Advancing Fixed Point Theory in Elliptic-Valued Suprametric Spaces

Abstract

The purpose of this research article is to establish common fixed point results for generalized contractions defined with control functions in the framework of elliptic-valued suprametric spaces. By utilizing the structure and properties of these spaces, we develop new fixed point results that extend and unify several known theorems in the literature. The theoretical findings are illustrated through a series of non-trivial examples, demonstrating the applicability and robustness of the proposed approach. As a concrete application, we employ the established fixed point theorems to analyze the existence and uniqueness of solutions for nonlinear Volterra integral equations of the second kind. Furthermore, the results obtained herein naturally encompass a wide range of fixed point theorems previously reported in the literature as direct corollaries, emphasizing the broad applicability and flexibility of the proposed framework.

1. Introduction

The real number system forms the basis of conventional mathematics and provides a scheme for measuring quantities and modeling physical phenomena. However, the need to solve algebraic equations beyond the real domain led to the introduction of complex numbers. Complex numbers, involving the imaginary unit i with $i^2 = -1,$ were first studied by Gerolamo Cardano (1501–1576) and later formalized by Rafael Bombelli (1526–1572). Later developments introduced other generalized number systems. In particular, by specifying the imaginary unit i with the property $i^2 = \wp \in \mathbb{R}$ yields elliptic numbers when $\wp < 0,$ dual numbers when $\wp = 0,$ and hyperbolic numbers when $\wp > 0.$ This unified framework provides a natural extension of the complex number system and is useful in various mathematical models.

The concept of a metric space (MS) was first developed by Fréchet [1]. An MS is formed by a set, together with an associated distance function that satisfies three key axioms—(i) the distance between different elements is positive, while an element has zero distance from itself; (ii) distances are symmetric, so the order of points does not affect the measurement; and (iii) the distance function obeys the triangle inequality, ensuring that the distance connecting two points directly does not exceed the aggregate distance measured via an intermediate point. This set of rules offers a precise foundation for investigating continuity, convergence, and geometric frameworks in pure and applied mathematics. Within MSs, the Banach Contraction Principle (BCP) [2] stands out as a fundamental result, guaranteeing that every contraction mapping on a complete MS possesses a unique fixed point (FP). Fisher [3] extended this theory by introducing rational contractions, while Azam et al. [4] further generalized the concept to complex-valued metric spaces (C-VMSs), where distances take values in the complex numbers and satisfy suitably adapted metric axioms. This extension broadens the applicability of MS theory to more sophisticated mathematical frameworks and real-world problems. Within this framework, Azam et al. [4] proved common FP theorems using Fisher’s rational contraction methodology, and subsequent work by Rouzkard et al. [5] incorporated additional terms into the contractive condition, further generalizing these results. Sintunavarat et al. [6] later employed control functions to refine the contractive conditions, extending the foundational results of Azam et al. [4]. Shammaky et al. [7] applied common FP results in C-VMSs to Homotopy theory, while in a related work, they further generalized the framework by introducing complex-valued bipolar MSs with applications [8]. Moussaoui et al. [9] provided a comprehensive survey of b-metric-like spaces, highlighting their structural properties, FP results, and diverse applications across nonlinear analysis and applied mathematics. Such spaces generalize classical b-metric settings and offer greater flexibility in modeling nonlinear phenomena. Moussaoui et al. [10] introduced a further generalization of parametric MSs through $\mathcal{B}$-actions, establishing a unified framework that encompasses various existing metric-type structures and enables new FP results in a broader setting.

The introduction of elliptic-valued metric spaces (E-VMSs) by Ozturk et al. [11] marked a further development in FP theory. In this setting, the metric is assigned using elliptic numbers, a generalization motivated by elliptic function theory which provides a richer structure than traditional real or complex numbers. Alamri et al. [12,13] have expanded FP theory within this context, proving the new FP theorems for a variety of mappings and illustrating their applications to nonlinear Fredholm integral equations, including models pertinent to climate dynamics.

Additionally, Berzig [14] developed SMSs by extending the classical idea of an MS. In SMSs, a supplementary term represents the interactions mediated through intermediate points, allowing these spaces to capture more complex relationships while retaining fundamental conditions, including positivity, symmetry, and zero distance holding only between the same elements. Panda et al. [15] further broadened this notion to complex numbers, introducing C-VSMSs. Their work not only generalized FP theory but also showed practical applications in generating Barnsley Fern fractals and solving mixed Volterra–Fredholm integral equations, thereby linking abstract mathematical concepts to their computational and applied implementations. Later, Abdou [16] and Shammaky et al. [17] employed C-VSMSs to prove new FP results under generalized contraction conditions.

In this study, we develop some new common FP (CFP) theorems within the framework of E-VSMSs, under generalized contractive conditions involving single-variable control functions. The obtained results broaden and harmonize several significant advancements reported in the literature, incorporating the core theorems developed by Panda et al. [15] and Abdou [16] in the setting of C-VSMSs; the core theoretical results of Alamri [13] and Ozturk et al. [11] in the established setting of E-VMSs; and the theorems of Azam et al. [4], Rouzkard et al. [5], and Sintunavarat et al. [6] in C-VMSs. To highlight the practical relevance of the formulated framework, we describe a comprehensive example demonstrating its applicability and benefits. Furthermore, we utilize our FP results to solve the NVIEs of the second kind.

The remainder of this paper is organized as follows. In the Preliminaries Section, we recall the necessary background material, including elliptic numbers and their basic properties, C-VMSs, E-VMSs, SMSs, and E-VSMSs, together with some well-known auxiliary results. In the Main Results Section, we establish some new common FP theorems in E-VSMSs under generalized contractive conditions involving single-variable control functions, and we show that a number of existing FP results can be derived as particular cases of our main theorem. In the Applications Section, we apply the obtained FP results to solve the NVIEs of the second kinds. Finally, the Conclusion Section summarizes the main contributions of the paper and outlines possible directions for future research.

2. Preliminaries

We begin by introducing the fundamental notation and key definitions required for the development of this study. For comprehensive background material, the reader may consult . Let ${\mathbb{E}}_{\wp }$ represent the set of elliptic numbers, defined as follows:

$${\mathbb{E}}_{\wp }=\left\{z=\nu +i\omega :\nu ,\omega \in \mathbb{R},i^2=\wp <0\right\}.$$

Let $z\in {\mathbb{E}}_{\wp }$ be written as $\nu +i\omega$, where $\nu$ and $\omega$ denote the real and imaginary parts of $z,$ respectively.

The fundamental arithmetic operations on elliptic numbers including addition, scalar multiplication, elliptic multiplication, and conjugation are defined as follows.

For, $z_1={\nu }_1+i{\omega }_1$ and $z_2={\nu }_2+i{\omega }_2,$ their sum is defined by

$$z_1+z_2={\nu }_1+{\nu }_2+i\left({\omega }_1+{\omega }_2\right).$$

For $z_1\in {\mathbb{E}}_{\wp }$ and $\lambda \in \mathbb{R}$, their product is defined as

$$\lambda z_1=\lambda \left({\nu }_1+i{\omega }_1\right)=\lambda {\nu }_1+i\lambda {\omega }_1.$$

In addition, for $z_1,z_2\in {\mathbb{E}}_{\wp },$ their elliptic multiplication is given as

$$z_1{z}_2=\left({\nu }_1+i{\omega }_1\right)\left({\nu }_2+i{\omega }_2\right)=\left({\nu }_1{\nu }_2+\wp {\omega }_1{\omega }_2\right)+i\left({\nu }_1{\omega }_2+{\nu }_2{\omega }_1\right).$$

The elliptic conjugate of $z,$ written as $\overline{z},$ is expressed as

$$\overline{z}=Re\left(z\right)-Im\left(z\right)=\nu -i\omega .$$

For $z\in {\mathbb{E}}_{\wp }$, we have

$${\parallel z\parallel }_{\mathbb{E}}=\sqrt{z\overline{z}}=\sqrt{{\nu }^2-\wp {\omega }^2}.$$

It is evident that ${\mathbb{E}}_{\wp }$ forms a two-dimensional vector space over the real numbers $\mathbb{R}$, with addition and scalar multiplication as the defining operations. Moreover, ${\mathbb{E}}_{\wp }$ fulfills the axioms of a field. Consequently, there exists a bijective correspondence between ${\mathbb{E}}_{\wp }$ and ${\mathbb{R}}^2$, which allows each elliptic number $z=\nu +i\omega$ to be uniquely represented in the plane. This structure, referred to as the elliptic plane, defines the distance between $z_1=\left({\nu }_1,{\omega }_1\right)$ and $z_2=\left({\nu }_2,{\omega }_2\right)\in {\mathbb{E}}_{\wp }$ by

$${∥z_1-z_2∥}_{\mathbb{E}}=\sqrt{{\left({\nu }_1-{\nu }_2\right)}^2-\wp {\left({\omega }_1-{\omega }_2\right)}^2},$$

under the constraints ${\left({\nu }_1-{\nu }_2\right)}^2-\wp {\left({\omega }_1-{\omega }_2\right)}^2>0$ and $\wp <0.$

Operating in the elliptic geometric framework, the set of points at unit distance from the origin forms an ellipse, described by

$${\nu }^2-\wp {\omega }^2=1.$$

Hereafter, the zero vector in ${\mathbb{E}}_{\wp }$ is denoted by $\theta .$ A partial order, indicated by $⪯,$ is introduced on ${\mathbb{E}}_{\wp }$ in the following manner: for all $z_1={\nu }_1+i{\omega }_1$, $z_2={\nu }_2+i{\omega }_2\in {\mathbb{E}}_{\wp }$, we define

$$z_1⪯z_2\mathrm{iff}Re\left(z_1\right)\le Re\left(z_2\right)\mathrm{and}Im\left(z_1\right)\le Im\left(z_2\right).$$

Accordingly, $z_1⪯z_2$ is valid if one or more of the following criteria are fulfilled:

$$\begin{array}{c}\left(\mathrm{a}\right)Re\left(z_1\right) = Re\left(z_2\right)\mathrm{and}Im\left(z_1\right)<Im\left(z_2\right),\\ \left(\mathrm{b}\right)Re\left(z_1\right) < Re\left(z_2\right)\mathrm{and}Im\left(z_1\right)=Im\left(z_2\right),\\ \left(\mathrm{c}\right)Re\left(z_1\right) < Re\left(z_2\right)\mathrm{and}Im\left(z_1\right)<Im\left(z_2\right),\\ \left(\mathrm{d}\right)Re\left(z_1\right) = Re\left(z_2\right)\mathrm{and}Im\left(z_1\right)=Im\left(z_2\right).\end{array}$$

To be precise, $z_1\precsim z_2 \left(\mathrm{with}{z}_1\ne z_2\right)$ shows that at minimum, one of the specified axioms (a), (b) or (c) hold, while $z_1\prec z_2$ is employed solely when axioms (d) is true. The following outlines some basic properties of the partial order ⪯ on ${\mathbb{E}}_{\wp }$:

(i)

If $\theta ⪯z_1\precsim z_2,$ then $z_1$ has a strictly smaller elliptic norm than $z_2,$ i.e., ${∥z_1∥}_{\mathbb{E}}<{∥z_2∥}_{\mathbb{E}}.$

(ii)

The relation $z_1⪯z_2$ holds if and only if $z_1-z_2⪯\theta ,$

(iii)

If $z_1⪯z_2$ and $z_2⪯z_3,$ then $z_1⪯z_3$ (transitivity).

(iv)

For $z_1⪯z_2$ and $\lambda >0,$ the following $\lambda z_1⪯\lambda z_2,$ holds.

(v)

Having $\theta ⪯z_1$ and $\theta ⪯z_2$ does not ensure that $\theta ⪯z_1{z}_2.$

Ozturk et al. [11] introduced a new class of MSs, termed E-VMSs, which extends the conventional framework by allowing the distance function to take values in the set of elliptic numbers.

Definition 1

([11]). Let $\mathcal{Y}\ne \varnothing$ and $ð:\mathcal{Y}\times \mathcal{Y}\to {\mathbb{E}}_{\wp }$ be an elliptic-valued distance function satisfying

• $\left(E_1\right):$$\theta ⪯ð(\varsigma ,\sigma )$, and $ð(\varsigma ,\sigma )=\theta$⇔$\varsigma =\sigma$,
• $\left(E_2\right):$$ð(\varsigma ,\sigma )=ð(\sigma ,\varsigma )$,
• $\left(E_3\right):$$ð(\varsigma ,\sigma )⪯ð(\varsigma ,\varrho )+ð(\varrho ,\sigma ),$

for all $\varsigma ,\sigma ,\varrho \in \mathcal{Y}$, then $(\mathcal{Y},ð)$ is called an E-VMS.

Example 1

([11]). Let $\mathcal{Y}={\mathbb{E}}_{\wp }$. We define $ð:{\mathbb{E}}_{\wp }\times {\mathbb{E}}_{\wp }\to$${\mathbb{E}}_{\wp }$ in this way

$$ð\left(z_1,z_2\right)=\sqrt{{\left({\nu }_1-{\nu }_2\right)}^2-\wp {\left({\omega }_1-{\omega }_2\right)}^2,}$$

for $z_1={\nu }_1+i{\omega }_1,z_2={\nu }_2+i{\omega }_2$ in ${\mathbb{E}}_{\wp }$ with ${\left({\nu }_1-{\nu }_2\right)}^2-\wp {\left({\omega }_1-{\omega }_2\right)}^2>0$ and $i^2=\wp <0$. Consequently, $\left({\mathbb{E}}_{\wp },ð\right)$ fulfills the properties of an E-VMS.

The next fixed point theorem, as presented by Ozturk et al. [11] is stated below.

Theorem 1

([11]). Let $\mathcal{W}$ be a self mapping on a complete E-VMS $(\mathcal{Y},ð)$ and there is a constant $\varpi \in [0,1)$ such that

$$ð\left(\mathcal{W}\varsigma ,\mathcal{W}\sigma \right)⪯\varpi ð\left(\varsigma ,\sigma \right),$$

∀$\varsigma ,\sigma \in$$\mathcal{Y},$ then $\mathcal{W}$ has a unique FP.

The concept of SMS, as defined by Berzig [14], is described below

Definition 2

([14]). Let $\mathcal{Y}\ne \varnothing$ and $ð:\mathcal{Y}\times \mathcal{Y}\to {\mathbb{R}}^{+}$ be a function satisfying

(i) 

$0\le ð(\varsigma ,\sigma )$ and $ð(\varsigma ,\sigma )=0$⟺$\varsigma =\sigma$,

(ii) 

$ð(\varsigma ,\sigma )=ð(\sigma ,\varsigma ),$

(iii) 

$ð(\varsigma ,\sigma )\le ð(\varsigma ,\varrho )+ð(\varrho ,\sigma )+\vartheta ð(\varsigma ,\varrho )ð(\varrho ,\sigma ),$

for all $\varsigma ,\sigma ,\varrho \in \mathcal{Y},$ where $\vartheta \ge 0.$ Then $(\mathcal{Y},ð)$ is said to be a SMS.

Recently, Alamri [18] defined the notion of E-VSMSs as follows:

Definition 3

([18]). Let $\mathcal{Y}\ne \varnothing ,$$\vartheta \in {\mathbb{E}}_{\wp }$ with $\vartheta ⪰\theta$ and $ð:\mathcal{Y}\times \mathcal{Y}\to {\mathbb{E}}_{\wp }$ be an elliptic-valued function satisfying

$\left(E_{S_1}\right):$$\theta ⪯ð(\varsigma ,\sigma )$, and $ð(\varsigma ,\sigma )=\theta$⇔$\varsigma =\sigma$,

$\left(E_{S_2}\right):ð(\varsigma ,\sigma )=ð(\sigma ,\varsigma )$,

$\left(E_{S_3}\right):ð(\varsigma ,\sigma )⪯ð(\varsigma ,\varrho )+ð(\varrho ,\sigma )+\vartheta ð(\varsigma ,\varrho )ð(\varrho ,\sigma ),$

∀$\varsigma ,\sigma ,\varrho \in \mathcal{Y}$, then $(\mathcal{Y},ð)$ is said to be an E-VSMS.

Remark 1

([18]). If we set $\vartheta =\theta$ in the above definition, condition $\left(E_{S_3}\right)$ reduces to the usual triangle inequality. In this case, the structure $(\mathcal{Y},ð)$ becomes an E-VMS. Hence, every E-VMS can be regarded as a particular case of an E-VSMS. However, the converse does not hold in general, since an E-VSMS may involve a nontrivial parameter $\vartheta \succ \theta .$

Example 2.

Let $\mathcal{Y}=\mathbb{R}$ and

$${\mathbb{E}}_{\wp }=\left\{z=\nu +i\omega :\nu ,\omega \in \mathbb{R},i^2=\wp <0\right\}.$$

Define $ð:\mathcal{Y}\times \mathcal{Y}\to$${\mathbb{E}}_{\wp }$ by

$$ð(\varsigma ,\sigma )=(1+i)\left|\varsigma -\sigma \right|,$$

for $\varsigma ,\sigma \in \mathcal{Y}$. Then, $\left(\mathcal{Y},ð\right)$ is an E-VSMS.

Example 3.

Let ${\mathbb{E}}_{\wp }=\left\{z=\nu +i\omega :\nu ,\omega \in \mathbb{R},i^2=\wp <0\right\}$ be the set of elliptic numbers and consider the non-empty set $\mathcal{Y}=\{x,y,z\}$. Define the mapping $ð:\mathcal{Y}\times \mathcal{Y}\to$${\mathbb{E}}_{\wp }$ by

$$ð(x,x)=ð(y,y)=ð(z,z)=\theta ,$$

where θ denotes the zero element of ${\mathbb{E}}_{\wp },$ and

$$ð(x,y)=ð(y,x)=3,$$

$$ð(y,z)=ð(z,y)=3,$$

$$ð(x,z)=ð(z,x)=7.$$

The axioms $\left(E_{S_1}\right)$ and $\left(E_{S_2}\right)$ are immediately fulfilled. We verify the axiom

• $\left(E_{S_3}\right)$ Choose $\vartheta =1\in {\mathbb{E}}_{\wp }$ (note that $\vartheta ⪰\theta$). For $x,y,z\in {\mathbb{E}}_{\wp }$, we compute

  $$ð(x,y)+ð(y,z)+\vartheta ð(x,y)ð(y,z)=3+3+1·(3·3)=15.$$
  Since

  $$ð(x,z)=7⪯15.$$
  So condition $\left(E_{S_3}\right)$ holds. A similar computation is valid for any rearrangement of {$x,y,z$}. Thus $\left(\mathcal{Y},ð\right)$ is an E-VSMS. However, the usual E-VM triangle inequality fails, since

  $$ð(x,z)=7 /⪯6=3+3=ð(x,y)+ð(y,z).$$
  Therefore ($\mathcal{Y},ð$) is not an E-VMS.

Example 4.

Let $\mathcal{Y}=C\left(\right[0,\tau ],\mathbb{R})$ with $\tau >0,$ representing the space of continuous real-valued functions on $[0,\tau ].$ A function $ð:\mathcal{Y}\times \mathcal{Y}\to {\mathbb{E}}_{\wp }$ is defined as

$$ð(\varsigma ,\sigma )=\left({∥\varsigma -\sigma ∥}_{\infty }\right)e^{i\theta }$$

where

$${∥\varsigma -\sigma ∥}_{\infty }=\underset{t\in [0,\tau ]}{max}∥\varsigma \left(t\right)-\sigma \left(t\right)∥\mathit{and}\theta \in \left(0,{\displaystyle \frac{\pi }{2}}\right)\mathit{is}\mathit{fixed}.$$

Then, $(\mathcal{Y},ð)$ is an E-VSMS.

Definition 4

([18]). Let ${\left\{{\varsigma }_J\right\}}_{J\in \mathbb{N}}$ be a sequence in an E-VSMS $(\mathcal{Y},ð)$.

(i) A sequence ${\left\{{\varsigma }_J\right\}}_{J\in \mathbb{N}}$ is said to converge to $\varsigma \in \mathcal{Y}$ if, for each $\delta \in {\mathbb{E}}_{\wp }$ with $\theta \prec \delta$, there exists an integer $J_0\in \mathbb{N}$ such that

$$ð\left({\varsigma }_J,\varsigma \right)\prec \delta ,\forall J>J_0.$$

This convergence is denoted as

$${\varsigma }_J\to \varsigma \mathit{as}J\to \infty ,$$

or equivalently,

$$\underset{J\to \infty }{lim}{\varsigma }_J=\varsigma .$$

(ii) A sequence ${\left\{{\varsigma }_J\right\}}_{J\in \mathbb{N}}$ in ($\mathcal{Y},ð)$ is called a Cauchy sequence if, for each nonzero $\delta \in {\mathbb{E}}_{\wp }$ with $\theta \prec \delta$, there exists $J_0\in \mathbb{N}$ such that $ð\left({\varsigma }_J,{\varsigma }_{J+m}\right)\prec \delta$, for all $J>J_0$ and $m\in \mathbb{N}$.

(iii) The E-VSMS ($\mathcal{Y},ð$) is said to be complete if every Cauchy sequence in $\mathcal{Y}$ converges to an member of $\mathcal{Y}.$

The following lemma will be used in subsequent results:

Lemma 1

([18]). Let $(\mathcal{Y},ð)$ be an E-VSMS and let ${\left\{{\varsigma }_J\right\}}_{J\in \mathbb{N}}$ be a sequence in $\mathcal{Y}$. The sequence ${\left\{{\varsigma }_J\right\}}_{J\in \mathbb{N}}$ converges to $\varsigma \in \mathcal{Y},$$J\to \infty$ (or equivalently $ð\left({\varsigma }_J,\varsigma \right)\to \theta$ as $J\to \infty$) iff

$${∥ð\left({\varsigma }_J,\varsigma \right)∥}_{\mathbb{E}}\to 0,$$

as $J\to \infty$.

Lemma 2

([18]). A sequence ${\left\{{\varsigma }_J\right\}}_{J\in \mathbb{N}}$ is a Cauchy sequence in $(\mathcal{Y},ð)$ if and only if the elliptic norm of $ð\left({\varsigma }_J,{\varsigma }_{J+m}\right)$ approaches zero as $J\to \infty$. Formally,

$${∥ð\left({\varsigma }_J,{\varsigma }_{J+m}\right)∥}_{\mathbb{E}}\to 0,$$

as $J\to \infty$, with $m\in \mathbb{N}$.

Lemma 3

([18]). Every convergent sequence in an E-VSMS has a unique limit; in other words, a sequence cannot converge to more than one element of $\mathcal{Y}$.

3. Main Results

In this section, we present the main FP theorems that underpin our investigation. For the purposes of this section, $(\mathcal{Y},ð)$ is assumed to be a complete E-VSMS. By exploiting the distinctive properties of E-VSMSs, we establish new results for FPs subject to an extended contraction criterion.

Theorem 2.

Let ${\mathcal{W}}_1,{\mathcal{W}}_2:\mathcal{Y}$$\to \mathcal{Y}$. Assume that there exist the functions

$${\varpi }_1,{\varpi }_2,{\varpi }_3:\mathcal{Y}\to [0,1)$$

satisfying the following conditions:

(i) 

${\varpi }_1\left({\mathcal{W}}_1\varsigma \right)\le {\varpi }_1\left(\varsigma \right)$ and ${\varpi }_1\left({\mathcal{W}}_2\varsigma \right)\le {\varpi }_1\left(\varsigma \right)$

${\varpi }_2\left({\mathcal{W}}_1\varsigma \right)\le {\varpi }_2\left(\varsigma \right)$ and ${\varpi }_2\left({\mathcal{W}}_2\varsigma \right)\le {\varpi }_2\left(\varsigma \right)$

${\varpi }_3\left({\mathcal{W}}_1\varsigma \right)\le {\varpi }_3\left(\varsigma \right)$ and ${\varpi }_3\left({\mathcal{W}}_2\varsigma \right)\le {\varpi }_3\left(\varsigma \right),$

(ii) 

${\varpi }_1\left(\varsigma \right)+{\varpi }_2\left(\varsigma \right)+{\varpi }_3\left(\varsigma \right)<1,$

(iii) 

$$ð\left({\mathcal{W}}_1\varsigma ,{\mathcal{W}}_2\sigma \right)⪯{\varpi }_1\left(\varsigma \right)ð\left(\varsigma ,\sigma \right)+{\varpi }_2\left(\varsigma \right){\displaystyle \frac{ð\left(\varsigma ,{\mathcal{W}}_1\varsigma \right)ð\left(\sigma ,{\mathcal{W}}_2\sigma \right)}{1+ð\left(\varsigma ,\sigma \right)}}+{\varpi }_3\left(\varsigma \right){\displaystyle \frac{ð\left(\sigma ,{\mathcal{W}}_1\varsigma \right)ð\left(\varsigma ,{\mathcal{W}}_2\sigma \right)}{1+ð\left(\varsigma ,\sigma \right)}},$$

(1)

for all $\varsigma ,\sigma \in \mathcal{Y}$ with $\varsigma \ne \sigma .$ Then ${\mathcal{W}}_1$ and ${\mathcal{W}}_2$ have a unique CFP.

Proof. 

Now, let ${\varsigma }_0\in \mathcal{Y}$ be an arbitrary point. Define

$${\varsigma }_{2J+1}={\mathcal{W}}_2{\varsigma }_{2J}\mathrm{and}{\varsigma }_{2J+2}={\mathcal{W}}_1{\varsigma }_{2J+1},$$

for all $J\ge 0.$ Assuming the existence of an integer J such that ${\varsigma }_{2J}={\varsigma }_{2J+1}.$ Then

$${\varsigma }_{2J}={\mathcal{W}}_2{\varsigma }_{2J},$$

and ${\varsigma }_{2J}$ is an FP of ${\mathcal{W}}_2$, hence a FP of ${\mathcal{W}}_1$. Furthermore, if ${\varsigma }_{2J+1}={\varsigma }_{2J+2}$ for a particular value of $J,$ it follows that ${\varsigma }_{2J+1}$ represents a CFP of the mappings ${\mathcal{W}}_2$ and ${\mathcal{W}}_1$. By (1), we have

$$\begin{array}{ccc}\hfill ð({\varsigma }_{2J},{\varsigma }_{2J+1})& =& ð({\mathcal{W}}_1{\varsigma }_{2J-1},{\mathcal{W}}_2{\varsigma }_{2J})\hfill \\ & ⪯& {\varpi }_1\left({\varsigma }_{2J-1}\right)ð({\varsigma }_{2J-1},{\varsigma }_{2J})\hfill \\ & & +{\varpi }_2\left({\varsigma }_{2J-1}\right){\displaystyle \frac{ð\left({\varsigma }_{2J-1},{\mathcal{W}}_1{\varsigma }_{2J-1}\right)ð\left({\varsigma }_{2J},{\mathcal{W}}_2{\varsigma }_{2J}\right)}{1+ð({\varsigma }_{2J-1},{\varsigma }_{2J})}}\hfill \\ & & +{\varpi }_3\left({\varsigma }_{2J-1}\right){\displaystyle \frac{ð\left({\varsigma }_{2J},{\mathcal{W}}_1{\varsigma }_{2J-1}\right)ð\left({\varsigma }_{2J-1},{\mathcal{W}}_2{\varsigma }_{2J}\right)}{1+ð({\varsigma }_{2J-1},{\varsigma }_{2J})}}.\hfill \end{array}$$

Since ${\mathcal{W}}_1{\varsigma }_{2J-1}={\varsigma }_{2J}$ and ${\mathcal{W}}_2{\varsigma }_{2J}={\varsigma }_{2J+1},$ we have

$$\begin{array}{ccc}\hfill ð({\varsigma }_{2J},{\varsigma }_{2J+1})& ⪯& {\varpi }_1\left({\varsigma }_{2J-1}\right)ð({\varsigma }_{2J-1},{\varsigma }_{2J})\hfill \\ & & +{\varpi }_2\left({\varsigma }_{2J-1}\right){\displaystyle \frac{ð\left({\varsigma }_{2J-1},{\varsigma }_{2J}\right)ð\left({\varsigma }_{2J},{\varsigma }_{2J+1}\right)}{1+ð({\varsigma }_{2J-1},{\varsigma }_{2J})}}\hfill \\ & & +{\varpi }_3\left({\varsigma }_{2J-1}\right){\displaystyle \frac{ð\left({\varsigma }_{2J},{\varsigma }_{2J}\right)ð\left({\varsigma }_{2J-1},{\varsigma }_{2J+1}\right)}{1+ð({\varsigma }_{2J-1},{\varsigma }_{2J})}}.\hfill \end{array}$$

Using ${\varpi }_k\left({\varsigma }_{2J-1}\right)={\varpi }_k\left({\mathcal{W}}_2{\varsigma }_{2J-2}\right)$ for $k=1,2,3$, and subsequently applying assumption (i), we get

$$\begin{array}{ccc}\hfill ð({\varsigma }_{2J},{\varsigma }_{2J+1})& ⪯& {\varpi }_1\left({\mathcal{W}}_2{\varsigma }_{2J-2}\right)ð({\varsigma }_{2J-1},{\varsigma }_{2J})\hfill \\ & & +{\varpi }_2\left({\mathcal{W}}_2{\varsigma }_{2J-2}\right){\displaystyle \frac{ð\left({\varsigma }_{2J-1},{\varsigma }_{2J}\right)ð\left({\varsigma }_{2J},{\varsigma }_{2J+1}\right)}{1+ð({\varsigma }_{2J-1},{\varsigma }_{2J})}}\hfill \\ & ⪯& {\varpi }_1\left({\varsigma }_{2J-2}\right)ð({\varsigma }_{2J-1},{\varsigma }_{2J})+{\varpi }_2\left({\varsigma }_{2J-2}\right){\displaystyle \frac{ð\left({\varsigma }_{2J-1},{\varsigma }_{2J}\right)ð\left({\varsigma }_{2J},{\varsigma }_{2J+1}\right)}{1+ð({\varsigma }_{2J-1},{\varsigma }_{2J})}}\hfill \\ & ⪯& ···⪯{\varpi }_1\left({\varsigma }_0\right)ð({\varsigma }_{2J-1},{\varsigma }_{2J})\hfill \\ & & +{\varpi }_2\left({\varsigma }_0\right){\displaystyle \frac{ð\left({\varsigma }_{2J-1},{\varsigma }_{2J}\right)ð\left({\varsigma }_{2J},{\varsigma }_{2J+1}\right)}{1+ð({\varsigma }_{2J-1},{\varsigma }_{2J})}},\hfill \end{array}$$

which implies that

$$\begin{array}{ccc}\hfill {∥ð({\varsigma }_{2J},{\varsigma }_{2J+1})∥}_{\mathbb{E}}& \le & {\varpi }_1\left({\varsigma }_0\right){∥ð({\varsigma }_{2J-1},{\varsigma }_{2J})∥}_{\mathbb{E}}\hfill \\ & & +{\varpi }_2\left({\varsigma }_0\right){\displaystyle \frac{{∥ð\left({\varsigma }_{2J-1},{\varsigma }_{2J}\right)∥}_{\mathbb{E}}}{{∥1+ð({\varsigma }_{2J-1},{\varsigma }_{2J})∥}_{\mathbb{E}}}}{∥ð\left({\varsigma }_{2J},{\varsigma }_{2J+1}\right)∥}_{\mathbb{E}}\hfill \\ & \le & {\varpi }_1\left({\varsigma }_0\right){∥ð({\varsigma }_{2J-1},{\varsigma }_{2J})∥}_{\mathbb{E}}\hfill \\ & & +{\varpi }_2\left({\varsigma }_0\right){∥ð\left({\varsigma }_{2J},{\varsigma }_{2J+1}\right)∥}_{\mathbb{E}}\hfill \end{array}$$

since ${\displaystyle \frac{{∥ð\left({\varsigma }_{2J-1},{\varsigma }_{2J}\right)∥}_{\mathbb{E}}}{{∥1+ð({\varsigma }_{2J-1},{\varsigma }_{2J})∥}_{\mathbb{E}}}}<1.$ Thus, from the above inequality, we have

$${∥ð({\varsigma }_{2J},{\varsigma }_{2J+1})∥}_{\mathbb{E}}\le {\displaystyle \frac{{\varpi }_1\left({\varsigma }_0\right)}{1-{\varpi }_2\left({\varsigma }_0\right)}}{∥ð\left({\varsigma }_{2J},{\varsigma }_{2J+1}\right)∥}_{\mathbb{E}}.$$

(2)

Similarly, by (1), we have

$$\begin{array}{ccc}\hfill ð({\varsigma }_{2J+1},{\varsigma }_{2J+2})& =& ð({\mathcal{W}}_2{\varsigma }_{2J},{\mathcal{W}}_1{\varsigma }_{2J+1})=ð({\mathcal{W}}_1{\varsigma }_{2J+1},{\mathcal{W}}_2{\varsigma }_{2J})\hfill \\ & ⪯& {\varpi }_1\left({\varsigma }_{2J+1}\right)ð({\varsigma }_{2J+1},{\varsigma }_{2J})\hfill \\ & & +{\varpi }_2\left({\varsigma }_{2J+1}\right){\displaystyle \frac{ð\left({\varsigma }_{2J+1},{\mathcal{W}}_1{\varsigma }_{2J+1}\right)ð\left({\varsigma }_{2J},{\mathcal{W}}_2{\varsigma }_{2J}\right)}{1+ð({\varsigma }_{2J+1},{\varsigma }_{2J})}}\hfill \\ & & +{\varpi }_3\left({\varsigma }_{2J+1}\right){\displaystyle \frac{ð\left({\varsigma }_{2J},{\mathcal{W}}_1{\varsigma }_{2J+1}\right)ð\left({\varsigma }_{2J+1},{\mathcal{W}}_2{\varsigma }_{2J}\right)}{1+ð({\varsigma }_{2J+1},{\varsigma }_{2J})}}.\hfill \end{array}$$

Since ${\mathcal{W}}_1{\varsigma }_{2J+1}={\varsigma }_{2J+2}$ and ${\mathcal{W}}_2{\varsigma }_{2J}={\varsigma }_{2J+1},$ we obtain

$$\begin{array}{ccc}\hfill ð({\varsigma }_{2J+1},{\varsigma }_{2J+2})& ⪯& {\varpi }_1\left({\varsigma }_{2J+1}\right)ð({\varsigma }_{2J+1},{\varsigma }_{2J})\hfill \\ & & +{\varpi }_2\left({\varsigma }_{2J+1}\right){\displaystyle \frac{ð\left({\varsigma }_{2J+1},{\varsigma }_{2J+2}\right)ð\left({\varsigma }_{2J},{\varsigma }_{2J+1}\right)}{1+ð({\varsigma }_{2J+1},{\varsigma }_{2J})}}\hfill \\ & & +{\varpi }_3\left({\varsigma }_{2J+1}\right){\displaystyle \frac{ð\left({\varsigma }_{2J},{\varsigma }_{2J+2}\right)ð\left({\varsigma }_{2J+1},{\varsigma }_{2J+1}\right)}{1+ð({\varsigma }_{2J+1},{\varsigma }_{2J})}}.\hfill \end{array}$$

By first substituting ${\varpi }_k\left({\varsigma }_{2J+1}\right)={\varpi }_k\left({\mathcal{W}}_2{\varsigma }_{2J}\right)$ for $k=1,2,3$, and applying assumption (i) thereafter, we obtain

$$\begin{array}{ccc}\hfill ð({\varsigma }_{2J+1},{\varsigma }_{2J+2})& ⪯& {\varpi }_1\left({\mathcal{W}}_2{\varsigma }_{2J}\right)ð({\varsigma }_{2J+1},{\varsigma }_{2J})\hfill \\ & & +{\varpi }_2\left({\mathcal{W}}_2{\varsigma }_{2J}\right){\displaystyle \frac{ð\left({\varsigma }_{2J+1},{\varsigma }_{2J+2}\right)ð\left({\varsigma }_{2J},{\varsigma }_{2J+1}\right)}{1+ð({\varsigma }_{2J+1},{\varsigma }_{2J})}}\hfill \\ & ⪯& {\varpi }_1\left({\varsigma }_{2J}\right)ð({\varsigma }_{2J+1},{\varsigma }_{2J})\hfill \\ & & +{\varpi }_2\left({\varsigma }_{2J}\right){\displaystyle \frac{ð\left({\varsigma }_{2J+1},{\varsigma }_{2J+2}\right)ð\left({\varsigma }_{2J},{\varsigma }_{2J+1}\right)}{1+ð({\varsigma }_{2J+1},{\varsigma }_{2J})}}\hfill \\ & ⪯& ···⪯{\varpi }_1\left({\varsigma }_0\right)ð({\varsigma }_{2J+1},{\varsigma }_{2J})\hfill \\ & & +{\varpi }_2\left({\varsigma }_0\right){\displaystyle \frac{ð\left({\varsigma }_{2J+1},{\varsigma }_{2J+2}\right)ð\left({\varsigma }_{2J},{\varsigma }_{2J+1}\right)}{1+ð({\varsigma }_{2J+1},{\varsigma }_{2J})}},\hfill \end{array}$$

which implies that

$$\begin{array}{ccc}\hfill {∥ð({\varsigma }_{2J+1},{\varsigma }_{2J+2})∥}_{\mathbb{E}}& \le & {\varpi }_1\left({\varsigma }_0\right){∥ð({\varsigma }_{2J+1},{\varsigma }_{2J})∥}_{\mathbb{E}}\hfill \\ & & +{\varpi }_2\left({\varsigma }_0\right){∥ð({\varsigma }_{2J+1},{\varsigma }_{2J+2})∥}_{\mathbb{E}}{\displaystyle \frac{{∥ð\left({\varsigma }_{2J},{\varsigma }_{2J+1}\right)∥}_{\mathbb{E}}}{{∥1+ð({\varsigma }_{2J+1},{\varsigma }_{2J})∥}_{\mathbb{E}}}}\hfill \\ & \le & {\varpi }_1\left({\varsigma }_0\right){∥ð({\varsigma }_{2J},{\varsigma }_{2J+1})∥}_{\mathbb{E}}+{\varpi }_2\left({\varsigma }_0\right){∥ð({\varsigma }_{2J+1},{\varsigma }_{2J+2})∥}_{\mathbb{E}}\hfill \end{array}$$

since ${\displaystyle \frac{{∥ð\left({\varsigma }_{2J},{\varsigma }_{2J+1}\right)∥}_{\mathbb{E}}}{{∥1+ð({\varsigma }_{2J+1},{\varsigma }_{2J})∥}_{\mathbb{E}}}}<1.$ Thus, from the above inequality, we have

$${∥ð({\varsigma }_{2J+1},{\varsigma }_{2J+2})∥}_{\mathbb{E}}\le {\displaystyle \frac{{\varpi }_1\left({\varsigma }_0\right)}{1-{\varpi }_2\left({\varsigma }_0\right)}}{∥ð({\varsigma }_{2J},{\varsigma }_{2J+1})∥}_{\mathbb{E}}.$$

(3)

Let $\lambda =$${\displaystyle \frac{{\varpi }_1\left({\varsigma }_0\right)}{1-{\varpi }_2\left({\varsigma }_0\right)}}<1.$ Then, from (2) and (3), we have

$${∥ð\left({\varsigma }_J,{\varsigma }_{J+1}\right)∥}_{\mathbb{E}}\le \lambda {∥ð\left({\varsigma }_{J-1},{\varsigma }_J\right)∥}_{\mathbb{E}},$$

for all $J\in \mathbb{N}.$ Inductively, we can construct a sequence $\left\{{\varsigma }_J\right\}$ in $\mathcal{Y}$ such that

$${∥ð\left({\varsigma }_J,{\varsigma }_{J+1}\right)∥}_{\mathbb{E}}\le \lambda {∥ð\left({\varsigma }_{J-1},{\varsigma }_J\right)∥}_{\mathbb{E}}\le ···\le {\lambda }^J{∥ð\left({\varsigma }_0,{\varsigma }_1\right)∥}_{\mathbb{E}},$$

(4)

∀$J\in \mathbb{N}.$ Now, for any natural number m greater than n, we have

$$\begin{array}{ccc}\hfill {∥ð\left({\varsigma }_n,{\varsigma }_m\right)∥}_{\mathbb{E}}& \le & {∥ð\left({\varsigma }_n,{\varsigma }_{n+1}\right)∥}_{\mathbb{E}}+{∥ð\left({\varsigma }_{n+1},{\varsigma }_m\right)∥}_{\mathbb{E}}\hfill \\ & & +\vartheta {∥ð\left({\varsigma }_n,{\varsigma }_{n+1}\right)∥}_{\mathbb{E}}{∥ð\left({\varsigma }_{n+1},{\varsigma }_m\right)∥}_{\mathbb{E}}\hfill \\ & =& {∥ð\left({\varsigma }_n,{\varsigma }_{n+1}\right)∥}_{\mathbb{E}}\hfill \\ & & +\left(1+\vartheta {∥ð\left({\varsigma }_n,{\varsigma }_{n+1}\right)∥}_{\mathbb{E}}\right){∥ð\left({\varsigma }_{n+1},{\varsigma }_m\right)∥}_{\mathbb{E}}.\hfill \end{array}$$

(5)

Considering the second term ${∥ð\left({\varsigma }_{n+1},{\varsigma }_m\right)∥}_{\mathbb{E}}$ and applying the triangle inequality, we have

$$\begin{array}{ccc}\hfill {∥ð\left({\varsigma }_{n+1},{\varsigma }_m\right)∥}_{\mathbb{E}}& \le & {∥ð\left({\varsigma }_{n+1},{\varsigma }_{n+2}\right)∥}_{\mathbb{E}}+{∥ð\left({\varsigma }_{n+2},{\varsigma }_m\right)∥}_{\mathbb{E}}\hfill \\ & & +\vartheta {∥ð\left({\varsigma }_{n+1},{\varsigma }_{n+2}\right)∥}_{\mathbb{E}}{∥ð\left({\varsigma }_{n+2},{\varsigma }_m\right)∥}_{\mathbb{E}}\hfill \\ & =& {∥ð\left({\varsigma }_{n+1},{\varsigma }_{n+2}\right)∥}_{\mathbb{E}}\hfill \\ & & +\left(1+\vartheta {∥ð\left({\varsigma }_{n+1},{\varsigma }_{n+2}\right)∥}_{\mathbb{E}}\right){∥ð\left({\varsigma }_{n+2},{\varsigma }_m\right)∥}_{\mathbb{E}}.\hfill \end{array}$$

We again invoke the triangle inequality for the term ${∥ð\left({\varsigma }_{n+2},{\varsigma }_m\right)∥}_{\mathbb{E}},$ resulting in

$$\begin{array}{ccc}\hfill {∥ð\left({\varsigma }_{n+2},{\varsigma }_m\right)∥}_{\mathbb{E}}& \le & {∥ð\left({\varsigma }_{n+2},{\varsigma }_{n+3}\right)∥}_{\mathbb{E}}+{∥ð\left({\varsigma }_{n+3},{\varsigma }_m\right)∥}_{\mathbb{E}}\hfill \\ & & +\vartheta {∥ð\left({\varsigma }_{n+2},{\varsigma }_{n+3}\right)∥}_{\mathbb{E}}{∥ð\left({\varsigma }_{n+3},{\varsigma }_m\right)∥}_{\mathbb{E}}\hfill \\ & =& {∥ð\left({\varsigma }_{n+2},{\varsigma }_{n+3}\right)∥}_{\mathbb{E}}\hfill \\ & & +\left(1+\vartheta {∥ð\left({\varsigma }_{n+2},{\varsigma }_{n+3}\right)∥}_{\mathbb{E}}\right){∥ð\left({\varsigma }_{n+3},{\varsigma }_m\right)∥}_{\mathbb{E}},\hfill \end{array}$$

and so on

$$\begin{array}{ccc}\hfill {∥ð\left({\varsigma }_{m-2},{\varsigma }_m\right)∥}_{\mathbb{E}}& \le & {∥ð\left({\varsigma }_{m-2},{\varsigma }_{m-1}\right)∥}_{\mathbb{E}}+{∥ð\left({\varsigma }_{m-1},{\varsigma }_m\right)∥}_{\mathbb{E}}\hfill \\ & & +\vartheta {∥ð\left({\varsigma }_{m-2},{\varsigma }_{m-1}\right)∥}_{\mathbb{E}}{∥ð\left({\varsigma }_{m-1},{\varsigma }_m\right)∥}_{\mathbb{E}}\hfill \\ & =& {∥ð\left({\varsigma }_{m-2},{\varsigma }_{m-1}\right)∥}_{\mathbb{E}}+\left(1+\vartheta {∥ð\left({\varsigma }_{m-2},{\varsigma }_{m-1}\right)∥}_{\mathbb{E}}\right){∥ð\left({\varsigma }_{m-1},{\varsigma }_m\right)∥}_{\mathbb{E}}.\hfill \end{array}$$

Through successive substitution of each inequality into the prior one (5) and simplification, it follows that

$${∥ð\left({\varsigma }_n,{\varsigma }_m\right)∥}_{\mathbb{E}}\le \sum _{k=n}^{m-1}∥ð\left({\varsigma }_n,{\varsigma }_{n+1}\right)∥\prod _{i=n}^{k-1}\left(1+\vartheta ∥ð{\varsigma }_i,{\varsigma }_{i+1}∥\right).$$

In view of inequality (4), we deduce that

$${∥ð\left({\varsigma }_n,{\varsigma }_m\right)∥}_{\mathbb{E}}\le {∥ð\left({\varsigma }_0,{\varsigma }_1\right)∥}_{\mathbb{E}}\sum _{k=n}^{m-1}{\varpi }^n\prod _{i=n}^{k-1}\left(1+\vartheta {\lambda }^i{∥ð\left({\varsigma }_0,{\varsigma }_1\right)∥}_{\mathbb{E}}\right).$$

(6)

Now, observe that $\left(1+\vartheta {\lambda }^i{∥ð\left({\varsigma }_0,{\varsigma }_1\right)∥}_{\mathbb{E}}\right)\ge 1,$ for all $i.$ Therefore,

$$\prod _{i=n}^{k-1}\left(1+\vartheta {\lambda }^i{∥ð\left({\varsigma }_0,{\varsigma }_1\right)∥}_{\mathbb{E}}\right)\ge 1.$$

Now, with the product term bounded below by 1, we can simplify the summation

$${∥ð\left({\varsigma }_0,{\varsigma }_1\right)∥}_{\mathbb{E}}\sum _{k=n}^{m-1}{\lambda }^k\prod _{i=n}^{k-1}\left(1+\vartheta {\lambda }^i{∥ð\left({\varsigma }_0,{\varsigma }_1\right)∥}_{\mathbb{E}}\right)\ge {∥ð\left({\varsigma }_0,{\varsigma }_1\right)∥}_{\mathbb{E}}\sum _{k=n}^{m-1}{\lambda }^k.$$

Remark that ${\displaystyle \sum _{k=n}^{m-1}}{\lambda }^k$ is a finite geometric series with first term ${\lambda }^n$ and the common ratio $\lambda .$ The sum of a finite geometric series is given by

$$\sum _{k=n}^{m-1}{\lambda }^k={\lambda }^n{\displaystyle \frac{\left(1-{\lambda }^{m-n}\right)}{1-\lambda }}.$$

As $m\to \infty ,$ ${\lambda }^{m-n}\to 0.$ Hence, the series converges to

$${\displaystyle \frac{{\lambda }^n}{1-\lambda }},\mathrm{that}\mathrm{is},\underset{m\to \infty }{lim}{\lambda }^n{\displaystyle \frac{\left(1-{\lambda }^{m-n}\right)}{1-\lambda }}={\displaystyle \frac{{\lambda }^n}{1-\lambda }}.$$

Since $\lambda <1,$ ${\displaystyle \frac{{\lambda }^n}{1-\lambda }}$ vanishes in the limit $n\to \infty .$ Applying this limit to the inequality (6) and utilizing the established facts, it follows that

$$\underset{n\to \infty }{lim}{∥ð\left({\varsigma }_n,{\varsigma }_m\right)∥}_{\mathbb{E}}=0.$$

This shows that the sequence $\left\{{\varsigma }_n\right\}$ is Cauchy. Since $\mathcal{Y}$ is complete, there exists an element ${\varsigma }^{*}\in \mathcal{Y}$ such that ${\varsigma }_n\to {\varsigma }_n^{*}$ as $n\to \infty .$ Thus,

$$\underset{n\to \infty }{lim}{\varsigma }_n={\varsigma }^{*}.$$

From the inequality (1), we have

$$\begin{array}{ccc}\hfill ð\left({\varsigma }^{*},{\mathcal{W}}_1{\varsigma }^{*}\right)& ⪯& ð\left({\varsigma }^{*},{\varsigma }_{2J+1}\right)+ð\left({\varsigma }_{2J+1},{\mathcal{W}}_1{\varsigma }^{*}\right)+\vartheta ð\left({\varsigma }^{*},{\varsigma }_{2J+1}\right)ð\left({\varsigma }_{2J+1},{\mathcal{W}}_1{\varsigma }^{*}\right)\hfill \\ & =& ð\left({\varsigma }^{*},{\varsigma }_{2J+1}\right)+ð\left({\mathcal{W}}_2{\varsigma }_{2J},{\mathcal{W}}_1{\varsigma }^{*}\right)+\vartheta ð\left({\varsigma }^{*},{\varsigma }_{2J+1}\right)ð\left({\mathcal{W}}_2{\varsigma }_{2J},{\mathcal{W}}_1{\varsigma }^{*}\right)\hfill \\ & =& ð\left({\varsigma }^{*},{\varsigma }_{2J+1}\right)+ð\left({\mathcal{W}}_1{\varsigma }^{*},{\mathcal{W}}_2{\varsigma }_{2J}\right)+\vartheta ð\left({\varsigma }^{*},{\varsigma }_{2J+1}\right)ð\left({\mathcal{W}}_1{\varsigma }^{*},{\mathcal{W}}_2{\varsigma }_{2J}\right)\hfill \\ & ⪯& ð\left({\varsigma }^{*},{\varsigma }_{2J+1}\right)+\left(\begin{array}{c}{\varpi }_1\left({\varsigma }^{*}\right)ð\left({\varsigma }^{*},{\varsigma }_{2J}\right)\\ +{\varpi }_2\left({\varsigma }^{*}\right){\displaystyle \frac{ð\left({\varsigma }^{*},{\mathcal{W}}_1{\varsigma }^{*}\right)ð\left({\varsigma }_{2J},{\mathcal{W}}_2{\varsigma }_{2J}\right)}{1+ð\left({\varsigma }^{*},{\varsigma }_{2J}\right)}}\\ +{\varpi }_3\left({\varsigma }^{*}\right){\displaystyle \frac{ð\left({\varsigma }_{2J},{\mathcal{W}}_1{\varsigma }^{*}\right)ð\left({\varsigma }^{*},{\mathcal{W}}_2{\varsigma }_{2J}\right)}{1+ð\left({\varsigma }^{*},{\varsigma }_{2J}\right)}}\end{array}\right)\hfill \\ & & +\vartheta ð\left({\varsigma }^{*},{\varsigma }_{2J+1}\right)\left(\begin{array}{c}{\varpi }_1\left({\varsigma }^{*}\right)ð\left({\varsigma }^{*},{\varsigma }_{2J}\right)\\ +{\varpi }_2\left({\varsigma }^{*}\right){\displaystyle \frac{ð\left({\varsigma }^{*},{\mathcal{W}}_1{\varsigma }^{*}\right)ð\left({\varsigma }_{2J},{\mathcal{W}}_2{\varsigma }_{2J}\right)}{1+ð\left({\varsigma }^{*},{\varsigma }_{2J}\right)}}\\ +{\varpi }_3\left({\varsigma }^{*}\right){\displaystyle \frac{ð\left({\varsigma }_{2J},{\mathcal{W}}_1{\varsigma }^{*}\right)ð\left({\varsigma }^{*},{\mathcal{W}}_2{\varsigma }_{2J}\right)}{1+ð\left({\varsigma }^{*},{\varsigma }_{2J}\right)}}\end{array}\right)\hfill \\ & =& ð\left({\varsigma }^{*},{\varsigma }_{2J+1}\right)+\left(\begin{array}{c}{\varpi }_1\left({\varsigma }^{*}\right)ð\left({\varsigma }^{*},{\varsigma }_{2J}\right)+{\varpi }_2\left({\varsigma }^{*}\right){\displaystyle \frac{ð\left({\varsigma }^{*},{\mathcal{W}}_1{\varsigma }^{*}\right)·ð\left({\varsigma }_{2J},{\varsigma }_{2J}\right)}{1+ð\left({\varsigma }^{*},{\varsigma }_{2J}\right)}}\\ +{\varpi }_3\left({\varsigma }^{*}\right){\displaystyle \frac{ð\left({\varsigma }_{2J},{\mathcal{W}}_1{\varsigma }^{*}\right)ð\left({\varsigma }^{*},{\varsigma }_{2J+1}\right)}{1+ð\left({\varsigma }^{*},{\varsigma }_{2J}\right)}}\end{array}\right)\hfill \\ & & +\vartheta ð\left({\varsigma }^{*},{\varsigma }_{2J+1}\right)\left(\begin{array}{c}{\varpi }_1\left({\varsigma }^{*}\right)ð\left({\varsigma }^{*},{\varsigma }_{2J}\right)+{\varpi }_2\left({\varsigma }^{*}\right){\displaystyle \frac{ð\left({\varsigma }^{*},{\mathcal{W}}_1{\varsigma }^{*}\right)·ð\left({\varsigma }_{2J},{\varsigma }_{2J}\right)}{1+ð\left({\varsigma }^{*},{\varsigma }_{2J}\right)}}\\ +{\varpi }_3\left({\varsigma }^{*}\right){\displaystyle \frac{ð\left({\varsigma }_{2J},{\mathcal{W}}_1{\varsigma }^{*}\right)ð\left({\varsigma }^{*},{\varsigma }_{2J+1}\right)}{1+ð\left({\varsigma }^{*},{\varsigma }_{2J}\right)}}\end{array}\right),\hfill \end{array}$$

which implies that

$$\begin{array}{ccc}\hfill {∥ð\left({\varsigma }^{*},{\mathcal{W}}_1{\varsigma }^{*}\right)∥}_{\mathbb{E}}& \le & {∥ð\left({\varsigma }^{*},{\varsigma }_{2J+1}\right)∥}_{\mathbb{E}}+{\varpi }_1\left({\varsigma }^{*}\right){∥ð\left({\varsigma }^{*},{\varsigma }_{2J}\right)∥}_{\mathbb{E}}\hfill \\ & & +{\varpi }_2\left({\varsigma }^{*}\right){\displaystyle \frac{{∥ð\left({\varsigma }^{*},{\mathcal{W}}_1{\varsigma }^{*}\right)∥}_{\mathbb{E}}·{∥ð\left({\varsigma }_{2J},{\varsigma }_{2J+1}\right)∥}_{\mathbb{E}}}{{∥1+ð\left({\varsigma }^{*},{\varsigma }_{2J}\right)∥}_{\mathbb{E}}}}\hfill \\ & & +{\varpi }_3\left({\varsigma }^{*}\right){\displaystyle \frac{{∥ð\left({\varsigma }_{2J},{\mathcal{W}}_1{\varsigma }^{*}\right)∥}_{\mathbb{E}}{∥ð\left({\varsigma }^{*},{\varsigma }_{2J+1}\right)∥}_{\mathbb{E}}}{{∥1+ð\left({\varsigma }^{*},{\varsigma }_{2J}\right)∥}_{\mathbb{E}}}}\hfill \\ & & +\vartheta {∥ð\left({\varsigma }^{*},{\varsigma }_{2J+1}\right)∥}_{\mathbb{E}}\left(\begin{array}{c}{\varpi }_1\left({\varsigma }^{*}\right){∥ð\left({\varsigma }^{*},{\varsigma }_{2J}\right)∥}_{\mathbb{E}}\\ +{\varpi }_2\left({\varsigma }^{*}\right){\displaystyle \frac{{∥ð\left({\varsigma }^{*},{\mathcal{W}}_1{\varsigma }^{*}\right)∥}_{\mathbb{E}}·{∥ð\left({\varsigma }_{2J},{\varsigma }_{2J+1}\right)∥}_{\mathbb{E}}}{{∥1+ð\left({\varsigma }^{*},{\varsigma }_{2J}\right)∥}_{\mathbb{E}}}}\\ +{\varpi }_3\left({\varsigma }^{*}\right){\displaystyle \frac{{∥ð\left({\varsigma }_{2J},{\mathcal{W}}_1{\varsigma }^{*}\right)∥}_{\mathbb{E}}{∥ð\left({\varsigma }^{*},{\varsigma }_{2J+1}\right)∥}_{\mathbb{E}}}{{∥1+ð\left({\varsigma }^{*},{\varsigma }_{2J}\right)∥}_{\mathbb{E}}}}\end{array}\right).\hfill \end{array}$$

By passing to the limit $J\to \infty$ in the preceding inequality and considering that ${\varsigma }_J\to {\varsigma }^{*}\in \mathcal{Y}$ as $J\to \infty ,$ we arrive at

$${∥ð\left({\varsigma }^{*},{\mathcal{W}}_1{\varsigma }^{*}\right)∥}_{\mathbb{E}}\le 0.$$

It follows that ${∥ð\left({\varsigma }^{*},{\mathcal{W}}_1{\varsigma }^{*}\right)∥}_{\mathbb{E}}=0,$ leading to ${\varsigma }^{*}={\mathcal{W}}_1{\varsigma }^{*},$ and thus, ${\varsigma }^{*}$ is a FP of ${\mathcal{W}}_1$. From (1), we have

$$\begin{array}{ccc}\hfill ð\left({\varsigma }^{*},{\mathcal{W}}_2{\varsigma }^{*}\right)& ⪯& ð\left({\varsigma }^{*},{\varsigma }_{2J+2}\right)+ð\left({\varsigma }_{2J+2},{\mathcal{W}}_2{\varsigma }^{*}\right)+\vartheta ð\left({\varsigma }^{*},{\varsigma }_{2J+2}\right)ð\left({\varsigma }_{2J+2},{\mathcal{W}}_2{\varsigma }^{*}\right)\hfill \\ & =& ð\left({\varsigma }^{*},{\varsigma }_{2J+2}\right)+ð\left({\mathcal{W}}_1{\varsigma }_{2J+1},{\mathcal{W}}_2{\varsigma }^{*}\right)+\vartheta ð\left({\varsigma }^{*},{\varsigma }_{2J+2}\right)ð\left({\mathcal{W}}_1{\varsigma }_{2J+1},{\mathcal{W}}_2{\varsigma }^{*}\right)\hfill \\ & ⪯& ð\left({\varsigma }^{*},{\varsigma }_{2J+2}\right)+\left(\begin{array}{c}{\varpi }_1\left({\varsigma }_{2J+1}\right)ð\left({\varsigma }_{2J+1},{\varsigma }^{*}\right)\\ +{\varpi }_2\left({\varsigma }_{2J+1}\right){\displaystyle \frac{ð\left({\varsigma }_{2J+1},{\mathcal{W}}_1{\varsigma }_{2J+1}\right)ð\left({\varsigma }^{*},{\mathcal{W}}_2{\varsigma }^{*}\right)}{1+ð\left({\varsigma }_{2J+1},{\varsigma }^{*}\right)}}\\ +{\varpi }_3\left({\varsigma }_{2J+1}\right){\displaystyle \frac{ð\left({\varsigma }^{*},{\mathcal{W}}_1{\varsigma }_{2J+1}\right)ð\left({\varsigma }_{2J+1},{\mathcal{W}}_2{\varsigma }^{*}\right)}{1+ð\left({\varsigma }_{2J+1},{\varsigma }^{*}\right)}}\end{array}\right)\hfill \\ & & +\vartheta ð\left({\varsigma }^{*},{\varsigma }_{2J+2}\right)\left(\begin{array}{c}{\varpi }_1\left({\varsigma }_{2J+1}\right)ð\left({\varsigma }_{2J+1},{\varsigma }^{*}\right)\\ +{\varpi }_2\left({\varsigma }_{2J+1}\right){\displaystyle \frac{ð\left({\varsigma }_{2J+1},{\mathcal{W}}_1{\varsigma }_{2J+1}\right)ð\left({\varsigma }^{*},{\mathcal{W}}_2{\varsigma }^{*}\right)}{1+ð\left({\varsigma }_{2J+1},{\varsigma }^{*}\right)}}\\ +{\varpi }_3\left({\varsigma }_{2J+1}\right){\displaystyle \frac{ð\left({\varsigma }^{*},{\mathcal{W}}_1{\varsigma }_{2J+1}\right)ð\left({\varsigma }_{2J+1},{\mathcal{W}}_2{\varsigma }^{*}\right)}{1+ð\left({\varsigma }_{2J+1},{\varsigma }^{*}\right)}}\end{array}\right)\hfill \\ & =& ð\left({\varsigma }^{*},{\varsigma }_{2J+2}\right)+\left(\begin{array}{c}{\varpi }_1\left({\varsigma }_{2J+1}\right)ð\left({\varsigma }_{2J+1},{\varsigma }^{*}\right)\\ +{\varpi }_2\left({\varsigma }_{2J+1}\right){\displaystyle \frac{ð\left({\varsigma }_{2J+1},{\varsigma }_{2J+2}\right)ð\left({\varsigma }^{*},{\mathcal{W}}_2{\varsigma }^{*}\right)}{1+ð\left({\varsigma }_{2J+1},{\varsigma }^{*}\right)}}\\ +{\varpi }_3\left({\varsigma }_{2J+1}\right){\displaystyle \frac{ð\left({\varsigma }^{*},{\varsigma }_{2J+2}\right)ð\left({\varsigma }_{2J+1},{\mathcal{W}}_2{\varsigma }^{*}\right)}{1+ð\left({\varsigma }_{2J+1},{\varsigma }^{*}\right)}}\end{array}\right)\hfill \\ & & +\vartheta ð\left({\varsigma }^{*},{\varsigma }_{2J+2}\right)\left(\begin{array}{c}{\varpi }_1\left({\varsigma }_{2J+1}\right)ð\left({\varsigma }_{2J+1},{\varsigma }^{*}\right)\\ +{\varpi }_2\left({\varsigma }_{2J+1}\right){\displaystyle \frac{ð\left({\varsigma }_{2J+1},{\varsigma }_{2J+2}\right)ð\left({\varsigma }^{*},{\mathcal{W}}_2{\varsigma }^{*}\right)}{1+ð\left({\varsigma }_{2J+1},{\varsigma }^{*}\right)}}\\ +{\varpi }_3\left({\varsigma }_{2J+1}\right){\displaystyle \frac{ð\left({\varsigma }^{*},{\varsigma }_{2J+2}\right)ð\left({\varsigma }_{2J+1},{\mathcal{W}}_2{\varsigma }^{*}\right)}{1+ð\left({\varsigma }_{2J+1},{\varsigma }^{*}\right)}}\end{array}\right),\hfill \end{array}$$

which implies that

$$\begin{array}{ccc}\hfill {∥ð\left({\varsigma }^{*},{\mathcal{W}}_2{\varsigma }^{*}\right)∥}_{\mathbb{E}}& \le & {∥ð\left({\varsigma }^{*},{\varsigma }_{2J+2}\right)∥}_{\mathbb{E}}+{∥ð\left({\varsigma }_{2J+2},{\mathcal{W}}_2{\varsigma }^{*}\right)∥}_{\mathbb{E}}\hfill \\ & \le & {∥ð\left({\varsigma }^{*},{\varsigma }_{2J+2}\right)∥}_{\mathbb{E}}+{∥ð\left({\mathcal{W}}_1{\varsigma }_{2J+1},{\mathcal{W}}_2{\varsigma }^{*}\right)∥}_{\mathbb{E}}\hfill \\ & \le & {∥ð\left({\varsigma }^{*},{\varsigma }_{2J+2}\right)∥}_{\mathbb{E}}+{\varpi }_1\left({\varsigma }_{2J+1}\right){∥ð\left({\varsigma }_{2J+1},{\varsigma }^{*}\right)∥}_{\mathbb{E}}\hfill \\ & & +{\varpi }_2\left({\varsigma }_{2J+1}\right){\displaystyle \frac{{∥ð\left({\varsigma }_{2J+1},{\varsigma }_{2J+2}\right)∥}_{\mathbb{E}}·{∥ð\left({\varsigma }^{*},{\mathcal{W}}_2{\varsigma }^{*}\right)∥}_{\mathbb{E}}}{{∥1+ð\left({\varsigma }_{2J+1},{\varsigma }^{*}\right)∥}_{\mathbb{E}}}}\hfill \\ & & +{\varpi }_3\left({\varsigma }_{2J+1}\right){\displaystyle \frac{{∥ð\left({\varsigma }^{*},{\varsigma }_{2J+2}\right)∥}_{\mathbb{E}}{∥ð\left({\varsigma }_{2J+1},{\mathcal{W}}_2{\varsigma }^{*}\right)∥}_{\mathbb{E}}}{{∥1+ð\left({\varsigma }_{2J+1},{\varsigma }^{*}\right)∥}_{\mathbb{E}}}}\hfill \\ & & +\vartheta {∥ð\left({\varsigma }^{*},{\varsigma }_{2J+2}\right)∥}_{\mathbb{E}}\left(\begin{array}{c}{\varpi }_1\left({\varsigma }_{2J+1}\right){∥ð\left({\varsigma }_{2J+1},{\varsigma }^{*}\right)∥}_{\mathbb{E}}\\ +{\varpi }_2\left({\varsigma }_{2J+1}\right){\displaystyle \frac{{∥ð\left({\varsigma }_{2J+1},{\varsigma }_{2J+2}\right)∥}_{\mathbb{E}}·{∥ð\left({\varsigma }^{*},{\mathcal{W}}_2{\varsigma }^{*}\right)∥}_{\mathbb{E}}}{{∥1+ð\left({\varsigma }_{2J+1},{\varsigma }^{*}\right)∥}_{\mathbb{E}}}}\\ +{\varpi }_3\left({\varsigma }_{2J+1}\right){\displaystyle \frac{{∥ð\left({\varsigma }^{*},{\varsigma }_{2J+2}\right)∥}_{\mathbb{E}}{∥ð\left({\varsigma }_{2J+1},{\mathcal{W}}_2{\varsigma }^{*}\right)∥}_{\mathbb{E}}}{{∥1+ð\left({\varsigma }_{2J+1},{\varsigma }^{*}\right)∥}_{\mathbb{E}}}}\end{array}\right).\hfill \end{array}$$

Taking the limit $J\to \infty$ in the preceding inequality and using the fact that ${\varsigma }_J\to {\varsigma }^{*}\in \mathcal{Y}$ as $J\to \infty ,$ we obtain

$${∥ð\left({\varsigma }^{*},{\mathcal{W}}_2{\varsigma }^{*}\right)∥}_{\mathbb{E}}=0.$$

This implies ${\varsigma }^{*}={\mathcal{W}}_2{\varsigma }^{*},$ so ${\varsigma }^{*}$ is a FP of ${\mathcal{W}}_2$. Consequently, ${\varsigma }^{*}$ is CFP of ${\mathcal{W}}_1$ and ${\mathcal{W}}_2.$ To establish the uniqueness of ${\varsigma }^{*}$, assume, for contradiction, that ${\mathcal{W}}_2$ and ${\mathcal{W}}_1$ have a distinct CFP denoted as ${\varsigma }^{/}$. Then

$${\varsigma }^{/}={\mathcal{W}}_1{\varsigma }^{/}={\mathcal{W}}_2{\varsigma }^{/},$$

but ${\varsigma }^{*}\ne {\varsigma }^{/}.$ Now, from (1), we have

$$\begin{array}{ccc}\hfill ð\left({\varsigma }^{*},{\varsigma }^{/}\right)& =& ð\left({\mathcal{W}}_1{\varsigma }^{*},{\mathcal{W}}_2{\varsigma }^{/}\right)\hfill \\ & ⪯& {\varpi }_1\left({\varsigma }^{*}\right)ð\left({\varsigma }^{*},{\varsigma }^{/}\right)+{\varpi }_2\left({\varsigma }^{*}\right){\displaystyle \frac{ð\left({\varsigma }^{*},{\mathcal{W}}_1{\varsigma }^{*}\right)ð\left({\varsigma }^{/},{\mathcal{W}}_2{\varsigma }^{/}\right)}{1+ð\left(\varsigma ,\sigma \right)}}\hfill \\ & & +{\varpi }_3\left({\varsigma }^{*}\right){\displaystyle \frac{ð\left({\varsigma }^{/},{\mathcal{W}}_1{\varsigma }^{*}\right)ð\left({\varsigma }^{*},{\mathcal{W}}_2{\varsigma }^{/}\right)}{1+ð\left({\varsigma }^{*},{\varsigma }^{/}\right)}}\hfill \\ & =& {\varpi }_1\left({\varsigma }^{*}\right)ð\left({\varsigma }^{*},{\varsigma }^{/}\right)+{\varpi }_3\left({\varsigma }^{*}\right){\displaystyle \frac{ð\left({\varsigma }^{/},{\varsigma }^{*}\right)ð\left({\varsigma }^{*},{\varsigma }^{/}\right)}{1+ð\left({\varsigma }^{*},{\varsigma }^{/}\right)}},\hfill \end{array}$$

which implies that

$$\begin{array}{ccc}\hfill {∥ð\left({\varsigma }^{*},{\varsigma }^{/}\right)∥}_{\mathbb{E}}& \le & {\varpi }_1\left({\varsigma }^{*}\right){∥ð\left({\varsigma }^{*},{\varsigma }^{/}\right)∥}_{\mathbb{E}}+{\varpi }_3\left({\varsigma }^{*}\right){\displaystyle \frac{{∥ð\left({\varsigma }^{/},{\varsigma }^{*}\right)∥}_{\mathbb{E}}{∥ð\left({\varsigma }^{*},{\varsigma }^{/}\right)∥}_{\mathbb{E}}}{{∥1+ð\left({\varsigma }^{*},{\varsigma }^{/}\right)∥}_{\mathbb{E}}}}\hfill \\ & \le & {\varpi }_1\left({\varsigma }^{*}\right){∥ð\left({\varsigma }^{*},{\varsigma }^{/}\right)∥}_{\mathbb{E}}+{\varpi }_3\left({\varsigma }^{*}\right){∥ð\left({\varsigma }^{/},{\varsigma }^{*}\right)∥}_{\mathbb{E}},\hfill \end{array}$$

that is,

$$\left(1-{\varpi }_1\left({\varsigma }^{*}\right)-{\varpi }_3\left({\varsigma }^{*}\right)\right){∥ð\left({\varsigma }^{*},{\varsigma }^{/}\right)∥}_{\mathbb{E}}\le 0.$$

Since $\left(1-{\varpi }_1\left({\varsigma }^{*}\right)-{\varpi }_3\left({\varsigma }^{*}\right)\right)\ne 0,$ so ${∥ð\left({\varsigma }^{*},{\varsigma }^{/}\right)∥}_{\mathbb{E}}=0$, it is implied that ${\varsigma }^{*}={\varsigma }^{/}.$ Hence, the CFP of ${\mathcal{W}}_2$ and ${\mathcal{W}}_1$ is unique. □

Example 5.

Let $\mathcal{Y}=[0,1]$ and

$${\mathbb{E}}_{\wp }=\left\{z=\nu +i\omega :\nu ,\omega \in \mathbb{R},i^2=\wp =-0.5<0\right\}.$$

Define $ð:\mathcal{Y}\times \mathcal{Y}\to$${\mathbb{E}}_{\wp }$ by

$$ð(\varsigma ,\sigma )=(1+i)\left|\varsigma -\sigma \right|,$$

for $\varsigma ,\sigma \in \mathcal{Y}$. With this definition, $\left(\mathcal{Y},ð\right)$ forms a complete E-VSMS. Next, define the self-mappings ${\mathcal{W}}_1,{\mathcal{W}}_2:\mathcal{Y}$$\to \mathcal{Y}$ as

$${\mathcal{W}}_1\varsigma ={\displaystyle \frac{1}{6}}\varsigma \mathit{and}{\mathcal{W}}_2\varsigma ={\displaystyle \frac{1}{12}}\varsigma ,$$

∀$\varsigma \in \mathcal{Y}.$ Define ${\varpi }_1,{\varpi }_2,$${\varpi }_3:\mathcal{Y}\to [0,1)$ by

$${\varpi }_1\left(\varsigma \right)={\displaystyle \frac{1}{6}},{\varpi }_2\left(\varsigma \right)={\displaystyle \frac{5}{12}}\mathit{and}{\varpi }_3\left(\varsigma \right)={\displaystyle \frac{2}{7}}.$$

Then this directly satisfies the following

${\varpi }_1\left({\mathcal{W}}_1\varsigma \right)\le {\varpi }_1\left(\varsigma \right)$ and ${\varpi }_1\left({\mathcal{W}}_2\varsigma \right)\le {\varpi }_1\left(\varsigma \right)$

${\varpi }_2\left({\mathcal{W}}_1\varsigma \right)\le {\varpi }_2\left(\varsigma \right)$ and ${\varpi }_2\left({\mathcal{W}}_2\varsigma \right)\le {\varpi }_2\left(\varsigma \right)$

${\varpi }_3\left({\mathcal{W}}_1\varsigma \right)\le {\varpi }_3\left(\varsigma \right)$ and ${\varpi }_3\left({\mathcal{W}}_2\varsigma \right)\le {\varpi }_3\left(\varsigma \right).$

Thus, condition (i) is satisfied.

Moreover

$${\varpi }_1\left(\varsigma \right)+{\varpi }_2\left(\varsigma \right)+{\varpi }_3\left(\varsigma \right)={\displaystyle \frac{1}{6}}+{\displaystyle \frac{5}{12}}+{\displaystyle \frac{2}{7}}={\displaystyle \frac{73}{84}}<1.$$

Thus condition (ii) holds. Now,

$$ð\left(\varsigma ,\sigma \right)=(1+i)\left|\varsigma -\sigma \right|,$$

$$ð\left(\varsigma ,{\mathcal{W}}_1\varsigma \right)={\displaystyle \frac{5}{6}}(1+i)\left|\varsigma \right|,$$

$$ð\left(\sigma ,{\mathcal{W}}_2\sigma \right)={\displaystyle \frac{11}{12}}(1+i)\left|\sigma \right|,$$

$$ð\left(\sigma ,{\mathcal{W}}_1\varsigma \right)=(1+i)\left|\sigma -{\displaystyle \frac{1}{6}}\varsigma \right|,$$

$$ð\left(\varsigma ,{\mathcal{W}}_2\sigma \right)=(1+i)\left|\varsigma -{\displaystyle \frac{1}{12}}\sigma \right|.$$

Then, for all $\varsigma ,\sigma \in \mathcal{Y}$ with $\varsigma \ne \sigma ,$ we have

$$\begin{array}{ccc}\hfill ð\left({\mathcal{W}}_1\varsigma ,{\mathcal{W}}_2\sigma \right)& =& (1+i)\left|{\displaystyle \frac{1}{6}}\varsigma -{\displaystyle \frac{1}{12}}\sigma \right|⪯(1+i)max\left\{{\displaystyle \frac{1}{6}},{\displaystyle \frac{1}{12}}\right\}\left|\varsigma -\sigma \right|\hfill \\ & =& (1+i){\displaystyle \frac{1}{6}}\left|\varsigma -\sigma \right|\hfill \\ & \prec & (1+i){\displaystyle \frac{1}{6}}\left|\varsigma -\sigma \right|+{\displaystyle \frac{5}{12}}{\displaystyle \frac{\frac{55}{72}{(1+i)}^2\left|\varsigma \right|\left|\sigma \right|}{1+(1+i)\left|\varsigma -\sigma \right|}}\hfill \\ & & +{\displaystyle \frac{2}{7}}{\displaystyle \frac{{(1+i)}^2\left|\sigma -\frac{1}{6}\varsigma \right|\left|\varsigma -\frac{1}{12}\sigma \right|}{1+(1+i)\left|\varsigma -\sigma \right|}}\hfill \\ & =& {\displaystyle \frac{1}{6}}ð\left(\varsigma ,\sigma \right)+{\displaystyle \frac{5}{12}}{\displaystyle \frac{ð\left(\varsigma ,{\mathcal{W}}_1\varsigma \right)ð\left(\sigma ,{\mathcal{W}}_2\sigma \right)}{1+ð\left(\varsigma ,\sigma \right)}}\hfill \\ & & +{\displaystyle \frac{2}{7}}{\displaystyle \frac{ð\left(\sigma ,{\mathcal{W}}_1\varsigma \right)ð\left(\varsigma ,{\mathcal{W}}_2\sigma \right)}{1+ð\left(\varsigma ,\sigma \right)}}\hfill \\ & =& {\varpi }_1ð\left(\varsigma ,\sigma \right)+{\varpi }_2{\displaystyle \frac{ð\left(\varsigma ,{\mathcal{W}}_1\varsigma \right)ð\left(\sigma ,{\mathcal{W}}_2\sigma \right)}{1+ð\left(\varsigma ,\sigma \right)}}+{\varpi }_3{\displaystyle \frac{ð\left(\sigma ,{\mathcal{W}}_1\varsigma \right)ð\left(\varsigma ,{\mathcal{W}}_2\sigma \right)}{1+ð\left(\varsigma ,\sigma \right)}}.\hfill \end{array}$$

It follows that the contractive condition (1) specified in Theorem 2 is met, ensuring that 0 is the unique CFP of mappings ${\mathcal{W}}_1$ and ${\mathcal{W}}_2.$

Corollary 1.

Let $\mathcal{W}:\mathcal{Y}$$\to \mathcal{Y}$. Assume that there exist the functions ${\varpi }_1, {\varpi }_2,$${\varpi }_3:\mathcal{Y}\to [0, 1)$ satisfying the following conditions:

(i) 

${\varpi }_1\left(\mathcal{W}\varsigma \right)\le {\varpi }_1\left(\varsigma \right)$

${\varpi }_2\left(\mathcal{W}\varsigma \right)\le {\varpi }_2\left(\varsigma \right)$

${\varpi }_3\left(\mathcal{W}\varsigma \right)\le {\varpi }_3\left(\varsigma \right),$

(ii) 

${\varpi }_1\left(\varsigma \right)+{\varpi }_2\left(\varsigma \right)+{\varpi }_3\left(\varsigma \right)<1,$

(iii) 

$$ð\left(\mathcal{W}\varsigma ,\mathcal{W}\sigma \right)⪯{\varpi }_1\left(\varsigma \right)ð\left(\varsigma ,\sigma \right)+{\varpi }_2\left(\varsigma \right){\displaystyle \frac{ð\left(\varsigma ,\mathcal{W}\varsigma \right)ð\left(\sigma ,\mathcal{W}\sigma \right)}{1+ð\left(\varsigma ,\sigma \right)}}+{\varpi }_3\left(\varsigma \right){\displaystyle \frac{ð\left(\sigma ,\mathcal{W}\varsigma \right)ð\left(\varsigma ,\mathcal{W}\sigma \right)}{1+ð\left(\varsigma ,\sigma \right)}},$$

∀$\varsigma ,\sigma \in \mathcal{Y}$ with $\varsigma \ne \sigma .$ Then $\mathcal{W}$ admits a unique FP.

Proof. 

For the purpose of Theorem 2, let ${\mathcal{W}}_1={\mathcal{W}}_2=\mathcal{W}$. □

Example 6.

Let $\mathcal{Y}=[0, 1]$ and

$${\mathbb{E}}_{\wp }=\left\{z=\nu +i\omega :\nu ,\omega \in \mathbb{R},i^2=\wp <0\right\}.$$

Define $ð:\mathcal{Y}\times \mathcal{Y}\to$${\mathbb{E}}_{\wp }$ by

$$ð(\varsigma ,\sigma )=(1+i)\left|\varsigma -\sigma \right|,$$

for $\varsigma ,\sigma \in \mathcal{Y}$. Consequently, $\left(\mathcal{Y},ð\right)$ is a complete E-VSMS. Let us define $\mathcal{W}:\mathcal{Y}$ $\to \mathcal{Y}$ as

$$\mathcal{W}\left(\varsigma \right)={\displaystyle \frac{1}{3}}\varsigma$$

for $\varsigma \in \mathcal{Y}.$ Define ${\varpi }_1, {\varpi }_2,$${\varpi }_3:\mathcal{Y}\to [0, 1)$ by ${\varpi }_1\left(\varsigma \right)={\displaystyle \frac{1}{3}}, {\varpi }_2\left(\varsigma \right)={\displaystyle \frac{1}{5}}, {\varpi }_3\left(\varsigma \right)={\displaystyle \frac{1}{7}}.$ Then

$${\varpi }_1\left(\varsigma \right)+{\varpi }_2\left(\varsigma \right)+{\varpi }_3\left(\varsigma \right)={\displaystyle \frac{71}{105}}<1.$$

Then, conditions (i) and (ii) of the above corollary are satisfied. For any $\varsigma ,\sigma \in [0, 1],$ we have

$$\begin{array}{ccc}\hfill \left|\mathcal{W}\left(\varsigma \right)-\mathcal{W}\left(\sigma \right)\right|& =& \left|{\displaystyle \frac{1}{3}}\varsigma -{\displaystyle \frac{1}{3}}\sigma \right|\le {\displaystyle \frac{1}{3}}\left|\varsigma -\sigma \right|\hfill \\ & & +{\displaystyle \frac{1}{5}}{\displaystyle \frac{\left|\varsigma -\frac{1}{3}\varsigma \right|\left|\sigma -\frac{1}{3}\sigma \right|}{1+\left|\varsigma -\sigma \right|}}\hfill \\ & & +{\displaystyle \frac{1}{7}}{\displaystyle \frac{\left|\sigma -\frac{1}{3}\varsigma \right|\left|\varsigma -\frac{1}{3}\sigma \right|}{1+\left|\varsigma -\sigma \right|}}.\hfill \end{array}$$

Then,

$$\begin{array}{ccc}\hfill ð\left(\mathcal{W}\right(\varsigma ),\mathcal{W}(\sigma \left)\right)& =& \left|\mathcal{W}\left(\varsigma \right)-\mathcal{W}\left(\sigma \right)\right|(1+i)=\left|{\displaystyle \frac{1}{3}}\varsigma -{\displaystyle \frac{1}{3}}\sigma \right|(1+i)\le {\displaystyle \frac{1}{3}}\left|\varsigma -\sigma \right|(1+i)\hfill \\ & & +{\displaystyle \frac{1}{5}}{\displaystyle \frac{\left|\varsigma -\frac{1}{3}\varsigma \right|(1+i)\left|\sigma -\frac{1}{3}\sigma \right|(1+i)}{1+\left|\varsigma -\sigma \right|(1+i)}}\hfill \\ & & +{\displaystyle \frac{1}{7}}{\displaystyle \frac{\left|\sigma -\frac{1}{3}\varsigma \right|(1+i)\left|\varsigma -\frac{1}{3}\sigma \right|(1+i)}{1+\left|\varsigma -\sigma \right|(1+i)}}\hfill \\ & =& {\displaystyle \frac{1}{3}}ð(\varsigma ,\sigma )+{\displaystyle \frac{1}{5}}{\displaystyle \frac{ð\left(\varsigma ,{\mathcal{W}}_2\varsigma \right)ð\left(\sigma ,{\mathcal{W}}_2\sigma \right)}{1+ð\left(\varsigma ,\sigma \right)}}+{\displaystyle \frac{1}{7}}{\displaystyle \frac{ð\left(\sigma ,{\mathcal{W}}_2\varsigma \right)ð\left(\varsigma ,{\mathcal{W}}_2\sigma \right)}{1+ð\left(\varsigma ,\sigma \right)}}.\hfill \end{array}$$

Hence, the requirements of Corollary 1 hold, and 0 serves as an FP of $\mathcal{W}$.

Consequently, the principal result of Alamri [18] follows as a special case of Theorem 2.

Corollary 2.

Let ${\mathcal{W}}_1, {\mathcal{W}}_2:$$\mathcal{Y}$$\to \mathcal{Y}$. Assume that there are ${\varpi }_1, {\varpi }_2,$${\varpi }_3\in [0, 1)$ such that ${\varpi }_1+{\varpi }_2+{\varpi }_3<1$ and

$$ð\left({\mathcal{W}}_1\varsigma ,{\mathcal{W}}_2\sigma \right)⪯{\varpi }_1ð\left(\varsigma ,\sigma \right)+{\varpi }_2{\displaystyle \frac{ð\left(\varsigma ,{\mathcal{W}}_1\varsigma \right)ð\left(\sigma ,{\mathcal{W}}_2\sigma \right)}{1+ð\left(\varsigma ,\sigma \right)}}+{\varpi }_3{\displaystyle \frac{ð\left(\sigma ,{\mathcal{W}}_1\varsigma \right)ð\left(\varsigma ,{\mathcal{W}}_2\sigma \right)}{1+ð\left(\varsigma ,\sigma \right)}},$$

∀$\varsigma ,\sigma \in \mathcal{Y}$ with $\varsigma \ne \sigma .$ Then ${\mathcal{W}}_1$ and ${\mathcal{W}}_2$ has a unique CFP.

Proof. 

Define ${\varpi }_1, {\varpi }_2, {\varpi }_3:\mathcal{Y}\to [0,1)$ by ${\varpi }_1\left(\varsigma \right)=\varsigma , {\varpi }_2\left(\varsigma \right)={\varpi }_2$ and ${\varpi }_3\left(\varsigma \right)=\varsigma$ to become Theorem 2. □

Corollary 3.

Let $\mathcal{W}:\mathcal{Y}$$\to \mathcal{Y}$. Assume that there is ${\varpi }_1\in [0, 1)$ such that

$$ð\left(\mathcal{W}\varsigma ,\mathcal{W}\sigma \right)⪯{\varpi }_1ð\left(\varsigma ,\sigma \right),$$

for all $\varsigma ,\sigma \in \mathcal{Y}$ with $\varsigma \ne \sigma .$ Then $\mathcal{W}$ have a unique FP.

Proof. 

Take ${\varpi }_2={\varpi }_3=0$ and ${\mathcal{W}}_1={\mathcal{W}}_2=\mathcal{W}$ in Corollary 2. □

Remark 2. 

(i) By choosing $\wp =-1$ in Definition 3, the notion of an E-VSMS reduces to a C-VSMS. Under this choice, Corollary 2 naturally reproduces the principal result obtained by Panda et al. [15].

(ii) If $\wp =-1$ in Definition 3 and ${\varpi }_3=0,$ then Corollary 2 coincides with the central result of Abdou [16].

(iii) Setting $\vartheta =\theta$ in Definition 3, reduces the E-VSMS structure to that of an E-VMS, whereby Theorem 2 directly reproduces the primary theorem of [13].

(iv) By taking $\wp =-1$ and $\vartheta =\theta$ in Definition 3, the E-VSMS simplifies to a C-VMS, and Theorem 2 recovers the major result of Sintunavarat et al. [6] by defining ${\varpi }_3:\mathcal{Y}\to [0,1)$ as ${\varpi }_3\left(\varsigma \right)=0.$

(v) Similarly, if $\wp =-1$ and $\vartheta =\theta$, then the E-VSMS reduces to a C-VMS, and Corollary 2 reproduces the foremost result of Rouzkard et al. [5].

(vi) Furthermore, under the assumptions in (iv), setting ${\varpi }_2=0$ leads Corollary 2 to coincide with the chief result of Azam et al. [4].

4. Application of Fixed Point Theory to Nonlinear Volterra Integral Equations

NVIEs of the second kind appear frequently in many branches of science and engineering, covering areas such as physics, control systems, biology and climate dynamics. A general form of such an equation is

$$\varsigma \left(t\right)=g\left(t\right)+{\int }_a^t\mathcal{F}(t,s,\varsigma \left(s\right))ðs,t\in [a,b].$$

Here, $g\left(t\right)$ denotes a given function, $\mathcal{F}(t,s,\varsigma (s\left)\right)$ represents the kernel, which may depend nonlinearly on $\varsigma \left(s\right),$ and the unknown function $\varsigma$ appears both inside and outside the integral.

A powerful technique for studying such equations is the FP approach. By rewriting the NVIE as an operator equation

$$\left(\mathcal{W}\varsigma \right)\left(t\right)=g\left(t\right)+{\int }_a^t\mathcal{F}(t,s,\varsigma \left(s\right))ðs,$$

and seeks a function $\varsigma$ satisfying $\mathcal{W}\varsigma =\varsigma$. Under suitable assumptions, FP methods, especially those based on contraction principles, guarantee both the existence and uniqueness of the solution. The BCP and its extensions provide a robust strategy for confirming these properties and justify the use of iterative schemes that converge to the exact solution from an arbitrary initial guess.

This approach is particularly effective when the kernel is nonlinear or exhibits complex dependence on past states, frequently arises in practical applications. Along with this, to providing theoretical assurance, the FP methodology naturally yields practical numerical procedures that can be implemented efficiently.

In this section, we apply an FP theorem in the setting of E-VSMSs to investigate NVIEs. This framework is well-suited for handling nonlinearities and intricate kernel structures, offering a rigorous basis for proving existence and uniqueness of solutions, as well as supporting iterative computational techniques.

Let $\mathcal{Y}=C\left(\right[0,T],\mathbb{R})$ with $T>0,$ denote the set of all real-valued continuous functions defined on $[0,T].$ Define $ð:\mathcal{Y}\times \mathcal{Y}\to {\mathbb{E}}_{\wp }$ by

$$ð(\varsigma ,\sigma )=\left({∥\varsigma -\sigma ∥}_{\infty }\right)e^{i{\displaystyle \frac{\pi }{4}}}=\left(\underset{t\in [0,T]}{max}∥\varsigma \left(t\right)-\sigma \left(t\right)∥\right)e^{i{\displaystyle \frac{\pi }{4}}},$$

where $i^2=\wp <0.$ Consequently, $\left(C([0,T],\mathbb{R}),∥·∥\right)$ is complete E-VSMS.

Let us examine the NVIE

$$\varsigma \left(t\right)=g\left(t\right)+{\int }_0^t\mathcal{F}(t,s,\varsigma \left(s\right))ðs,t\in [0,T]$$

(7)

in which

1.

$\varsigma \left(t\right)$ denotes the function to be found.

2.

$g\left(t\right)$ is a prescribed function, commonly called the free term.

3.

$\mathcal{F}(t,s,\varsigma (s\left)\right)$ represents the kernel, which may exhibit nonlinear dependence on $\varsigma \left(s\right).$

Theorem 3.

Assume that $\left(C\left(\right[0,T],\mathbb{R}),ð\right)$ be a complete E-VSMS and $T>0.$ Let

(a) 

$\mathcal{F}:[0,T]\times [0,T]\times \mathbb{R}\to \mathbb{R}$ and $g:[0,T]\to \mathbb{R}$ are continuous functions;

(b) 

let us take a constant $L\ge 0$ such that

$$∥\mathcal{F}(t,s,\varsigma )-\mathcal{F}(t,s,\sigma )∥\le L∥\varsigma -\sigma ∥,$$

∀$t,s\in [0,T]$ and $\varsigma ,\sigma \in \mathbb{R};$

(c) 

assume

$$LT<1.$$

Then, the NVIE (7) admits a unique solution in $C\left(\right[0,T],\mathbb{R})$.

Proof. 

Define $\mathcal{W}:\mathcal{Y}\to \mathcal{Y}$ by

$$\left(\mathcal{W}\varsigma \right)\left(t\right)=g\left(t\right)+{\int }_0^t\mathcal{F}(t,s,\varsigma \left(s\right))ðs,t\in [0,T].$$

Since g and $\mathcal{F}$ are continuous functions. So it ensures that, for each fixed $\varsigma \in \mathcal{Y},$ the integrand $\mathcal{F}(t,s,\varsigma (s\left)\right)$ is continuous in s and thus the integral ${\int }_0^t\mathcal{F}(t,s,\varsigma \left(s\right))ðs$ is continuous. Therefore, $\mathcal{W}\varsigma \in \mathcal{Y},$ so $\mathcal{W}$ maps $\mathcal{Y}$ into itself. For any $\varsigma ,\sigma \in C\left(\right[0,T],\mathbb{R})$ and $t\in [0,T],$ we have

$$\begin{array}{ccc}\hfill \left|\left(\mathcal{W}\varsigma \right)\left(t\right)-\left(\mathcal{W}\sigma \right)\left(t\right)\right|& =& \left|{\int }_0^t\left(\mathcal{F}(t,s,\varsigma (s\left)\right)-\mathcal{F}(t,s,\sigma (s\left)\right)\right)ðs\right|\hfill \\ & \le & {\int }_0^t\left|\left(\mathcal{F}(t,s,\varsigma (s\left)\right)-\mathcal{F}(t,s,\sigma (s\left)\right)\right)\right|ðs\hfill \\ & \le & {\int }_0^{t}L\left|\varsigma \left(s\right)-\sigma \left(s\right)\right|ðs\hfill \\ & \le & L{\int }_0^t{∥\varsigma -\sigma ∥}_{\infty }ðs=Lt{∥\varsigma -\sigma ∥}_{\infty }.\hfill \end{array}$$

Evaluating the supremum across $t\in [0,T],$ yields

$${∥\mathcal{W}\varsigma -\mathcal{W}\sigma ∥}_{\infty }\le LT{∥\varsigma -\sigma ∥}_{\infty }.$$

Consequently, in the E-VSMS metric ð

$$ð(\mathcal{W}\varsigma ,\mathcal{W}\sigma )=\left({∥\mathcal{W}\varsigma -\mathcal{W}\sigma ∥}_{\infty }\right)e^{i{\displaystyle \frac{\pi }{4}}}⪯LT\left({∥\varsigma -\sigma ∥}_{\infty }\right)e^{i{\displaystyle \frac{\pi }{4}}}=\varpi ð(\varsigma ,\sigma ),$$

where $\varpi =$$LT\in [0, 1)$ by assumption (c). Define ${\varpi }_1, {\varpi }_2, {\varpi }_3:\mathcal{Y}\to [0, 1)$ by

$${\varpi }_1\left(\varsigma \right)=\varpi =LT,{\varpi }_2\left(\varsigma \right)={\varpi }_3\left(\varsigma \right)=0.$$

Thus, all the hypotheses of Theorem 2 are satisfied for the self-mapping $\mathcal{W}$ on the complete E-VSMS $\left(C\left(\right[0,T],\mathbb{R}),ð\right).$ Hence, by Theorem 2, the operator $\mathcal{W}$ has a unique FP $\varsigma \in \mathcal{Y},$ that is

$$\varsigma =\mathcal{W}\varsigma .$$

Therefore, the NVIE (7) admits a unique solution in $\left(C\left(\right[0,T],\mathbb{R}),ð\right).$ □

Example 7.

We analyze the NVIE

$$\varsigma \left(t\right)=g\left(t\right)+{\int }_0^{t}a(t,s)\varsigma \left(s\right)ðs,t\in [0,T],$$

(8)

take $g\in C\left(\right[0,T],\mathbb{R})$ and $\mathcal{F}(t,s,\varsigma (s\left)\right)=a(t,s)\varsigma \left(s\right)$, where $a\in C\left(\right[0,T]\times [0,T],\mathbb{R}).$

• Lipschitz property in $x:$

  $$\left|\mathcal{F}(t,s,\varsigma )-\mathcal{F}(t,s,\sigma )\right|=\left|a(t,s)\right|\left|\varsigma -\sigma \right|.$$
• Set

  $$L=\underset{(t,s)\in {[0,T]}^2}{sup}\left|a(t,s)\right|,$$
  which is finite by the continuity of $a(t,s).$
• Condition (c): select T such that $LT<1$ $(\mathit{for}\mathit{instance}, \mathit{if} L=2,$ choose any $0<T<{\displaystyle \frac{1}{2}})$.
• Condition (a) is satisfied because the functions a and g are continuous.

Hence, all the hypotheses of the previously stated theorem are fulfilled, and consequently, the NVIE admits a unique solution.

5. Conclusions

In this study, we developed several new common FP theorems within the framework of E-VSMSs, using generalized contractive mappings governed by single-variable control functions. In this study, we utilized a new generalized metric-type structure, namely the E-VSMS, which extends E-VMSs by incorporating a suprametric inequality involving an elliptic-valued control parameter. Within this newly established framework, we developed some common FP theorems for pairs of self-mappings satisfying generalized contractive conditions governed by single-variable control functions.

The obtained results significantly extend and unify a wide range of existing FP theorems in the literature, including those established in E-VSMSs by Alamri [18], the key theorems in C-VSMSs outlined by Panda et al. [15] and Abdou [16], and the main results in E-VMSs described by Alamri [13] and Ozturk et al. [11]. As particular cases, our theorems also recover some classical FP results in C-VMSs reported by Azam et al. [4], Rouzkard et al. [5], and Sintunavarat et al. [6].

To demonstrate the effectiveness of the proposed theoretical framework, we utilized the constructed FP results to investigate the existence and uniqueness of solutions for nonlinear Volterra integral equations of the second kind.

Future research directions may include extending the concept of E-VSMSs to multivalued and hybrid mappings, studying stability and data-dependence results, and exploring further applications to fractional differential equations, integro-differential equations, and mathematical models arising in physics and engineering.