New α-ɛ-Suzuki-Type Contraction Mapping Methods on Fractional Differential and Integral Equations

Abstract

This paper introduces novel formulations in the framework of $\alpha -\mathcal{E}$-contractions within the context of admissible mappings. We establish new fixed point theorems for $\alpha -\mathcal{E}$-Suzuki-type contractions, thereby generalizing and extending the foundational work of Hossein Piri and Poom Kumam. The principal objective of this research is to investigate the existence and uniqueness of solutions to a class of integral equations by leveraging fixed point methodologies in complete metric spaces. By developing these advanced $\alpha -\mathcal{E}$-contraction concepts and analyzing their implications for admissible mappings, this work contributes to the theoretical advancement of fixed point theory. To support our new definition, we illustrate examples. The results demonstrate the efficacy of this approach for addressing nonlinear problems in analysis, specifically enriching the methodology for solving integral equations. The overarching aim is to consolidate the theoretical underpinnings and provide a rigorous analytical framework for the application of $\alpha -\mathcal{E}$-contraction mappings, thereby fostering further progress in mathematical analysis.

1. Introduction

The study of fixed point theory in metric spaces has been a principal focus of mathematical research for decades. This field has expanded considerably through investigations in diverse abstract spaces and the development of numerous generalizations of the standard metric space, which have provided new frameworks for analysis. A significant corpus of work, exemplified by various authors, demonstrates the field’s extensive scope and the sophistication of its methodologies. The theory of fixed points has evolved through a series of fundamental contributions that have significantly influenced modern nonlinear and fractional analysis. The classical Banach (1922) contraction principle laid the groundwork for fixed point theory by guaranteeing the existence and uniqueness of fixed points in complete metric spaces.

This foundational idea was further enriched by Edelstein [1] (1962), who explored fixed and periodic points under contractive mappings, and by $\check{\mathrm{C}}$iri$\check{\mathrm{c}}$ (1971) [2], who extended Banach’s [3] principle to generalized contractions, thereby widening its applicability to broader classes of nonlinear operators. Later in 2009, Suzuki [4] introduced a new type of fixed point theorem in metric spaces by relaxing the classical contraction conditions, providing a more flexible analytical framework. Advancing these developments, Secelean [5] (2013) employed F-contractions in iterated function systems, linking fixed point theory with fractal structures and self-similar dynamics. Further generalizations appeared in the works of Gordji et al. [6] (2017), who examined fixed points in orthogonal sets, and Cho [7] (2017), who provided a comprehensive survey outlining the progress and applications of metric fixed point theory across real and complex analysis.

More recently, researchers have extended these classical concepts to advanced analytical frameworks such as fractional differential systems and generalized metric spaces. For instance, Etemad et al. [8] (2020, 2022) investigated $(\alpha -\psi )$-contractions and tripled fixed point results in fractional and impulsive differential systems, highlighting the intersection between fixed point theory and fractional calculus [9]. Similarly, Joshi et al. [10] (2023) applied geometric interpretations of fixed points to satellite web coupling problems in S-metric spaces, while Rezazgui et al. [11] (2023) and Shatanawi and Shatnawi [12] (2023) developed novel contractive conditions in extended quasi b-metric- and controlled metric-type spaces, respectively. Collectively, these works illustrate the continuous expansion and versatility of fixed point theory—from classical contraction principles to modern fractional and generalized frameworks—underscoring its central role in advancing nonlinear analysis and applied mathematics.

A significant development in the theory of $\mathcal{E}$-contractions was introduced by Piri and Kumam [13], who refined Wardowski’s original framework [14] by replacing axiom $\left(\mathcal{E}3\right)$ with a continuity condition $\left(\mathcal{E}{3}^{\prime }\right)$, requiring $\mathcal{E}$ to be continuous on $(0,\infty )$. Concurrently, Samet et al. [15] established the concept of $\alpha$-admissible mappings, which facilitated the generalization of contractive conditions via $\alpha -\psi$-contractions in metric spaces. Some fixed point theorems using $\mathrm{O}-\alpha -\psi$-Geraghty-type mapping were proved by Prakasam and associates [16]. Monfared et al. [17] demonstrated a class of $\alpha$-admissible $\mathrm{F}(\psi ,\varphi )$-contractions in $\mathrm{M}$-metric spaces and establish new fixed point results. These dual advances precipitated extensive research into fixed points for various contractions defined with respect to $\alpha$-admissible mappings, as documented by many authors.

The concept of $(\alpha -\psi )$-contractive mappings has attracted considerable attention for its ability to unify and extend various contraction principles in fixed point theory. Karapinar et al. [18] (2013) introduced the notion of $(\alpha -\psi )$-Meir–Keeler contractive mappings, establishing new fixed point results that generalized classical contraction frameworks. Building upon this foundation, Karapinar (2014) [19] further developed $(\alpha -\psi )$-Geraghty-type contractions, offering broader applicability to nonlinear mappings in abstract spaces. Subsequently, Alsulami et al. (2015) [20] extended these ideas to $(\alpha -\psi )$-rational-type contractive mappings, thereby enriching the structure of fixed point theory and demonstrating its potential in more generalized metric and analytical settings. Collectively, these studies have significantly advanced the theoretical depth and flexibility of contractive mapping approaches in modern fixed point analysis.

Building upon the foundational results of Piri and Kumam [13], this work presents a generalization of their principal theorem. The novel concept of $\theta -\varphi$-multivalued contractions is established for an $\alpha$-admissible mapping in the context of $\mathrm{b}$-metric space by Taqbibt and associates [21]. Kumar et al. [22] defined $\alpha$-admissible $\mathrm{E}$-type contractions via simulation functions and established related fixed point theorems in complete supra metric spaces. We achieve this by defining a novel class of $\alpha -\mathcal{E}$-contractions for $\alpha$-admissible mappings in metric spaces. The objective of this study is to advance the theoretical understanding of the synthesis between $\alpha$-admissibility and $\mathcal{E}$-contractions, thereby contributing new fixed point theorems to the literature.

This manuscript is organized as follows: In Section 1, we discuss the existing literature relevant to our work. In Section 2, we provide the basic definitions and existing results we need to understand our main work. Section 3 contains the novel work of this manuscript. In Section 4, we discuss the uniqueness and existence of a solution for an integral equation. In Section 5, we apply our result to find the solution of a fractional-order derivative.

2. Preliminaries

Throughout this paper, the symbols $\mathcal{R}$, ${\mathcal{R}}^{+}$ and $\mathbb{N}$ represent the set of all real numbers, all positive real numbers, and all natural numbers, respectively.

Wardowski [14] established the concept of $\mathcal{E}$-contraction as follows.

Definition 1 

([14]). A collection of functions $\mathcal{E}\in \mathcal{F}$ mapping from $[0,\infty )$ into $(-\infty ,+\infty )$ has the following conditions:

($\mathcal{E}$₁)

$\mathcal{E}$ is strictly increasing; i.e., $\forall \eta ,\mu \in [0,\infty )$ such that $\eta <\mu ,\mathcal{E}\left(\eta \right)<\mathcal{E}\left(\mu \right)$.

($\mathcal{E}$₂)

For each sequence ${\left\{{\eta }_{\ell }\right\}}_{\ell \in \mathbb{N}}$ of $[0,\infty )$, $\underset{\ell \to \infty }{lim}{\eta }_{\ell }=0⟺\underset{\ell \to \infty }{lim}\mathcal{E}\left({\eta }_{\ell }\right)=-\infty$.

($\mathcal{E}$₃)

For every $\kappa \in (0,1)$, $\underset{\eta \to 0^{+}}{lim}{\eta }^{\kappa }\mathcal{E}\left(\eta \right)=0$.

Definition 2 

([14]). A self-map $\mathcal{Q}$ on $\mathcal{S}$ is called an $\mathcal{E}$-contraction if $\exists \mathcal{E}\in \mathcal{F}$ and $\mathfrak{b}>0$ s.t. $\forall \mathrm{x},\mathrm{y}\in \mathcal{S}$, $\mathrm{d}(\mathcal{Q}\mathrm{x},\mathcal{Q}\mathrm{y})>0$ gives $\mathfrak{b}+\mathcal{E}\left(\mathrm{d}\right(\mathcal{Q}\mathrm{x},\mathcal{Q}\mathrm{y}\left)\right)\le \mathcal{E}\left(\mathrm{d}\right(\mathrm{x},\mathrm{y}\left)\right)$.

Edelstein [1] demonstrated a version of the following theorem in 1962.

Theorem 1 

([1]). Consider a self-map $\mathcal{Q}$ on $\mathcal{S}$. Assume that $\mathrm{d}(\mathcal{Q}\mathrm{x},\mathcal{Q}\mathrm{y})<\mathrm{d}(\mathrm{x},\mathrm{y})$ is true $\forall \mathrm{x},\mathrm{y}\in \mathcal{S}$ with $\mathrm{x}\ne \mathrm{y}$. Then $\mathcal{Q}$ has a unique fixed point (shortly, $\mathit{UFP}$).

Suzuki, in 2008 [4], showed that Edelstein’s conclusions could be generalized in a compact metric space.

Theorem 2 

([4]). Let $\mathcal{Q}$ be a self-map on a compact metric space $\mathcal{S}$. Assume that $\mathrm{d}(\mathcal{Q}\mathrm{x},\mathcal{Q}\mathrm{y})<\mathrm{d}(\mathrm{x},\mathrm{y})$ is true $\forall \mathrm{x},\mathrm{y}\in \mathcal{S}$ with $\mathrm{x}\ne \mathrm{y}$,

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}\mathrm{d}(\mathrm{x},\mathcal{Q}\mathrm{x})<\mathrm{d}(\mathrm{x},\mathrm{y})\Rightarrow \mathrm{d}(\mathcal{Q}\mathrm{x},\mathcal{Q}\mathrm{y})<\mathrm{d}(\mathrm{x},\mathrm{y}).\end{array}$$

Then $\mathcal{Q}$ has a $\mathit{UFP}$ in $\mathcal{S}$.

Wardowski [14] demonstrated a version of the following theorem in 2012.

Theorem 3 

([14]). Consider an $\mathcal{E}$-contractive self-map $\mathcal{Q}$ on a complete metric space $\mathcal{S}$. Then, we have the following:

1. 

$\mathcal{Q}$ has a $\mathit{UFP}$ φ in $\mathcal{S}$;

2. 

For all $\mathrm{x}\in \mathcal{S}$, the sequence $\left\{{\mathcal{Q}}^{\ell }\left(x\right)\right\}$ is convergent to φ in $\mathcal{S}$.

Secelean [5] demonstrated the following lemma.

Lemma 1 

([5]). Consider an increasing mapping $\mathcal{E}:{\mathcal{R}}^{+}\to \mathcal{R}$ and a sequence ${\left\{{\eta }_{\ell }\right\}}_{\ell =1}^{\infty }$ of ${\mathcal{R}}^{+}$. Then the following conditions hold:

1. 

If $\underset{\ell \to \infty }{lim}\mathcal{E}\left({\eta }_{\ell }\right)=-\infty$, then $\underset{\ell \to \infty }{lim}{\eta }_{\ell }=0;$

2. 

If infimum $\left(\mathcal{E}\right)=-\infty$, and $\underset{\ell \to \infty }{lim}{\eta }_{\ell }=0$, then $\underset{\ell \to \infty }{lim}\mathcal{E}\left({\eta }_{\ell }\right)=-\infty$.

Secelean established that the condition (${\mathcal{E}}_2$) in Definition 1 may be retrieved by establishing Lemma 1, and Hossan Piri and Poom Kumam demonstrated that the condition (${\mathcal{E}}_3$) in Definition 1 may be retrieved by a condition that is comparable but simpler.

Definition 3 

([13]). The set of all functions $\mathcal{E}\in \mathrm{F}:{\mathcal{R}}^{+}\to \mathcal{R}$ fulfills the following:

($\mathcal{E}$₁)

$\mathcal{E}$ is strictly increasing; i.e., $\forall \eta ,\mu \in {\mathcal{R}}^{+}$ such that $\eta <\mu ,\mathcal{E}\left(\eta \right)<\mathcal{E}\left(\mu \right)$.

(${\mathcal{E}}_2^{\prime }$)

Infimum $\left(\mathcal{E}\right)=-\infty$.

(or)

(${\mathcal{E}}_2^{″}$)

There exists a sequence ${\left\{{\eta }_{\ell }\right\}}_{\ell =1}^{\infty }$ of ${\mathcal{R}}^{+}$ such that $\underset{\ell \to \infty }{lim}\mathcal{E}\left({\eta }_{\ell }\right)=-\infty$.

(${\mathcal{E}}_3^{\prime }$)

$\mathcal{E}$ is continuous on ${\mathcal{R}}^{+}$.

Example 1 

([13]). Let ${\mathcal{E}}_1\left(\eta \right)={\displaystyle \frac{-1}{\eta }},{\mathcal{E}}_2\left(\eta \right)={\displaystyle \frac{-1}{\eta }}+\eta ,{\mathcal{E}}_3\left(\eta \right)={\displaystyle \frac{1}{1-{\mathfrak{e}}^{\eta }}}$. Then ${\mathcal{E}}_1,{\mathcal{E}}_2,{\mathcal{E}}_3\in \mathrm{F}$.

Definition 4 

([2]). Let $\mathcal{Q}$ be a self-map on a metric space $(\mathcal{S},\mathrm{d})$, which is known as orbitally continuous on $\mathcal{S}$ if $\underset{\mathrm{n}\to \infty }{lim}{\mathcal{Q}}^{{\ell }_{\mathrm{n}}}\left(\mathrm{x}\right)=\phi \Rightarrow \underset{\mathrm{n}\to \infty }{lim}{\mathcal{Q}}^{{\ell }_{\mathrm{n}}}\left(\mathrm{x}\right)=\mathcal{Q}\phi$.

Let $\mathcal{Q}$ be a self-map on $\mathcal{S}\{\ne \varnothing \}$. We denote $Fix\left(\mathcal{Q}\right)=\{\mathrm{x}:\mathcal{Q}\mathrm{x}=\mathrm{x}\forall \mathrm{x}\in \mathcal{S}\}$.

Definition 5 

([15]). Consider a self-map $\mathcal{Q}$ on $\mathcal{S}\{\ne \varnothing \}$ and a mapping $\alpha :\mathcal{S}\times \mathcal{S}\to [0,\infty )$; then $\mathcal{Q}$ is α-admissible if $\mathrm{x},\mathrm{y}\in \mathcal{S},\alpha (\mathrm{x},\mathrm{y})\ge 1\Rightarrow \alpha (\mathcal{Q}\mathrm{x},\mathcal{Q}\mathrm{y})\ge 1$.

Definition 6 

([18]). Let $\mathcal{Q}$ be a self-map on $\mathcal{S}$ and consider a mapping $\alpha :\mathcal{S}\times \mathcal{S}\to (-\infty ,+\infty )$. Then $\mathcal{Q}$ is triangular α-admissible if

1. 

$\alpha (\mathrm{x},\mathrm{y})\ge 1\Rightarrow \alpha (\mathcal{Q}\mathrm{x},\mathcal{Q}\mathrm{y})\ge 1$;

2. 

$\alpha (\mathrm{x},\phi )\ge 1$, and $\alpha (\phi ,\mathrm{y})\ge 1\Rightarrow \alpha (\mathrm{x},\mathrm{y})\ge 1\forall \mathrm{x},\mathrm{y},\phi \in \mathcal{S}$.

Example 2.

Let $\mathcal{Q}:[0,\infty )\to [0,\infty )$ and $\alpha :[0,\infty )\times [0,\infty )\to (-\infty ,+\infty )$ defined by $\mathcal{Q}\mathrm{x}=In(1+\mathrm{x})\forall \mathrm{x}\in [0,\infty )$ and

$$\alpha (\mathrm{x},\mathrm{y})=\left\{\begin{array}{c}1+\mathrm{x},if\mathrm{x}\ge \mathrm{y};\hfill \\ 0,else.\hfill \end{array}\right.$$

Then, by Definition 5, $\alpha (\mathrm{x},\mathrm{y})\ge 1\Rightarrow \alpha (\mathcal{Q}\mathrm{x},\mathcal{Q}\mathrm{y})\ge 1\forall \mathrm{x}\ge \mathrm{y}$ and $\alpha (\mathrm{x},\mathrm{y})=\alpha (\mathrm{y},\mathrm{x})\forall \mathrm{x}=\mathrm{y}$.

Definition 7 

([19]). Let $\mathcal{S}$ be a non-void set and let $\mathcal{Q}$ be an α-admissible map on $\mathcal{S}$. Then, $\mathcal{S}$ has the following condition:

• (H) if each $\mathrm{x},\mathrm{y}\in Fix\left(\mathcal{Q}\right),\exists \phi \in \mathcal{S}$ such that $\alpha (\mathrm{x},\phi )\ge 1$ and $\alpha (\mathrm{y},\phi )\ge 1$.

Definition 8 

([23]). Let $\mathcal{S}$ be a non-void set and let $\mathcal{Q}$ be an α-admissible map on $\mathcal{S}$. Then $\mathcal{Q}$ is ${\alpha }^{*}$-admissible if $\forall \mathrm{x},\mathrm{y}\in Fix\left(\mathcal{Q}\right)\ne \varnothing$, we have $\alpha (\mathrm{x},\mathrm{y})\ge 1$. If $Fix\left(\mathcal{Q}\right)=\varnothing$, we say that $\mathcal{Q}$ is vacuously ${\alpha }^{*}$-admissible.

Some authors, such as Alsulami et al. [20], Khan et al. [24], and others, have used the idea above without its nomenclature concerning the uniqueness of the fixed point.

Example 3.

Define $\mathcal{Q}:[0,\infty )\to [0,\infty )$ and $\alpha :[0,\infty )\times [0,\infty )\to (-\infty ,+\infty )$ by $\mathcal{Q}\mathrm{x}=1+\mathrm{x}\forall \mathrm{x}\in [0,\infty )$ and 

$$\alpha (\mathrm{x},\mathrm{y})=\left\{\begin{array}{c}{\mathfrak{e}}^{2(\mathrm{x}-\mathrm{y})},if\mathrm{x}\ge \mathrm{y};\hfill \\ 0,else.\hfill \end{array}\right.$$

Then, $\mathcal{Q}$ is α-admissible. Here, $\mathcal{Q}$ is vacuously ${\alpha }^{*}$-admissible because $\mathcal{Q}$ has no fixed point; i.e., $Fix\left(\mathcal{Q}\right)=\varnothing$.

In the first part of this section, the idea of an $\alpha -\mathcal{E}$-contraction is defined. The second part introduces the notion of an $\alpha -\mathcal{E}$-Suzuki contraction. Some fixed point theorems are made for these contractions, and relevant applications are given to show that the hypotheses that led to our results and our generalizations are realistic. We commence our main result with the following definition.

3. Main Results

In this part, we define the class of $\alpha$-$\mathcal{E}$-contractions.

Definition 9.

We say that a self-map $\mathcal{Q}$ on $\mathcal{S}$ is an α-$\mathcal{E}$-contraction if a function $\alpha :\mathcal{S}\times \mathcal{S}\to [0,\infty )$ exists and $\mathcal{E}\in \mathcal{F}$ such that

$$\begin{array}{c}\hfill \mathrm{d}(\mathcal{Q}\mathrm{x},\mathcal{Q}\mathrm{y})>0\Rightarrow \mathfrak{b}+\mathcal{E}\left(\alpha \right(\mathrm{x},\mathrm{y}\left)\mathrm{d}\right(\mathcal{Q}\mathrm{x},\mathcal{Q}\mathrm{y}\left)\right)\le \mathcal{E}\left(\mathrm{d}\right(\mathrm{x},\mathrm{y}\left)\right),\forall \mathrm{x},\mathrm{y}\in \mathcal{S}.\end{array}$$

(1)

Lemma 2.

For an α-$\mathcal{E}$-contractive self-map $\mathcal{Q}$ on metric space $(\mathcal{S},\mathrm{d})$,the given conditions hold:

(i)

$\mathcal{Q}$ is α-admissible;

(ii)

$\exists {\mathrm{x}}_0\in \mathcal{S}$ such that $\alpha ({\mathrm{x}}_0,\mathcal{Q}{\mathrm{x}}_0)\ge 1$.

Also, construct a sequence $\left\{{\mathrm{x}}_{\ell }\right\}\in \mathcal{S}$ by ${\mathrm{x}}_{\ell +1}={\mathcal{Q}}^{\ell +1}{\mathrm{x}}_0=\mathcal{Q}{\mathrm{x}}_{\ell },\forall \ell \in \mathbb{N}\cup \left\{0\right\}$. Then, $\alpha ({\mathrm{x}}_{\ell },{\mathrm{x}}_{\ell +1})>1,\forall \ell \ge 0$ and

$$\mathcal{E}\left(\mathrm{d}({\mathrm{x}}_{\ell },\mathcal{Q}{\mathrm{x}}_{\ell })\right)=\mathcal{E}\left(\alpha ({\mathrm{x}}_{\ell -1},{\mathrm{x}}_{\ell })\mathrm{d}({\mathrm{x}}_{\ell },\mathcal{Q}{\mathrm{x}}_{\ell })\right)\le \mathcal{E}\left(\mathrm{d}({\mathrm{x}}_0,\mathcal{Q}{\mathrm{x}}_0)\right)-\ell \mathfrak{b}.$$

Proof. 

Let ${\mathrm{x}}_0\in \mathcal{S}$. It follows that $\alpha (\mathcal{Q}{\mathrm{x}}_0,{\mathrm{x}}_0)\ge 1$, and we construct a sequence $\left\{{\mathrm{x}}_{\ell }\right\}$ by ${\mathrm{x}}_{\ell +1}=\mathcal{Q}{\mathrm{x}}_{\ell }={\mathcal{Q}}^{\ell +1}{\mathrm{x}}_0\forall \ell \in \mathbb{N}\cup \left\{0\right\}$.

By Definition 5, $\alpha ({\mathrm{x}}_0,{\mathrm{x}}_1)=\alpha ({\mathrm{x}}_0,\mathcal{Q}{\mathrm{x}}_0)\ge 1\Rightarrow \alpha ({\mathrm{x}}_1,{\mathrm{x}}_2)=\alpha (\mathcal{Q}{\mathrm{x}}_0,{\mathcal{Q}}^2{\mathrm{x}}_0)\ge 1$. Therefore, we obtain inductively that $\alpha ({\mathrm{x}}_{\ell },{\mathrm{x}}_{\ell +1})\ge 1\forall \ell \ge 0$. Assume that ${\mathrm{x}}_{\ell }\ne {\mathrm{x}}_{\ell +1}\forall \ell \ge 0$. Then, $\mathrm{d}({\mathrm{x}}_{\ell },{\mathrm{x}}_{\ell +1})>0\forall \ell \ge 0$.

According to condition ${\mathcal{E}}_1$ and $\alpha ({\mathrm{x}}_0,{\mathrm{x}}_1)\ge 1$, by Equation (1), we get

$$\begin{array}{ccc}\hfill \mathcal{E}\left(\mathrm{d}({\mathrm{x}}_2,{\mathrm{x}}_3)\right)& =& \mathcal{E}\left(\mathrm{d}(\mathcal{Q}{\mathrm{x}}_1,\mathcal{Q}{\mathrm{x}}_2)\right)\hfill \\ & \le & \mathcal{E}\left(\alpha ({\mathrm{x}}_1,{\mathrm{x}}_2)\mathrm{d}(\mathcal{Q}{\mathrm{x}}_1,\mathcal{Q}{\mathrm{x}}_2)\right)\hfill \\ & \le & \mathcal{E}\left(\mathrm{d}({\mathrm{x}}_1,{\mathrm{x}}_2)\right)-\mathfrak{b}<\mathcal{E}\left(\mathrm{d}({\mathrm{x}}_1,{\mathrm{x}}_2)\right).\hfill \end{array}$$

(2)

Since $\mathfrak{b}>0$, and by condition ${\mathcal{E}}_1$, we get

$$\begin{array}{c}\hfill \mathrm{d}({\mathrm{x}}_2,{\mathrm{x}}_3)<\mathrm{d}({\mathrm{x}}_1,{\mathrm{x}}_2).\end{array}$$

Again by Equation (1), we have

$$\begin{array}{ccc}\hfill \mathcal{E}\left(\mathrm{d}({\mathrm{x}}_3,{\mathrm{x}}_4)\right)& =& \mathcal{E}\left(\mathrm{d}(\mathcal{Q}{\mathrm{x}}_2,\mathcal{Q}{\mathrm{x}}_3)\right)\hfill \\ & \le & \mathcal{E}\left(\alpha ({\mathrm{x}}_2,{\mathrm{x}}_3)\mathrm{d}(\mathcal{Q}{\mathrm{x}}_2,\mathcal{Q}{\mathrm{x}}_3)\right)\hfill \\ & \le & \mathcal{E}\left(\mathrm{d}({\mathrm{x}}_2,{\mathrm{x}}_3)\right)-\mathfrak{b}<\mathcal{E}\left(\mathrm{d}({\mathrm{x}}_2,{\mathrm{x}}_3)\right).\hfill \end{array}$$

(3)

Since $\mathfrak{b}>0$, and by condition ${\mathcal{E}}_1$, it follows that

$$\begin{array}{c}\hfill \mathrm{d}({\mathrm{x}}_3,{\mathrm{x}}_4)<\mathrm{d}({\mathrm{x}}_2,{\mathrm{x}}_3)<\mathrm{d}({\mathrm{x}}_1,{\mathrm{x}}_2).\end{array}$$

In this manner, we show inductively that $\mathcal{S}$ has a strictly non-increasing sequence $\left\{\mathrm{d}({\mathrm{x}}_{\ell },{\mathrm{x}}_{\ell +1})\right\}$ in $\mathcal{S}$. □

In the framework of $\alpha$-admissible mapping, we prove the given $\mathcal{E}$-contraction theorem, which is inspired by the work of Hossein Piri and Poom Kumam [13].

Theorem 4.

Let $(\mathcal{S},\mathrm{d})$ be a complete metric space and $\mathcal{Q}:\mathcal{S}\to \mathcal{S}$ be an α-$\mathcal{E}$-contraction; the subsequent assertions hold:

(i)

$\mathcal{Q}$ is α-admissible;

(ii)

$\exists {\mathrm{x}}_0\in \mathcal{S}$ such that $\alpha ({\mathrm{x}}_0,\mathcal{Q}{\mathrm{x}}_0)\ge 1$;

(iii)

$\mathcal{Q}$ is continuous or orbitally continuous on $\mathcal{S}$.

Then, the point $\phi \in \mathcal{S}$ is a fixed point of $\mathcal{Q}$. Moreover, if $\mathcal{Q}$ is ${\alpha }^{*}$-admissible, then the point $\phi \in \mathcal{S}$ is a $\mathit{UFP}$ of $\mathcal{Q}$. Furthermore, for every ${\mathrm{x}}_0\in \mathcal{S}$ if ${\mathrm{x}}_{\ell +1}={\mathcal{Q}}^{\ell +1}{\mathrm{x}}_0\ne \mathcal{Q}{\mathrm{x}}_{\ell }\forall \ell \ge 0$, then $\underset{\ell \to \infty }{lim}{\mathcal{Q}}^{\ell }{\mathrm{x}}_0=\phi$.

Proof. 

Let ${\mathrm{x}}_0\in \mathcal{S}$ be such that $\alpha (\mathcal{Q}{\mathrm{x}}_0,{\mathrm{x}}_0)\ge 1$ and construct a sequence $\left\{{\mathrm{x}}_{\ell }\right\}$ by ${\mathrm{x}}_{\ell +1}=\mathcal{Q}{\mathrm{x}}_{\ell }={\mathcal{Q}}^{\ell +1}{\mathrm{x}}_0\forall \ell \ge 0$. If ${\mathrm{x}}_{{\ell }_0}={\mathrm{x}}_{{\ell }_0+1}$, it follows that $\mathcal{Q}{\mathrm{x}}_{{\ell }_0}={\mathrm{x}}_{{\ell }_0}$ for some ${\ell }_0\ge 0$; then ${\mathrm{x}}_{{\ell }_0}$ in $\mathcal{S}$ is a fixed point of $\mathcal{Q}$.

Now, suppose ${\mathrm{x}}_{\ell }\ne {\mathrm{x}}_{\ell +1}\forall \ell \ge 0$. Then, $\mathrm{d}({\mathrm{x}}_{\ell },{\mathrm{x}}_{\ell +1})>0\forall \ell \ge 0$. By Lemma 2, we get $\alpha ({\mathrm{x}}_{\ell },{\mathrm{x}}_{\ell +1})>1\forall \ell \ge 0$ Therefore, $\alpha ({\mathrm{x}}_{\ell },{\mathrm{x}}_{\ell +1})=\alpha ({\mathcal{Q}}^{\ell }{\mathrm{x}}_0,{\mathcal{Q}}^{\ell +1}{\mathrm{x}}_0)\ge 1\forall \ell \ge 0$. According to ${\mathcal{E}}_1$ and Definition 5,

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}\mathrm{d}({\mathrm{x}}_{\ell },\mathcal{Q}{\mathrm{x}}_{\ell })={\displaystyle \frac{1}{2}}\mathrm{d}({\mathrm{x}}_{\ell },{\mathrm{x}}_{\ell +1})<\mathrm{d}({\mathrm{x}}_{\ell },{\mathrm{x}}_{\ell +1}).\end{array}$$

Now, by Equation (14), we get

$$\begin{array}{cc}\hfill \mathcal{E}\left(\mathrm{d}({\mathrm{x}}_{\ell },{\mathrm{x}}_{\ell +1})\right)& =\mathcal{E}\left(\mathrm{d}(\mathcal{Q}{\mathrm{x}}_{\ell -1},\mathcal{Q}{\mathrm{x}}_{\ell })\right)\hfill \\ & =\mathcal{E}\left(\alpha ({\mathrm{x}}_{\ell -1},{\mathrm{x}}_{\ell })\mathrm{d}(\mathcal{Q}{\mathrm{x}}_{\ell -1},\mathcal{Q}{\mathrm{x}}_{\ell })\right)\hfill \\ & \le \mathcal{E}\left(\mathrm{d}({\mathrm{x}}_{\ell -1},{\mathrm{x}}_{\ell })\right)-\mathfrak{b}.\hfill \end{array}$$

(4)

Repeating this process, we get

$$\begin{array}{cc}\hfill \mathcal{E}\left(\mathrm{d}(\mathcal{Q}{\mathrm{x}}_{\ell -1},\mathcal{Q}{\mathrm{x}}_{\ell })\right)& \le \mathcal{E}\left(\mathrm{d}({\mathrm{x}}_{\ell -1},{\mathrm{x}}_{\ell })\right)-\mathfrak{b}\hfill \\ & =\mathcal{E}\left(\mathrm{d}(\mathcal{Q}{\mathrm{x}}_{\ell -2},\mathcal{Q}{\mathrm{x}}_{\ell -1})\right)-\mathfrak{b}\hfill \\ & =\mathcal{E}\left(\alpha ({\mathrm{x}}_{\ell -2},{\mathrm{x}}_{\ell -1})\mathrm{d}(\mathcal{Q}{\mathrm{x}}_{\ell -2},\mathcal{Q}{\mathrm{x}}_{\ell -1})\right)-\mathfrak{b}\hfill \\ & \le \mathcal{E}\left(\mathrm{d}({\mathrm{x}}_{\ell -2},{\mathrm{x}}_{\ell -1})\right)-2\mathfrak{b}\hfill \\ & =\mathcal{E}\left(\mathrm{d}(\mathcal{Q}{\mathrm{x}}_{\ell -3},\mathcal{Q}{\mathrm{x}}_{\ell -2})\right)-2\mathfrak{b}\hfill \\ & =\mathcal{E}\left(\alpha (x_{n-3},{\mathrm{x}}_{\ell -2})\mathrm{d}(\mathcal{Q}{\mathrm{x}}_{\ell -3},\mathcal{Q}{\mathrm{x}}_{\ell -2})\right)-2\mathfrak{b}\hfill \\ & \le \mathcal{E}\mathrm{d}({\mathrm{x}}_{\ell -3},{\mathrm{x}}_{\ell -2})-3\mathfrak{b}\hfill \\ & .\hfill \\ & .\hfill \\ & .\hfill \\ & \le \mathcal{E}\left(\mathrm{d}({\mathrm{x}}_0,{\mathrm{x}}_1)\right)-\ell \mathfrak{b}.\hfill \end{array}$$

(5)

Taking the limit on both sides, we get

$$\begin{array}{c}\hfill \underset{\ell \to \infty }{lim}\mathcal{E}\left(\mathrm{d}(\mathcal{Q}{\mathrm{x}}_{\ell -1},\mathcal{Q}{\mathrm{x}}_{\ell })\right)=-\infty .\end{array}$$

(6)

Therefore, by (${\mathcal{E}}_2$) of Definition 1 with Equation (6), we get

$$\begin{array}{c}\hfill \underset{\ell \to \infty }{lim}\mathrm{d}(\mathcal{Q}{\mathrm{x}}_{\ell -1},\mathcal{Q}{\mathrm{x}}_{\ell })=0.\end{array}$$

(7)

By (${\mathcal{E}}_3$) of Definition 1, $\exists \kappa$ in $(0,1)$ such that

$$\begin{array}{c}\hfill \underset{\ell \to \infty }{lim}{\left(\mathrm{d}({\mathrm{x}}_{\ell },{\mathrm{x}}_{\ell +1})\right)}^{\kappa }\mathcal{E}\left(\mathrm{d}({\mathrm{x}}_{\ell },{\mathrm{x}}_{\ell +1})\right)=0.\end{array}$$

(8)

Also, by Equation (5), we have

$$\begin{array}{c}\hfill {\left[\mathrm{d}({\mathrm{x}}_{\ell },{\mathrm{x}}_{\ell +1})\right]}^{\kappa }[\mathcal{E}\left(\mathrm{d}({\mathrm{x}}_{\ell },{\mathrm{x}}_{\ell +1})\right)-\mathcal{E}\left(\mathrm{d}({\mathrm{x}}_0,{\mathrm{x}}_1)\right)]\le -{\left[\mathrm{d}({\mathrm{x}}_{\ell },{\mathrm{x}}_{\ell +1})\right]}^{\kappa }\ell \mathfrak{b}\le 0.\end{array}$$

(9)

Taking $\ell \to \infty$ in the above equation along with Equations (7) and (8), we get

$$\begin{array}{c}\hfill \underset{\ell \to \infty }{lim}\ell {\left[\mathrm{d}({\mathrm{x}}_{\ell },{\mathrm{x}}_{\ell +1})\right]}^{\kappa }=0.\end{array}$$

(10)

Now, we will present two cases.

• Case-(i): Suppose ℓ is even; by Equation (10), we get

  $$\begin{array}{c}\hfill \underset{\ell \to \infty }{lim}\ell {\left[\mathrm{d}({\mathrm{x}}_{\ell },{\mathrm{x}}_{\ell +1})\right]}^{\kappa }=0.\end{array}$$

  (11)
• Case-(ii): Suppose ℓ is odd; by Equation (10), we get

  $$\begin{array}{c}\hfill \underset{\ell \to \infty }{lim}(\ell -1){\left[\mathrm{d}({\mathrm{x}}_{\ell },{\mathrm{x}}_{\ell +1})\right]}^{\kappa }=0.\end{array}$$

  (12)

By Equations (7) and (12), we have

$$\begin{array}{c}\hfill \underset{\ell \to \infty }{lim}\ell {\left[\mathrm{d}({\mathrm{x}}_{\ell },{\mathrm{x}}_{\ell +1})\right]}^{\kappa }=0.\end{array}$$

(13)

From the above we observe that $\exists {\ell }_1\in \mathbb{N}$ such that

$$\begin{array}{c}\hfill \ell {\left[\mathrm{d}({\mathrm{x}}_{\ell },{\mathrm{x}}_{\ell +1})\right]}^{\kappa }\le 1\forall \ell \ge {\ell }_1.\end{array}$$

Therefore, we have

$$\begin{array}{c}\hfill \mathrm{d}({\mathrm{x}}_{\ell },{\mathrm{x}}_{\ell +1})\le {\displaystyle \frac{1}{{\ell }^{\frac{1}{\kappa }}}}\forall \ell \ge {\ell }_1.\end{array}$$

Next, we prove that the sequence $\left\{{\mathrm{x}}_{\ell }\right\}$ is Cauchy. Now, $\forall \wp >\mathrm{r}\ge {\ell }_1$, we have

$$\begin{array}{cc}\hfill \mathrm{d}({\mathrm{x}}_{\wp },{\mathrm{x}}_{\mathrm{r}})& \le \mathrm{d}({\mathrm{x}}_{\wp },{\mathrm{x}}_{\wp -1})+\mathrm{d}({\mathrm{x}}_{\wp -1},{\mathrm{x}}_{\wp -2})+\mathrm{d}({\mathrm{x}}_{\wp -2},{\mathrm{x}}_{\wp -3})+\dots \dots +\mathrm{d}({\mathrm{x}}_{\mathrm{r}+1},{\mathrm{x}}_{\mathrm{r}})\hfill \\ & <\sum _{\mathrm{n}=\mathrm{r}}^{\infty }\mathrm{d}({\mathrm{x}}_{\mathrm{n}},{\mathrm{x}}_{\mathrm{n}+1})\hfill \\ & \le \sum _{\mathrm{n}=\mathrm{r}}^{\infty }{\displaystyle \frac{1}{{\mathrm{n}}^{\frac{1}{\kappa }}}}.\hfill \end{array}$$

Taking ${lim}_{\mathrm{r}\to \infty }$, we get $\underset{\wp ,\mathrm{r}\to \infty }{lim}\mathrm{d}({\mathrm{x}}_{\wp },{\mathrm{x}}_{\mathrm{r}})=0$, since ${\sum }_{\mathrm{n}=\mathrm{r}}^{\infty }{\displaystyle \frac{1}{{\mathrm{n}}^{\frac{1}{\kappa }}}}$ is convergent if $\kappa <1$. This shows that the sequence $\left\{{\mathrm{x}}_{\ell }\right\}$ in $\mathcal{S}$ is Cauchy. Since $\mathcal{S}$ is complete, there exists $\phi \in \mathcal{S}$ such that $\underset{\ell \to \infty }{lim}{\mathrm{x}}_{\ell }=\phi$. Next, prove that $\phi$ is a fixed point of $\mathcal{Q}$. By continuity, then, we have

$$\begin{array}{c}\hfill \mathrm{d}(\phi ,\mathcal{Q}\phi )=\underset{\ell \to \infty }{lim}\mathrm{d}({\mathrm{x}}_{\ell },\mathcal{Q}{\mathrm{x}}_{\ell })=\underset{\ell \to \infty }{lim}\mathrm{d}({\mathrm{x}}_{\ell },{\mathrm{x}}_{\ell +1})=0.\end{array}$$

Therefore, $\mathcal{Q}$ has a fixed point $\phi$ in $\mathcal{Q}$.

Now, assume $\mathcal{Q}$ is orbitally continuous on $\mathcal{S}$; then ${\mathrm{x}}_{\ell +1}=\mathcal{Q}{\mathrm{x}}_{\ell }=\mathcal{Q}\left(\mathcal{Q}{\mathrm{x}}_0\right)\to \mathcal{Q}\phi$ as $\ell \to \infty$.

By the completeness property, we have $\mathcal{Q}\phi =\phi$. Therefore, $Fix\left(\mathcal{Q}\right)\ne \varnothing$.

Again, assume that $\mathcal{Q}$ is ${\alpha }^{*}$-admissible; this implies that $\forall \phi ,{\phi }^{*}\in Fix\left(\mathcal{Q}\right)$, we have $\alpha (\phi ,{\phi }^{*})\ge 1$. Therefore, $\mathrm{d}(\mathcal{Q}\phi ,\mathcal{Q}{\phi }^{*})=\mathrm{d}(\phi ,{\phi }^{*})>0$. From (1), we obtain

$$\begin{array}{c}\hfill \mathcal{E}\left(\mathrm{d}(\phi ,{\phi }^{*})\right)=\mathcal{E}\left(\mathrm{d}(\mathcal{Q}\phi ,\mathcal{Q}{\phi }^{*})\right)=\mathcal{E}\left(\alpha (\phi ,{\phi }^{*})\mathrm{d}(\mathcal{Q}\phi ,\mathcal{Q}{\phi }^{*})\right)\le \mathcal{E}\left(\mathrm{d}(\phi ,{\phi }^{*})\right)-\mathfrak{b}.\end{array}$$

Since $\mathfrak{b}>0$, and using $\left({\mathcal{E}}_1\right)$, we have

$$\begin{array}{c}\hfill \mathrm{d}(\phi ,{\phi }^{*})<\mathrm{d}(\phi ,{\phi }^{*}).\end{array}$$

which contradicts by our assumption. Therefore, $\mathcal{Q}$ has a $\mathit{UFP}$ in $\mathcal{S}$. □

Here we define $\alpha$-$\mathcal{E}$-Suzuki contraction as follows.

Definition 10.

A self-map$\mathcal{Q}$ on $\mathcal{S}$ is called an α-$\mathcal{E}$-Suzuki contraction if $\mathfrak{b}>0$ exists such that $\forall \mathrm{x},\mathrm{y}\in \mathcal{S}$ with $\mathcal{Q}\mathrm{x}\ne \mathcal{Q}\mathrm{y}$

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}\mathrm{d}(\mathrm{x},\mathcal{Q}\mathrm{x})<\mathrm{d}(\mathrm{x},\mathrm{y})\Rightarrow \mathfrak{b}+\mathcal{E}\left(\alpha (\mathrm{x},\mathrm{y})\mathrm{d}(\mathcal{Q}\mathrm{x},\mathcal{Q}\mathrm{y})\right)\le \mathcal{E}\left(\mathrm{d}(\mathrm{x},\mathrm{y})\right),\end{array}$$

(14)

where $\mathcal{E}\in \mathcal{F}$. 

Example 4.

Let $\mathcal{S}=[0,3]$, $\mathrm{d}=|\mathrm{x}-\mathrm{y}|$, $\mathcal{E}\left(\mathrm{t}\right)=ln(1+\mathrm{t}),\alpha (\mathrm{x},\mathrm{y})={\displaystyle \frac{1}{1+|\mathrm{x}-\mathrm{y}|}}$, and $\mathrm{b}=0.2$ and define

$$\mathcal{Q}\left(\mathrm{x}\right)=\left\{\begin{array}{c}2on[0,1]\hfill \\ 2-(\mathrm{x}-1)on[1,2]\hfill \\ 3on[2,3].\hfill \end{array}\right.$$

Clearly, it is continuous. Let us check ${\displaystyle \frac{1}{2}}\mathrm{d}(\mathrm{x},\mathcal{Q}\mathrm{x})<\mathrm{d}(\mathrm{x},\mathrm{y})$ first.

• case-i: When $\mathrm{x},\mathrm{y}\in [0,1]$,

  $$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}\mathrm{d}(\mathrm{x},\mathcal{Q}\mathrm{x})=& {\displaystyle \frac{1}{2}}|\mathrm{x}-2|\\ \hfill =& {\displaystyle \frac{|x-2|}{2}}\le |\mathrm{x}-\mathrm{y}|.\end{array}$$
• case-ii: When $\mathrm{x},\mathrm{y}\in [1,2]$,

  $$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}\mathrm{d}(\mathrm{x},\mathcal{Q}\mathrm{x})=& {\displaystyle \frac{1}{2}}|\mathrm{x}-2+(\mathrm{x}-1)|\\ \hfill =& {\displaystyle \frac{|2\mathrm{x}-3|}{2}}\le |\mathrm{x}-\mathrm{y}|.\end{array}$$
• case-iii: When $\mathrm{x},\mathrm{y}\in [2,3]$,

  $$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}\mathrm{d}(\mathrm{x},\mathcal{Q}\mathrm{x})=& {\displaystyle \frac{1}{2}}|\mathrm{x}-3|\\ \hfill =& {\displaystyle \frac{|\mathrm{x}-3|}{2}}\le |\mathrm{x}-\mathrm{y}|.\end{array}$$

Now, let us check $\mathfrak{b}+\mathcal{E}\left(\alpha \right(\mathrm{x},\mathrm{y}\left)\mathrm{d}\right(\mathcal{Q}\mathrm{x},\mathcal{Q}\mathrm{y}\left)\right)\le \mathcal{E}\left(\mathrm{d}\right(\mathrm{x},\mathrm{y}\left)\right)$ on $\mathrm{x}\in [0,1]$ and $\mathrm{y}\in [2,3]$. In this case, $\mathcal{Q}\left(\mathrm{x}\right)=2,\mathcal{Q}\left(\mathrm{y}\right)=3$. Also, $\mathrm{d}(\mathcal{Q}\mathrm{x},\mathcal{Q}\mathrm{y})=|2-3|=1$ and $\mathrm{d}(\mathrm{x},\mathrm{y})=|\mathrm{x}-\mathrm{y}|\ge 1$. Now,

$$\begin{array}{cc}\hfill \mathfrak{b}+\mathcal{E}\left(\alpha \right(\mathrm{x},\mathrm{y}\left)\mathrm{d}\right(\mathcal{Q}\mathrm{x},\mathcal{Q}\mathrm{y}\left)\right)=& 0.2+\mathcal{E}\left({\displaystyle \frac{1}{1+|\mathrm{x}-\mathrm{y}|}}·1\right)\\ \hfill =& 0.2+ln\left(1+{\displaystyle \frac{1}{1+|\mathrm{x}-\mathrm{y}|}}\right)\\ \hfill =& 0.2+ln\left({\displaystyle \frac{2+|\mathrm{x}-\mathrm{y}|}{1+|\mathrm{x}-\mathrm{y}|}}\right).\end{array}$$

Also,

$$\begin{array}{c}\hfill \mathcal{E}\left(\mathrm{d}(\mathrm{x},\mathrm{y})\right)=ln\left(1+\mathrm{d}(\mathrm{x},\mathrm{y})\right)=ln\left(1+|\mathrm{x}-\mathrm{y}|\right).\end{array}$$

It is enough if we show that

$$\begin{array}{cc}\hfill 0.2+& ln\left({\displaystyle \frac{2+|\mathrm{x}-\mathrm{y}|}{1+|\mathrm{x}-\mathrm{y}|}}\right)\le ln\left(1+|\mathrm{x}-\mathrm{y}|\right).\\ \hfill (i.e)0.2\le & ln\left(1+|\mathrm{x}-\mathrm{y}|\right)-ln\left({\displaystyle \frac{2+|\mathrm{x}-\mathrm{y}|}{1+|\mathrm{x}-\mathrm{y}|}}\right)\\ \hfill \le & ln\left({\displaystyle \frac{{(1+|\mathrm{x}-\mathrm{y}\left|\right)}^2}{2+|\mathrm{x}-\mathrm{y}|}}\right)\\ \hfill \Rightarrow e^{0.2}\le {\displaystyle \frac{{(1+|\mathrm{x}-\mathrm{y}\left|\right)}^2}{2+|\mathrm{x}-\mathrm{y}|}}.\end{array}$$

Since $\mathrm{d}(\mathrm{x},\mathrm{y})\ge 1$, we get

$$\begin{array}{c}\hfill {\displaystyle \frac{4}{3}}\ge e^{0.2}\Rightarrow 1.333\ge 1.22.\end{array}$$

Hence, the contraction inequality holds well. Similarly, one can prove for other cases also.

Let us discuss the importance of our new contraction in the following example.

Example 5.

Let $\mathcal{S}=[0,3]$, $\mathrm{d}(\mathrm{x},\mathrm{y})=|\mathrm{x}-\mathrm{y}|$, $\mathcal{E}\left(\mathrm{t}\right)=ln(1+\mathrm{t})$, $\alpha (\mathrm{x},\mathrm{y})={\displaystyle \frac{1}{1+\mathrm{d}{(\mathrm{x},\mathrm{y})}^2}}$, and $\mathrm{b}=0.1$; $\mathcal{Q}$ is defined by

$$\begin{array}{c}\hfill \mathcal{Q}\left(\mathrm{x}\right)=\left\{\begin{array}{c}2.5,0\le \mathrm{x}\le 0.2,\hfill \\ 2.5-3(\mathrm{x}-0.2),0.2\le \mathrm{x}\le 0.7\hfill \\ 1,0.7\le \mathrm{x}\le 3\hfill \\ 1+2(\mathrm{x}-3),3\le \mathrm{x}\le 3.5\hfill \\ 4,3.5\le \mathrm{x}\le 4.\hfill \end{array}\right.\end{array}$$

Clearly $\mathcal{Q}$ is continuous. Let $\mathrm{x}\in [0,0.2],\mathrm{y}\in [3.5,4]$; then $\mathrm{d}(\mathrm{x},\mathcal{Q}\mathrm{x})=|x-2.5|\ge 2.3$. Thus, ${\displaystyle \frac{1}{2}}\mathrm{d}(\mathrm{x},\mathcal{Q}\mathrm{x})\ge 1.15$ and also $\mathrm{d}(\mathrm{x},\mathrm{y})\ge 3.3$. Hence, ${\displaystyle \frac{1}{2}}\mathrm{d}(\mathrm{x},\mathcal{Q}\mathrm{x})<\mathrm{d}(\mathrm{x},\mathrm{y})$.

• Now,

  $$\begin{array}{cc}\hfill \mathfrak{b}+\mathcal{E}\left(\alpha \right(\mathrm{x},\mathrm{y}\left)\mathrm{d}\right(\mathcal{Q}\mathrm{x},\mathcal{Q}\mathrm{y}\left)\right)=& 0.1+ln\left(1+{\displaystyle \frac{0.5}{1+10.89}}\right)\\ \hfill =& 0.1+{\displaystyle \frac{12.39}{11.89}}=0.1+ln\left(1.0421\right)\approx 0.1412.\end{array}$$

And

$$\begin{array}{c}\hfill \mathcal{E}\left(\mathrm{d}\right(\mathrm{x},\mathrm{y}\left)\right)=ln(1+\mathrm{d}(\mathrm{x},\mathrm{y}\left)\right)\approx ln(1+3.3)=ln\left(4.3\right)\approx 1.458.\end{array}$$

Therefore,

$$\begin{array}{c}\hfill \mathfrak{b}+\mathcal{E}\left(\alpha \right(\mathrm{x},\mathrm{y}\left)\mathrm{d}\right(\mathcal{Q}\mathrm{x},\mathcal{Q}\mathrm{y}\left)\right)\le \mathcal{E}\left(\mathrm{d}\right(\mathrm{x},\mathrm{y}\left)\right)\end{array}$$

holds well. Similarly one can check the applicability for other cases also. Suppose, $\mathrm{x}=0.2$ and $\mathrm{y}=0.7$: $\mathcal{Q}\mathrm{x}=2.5,\mathcal{Q}\mathrm{y}=1\Rightarrow \mathcal{Q}\mathrm{x}\ne \mathcal{Q}\mathrm{y}$. Furthermore, $\mathrm{d}(\mathrm{x},\mathcal{Q}\mathrm{x})=2.3>0$. Hence, we can verify the contraction condition presented in Equation (1). Thus,

$$\begin{array}{cc}\hfill \mathfrak{b}+\mathcal{E}\left(\alpha \right(\mathrm{x},\mathrm{y}\left)\mathrm{d}\right(\mathcal{Q}\mathrm{x},\mathcal{Q}\mathrm{y}\left)\right)=& 0.1+ln\left(1+{\displaystyle \frac{|2.5-1|}{{1+|2.5-1|}^2}}\right)\\ \hfill =& 0.1+{\displaystyle \frac{4.25}{3.25}}=0.1+ln\left(1.307\right)\approx 0.2162.\end{array}$$

And

$$\begin{array}{c}\hfill \mathcal{E}\left(\mathrm{d}\right(\mathrm{x},\mathrm{y}\left)\right)=ln(1+\mathrm{d}(\mathrm{x},\mathrm{y}\left)\right)\approx ln(1+0.5)=ln\left(1.5\right)\approx 0.176.\end{array}$$

Here

$$\begin{array}{c}\hfill \mathfrak{b}+\mathcal{E}\left(\alpha \right(\mathrm{x},\mathrm{y}\left)\mathrm{d}\right(\mathcal{Q}\mathrm{x},\mathcal{Q}\mathrm{y}\left)\right)\le \mathcal{E}\left(\mathrm{d}\right(\mathrm{x},\mathrm{y}\left)\right)\end{array}$$

does not hold. This shows the need for our contraction.

We prove the following theorem of $\mathcal{E}$-Suzuki contraction inspired by Hossein Piri and Poom Kumam [13] to extend this into $\alpha$-$\mathcal{E}$-Suzuki contraction with the notion of $\alpha$-admissibility.

Theorem 5.

Let $(\mathcal{S},\mathrm{d})$ be a complete metric space and $\mathcal{Q}:\mathcal{S}\to \mathcal{S}$ be an α-$\mathcal{E}$-Suzuki contraction such that the given hypotheses hold:

(i)

$\mathcal{Q}$ is α-admissible;

(ii)

$\exists {\mathrm{x}}_0\in \mathcal{S}$ such that $\alpha ({\mathrm{x}}_0,\mathcal{Q}{\mathrm{x}}_0)\ge 1$;

(iii)

$\mathcal{Q}$ is continuous or orbitally continuous on $\mathcal{S}$.

Then, the point $\phi \in \mathcal{S}$ is a fixed point of $\mathcal{Q}$. Moreover, if $\mathcal{Q}$ is ${\alpha }^{*}$-admissible, then the point $\phi \in \mathcal{S}$ is a $\mathit{UFP}$ of $\mathcal{Q}$. Furthermore, for every ${\mathrm{x}}_0\in \mathcal{S}$, if ${\mathrm{x}}_{\ell +1}={\mathcal{Q}}^{\ell +1}{\mathrm{x}}_0\ne \mathcal{Q}{\mathrm{x}}_{\ell }\forall \ell \ge 0$, then $\underset{\ell \to \infty }{lim}{\mathcal{Q}}^{\ell }{\mathrm{x}}_0=\phi$.

Proof. 

Let ${\mathrm{x}}_0\in \mathcal{S}$. It follows that $\alpha (\mathcal{Q}{\mathrm{x}}_0,{\mathrm{x}}_0)\ge 1$, and we construct a sequence $\left\{{\mathrm{x}}_{\ell }\right\}$ by ${\mathrm{x}}_{\ell +1}=\mathcal{Q}{\mathrm{x}}_{\ell }={\mathcal{Q}}^{\ell +1}{\mathrm{x}}_0\forall \ell \ge 0$. If ${\mathrm{x}}_{{\ell }_0}={\mathrm{x}}_{{\ell }_0+1}$, it follows that $\mathcal{Q}{\mathrm{x}}_{{\ell }_0}={\mathrm{x}}_{{\ell }_0}$ for some ${\ell }_0\ge 0$; then $\mathcal{Q}$ has a fixed point ${\mathrm{x}}_{{\ell }_0}$ in $\mathcal{S}$.

Now, suppose ${\mathrm{x}}_{\ell }\ne {\mathrm{x}}_{\ell +1}\forall \ell \ge 0$. Then, $\mathrm{d}({\mathrm{x}}_{\ell },{\mathrm{x}}_{\ell +1})>0\forall \ell \ge 0$. By condition $\left(ii\right)$, we get $\alpha ({\mathrm{x}}_{\ell },{\mathrm{x}}_{\ell +1})>1\forall \ell \ge 0$. Therefore, $\alpha ({\mathrm{x}}_{\ell },{\mathrm{x}}_{\ell +1})=\alpha ({\mathcal{Q}}^{\ell }{\mathrm{x}}_0,{\mathcal{Q}}^{\ell +1}{\mathrm{x}}_0)\ge 1\forall \ell \ge 0$. According to $\left({\mathcal{E}}_1\right)$ and Definition 5,

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}\mathrm{d}({\mathrm{x}}_{\ell },\mathcal{Q}{\mathrm{x}}_{\ell })={\displaystyle \frac{1}{2}}\mathrm{d}({\mathrm{x}}_{\ell },{\mathrm{x}}_{\ell +1})<\mathrm{d}({\mathrm{x}}_{\ell },{\mathrm{x}}_{\ell +1}).\end{array}$$

Now, by Equation (14), we get

$$\begin{array}{cc}\hfill \mathcal{E}\left(\mathrm{d}(\mathcal{Q}{\mathrm{x}}_{\ell },{\mathcal{Q}}^2{\mathrm{x}}_{\ell })\right)& =\mathcal{E}\left(\alpha ({\mathrm{x}}_{\ell },\mathcal{Q}{\mathrm{x}}_{\ell })\mathrm{d}(\mathcal{Q}{\mathrm{x}}_{\ell },{\mathcal{Q}}^2{\mathrm{x}}_{\ell })\right)\hfill \\ & \le \mathcal{E}\left(\mathrm{d}({\mathrm{x}}_{\ell },\mathcal{Q}{\mathrm{x}}_{\ell })\right)-\mathfrak{b}.\hfill \end{array}$$

(15)

Repeating this process, we get

$$\begin{array}{cc}\hfill \mathcal{E}\left(\mathrm{d}({\mathrm{x}}_{\ell },\mathcal{Q}{\mathrm{x}}_{\ell })\right)& =\mathcal{E}\left(\mathrm{d}(\mathcal{Q}{\mathrm{x}}_{\ell -1},\mathcal{Q}{\mathrm{x}}_{\ell })\right)\hfill \\ & =\mathcal{E}\left(\alpha ({\mathrm{x}}_{\ell -1},{\mathrm{x}}_{\ell })\mathrm{d}(\mathcal{Q}{\mathrm{x}}_{\ell -1},\mathcal{Q}{\mathrm{x}}_{\ell })\right)\hfill \\ & \le \mathcal{E}\left(\mathrm{d}({\mathrm{x}}_{\ell -1},\mathcal{Q}{\mathrm{x}}_{\ell -1})\right)-\mathfrak{b}\hfill \\ & =\mathcal{E}\left(\mathrm{d}(\mathcal{Q}{\mathrm{x}}_{\ell -2},\mathcal{Q}{\mathrm{x}}_{\ell -1})\right)-\mathfrak{b}\hfill \\ & =\mathcal{E}\left(\alpha ({\mathrm{x}}_{\ell -2},{\mathrm{x}}_{\ell -1})\mathrm{d}(\mathcal{Q}{\mathrm{x}}_{\ell -2},\mathcal{Q}{\mathrm{x}}_{\ell -1})\right)-\mathfrak{b}\hfill \\ & \le \mathcal{E}\left(\mathrm{d}({\mathrm{x}}_{\ell -2},\mathcal{Q}{\mathrm{x}}_{\ell -2})\right)-2\mathfrak{b}\hfill \\ & =\mathcal{E}\left(\mathrm{d}(\mathcal{Q}{\mathrm{x}}_{\ell -3},\mathcal{Q}{\mathrm{x}}_{\ell -2})\right)-2\mathfrak{b}\hfill \\ & =\mathcal{E}\left(\alpha ({\mathrm{x}}_{\ell -3},{\mathrm{x}}_{\ell -2})\mathrm{d}(\mathcal{Q}{\mathrm{x}}_{\ell -3},\mathcal{Q}{\mathrm{x}}_{\ell -2})\right)-2\mathfrak{b}\hfill \\ & \le \mathcal{E}\left(\mathrm{d}({\mathrm{x}}_{\ell -3},\mathcal{Q}{\mathrm{x}}_{\ell -3})\right)-3\mathfrak{b}\hfill \\ & .\hfill \\ & .\hfill \\ & .\hfill \\ & \le \mathcal{E}\left(\mathrm{d}({\mathrm{x}}_0,\mathcal{Q}{\mathrm{x}}_0)\right)-\ell \mathfrak{b}.\hfill \end{array}$$

(16)

Taking the limit on both sides, we get

$$\begin{array}{c}\hfill \underset{\ell \to \infty }{lim}\mathcal{E}\left(\mathrm{d}(\mathcal{Q}{\mathrm{x}}_{\ell -1},\mathcal{Q}{\mathrm{x}}_{\ell })\right)=-\infty .\end{array}$$

(17)

Therefore, by (${\mathcal{E}}_2$) of Definition 1 with Equation (17), we get

$$\begin{array}{c}\hfill \underset{\ell \to \infty }{lim}\mathrm{d}(\mathcal{Q}{\mathrm{x}}_{\ell -1},\mathcal{Q}{\mathrm{x}}_{\ell })=0.\end{array}$$

(18)

By (${\mathcal{E}}_3$), $\exists \kappa \in (0,1)$ such that

$$\begin{array}{c}\hfill \underset{\ell \to \infty }{lim}{\left(\mathrm{d}({\mathrm{x}}_{\ell },{\mathrm{x}}_{\ell +1})\right)}^{\kappa }\mathcal{E}\left(\mathrm{d}({\mathrm{x}}_{\ell },{\mathrm{x}}_{\ell +1})\right)=0.\end{array}$$

(19)

From (16), we have

$$\begin{array}{c}\hfill {\left[\mathrm{d}({\mathrm{x}}_{\ell },{\mathrm{x}}_{\ell +1})\right]}^{\kappa }[\mathcal{E}\left(\mathrm{d}({\mathrm{x}}_{\ell },{\mathrm{x}}_{\ell +1})\right)-\mathcal{E}\left(\mathrm{d}({\mathrm{x}}_0,{\mathrm{x}}_1)\right)]\le -{\left[\mathrm{d}({\mathrm{x}}_{\ell },{\mathrm{x}}_{\ell +1})\right]}^{\kappa }\ell \mathfrak{b}\le 0.\end{array}$$

(20)

Taking ${lim}_{\ell \to \infty }$ in the above equation along with Equations (18) and (19), we have

$$\begin{array}{c}\hfill \underset{\ell \to \infty }{lim}\ell {\left[\mathrm{d}({\mathrm{x}}_{\ell },{\mathrm{x}}_{\ell +1})\right]}^{\kappa }=0.\end{array}$$

(21)

Now, we will present two cases.

Case-(i): Suppose ℓ is even; we get, by (21),

$$\begin{array}{c}\hfill \underset{\ell \to \infty }{lim}\ell {\left[\mathrm{d}({\mathrm{x}}_{\ell },{\mathrm{x}}_{\ell +1})\right]}^{\kappa }=0.\end{array}$$

(22)

Case-(ii): Suppose ℓ is odd; we get, by (21),

$$\begin{array}{c}\hfill \underset{\ell \to \infty }{lim}(\ell -1){\left[\mathrm{d}({\mathrm{x}}_{\ell },{\mathrm{x}}_{\ell +1})\right]}^{\kappa }=0.\end{array}$$

(23)

Using Equations (18) and (23), we get

$$\begin{array}{c}\hfill \underset{\ell \to \infty }{lim}\ell {\left[\mathrm{d}({\mathrm{x}}_{\ell },{\mathrm{x}}_{\ell +1})\right]}^{\kappa }=0.\end{array}$$

(24)

From the above we observe that $\exists {\ell }_1\in \mathbb{N}$ such that

$$\begin{array}{c}\hfill \ell {\left[\mathrm{d}({\mathrm{x}}_{\ell },{\mathrm{x}}_{\ell +1})\right]}^{\kappa }\le 1\forall \ell \ge {\ell }_1.\end{array}$$

Therefore, we have

$$\begin{array}{c}\hfill \mathrm{d}({\mathrm{x}}_{\ell },{\mathrm{x}}_{\ell +1})\le {\displaystyle \frac{1}{{\ell }^{\frac{1}{\kappa }}}}\forall \ell \ge {\ell }_1.\end{array}$$

Next, we show that $\left\{{\mathrm{x}}_{\ell }\right\}$ is a Cauchy sequence. Now, $\forall \wp >\mathrm{r}\ge {\ell }_1$, we have

$$\begin{array}{cc}\hfill \mathrm{d}({\mathrm{x}}_{\wp },{\mathrm{x}}_{\mathrm{r}})& \le \mathrm{d}({\mathrm{x}}_{\wp },{\mathrm{x}}_{\wp -1})+\mathrm{d}({\mathrm{x}}_{\wp -1},{\mathrm{x}}_{\wp -2})+\mathrm{d}({\mathrm{x}}_{\wp -2},{\mathrm{x}}_{\wp -3})+\dots \dots +\mathrm{d}({\mathrm{x}}_{\mathrm{r}+1},{\mathrm{x}}_{\mathrm{r}})\hfill \\ & <\sum _{\mathrm{n}=\mathrm{r}}^{\infty }\mathrm{d}({\mathrm{x}}_{\mathrm{n}},{\mathrm{x}}_{\mathrm{n}+1})\hfill \\ & \le \sum _{\mathrm{n}=\mathrm{r}}^{\infty }{\displaystyle \frac{1}{{\mathrm{n}}^{\frac{1}{\kappa }}}}.\hfill \end{array}$$

Let ${lim}_{\mathrm{r}\to \infty }$; we get $\underset{\wp ,\mathrm{r}\to \infty }{lim}\mathrm{d}({\mathrm{x}}_{\wp },{\mathrm{x}}_{\mathrm{r}})=0$, since ${\sum }_{\mathrm{n}=\mathrm{r}}^{\infty }{\displaystyle \frac{1}{{\mathrm{n}}^{\frac{1}{\kappa }}}}$ is convergent if $\kappa <1$. This shows that the sequence $\left\{{\mathrm{x}}_{\ell }\right\}$ in $\mathcal{S}$ is Cauchy. By completeness, $\exists \phi \in \mathcal{S}$ such that $\underset{\ell \to \infty }{lim}{\mathrm{x}}_{\ell }=\phi$. Therefore,

$$\begin{array}{c}\hfill \underset{\ell \to \infty }{lim}\mathrm{d}({\mathrm{x}}_{\ell },\phi )=0.\end{array}$$

(25)

Now, we prove that $\mathcal{Q}$ has a fixed point $\phi$ in $\mathcal{Q}$. Now, we claim that

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}\mathrm{d}({\mathrm{x}}_{\ell },\mathcal{Q}{\mathrm{x}}_{\ell })<\mathrm{d}({\mathrm{x}}_{\ell },\phi )or{\displaystyle \frac{1}{2}}\mathrm{d}(\mathcal{Q}{\mathrm{x}}_{\ell },{\mathcal{Q}}^2{\mathrm{x}}_{\ell })<\mathrm{d}(\mathcal{Q}{\mathrm{x}}_{\ell },\phi ),\ell \in \mathbb{N}.\end{array}$$

(26)

Again, suppose that $\exists \mathfrak{m}\in \mathbb{N}$ such that

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}\mathrm{d}({\mathrm{x}}_{\mathfrak{m}},\mathcal{Q}{\mathrm{x}}_{\mathfrak{m}})\ge \mathrm{d}({\mathrm{x}}_{\mathfrak{m}},\phi )and{\displaystyle \frac{1}{2}}\mathrm{d}(\mathcal{Q}{\mathrm{x}}_{\mathfrak{m}},{\mathcal{Q}}^2{\mathrm{x}}_{\mathfrak{m}})\ge \mathrm{d}(\mathcal{Q}{\mathrm{x}}_{\mathfrak{m}},\phi ).\end{array}$$

(27)

Therefore,

$$\begin{array}{c}\hfill 2\mathrm{d}({\mathrm{x}}_{\mathfrak{m}},\phi )\le \mathrm{d}({\mathrm{x}}_{\mathfrak{m}},\mathcal{Q}{\mathrm{x}}_{\mathfrak{m}})\le \mathrm{d}({\mathrm{x}}_{\mathfrak{m}},\phi )+\mathrm{d}(\phi ,\mathcal{Q}{\mathrm{x}}_{\mathfrak{m}}),\end{array}$$

which implies that

$$\begin{array}{c}\hfill \mathrm{d}({\mathrm{x}}_{\mathfrak{m}},\phi )\le \mathrm{d}(\phi ,\mathcal{Q}{\mathrm{x}}_{\mathfrak{m}}).\end{array}$$

(28)

It follows from Equations (27) and (28) that

$$\begin{array}{c}\hfill \mathrm{d}({\mathrm{x}}_{\mathfrak{m}},\phi )\le \mathrm{d}(\phi ,\mathcal{Q}{\mathrm{x}}_{\mathfrak{m}})\le {\displaystyle \frac{1}{2}}\mathrm{d}(\mathcal{Q}{\mathrm{x}}_{\mathfrak{m}},{\mathcal{Q}}^2{\mathrm{x}}_{\mathfrak{m}}).\end{array}$$

(29)

Since ${\displaystyle \frac{1}{2}}\mathrm{d}({\mathrm{x}}_{\mathfrak{m}},\mathcal{Q}{\mathrm{x}}_{\mathfrak{m}})<\mathrm{d}({\mathrm{x}}_{\mathfrak{m}},\mathcal{Q}{\mathrm{x}}_{\mathfrak{m}})$, by the assumption of the theorem, we get

$$\begin{array}{c}\hfill \mathfrak{b}+\mathcal{E}\left(\alpha ({\mathrm{x}}_{\mathfrak{m}},\mathcal{Q}{\mathrm{x}}_{\mathfrak{m}})\mathrm{d}(\mathcal{Q}{\mathrm{x}}_{\mathfrak{m}},{\mathcal{Q}}^2{\mathrm{x}}_{\mathfrak{m}})\right)\le \mathcal{E}\left(\mathrm{d}({\mathrm{x}}_{\mathfrak{m}},\mathcal{Q}{\mathrm{x}}_{\mathfrak{m}})\right).\end{array}$$

Since $\mathfrak{b}>0$, it follows that

$$\begin{array}{c}\hfill \mathcal{E}\left(\alpha ({\mathrm{x}}_{\mathfrak{m}},\mathcal{Q}{\mathrm{x}}_{\mathfrak{m}})\mathrm{d}(\mathcal{Q}{\mathrm{x}}_{\mathfrak{m}},{\mathcal{Q}}^2{\mathrm{x}}_{\mathfrak{m}})\right)<\mathcal{E}\left(\mathrm{d}({\mathrm{x}}_{\mathfrak{m}},\mathcal{Q}{\mathrm{x}}_{\mathfrak{m}})\right).\end{array}$$

By Definition 5, we get

$$\begin{array}{c}\hfill \mathcal{E}\left(\mathrm{d}(\mathcal{Q}{\mathrm{x}}_{\mathfrak{m}},{\mathcal{Q}}^2{\mathrm{x}}_{\mathfrak{m}})\right)<\mathcal{E}\left(\mathrm{d}({\mathrm{x}}_{\mathfrak{m}},\mathcal{Q}{\mathrm{x}}_{\mathfrak{m}})\right).\end{array}$$

So, by (${\mathcal{E}}_1$), we get

$$\begin{array}{c}\hfill \mathrm{d}(\mathcal{Q}{\mathrm{x}}_{\mathfrak{m}},{\mathcal{Q}}^2{\mathrm{x}}_{\mathfrak{m}})<\mathrm{d}({\mathrm{x}}_{\mathfrak{m}},\mathcal{Q}{\mathrm{x}}_{\mathfrak{m}}).\end{array}$$

(30)

It follows from Equations (27), (29) and (30) that

$$\begin{array}{cc}\hfill \mathrm{d}(\mathcal{Q}{\mathrm{x}}_{\mathfrak{m}},{\mathcal{Q}}^2{\mathrm{x}}_{\mathfrak{m}})& <\mathrm{d}({\mathrm{x}}_{\mathfrak{m}},\mathcal{Q}{\mathrm{x}}_{\mathfrak{m}})\hfill \\ & \le \mathrm{d}({\mathrm{x}}_{\mathfrak{m}},\phi )+d(\phi ,\mathcal{Q}{\mathrm{x}}_{\mathfrak{m}})\hfill \\ & \le {\displaystyle \frac{1}{2}}\mathrm{d}(\mathcal{Q}{\mathrm{x}}_{\mathfrak{m}},{\mathcal{Q}}^2{\mathrm{x}}_{\mathfrak{m}})+{\displaystyle \frac{1}{2}}\mathrm{d}(\mathcal{Q}{\mathrm{x}}_{\mathfrak{m}},{\mathcal{Q}}^2{\mathrm{x}}_{\mathfrak{m}})\hfill \end{array}$$

$$\begin{array}{c}\hfill =\mathrm{d}(\mathcal{Q}{\mathrm{x}}_{\mathfrak{m}},{\mathcal{Q}}^2{\mathrm{x}}_{\mathfrak{m}}).\end{array}$$

(31)

which contradicts our assumption. Hence, Equation (26) holds. So, by Equation (26), for each $\ell \in \mathbb{N}$,

$$\begin{array}{c}\hfill \mathfrak{b}+\mathcal{E}\left(\alpha ({\mathrm{x}}_{\ell },\phi )\mathrm{d}(\mathcal{Q}{\mathrm{x}}_{\ell },\mathcal{Q}\phi )\right)\le \mathcal{E}\left(\mathrm{d}({\mathrm{x}}_{\ell },\phi )\right),\end{array}$$

or

$$\begin{array}{c}\hfill \mathfrak{b}+\mathcal{E}\left(\alpha ({\mathrm{x}}_{\ell },\phi )\mathrm{d}({\mathcal{Q}}^2{\mathrm{x}}_{\ell },\mathcal{Q}\phi )\right)\le \mathcal{E}\left(\mathrm{d}(\mathcal{Q}{\mathrm{x}}_{\ell },\phi )\right)=\mathcal{E}\left(\mathrm{d}({\mathrm{x}}_{\ell +1},\phi )\right)\end{array}$$

holds. In case-(i), from Equation (25), by condition (${\mathcal{E}}_2$) of Definition 1, we have

$$\begin{array}{c}\hfill \underset{\ell \to \infty }{lim}\mathcal{E}\left(\mathrm{d}(\mathcal{Q}{\mathrm{x}}_{\ell },\mathcal{Q}\phi )\right)=-\infty .\end{array}$$

This implies by condition (${\mathcal{E}}_2$) of Definition 1 that ${lim}_{\ell \to \infty }\mathrm{d}(\mathcal{Q}{\mathrm{x}}_{\ell },\mathcal{Q}\phi ))=0$. Therefore,

$$\begin{array}{c}\hfill \mathrm{d}(\phi ,\mathcal{Q}\phi )=\underset{\ell \to \infty }{lim}\mathrm{d}({\mathrm{x}}_{\ell +1},\mathcal{Q}\phi )=\underset{\ell \to \infty }{lim}\mathrm{d}(\mathcal{Q}{\mathrm{x}}_{\ell },\mathcal{Q}\phi )=0.\end{array}$$

In case-(ii), by Equation (25) and condition (${\mathcal{E}}_2$) of Definition 1, we have

$$\begin{array}{c}\hfill \underset{\ell \to \infty }{lim}\mathcal{E}\left(\mathrm{d}({\mathcal{Q}}^2{\mathrm{x}}_{\ell },\mathcal{Q}\phi )\right)=-\infty .\end{array}$$

This implies by condition (1) and Lemma 1 that ${lim}_{\ell \to \infty }\mathrm{d}(\mathcal{Q}{\mathrm{x}}_{\ell },\mathcal{Q}\phi ))=0$. Therefore,

$$\begin{array}{c}\hfill \mathrm{d}(\phi ,\mathcal{Q}\phi )=\underset{\ell \to \infty }{lim}\mathrm{d}({\mathrm{x}}_{\ell +2},\mathcal{Q}\phi )=\underset{\ell \to \infty }{lim}\mathrm{d}({\mathcal{Q}}^2{\mathrm{x}}_{\ell },\mathcal{Q}\phi )=0.\end{array}$$

Hence, $\phi$ is a fixed point of $\mathcal{Q}$. Now, assume $\mathcal{Q}$ is orbitally continuous on $\mathcal{S}$; then ${\mathrm{x}}_{\ell +1}=\mathcal{Q}{\mathrm{x}}_{\ell }=\mathcal{Q}\left(\mathcal{Q}{\mathrm{x}}_0\right)\to \mathcal{Q}\phi$ as $\ell \to \infty$. By the completeness property, we obtain $\mathcal{Q}\phi =\phi$. Therefore, $Fix\left(\mathcal{Q}\right)\ne \varnothing$. Since $\mathcal{Q}$ is ${\alpha }^{*}$-admissible, this implies that $\forall \phi ,{\phi }^{*}\in Fix\left(\mathcal{Q}\right)$, we have $\alpha (\phi ,{\phi }^{*})\ge 1$. Therefore, $\mathrm{d}(\mathcal{Q}\phi ,\mathcal{Q}{\phi }^{*})=\mathrm{d}(\phi ,{\phi }^{*})>0$. So, we have $0={\displaystyle \frac{1}{2}}\mathrm{d}(\phi ,\mathcal{Q}\phi )<\mathrm{d}(\phi ,{\phi }^{*})$, and by our assertion of the theorem, we have

$$\begin{array}{c}\hfill \mathcal{E}\left(\mathrm{d}(\phi ,{\phi }^{*})\right)=\mathcal{E}\left(\mathrm{d}(\mathcal{Q}\phi ,\mathcal{Q}{\phi }^{*})\right)=\mathcal{E}\left(\alpha (\phi ,{\phi }^{*})\mathrm{d}(\mathcal{Q}\phi ,\mathcal{Q}{\phi }^{*})\right)\le \mathcal{E}\left(\mathrm{d}(\phi ,{\phi }^{*})\right)-\mathfrak{b}.\end{array}$$

Since $\mathfrak{b}>0$, and using $\left({\mathcal{E}}_1\right)$, we have

$$\begin{array}{c}\hfill \mathrm{d}(\phi ,{\phi }^{*})<\mathrm{d}(\phi ,{\phi }^{*}).\end{array}$$

This is conflicting. Therefore, $\phi$ in $\mathcal{S}$ is a $\mathit{UFP}$ of $\mathcal{Q}$. □

4. Application

This section presents an application of our main results to a nonlinear integral equation.

Consider a real-valued continuous function $\mathcal{S}=\mathcal{C}[\mathfrak{c},\mathfrak{d}]$ defined on $[\mathfrak{c},\mathfrak{d}]$ with metric $\mathrm{d}(\mathfrak{e},\mathfrak{h})={max}_{\mathit{p}\in [\mathfrak{c},\mathfrak{d}]}|\mathfrak{e}\left(\mathit{p}\right)-\mathfrak{h}\left(\mathit{p}\right)|\forall \mathfrak{e},\mathfrak{h}\in \mathcal{C}[\mathfrak{c},\mathfrak{d}]$. Then, $(\mathcal{S},\mathrm{d})$ is a complete metric space.

Consider the nonlinear integral equation

$$\begin{array}{c}\hfill \mathrm{x}\left(\mathit{p}\right)=\epsilon \left(\mathit{p}\right)+{\displaystyle \frac{1}{\mathfrak{d}-\mathfrak{c}}}{\int }_{\mathfrak{c}}^{\mathfrak{d}}\mathcal{K}(\mathit{p},\mathrm{u},\mathrm{x}\left(\mathrm{u}\right))d\mathrm{u},\end{array}$$

(32)

where $\mathit{p},\mathrm{u}\in [\mathfrak{c},\mathfrak{d}]$, $\epsilon \left(\mathit{p}\right)$ is a given function in $\mathcal{S}$, and $\mathcal{K}:[\mathfrak{c},\mathfrak{d}]\times [\mathfrak{c},\mathfrak{d}]\times \mathcal{S}\to \mathcal{R},\epsilon :[\mathfrak{c},\mathfrak{d}]\to \mathcal{R}$ are given continuous functions.

Theorem 6.

Let $(\mathcal{S},\mathrm{d})$ be a complete metric space with metric $\mathrm{d}(\mathfrak{e},\mathfrak{h})={max}_{\mathit{p}\in [\mathfrak{c},\mathfrak{d}]}|\mathfrak{e}\left(\mathit{p}\right)-\mathfrak{h}\left(\mathit{p}\right)|\forall \mathfrak{e},\mathfrak{h}\in \mathcal{C}[\mathfrak{c},\mathfrak{d}]$ and define a continuous self-map $\mathcal{Q}$ on $\mathcal{S}$ by

$$\begin{array}{c}\hfill \mathcal{Q}\mathrm{x}\left(\mathit{p}\right)=\epsilon \left(\mathit{p}\right)+{\displaystyle \frac{1}{\mathfrak{d}-\mathfrak{c}}}{\int }_{\mathfrak{c}}^{\mathfrak{d}}\mathcal{K}(\mathit{p},\mathrm{u},\mathrm{x}\left(\mathrm{u}\right))\mathit{d}\mathrm{u}.\end{array}$$

(33)

If $\exists \theta \in [0,1)$ such that $\forall \mathrm{x},\mathrm{y}\in \mathcal{S}$ with $\mathrm{x}\ne \mathrm{y}$ and $\mathrm{u},\mathit{p}\in [\mathfrak{c},\mathfrak{d}]$ satisfying the following inequality:

$$\begin{array}{c}\hfill \left|\mathcal{K}\right(\mathit{p},\mathrm{u},\mathcal{Q}\mathrm{x}\left(\mathrm{u}\right))-\mathcal{K}(\mathit{p},\mathrm{u},\mathcal{Q}\mathrm{y}\left(\mathrm{u}\right)\left)\right|\le \theta \left|\mathrm{x}\right(\mathrm{u})-\mathrm{y}(\mathrm{u}\left)\right|\end{array}$$

(34)

then $\mathcal{Q}$ has a unique solution $\phi \in \mathcal{S}$ and $\forall {\mathrm{x}}_0\in \mathcal{S},\mathcal{Q}{\mathrm{x}}_{\ell }\ne {\mathrm{x}}_{\ell }\forall \ell \ge 0$, we obtain $\underset{\ell \to \infty }{lim}{\mathrm{x}}_{\ell }=\phi$.

Proof. 

Define a map $\alpha :\mathcal{S}\times \mathcal{S}\to {\mathcal{R}}^{+}$ by $\alpha (\mathrm{x},\mathrm{y})=1\forall \mathrm{x},\mathrm{y}\in \mathcal{S}$. Thus, $\mathcal{Q}$ is $\alpha$-admissible. Let $\mathcal{E}\in \mathcal{F}$ such that $\mathcal{E}\left(\lambda \right)=ln\lambda ,\lambda >0$. Let ${\mathrm{x}}_0\in \mathcal{S}$ and consider a sequence $\left\{{\mathrm{x}}_{\ell }\right\}$ in $\mathcal{S}$ defined by ${\mathrm{x}}_{\ell +1}=\mathcal{Q}{\mathrm{x}}_{\ell }={\mathcal{Q}}^{\ell +1}{\mathrm{x}}_0\forall \ell \ge 0$. By Equation (33), we have

$$\begin{array}{c}\hfill {\mathrm{x}}_{\ell +1}=\mathcal{Q}{\mathrm{x}}_{\ell }\left(\mathit{p}\right)=\epsilon \left(\mathit{p}\right)+{\displaystyle \frac{1}{\mathfrak{d}-\mathfrak{c}}}{\int }_{\mathfrak{c}}^{\mathfrak{d}}\mathcal{K}(\mathit{p},\mathrm{u},{\mathrm{x}}_{\ell }\left(\mathrm{u}\right))\mathit{d}\mathrm{u}.\end{array}$$

(35)

We demonstrate that $\mathcal{Q}$ is an $\alpha$-$\mathcal{E}$-contraction on $\mathcal{C}[\mathfrak{c},\mathfrak{d}]$. By Equation (33), we obtain

$$\begin{array}{cc}\hfill \left|\mathcal{Q}\mathrm{x}\right(\mathit{p})-\mathcal{Q}\mathrm{y}(\mathit{p}\left)\right|& ={\displaystyle \frac{1}{\mathfrak{d}-\mathfrak{c}}}{\int }_{\mathfrak{c}}^{\mathfrak{d}}|\mathcal{K}(\mathit{p},\mathrm{u},\mathrm{x}\left(\mathrm{u}\right))-\mathcal{K}(\mathit{p},\mathrm{u},\mathrm{y}\left(\mathrm{u}\right))|\mathit{d}\mathrm{u}\hfill \\ & \le {\displaystyle \frac{\theta }{\mathfrak{d}-\mathfrak{c}}}{\int }_{\mathfrak{c}}^{\mathfrak{d}}|\mathrm{x}\left(\mathrm{u}\right)-\mathrm{y}\left(\mathrm{u}\right)|\mathit{d}\mathrm{u}.\hfill \end{array}$$

Taking the maximum $\forall \mathit{p}\in [\mathfrak{c},\mathfrak{d}]$, we have

$$\begin{array}{cc}\hfill \mathrm{d}(\mathcal{Q}\mathrm{x},\mathcal{Q}\mathrm{y})& =\underset{\mathit{p}\in [\mathfrak{c},\mathrm{d}]}{max}|\mathcal{Q}\mathrm{x}\left(\mathit{p}\right)-\mathcal{Q}\mathrm{y}\left(\mathit{p}\right)|\hfill \\ & \le {\displaystyle \frac{\theta }{\mathfrak{d}-\mathfrak{c}}}\underset{\mathit{p}\in [\mathfrak{c},\mathfrak{d}]}{max}{\int }_{\mathfrak{c}}^{\mathfrak{d}}|\mathrm{x}\left(\mathrm{u}\right)-\mathrm{y}\left(\mathrm{u}\right)|\mathrm{d}\mathrm{u}\hfill \\ & \le {\displaystyle \frac{\theta }{\mathfrak{d}-\mathfrak{c}}}\underset{\mathit{p}\in [\mathfrak{c},\mathfrak{d}]}{max}|\mathrm{x}\left(\mathrm{u}\right)-\mathrm{y}\left(\mathrm{u}\right)|{\int }_{\mathfrak{c}}^{\mathfrak{d}}\mathit{d}\mathrm{u}\hfill \\ & =\theta \underset{\mathit{p}\in [\mathfrak{c},\mathfrak{d}]}{max}|\mathrm{x}\left(\mathrm{u}\right)-\mathrm{y}\left(\mathrm{u}\right)|\hfill \\ & =\theta \mathrm{d}(\mathrm{x},\mathrm{y}).\hfill \end{array}$$

Therefore

$$\begin{array}{c}\hfill \alpha (\mathrm{x},\mathrm{y})\mathrm{d}(\mathcal{Q}\mathrm{x},\mathcal{Q}\mathrm{y})\le \theta \mathrm{d}(\mathrm{x},\mathrm{y}).\end{array}$$

Taking log on both sides, we get

$$\begin{array}{c}\hfill -ln\theta +ln\left[\alpha \right(\mathrm{x},\mathrm{y}\left)\mathrm{d}\right(\mathcal{Q}\mathrm{x},\mathcal{Q}\mathrm{y}\left)\right]\le ln\mathrm{d}(\mathrm{x},\mathrm{y}).\end{array}$$

Therefore, we have

$$\begin{array}{c}\hfill \mathfrak{b}+\mathcal{E}\left(\alpha \right(\mathrm{x},\mathrm{y}\left)\mathrm{d}\right(\mathcal{Q}\mathrm{x},\mathcal{Q}\mathrm{y}\left)\right)\le \mathcal{E}\left(\mathrm{d}\right(\mathrm{x},\mathrm{y}),\end{array}$$

where $-ln\theta =\mathfrak{b}$. It follows that $\mathcal{Q}$ is an $\alpha$-$\mathcal{E}$-contraction. According to Definition 5 and by the completeness property, we have that the sequence $\left\{{\mathrm{x}}_{\ell }\right\}$ converges at $\phi \in \mathcal{S}$; i.e., $\underset{\ell \to \infty }{lim}{\mathrm{x}}_{\ell }=\phi$. From the continuity, we show that $\phi$ is a fixed point of $\mathcal{Q}$ as follows: $\mathcal{Q}\phi =\phi$. Certainly, Fix($\mathcal{Q}$) $\ne \varnothing$. Now, $\forall \mathrm{x},\mathrm{y}\in Fix\left(\mathcal{Q}\right),\alpha (\phi ,{\phi }^{*})=1$. As a result, $\mathcal{Q}$ is ${\alpha }^{*}$-admissible. Therefore all the assertions of Theorem 4 are satisfied and hence $\mathcal{Q}$ has a unique fixed point (solution) $\phi \in \mathcal{S}$. □

5. An Application to Fractional Differential Equations

Considering all the analytical approaches that have been used to study different aspects of fractional-order chaotic systems, Chauhan et al. [25] analyzed the intricate dynamics of a fractional-order King Cobra model. Sene [26] analyzed a chaotic system using the Caputo fractional derivative and identified the bifurcation structures and Lyapunov exponents to assess the system’s stability. Muthukumar et al. [27] studied the practical implications of the synchronization behavior of their new fractional-order King Cobra system. Petras et al. [28] examined and analyzed the modeling and simulation of fractal-order chaotic systems and their complex dynamics. Haroon et al. [29] analyzed the Reich–Rus–Cirić, Gupta–Saxena, and Kannan contractions, along with a refined simple $\Theta$ contraction, and studied the existence and uniqueness of solutions to the fractional-order King Cobra model using the Atangana–Baleanu–Caputo derivative ($\mathcal{ABC}$), which has non-local and non-singular kernels. This body of work enhances our understanding of fractional-order chaotic systems and their dynamics.

Chua et al. [30] discussed a set of complex nonlinear ordinary differential equations, which describes the dynamics of the three state variables $\mathrm{x}$, $\mathrm{y}$, and $\mathrm{z}$. These evolving variables exhibit complex behavior over time, as they are interdependent through linear and complex nonlinear relationships. This leads to a complex structure from which different dynamical patterns can be studied. These differential equations are formulated as follows,

$$\begin{array}{cc}\hfill & {\displaystyle \frac{d\mathrm{x}}{d\mathit{p}}}=\alpha (\mathrm{y}\left(\mathit{p}\right)-h\left(\mathrm{x}\left(\mathit{p}\right)\right))\hfill \\ \hfill & {\displaystyle \frac{d\mathrm{y}}{d\mathit{p}}}=\mathrm{x}\left(\mathit{p}\right)-\mathrm{y}\left(\mathit{p}\right)+\mathrm{z}\left(\mathit{p}\right)\hfill \\ \hfill & {\displaystyle \frac{d\mathrm{z}}{d\mathit{p}}}=-\beta \mathrm{y}\left(\mathit{p}\right)\hfill \end{array}$$

where the piecewise linear function $h\left(\mathrm{x}\right)$ is defined as

$$\begin{array}{c}\hfill h\left(\mathrm{x}\right)=m_1\mathrm{x}+{\displaystyle \frac{1}{2}}(m_0-m_1)\left[|\mathrm{x}+1|-|\mathrm{x}-1|\right].\end{array}$$

This can be equivalently written in piecewise form

$$\begin{array}{c}\hfill h\left(\mathrm{x}\right)=\left\{\begin{array}{cc}{m}_1\mathrm{x}+(m_0-m_1),\hfill & \mathrm{x}>1\hfill \\ m_0\mathrm{x},\hfill & \left|\mathrm{x}\right|\le 1\hfill \\ m_1\mathrm{x}-(m_0-m_1),\hfill & \mathrm{x}<-1\hfill \end{array}\right..\end{array}$$

The basic structure of the model consists of several interlinked coupled devices. The nonlinear piecewise linear function h(x) adds more complication and unsymmetrical asymmetry to the system after the linear components create the basic framework. The model exhibits nonlinearity and, hence, can produce different results featuring, among other things, stable periodic phases and chaotic states for a given parameter. Nonlinear basic-style elements collectively form a feedback system that magnifies the primary systemic shifts by several orders of magnitude in response to a relatively minor change in the variables.

Each parameter within Chua’s circuit serves the purpose of producing particular behaviors of the system. One such parameter, $\alpha$, decides how fast $\mathrm{x}$ changes and how strongly $\mathrm{y}$ is coupled with $\mathrm{x}$. Another parameter, $\beta$, mitigates the effect of damping in the $\mathrm{z}$ dynamics, and thus it controls the oscillations of the system. The system’s nonlinearity, which is necessary for the system to become chaotic, is created by parameters $m_0$ and $m_1$, which correspond to the slopes of the piecewise linear characteristic of Chua’s diode. The $h\left(\mathrm{x}\right)$ function’s absolute value components contribute to the system’s dynamics by the formation of breakpoints, prompt threshold changes, and the imposition of asymmetry through the system’s dynamics.

Chua’s circuit demonstrates its versatility by modeling various systems in the real world by functioning in different operational frameworks. Values of the circuit components (capacitances and inductances) result in the parameters $\alpha$ and $\beta$. Physical parameters $m_0$ and $m_1$ determine the characteristics of the nonlinear resistor. These describe the feedback function and the resistance along with the capacitive discharge. The piecewise linear function $h\left(\mathrm{x}\right)$ introduces realism by responding to changes in the values of variables due to threshold and sign restrictions. Nonlinear dynamics associated with Chua’s circuit help scientists as they capture the entirety of the phenomenon of chaotic behavior. Chua’s circuit permits researchers to study the fine-ordered states of chaos and the dynamics of the circuit owing to its design with several adjustable parameters. These reveal the principles underpinning complex systems. Chua’s circuit connects intricate dynamics to elegant mathematics, offering an explanation of the behavior of disparate systems and the patterns in order that characterize systems in nature.

Subsequently, we substitute the provided Chua’s circuit model with a classical time derivative utilizing the $\mathcal{ABC}$ operator as

$$\begin{array}{cc}\hfill & \stackrel{ABC}{0}{\mathfrak{D}}_{\mathit{p}}^{\widehat{s}}\left(\mathrm{x}\left(\mathit{p}\right)\right)=\alpha (\mathrm{y}\left(\mathit{p}\right)-h\left(\mathrm{x}\left(\mathit{p}\right)\right))\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill & \stackrel{ABC}{0}{\mathfrak{D}}_{\mathit{p}}^{\widehat{s}}\left(\mathrm{y}\left(\mathit{p}\right)\right)=\mathrm{x}\left(\mathit{p}\right)-\mathrm{y}\left(\mathit{p}\right)+\mathrm{z}\left(\mathit{p}\right)\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill & \stackrel{ABC}{0}{\mathfrak{D}}_{\mathit{p}}^{\widehat{s}}\left(\mathrm{z}\left(\mathit{p}\right)\right)=-\beta \mathrm{y}\left(\mathit{p}\right)\hfill \end{array}$$

(38)

Physical significance of fractional-order Chua’s circuit: The introduction of fractional-order analysis through Atangana–Baleanu–Caputo ($\mathcal{ABC}$) fractional derivatives transforms Chua’s circuit into a system that embodies a physical reality with memory effects, non-local interactions, and hereditary attributes. The $\mathcal{ABC}$ fractional derivative performs its operations with a Mittag–Leffler function-based non-singular and non-local kernel that eliminates singularities. The non-singularity of this model resolves mathematical and computational challenges that singular kernels pose and enhances its representation of real physical processes. This improvement extends to viscoelastic materials, biological tissues, and ecological systems as well as financial market analysis. The $\mathcal{ABC}$ derivative integrates a memory kernel to model historic $\mathrm{x}$, $\mathrm{y}$, and $\mathrm{z}$ state variable coupling to explain the long periods of dependence and delayed feedback characteristic of electronic circuits, anomalous diffusion, and fractional-order Chua circuits.

Employing Chua’s fractional circuit is useful for describing intricate dynamical behaviors due to heterogeneity or long-range interactions. Aspects of the fractional framework and the nonlinear piecewise linear function $h\left(\mathrm{x}\right)$ result in convoluted feedback systems that produce multiple unstable configurations and chaotic dynamics beyond the scope of classical integer-order systems. The fractional-order perspective enhances the approach to stability, the relative sequence of time, and the smoothing of sharp transitions. The enhanced fractional-order Chua’s circuit with Chua’s $\mathcal{ABC}$ derivative expanded the usefulness of the circuit for designers. The circuit’s performance on systems with hereditary characteristics, viscoelastic damping and dynamics, and viscoelastic population control is outstanding. The fractional-order Chua’s circuit integrated with the $\mathcal{ABC}$ derivative provides a composite system for exploring the interplay of memory, nonlinear dynamics, and complicated systems, making it a versatile composite system for various mathematical, physical, and engineering problems.

To facilitate easier comprehension, the fractional Chua’s circuit system is presented in the following manner:

$$\begin{array}{cc}\hfill & {\mathcal{H}}_1(\mathrm{x},\mathit{p})=\alpha (\mathrm{y}\left(\mathit{p}\right)-h\left(\mathrm{x}\left(\mathit{p}\right)\right))\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill & {\mathcal{H}}_2(\mathrm{y},\mathit{p})=\mathrm{x}\left(\mathit{p}\right)-\mathrm{y}\left(\mathit{p}\right)+\mathrm{z}\left(\mathit{p}\right)\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill & {\mathcal{H}}_3(\mathrm{z},\mathit{p})=-\beta \mathrm{y}\left(\mathit{p}\right)\hfill \end{array}$$

(41)

where $h\left(\mathrm{x}\right)$ remains defined as

$$\begin{array}{c}\hfill h\left(\mathrm{x}\right)=m_1\mathrm{x}+{\displaystyle \frac{1}{2}}(m_0-m_1)\left[|\mathrm{x}+1|-|\mathrm{x}-1|\right].\end{array}$$

(42)

Based on the $\mathcal{ABC}$ derivative’s established definition, a reformulation of the system into a fractional Volterra integral equation is presented below:

$$\begin{array}{cc}\hfill & \mathrm{x}\left(\mathit{p}\right)-v_1\left(\mathit{p}\right)={\displaystyle \frac{1-\widehat{s}}{\aleph \left(\widehat{s}\right)}}\left\{{\mathcal{H}}_1(\mathit{p},\mathrm{x})\right\}+{\displaystyle \frac{\widehat{s}}{\aleph \left(\widehat{s}\right)\Gamma \left(\widehat{s}\right)}}{\int }_0^{\mathit{p}}{(\mathit{p}-h)}^{\widehat{s}-1}\left\{{\mathcal{H}}_1(h,\mathrm{x})\right\}dh\hfill \\ \hfill & \mathrm{y}\left(\mathit{p}\right)-v_2\left(\mathit{p}\right)={\displaystyle \frac{1-\widehat{s}}{\aleph \left(\widehat{s}\right)}}\left\{{\mathcal{H}}_2(\mathit{p},\mathrm{y})\right\}+{\displaystyle \frac{\widehat{s}}{\aleph \left(\widehat{s}\right)\Gamma \left(\widehat{s}\right)}}{\int }_0^{\mathit{p}}{(\mathit{p}-h)}^{\widehat{s}-1}\left\{{\mathcal{H}}_2(h,\mathrm{y})\right\}dh\hfill \\ \hfill & \mathrm{z}\left(\mathit{p}\right)-v_3\left(\mathit{p}\right)={\displaystyle \frac{1-\widehat{s}}{\aleph \left(\widehat{s}\right)}}\left\{{\mathcal{H}}_3(\mathit{p},\mathrm{z})\right\}+{\displaystyle \frac{\widehat{s}}{\aleph \left(\widehat{s}\right)\Gamma \left(\widehat{s}\right)}}{\int }_0^{\mathit{p}}{(\mathit{p}-h)}^{\widehat{s}-1}\left\{{\mathcal{H}}_3(h,\mathrm{z})\right\}dh.\hfill \end{array}$$

The iterative procedure outlined previously is presented in the following equations:

$$\begin{array}{c}\hfill \mathrm{x}\left(0\right)=v_1\left(\mathit{p}\right),\mathrm{y}\left(0\right)=v_2\left(\mathit{p}\right)\mathrm{and}\mathrm{z}\left(0\right)=v_3\left(\mathit{p}\right).\end{array}$$

Consequently, the system in (11) takes the following form:

$$\begin{array}{cc}\hfill & \mathrm{x}\left(\mathit{p}\right)=\mathrm{x}\left(0\right)+{\displaystyle \frac{1-\widehat{s}}{\aleph \left(\widehat{s}\right)}}{\mathcal{H}}_1(\mathit{p},\mathrm{x})+{\displaystyle \frac{\widehat{s}}{\aleph \left(\widehat{s}\right)\Gamma \left(\widehat{s}\right)}}{\int }_0^{\mathit{p}}{(\mathit{p}-h)}^{\widehat{s}-1}{\mathcal{H}}_1(h,\mathrm{x})dh\hfill \\ \hfill & \mathrm{y}\left(\mathit{p}\right)=\mathrm{y}\left(0\right)+{\displaystyle \frac{1-\widehat{s}}{\aleph \left(\widehat{s}\right)}}{\mathcal{H}}_2(\mathit{p},\mathrm{y})+{\displaystyle \frac{\widehat{s}}{\aleph \left(\widehat{s}\right)\Gamma \left(\widehat{s}\right)}}{\int }_0^{\mathit{p}}{(\mathit{p}-h)}^{\widehat{s}-1}{\mathcal{H}}_2(h,\mathrm{y})dh\hfill \\ \hfill & \mathrm{z}\left(\mathit{p}\right)=\mathrm{z}\left(0\right)+{\displaystyle \frac{1-\widehat{s}}{\aleph \left(\widehat{s}\right)}}{\mathcal{H}}_3(\mathit{p},\mathrm{z})+{\displaystyle \frac{\widehat{s}}{\aleph \left(\widehat{s}\right)\Gamma \left(\widehat{s}\right)}}{\int }_0^{\mathit{p}}{(\mathit{p}-h)}^{\widehat{s}-1}{\mathcal{H}}_3(h,\mathrm{z})dh.\hfill \end{array}$$

Through the process of defining the iterative process, we arrive at

$$\begin{array}{cc}\hfill & {\mathrm{x}}_n\left(\mathit{p}\right)={\displaystyle \frac{1-\widehat{s}}{\aleph \left(\widehat{s}\right)}}{\mathcal{H}}_1(\mathit{p},{\mathrm{x}}_{n-1})+{\displaystyle \frac{\widehat{s}}{\aleph \left(\widehat{s}\right)\Gamma \left(\widehat{s}\right)}}{\int }_0^{\mathit{p}}{(\mathit{p}-h)}^{\widehat{s}-1}{\mathcal{H}}_1(h,{\mathrm{x}}_{n-1})dh\hfill \\ \hfill & {\mathrm{y}}_n\left(\mathit{p}\right)={\displaystyle \frac{1-\widehat{s}}{\aleph \left(\widehat{s}\right)}}{\mathcal{H}}_2(\mathit{p},{\mathrm{y}}_{n-1})+{\displaystyle \frac{\widehat{s}}{\aleph \left(\widehat{s}\right)\Gamma \left(\widehat{s}\right)}}{\int }_0^{\mathit{p}}{(\mathit{p}-h)}^{\widehat{s}-1}{\mathcal{H}}_2(h,{\mathrm{y}}_{n-1})dh\hfill \\ \hfill & {\mathrm{z}}_n\left(\mathit{p}\right)={\displaystyle \frac{1-\widehat{s}}{\aleph \left(\widehat{s}\right)}}{\mathcal{H}}_3(\mathit{p},{\mathrm{z}}_{n-1})+{\displaystyle \frac{\widehat{s}}{\aleph \left(\widehat{s}\right)\Gamma \left(\widehat{s}\right)}}{\int }_0^{\mathit{p}}{(\mathit{p}-h)}^{\widehat{s}-1}{\mathcal{H}}_3(h,{\mathrm{z}}_{n-1})dh.\hfill \end{array}$$

Suppose $\mathrm{d}:\mathcal{S}\times \mathcal{S}\to [0,\infty )$ is a mapping on $\mathcal{S}=\{{\mathcal{H}}_1,{\mathcal{H}}_2,{\mathcal{H}}_3\in \mathbb{C}(\mathbb{I},{\mathbb{R}}^3)\mid {\mathcal{H}}_1\left(\mathit{p}\right),{\mathcal{H}}_2\left(\mathit{p}\right),{\mathcal{H}}_3\left(\mathit{p}\right)>0,\forall \mathit{p}\in \mathbb{I}=[0,{\mathit{p}}_{max}]\}$, where ${\mathit{p}}_{max}>0$, in such a way that $\mathrm{d}({\mathrm{x}}_1\left(\mathit{p}\right),{\mathrm{x}}_2\left(\mathit{p}\right))={sup}_{\mathit{p}\in [0,{\mathit{p}}_{max}]}|{\mathrm{x}}_1\left(\mathit{p}\right)-{\mathrm{x}}_2\left(\mathit{p}\right)|$. Obviously, $(\mathcal{S},\mathrm{d})$ is a complete metric space. Define a mapping $\mathcal{Q}:\mathbb{C}(\mathbb{I},{\mathbb{R}}^3)\to \mathbb{C}(\mathbb{I},{\mathbb{R}}^3)$ by

$$\begin{array}{cc}\hfill & \mathcal{Q}\mathrm{x}\left(\mathit{p}\right)=\mathrm{x}\left(0\right)+{\displaystyle \frac{1-\widehat{s}}{\aleph \left(\widehat{s}\right)}}{\mathcal{H}}_1(\mathit{p},\mathrm{x})+{\displaystyle \frac{\widehat{s}}{\aleph \left(\widehat{s}\right)\Gamma \left(\widehat{s}\right)}}{\int }_0^{\mathit{p}}{(\mathit{p}-h)}^{\widehat{s}-1}{\mathcal{H}}_1(\mathrm{x},h)dh,\hfill \\ \hfill & \mathcal{Q}\mathrm{y}\left(\mathit{p}\right)=\mathrm{y}\left(0\right)+{\displaystyle \frac{1-\widehat{s}}{\aleph \left(\widehat{s}\right)}}{\mathcal{H}}_2(\mathit{p},\mathrm{y})+{\displaystyle \frac{\widehat{s}}{\aleph \left(\widehat{s}\right)\Gamma \left(\widehat{s}\right)}}{\int }_0^{\mathit{p}}{(\mathit{p}-h)}^{\widehat{s}-1}{\mathcal{H}}_2(\mathrm{y},h)dh,\hfill \\ \hfill & \mathcal{Q}\mathrm{z}\left(\mathit{p}\right)=\mathrm{z}\left(0\right)+{\displaystyle \frac{1-\widehat{s}}{\aleph \left(\widehat{s}\right)}}{\mathcal{H}}_3(\mathit{p},\mathrm{z})+{\displaystyle \frac{\widehat{s}}{\aleph \left(\widehat{s}\right)\Gamma \left(\widehat{s}\right)}}{\int }_0^{\mathit{p}}{(\mathit{p}-h)}^{\widehat{s}-1}{\mathcal{H}}_3(\mathrm{z},h)dh.\hfill \end{array}$$

Theorem 7.

Under the conditions

1. 

$\left({\displaystyle \frac{\Gamma \left(\widehat{s}\right)(1-\widehat{s})+{\mathit{p}}^{\widehat{s}}}{\aleph \left(\widehat{s}\right)\Gamma \left(\widehat{s}\right)}}\right)<1$,

2. 

there exists ${\displaystyle \frac{1}{3}}\in [0,1)$ such that $\left|{\mathcal{H}}_1(\mathit{p},{\mathrm{x}}_1\left(\mathit{p}\right))-{\mathcal{H}}_1(\mathit{p},{\mathrm{x}}_2\left(\mathit{p}\right))\right|\le {\displaystyle \frac{1}{3}}\left[|{\mathrm{x}}_1\left(\mathit{p}\right)-{\mathrm{x}}_2\left(\mathit{p}\right)|\right]$, $\left|{\mathcal{H}}_2(\mathit{p},{\mathrm{y}}_1\left(\mathit{p}\right))-{\mathcal{H}}_2(\mathit{p},{\mathrm{y}}_2\left(\mathit{p}\right))\right|\le {\displaystyle \frac{1}{3}}\left[|{\mathrm{y}}_1\left(\mathit{p}\right)-{\mathrm{y}}_2\left(\mathit{p}\right)|\right]$, $\left|{\mathcal{H}}_3(\mathit{p},{\mathrm{z}}_1\left(\mathit{p}\right))-{\mathcal{H}}_3(\mathit{p},{\mathrm{z}}_2\left(\mathit{p}\right))\right|\le {\displaystyle \frac{1}{3}}\left[|{\mathrm{z}}_1\left(\mathit{p}\right)-{\mathrm{z}}_2\left(\mathit{p}\right)|\right]$,

for all ${\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{y}}_1,{\mathrm{y}}_2,{\mathrm{z}}_1,{\mathrm{z}}_2\in \mathcal{S}$. Then, the fractional Chua’s circuit system has a unique solution.

Proof. 

Define a map $\alpha :\mathcal{S}\times \mathcal{S}\to {\mathcal{R}}^{+}$ by $\alpha ({\mathrm{x}}_1,{\mathrm{x}}_2)=1\forall {\mathrm{x}}_1,{\mathrm{x}}_2\in \mathcal{S}$. Thus, $\mathcal{Q}$ is ${\alpha }^{*}$-admissible. Let $\mathcal{E}\in \mathcal{F}$ such that $\mathcal{E}\left(\lambda \right)=ln\lambda ,\lambda >0$. Consider the operator $\mathcal{Q}$ defined by

$$\begin{array}{cc}\hfill & \mathcal{Q}\mathrm{x}\left(\mathit{p}\right)=\mathrm{x}\left(0\right)+{\displaystyle \frac{1-\widehat{s}}{\aleph \left(\widehat{s}\right)}}{\mathcal{H}}_1(\mathit{p},\mathrm{x})+{\displaystyle \frac{\widehat{s}}{\aleph \left(\widehat{s}\right)\Gamma \left(\widehat{s}\right)}}{\int }_0^{\mathit{p}}{(\mathit{p}-h)}^{\widehat{s}-1}{\mathcal{H}}_1(\mathrm{x},h)dh\hfill \end{array}$$

We show that $\mathcal{Q}$ is a contraction mapping. For any ${\mathrm{x}}_1,{\mathrm{x}}_2\in \mathcal{S}$, we have

$$\begin{array}{cc}\hfill \left|\mathcal{Q}{\mathrm{x}}_1\left(\mathit{p}\right)-\mathcal{Q}{\mathrm{x}}_2\left(\mathit{p}\right)\right|& =\left|{\displaystyle \frac{1-\widehat{s}}{\aleph \left(\widehat{s}\right)}}\left[{\mathcal{H}}_1(\mathit{p},{\mathrm{x}}_1\left(\mathit{p}\right))-{\mathcal{H}}_1(\mathit{p},{\mathrm{x}}_2\left(\mathit{p}\right))\right]\right.\hfill \\ \hfill & +\left.{\displaystyle \frac{\widehat{s}}{\aleph \left(\widehat{s}\right)\Gamma \left(\widehat{s}\right)}}{\int }_0^{\mathit{p}}{(\mathit{p}-h)}^{\widehat{s}-1}\left[{\mathcal{H}}_1(\mathit{p},{\mathrm{x}}_1\left(h\right))-{\mathcal{H}}_1(\mathit{p},{\mathrm{x}}_2\left(h\right))\right]dh\right|\hfill \\ \hfill & \le {\displaystyle \frac{1-\widehat{s}}{\aleph \left(\widehat{s}\right)}}\left|{\mathcal{H}}_1(\mathit{p},{\mathrm{x}}_1\left(\mathit{p}\right))-{\mathcal{H}}_1(\mathit{p},{\mathrm{x}}_2\left(\mathit{p}\right))\right|\hfill \\ \hfill & +{\displaystyle \frac{\widehat{s}}{\aleph \left(\widehat{s}\right)\Gamma \left(\widehat{s}\right)}}{\int }_0^{\mathit{p}}{(\mathit{p}-h)}^{\widehat{s}-1}\left|{\mathcal{H}}_1(\mathit{p},{\mathrm{x}}_1\left(h\right))-{\mathcal{H}}_1(\mathit{p},{\mathrm{x}}_2\left(h\right))\right|dh\hfill \\ \hfill & \le {\displaystyle \frac{1-\widehat{s}}{\aleph \left(\widehat{s}\right)}}·{\displaystyle \frac{1}{3}}|{\mathrm{x}}_1\left(\mathit{p}\right)-{\mathrm{x}}_2\left(\mathit{p}\right)|\hfill \\ \hfill & +{\displaystyle \frac{\widehat{s}}{\aleph \left(\widehat{s}\right)\Gamma \left(\widehat{s}\right)}}{\int }_0^{\mathit{p}}{(\mathit{p}-h)}^{\widehat{s}-1}·{\displaystyle \frac{1}{3}}|{\mathrm{x}}_1\left(h\right)-{\mathrm{x}}_2\left(h\right)|dh.\hfill \end{array}$$

Since $|{\mathrm{x}}_1\left(h\right)-{\mathrm{x}}_2\left(h\right)|\le {sup}_{h\in [0,{\mathit{p}}_{max}]}|{\mathrm{x}}_1\left(h\right)-{\mathrm{x}}_2\left(h\right)|=\parallel {\mathrm{x}}_1-{\mathrm{x}}_2{\parallel }_{\infty }$, we have

$$\begin{array}{cc}\hfill & \le {\displaystyle \frac{1-\widehat{s}}{3\aleph \left(\widehat{s}\right)}}\parallel {\mathrm{x}}_1-{\mathrm{x}}_2{\parallel }_{\infty }+{\displaystyle \frac{\widehat{s}}{3\aleph \left(\widehat{s}\right)\Gamma \left(\widehat{s}\right)}}{\parallel {\mathrm{x}}_1-{\mathrm{x}}_2\parallel }_{\infty }{\int }_0^{\mathit{p}}{(\mathit{p}-h)}^{\widehat{s}-1}dh\hfill \\ \hfill & ={\displaystyle \frac{1-\widehat{s}}{3\aleph \left(\widehat{s}\right)}}\parallel {\mathrm{x}}_1-{\mathrm{x}}_2{\parallel }_{\infty }+{\displaystyle \frac{\widehat{s}}{3\aleph \left(\widehat{s}\right)\Gamma \left(\widehat{s}\right)}}{\parallel {\mathrm{x}}_1-{\mathrm{x}}_2\parallel }_{\infty }·{\displaystyle \frac{{\mathit{p}}^{\widehat{s}}}{\widehat{s}}}\hfill \\ \hfill & =\left({\displaystyle \frac{1-\widehat{s}}{3\aleph \left(\widehat{s}\right)}}+{\displaystyle \frac{{\mathit{p}}^{\widehat{s}}}{3\aleph \left(\widehat{s}\right)\Gamma \left(\widehat{s}\right)}}\right){\parallel {\mathrm{x}}_1-{\mathrm{x}}_2\parallel }_{\infty }\hfill \\ \hfill & ={\displaystyle \frac{1}{3\aleph \left(\widehat{s}\right)}}\left(1-\widehat{s}+{\displaystyle \frac{{\mathit{p}}^{\widehat{s}}}{\Gamma \left(\widehat{s}\right)}}\right){\parallel {\mathrm{x}}_1-{\mathrm{x}}_2\parallel }_{\infty }.\hfill \end{array}$$

Since $\left({\displaystyle \frac{\Gamma \left(\widehat{s}\right)(1-\widehat{s})+{\mathit{p}}^{\widehat{s}}}{\aleph \left(\widehat{s}\right)\Gamma \left(\widehat{s}\right)}}\right)<1$, we can write

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{3\aleph \left(\widehat{s}\right)}}\left(1-\widehat{s}+{\displaystyle \frac{{\mathit{p}}^{\widehat{s}}}{\Gamma \left(\widehat{s}\right)}}\right)& ={\displaystyle \frac{1}{3}}·{\displaystyle \frac{\Gamma \left(\widehat{s}\right)(1-\widehat{s})+{\mathit{p}}^{\widehat{s}}}{\aleph \left(\widehat{s}\right)\Gamma \left(\widehat{s}\right)}}<{\displaystyle \frac{1}{3}}\hfill \end{array}$$

Therefore

$$\begin{array}{c}\hfill \left|\mathcal{Q}{\mathrm{x}}_1\left(\mathit{p}\right)-\mathcal{Q}{\mathrm{x}}_2\left(\mathit{p}\right)\right|\le {\displaystyle \frac{1}{3}}{\parallel {\mathrm{x}}_1-{\mathrm{x}}_2\parallel }_{\infty }.\end{array}$$

Taking the supremum over $\mathit{p}\in [0,{\mathit{p}}_{max}]$, we obtain

$$\begin{array}{c}\hfill \parallel \mathcal{Q}{\mathrm{x}}_1-\mathcal{Q}{\mathrm{x}}_2{\parallel }_{\infty }\le e^{-\mathfrak{b}}{\parallel {\mathrm{x}}_1-{\mathrm{x}}_2\parallel }_{\infty },\end{array}$$

where $ln\left({\displaystyle \frac{1}{3}}\right)=-\mathfrak{b}$. Therefore

$$\begin{array}{c}\hfill \alpha (\mathrm{x},\mathrm{y})\mathrm{d}(\mathcal{Q}\mathrm{x},\mathcal{Q}\mathrm{y})\le e^{-\mathfrak{b}}\mathrm{d}(\mathrm{x},\mathrm{y}).\end{array}$$

Taking log on both sides, we get

$$\begin{array}{c}\hfill \mathfrak{b}+ln\left[\alpha \right(\mathrm{x},\mathrm{y}\left)\mathrm{d}\right(\mathcal{Q}\mathrm{x},\mathcal{Q}\mathrm{y}\left)\right]\le ln\mathrm{d}(\mathrm{x},\mathrm{y}).\end{array}$$

Therefore, we have

$$\begin{array}{c}\hfill \mathfrak{b}+\mathcal{E}\left(\alpha \right(\mathrm{x},\mathrm{y}\left)\mathrm{d}\right(\mathcal{Q}\mathrm{x},\mathcal{Q}\mathrm{y}\left)\right)\le \mathcal{E}\left(\mathrm{d}\right(\mathrm{x},\mathrm{y}),\end{array}$$

where $\mathfrak{b}=-ln\left({\displaystyle \frac{1}{3}}\right)$. Therefore all the assertions of Theorem 5 are satisfied, and hence $\mathcal{Q}$ has a unique fixed point. Similarly, we can prove for $\mathrm{y}\left(\mathit{p}\right)$ and $\mathrm{z}\left(\mathit{p}\right)$. □

6. Conclusions

In this paper, we initiated a new idea of $\alpha$-$\mathcal{E}$-contraction with $\alpha$-admissibility in the surrounding of metric space. We developed here an $\alpha$-$\mathcal{E}$-contraction and $\alpha$-$\mathcal{E}$-Suzuki-type contraction of fixed point results in orbitally continuous mappings. To support our result, we presented an example illustrating how our contraction operates in comparison to existing contractions. These new investigations and applications would improve the impact of a new setup. As an application of our newly defined contraction, we solved an integral equation and a fractional differential equation.