\delta }\mho ),\underset{{\varrho }^o\in \mho }{sup}\mathfrak{D}({\mathrm{\Theta }}_{\delta }{\varrho }^o,{\mathrm{\Theta }}_{\delta }{\mathrm{{\rm Y}}}^o)\}\hfill \\ \hfill & <max\{\underset{{\iota }^o\in {\mathrm{{\rm Y}}}^o}{sup}\mathfrak{D}({\iota }^o,\mho ),\underset{{\varrho }^o\in \mho }{sup}D({\varrho }^o,{\mathrm{{\rm Y}}}^o)\}\hfill \\ \hfill & =\mathring{\partial }({\mathrm{{\rm Y}}}^o,\mho ).\hfill \end{array}$$

(15)

Using assumption ${\lambda }_2$, we obtain

$$\mathfrak{Z}({\displaystyle \frac{\mathring{\partial }(\mathrm{d}{\mathrm{{\rm Y}}}^o+\tilde{\mathrm{\Theta }}{\mathrm{{\rm Y}}}^o,\mathrm{d}\mho +\tilde{\mathrm{\Theta }}\mho )}{\mathrm{d}+1}},\mathring{\partial }({\mathrm{{\rm Y}}}^o,\mho ))\ge 0.$$

This shows that $\tilde{\mathrm{\Theta }}$ is an enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-contraction on $(\mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right),\mathring{\partial })$.  □

Definition 7.

Suppose a normed space $(\widetilde{\mathrm{II}},||·|\left|\right)$ together with a finite class $\{{\mathrm{\Theta }}^{\left(i\right)},i=1,2,\cdots ,\mathsf{ȷ}\}$ of enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-contractions. Then, the operator $\mathcal{Y}:\mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right)\to \mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right)$ defined by

$$\mathcal{Y}\left({\mathrm{{\rm Y}}}^o\right)=\bigcup _{i=1}^{\mathsf{ȷ}}{\mathrm{\Theta }}^{\left(i\right)}\left({\mathrm{{\rm Y}}}^o\right),\forall {\mathrm{{\rm Y}}}^o\in \mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right),$$

is called the $\mathcal{Z}$-Hutchinson contraction.

Definition 8.

Consider a normed space $(\widetilde{\mathrm{II}},||·|\left|\right)$ with a class $\{{\mathrm{\Theta }}^{\left(i\right)},i=1,2,\cdots ,\mathsf{ȷ}\}$ of enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-contractions that is said to be a $\mathcal{Z}$-IFS, and it is denoted by $(\widetilde{\mathrm{II}};{\mathrm{\Theta }}^{\left(i\right)},i=1,2,\cdots ,\mathsf{ȷ})$.

Lemma 6.

Let $(\widetilde{\mathrm{II}},||·|\left|\right)$ be a normed space together with a finite class $\{{\mathrm{\Theta }}^{\left(i\right)},i=1,2,\cdots ,\mathsf{ȷ}\}$ of enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-contractions. Then, the $\mathcal{Z}$-Hutchinson operator is also an enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-contraction.

Proof. 

For some given $\mathsf{ȷ}\in \mathbb{N}$ with $\mathsf{ȷ}\ge 2$, let $\{{\mathrm{\Theta }}^{\left(\mathrm{i}\right)}:\widetilde{\mathrm{II}}\to \widetilde{\mathrm{II}}:\mathrm{i}=1,2,\dots \mathsf{ȷ}\}$ be a family of enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-contractions and ${\mathrm{{\rm Y}}}^o,\mho \in \mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right)$. Then, from Lemma 1, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathring{\partial }(\mathrm{d}{\mathrm{{\rm Y}}}^o+\tilde{\mathrm{\Theta }}{\mathrm{{\rm Y}}}^o,\mathrm{d}\mho +\tilde{\mathrm{\Theta }}\mho )}{\mathrm{d}+1}}& =\mathring{\partial }({\mathrm{\Theta }}_{\delta }{\mathrm{{\rm Y}}}^o,{\mathrm{\Theta }}_{\delta }\mho )\hfill \\ \hfill & =\mathring{\partial }(\bigcup _{\mathrm{i}=1}^{\mathsf{ȷ}}{{\mathrm{\Theta }}^{\left(\mathrm{i}\right)}}_{\delta }{\mathrm{{\rm Y}}}^o,\bigcup _{\mathrm{i}=1}^{\mathsf{ȷ}}{{\mathrm{\Theta }}^{\left(\mathrm{i}\right)}}_{\delta }\mho )\hfill \\ \hfill & \le \underset{1\le \mathrm{i}\le j}{max}\left\{\mathring{\partial }({{\mathrm{\Theta }}^{\left(\mathrm{i}\right)}}_{\delta }{\mathrm{{\rm Y}}}^o,{{\mathrm{\Theta }}^{\left(\mathrm{i}\right)}}_{\delta }\mho )\right\}\hfill \\ \hfill & \le \mathring{\partial }({\mathrm{{\rm Y}}}^o,\mho ).\hfill \end{array}$$

Therefore, using $\left({\lambda }_2\right)$, we obtain

$$\mathfrak{Z}({\displaystyle \frac{\mathring{\partial }(\mathrm{d}{\mathrm{{\rm Y}}}^o+\tilde{\mathrm{\Theta }}{\mathrm{{\rm Y}}}^o,\mathrm{d}\mho +\tilde{\mathrm{\Theta }}\mho )}{\mathrm{d}+1}},\mathring{\partial }({\mathrm{{\rm Y}}}^o,\mho ))\ge 0,\forall {\mathrm{{\rm Y}}}^o,\mho \in \mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right).$$

Accordingly, the proof is complete.  □

Theorem 5.

Let $(\widetilde{\mathrm{II}},||·|\left|\right)$ be a linear normed space with a finite class $\{{\mathrm{\Theta }}^{\left(i\right)},i=1,2,\cdots ,\mathsf{ȷ}\}$ of enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-contractions. Then,

1. 

$\mathcal{Y}$ also maps $\mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right)$ to itself;

2. 

the $\mathcal{Z}$-Hutchinson operator has a unique FP, say ${{\mathrm{{\rm Y}}}^o}^{*}\in \mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right)$;

3. 

the sequence $\left({{\mathrm{{\rm Y}}}^o}_q\right),\forall q\in \mathbb{N}$, and $\delta ={\displaystyle \frac{1}{\mathrm{d}+1}}$, as defined by ${{\mathrm{{\rm Y}}}^o}_{q+1}=(1-\delta ){{\mathrm{{\rm Y}}}^o}_q+\delta \mathcal{Y}{{\mathrm{{\rm Y}}}^o}_q$, converges to ${{\mathrm{{\rm Y}}}^o}^{*}\in \mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right)$.

Proof. 

Since each ${\mathrm{\Theta }}^{\left(\mathrm{i}\right)}$ for $\mathrm{i}=1,2,\dots ,\mathsf{ȷ}$ is an enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-contraction, conclusion (1) can be directly deduced from the definition of $\mathrm{\Theta }$ and Theorem (4). In addition, conclusions (2) and (3) follow from Lemma (3) and Theorem (2).  □

Definition 9.

An operator $\mathrm{\Theta }:\mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right)\to \mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right)$, where $(\widetilde{\mathrm{II}},||·|\left|\right)$ is a normed space, is said to be a generalized enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-Hutchinson operator or simply generalized enriched $\mathcal{Z}$-Hutchinson operator if there exists a constant $\mathrm{d}\in [0,\infty )$ and a SF $\mathfrak{Z}\in \mathcal{Z}$ such that

$$\mathfrak{Z}(\mathring{\partial }({\displaystyle \frac{\mathrm{d}{\mathrm{{\rm Y}}}^o+\mathrm{\Theta }{\mathrm{{\rm Y}}}^o}{\mathrm{d}+1}},{\displaystyle \frac{\mathrm{d}\mho +\mathrm{\Theta }\mho }{\mathrm{d}+1}}),{\mathcal{N}}_{\mathrm{\Theta }}({\mathrm{{\rm Y}}}^o,\mho )\ge 0,{\mathrm{{\rm Y}}}^o,\mho \in \mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right),$$

(16)

where

$$\begin{array}{cc}\hfill {\mathcal{N}}_{\mathrm{\Theta }}({\mathrm{{\rm Y}}}^o,\mho )=& max\left\{\mathring{\partial }({\mathrm{{\rm Y}}}^o,\mho ),\mathring{\partial }\left({\mathrm{{\rm Y}}}^o,{\displaystyle \frac{\mathrm{d}{\mathrm{{\rm Y}}}^o+\mathrm{\Theta }{\mathrm{{\rm Y}}}^o}{\mathrm{d}+1}}\right),\mathring{\partial }\left(\mho ,{\displaystyle \frac{\mathrm{d}\mho +\mathrm{\Theta }\mho }{\mathrm{d}+1}}\right),\right.\hfill \\ \hfill & {\displaystyle \frac{1}{2}}\left[\mathring{\partial }\left({\mathrm{{\rm Y}}}^o,{\displaystyle \frac{\mathrm{d}\mho +\mathrm{\Theta }\mho }{\mathrm{d}+1}}\right)+\mathring{\partial }\left(\mho ,{\displaystyle \frac{\mathrm{d}{\mathrm{{\rm Y}}}^o+\mathrm{\Theta }{\mathrm{{\rm Y}}}^o}{\mathrm{d}+1}}\right)\right],\hfill \\ \hfill & \mathring{\partial }\left({\displaystyle \frac{1}{\mathrm{d}+1}}\left[{\displaystyle \frac{\mathrm{d}(\mathrm{d}{\mathrm{{\rm Y}}}^o+\mathrm{\Theta }{\mathrm{{\rm Y}}}^o)}{\mathrm{d}+1}}+\mathrm{\Theta }\left({\displaystyle \frac{\mathrm{d}{\mathrm{{\rm Y}}}^o+\mathrm{\Theta }{\mathrm{{\rm Y}}}^o}{\mathrm{d}+1}}\right)\right],{\displaystyle \frac{\mathrm{d}{\mathrm{{\rm Y}}}^o+\mathrm{\Theta }{\mathrm{{\rm Y}}}^o}{\mathrm{d}+1}}\right),\hfill \\ \hfill & \mathring{\partial }\left({\displaystyle \frac{1}{\mathrm{d}+1}}\left[{\displaystyle \frac{\mathrm{d}(\mathrm{d}{\mathrm{{\rm Y}}}^o+\mathrm{\Theta }{\mathrm{{\rm Y}}}^o)}{\mathrm{d}+1}}+\mathrm{\Theta }\left({\displaystyle \frac{\mathrm{d}{\mathrm{{\rm Y}}}^o+\mathrm{\Theta }{\mathrm{{\rm Y}}}^o}{\mathrm{d}+1}}\right)\right],\mho \right),\hfill \\ \hfill & \left.\mathring{\partial }\left({\displaystyle \frac{1}{\mathrm{d}+1}}\left[{\displaystyle \frac{\mathrm{d}(\mathrm{d}{\mathrm{{\rm Y}}}^o+\mathrm{\Theta }{\mathrm{{\rm Y}}}^o)}{\mathrm{d}+1}}+\mathrm{\Theta }\left({\displaystyle \frac{\mathrm{d}{\mathrm{{\rm Y}}}^o+\mathrm{\Theta }{\mathrm{{\rm Y}}}^o}{\mathrm{d}+1}}\right)\right],{\displaystyle \frac{\mathrm{d}\mho +\mathrm{\Theta }\mho }{\mathrm{d}+1}}\right)\right\}.\hfill \end{array}$$

Lemma 7.

Let $(\widetilde{\mathrm{II}},||·|\left|\right)$ be a normed space and $\mathrm{\Theta }:\mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right)\to \mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right)$ be a generalized enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-Hutchinson operator. Then, the Krasnoselskii iteration scheme $\left\{{{\mathrm{{\rm Y}}}^o}_n\right\}$ obtained by Θ with initial guess ${{\mathrm{{\rm Y}}}^o}_0\in \delta \left({\iota }^o\right)$ is a bounded sequence, where ${{\mathrm{{\rm Y}}}^o}_q=(1-\delta ){{\mathrm{{\rm Y}}}^o}_{q-1}+\delta \mathrm{\Theta }{{\mathrm{{\rm Y}}}^o}_{q-1},\forall q\in \mathbb{N}$, and $\delta ={\displaystyle \frac{1}{\mathrm{d}+1}}$.

Proof. 

By the definition of the generalized enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-Hutchinson operator, we have, for $\forall {\mathrm{{\rm Y}}}^o,\mho \in \mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right)$,

$$\mathfrak{Z}\left(\mathring{\partial }({\mathrm{\Theta }}_{\delta }{\mathrm{{\rm Y}}}^o,{\mathrm{\Theta }}_{\delta }\mho \right),{\mathcal{N}}_{{\mathrm{\Theta }}_{\delta }}({\mathrm{{\rm Y}}}^o,\mho ))\ge 0,$$

(17)

where

$$\begin{array}{cc}\hfill {\mathcal{N}}_{{\mathrm{\Theta }}_{\delta }}({\mathrm{{\rm Y}}}^o,\mho )=max\{& \mathring{\partial }({\mathrm{{\rm Y}}}^o,\mho ),\mathring{\partial }\left({\mathrm{{\rm Y}}}^o,{\mathrm{\Theta }}_{\delta }{\mathrm{{\rm Y}}}^o\right),\mathring{\partial }\left(\mho ,{\mathrm{\Theta }}_{\delta }\mho \right),{\displaystyle \frac{1}{2}}\left[\mathring{\partial }\left({\mathrm{{\rm Y}}}^o,{\mathrm{\Theta }}_{\delta }\mho \right)+\mathring{\partial }\left(\mho ,{\mathrm{\Theta }}_{\delta }{\mathrm{{\rm Y}}}^o\right)\right],\hfill \\ & \mathring{\partial }({\mathrm{\Theta }}_{\delta }^2{\mathrm{{\rm Y}}}^o,{\mathrm{\Theta }}_{\delta }{\mathrm{{\rm Y}}}^o),\mathring{\partial }({\mathrm{\Theta }}_{\delta }^2{\mathrm{{\rm Y}}}^o,\mho ),\mathring{\partial }({\mathrm{\Theta }}_{\delta }^2{\mathrm{{\rm Y}}}^o,{\mathrm{\Theta }}_{\delta }\mho )\}.\hfill \end{array}$$

Let ${{\mathrm{{\rm Y}}}^o}_0\in \mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right)$ be any arbitrary element and generate the sequence as ${{\mathrm{{\rm Y}}}^o}_{q+1}=(1-\delta ){{\mathrm{{\rm Y}}}^o}_q+\delta \mathrm{\Theta }{{\mathrm{{\rm Y}}}^o}_q,\forall q\ge 0$. Assume that $\left\{{{\mathrm{{\rm Y}}}^o}_q\right\}$ is not bounded. Then, without loss of generality, we can suppose that ${{\mathrm{{\rm Y}}}^o}_{q+p}\ne {{\mathrm{{\rm Y}}}^o}_q,\forall q,p\in \mathbb{N}$. As the sequence $\left\{{{\mathrm{{\rm Y}}}^o}_q\right\}$ is not bounded, we must find a sub-sequence $\left\{{{\mathrm{{\rm Y}}}^o}_{q_k}\right\}$ such that $q_1=1$ and, for each $k\in \mathbb{N},q_{k+1}$ is the minimum integer such that $\mathring{\partial }({{\mathrm{{\rm Y}}}^o}_{q_{k+1}},{{\mathrm{{\rm Y}}}^o}_{q_k})>1$. Also, we obtain

$$\mathring{\partial }({{\mathrm{{\rm Y}}}^o}_m,{{\mathrm{{\rm Y}}}^o}_{q_k})\le 1,q_k\le m\le q_{k+1}-1.$$

(18)

Therefore, by utilizing inequality (18) and the triangular inequality, we have

$$\begin{array}{cc}\hfill 1<\mathring{\partial }({{\mathrm{{\rm Y}}}^o}_{q_{k+1}},{{\mathrm{{\rm Y}}}^o}_{q_k})& \le \mathring{\partial }({{\mathrm{{\rm Y}}}^o}_{q_{k+1}},{{\mathrm{{\rm Y}}}^o}_{q_{k+1}-1})+\mathring{\partial }({{\mathrm{{\rm Y}}}^o}_{q_{k+1},1}-{{\mathrm{{\rm Y}}}^o}_{q_k})\hfill \\ & \le \mathring{\partial }({{\mathrm{{\rm Y}}}^o}_{q_{k+1}},{{\mathrm{{\rm Y}}}^o}_{q_{k+1}-1})+1.\hfill \end{array}$$

Taking $k\to \infty$ and using Lemma (4), we obtain

$$\underset{k\to \infty }{lim}\mathring{\partial }({{\mathrm{{\rm Y}}}^o}_{q_{k+1}},{{\mathrm{{\rm Y}}}^o}_{q_k})=1.$$

(19)

Substituting ${\mathrm{{\rm Y}}}^o={{\mathrm{{\rm Y}}}^o}_q$ and $\mho ={{\mathrm{{\rm Y}}}^o}_{q+1}$ in the inequality (17), we obtain

$$0\le \mathfrak{Z}(\mathring{\partial }({\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}_k,{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}_{k+1}),{\mathcal{N}}_{{\mathrm{\Theta }}_{\delta }}({{\mathrm{{\rm Y}}}^o}_k,{{\mathrm{{\rm Y}}}^o}_{k+1}))=\mathfrak{Z}(\mathring{\partial }({{\mathrm{{\rm Y}}}^o}_{k+1},{{\mathrm{{\rm Y}}}^o}_{k+2}),{\mathcal{N}}_{{\mathrm{\Theta }}_{\delta }}({{\mathrm{{\rm Y}}}^o}_k,{{\mathrm{{\rm Y}}}^o}_{k+1})),$$

(20)

where

$$\begin{array}{cc}\hfill {\mathcal{N}}_{{\mathrm{\Theta }}_{\delta }}({{\mathrm{{\rm Y}}}^o}_k,{{\mathrm{{\rm Y}}}^o}_{k+1})& =max\left\{\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}_k,{{\mathrm{{\rm Y}}}^o}_{k+1}\right),\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}_k,{\mathrm{\Theta }}_{\delta }\left({{\mathrm{{\rm Y}}}^o}_k\right)\right),\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}_{k+1},{\mathrm{\Theta }}_{\delta }\left({{\mathrm{{\rm Y}}}^o}_{k+1}\right)\right)\right.\hfill \\ & {\displaystyle \frac{\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}_k,{\mathrm{\Theta }}_{\delta }\left({{\mathrm{{\rm Y}}}^o}_{k+1}\right)\right)+\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}_{k+1},{\mathrm{\Theta }}_{\delta }\left({{\mathrm{{\rm Y}}}^o}_k\right)\right)}{2}}\hfill \\ & \left.\mathring{\partial }\left({\mathrm{\Theta }}_{\delta }^2\left({{\mathrm{{\rm Y}}}^o}_k\right),{\mathrm{\Theta }}_{\delta }\left({{\mathrm{{\rm Y}}}^o}_k\right)\right),\mathring{\partial }\left({\mathrm{\Theta }}_{\delta }^2\left({{\mathrm{{\rm Y}}}^o}_k\right),{{\mathrm{{\rm Y}}}^o}_{k+1}\right),\mathring{\partial }\left({\mathrm{\Theta }}_{\delta }^2\left({{\mathrm{{\rm Y}}}^o}_k\right),{\mathrm{\Theta }}_{\delta }\left({{\mathrm{{\rm Y}}}^o}_{k+1}\right)\right)\right\}\hfill \\ & =max\left\{\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}_k,{{\mathrm{{\rm Y}}}^o}_{k+1}\right),\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}_k,{{\mathrm{{\rm Y}}}^o}_{k+1}\right),\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}_{k+1},{{\mathrm{{\rm Y}}}^o}_{k+2}\right)\right.\hfill \\ & {\displaystyle \frac{\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}_k,{{\mathrm{{\rm Y}}}^o}_{k+2}\right)+\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}_{k+1},{{\mathrm{{\rm Y}}}^o}_{k+1}\right)}{2}},\hfill \\ & \left.\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}_{k+2},{{\mathrm{{\rm Y}}}^o}_{k+1}\right),\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}_{k+2},{{\mathrm{{\rm Y}}}^o}_{k+1}\right),\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}_{k+2},{{\mathrm{{\rm Y}}}^o}_{k+2}\right)\right\}\hfill \\ & \le max\left\{\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}_k,{{\mathrm{{\rm Y}}}^o}_{k+1}\right),\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}_{k+1},{{\mathrm{{\rm Y}}}^o}_{k+2}\right),{\displaystyle \frac{\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}_k,{{\mathrm{{\rm Y}}}^o}_{k+1}\right)+\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}_{k+1},{{\mathrm{{\rm Y}}}^o}_{k+2}\right)}{2}}\right\}\hfill \\ & =max\left\{\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}_k,{{\mathrm{{\rm Y}}}^o}_{k+1}\right),\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}_{k+1},{{\mathrm{{\rm Y}}}^o}_{k+2}\right)\right\}.\hfill \end{array}$$

Therefore, using inequality (20) and ${\lambda }_2$, we obtain

$$\begin{array}{ccc}\hfill 0& \le & \mathfrak{Z}\left(\mathring{\partial }({{\mathrm{{\rm Y}}}^o}_{k+1},{{\mathrm{{\rm Y}}}^o}_{k+2})\right),{\mathcal{N}}_{{\mathrm{\Theta }}_{\delta }}({{\mathrm{{\rm Y}}}^o}_k,{{\mathrm{{\rm Y}}}^o}_{k+1}))\hfill \\ & \le & \mathfrak{Z}\left(\mathring{\partial }({{\mathrm{{\rm Y}}}^o}_{k+1},{{\mathrm{{\rm Y}}}^o}_{k+2})\right),max\left\{\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}_k,{{\mathrm{{\rm Y}}}^o}_{k+1}\right),\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}_{k+1},{{\mathrm{{\rm Y}}}^o}_{k+2}\right)\right\})\hfill \\ & <& max\left\{\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}_k,{{\mathrm{{\rm Y}}}^o}_{k+1}\right),\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}_{k+1},{{\mathrm{{\rm Y}}}^o}_{k+2}\right)\right\}-\mathring{\partial }({{\mathrm{{\rm Y}}}^o}_{k+1},{{\mathrm{{\rm Y}}}^o}_{k+2}))\hfill \\ \hfill ⟹\mathring{\partial }({{\mathrm{{\rm Y}}}^o}_{k+1},{{\mathrm{{\rm Y}}}^o}_{k+2})& <& max\left\{\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}_k,{{\mathrm{{\rm Y}}}^o}_{k+1}\right),\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}_{k+1},{{\mathrm{{\rm Y}}}^o}_{k+2}\right)\right\}.\hfill \end{array}$$

This implies that

$$\mathring{\partial }({{\mathrm{{\rm Y}}}^o}_{k+1},{{\mathrm{{\rm Y}}}^o}_{k+2})<\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}_k,{{\mathrm{{\rm Y}}}^o}_{k+1}\right).$$

(21)

Thus, by inequality (21), we conclude that $\mathring{\partial }({{\mathrm{{\rm Y}}}^o}_{q_{k+1}},{{\mathrm{{\rm Y}}}^o}_{q_k})\le \mathring{\partial }({{\mathrm{{\rm Y}}}^o}_{q_{k+1}-1},{{\mathrm{{\rm Y}}}^o}_{q_k-1})$. Further, using (21), (18), (19), and the triangular inequality, we have

$$\begin{array}{cc}\hfill 1<\mathring{\partial }({{\mathrm{{\rm Y}}}^o}_{q_{k+1}},{{\mathrm{{\rm Y}}}^o}_{q_k})& \le \mathring{\partial }({{\mathrm{{\rm Y}}}^o}_{q_{k+1}-1},{{\mathrm{{\rm Y}}}^o}_{q_k-1})\hfill \\ & \le \mathring{\partial }({{\mathrm{{\rm Y}}}^o}_{q_{k+1}-1},{\mathrm{{\rm Y}}}_{q_k}^{\mathfrak{o}})+\mathring{\partial }({{\mathrm{{\rm Y}}}^o}_{q_k},{{\mathrm{{\rm Y}}}^o}_{q_k-1})\hfill \\ & \le 1+\mathring{\partial }({{\mathrm{{\rm Y}}}^o}_{q_k},{{\mathrm{{\rm Y}}}^o}_{q_k-1}).\hfill \end{array}$$

Taking $k\to \infty$ and using Lemma (4), we obtain

$$\underset{k\to \infty }{lim}\mathring{\partial }({{\mathrm{{\rm Y}}}^o}_{q_{k+1}-1},{{\mathrm{{\rm Y}}}^o}_{q_k-1})=1.$$

Now, since $\mathrm{\Theta }$ is a generalized enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-Hutchinson operator, from condition ${\mathfrak{Z}}_3$, we have

$$\begin{array}{cc}\hfill 0& \le \underset{k\to \infty }{lim\; sup}\mathfrak{Z}\left(\mathring{\partial }({\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}_{q_{k+1}-1},{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}_{q_k-1}),\mathring{\partial }({{\mathrm{{\rm Y}}}^o}_{q_{k+1}-1},{{\mathrm{{\rm Y}}}^o}_{q_k-1})\right)\hfill \\ & =\underset{k\to \infty }{lim\; sup}\mathfrak{Z}\left(\mathring{\partial }({{\mathrm{{\rm Y}}}^o}_{q_{k+1}},{{\mathrm{{\rm Y}}}^o}_{q_k}),\mathring{\partial }({{\mathrm{{\rm Y}}}^o}_{q_{k+1}-1},{{\mathrm{{\rm Y}}}^o}_{q_k-1})\right)<0,\hfill \end{array}$$

which is a contradiction. This completes the proof.  □

Theorem 6.

Let $(\widetilde{\mathrm{II}},||·|\left|\right)$ be Banach space and $\mathrm{\Theta }:\mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right)\to \mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right)$ be a generalized enriched $\mathcal{Z}$-Hutchinson operator. Then,

1. 

the attractor of Θ is unique, say ${{\mathrm{{\rm Y}}}^o}^{*}\in \mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right)$;

2. 

the sequence $\left({{\mathrm{{\rm Y}}}^o}_q\right)$ defined by

$${{\mathrm{{\rm Y}}}^o}_{q+1}=(1-\delta ){{\mathrm{{\rm Y}}}^o}_q+\delta \mathrm{\Theta }{{\mathrm{{\rm Y}}}^o}_q,\forall q\ge 0,$$

(22)

converges to ${{\mathrm{{\rm Y}}}^o}^{*}$ for any initial point ${{\mathrm{{\rm Y}}}^o}_0\in \mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right)$,

where $\delta ={\displaystyle \frac{1}{\mathrm{d}+1}}.$

Proof. 

By the definition of the generalized enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-Hutchinson operator, we have, for $\forall {\mathrm{{\rm Y}}}^o,\mho \in \mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right)$,

$$\mathfrak{Z}\left(\mathring{\partial }({\mathrm{\Theta }}_{\delta }{\mathrm{{\rm Y}}}^o,{\mathrm{\Theta }}_{\delta }\mho \right),{\mathcal{N}}_{{\mathrm{\Theta }}_{\delta }}({\mathrm{{\rm Y}}}^o,\mho ))\ge 0,$$

(23)

where

$$\begin{array}{cc}\hfill {\mathcal{N}}_{{\mathrm{\Theta }}_{\delta }}({\mathrm{{\rm Y}}}^o,\mho )=max\{& \mathring{\partial }({\mathrm{{\rm Y}}}^o,\mho ),\mathring{\partial }\left({\mathrm{{\rm Y}}}^o,{\mathrm{\Theta }}_{\delta }{\mathrm{{\rm Y}}}^o\right),\mathring{\partial }\left(\mho ,{\mathrm{\Theta }}_{\delta }\mho \right),{\displaystyle \frac{1}{2}}\left[\mathring{\partial }\left({\mathrm{{\rm Y}}}^o,{\mathrm{\Theta }}_{\delta }\mho \right)+\mathring{\partial }\left(\mho ,{\mathrm{\Theta }}_{\delta }{\mathrm{{\rm Y}}}^o\right)\right],\hfill \\ & \mathring{\partial }({\mathrm{\Theta }}_{\delta }^2{\mathrm{{\rm Y}}}^o,{\mathrm{\Theta }}_{\delta }{\mathrm{{\rm Y}}}^o),\mathring{\partial }({\mathrm{\Theta }}_{\delta }^2{\mathrm{{\rm Y}}}^o,\mho ),\mathring{\partial }({\mathrm{\Theta }}_{\delta }^2{\mathrm{{\rm Y}}}^o,{\mathrm{\Theta }}_{\delta }\mho )\}.\hfill \end{array}$$

Let ${{\mathrm{{\rm Y}}}^o}_0\in \mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right)$ be any arbitrary element and define the sequence as given in (22). Our aim is to show that this sequence is Cauchy. To do this, take

$${\mathfrak{C}}_q=sup\left\{\mathring{\partial }({{\mathrm{{\rm Y}}}^o}_l,{{\mathrm{{\rm Y}}}^o}_m):l,m\ge q\right\}.$$

Observe that the sequence $\left\{{\mathfrak{C}}_q\right\}$ is a monotonically decreasing sequence of positive real numbers. According to Lemma (7), the sequence $\left\{{{\mathrm{{\rm Y}}}^o}_q\right\}$ is bounded, which implies that ${\mathfrak{C}}_q<\infty$ for all $q\in \mathbb{N}$. Therefore, the sequence $\left\{{\mathfrak{C}}_q\right\}$ is monotonic and bounded, which implies it is convergent. This means there exists a non-negative real number $\mathfrak{C}\ge 0$ such that ${lim}_{q\to \infty }{\mathfrak{C}}_q=\mathfrak{C}$. We aim to prove that $\mathfrak{C}=0$. If $\mathfrak{C}>0$, then according to the definition of ${\mathfrak{C}}_q$, for every $k\in \mathbb{N}$, there exist indices $q_k$ and $m_k$ such that $m_k>q_k\ge k$ and

$${\mathfrak{C}}_k-{\displaystyle \frac{1}{k}}<\mathring{\partial }({{\mathrm{{\rm Y}}}^o}_{m_k},{{\mathrm{{\rm Y}}}^o}_{q_k})\le {\mathfrak{C}}_k.$$

Hence,

$$\underset{k\to \infty }{lim}\mathring{\partial }({{\mathrm{{\rm Y}}}^o}_{m_k},{{\mathrm{{\rm Y}}}^o}_{q_k})=\mathfrak{C}.$$

(24)

Using inequality (21) and the triangular inequality, we have

$$\begin{array}{cc}\hfill \mathring{\partial }({{\mathrm{{\rm Y}}}^o}_{m_k},{{\mathrm{{\rm Y}}}^o}_{q_k})& <\mathring{\partial }({{\mathrm{{\rm Y}}}^o}_{m_k-1},{{\mathrm{{\rm Y}}}^o}_{q_k-1})\hfill \\ & <\mathring{\partial }({{\mathrm{{\rm Y}}}^o}_{m_k-1},{{\mathrm{{\rm Y}}}^o}_{m_k})+\mathring{\partial }({{\mathrm{{\rm Y}}}^o}_{m_k},{{\mathrm{{\rm Y}}}^o}_{q_k})+\mathring{\partial }({{\mathrm{{\rm Y}}}^o}_{q_k},{{\mathrm{{\rm Y}}}^o}_{q_k-1}).\hfill \end{array}$$

Using Lemma (4) and inequality (24) and letting $k\to \infty$ in the above inequality, we obtain

$$\underset{k\to \infty }{lim}\mathring{\partial }({{\mathrm{{\rm Y}}}^o}_{m_k-1},{{\mathrm{{\rm Y}}}^o}_{q_k-1})=\mathfrak{C}.$$

(25)

Since $\mathrm{\Theta }$ is an enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-Hutchinson operator, using inequalities (21), (24), (25), and (${\lambda }_3$), we therefore have

$$0\le \underset{k\to \infty }{lim\; sup}\mathfrak{Z}\left(\mathring{\partial }({{\mathrm{{\rm Y}}}^o}_{m_k-1},{{\mathrm{{\rm Y}}}^o}_{q_k-1}),\mathring{\partial }({{\mathrm{{\rm Y}}}^o}_{m_k},{{\mathrm{{\rm Y}}}^o}_{q_k})\right)<0,$$

which is a contradiction and proves that $\mathfrak{C}=0$. So, $\left\{{{\mathrm{{\rm Y}}}^o}_q\right\}$ is a Cauchy sequence. Since $\widetilde{\mathrm{II}}$ is a Banach space, there exists ${{\mathrm{{\rm Y}}}^o}^{*}\in \widetilde{\mathrm{II}}$ such that ${lim}_{q\to \infty }{{\mathrm{{\rm Y}}}^o}_q={{\mathrm{{\rm Y}}}^o}^{*}$. Next, we show that ${{\mathrm{{\rm Y}}}^o}^{*}$ is a unique FP of $\mathrm{\Theta }$. For this purpose, suppose to the contrary that ${{\mathrm{{\rm Y}}}^o}^{*}$ is not the FP of $\mathrm{\Theta }$. Thus, ${{\mathrm{{\rm Y}}}^o}^{*}$ will not be the FP of ${\mathrm{\Theta }}_{\delta }$. By utilizing inequality (23), we have

$$0\le \mathfrak{Z}(\mathring{\partial }({\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}_k,{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}^{*}),{\mathcal{N}}_{{\mathrm{\Theta }}_{\delta }}({{\mathrm{{\rm Y}}}^o}_k,{{\mathrm{{\rm Y}}}^o}^{*})),$$

(26)

where

$$\begin{array}{cc}\hfill {\mathcal{N}}_{{\mathrm{\Theta }}_{\delta }}({{\mathrm{{\rm Y}}}^o}_k,{{\mathrm{{\rm Y}}}^o}^{*})=max& \left\{\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}_k,{{\mathrm{{\rm Y}}}^o}^{*}\right),\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}_k,{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}_k\right),\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}^{*},{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}^{*}\right)\right.,\hfill \\ \hfill & {\displaystyle \frac{\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}_k,{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}^{*}\right)+\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}^{*},{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}_k\right)}{2}},\hfill \\ \hfill & \left.\mathring{\partial }\left({\mathrm{\Theta }}_{\delta }^2{{\mathrm{{\rm Y}}}^o}_k,{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}_k\right),\mathring{\partial }\left({\mathrm{\Theta }}_{\delta }^2{{\mathrm{{\rm Y}}}^o}_k,{{\mathrm{{\rm Y}}}^o}^{*}\right),\mathring{\partial }\left({\mathrm{\Theta }}_{\delta }^2{{\mathrm{{\rm Y}}}^o}_k,{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}^{*}\right)\right\}\hfill \\ \hfill =max& \left\{\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}_k,{{\mathrm{{\rm Y}}}^o}^{*}\right),\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}_k,{{\mathrm{{\rm Y}}}^o}_{k+1}\right),\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}^{*},{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}^{*}\right)\right.,\hfill \\ \hfill & {\displaystyle \frac{\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}_k,{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}^{*}\right)+\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}^{*},{{\mathrm{{\rm Y}}}^o}_{k+1}\right)}{2}},\hfill \\ \hfill & \left.\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}_{k+2},{{\mathrm{{\rm Y}}}^o}_{k+1}\right),\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}_{k+2},{{\mathrm{{\rm Y}}}^o}^{*}\right),\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}_{k+2},{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}^{*}\right)\right\}.\hfill \end{array}$$

We now have the following cases:

• If ${\mathcal{N}}_{{\mathrm{\Theta }}_{\delta }}({{\mathrm{{\rm Y}}}^o}_k,{{\mathrm{{\rm Y}}}^o}^{*})=\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}_k,{{\mathrm{{\rm Y}}}^o}^{*}\right)$, then using the limit as $k\to \infty$ in (26) and ${\lambda }_2$, we obtain

  $$\begin{array}{ccc}\hfill 0& \le & \mathfrak{Z}(\mathring{\partial }({{\mathrm{{\rm Y}}}^o}^{*},{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}^{*}),\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}^{*},{{\mathrm{{\rm Y}}}^o}^{*}\right))\hfill \\ & <& \mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}^{*},{{\mathrm{{\rm Y}}}^o}^{*}\right)-\mathring{\partial }({{\mathrm{{\rm Y}}}^o}^{*},{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}^{*})\hfill \\ & =& -\mathring{\partial }({{\mathrm{{\rm Y}}}^o}^{*},{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}^{*}),\hfill \end{array}$$

  which is a contradiction.
• If ${\mathcal{N}}_{{\mathrm{\Theta }}_{\delta }}({{\mathrm{{\rm Y}}}^o}_k,{{\mathrm{{\rm Y}}}^o}^{*})=\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}_k,{{\mathrm{{\rm Y}}}^o}_{k+1}\right)$, then using the limit as $k\to \infty$ in (26) and ${\lambda }_2$, we obtain

  $$\begin{array}{ccc}\hfill 0& \le & \mathfrak{Z}(\mathring{\partial }({{\mathrm{{\rm Y}}}^o}^{*},{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}^{*}),\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}^{*},{{\mathrm{{\rm Y}}}^o}^{*}\right))\hfill \\ & <& \mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}^{*},{{\mathrm{{\rm Y}}}^o}^{*}\right)-\mathring{\partial }({{\mathrm{{\rm Y}}}^o}^{*},{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}^{*})\hfill \\ & =& -\mathring{\partial }({{\mathrm{{\rm Y}}}^o}^{*},{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}^{*}),\hfill \end{array}$$

  which is a contradiction.
• If ${\mathcal{N}}_{{\mathrm{\Theta }}_{\delta }}({{\mathrm{{\rm Y}}}^o}_k,{{\mathrm{{\rm Y}}}^o}^{*})=\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}^{*},{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}^{*}\right)$, then using the limit as $k\to \infty$ in (26) and ${\lambda }_2$, we obtain

  $$\begin{array}{ccc}\hfill 0& \le & \mathfrak{Z}(\mathring{\partial }({{\mathrm{{\rm Y}}}^o}^{*},{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}^{*}),\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}^{*},{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}^{*}\right))\hfill \\ & <& \mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}^{*},{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}^{*}\right)-\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}^{*},{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}^{*}\right)\hfill \\ & =& 0,\hfill \end{array}$$

  which further implies $\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}^{*},{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}^{*}\right)=0$ by the aid of ${\lambda }_1$ and is thus a contradiction.
• If ${\mathcal{N}}_{{\mathrm{\Theta }}_{\delta }}({{\mathrm{{\rm Y}}}^o}_k,{{\mathrm{{\rm Y}}}^o}^{*})={\displaystyle \frac{\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}_k,{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}^{*}\right)+\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}^{*},{{\mathrm{{\rm Y}}}^o}_{k+1}\right)}{2}}$, then using the limit as $k\to \infty$ in (26) and ${\lambda }_2$, we obtain

  $$\begin{array}{ccc}\hfill 0& \le & \mathfrak{Z}(\mathring{\partial }({{\mathrm{{\rm Y}}}^o}^{*},{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}^{*}),{\displaystyle \frac{\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}^{*},{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}^{*}\right)+\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}^{*},{{\mathrm{{\rm Y}}}^o}^{*}\right)}{2}})\hfill \\ & =& \mathfrak{Z}(\mathring{\partial }({{\mathrm{{\rm Y}}}^o}^{*},{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}^{*}),{\displaystyle \frac{\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}^{*},{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}^{*}\right)}{2}})\hfill \\ & <& \mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}^{*},{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}^{*}\right)-{\displaystyle \frac{\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}^{*},{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}^{*}\right)}{2}}\hfill \\ & =& -{\displaystyle \frac{\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}^{*},{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}^{*}\right)}{2}},\hfill \end{array}$$

  which is a contradiction.
• If ${\mathcal{N}}_{{\mathrm{\Theta }}_{\delta }}({{\mathrm{{\rm Y}}}^o}_k,{{\mathrm{{\rm Y}}}^o}^{*})=\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}_{k+2},{{\mathrm{{\rm Y}}}^o}_{k+1}\right)$, then using the limit as $k\to \infty$ in (26) and ${\lambda }_2$, we get

  $$\begin{array}{ccc}\hfill 0& \le & \mathfrak{Z}(\mathring{\partial }({{\mathrm{{\rm Y}}}^o}^{*},{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}^{*}),\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}^{*},{{\mathrm{{\rm Y}}}^o}^{*}\right))\hfill \\ & <& \mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}^{*},{{\mathrm{{\rm Y}}}^o}^{*}\right)-\mathring{\partial }({{\mathrm{{\rm Y}}}^o}^{*},{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}^{*})\hfill \\ & =& -\mathring{\partial }({{\mathrm{{\rm Y}}}^o}^{*},{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}^{*}),\hfill \end{array}$$

  a contradiction.
• If ${\mathcal{N}}_{{\mathrm{\Theta }}_{\delta }}({{\mathrm{{\rm Y}}}^o}_k,{{\mathrm{{\rm Y}}}^o}^{*})=\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}_{k+2},{{\mathrm{{\rm Y}}}^o}^{*}\right)$, then using the limit as $k\to \infty$ in (26) and ${\lambda }_2$, we obtain

  $$\begin{array}{ccc}\hfill 0& \le & \mathfrak{Z}(\mathring{\partial }({{\mathrm{{\rm Y}}}^o}^{*},{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}^{*}),\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}^{*},{{\mathrm{{\rm Y}}}^o}^{*}\right))\hfill \\ & <& \mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}^{*},{{\mathrm{{\rm Y}}}^o}^{*}\right)-\mathring{\partial }({{\mathrm{{\rm Y}}}^o}^{*},{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}^{*})\hfill \\ & =& -\mathring{\partial }({{\mathrm{{\rm Y}}}^o}^{*},{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}^{*}),\hfill \end{array}$$

  which is a contradiction.
• If ${\mathcal{N}}_{{\mathrm{\Theta }}_{\delta }}({{\mathrm{{\rm Y}}}^o}_k,{{\mathrm{{\rm Y}}}^o}^{*})=\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}_{k+2},{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}^{*}\right)$, then using the limit as $k\to \infty$ in (26) and ${\lambda }_2$, we obtain

  $$\begin{array}{ccc}\hfill 0& \le & \mathfrak{Z}(\mathring{\partial }({{\mathrm{{\rm Y}}}^o}^{*},{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}^{*}),\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}^{*},{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}^{*}\right))\hfill \\ & <& \mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}^{*},{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}^{*}\right)-\mathring{\partial }({{\mathrm{{\rm Y}}}^o}^{*},{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}^{*})\hfill \\ & =& 0,\hfill \end{array}$$

  which further implies $\mathring{\partial }\left({{\mathrm{{\rm Y}}}^o}^{*},{\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}^{*}\right)=0$ by the aid of ${\lambda }_1$ and thus is a contradiction.

Thus, in all cases, $\mathring{\partial }({\mathrm{\Theta }}_{\delta }{{\mathrm{{\rm Y}}}^o}^{*},{{\mathrm{{\rm Y}}}^o}^{*})=0$. This is to say, ${{\mathrm{{\rm Y}}}^o}^{*}$ is the FP of ${\mathrm{\Theta }}_{\delta }$ and, as such, the FP of $\mathrm{\Theta }$. For the purpose of uniqueness, suppose that ${\mathrm{{\rm Y}}}^o,\mho \in \mathrm{\Lambda }\left(\widetilde{\mathrm{II}}\right)$ are two distinct FPs of $\mathrm{\Theta }$. Then, from (26), we have

$$\begin{array}{cc}\hfill 0& \le \mathfrak{Z}(\mathring{\partial }({\mathrm{{\rm Y}}}^o,\mho ),\mathring{\partial }({\mathrm{\Theta }}_{\delta }{\mathrm{{\rm Y}}}^o,{\mathrm{\Theta }}_{\delta }\mho ))\hfill \\ \hfill & \le \mathfrak{Z}(\mathring{\partial }({\mathrm{{\rm Y}}}^o,\mho ),max\{\mathring{\partial }({\mathrm{{\rm Y}}}^o,\mho ),\mathring{\partial }({\mathrm{{\rm Y}}}^o,{\mathrm{\Theta }}_{\delta }{\mathrm{{\rm Y}}}^o),\mathring{\partial }(\mho ,{\mathrm{\Theta }}_{\delta }\mho ),{\displaystyle \frac{\mathring{\partial }({\mathrm{{\rm Y}}}^o,{\mathrm{\Theta }}_{\delta }\mho )+\mathring{\partial }(\mho ,{\mathrm{\Theta }}_{\delta }{\mathrm{{\rm Y}}}^o)}{2}},\hfill \\ \hfill & \mathring{\partial }({\mathrm{\Theta }}_{\delta }^2{\mathrm{{\rm Y}}}^o,{\mathrm{\Theta }}_{\delta }{\mathrm{{\rm Y}}}^o),\mathring{\partial }({\mathrm{\Theta }}_{\delta }^2{\mathrm{{\rm Y}}}^o,\mho ),\mathring{\partial }({\mathrm{\Theta }}_{\delta }^2{\mathrm{{\rm Y}}}^o,{\mathrm{\Theta }}_{\delta }\mho )\}\hfill \\ \hfill & =\mathfrak{Z}(\mathring{\partial }({\mathrm{{\rm Y}}}^o,\mho ),\mathring{\partial }({\mathrm{{\rm Y}}}^o,\mho ))\hfill \\ \hfill & <\mathring{\partial }({\mathrm{{\rm Y}}}^o,\mho )-\mathring{\partial }({\mathrm{{\rm Y}}}^o,\mho )\hfill \\ \hfill & =0,\hfill \end{array}$$

which is a contradiction to the supposition. Accordingly, the FP of $\mathrm{\Theta }$ is unique.  □

Example 3.

Take $\mathbb{R}$ as the usual Banach space $\left|\right|·\left|\right|$ and a system of finite mappings $\{{\mathrm{\Theta }}^{\left(i\right)}:\mathbb{R}\to \mathbb{R},i=3,4,5\}$ by

$${\mathrm{\Theta }}^{\left(i\right)}\left({\iota }^o\right)=2-{\iota }^o-{\displaystyle \frac{2{\iota }^o}{i}},\forall i=3,4,5.$$

Then, for $\mathrm{d}=1$, we obtain $\delta ={\displaystyle \frac{1}{2}}$ and $\forall i=3,4,5$,

$${\mathrm{\Theta }}_{\delta }^{\left(i\right)}\left({\iota }^o\right)=1-{\displaystyle \frac{{\iota }^o}{i}},\forall {\iota }^o\in \mathbb{R}.$$

(27)

Therefore, $|{\mathrm{\Theta }}_{\delta }^{\left(i\right)}\left({\iota }^o\right)-{\mathrm{\Theta }}_{\delta }^{\left(i\right)}\left({\varrho }^o\right)|\le \pi |{\iota }^o-{\varrho }^o|,\forall {\iota }^o,{\varrho }^o\in \mathcal{X}$, where $\pi =max\{{\displaystyle \frac{1}{i}}:i=3,4,5\}={\displaystyle \frac{1}{3}}$. Considering $\mathfrak{Z}={\mathfrak{Z}}_E$ as defined in Corollary (2) by ${\mathfrak{Z}}_E({\iota }^o,{\varrho }^o)={\displaystyle \frac{1}{2}}{\varrho }^o-{\iota }^o,\forall {\varrho }^o,{\iota }^o\in [0,\infty ),$ then we obtain for $i=3,4,5$,

$$\begin{array}{ccc}\hfill \mathfrak{Z}\left(\right||{\mathrm{\Theta }}_{\delta }^{\left(i\right)}\left({\iota }^o\right)-{\mathrm{\Theta }}_{\delta }^{\left(i\right)}\left({\varrho }^o\right)||,||{\iota }^o-{\varrho }^o|\left|\right)& =& \mathfrak{Z}({\displaystyle \frac{1}{i}}|{\iota }^o-{\varrho }^o|,|{\iota }^o-{\varrho }^o|)\hfill \\ & =& {\displaystyle \frac{1}{2}}|{\iota }^o-{\varrho }^o|-{\displaystyle \frac{1}{i}}|{\iota }^o-{\varrho }^o|\hfill \\ & \ge & {\displaystyle \frac{1}{2}}|{\iota }^o-{\varrho }^o|-{\displaystyle \frac{1}{3}}|{\iota }^o-{\varrho }^o|\hfill \\ & =& {\displaystyle \frac{1}{6}}|{\iota }^o-{\varrho }^o|\hfill \\ \hfill \Rightarrow \zeta \left(\right||{\mathrm{\Theta }}_{\delta }^{\left(i\right)}\left({\iota }^o\right)-{\mathrm{\Theta }}_{\delta }^{\left(i\right)}\left({\varrho }^o\right)||,||{\iota }^o-{\varrho }^o|\left|\right)& \ge & 0.\hfill \end{array}$$

Thus, $(\mathbb{R},{\mathrm{\Theta }}^{\left(i\right)}:i=3,4,5)$ is an IFS via the enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-contractions. Therefore, the mapping $\mathrm{\Theta }:\mathrm{\Lambda }\left(\mathbb{R}\right)\to \mathrm{\Lambda }\left(\mathbb{R}\right)$ given by

$$\mathrm{\Theta }\left({\mathrm{{\rm Y}}}^o\right)={\cup }_{i=3}^5{\mathrm{\Theta }}^{\left(i\right)}\left({\mathrm{{\rm Y}}}^o\right),\forall {\mathrm{{\rm Y}}}^o\in \mathrm{\Lambda }\left(\mathbb{R}\right)$$

must satisfy the following by Theorem (4):

$$\mathfrak{Z}(\mathring{\partial }({\mathrm{\Theta }}_{\delta }^{\left(i\right)}\left({\mathrm{{\rm Y}}}^o\right),{\mathrm{\Theta }}_{\delta }^{\left(i\right)}(\mho )),\mathring{\partial }({\mathrm{{\rm Y}}}^o,\mho ))\ge 0,\forall {\mathrm{{\rm Y}}}^o,\mho \in \mathrm{\Lambda }\left(\mathcal{X}\right),\forall i=3,4,5.$$

Therefore, by Theorem 6, Θ has a unique FP, as Θ meets all of its requirements.

The intricate construction of enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-contractions is essential for deriving existence and uniqueness results because it broadens classical methods to address more complex scenarios and offers a more profound theoretical framework. Although classical methods may work for simpler cases, generalized contractions and their associated IFSs provide enhanced insights and solutions for more complex problems. This approach enables a more thorough analysis and application, especially for specialized integral equations and advanced operators such as the enriched $\mathcal{Z}$-Hutchinson operator.

5. Conclusions and Future Directions

In conclusion, we introduced a wide class of enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-contractions defined on Banach spaces and established the existence and uniqueness of their FPs. To validate our findings, we gave a concrete example. In addition, we demonstrated an existence condition confirming the uniqueness of the solution to an integral equation. Moreover, we defined the IFS associated with enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-contractions in Banach spaces and defined the enriched $\mathcal{Z}$-Hutchinson operator. We also established a result on the convergence of Krasnoselskii’s iteration method and the uniqueness of the attractor via enriched $(\mathrm{d},\mathfrak{Z})$-$\mathcal{Z}$-contractions. As a result, our findings not only confirm but also significantly build upon and broaden several established results.

In future work, it would be interesting to examine whether it is possible to deduce Kannan, Chatterjea, interpolative Kannan, and interpolative Chatterjea-type contractions and their FP results in the context of $\mathcal{Z}$-type contractions via enriched techniques. Additionally, investigating the same task for cyclic contractions via enriched techniques could provide valuable insights and further extend the applicability of enriched contractions in various mathematical contexts.