Solving Riemann–Liouville Fractional Integral Equations by Fixed Point Results in Complex-Valued Suprametric Spaces

Abstract

Theaim of this research is to establish existence and uniqueness results for the Riemann–Liouville fractional integral equation of order $\alpha$ $\varkappa \left(t\right)=f\left(t\right)+{\displaystyle \frac{\lambda }{\Gamma \left(\alpha \right)}}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{\alpha -1}g\left(s,\varkappa \left(s\right)\right)ds,t\in [0,1],$ by developing common fixed point theorems for generalized contractions involving control functions of two variables in the framework of complex valued suprametric spaces. The proposed results extend and generalize several existing findings in the literature, and some illustrative examples are provided to demonstrate the novelty and applicability of the main theorem.

1. Introduction

Metric fixed point (FP) theory provides a natural setting for analyzing FPs because it is grounded in the notion of metric spaces (MSs). Within this framework, the behavior of iterative sequences, contractive mappings, and continuity can be precisely controlled, allowing researchers to prove when a self-mapping must possess a point that maps to itself. The idea of an MS was first introduced by Maurice Fréchet [1] in 1906, describing a set supplied with a distance function that measures the separation between any two elements of the space. This study of metric FP theory was revolutionized by the seminal work of Banach [2], who introduced the celebrated Banach contraction principle (BCP). This fundamental result asserts that a contraction mapping on a complete MS possesses a unique FP, and successive iterations of the mapping converge to that point. The simplicity and elegance of this theorem made it a cornerstone for numerous developments in nonlinear analysis and its applications to integral and differential equations. Later, Kannan [3] proposed a significant generalization of Banach’s result by relaxing the contraction condition. Instead of depending solely on the distance between two points, Kannan’s condition involves the distances between each point and its image under the mapping, thereby providing a broader class of mappings that still guarantee the existence of a unique FP. Subsequently, Fisher [4] extended these ideas further by introducing rational-type contractive conditions that encompass both Banach and Kannan contractions as special cases. These progressive generalizations have deepened the scope of FP theory, paving the way for modern extensions in generalized metric frameworks.

In an effort to extend the classical framework of MSs, Matthews [5] introduced the concept of partial MSs, in which the self-distance of a point need not be zero. This notion provides a more flexible structure that captures the behavior of non-disjoint self-mappings and is particularly useful in computer science, where it models computations that are only partially completed. Subsequently, Czerwik [6] proposed the concept of b-MSs by relaxing the triangle inequality through a constant factor $s\ge 1,$ thereby generalizing the classical metric structure while preserving completeness and contraction principles. Furthermore, Huang et al. [7] extended the idea to cone MSs, where the distance function takes values in an ordered Banach space rather than in the real numbers, thus enabling the study of FPs in more abstract settings. Building on these generalizations, Berzig [8] introduced the notion of suprametric spaces (SMSs), which unify and extend various existing metric-type structures by relaxing the distance axioms while retaining enough structure to apply powerful FP theorems. In a later development, Berzig [9,10,11] further generalized the notion of SMSs by modifying the triangle inequality axiom, which led to the formulation of two new classes of spaces, namely the generalized SMSs and the b-SMSs.

Motivated by the versatility of complex numbers, Azam et al. [12] introduced complex valued MS (CVMS) as an extension of classical MSs, where the distance between points is a complex number rather than a real one. A CVMS is defined on a set $\Omega$ with a mapping $d:\Omega \times \Omega \to \mathbb{C}$ satisfying properties analogous to those of real-valued metrics, including properties like non-negativity, symmetry, and an extended form of the triangle inequality. These spaces have proven useful in FP theory, offering a broader framework for studying contraction mappings. Rouzkard et al. [13] broadened the findings of Azam et al. [12] through the inclusion of a rational component embedded in the contractive framework, thereby broadening the reach of complex valued FP results. Later, Sintunavarat et al. [14] refined these results by introducing control functions of a single variable, while Sitthikul et al. [15] further advanced the theory through the use of two-variable control functions within the structure of CVMSs. In contemporary works, Panda et al. [16] unified the ideas of CVMSs and SMSs to establish the idea of complex-valued suprametric spaces (CVSMSs).

In recent years, FP methods have been widely applied to solve nonlinear integral equations, fractional models, and functional equations. For example, Fredholm integral equations have been investigated via FP techniques in [17], stability of functional equations in quasi-Banach spaces has been analyzed using FP methods in [18], and mixed integer–fractional dynamical models have been explored through FP theory in [19]. Earlier significant contributions include the work of Gülyaz et al. [20], who addressed integral equations using classical FP theorems. Abdou [21] utilized a FP approach within CVSMSs to investigate predator-prey dynamics modeled by nonlinear mixed Volterra–Fredholm integral equations, demonstrating the existence and uniqueness of solutions under suitable conditions. Similarly, Shammaky et al. [22] addressed nonlinear Volterra–Hammerstein integral equations in the setting of CVSMSs, employing FP theory to establish solution existence and uniqueness results. These studies exemplify the power of CVSMSs in handling complex nonlinear integral equations and provide a strong motivation for further research in this area. For further details, see references [23,24,25,26].

In this research article, we explore the concept of CVSMSs and investigate common FP theorems for generalized contractions involving control functions of two variables. As consequences of our main results, we recover the leading theorems of Abdou [21], Shammaky et al. [22] and Panda et al. [16] within the framework of CVSMSs, the main results of Azam et al. [12], Rouzkard et al. [13], Sintunavarat et al. [14] and Sitthikul et al. [15] in CVMSs, and the prime result of Berzig [8] in SMSs. To highlight the novelty of our approach, an illustrative example is provided, and the results are applied to solve a Riemann–Liouville fractional integral equation.

2. Preliminaries

Fréchet [1] first proposed the notion of an MS in 1906, which is defined as follows:

Definition 1

([8]). Let $\Omega \ne \varnothing$, and define a function $d:\Omega \times \Omega \to {\mathbb{R}}^{+}$ as follows

(m₁)

$0\le d(\varkappa ,\mathfrak{l})$ and $d(\varkappa ,\mathfrak{l})=0$$⟺\varkappa =\mathfrak{l}$,

(m₂)

$d(\varkappa ,\mathfrak{l})=d(\mathfrak{l},\varkappa ),$

(m₃)

$d(\varkappa ,\mathfrak{l})\le d(\varkappa ,\rho )+d(\rho ,\mathfrak{l}),$

∀$\varkappa ,\mathfrak{l},\rho \in \Omega ,$ then $(\Omega ,d)$ is referred to as an MS.

The concept of SMS was introduced by Berzig [8] as described below.

Definition 2

([8]). Let $\Omega \ne \varnothing$ and $\aleph \ge 0$. Define $d:\Omega \times \Omega \to {\mathbb{R}}^{+}$ that adheres to these conditions:

(s₁)

$0\le d(\varkappa ,\mathfrak{l})$ and $d(\varkappa ,\mathfrak{l})=0$⟺$\varkappa =\mathfrak{l}$,

(s₂)

$d(\varkappa ,\mathfrak{l})=d(\mathfrak{l},\varkappa ),$

(s₃)

$d(\varkappa ,\mathfrak{l})\le d(\varkappa ,\rho )+d(\rho ,\mathfrak{l})+\aleph d(\varkappa ,\rho )d(\rho ,\mathfrak{l}),$

∀$\varkappa ,\mathfrak{l},\rho \in \Omega ,$ then $(\Omega ,d)$ is called a SMS.

Example 1.

Let $\Omega =\left\{0,1,2\right\}.$ Define $d:\Omega \times \Omega \to {\mathbb{R}}^{+}$ as follows:

$$d(0,0)=d(1,1)=d(2,2)=0,$$

$$d(1,2)=d(2,1)=2,$$

$$d(0,1)=d(1,0)=0.5,$$

and

$$d(0,2)=d(2,0)=1.$$

Let us take $\aleph =1.5.$ Then, $\left(\Omega ,d\right)$ forms a SMS but does not qualify as an MS, since the triangle inequality of an MS is not satisfied, namely,

$$2=d(1,2)>d(1,0)+d(0,2)=0.5+1.$$

Let $z_1,z_2\in \mathbb{C}$. It is well-known that

$$z_1\precsim z_2\iff \mathrm{Re}\left(z_1\right)⩽\mathrm{Re}\left(z_2\right),\mathrm{Im}\left(z_1\right)⩽\mathrm{Im}\left(z_2\right),$$

where ≾ is a partial order on $\mathbb{C}$. Therefore,

$$z_1\precsim z_2$$

if one of these axioms is met:

$$\begin{array}{ccc}\hfill \left(\mathrm{a}\right)\mathrm{R}\mathrm{e}\left(z_1\right)& =& \mathrm{R}\mathrm{e}\left(z_2\right),\mathrm{I}\mathrm{m}\left(z_1\right)<\mathrm{I}\mathrm{m}\left(z_2\right),\hfill \\ \hfill \left(\mathrm{b}\right)\mathrm{R}\mathrm{e}\left(z_1\right)& <& \mathrm{R}\mathrm{e}\left(z_2\right),\mathrm{I}\mathrm{m}\left(z_1\right)=\mathrm{I}\mathrm{m}\left(z_2\right),\hfill \\ \hfill \left(\mathrm{c}\right)\mathrm{R}\mathrm{e}\left(z_1\right)& <& \mathrm{R}\mathrm{e}\left(z_2\right),\mathrm{I}\mathrm{m}\left(z_1\right)<\mathrm{I}\mathrm{m}\left(z_2\right),\hfill \\ \hfill \left(\mathrm{d}\right)\mathrm{R}\mathrm{e}\left(z_1\right)& =& \mathrm{R}\mathrm{e}\left(z_2\right),\mathrm{I}\mathrm{m}\left(z_1\right)=\mathrm{I}\mathrm{m}\left(z_2\right).\hfill \end{array}$$

Azam et al. [12] defined the idea of CVMS in the manner described:

Definition 3

([12]). Let $\Omega \ne \varnothing .$ Define a function $d:\Omega \times \Omega \to \mathbb{C}$ such that

(c₁)

$0\precsim d(\varkappa ,\mathfrak{l})$ and $d(\varkappa ,\mathfrak{l})=0$⟺$\varkappa =\mathfrak{l}$,

(c₂)

$d(\varkappa ,\mathfrak{l})=d(\mathfrak{l},\varkappa ),$

(c₃)

$d(\varkappa ,\mathfrak{l})\precsim d(\varkappa ,\nu )+d(\nu ,\mathfrak{l}),$

∀$\varkappa ,\mathfrak{l},\nu \in \Omega$, then $(\Omega ,d)$ is said to be a CVMS.

Example 2

([12]). Let $\Omega =[0,1]$ and $\varkappa ,\mathfrak{l}\in \Omega .$ Define $d:\Omega \times \Omega \to \mathbb{C}$ by

$$d(\varkappa ,\mathfrak{l})=\left\{\begin{array}{c}0,\mathrm{if}\varkappa =\mathfrak{l},\\ {\displaystyle \frac{i}{2}},\mathrm{if}\varkappa \ne \mathfrak{l}.\end{array}\right.$$

Then $(\Omega ,d)$ is a CVMS.

In contemporary studies, Panda et al. [16] defined the notion of CVSMS through this approach.

Definition 4

([16]). Let $\Omega \ne \varnothing$ and $\aleph \ge 0$. Define a complex function $d:\Omega \times \Omega \to \mathbb{C}$ such that

(D₁)

$0\precsim d(\varkappa ,\mathfrak{l})$ and $d(\varkappa ,\mathfrak{l})=0$⟺$\varkappa =\mathfrak{l}$,

(D₂)

$d(\varkappa ,\mathfrak{l})=d(\mathfrak{l},\varkappa ),$

(D₃)

$d(\varkappa ,\mathfrak{l})\precsim d(\varkappa ,\rho )+d(\rho ,\mathfrak{l})+\aleph d(\varkappa ,\rho )d(\rho ,\mathfrak{l}),$

∀$\varkappa ,\mathfrak{l},\rho \in \Omega ,$ then $(\Omega ,d)$ is considered as a CVSMS.

Example 3.

Let $\Omega =\mathbb{R}$. Define a function $d:\Omega \times \Omega \to \mathbb{C}$ by

$$d(\varkappa ,\mathfrak{l})=\left(1+i\right)\left|\varkappa -\mathfrak{l}\right|,$$

Then $\left(\Omega ,d\right)$ is a CVSMS.

Lemma 1

([16]). Let $(\Omega ,d)$ be a CVSMS and let $\left\{{\varkappa }_{\mathfrak{p}}\right\}$$\subseteq \Omega$. Then $\left\{{\varkappa }_{\mathfrak{p}}\right\}$ converges to ϰ if and only if $\left|d({\varkappa }_{\mathfrak{p}},\varkappa )\right|\to 0$ as $\mathfrak{p}\to \infty .$

Lemma 2

([16]). Let $(\Omega ,d)$ be a CVSMS and let $\left\{{\varkappa }_{\mathfrak{p}}\right\}$$\subseteq \Omega$. Then $\left\{{\varkappa }_{\mathfrak{p}}\right\}$ is a Cauchy sequence iff $\left|d({\varkappa }_{\mathfrak{p}},{\varkappa }_{\mathfrak{p}+m})\right|\to 0$ as $\mathfrak{p}\to \infty ,$ where $m\in \mathbb{N}.$

3. Main Results

In the present section, we establish several CFP results in the framework of CVSMSs, which extend the concept of CVMSs. The obtained results are generalizations of some previous results of literature. To facilitate the proofs of the main theorems, we first present a proposition and a lemma that serve as key tools in the subsequent analysis. Thereafter, we proceed to demonstrate the CFP theorems in this generalized setting.

Proposition 1.

Let $\mathfrak{I},\mathcal{O}:\Omega$$\to \Omega$ be two self-mappings, and let ${\varkappa }_0\in$ $\Omega .$ Define a sequence {${\varkappa }_{\mathfrak{p}}$} recursively as follows:

$${\varkappa }_{2\mathfrak{p}+1}=\mathfrak{I}{\varkappa }_{2\mathfrak{p}}\mathit{and}{\varkappa }_{2\mathfrak{p}+2}=\mathcal{O}{\varkappa }_{2\mathfrak{p}+1}$$

(1)

for all $\mathfrak{p}\in \mathbb{N}\cup \left\{0\right\}.$

Assume that there exists a function ${\varpi }_1:\Omega \times \Omega \to [0,1)$ satisfying

$${\varpi }_1\left(\mathcal{O}\mathfrak{I}\varkappa ,\mathfrak{l}\right)\le {\varpi }_1\left(\varkappa ,\mathfrak{l}\right)\mathrm{and}{\varpi }_1\left(\varkappa ,\mathfrak{I}\mathcal{O}\mathfrak{l}\right)\le {\varpi }_1\left(\varkappa ,\mathfrak{l}\right),$$

∀$\varkappa ,\mathfrak{l}\in \Omega .$ Then the following inequalities hold:

$${\varpi }_1\left({\varkappa }_{2\mathfrak{p}},\mathfrak{l}\right)\le {\varpi }_1\left({\varkappa }_0,\mathfrak{l}\right)\mathrm{and}{\varpi }_1\left(\varkappa ,{\varkappa }_{2\mathfrak{p}+1}\right)\le {\varpi }_1\left(\varkappa ,{\varkappa }_1\right),$$

∀$\varkappa ,\mathfrak{l}\in \Omega$ and $\mathfrak{p}\in \mathbb{N}\cup \left\{0\right\}.$

Before proceeding to the main results, we first establish the following lemma, which provides an essential inequality used in the proof of the subsequent theorems.

Lemma 3.

Let ${\varpi }_1,{\varpi }_2,{\varpi }_3,{\varpi }_4:\Omega \times \Omega \to [0,1).$ Suppose the mappings $\mathfrak{I},\mathcal{O}:\Omega$$\to \Omega$ satisfy the following conditions:

$$\begin{array}{ccc}\hfill d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{I}\varkappa \right)& ⪯& {\varpi }_1\left(\varkappa ,\mathfrak{I}\varkappa \right)d\left(\varkappa ,\mathfrak{I}\varkappa \right)+{\varpi }_2\left(\varkappa ,\mathfrak{I}\varkappa \right)\left(d\left(\varkappa ,\mathfrak{I}\varkappa \right)+d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{I}\varkappa \right)\right)\hfill \\ & & +{\varpi }_3\left(\varkappa ,\mathfrak{I}\varkappa \right){\displaystyle \frac{d\left(\varkappa ,\mathfrak{I}\varkappa \right)d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{I}\varkappa \right)}{1+d\left(\varkappa ,\mathfrak{I}\varkappa \right)}}\hfill \\ & & +{\varpi }_4\left(\varkappa ,\mathfrak{I}\varkappa \right){\displaystyle \frac{d\left(\mathfrak{I}\varkappa ,\mathfrak{I}\varkappa \right)d\left(\varkappa ,\mathcal{O}\mathfrak{I}\varkappa \right)}{1+d\left(\varkappa ,\mathfrak{I}\varkappa \right)}},\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill d\left(\mathfrak{I}\mathcal{O}\mathfrak{l},\mathcal{O}\mathfrak{l}\right)& ⪯& {\varpi }_1\left(\mathcal{O}\mathfrak{l},\mathfrak{l}\right)d\left(\mathcal{O}\mathfrak{l},\mathfrak{l}\right)+{\varpi }_2\left(\mathcal{O}\mathfrak{l},\mathfrak{l}\right)\left(d\left(\mathcal{O}\mathfrak{l},\mathfrak{I}\mathcal{O}\mathfrak{l}\right)+d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)\right)\hfill \\ & & +{\varpi }_3\left(\mathcal{O}\mathfrak{l},\mathfrak{l}\right){\displaystyle \frac{d\left(\mathcal{O}\mathfrak{l},\mathfrak{I}\mathcal{O}\mathfrak{l}\right)d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)}{1+d\left(\mathcal{O}\mathfrak{l},\mathfrak{l}\right)}}\hfill \\ & & +{\varpi }_4\left(\mathcal{O}\mathfrak{l},\mathfrak{l}\right){\displaystyle \frac{d\left(\mathfrak{l},\mathfrak{I}\mathcal{O}\mathfrak{l}\right)d\left(\mathcal{O}\mathfrak{l},\mathcal{O}\mathfrak{l}\right)}{1+d\left(\mathcal{O}\mathfrak{l},\mathfrak{l}\right)}},\hfill \end{array}$$

$\forall \varkappa ,\mathfrak{l}\in \Omega$. Then the following estimates hold:

$$\begin{array}{ccc}\hfill \left|d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{I}\varkappa \right)\right|& \le & \left({\varpi }_1\left(\varkappa ,\mathfrak{I}\varkappa \right)+{\varpi }_2\left(\varkappa ,\mathfrak{I}\varkappa \right)\right)\left|d\left(\varkappa ,\mathfrak{I}\varkappa \right)\right|\hfill \\ & & +\left({\varpi }_2\left(\varkappa ,\mathfrak{I}\varkappa \right)+{\varpi }_3\left(\varkappa ,\mathfrak{I}\varkappa \right)\right)\left|d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{I}\varkappa \right)\right|\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \left|d\left(\mathfrak{I}\mathcal{O}\mathfrak{l},\mathcal{O}\mathfrak{l}\right)\right|& \le & \left({\varpi }_1\left(\mathcal{O}\mathfrak{l},\mathfrak{l}\right)+{\varpi }_2\left(\mathcal{O}\mathfrak{l},\mathfrak{l}\right)\right)\left|d\left(\mathcal{O}\mathfrak{l},\mathfrak{l}\right)\right|\hfill \\ & & +\left({\varpi }_2\left(\mathcal{O}\mathfrak{l},\mathfrak{l}\right)+{\varpi }_3\left(\mathcal{O}\mathfrak{l},\mathfrak{l}\right)\right)\left|d\left(\mathcal{O}\mathfrak{l},\mathfrak{I}\mathcal{O}\mathfrak{l}\right)\right|.\hfill \end{array}$$

Proof. 

We can express the inequality as follows

$$\begin{array}{ccc}\hfill \left|d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{I}\varkappa \right)\right|& \le & \left|\begin{array}{c}{\varpi }_1\left(\varkappa ,\mathfrak{I}\varkappa \right)d\left(\varkappa ,\mathfrak{I}\varkappa \right)+{\varpi }_2\left(\varkappa ,\mathfrak{I}\varkappa \right)\left(d\left(\varkappa ,\mathfrak{I}\varkappa \right)+d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{I}\varkappa \right)\right)\\ +{\varpi }_3\left(\varkappa ,\mathfrak{I}\varkappa \right){\displaystyle \frac{d\left(\varkappa ,\mathfrak{I}\varkappa \right)d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{I}\varkappa \right)}{1+d\left(\varkappa ,\mathfrak{I}\varkappa \right)}}\\ +{\varpi }_4\left(\varkappa ,\mathfrak{I}\varkappa \right){\displaystyle \frac{d\left(\mathfrak{I}\varkappa ,\mathfrak{I}\varkappa \right)d\left(\varkappa ,\mathcal{O}\mathfrak{I}\varkappa \right)}{1+d\left(\varkappa ,\mathfrak{I}\varkappa \right)}}\end{array}\right|\hfill \\ & \le & {\varpi }_1\left(\varkappa ,\mathfrak{I}\varkappa \right)\left|d\left(\varkappa ,\mathfrak{I}\varkappa \right)\right|\hfill \\ & & +{\varpi }_2\left(\varkappa ,\mathfrak{I}\varkappa \right)\left|d\left(\varkappa ,\mathfrak{I}\varkappa \right)\right|+{\varpi }_2\left(\varkappa ,\mathfrak{I}\varkappa \right)\left|d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{I}\varkappa \right)\right|\hfill \\ & & +{\varpi }_3\left(\varkappa ,\mathfrak{I}\varkappa \right)\left|{\displaystyle \frac{d\left(\varkappa ,\mathfrak{I}\varkappa \right)}{1+d\left(\varkappa ,\mathfrak{I}\varkappa \right)}}\right|\left|d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{I}\varkappa \right)\right|\hfill \\ & & +{\varpi }_4\left(\varkappa ,\mathfrak{I}\varkappa \right)\left|d\left(\mathfrak{I}\varkappa ,\mathfrak{I}\varkappa \right)\right|\left|{\displaystyle \frac{d\left(\varkappa ,\mathcal{O}\mathfrak{I}\varkappa \right)}{1+d\left(\varkappa ,\mathfrak{I}\varkappa \right)}}\right|.\hfill \end{array}$$

Simplifying the above, we obtain

$$\begin{array}{ccc}\hfill \left|d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{I}\varkappa \right)\right|& \le & \left({\varpi }_1\left(\varkappa ,\mathfrak{I}\varkappa \right)+{\varpi }_2\left(\varkappa ,\mathfrak{I}\varkappa \right)\right)\left|d\left(\varkappa ,\mathfrak{I}\varkappa \right)\right|\hfill \\ & & +\left({\varpi }_2\left(\varkappa ,\mathfrak{I}\varkappa \right)+{\varpi }_3\left(\varkappa ,\mathfrak{I}\varkappa \right)\right)\left|d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{I}\varkappa \right)\right|.\hfill \end{array}$$

Similarly, one can derive

$$\begin{array}{ccc}\hfill \left|d\left(\mathfrak{I}\mathcal{O}\mathfrak{l},\mathcal{O}\mathfrak{l}\right)\right|& \le & \left|\begin{array}{c}{\varpi }_1\left(\mathcal{O}\mathfrak{l},\mathfrak{l}\right)d\left(\mathcal{O}\mathfrak{l},\mathfrak{l}\right)+{\varpi }_2\left(\mathcal{O}\mathfrak{l},\mathfrak{l}\right)\left(d\left(\mathcal{O}\mathfrak{l},\mathfrak{I}\mathcal{O}\mathfrak{l}\right)+d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)\right)\\ +{\varpi }_3\left(\mathcal{O}\mathfrak{l},\mathfrak{l}\right){\displaystyle \frac{d\left(\mathcal{O}\mathfrak{l},\mathfrak{I}\mathcal{O}\mathfrak{l}\right)d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)}{1+d\left(\mathcal{O}\mathfrak{l},\mathfrak{l}\right)}}\\ +{\varpi }_4\left(\mathcal{O}\mathfrak{l},\mathfrak{l}\right){\displaystyle \frac{d\left(\mathfrak{l},\mathfrak{I}\mathcal{O}\mathfrak{l}\right)d\left(\mathcal{O}\mathfrak{l},\mathcal{O}\mathfrak{l}\right)}{1+d\left(\mathcal{O}\mathfrak{l},\mathfrak{l}\right)}}\end{array}\right|\hfill \\ & \le & {\varpi }_1\left(\mathcal{O}\mathfrak{l},\mathfrak{l}\right)\left|d\left(\mathcal{O}\mathfrak{l},\mathfrak{l}\right)\right|\hfill \\ & & +{\varpi }_2\left(\mathcal{O}\mathfrak{l},\mathfrak{l}\right)\left|d\left(\mathcal{O}\mathfrak{l},\mathfrak{I}\mathcal{O}\mathfrak{l}\right)\right|++{\varpi }_2\left(\mathcal{O}\mathfrak{l},\mathfrak{l}\right)\left|d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)\right|\hfill \\ & & +{\varpi }_3\left(\mathcal{O}\mathfrak{l},\mathfrak{l}\right)\left|d\left(\mathcal{O}\mathfrak{l},\mathfrak{I}\mathcal{O}\mathfrak{l}\right)\right|\left|{\displaystyle \frac{d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)}{1+d\left(\mathcal{O}\mathfrak{l},\mathfrak{l}\right)}}\right|\hfill \\ & & +{\varpi }_4\left(\mathcal{O}\mathfrak{l},\mathfrak{l}\right)\left|{\displaystyle \frac{d\left(\mathfrak{l},\mathfrak{I}\mathcal{O}\mathfrak{l}\right)}{1+d\left(\mathcal{O}\mathfrak{l},\mathfrak{l}\right)}}\right|\left|d\left(\mathcal{O}\mathfrak{l},\mathcal{O}\mathfrak{l}\right)\right|.\hfill \end{array}$$

Thus, it follows that

$$\begin{array}{ccc}\hfill \left|d\left(\mathfrak{I}\mathcal{O}\mathfrak{l},\mathcal{O}\mathfrak{l}\right)\right|& \le & \left({\varpi }_1\left(\mathcal{O}\mathfrak{l},\mathfrak{l}\right)+{\varpi }_2\left(\mathcal{O}\mathfrak{l},\mathfrak{l}\right)\right)\left|d\left(\mathcal{O}\mathfrak{l},\mathfrak{l}\right)\right|\hfill \\ & & +\left({\varpi }_2\left(\mathcal{O}\mathfrak{l},\mathfrak{l}\right)+{\varpi }_3\left(\mathcal{O}\mathfrak{l},\mathfrak{l}\right)\right)\left|d\left(\mathcal{O}\mathfrak{l},\mathfrak{I}\mathcal{O}\mathfrak{l}\right)\right|.\hfill \end{array}$$

 □

In what follows, we present our main theorem, which establishes sufficient conditions ensuring the existence and uniqueness of a CFP for two self-mappings in a complete CVSMS under a generalized contractive framework involving four control functions of two variables.

Theorem 1.

Let $\left(\Omega ,d\right)$ be a complete CVSMS and $\mathfrak{I},\mathcal{O}:\Omega$$\to \Omega$. Assume that there exist functions ${\varpi }_1,{\varpi }_2,{\varpi }_3,{\varpi }_4:\Omega \times \Omega \to [0,1)$ satisfying the following conditions:

(a) 

$$\begin{array}{ccc}\hfill {\varpi }_1\left(\mathcal{O}\mathfrak{I}\varkappa ,\mathfrak{l}\right)& \le & {\varpi }_1\left(\varkappa ,\mathfrak{l}\right),{\varpi }_1\left(\varkappa ,\mathfrak{I}\mathcal{O}\mathfrak{l}\right)\le {\varpi }_1\left(\varkappa ,\mathfrak{l}\right)\hfill \\ \hfill {\varpi }_2\left(\mathcal{O}\mathfrak{I}\varkappa ,\mathfrak{l}\right)& \le & {\varpi }_2\left(\varkappa ,\mathfrak{l}\right),{\varpi }_2\left(\varkappa ,\mathfrak{I}\mathcal{O}\mathfrak{l}\right)\le {\varpi }_2\left(\varkappa ,\mathfrak{l}\right)\hfill \\ \hfill {\varpi }_3\left(\mathcal{O}\mathfrak{I}\varkappa ,\mathfrak{l}\right)& \le & {\varpi }_3\left(\varkappa ,\mathfrak{l}\right),{\varpi }_3\left(\varkappa ,\mathfrak{I}\mathcal{O}\mathfrak{l}\right)\le {\varpi }_3\left(\varkappa ,\mathfrak{l}\right)\hfill \\ \hfill {\varpi }_4\left(\mathcal{O}\mathfrak{I}\varkappa ,\mathfrak{l}\right)& \le & {\varpi }_4\left(\varkappa ,\mathfrak{l}\right),{\varpi }_4\left(\varkappa ,\mathfrak{I}\mathcal{O}\mathfrak{l}\right)\le {\varpi }_4\left(\varkappa ,\mathfrak{l}\right),\hfill \end{array}$$

(b) 

$${\varpi }_1\left(\varkappa ,\mathfrak{l}\right)+2{\varpi }_2\left(\varkappa ,\mathfrak{l}\right)+{\varpi }_3\left(\varkappa ,\mathfrak{l}\right)+{\varpi }_4\left(\varkappa ,\mathfrak{l}\right)<1,$$

(c) 

$$\begin{array}{ccc}\hfill d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{l}\right)& ⪯& {\varpi }_1\left(\varkappa ,\mathfrak{l}\right)d\left(\varkappa ,\mathfrak{l}\right)+{\varpi }_2\left(\varkappa ,\mathfrak{l}\right)\left(d\left(\varkappa ,\mathfrak{I}\varkappa \right)+d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)\right)\hfill \\ & & +{\varpi }_3\left(\varkappa ,\mathfrak{l}\right){\displaystyle \frac{d\left(\varkappa ,\mathfrak{I}\varkappa \right)d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}}+{\varpi }_4\left(\varkappa ,\mathfrak{l}\right){\displaystyle \frac{d\left(\mathfrak{l},\mathfrak{I}\varkappa \right)d\left(\varkappa ,\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}},\hfill \end{array}$$

(2)

for all $\varkappa ,\mathfrak{l}\in \Omega .$ Then $\mathfrak{I}$ and $\mathcal{O}$ admit a unique CFP.

Proof. 

Let ${\varkappa }_0\in \Omega$ be arbitrary. Consider the sequence {${\varkappa }_{\mathfrak{p}}$} constructed inductively following the approach in (1). Using (2) and Lemma 3, we obtain

$$\begin{array}{ccc}\hfill \left|d\left({\varkappa }_{2\mathfrak{p}+1},{\varkappa }_{2\mathfrak{p}}\right)\right|& =& \left|d\left(\mathfrak{I}\mathcal{O}{\varkappa }_{2\mathfrak{p}-1},\mathcal{O}{\varkappa }_{2\mathfrak{p}-1}\right)\right|\hfill \\ & \le & \left({\varpi }_1\left(\mathcal{O}{\varkappa }_{2\mathfrak{p}-1},{\varkappa }_{2\mathfrak{p}-1}\right)+{\varpi }_2\left(\mathcal{O}{\varkappa }_{2\mathfrak{p}-1},{\varkappa }_{2\mathfrak{p}-1}\right)\right)\left|d\left(\mathcal{O}{\varkappa }_{2\mathfrak{p}-1},{\varkappa }_{2\mathfrak{p}-1}\right)\right|\hfill \\ & & +\left({\varpi }_2\left(\mathcal{O}{\varkappa }_{2\mathfrak{p}-1},{\varkappa }_{2\mathfrak{p}-1}\right)+{\varpi }_3\left(\mathcal{O}{\varkappa }_{2\mathfrak{p}-1},{\varkappa }_{2\mathfrak{p}-1}\right)\right)\left|d\left(\mathcal{O}{\varkappa }_{2\mathfrak{p}-1},\mathfrak{I}\mathcal{O}{\varkappa }_{2\mathfrak{p}-1}\right)\right|\hfill \\ & =& \left({\varpi }_1\left({\varkappa }_{2\mathfrak{p}},{\varkappa }_{2\mathfrak{p}-1}\right)+{\varpi }_2\left({\varkappa }_{2\mathfrak{p}},{\varkappa }_{2\mathfrak{p}-1}\right)\right)\left|d\left({\varkappa }_{2\mathfrak{p}},{\varkappa }_{2\mathfrak{p}-1}\right)\right|\hfill \\ & & +\left({\varpi }_2\left({\varkappa }_{2\mathfrak{p}},{\varkappa }_{2\mathfrak{p}-1}\right)+{\varpi }_3\left({\varkappa }_{2\mathfrak{p}},{\varkappa }_{2\mathfrak{p}-1}\right)\right)\left|d\left({\varkappa }_{2\mathfrak{p}},{\varkappa }_{2\mathfrak{p}+1}\right)\right|.\hfill \end{array}$$

By Proposition 1, this yields

$$\begin{array}{ccc}\hfill \left|d\left({\varkappa }_{2\mathfrak{p}+1},{\varkappa }_{2\mathfrak{p}}\right)\right|& \le & \left({\varpi }_1\left({\varkappa }_0,{\varkappa }_{2\mathfrak{p}-1}\right)+{\varpi }_2\left({\varkappa }_0,{\varkappa }_{2\mathfrak{p}-1}\right)\right)\left|d\left({\varkappa }_{2\mathfrak{p}},{\varkappa }_{2\mathfrak{p}-1}\right)\right|\hfill \\ & & +\left({\varpi }_2\left({\varkappa }_0,{\varkappa }_{2\mathfrak{p}-1}\right)+{\varpi }_3\left({\varkappa }_0,{\varkappa }_{2\mathfrak{p}-1}\right)\right)\left|d\left({\varkappa }_{2\mathfrak{p}},{\varkappa }_{2\mathfrak{p}+1}\right)\right|\hfill \\ & \le & \left({\varpi }_1\left({\varkappa }_0,{\varkappa }_1\right)+{\varpi }_2\left({\varkappa }_0,{\varkappa }_1\right)\right)\left|d\left({\varkappa }_{2\mathfrak{p}},{\varkappa }_{2\mathfrak{p}-1}\right)\right|\hfill \\ & & +\left({\varpi }_2\left({\varkappa }_0,{\varkappa }_1\right)+{\varpi }_3\left({\varkappa }_0,{\varkappa }_1\right)\right)\left|d\left({\varkappa }_{2\mathfrak{p}},{\varkappa }_{2\mathfrak{p}+1}\right)\right|.\hfill \end{array}$$

Hence,

$$\left|d\left({\varkappa }_{2\mathfrak{p}+1},{\varkappa }_{2\mathfrak{p}}\right)\right|\le {\displaystyle \frac{{\varpi }_1\left({\varkappa }_0,{\varkappa }_1\right)+{\varpi }_2\left({\varkappa }_0,{\varkappa }_1\right)}{1-{\varpi }_2\left({\varkappa }_0,{\varkappa }_1\right)+{\varpi }_3\left({\varkappa }_0,{\varkappa }_1\right)}}\left|d\left({\varkappa }_{2\mathfrak{p}},{\varkappa }_{2\mathfrak{p}-1}\right)\right|.$$

(3)

Likewise, by (2) and Lemma 3, we get

$$\begin{array}{ccc}\hfill \left|d\left({\varkappa }_{2\mathfrak{p}+2},{\varkappa }_{2\mathfrak{p}+1}\right)\right|& =& \left|d\left(\mathcal{O}\mathfrak{I}{\varkappa }_{2\mathfrak{p}},\mathfrak{I}{\varkappa }_{2\mathfrak{p}}\right)\right|\hfill \\ & =& \left|d\left(\mathfrak{I}{\varkappa }_{2\mathfrak{p}},\mathcal{O}\mathfrak{I}{\varkappa }_{2\mathfrak{p}}\right)\right|\hfill \\ & \le & \left({\varpi }_1\left({\varkappa }_{2\mathfrak{p}},\mathfrak{I}{\varkappa }_{2\mathfrak{p}}\right)+{\varpi }_2\left({\varkappa }_{2\mathfrak{p}},\mathfrak{I}{\varkappa }_{2\mathfrak{p}}\right)\right)\left|d\left({\varkappa }_{2\mathfrak{p}},\mathfrak{I}{\varkappa }_{2\mathfrak{p}}\right)\right|\hfill \\ & & +\left({\varpi }_2\left({\varkappa }_{2\mathfrak{p}},\mathfrak{I}{\varkappa }_{2\mathfrak{p}}\right)+{\varpi }_3\left({\varkappa }_{2\mathfrak{p}},\mathfrak{I}{\varkappa }_{2\mathfrak{p}}\right)\right)\left|d\left(\mathfrak{I}{\varkappa }_{2\mathfrak{p}},\mathcal{O}\mathfrak{I}{\varkappa }_{2\mathfrak{p}}\right)\right|\hfill \\ & =& \left({\varpi }_1\left({\varkappa }_{2\mathfrak{p}},{\varkappa }_{2\mathfrak{p}+1}\right)+{\varpi }_2\left({\varkappa }_{2\mathfrak{p}},{\varkappa }_{2\mathfrak{p}+1}\right)\right)\left|d\left({\varkappa }_{2\mathfrak{p}},{\varkappa }_{2\mathfrak{p}+1}\right)\right|\hfill \\ & & +\left({\varpi }_2\left({\varkappa }_{2\mathfrak{p}},{\varkappa }_{2\mathfrak{p}+1}\right)+{\varpi }_3\left({\varkappa }_{2\mathfrak{p}},{\varkappa }_{2\mathfrak{p}+1}\right)\right)\left|d\left({\varkappa }_{2\mathfrak{p}+1},{\varkappa }_{2\mathfrak{p}+2}\right)\right|,\hfill \end{array}$$

Applying Proposition 1, we obtain

$$\begin{array}{ccc}\hfill \left|d\left({\varkappa }_{2\mathfrak{p}+2},{\varkappa }_{2\mathfrak{p}+1}\right)\right|& \le & \left({\varpi }_1\left({\varkappa }_0,{\varkappa }_{2\mathfrak{p}+1}\right)+{\varpi }_2\left({\varkappa }_0,{\varkappa }_{2\mathfrak{p}+1}\right)\right)\left|d\left({\varkappa }_{2\mathfrak{p}},{\varkappa }_{2\mathfrak{p}+1}\right)\right|\hfill \\ & & +\left({\varpi }_2\left({\varkappa }_0,{\varkappa }_{2\mathfrak{p}+1}\right)+{\varpi }_3\left({\varkappa }_0,{\varkappa }_{2\mathfrak{p}+1}\right)\right)\left|d\left({\varkappa }_{2\mathfrak{p}+1},{\varkappa }_{2\mathfrak{p}+2}\right)\right|\hfill \\ & \le & \left({\varpi }_1\left({\varkappa }_0,{\varkappa }_1\right)+{\varpi }_2\left({\varkappa }_0,{\varkappa }_1\right)\right)\left|d\left({\varkappa }_{2\mathfrak{p}},{\varkappa }_{2\mathfrak{p}+1}\right)\right|\hfill \\ & & +\left({\varpi }_2\left({\varkappa }_0,{\varkappa }_1\right)+{\varpi }_3\left({\varkappa }_0,{\varkappa }_1\right)\right)\left|d\left({\varkappa }_{2\mathfrak{p}+1},{\varkappa }_{2\mathfrak{p}+2}\right)\right|,\hfill \end{array}$$

which leads to

$$\left|d\left({\varkappa }_{2\mathfrak{p}+2},{\varkappa }_{2\mathfrak{p}+1}\right)\right|\le {\displaystyle \frac{{\varpi }_1\left({\varkappa }_0,{\varkappa }_1\right)+{\varpi }_2\left({\varkappa }_0,{\varkappa }_1\right)}{1-{\varpi }_2\left({\varkappa }_0,{\varkappa }_1\right)-{\varpi }_3\left({\varkappa }_0,{\varkappa }_1\right)}}\left|d\left({\varkappa }_{2\mathfrak{p}},{\varkappa }_{2\mathfrak{p}+1}\right)\right|.$$

(4)

Rewriting (3) and (4), we obtain

$$\left|d\left({\varkappa }_{2\mathfrak{p}},{\varkappa }_{2\mathfrak{p}+1}\right)\right|\le {\displaystyle \frac{{\varpi }_1\left({\varkappa }_0,{\varkappa }_1\right)+{\varpi }_2\left({\varkappa }_0,{\varkappa }_1\right)}{1-{\varpi }_2\left({\varkappa }_0,{\varkappa }_1\right)+{\varpi }_3\left({\varkappa }_0,{\varkappa }_1\right)}}\left|d\left({\varkappa }_{2\mathfrak{p}-1},{\varkappa }_{2\mathfrak{p}}\right)\right|,$$

(5)

and

$$\left|d\left({\varkappa }_{2\mathfrak{p}+1},{\varkappa }_{2\mathfrak{p}+2}\right)\right|\le {\displaystyle \frac{{\varpi }_1\left({\varkappa }_0,{\varkappa }_1\right)+{\varpi }_2\left({\varkappa }_0,{\varkappa }_1\right)}{1-{\varpi }_2\left({\varkappa }_0,{\varkappa }_1\right)-{\varpi }_3\left({\varkappa }_0,{\varkappa }_1\right)}}\left|d\left({\varkappa }_{2\mathfrak{p}+1},{\varkappa }_{2\mathfrak{p}}\right)\right|.$$

(6)

Let $\mu =$${\displaystyle \frac{{\varpi }_1\left({\varkappa }_0,{\varkappa }_1\right)+{\varpi }_2\left({\varkappa }_0,{\varkappa }_1\right)}{1-{\varpi }_2\left({\varkappa }_0,{\varkappa }_1\right)-{\varpi }_3\left({\varkappa }_0,{\varkappa }_1\right)}}<1.$ Then, from (5) and (6), we have

$$\left|d\left({\varkappa }_{2\mathfrak{p}},{\varkappa }_{2\mathfrak{p}+1}\right)\right|\le \mu \left|d\left({\varkappa }_{2\mathfrak{p}-1},{\varkappa }_{2\mathfrak{p}}\right)\right|$$

(7)

and

$$\left|d\left({\varkappa }_{2\mathfrak{p}+1},{\varkappa }_{2\mathfrak{p}+2}\right)\right|\le \mu \left|d\left({\varkappa }_{2\mathfrak{p}+1},{\varkappa }_{2\mathfrak{p}}\right)\right|.$$

(8)

Combining (7) and (8), we conclude that

$$\left|d\left({\varkappa }_{\mathfrak{p}},{\varkappa }_{\mathfrak{p}+1}\right)\right|\le \mu \left|d\left({\varkappa }_{\mathfrak{p}-1},{\varkappa }_{\mathfrak{p}}\right)\right|,$$

∀$\mathfrak{p}\in \mathbb{N}\cup \left\{0\right\}.$ Recursively, we can generate a sequence $\left\{{\varkappa }_{\mathfrak{p}}\right\}$ in $\Omega$ such that

$$\left|d\left({\varkappa }_{\mathfrak{p}},{\varkappa }_{\mathfrak{p}+1}\right)\right|\le \mu \left|d\left({\varkappa }_{\mathfrak{p}-1},{\varkappa }_{\mathfrak{p}}\right)\right|\le {\mu }^2\left|d\left({\varkappa }_{\mathfrak{p}-2},{\varkappa }_{\mathfrak{p}-1}\right)\right|\le ···\le {\mu }^{\mathfrak{p}}\left|d\left({\varkappa }_0,{\varkappa }_1\right)\right|,$$

that is,

$$\left|d\left({\varkappa }_{\mathfrak{p}},{\varkappa }_{\mathfrak{p}+1}\right)\right|\le {\mu }^{\mathfrak{p}}\left|d\left({\varkappa }_0,{\varkappa }_1\right)\right|,$$

(9)

∀$\mathfrak{p}\in \mathbb{N}\cup \left\{0\right\}.$ For $m>\mathfrak{p}$, we obtain

$$\begin{array}{ccc}\hfill \left|d\left({\varkappa }_{\mathfrak{p}},{\varkappa }_m\right)\right|& \le & \left|d\left({\varkappa }_{\mathfrak{p}},{\varkappa }_{\mathfrak{p}+1}\right)\right|+\left|d\left({\varkappa }_{\mathfrak{p}+1},{\varkappa }_m\right)\right|+\aleph \left|d\left({\varkappa }_{\mathfrak{p}},{\varkappa }_{\mathfrak{p}+1}\right)\right|\left|d\left({\varkappa }_{\mathfrak{p}+1},{\varkappa }_m\right)\right|\hfill \\ & =& \left|d\left({\varkappa }_{\mathfrak{p}},{\varkappa }_{\mathfrak{p}+1}\right)\right|+\left(1+\aleph \left|d\left({\varkappa }_{\mathfrak{p}},{\varkappa }_{\mathfrak{p}+1}\right)\right|\right)\left|d\left({\varkappa }_{\mathfrak{p}+1},{\varkappa }_m\right)\right|.\hfill \end{array}$$

(10)

Applying the triangle inequality ($D_3$) once again to $\left|d\left({\varkappa }_{\mathfrak{p}+1},{\varkappa }_m\right)\right|,$ we get

$$\begin{array}{ccc}\hfill \left|d\left({\varkappa }_{\mathfrak{p}+1},{\varkappa }_m\right)\right|& \le & \left|d\left({\varkappa }_{\mathfrak{p}+1},{\varkappa }_{\mathfrak{p}+2}\right)\right|+\left|d\left({\varkappa }_{\mathfrak{p}+2},{\varkappa }_m\right)\right|+\aleph \left|d\left({\varkappa }_{\mathfrak{p}+1},{\varkappa }_{\mathfrak{p}+2}\right)\right|\left|d\left({\varkappa }_{\mathfrak{p}+2},{\varkappa }_m\right)\right|\hfill \\ & =& \left|d\left({\varkappa }_{\mathfrak{p}+1},{\varkappa }_{\mathfrak{p}+2}\right)\right|+\left(1+\aleph \left|d\left({\varkappa }_{\mathfrak{p}+1},{\varkappa }_{\mathfrak{p}+2}\right)\right|\right)\left|d\left({\varkappa }_{\mathfrak{p}+2},{\varkappa }_m\right)\right|.\hfill \end{array}$$

Proceeding similarly for the subsequent terms, we have

$$\begin{array}{ccc}\hfill \left|d\left({\varkappa }_{\mathfrak{p}+2},{\varkappa }_m\right)\right|& \le & \left|d\left({\varkappa }_{\mathfrak{p}+2},{\varkappa }_{\mathfrak{p}+3}\right)\right|+\left|d\left({\varkappa }_{\mathfrak{p}+3},{\varkappa }_m\right)\right|+\aleph \left|d\left({\varkappa }_{\mathfrak{p}+2},{\varkappa }_{\mathfrak{p}+3}\right)\right|\left|d\left({\varkappa }_{\mathfrak{p}+3},{\varkappa }_m\right)\right|\hfill \\ & =& \left|d\left({\varkappa }_{\mathfrak{p}+2},{\varkappa }_{\mathfrak{p}+3}\right)\right|+\left(1+\aleph \left|d\left({\varkappa }_{\mathfrak{p}+2},{\varkappa }_{\mathfrak{p}+3}\right)\right|\right)\left|d\left({\varkappa }_{\mathfrak{p}+3},{\varkappa }_m\right)\right|.\hfill \end{array}$$

Continuing this recursive process, we obtain

$$\begin{array}{ccc}\hfill \left|d\left({\varkappa }_{m-2},{\varkappa }_m\right)\right|& \le & \left|d\left({\varkappa }_{m-2},{\varkappa }_{m-1}\right)\right|+\left|d\left({\varkappa }_{m-1},{\varkappa }_m\right)\right|+\aleph \left|d\left({\varkappa }_{m-2},{\varkappa }_{m-1}\right)\right|\left|d\left({\varkappa }_{m-1},{\varkappa }_m\right)\right|\hfill \\ & =& \left|d\left({\varkappa }_{m-2},{\varkappa }_{m-1}\right)\right|+\left(1+\aleph \left|d\left({\varkappa }_{m-2},{\varkappa }_{m-1}\right)\right|\right)\left|d\left({\varkappa }_{m-1},{\varkappa }_m\right)\right|.\hfill \end{array}$$

By successively substituting each inequality into its predecessor in (10) and simplifying, we derive

$$\left|d\left({\varkappa }_{\mathfrak{p}},{\varkappa }_m\right)\right|\le \sum _{k=\mathfrak{p}}^{m-1}\left|d\left({\varkappa }_{\mathfrak{p}},{\varkappa }_{\mathfrak{p}+1}\right)\right|\prod _{j=\mathfrak{p}}^{k-1}\left(1+\aleph \left|d\left({\varkappa }_j,{\varkappa }_{j+1}\right)\right|\right).$$

Using inequality (9), it follows that

$$\left|d\left({\varkappa }_{\mathfrak{p}},{\varkappa }_m\right)\right|\le \left|d\left({\varkappa }_0,{\varkappa }_1\right)\right|\sum _{k=\mathfrak{p}}^{m-1}{\mu }^{\mathfrak{p}}\prod _{j=\mathfrak{p}}^{k-1}\left(1+\aleph {\mu }^j\left|d\left({\varkappa }_0,{\varkappa }_1\right)\right|\right).$$

(11)

Since $\left(1+\aleph {\mu }^j\left|d\left({\varkappa }_0,{\varkappa }_1\right)\right|\right)\ge 1,$ for all $j,$ we have

$$\prod _{j=\mathfrak{p}}^{k-1}\left(1+\aleph {\mu }^j\left|d\left({\varkappa }_0,{\varkappa }_1\right)\right|\right)\ge 1.$$

With the product term having a lower bound of 1, the summation becomes

$$\left|d\left({\varkappa }_0,{\varkappa }_1\right)\right|\sum _{k=\mathfrak{p}}^{m-1}{\mu }^{\mathfrak{p}}\prod _{j=\mathfrak{p}}^{k-1}\left(1+\aleph {\mu }^j\left|d\left({\varkappa }_0,{\varkappa }_1\right)\right|\right)\ge \left|d\left({\varkappa }_0,{\varkappa }_1\right)\right|\sum _{k=\mathfrak{p}}^{m-1}{\mu }^{\mathfrak{p}}.$$

(12)

The series ${\displaystyle \sum _{k=\mathfrak{p}}^{m-1}}{\mu }^{\mathfrak{p}}$ forms a finite geometric progression with initial term ${\mu }^{\mathfrak{p}}$ and the common ratio $\mu .$ The sum of a finite geometric progression is expressed as

$$\sum _{k=\mathfrak{p}}^{m-1}{\mu }^{\mathfrak{p}}={\mu }^{\mathfrak{p}}{\displaystyle \frac{\left(1-{\mu }^{m-\mathfrak{p}}\right)}{1-\mu }}.$$

As $m\to \infty ,$${\mu }^{m-\mathfrak{p}}\to 0.$ Hence, the series converges to ${\displaystyle \frac{{\mu }^{\mathfrak{p}}}{1-\mu }},$ i.e.,

$$\underset{m\to \infty }{lim}{\mu }^{\mathfrak{p}}{\displaystyle \frac{\left(1-{\mu }^{m-\mathfrak{p}}\right)}{1-\mu }}={\displaystyle \frac{{\mu }^{\mathfrak{p}}}{1-\mu }}.$$

Because $\mu <1,$ the term ${\displaystyle \frac{{\mu }^{\mathfrak{p}}}{1-\mu }}$ tends to 0 as $\mathfrak{p}\to \infty .$ Taking the limit $\mathfrak{p}\to \infty$ in inequality (11) andand using the previous facts, we obtain

$$\underset{\mathfrak{p}\to \infty }{lim}\left|d\left({\varkappa }_{\mathfrak{p}},{\varkappa }_m\right)\right|=0.$$

Therefore by Lemma 2 $\left\{{\varkappa }_{\mathfrak{p}}\right\}$ is Cauchy. The completeness of $\Omega$ implies the existence of ${\varkappa }^{*}$ such that ${\varkappa }_{\mathfrak{p}}\to {\varkappa }^{*}\in \Omega$ as $\mathfrak{p}\to \infty .$ Thus,

$$\underset{\mathfrak{p}\to \infty }{lim}{\varkappa }_{\mathfrak{p}}={\varkappa }^{*}.$$

Next, we demonstrate that ${\varkappa }^{*}$ is a FP of $\mathfrak{I}.$ Using (2), we have

$$\begin{array}{ccc}\hfill d\left({\varkappa }^{*},\mathfrak{I}{\varkappa }^{*}\right)& ⪯& d\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+2}\right)+d\left({\varkappa }_{2\mathfrak{p}+2},\mathfrak{I}{\varkappa }^{*}\right)+\aleph d\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+2}\right)d\left({\varkappa }_{2\mathfrak{p}+2},\mathfrak{I}{\varkappa }^{*}\right)\hfill \\ & =& d\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+2}\right)+d\left(\mathcal{O}{\varkappa }_{2\mathfrak{p}+1},\mathfrak{I}{\varkappa }^{*}\right)+\aleph d\left({\varkappa }^{*},\mathcal{O}{\varkappa }_{2\mathfrak{p}+1}\right)d\left(\mathcal{O}{\varkappa }_{2\mathfrak{p}+1},\mathfrak{I}{\varkappa }^{*}\right)\hfill \\ & =& d\left({\varkappa }^{*},\mathcal{O}{\varkappa }_{2\mathfrak{p}+1}\right)+d\left(\mathfrak{I}{\varkappa }^{*},\mathcal{O}{\varkappa }_{2\mathfrak{p}+1}\right)+\aleph d\left({\varkappa }^{*},\mathcal{O}{\varkappa }_{2\mathfrak{p}+1}\right)d\left({\varkappa }_{2\mathfrak{p}+2},\mathfrak{I}{\varkappa }^{*}\right)\hfill \\ & ⪯& d\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+2}\right)+\left(\begin{array}{c}{\varpi }_1\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+1}\right)d\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+1}\right)\\ +{\varpi }_2\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+1}\right)\left(d\left({\varkappa }^{*},\mathfrak{I}{\varkappa }^{*}\right)+d\left({\varkappa }_{2\mathfrak{p}+1},\mathcal{O}{\varkappa }_{2\mathfrak{p}+1}\right)\right)\\ +{\varpi }_3\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+1}\right){\displaystyle \frac{d\left({\varkappa }^{*},\mathfrak{I}{\varkappa }^{*}\right)d\left({\varkappa }_{2\mathfrak{p}+1},\mathcal{O}{\varkappa }_{2\mathfrak{p}+1}\right)}{1+d\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+1}\right)}}\\ +{\varpi }_4\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+1}\right){\displaystyle \frac{d\left({\varkappa }_{2\mathfrak{p}+1},\mathfrak{I}{\varkappa }^{*}\right)d\left({\varkappa }^{*},\mathcal{O}{\varkappa }_{2\mathfrak{p}+1}\right)}{1+d\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+1}\right)}}\end{array}\right)\hfill \\ & & +\aleph d\left({\varkappa }^{*},\mathcal{O}{\varkappa }_{2\mathfrak{p}+1}\right)d\left({\varkappa }_{2\mathfrak{p}+2},\mathfrak{I}{\varkappa }^{*}\right)\hfill \\ & =& d\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+2}\right)+\left(\begin{array}{c}{\varpi }_1\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+1}\right)d\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+1}\right)\\ +{\varpi }_2\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+1}\right)\left(d\left({\varkappa }^{*},\mathfrak{I}{\varkappa }^{*}\right)+d\left({\varkappa }_{2\mathfrak{p}+1},{\varkappa }_{2\mathfrak{p}+2}\right)\right)\\ +{\varpi }_3\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+1}\right){\displaystyle \frac{d\left({\varkappa }^{*},\mathfrak{I}{\varkappa }^{*}\right)d\left({\varkappa }_{2\mathfrak{p}+1},{\varkappa }_{2\mathfrak{p}+2}\right)}{1+d\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+1}\right)}}\\ +{\varpi }_4\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+1}\right){\displaystyle \frac{d\left({\varkappa }_{2\mathfrak{p}+1},\mathfrak{I}{\varkappa }^{*}\right)d\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+2}\right)}{1+d\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+1}\right)}}\end{array}\right)\hfill \\ & & +\aleph d\left({\varkappa }^{*},\mathcal{O}{\varkappa }_{2\mathfrak{p}+1}\right)d\left({\varkappa }_{2\mathfrak{p}+2},\mathfrak{I}{\varkappa }^{*}\right).\hfill \end{array}$$

which implies by Proposition 1 that

$$\begin{array}{ccc}\hfill \left|d\left({\varkappa }^{*},\mathfrak{I}{\varkappa }^{*}\right)\right|& \le & \left|d\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+2}\right)\right|+\left(\begin{array}{c}{\varpi }_1\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+1}\right)\left|d\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+1}\right)\right|\\ +{\varpi }_2\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+1}\right)\left(\left|d\left({\varkappa }^{*},\mathfrak{I}{\varkappa }^{*}\right)\right|+\left|d\left({\varkappa }_{2\mathfrak{p}+1},{\varkappa }_{2\mathfrak{p}+2}\right)\right|\right)\\ +{\varpi }_3\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+1}\right)\left|d\left({\varkappa }^{*},\mathfrak{I}{\varkappa }^{*}\right)\right|\left|{\displaystyle \frac{d\left({\varkappa }_{2\mathfrak{p}+1},{\varkappa }_{2\mathfrak{p}+2}\right)}{1+d\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+1}\right)}}\right|\\ +{\varpi }_4\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+1}\right)\left|{\displaystyle \frac{d\left({\varkappa }_{2\mathfrak{p}+1},\mathfrak{I}{\varkappa }^{*}\right)}{1+d\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+1}\right)}}\right|\left|d\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+2}\right)\right|\end{array}\right)\hfill \\ & & +\aleph \left|d\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+2}\right)\right|\left|d\left({\varkappa }_{2\mathfrak{p}+2},\mathfrak{I}{\varkappa }^{*}\right)\right|.\hfill \\ & \le & \left|d\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+2}\right)\right|+\left(\begin{array}{c}{\varpi }_1\left({\varkappa }^{*},{\varkappa }_1\right)\left|d\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+1}\right)\right|\\ +{\varpi }_2\left({\varkappa }^{*},{\varkappa }_1\right)\left(\left|d\left({\varkappa }^{*},\mathfrak{I}{\varkappa }^{*}\right)\right|+\left|d\left({\varkappa }_{2\mathfrak{p}+1},{\varkappa }_{2\mathfrak{p}+2}\right)\right|\right)\\ +{\varpi }_3\left({\varkappa }^{*},{\varkappa }_1\right)\left|d\left({\varkappa }^{*},\mathfrak{I}{\varkappa }^{*}\right)\right|\left|{\displaystyle \frac{d\left({\varkappa }_{2\mathfrak{p}+1},{\varkappa }_{2\mathfrak{p}+2}\right)}{1+d\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+1}\right)}}\right|\\ +{\varpi }_4\left({\varkappa }^{*},{\varkappa }_1\right)\left|{\displaystyle \frac{d\left({\varkappa }_{2\mathfrak{p}+1},\mathfrak{I}{\varkappa }^{*}\right)}{1+d\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+1}\right)}}\right|\left|d\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+2}\right)\right|\end{array}\right)\hfill \\ & & +\aleph \left|d\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+2}\right)\right|\left|d\left({\varkappa }_{2\mathfrak{p}+2},\mathfrak{I}{\varkappa }^{*}\right)\right|.\hfill \end{array}$$

Letting $\mathfrak{p}\to \infty ,$ we have $\left(1-{\varpi }_2\left({\varkappa }^{*},{\varkappa }_1\right)\right)\left|d\left({\varkappa }^{*},\mathfrak{I}{\varkappa }^{*}\right)\right|=0.$ But $\left(1-{\varpi }_2\left({\varkappa }^{*},{\varkappa }_1\right)\right)\ne 0,$ thus $\left|d\left({\varkappa }^{*},\mathfrak{I}{\varkappa }^{*}\right)\right|=0$ implies ${\varkappa }^{*}=\mathfrak{I}{\varkappa }^{*}.$ To demonstrate that ${\varkappa }^{*}$ is a FP of $\mathcal{O},$ we observe from (2) that

$$\begin{array}{ccc}\hfill d\left({\varkappa }^{*},\mathcal{O}{\varkappa }^{*}\right)& ⪯& d\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+1}\right)+d\left({\varkappa }_{2\mathfrak{p}+1},\mathcal{O}{\varkappa }^{*}\right)+\aleph d\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+1}\right)d\left({\varkappa }_{2\mathfrak{p}+1},\mathcal{O}{\varkappa }^{*}\right)\hfill \\ & ⪯& d\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+1}\right)+d\left(\mathfrak{I}{\varkappa }_{2\mathfrak{p}},\mathcal{O}{\varkappa }^{*}\right)+\aleph d\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+1}\right)d\left({\varkappa }_{2\mathfrak{p}+1},\mathcal{O}{\varkappa }^{*}\right)\hfill \\ & ⪯& d\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+1}\right)+\left(\begin{array}{c}{\varpi }_1\left({\varkappa }_{2\mathfrak{p}},{\varkappa }^{*}\right)d\left({\varkappa }_{2\mathfrak{p}},{\varkappa }^{*}\right)\\ +{\varpi }_2\left({\varkappa }_{2\mathfrak{p}},{\varkappa }^{*}\right)\left(d\left({\varkappa }_{2\mathfrak{p}},\mathfrak{I}{\varkappa }_{2\mathfrak{p}}\right)+d\left({\varkappa }^{*},\mathcal{O}{\varkappa }^{*}\right)\right)\\ +{\varpi }_3\left({\varkappa }_{2\mathfrak{p}},{\varkappa }^{*}\right){\displaystyle \frac{d\left({\varkappa }_{2\mathfrak{p}},\mathfrak{I}{\varkappa }_{2\mathfrak{p}}\right)d\left({\varkappa }^{*},\mathcal{O}{\varkappa }^{*}\right)}{1+d\left({\varkappa }_{2\mathfrak{p}},{\varkappa }^{*}\right)}}\\ +{\varpi }_4\left({\varkappa }_{2\mathfrak{p}},{\varkappa }^{*}\right){\displaystyle \frac{d\left({\varkappa }^{*},\mathfrak{I}{\varkappa }_{2\mathfrak{p}}\right)d\left({\varkappa }_{2\mathfrak{p}},\mathcal{O}{\varkappa }^{*}\right)}{1+d\left({\varkappa }_{2\mathfrak{p}},{\varkappa }^{*}\right)}}\end{array}\right)\hfill \\ & & +\aleph d\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+1}\right)d\left({\varkappa }_{2\mathfrak{p}+1},\mathcal{O}{\varkappa }^{*}\right)\hfill \\ & ⪯& d\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+1}\right)+\left(\begin{array}{c}{\varpi }_1\left({\varkappa }_{2\mathfrak{p}},{\varkappa }^{*}\right)d\left({\varkappa }_{2\mathfrak{p}},{\varkappa }^{*}\right)\\ +{\varpi }_2\left({\varkappa }_{2\mathfrak{p}},{\varkappa }^{*}\right)\left(d\left({\varkappa }_{2\mathfrak{p}},{\varkappa }_{2\mathfrak{p}+1}\right)+d\left({\varkappa }^{*},\mathcal{O}{\varkappa }^{*}\right)\right)\\ +{\varpi }_3\left({\varkappa }_{2\mathfrak{p}},{\varkappa }^{*}\right){\displaystyle \frac{d\left({\varkappa }_{2\mathfrak{p}},{\varkappa }_{2\mathfrak{p}+1}\right)d\left({\varkappa }^{*},\mathcal{O}{\varkappa }^{*}\right)}{1+d\left({\varkappa }_{2\mathfrak{p}},{\varkappa }^{*}\right)}}\\ +{\varpi }_4\left({\varkappa }_{2\mathfrak{p}},{\varkappa }^{*}\right){\displaystyle \frac{d\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+1}\right)d\left({\varkappa }_{2\mathfrak{p}},\mathcal{O}{\varkappa }^{*}\right)}{1+d\left({\varkappa }_{2\mathfrak{p}},{\varkappa }^{*}\right)}}\end{array}\right)\hfill \\ & & +\aleph d\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+1}\right)d\left({\varkappa }_{2\mathfrak{p}+1},\mathcal{O}{\varkappa }^{*}\right),\hfill \end{array}$$

which implies by Proposition 1 that

$$\begin{array}{ccc}\hfill \left|d\left({\varkappa }^{*},\mathcal{O}{\varkappa }^{*}\right)\right|& \le & \left|d\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+1}\right)\right|+\left(\begin{array}{c}{\varpi }_1\left({\varkappa }_{2\mathfrak{p}},{\varkappa }^{*}\right)\left|d\left({\varkappa }_{2\mathfrak{p}},{\varkappa }^{*}\right)\right|\\ +{\varpi }_2\left({\varkappa }_{2\mathfrak{p}},{\varkappa }^{*}\right)\left(\left|d\left({\varkappa }_{2\mathfrak{p}},{\varkappa }_{2\mathfrak{p}+1}\right)\right|+\left|d\left({\varkappa }^{*},\mathcal{O}{\varkappa }^{*}\right)\right|\right)\\ +{\varpi }_3\left({\varkappa }_{2\mathfrak{p}},{\varkappa }^{*}\right)\left|{\displaystyle \frac{d\left({\varkappa }_{2\mathfrak{p}},{\varkappa }_{2\mathfrak{p}+1}\right)}{1+d\left({\varkappa }_{2\mathfrak{p}},{\varkappa }^{*}\right)}}\right|\left|d\left({\varkappa }^{*},\mathcal{O}{\varkappa }^{*}\right)\right|\\ +{\varpi }_4\left({\varkappa }_{2\mathfrak{p}},{\varkappa }^{*}\right)\left|{\displaystyle \frac{d\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+1}\right)}{1+d\left({\varkappa }_{2\mathfrak{p}},{\varkappa }^{*}\right)}}\right|\left|d\left({\varkappa }_{2\mathfrak{p}},\mathcal{O}{\varkappa }^{*}\right)\right|\end{array}\right)\hfill \\ & & +\aleph \left|d\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+1}\right)\right|\left|d\left({\varkappa }_{2\mathfrak{p}+1},\mathcal{O}{\varkappa }^{*}\right)\right|\hfill \\ & \le & \left|d\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+1}\right)\right|+\left(\begin{array}{c}{\varpi }_1\left({\varkappa }_0,{\varkappa }^{*}\right)\left|d\left({\varkappa }_{2\mathfrak{p}},{\varkappa }^{*}\right)\right|\\ +{\varpi }_2\left({\varkappa }_0,{\varkappa }^{*}\right)\left(\left|d\left({\varkappa }_{2\mathfrak{p}},{\varkappa }_{2\mathfrak{p}+1}\right)\right|+\left|d\left({\varkappa }^{*},\mathcal{O}{\varkappa }^{*}\right)\right|\right)\\ +{\varpi }_3\left({\varkappa }_0,{\varkappa }^{*}\right)\left|{\displaystyle \frac{d\left({\varkappa }_{2\mathfrak{p}},{\varkappa }_{2\mathfrak{p}+1}\right)}{1+d\left({\varkappa }_{2\mathfrak{p}},{\varkappa }^{*}\right)}}\right|\left|d\left({\varkappa }^{*},\mathcal{O}{\varkappa }^{*}\right)\right|\\ +{\varpi }_4\left({\varkappa }_0,{\varkappa }^{*}\right)\left|{\displaystyle \frac{d\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+1}\right)}{1+d\left({\varkappa }_{2\mathfrak{p}},{\varkappa }^{*}\right)}}\right|\left|d\left({\varkappa }_{2\mathfrak{p}},\mathcal{O}{\varkappa }^{*}\right)\right|\end{array}\right)\hfill \\ & & +\aleph \left|d\left({\varkappa }^{*},{\varkappa }_{2\mathfrak{p}+1}\right)\right|\left|d\left({\varkappa }_{2\mathfrak{p}+1},\mathcal{O}{\varkappa }^{*}\right)\right|.\hfill \end{array}$$

Letting $\mathfrak{p}\to \infty ,$ we have $\left(1-{\varpi }_2\left({\varkappa }_0,{\varkappa }^{*}\right)\right)\left|d\left({\varkappa }^{*},\mathcal{O}{\varkappa }^{*}\right)\right|=0,$ but $\left(1-{\varpi }_2\left({\varkappa }_0\right)\right)\ne 0,$ hence $\left|d\left({\varkappa }^{*},\mathcal{O}{\varkappa }^{*}\right)\right|=0$ implies ${\varkappa }^{*}=\mathcal{O}{\varkappa }^{*}.$ Thus ${\varkappa }^{*}$ is a CFP of $\mathfrak{I}$ and $\mathcal{O}.$ To establish the uniqueness of ${\varkappa }^{*},$ assume, to the contrary, that there exists a different CFP ${\varkappa }^{/}$ of $\mathfrak{I}$ and $\mathcal{O}.$ This implies that

$${\varkappa }^{/}=\mathfrak{I}{\varkappa }^{/}=\mathcal{O}{\varkappa }^{/}$$

but ${\varkappa }^{*}\ne {\varkappa }^{/}.$ Now from (2), we have

$$\begin{array}{ccc}\hfill d\left({\varkappa }^{*},{\varkappa }^{/}\right)& =& d\left(\mathfrak{I}{\varkappa }^{*},\mathcal{O}{\varkappa }^{/}\right)\hfill \\ & ⪯& {\varpi }_1\left({\varkappa }^{*},{\varkappa }^{/}\right)d\left({\varkappa }^{*},{\varkappa }^{/}\right)+{\varpi }_2\left({\varkappa }^{*},{\varkappa }^{/}\right)\left(d\left({\varkappa }^{*},\mathfrak{I}{\varkappa }^{*}\right)+d\left({\varkappa }^{/},\mathcal{O}{\varkappa }^{/}\right)\right)\hfill \\ & & +{\varpi }_3\left({\varkappa }^{*},{\varkappa }^{/}\right){\displaystyle \frac{d\left({\varkappa }^{*},\mathfrak{I}{\varkappa }^{*}\right)d\left({\varkappa }^{/},\mathcal{O}{\varkappa }^{/}\right)}{1+d\left({\varkappa }^{*},{\varkappa }^{/}\right)}}\hfill \\ & & +{\varpi }_4\left({\varkappa }^{*},{\varkappa }^{/}\right){\displaystyle \frac{d\left({\varkappa }^{/},\mathfrak{I}{\varkappa }^{*}\right)d\left({\varkappa }^{*},\mathcal{O}{\varkappa }^{/}\right)}{1+d\left({\varkappa }^{*},{\varkappa }^{/}\right)}}\hfill \\ & =& {\varpi }_1\left({\varkappa }^{*}\right)d\left({\varkappa }^{*},{\varkappa }^{/}\right)+{\varpi }_4\left({\varkappa }^{*},{\varkappa }^{/}\right){\displaystyle \frac{d\left({\varkappa }^{/},{\varkappa }^{*}\right)d\left({\varkappa }^{*},{\varkappa }^{/}\right)}{1+d\left({\varkappa }^{*},{\varkappa }^{/}\right)}}\hfill \end{array}$$

This implies that, we have

$$\begin{array}{ccc}\hfill \left|d\left({\varkappa }^{*},{\varkappa }^{/}\right)\right|& \le & {\varpi }_1\left({\varkappa }^{*},{\varkappa }^{/}\right)\left|d\left({\varkappa }^{*},{\varkappa }^{/}\right)\right|+{\varpi }_4\left({\varkappa }^{*},{\varkappa }^{/}\right)\left|d\left({\varkappa }^{/},{\varkappa }^{*}\right)\right|\left|{\displaystyle \frac{d\left({\varkappa }^{*},{\varkappa }^{/}\right)}{1+d\left({\varkappa }^{*},{\varkappa }^{/}\right)}}\right|\hfill \\ & \le & {\varpi }_1\left({\varkappa }^{*},{\varkappa }^{/}\right)\left|d\left({\varkappa }^{*},{\varkappa }^{/}\right)\right|+{\varpi }_4\left({\varkappa }^{*},{\varkappa }^{/}\right)\left|d\left({\varkappa }^{/},{\varkappa }^{*}\right)\right|\hfill \end{array}$$

which implies $\left(1-{\varpi }_1\left({\varkappa }^{*},{\varkappa }^{/}\right)-{\varpi }_4\left({\varkappa }^{*},{\varkappa }^{/}\right)\right)\left|d\left({\varkappa }^{*},{\varkappa }^{/}\right)\right|=0.$

Since $\left(1-{\varpi }_1\left({\varkappa }^{*},{\varkappa }^{/}\right)-{\varpi }_4\left({\varkappa }^{*},{\varkappa }^{/}\right)\right)\ne 0,$ thus

$$\left|d\left({\varkappa }^{*},{\varkappa }^{/}\right)\right|=0.$$

Hence ${\varkappa }^{*}={\varkappa }^{/}.$ □

Example 4.

Let $\Omega =[0,1].$ Define $d:\Omega \times \Omega \to \mathbb{C}$ by

$$d(\varkappa ,\mathfrak{l})=(1+i)\left|\varkappa -\mathfrak{l}\right|$$

for all $\varkappa ,\mathfrak{l}\in$$\Omega .$ Then $(\Omega ,d)$ is a complete CVSMS with $\aleph \ge 0$. Define the self mappings $\mathfrak{I},\mathcal{O}:\Omega$$\to \Omega$ by

$$\mathfrak{I}\varkappa =\mathcal{O}\varkappa ={\displaystyle \frac{\varkappa +1}{2}}$$

for $\varkappa \in \Omega .$ Define ${\varpi }_1,{\varpi }_2,{\varpi }_3,{\varpi }_4:\Omega \times \Omega \to [0,1)$ by

$${\varpi }_1\left(\varkappa ,\mathfrak{l}\right)=0.6\left(1-{\displaystyle \frac{\varkappa }{10}}\right)$$

$${\varpi }_2\left(\varkappa ,\mathfrak{l}\right)={\varpi }_3\left(\varkappa ,\mathfrak{l}\right)={\varpi }_4\left(\varkappa ,\mathfrak{l}\right)=0.$$

Here

$$\mathcal{O}\mathfrak{I}=\mathcal{O}\circ \mathfrak{I}=\mathfrak{I}\circ \mathfrak{I}$$

and

$$\mathfrak{I}\mathcal{O}=\mathfrak{I}\circ \mathcal{O}=\mathfrak{I}\circ \mathfrak{I}.$$

Note that

$$\mathfrak{I}\varkappa ={\displaystyle \frac{\varkappa +1}{2}}\ge \varkappa$$

for all $\varkappa \in \Omega .$ Hence

$${\mathfrak{I}}^2\left(\varkappa \right)\ge \mathfrak{I}\left(\varkappa \right)\ge \varkappa .$$

Thus

$$\mathcal{O}\mathfrak{I}\left(\varkappa \right)={\mathfrak{I}}^2\left(\varkappa \right)\ge \varkappa$$

and

$$\mathfrak{I}\mathcal{O}\left(\mathfrak{l}\right)={\mathfrak{I}}^2\left(\mathfrak{l}\right)\ge \mathfrak{l}.$$

Since ${\varpi }_1\left(\varkappa ,\mathfrak{l}\right)=0.6\left(1-{\displaystyle \frac{\varkappa }{10}}\right)$ is monotone nonincreasing in the first argument $\varkappa ,$ we get

$${\varpi }_1\left(\mathcal{O}\mathfrak{I}\varkappa ,\mathfrak{l}\right)=0.6\left(1-{\displaystyle \frac{\mathcal{O}\mathfrak{I}\varkappa }{10}}\right)\le 0.6\left(1-{\displaystyle \frac{\varkappa }{10}}\right)={\varpi }_1\left(\varkappa ,\mathfrak{l}\right),$$

and similarly ${\varpi }_1\left(\varkappa ,\mathfrak{I}\mathcal{O}\mathfrak{l}\right)\le {\varpi }_1\left(\varkappa ,\mathfrak{l}\right).$ For $i=2,3,4$ the inequalities hold trivially because those ${\varpi }_i$ are zero. So condition (a) is satisfied. Moreover,

$${\varpi }_1(\varkappa ,\mathfrak{l})+2{\varpi }_2(\varkappa ,\mathfrak{l})+{\varpi }_3(\varkappa ,\mathfrak{l})+{\varpi }_4(\varkappa ,\mathfrak{l})\le 0.6<1,$$

hence (b) holds pointwise. Since

$${\displaystyle \frac{1}{2}}\le 0.6\left(1-{\displaystyle \frac{\varkappa }{10}}\right)$$

holds true for $\varkappa \in \Omega$ Then, it is very simple to verify

$$\begin{array}{ccc}\hfill d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{l}\right)& ⪯& {\varpi }_1\left(\varkappa ,\mathfrak{l}\right)d\left(\varkappa ,\mathfrak{l}\right)+{\varpi }_2\left(\varkappa ,\mathfrak{l}\right)\left(d\left(\varkappa ,\mathfrak{I}\varkappa \right)+d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)\right)\hfill \\ & & +{\varpi }_3\left(\varkappa ,\mathfrak{l}\right){\displaystyle \frac{d\left(\varkappa ,\mathfrak{I}\varkappa \right)d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}}+{\varpi }_4\left(\varkappa ,\mathfrak{l}\right){\displaystyle \frac{d\left(\mathfrak{l},\mathfrak{I}\varkappa \right)d\left(\varkappa ,\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}}.\hfill \end{array}$$

Hence (c) is satisfied. Since (a)–(c) hold, the Theorem applies and $\mathfrak{I}$ and $\mathcal{O}$ have a unique CFP $1\in \Omega .$

By setting $\mathfrak{I}=\mathcal{O}$ in Theorem 1, we deduce the following corollary.

Corollary 1.

Let $\left(\Omega ,d\right)$ be a complete CVSMS and $\mathcal{O}:\Omega$$\to \Omega$. Assume that there exist functions ${\varpi }_1,{\varpi }_2,{\varpi }_3,{\varpi }_4:\Omega \times \Omega \to [0,1)$ satisfying the following conditions:

(a) 

$$\begin{array}{ccc}\hfill {\varpi }_1\left(\mathcal{O}\varkappa ,\mathfrak{l}\right)& \le & {\varpi }_1\left(\varkappa ,\mathfrak{l}\right),{\varpi }_1\left(\varkappa ,\mathcal{O}\mathfrak{l}\right)\le {\varpi }_1\left(\varkappa ,\mathfrak{l}\right)\hfill \\ \hfill {\varpi }_2\left(\mathcal{O}\varkappa ,\mathfrak{l}\right)& \le & {\varpi }_2\left(\varkappa ,\mathfrak{l}\right),{\varpi }_2\left(\varkappa ,\mathcal{O}\mathfrak{l}\right)\le {\varpi }_2\left(\varkappa ,\mathfrak{l}\right)\hfill \\ \hfill {\varpi }_3\left(\mathcal{O}\varkappa ,\mathfrak{l}\right)& \le & {\varpi }_3\left(\varkappa ,\mathfrak{l}\right),{\varpi }_3\left(\varkappa ,\mathcal{O}\mathfrak{l}\right)\le {\varpi }_3\left(\varkappa ,\mathfrak{l}\right)\hfill \\ \hfill {\varpi }_4\left(\mathcal{O}\varkappa ,\mathfrak{l}\right)& \le & {\varpi }_4\left(\varkappa ,\mathfrak{l}\right),{\varpi }_4\left(\varkappa ,\mathcal{O}\mathfrak{l}\right)\le {\varpi }_4\left(\varkappa ,\mathfrak{l}\right),\hfill \end{array}$$

(b) 

$${\varpi }_1\left(\varkappa ,\mathfrak{l}\right)+2{\varpi }_2\left(\varkappa ,\mathfrak{l}\right)+{\varpi }_3\left(\varkappa ,\mathfrak{l}\right)+{\varpi }_4\left(\varkappa ,\mathfrak{l}\right)<1,$$

(c) 

$$\begin{array}{ccc}\hfill d\left(\mathcal{O}\varkappa ,\mathcal{O}\mathfrak{l}\right)& ⪯& {\varpi }_1\left(\varkappa ,\mathfrak{l}\right)d\left(\varkappa ,\mathfrak{l}\right)+{\varpi }_2\left(\varkappa ,\mathfrak{l}\right)\left(d\left(\varkappa ,\mathcal{O}\varkappa \right)+d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)\right)\hfill \\ & & +{\varpi }_3\left(\varkappa ,\mathfrak{l}\right){\displaystyle \frac{d\left(\varkappa ,\mathcal{O}\varkappa \right)d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}}+{\varpi }_4\left(\varkappa ,\mathfrak{l}\right){\displaystyle \frac{d\left(\mathfrak{l},\mathcal{O}\varkappa \right)d\left(\varkappa ,\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}},\hfill \end{array}$$

for all $\varkappa ,\mathfrak{l}\in \Omega .$

Then, $\mathcal{O}$ possesses a unique FP.

Example 5.

Let $\Omega =[0,1].$ Define $d:\Omega \times \Omega \to \mathbb{C}$ by

$$d(\varkappa ,\mathfrak{l})=(1+i)\left|\varkappa -\mathfrak{l}\right|$$

for all $\varkappa ,\mathfrak{l}\in$$\Omega .$ Then $(\Omega ,d)$ is a complete CVSMS with $\aleph \ge 0$. Define the self mapping $\mathcal{O}:\Omega$$\to \Omega$ by

$$\mathcal{O}\varkappa ={\displaystyle \frac{1}{2}}\varkappa .$$

Define ${\varpi }_1,{\varpi }_2,{\varpi }_3,{\varpi }_4:\Omega \times \Omega \to [0,1)$ by

$${\varpi }_1(\varkappa ,\mathfrak{l})={\displaystyle \frac{1}{5}}+{\displaystyle \frac{1}{10}}\left(\varkappa +\mathfrak{l}\right)$$

$${\varpi }_2(\varkappa ,\mathfrak{l})={\displaystyle \frac{1}{20}}+{\displaystyle \frac{1}{50}}\left(\varkappa +\mathfrak{l}\right)$$

$${\varpi }_3(\varkappa ,\mathfrak{l})={\displaystyle \frac{1}{20}}+{\displaystyle \frac{1}{50}}\left(\varkappa +\mathfrak{l}\right)$$

$${\varpi }_4(\varkappa ,\mathfrak{l})={\displaystyle \frac{1}{20}}+{\displaystyle \frac{1}{50}}\left(\varkappa +\mathfrak{l}\right).$$

Then condition (a) holds, since ${\varpi }_i$ are non-decreasing in each argument. Since

$${\varpi }_1(\varkappa ,\mathfrak{l})+2{\varpi }_2(\varkappa ,\mathfrak{l})+{\varpi }_3(\varkappa ,\mathfrak{l})+{\varpi }_4(\varkappa ,\mathfrak{l})\le 0.96<1,$$

so the condition (b) holds. Now for $\varkappa ,\mathfrak{l}\in \Omega$ with $\varkappa \ne \mathfrak{l},$ we have

$$\begin{array}{ccc}\hfill d\left(\mathcal{O}\varkappa ,\mathcal{O}\mathfrak{l}\right)& =& (1+i)\left|\mathcal{O}\varkappa -\mathcal{O}\mathfrak{l}\right|\hfill \\ & =& (1+i)\left|{\displaystyle \frac{1}{2}}\varkappa -{\displaystyle \frac{1}{2}}\mathfrak{l}\right|\hfill \\ & ⪯& {\varpi }_1(1+i)\left|\varkappa -\mathfrak{l}\right|+{\varpi }_2\left({\displaystyle \frac{\varkappa +\mathfrak{l}}{2}}\right)+{\varpi }_3{\displaystyle \frac{\left(\varkappa \mathfrak{l}\right)/4}{1+\left|\varkappa -\mathfrak{l}\right|}}\hfill \\ & & +{\varpi }_4{\displaystyle \frac{\left|\mathfrak{l}-\frac{\varkappa }{2}\right|\left|\varkappa -\frac{\mathfrak{l}}{2}\right|}{1+\left|\varkappa -\mathfrak{l}\right|}}\hfill \end{array}$$

holds, where

$${\varpi }_1(\varkappa ,\mathfrak{l})={\displaystyle \frac{1}{5}}+{\displaystyle \frac{1}{10}}\left(\varkappa +\mathfrak{l}\right)$$

and

$${\varpi }_2(\varkappa ,\mathfrak{l})={\varpi }_3(\varkappa ,\mathfrak{l})={\varpi }_4(\varkappa ,\mathfrak{l})={\displaystyle \frac{1}{20}}+{\displaystyle \frac{1}{50}}\left(\varkappa +\mathfrak{l}\right).$$

Thus, the condition (c) of Corollary 1 is also satisfied. Consequently, 0 is the unique FP of the mapping $\mathcal{O}.$

Thus, the principal result of Abdou et al. [21] can be obtained as a particular case of our main theorem.

Corollary 2

([21]). Let $\left(\Omega ,d\right)$ be a complete CVSMS and $\mathfrak{I},\mathcal{O}:\Omega$$\to \Omega$. Assume that there exist functions ${\varpi }_1,{\varpi }_3,{\varpi }_4:\Omega \times \Omega \to [0,1)$ satisfying the following conditions:

(a) 

$$\begin{array}{ccc}\hfill {\varpi }_1\left(\mathcal{O}\mathfrak{I}\varkappa ,\mathfrak{l}\right)& \le & {\varpi }_1\left(\varkappa ,\mathfrak{l}\right),{\varpi }_1\left(\varkappa ,\mathfrak{I}\mathcal{O}\mathfrak{l}\right)\le {\varpi }_1\left(\varkappa ,\mathfrak{l}\right)\hfill \\ \hfill {\varpi }_3\left(\mathcal{O}\mathfrak{I}\varkappa ,\mathfrak{l}\right)& \le & {\varpi }_3\left(\varkappa ,\mathfrak{l}\right),{\varpi }_3\left(\varkappa ,\mathfrak{I}\mathcal{O}\mathfrak{l}\right)\le {\varpi }_3\left(\varkappa ,\mathfrak{l}\right)\hfill \\ \hfill {\varpi }_4\left(\mathcal{O}\mathfrak{I}\varkappa ,\mathfrak{l}\right)& \le & {\varpi }_4\left(\varkappa ,\mathfrak{l}\right),{\varpi }_4\left(\varkappa ,\mathfrak{I}\mathcal{O}\mathfrak{l}\right)\le {\varpi }_4\left(\varkappa ,\mathfrak{l}\right),\hfill \end{array}$$

(b) 

$${\varpi }_1\left(\varkappa ,\mathfrak{l}\right)+{\varpi }_3\left(\varkappa ,\mathfrak{l}\right)+{\varpi }_4\left(\varkappa ,\mathfrak{l}\right)<1,$$

(c) 

$$\begin{array}{ccc}\hfill d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{l}\right)⪯{\varpi }_1\left(\varkappa ,\mathfrak{l}\right)d\left(\varkappa ,\mathfrak{l}\right)& & +{\varpi }_3\left(\varkappa ,\mathfrak{l}\right){\displaystyle \frac{d\left(\varkappa ,\mathfrak{I}\varkappa \right)d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}}\hfill \\ & & +{\varpi }_4\left(\varkappa ,\mathfrak{l}\right){\displaystyle \frac{d\left(\mathfrak{l},\mathfrak{I}\varkappa \right)d\left(\varkappa ,\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}},\hfill \end{array}$$

for all $\varkappa ,\mathfrak{l}\in \Omega .$ Then, $\mathfrak{I}$ and $\mathcal{O}$ possess a unique CFP.

Proof. 

Define ${\varpi }_2:\Omega \times \Omega \to [0,1)$ by ${\varpi }_2\left(\varkappa ,\mathfrak{l}\right)=0$ in the Theorem 1. □

Corollary 3.

Let $\left(\Omega ,d\right)$ be a complete CVSMS and $\mathfrak{I},\mathcal{O}:\Omega$$\to \Omega$. Assume that there exist functions ${\varpi }_1,{\varpi }_2,{\varpi }_3:\Omega \times \Omega \to [0,1)$ satisfying the following conditions:

(a) 

$$\begin{array}{ccc}\hfill {\varpi }_1\left(\mathcal{O}\mathfrak{I}\varkappa ,\mathfrak{l}\right)& \le & {\varpi }_1\left(\varkappa ,\mathfrak{l}\right),{\varpi }_1\left(\varkappa ,\mathfrak{I}\mathcal{O}\mathfrak{l}\right)\le {\varpi }_1\left(\varkappa ,\mathfrak{l}\right)\hfill \\ \hfill {\varpi }_2\left(\mathcal{O}\mathfrak{I}\varkappa ,\mathfrak{l}\right)& \le & {\varpi }_2\left(\varkappa ,\mathfrak{l}\right),{\varpi }_2\left(\varkappa ,\mathfrak{I}\mathcal{O}\mathfrak{l}\right)\le {\varpi }_2\left(\varkappa ,\mathfrak{l}\right)\hfill \\ \hfill {\varpi }_3\left(\mathcal{O}\mathfrak{I}\varkappa ,\mathfrak{l}\right)& \le & {\varpi }_3\left(\varkappa ,\mathfrak{l}\right),{\varpi }_3\left(\varkappa ,\mathfrak{I}\mathcal{O}\mathfrak{l}\right)\le {\varpi }_3\left(\varkappa ,\mathfrak{l}\right),\hfill \end{array}$$

(b) 

$${\varpi }_1\left(\varkappa ,\mathfrak{l}\right)+2{\varpi }_2\left(\varkappa ,\mathfrak{l}\right)+{\varpi }_3\left(\varkappa ,\mathfrak{l}\right)<1,$$

(c) 

$$\begin{array}{ccc}\hfill d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{l}\right)& ⪯& {\varpi }_1\left(\varkappa ,\mathfrak{l}\right)d\left(\varkappa ,\mathfrak{l}\right)+{\varpi }_2\left(\varkappa ,\mathfrak{l}\right)\left(d\left(\varkappa ,\mathfrak{I}\varkappa \right)+d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)\right)\hfill \\ & & +{\varpi }_3\left(\varkappa ,\mathfrak{l}\right){\displaystyle \frac{d\left(\varkappa ,\mathfrak{I}\varkappa \right)d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}},\hfill \end{array}$$

for all $\varkappa ,\mathfrak{l}\in \Omega .$

Then $\mathfrak{I}$ and $\mathcal{O}$ possess a unique CFP.

Proof. 

Define ${\varpi }_4:\Omega \times \Omega \to [0,1)$ by ${\varpi }_4\left(\varkappa ,\mathfrak{l}\right)=0$ in Theorem 1. □

Corollary 4.

Let $\left(\Omega ,d\right)$ be a complete CVSMS and $\mathfrak{I},\mathcal{O}:\Omega$$\to \Omega$. Assume that there exist functions ${\varpi }_1,{\varpi }_2:\Omega \times \Omega \to [0,1)$ satisfying the following conditions:

(a) 

$$\begin{array}{ccc}\hfill {\varpi }_1\left(\mathcal{O}\mathfrak{I}\varkappa ,\mathfrak{l}\right)& \le & {\varpi }_1\left(\varkappa ,\mathfrak{l}\right),{\varpi }_1\left(\varkappa ,\mathfrak{I}\mathcal{O}\mathfrak{l}\right)\le {\varpi }_1\left(\varkappa ,\mathfrak{l}\right)\hfill \\ \hfill {\varpi }_2\left(\mathcal{O}\mathfrak{I}\varkappa ,\mathfrak{l}\right)& \le & {\varpi }_2\left(\varkappa ,\mathfrak{l}\right),{\varpi }_2\left(\varkappa ,\mathfrak{I}\mathcal{O}\mathfrak{l}\right)\le {\varpi }_2\left(\varkappa ,\mathfrak{l}\right),\hfill \end{array}$$

(b) 

$${\varpi }_1\left(\varkappa ,\mathfrak{l}\right)+2{\varpi }_2\left(\varkappa ,\mathfrak{l}\right)<1,$$

(c) 

$$d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{l}\right)⪯{\varpi }_1\left(\varkappa ,\mathfrak{l}\right)d\left(\varkappa ,\mathfrak{l}\right)+{\varpi }_2\left(\varkappa ,\mathfrak{l}\right)\left(d\left(\varkappa ,\mathfrak{I}\varkappa \right)+d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)\right),$$

for all $\varkappa ,\mathfrak{l}\in \Omega .$

Then $\mathfrak{I}$ and $\mathcal{O}$ possess a unique CFP.

4. Fixed Point Results via Control Functions of One Variable

This section focuses on deriving FP results through the use of one variable control functions, which serve as a crucial mechanism for analyzing the behavior of self mappings. For the purposes of this section, $\left(\Omega ,d\right)$ is assumed to be a complete CVSMS.

Corollary 5.

Let $\mathfrak{I},\mathcal{O}:\Omega$$\to \Omega$. Assume that there exist functions ${\omega }_1,{\omega }_2,{\omega }_3,{\omega }_4:\Omega \to [0,1)$ satisfying the following conditions:

(a) 

$$\begin{array}{ccc}\hfill {\omega }_1\left(\mathfrak{I}\varkappa \right)& \le & {\omega }_1\left(\varkappa \right),{\omega }_1\left(\mathcal{O}\varkappa \right)\le {\omega }_1\left(\varkappa \right)\hfill \\ \hfill {\omega }_2\left(\mathfrak{I}\varkappa \right)& \le & {\omega }_2\left(\varkappa \right),{\omega }_2\left(\mathcal{O}\varkappa \right)\le {\omega }_2\left(\varkappa \right)\hfill \\ \hfill {\omega }_3\left(\mathfrak{I}\varkappa \right)& \le & {\omega }_3\left(\varkappa \right),{\omega }_3\left(\mathcal{O}\varkappa \right)\le {\omega }_3\left(\varkappa \right)\hfill \\ \hfill {\omega }_4\left(\mathfrak{I}\varkappa \right)& \le & {\omega }_4\left(\varkappa \right),{\omega }_4\left(\mathcal{O}\varkappa \right)\le {\omega }_4\left(\varkappa \right),\hfill \end{array}$$

(b) 

$${\omega }_1\left(\varkappa \right)+2{\omega }_2\left(\varkappa \right)+{\omega }_3\left(\varkappa \right)+{\omega }_4\left(\varkappa \right)<1,$$

(c) 

$$\begin{array}{ccc}\hfill d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{l}\right)& ⪯& {\omega }_1\left(\varkappa \right)d\left(\varkappa ,\mathfrak{l}\right)+{\omega }_2\left(\varkappa \right)\left(d\left(\varkappa ,\mathfrak{I}\varkappa \right)+d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)\right)\hfill \\ & & +{\omega }_3\left(\varkappa \right){\displaystyle \frac{d\left(\varkappa ,\mathfrak{I}\varkappa \right)d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}}+{\omega }_4\left(\varkappa \right){\displaystyle \frac{d\left(\mathfrak{l},\mathfrak{I}\varkappa \right)d\left(\varkappa ,\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}},\hfill \end{array}$$

for all $\varkappa ,\mathfrak{l}\in \Omega .$

Then $\mathfrak{I}$ and $\mathcal{O}$ admit a unique CFP.

Proof. 

Define ${\varpi }_1,{\varpi }_2,{\varpi }_3,{\varpi }_4:\Omega \times \Omega \to [0,1)$ by

$${\varpi }_1\left(\varkappa ,\mathfrak{l}\right)={\omega }_1\left(\varkappa \right)$$

$${\varpi }_2\left(\varkappa ,\mathfrak{l}\right)={\omega }_2\left(\varkappa \right)$$

$${\varpi }_3\left(\varkappa ,\mathfrak{l}\right)={\omega }_3\left(\varkappa \right)$$

$${\varpi }_4\left(\varkappa ,\mathfrak{l}\right)={\omega }_4\left(\varkappa \right)$$

for all $\varkappa ,\mathfrak{l}\in \Omega$. Then for all $\varkappa ,\mathfrak{l}\in \Omega$, we have

(a)

$${\varpi }_1\left(\mathcal{O}\mathfrak{I}\varkappa ,\mathfrak{l}\right)={\omega }_1\left(\mathcal{O}\mathfrak{I}\varkappa \right)\le {\omega }_1\left(\mathfrak{I}\varkappa \right)\le {\omega }_1\left(\varkappa \right)={\varpi }_1\left(\varkappa ,\mathfrak{l}\right)$$

and

$${\varpi }_1\left(\varkappa ,\mathfrak{I}\mathcal{O}\mathfrak{l}\right)\le {\omega }_1\left(\varkappa \right)={\varpi }_1\left(\varkappa ,\mathfrak{l}\right).$$

Similarly, we can prove that

$$\begin{array}{c}\hfill {\varpi }_2\left(\mathcal{O}\mathfrak{I}\varkappa ,\mathfrak{l}\right)\le {\varpi }_2\left(\varkappa ,\mathfrak{l}\right)\mathrm{and}{\varpi }_2\left(\varkappa ,\mathfrak{I}\mathcal{O}\mathfrak{l}\right)\le {\varpi }_2\left(\varkappa ,\mathfrak{l}\right)\\ \hfill {\varpi }_3\left(\mathcal{O}\mathfrak{I}\varkappa ,\mathfrak{l}\right)\le {\varpi }_3\left(\varkappa ,\mathfrak{l}\right)\mathrm{and}{\varpi }_3\left(\varkappa ,\mathfrak{I}\mathcal{O}\mathfrak{l}\right)\le {\varpi }_3\left(\varkappa ,\mathfrak{l}\right)\\ \hfill {\varpi }_4\left(\mathcal{O}\mathfrak{I}\varkappa ,\mathfrak{l}\right)\le {\varpi }_4\left(\varkappa ,\mathfrak{l}\right)\mathrm{and}{\varpi }_4\left(\varkappa ,\mathfrak{I}\mathcal{O}\mathfrak{l}\right)\le {\varpi }_4\left(\varkappa ,\mathfrak{l}\right).\end{array}$$

(b)

$$\begin{array}{ccc}& & {\varpi }_1\left(\varkappa ,\mathfrak{l}\right)+2{\varpi }_2\left(\varkappa ,\mathfrak{l}\right)+{\varpi }_3\left(\varkappa ,\mathfrak{l}\right)+{\varpi }_4\left(\varkappa ,\mathfrak{l}\right)\hfill \\ & =& {\omega }_1\left(\varkappa \right)+2{\omega }_2\left(\varkappa \right)+{\omega }_3\left(\varkappa \right)+{\omega }_4\left(\varkappa \right)<1.\hfill \end{array}$$

(c)

$$\begin{array}{ccc}\hfill d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{l}\right)& ⪯& {\omega }_1\left(\varkappa \right)d\left(\varkappa ,\mathfrak{l}\right)+{\omega }_2\left(\varkappa \right)\left(d\left(\varkappa ,\mathfrak{I}\varkappa \right)+d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)\right)\hfill \\ & & +{\omega }_3\left(\varkappa \right){\displaystyle \frac{d\left(\varkappa ,\mathfrak{I}\varkappa \right)d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}}+{\omega }_4\left(\varkappa \right){\displaystyle \frac{d\left(\mathfrak{l},\mathfrak{I}\varkappa \right)d\left(\varkappa ,\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}}\hfill \\ & =& {\varpi }_1\left(\varkappa ,\mathfrak{l}\right)d\left(\varkappa ,\mathfrak{l}\right)+{\varpi }_2\left(\varkappa ,\mathfrak{l}\right)\left(d\left(\varkappa ,\mathfrak{I}\varkappa \right)+d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)\right)\hfill \\ & & +{\varpi }_3\left(\varkappa ,\mathfrak{l}\right){\displaystyle \frac{d\left(\varkappa ,\mathfrak{I}\varkappa \right)d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}}+{\varpi }_4\left(\varkappa ,\mathfrak{l}\right){\displaystyle \frac{d\left(\mathfrak{l},\mathfrak{I}\varkappa \right)d\left(\varkappa ,\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}}.\hfill \end{array}$$

Thus by Theorem 1, $\mathfrak{I}$ and $\mathcal{O}$ have a unique CFP. □

The subsequent result represents the principal finding of Shammaky et al. [22], which is obtained as a direct consequence of the preceding corollary.

Corollary 6.

Let $\mathfrak{I},\mathcal{O}:\Omega$$\to \Omega$. Assume that there exist functions ${\omega }_1,{\omega }_2,{\omega }_3:\Omega \to [0,1)$ satisfying the following conditions:

(a) 

$$\begin{array}{ccc}\hfill {\omega }_1\left(\mathfrak{I}\varkappa \right)& \le & {\omega }_1\left(\varkappa \right),{\omega }_1\left(\mathcal{O}\varkappa \right)\le {\omega }_1\left(\varkappa \right)\hfill \\ \hfill {\omega }_2\left(\mathfrak{I}\varkappa \right)& \le & {\omega }_2\left(\varkappa \right),{\omega }_2\left(\mathcal{O}\varkappa \right)\le {\omega }_2\left(\varkappa \right)\hfill \\ \hfill {\omega }_3\left(\mathfrak{I}\varkappa \right)& \le & {\omega }_3\left(\varkappa \right),{\omega }_3\left(\mathcal{O}\varkappa \right)\le {\omega }_3\left(\varkappa \right),\hfill \end{array}$$

(b) 

$${\omega }_1\left(\varkappa \right)+2{\omega }_2\left(\varkappa \right)+{\omega }_3\left(\varkappa \right)<1,$$

(c) 

$$\begin{array}{ccc}\hfill d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{l}\right)& ⪯& {\omega }_1\left(\varkappa \right)d\left(\varkappa ,\mathfrak{l}\right)+{\omega }_2\left(\varkappa \right)\left(d\left(\varkappa ,\mathfrak{I}\varkappa \right)+d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)\right)\hfill \\ & & +{\omega }_3\left(\varkappa \right){\displaystyle \frac{d\left(\varkappa ,\mathfrak{I}\varkappa \right)d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}},\hfill \end{array}$$

for all $\varkappa ,\mathfrak{l}\in \Omega .$ Then $\mathfrak{I}$ and $\mathcal{O}$ admit a unique CFP.

Proof. 

Define ${\omega }_4:\Omega \to [0,1)$ by ${\omega }_4\left(\varkappa \right)=0$ in Corollary 5. □

Corollary 7.

Let $\mathfrak{I},\mathcal{O}:\Omega$$\to \Omega$. Assume that there exist functions ${\omega }_1,{\omega }_2,{\omega }_4:\Omega \to [0,1)$ satisfying the following conditions:

(a) 

$$\begin{array}{ccc}\hfill {\omega }_1\left(\mathfrak{I}\varkappa \right)& \le & {\omega }_1\left(\varkappa \right),{\omega }_1\left(\mathcal{O}\varkappa \right)\le {\omega }_1\left(\varkappa \right)\hfill \\ \hfill {\omega }_2\left(\mathfrak{I}\varkappa \right)& \le & {\omega }_2\left(\varkappa \right),{\omega }_2\left(\mathcal{O}\varkappa \right)\le {\omega }_2\left(\varkappa \right)\hfill \\ \hfill {\omega }_4\left(\mathfrak{I}\varkappa \right)& \le & {\omega }_4\left(\varkappa \right),{\omega }_4\left(\mathcal{O}\varkappa \right)\le {\omega }_4\left(\varkappa \right),\hfill \end{array}$$

(b) 

$${\omega }_1\left(\varkappa \right)+2{\omega }_2\left(\varkappa \right)+{\omega }_4\left(\varkappa \right)<1,$$

(c) 

$$\begin{array}{ccc}\hfill d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{l}\right)& ⪯& {\omega }_1\left(\varkappa \right)d\left(\varkappa ,\mathfrak{l}\right)+{\omega }_2\left(\varkappa \right)\left(d\left(\varkappa ,\mathfrak{I}\varkappa \right)+d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)\right)\hfill \\ & & +{\omega }_4\left(\varkappa \right){\displaystyle \frac{d\left(\mathfrak{l},\mathfrak{I}\varkappa \right)d\left(\varkappa ,\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}},\hfill \end{array}$$

for all $\varkappa ,\mathfrak{l}\in \Omega .$

Then $\mathfrak{I}$ and $\mathcal{O}$ admit a unique CFP.

Proof. 

Take ${\omega }_3:\Omega \to [0,1)$ by ${\omega }_3\left(\varkappa \right)=0$ in Corollary 5. □

Corollary 8.

Let $\mathfrak{I},\mathcal{O}:\Omega$$\to \Omega$. Assume that there exist functions ${\omega }_1,{\omega }_3,{\omega }_4:\Omega \to [0,1)$ satisfying the following conditions:

(a) 

$$\begin{array}{ccc}\hfill {\omega }_1\left(\mathfrak{I}\varkappa \right)& \le & {\omega }_1\left(\varkappa \right),{\omega }_1\left(\mathcal{O}\varkappa \right)\le {\omega }_1\left(\varkappa \right)\hfill \\ \hfill {\omega }_3\left(\mathfrak{I}\varkappa \right)& \le & {\omega }_3\left(\varkappa \right),{\omega }_3\left(\mathcal{O}\varkappa \right)\le {\omega }_3\left(\varkappa \right)\hfill \\ \hfill {\omega }_4\left(\mathfrak{I}\varkappa \right)& \le & {\omega }_4\left(\varkappa \right),{\omega }_4\left(\mathcal{O}\varkappa \right)\le {\omega }_4\left(\varkappa \right),\hfill \end{array}$$

(b) 

$${\omega }_1\left(\varkappa \right)+{\omega }_3\left(\varkappa \right)+{\omega }_4\left(\varkappa \right)<1,$$

(c) 

$$\begin{array}{ccc}\hfill d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{l}\right)& ⪯& {\omega }_1\left(\varkappa \right)d\left(\varkappa ,\mathfrak{l}\right)\hfill \\ & & +{\omega }_3\left(\varkappa \right){\displaystyle \frac{d\left(\varkappa ,\mathfrak{I}\varkappa \right)d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}}+{\omega }_4\left(\varkappa \right){\displaystyle \frac{d\left(\mathfrak{l},\mathfrak{I}\varkappa \right)d\left(\varkappa ,\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}},\hfill \end{array}$$

for all $\varkappa ,\mathfrak{l}\in \Omega .$

Then $\mathfrak{I}$ and $\mathcal{O}$ admit a unique CFP.

Proof. 

Take ${\omega }_2:\Omega \to [0,1)$ by ${\omega }_2\left(\varkappa \right)=0$ in Corollary 5. □

Corollary 9.

Let $\mathfrak{I},\mathcal{O}:\Omega$$\to \Omega$. Assume that there exist functions ${\omega }_1,{\omega }_2:\Omega \to [0,1)$ satisfying the following conditions:

(a) 

$$\begin{array}{ccc}\hfill {\omega }_1\left(\mathfrak{I}\varkappa \right)& \le & {\omega }_1\left(\varkappa \right),{\omega }_1\left(\mathcal{O}\varkappa \right)\le {\omega }_1\left(\varkappa \right)\hfill \\ \hfill {\omega }_2\left(\mathfrak{I}\varkappa \right)& \le & {\omega }_2\left(\varkappa \right),{\omega }_2\left(\mathcal{O}\varkappa \right)\le {\omega }_2\left(\varkappa \right),\hfill \end{array}$$

(b) 

$${\omega }_1\left(\varkappa \right)+2{\omega }_2\left(\varkappa \right)<1,$$

(c) 

$$d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{l}\right)⪯{\omega }_1\left(\varkappa \right)d\left(\varkappa ,\mathfrak{l}\right)+{\omega }_2\left(\varkappa \right)\left(d\left(\varkappa ,\mathfrak{I}\varkappa \right)+d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)\right),$$

for all $\varkappa ,\mathfrak{l}\in \Omega .$

Then $\mathfrak{I}$ and $\mathcal{O}$ admit a unique CFP.

Proof. 

Take ${\omega }_3,{\omega }_4:\Omega \to [0,1)$ by ${\omega }_3\left(\varkappa \right)={\omega }_4\left(\varkappa \right)=0$ in Corollary 5. □

Corollary 10.

Let $\mathfrak{I},\mathcal{O}:\Omega$$\to \Omega$. Assume that there exist functions ${\omega }_1,{\omega }_3:\Omega \to [0,1)$ satisfying the following conditions:

(a) 

$$\begin{array}{ccc}\hfill {\omega }_1\left(\mathfrak{I}\varkappa \right)& \le & {\omega }_1\left(\varkappa \right),{\omega }_1\left(\mathcal{O}\varkappa \right)\le {\omega }_1\left(\varkappa \right)\hfill \\ \hfill {\omega }_3\left(\mathfrak{I}\varkappa \right)& \le & {\omega }_3\left(\varkappa \right),{\omega }_3\left(\mathcal{O}\varkappa \right)\le {\omega }_3\left(\varkappa \right),\hfill \end{array}$$

(b) 

$${\omega }_1\left(\varkappa \right)+{\omega }_3\left(\varkappa \right)<1,$$

(c) 

$$d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{l}\right)⪯{\omega }_1\left(\varkappa \right)d\left(\varkappa ,\mathfrak{l}\right)+{\omega }_3\left(\varkappa \right){\displaystyle \frac{d\left(\varkappa ,\mathfrak{I}\varkappa \right)d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}},$$

for all $\varkappa ,\mathfrak{l}\in \Omega .$

Then $\mathfrak{I}$ and $\mathcal{O}$ admit a unique CFP.

Proof. 

Take ${\omega }_2,{\omega }_4:\Omega \to [0,1)$ by ${\omega }_2\left(\varkappa \right)={\omega }_4\left(\varkappa \right)=0$ in Corollary 5. □

Remark 1.

It should be emphasized that the hypotheses (a) and (b) of Corollary 5 can be weakened by imposing the following assumption.

Corollary 11.

Let $\mathfrak{I},\mathcal{O}:\Omega$$\to \Omega$. Assume that there exist functions ${\omega }_1,{\omega }_2,{\omega }_3,{\omega }_4:\Omega \to [0,1)$ satisfying the following conditions:

(a) 

$$\begin{array}{ccc}\hfill {\omega }_1\left(\mathcal{O}\mathfrak{I}\varkappa \right)& \le & {\omega }_1\left(\varkappa \right)\hfill \\ \hfill {\omega }_2\left(\mathcal{O}\mathfrak{I}\varkappa \right)& \le & {\omega }_2\left(\varkappa \right)\hfill \\ \hfill {\omega }_3\left(\mathcal{O}\mathfrak{I}\varkappa \right)& \le & {\omega }_3\left(\varkappa \right)\hfill \\ \hfill {\omega }_4\left(\mathcal{O}\mathfrak{I}\varkappa \right)& \le & {\omega }_4\left(\varkappa \right),\hfill \end{array}$$

(b) 

$${\omega }_1\left(\varkappa \right)+2{\omega }_2\left(\varkappa \right)+{\omega }_3\left(\varkappa \right)+{\omega }_4\left(\varkappa \right)<1,$$

(c) 

$$\begin{array}{ccc}\hfill d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{l}\right)& ⪯& {\omega }_1\left(\varkappa \right)d\left(\varkappa ,\mathfrak{l}\right)+{\omega }_2\left(\varkappa \right)\left(d\left(\varkappa ,\mathfrak{I}\varkappa \right)+d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)\right)\hfill \\ & & +{\omega }_3\left(\varkappa \right){\displaystyle \frac{d\left(\varkappa ,\mathfrak{I}\varkappa \right)d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}}+{\omega }_4\left(\varkappa \right){\displaystyle \frac{d\left(\mathfrak{l},\mathfrak{I}\varkappa \right)d\left(\varkappa ,\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}},\hfill \end{array}$$

for all $\varkappa ,\mathfrak{l}\in \Omega .$

Then $\mathfrak{I}$ and $\mathcal{O}$ admit a unique CFP.

Proof. 

Define ${\varpi }_1,{\varpi }_2,{\varpi }_3,{\varpi }_4:\Omega \times \Omega \to [0,1)$ by

$${\varpi }_1\left(\varkappa ,\mathfrak{l}\right)={\omega }_1\left(\varkappa \right)$$

$${\varpi }_2\left(\varkappa ,\mathfrak{l}\right)={\omega }_2\left(\varkappa \right)$$

$${\varpi }_3\left(\varkappa ,\mathfrak{l}\right)={\omega }_3\left(\varkappa \right)$$

$${\varpi }_4\left(\varkappa ,\mathfrak{l}\right)={\omega }_4\left(\varkappa \right)$$

for all $\varkappa ,\mathfrak{l}\in \Omega$. Then for all $\varkappa ,\mathfrak{l}\in \Omega$, we have

(a)

$${\varpi }_1\left(\mathcal{O}\mathfrak{I}\varkappa ,\mathfrak{l}\right)={\omega }_1\left(\mathcal{O}\mathfrak{I}\varkappa \right)\le {\omega }_1\left(\varkappa \right)={\varpi }_1\left(\varkappa ,\mathfrak{l}\right)$$

and

$${\varpi }_1\left(\varkappa ,\mathfrak{I}\mathcal{O}\mathfrak{l}\right)\le {\omega }_1\left(\varkappa \right)={\varpi }_1\left(\varkappa ,\mathfrak{l}\right).$$

Similarly, we can prove that

$$\begin{array}{c}\hfill {\varpi }_2\left(\mathcal{O}\mathfrak{I}\varkappa ,\mathfrak{l}\right)\le {\varpi }_2\left(\varkappa ,\mathfrak{l}\right)\mathrm{and}{\varpi }_2\left(\varkappa ,\mathfrak{I}\mathcal{O}\mathfrak{l}\right)\le {\varpi }_2\left(\varkappa ,\mathfrak{l}\right)\\ \hfill {\varpi }_3\left(\mathcal{O}\mathfrak{I}\varkappa ,\mathfrak{l}\right)\le {\varpi }_3\left(\varkappa ,\mathfrak{l}\right)\mathrm{and}{\varpi }_3\left(\varkappa ,\mathfrak{I}\mathcal{O}\mathfrak{l}\right)\le {\varpi }_3\left(\varkappa ,\mathfrak{l}\right)\\ \hfill {\varpi }_4\left(\mathcal{O}\mathfrak{I}\varkappa ,\mathfrak{l}\right)\le {\varpi }_4\left(\varkappa ,\mathfrak{l}\right)\mathrm{and}{\varpi }_4\left(\varkappa ,\mathfrak{I}\mathcal{O}\mathfrak{l}\right)\le {\varpi }_4\left(\varkappa ,\mathfrak{l}\right).\end{array}$$

(b)

$${\varpi }_1\left(\varkappa ,\mathfrak{l}\right)+2{\varpi }_2\left(\varkappa ,\mathfrak{l}\right)+{\varpi }_3\left(\varkappa ,\mathfrak{l}\right)+{\varpi }_4\left(\varkappa ,\mathfrak{l}\right)={\omega }_1\left(\varkappa \right)+2{\omega }_2\left(\varkappa \right)+{\omega }_3\left(\varkappa \right)+{\omega }_4\left(\varkappa \right)<1.$$

(c)

$$\begin{array}{ccc}\hfill d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{l}\right)& ⪯& {\omega }_1\left(\varkappa \right)d\left(\varkappa ,\mathfrak{l}\right)+{\omega }_2\left(\varkappa \right)\left(d\left(\varkappa ,\mathfrak{I}\varkappa \right)+d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)\right)\hfill \\ & & +{\omega }_3\left(\varkappa \right){\displaystyle \frac{d\left(\varkappa ,\mathfrak{I}\varkappa \right)d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}}+{\omega }_4\left(\varkappa \right){\displaystyle \frac{d\left(\mathfrak{l},\mathfrak{I}\varkappa \right)d\left(\varkappa ,\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}}\hfill \\ & =& {\varpi }_1\left(\varkappa ,\mathfrak{l}\right)d\left(\varkappa ,\mathfrak{l}\right)+{\varpi }_2\left(\varkappa ,\mathfrak{l}\right)\left(d\left(\varkappa ,\mathfrak{I}\varkappa \right)+d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)\right)\hfill \\ & & +{\varpi }_3\left(\varkappa ,\mathfrak{l}\right){\displaystyle \frac{d\left(\varkappa ,\mathfrak{I}\varkappa \right)d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}}+{\varpi }_4\left(\varkappa ,\mathfrak{l}\right){\displaystyle \frac{d\left(\mathfrak{l},\mathfrak{I}\varkappa \right)d\left(\varkappa ,\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}}.\hfill \end{array}$$

Thus by Theorem 1, $\mathfrak{I}$ and $\mathcal{O}$ have a unique CFP. □

Corollary 12.

Let $\mathfrak{I},\mathcal{O}:\Omega$$\to \Omega$. Assume that there exist functions ${\omega }_1,{\omega }_2,{\omega }_3:\Omega \to [0,1)$ satisfying the following conditions:

(a) 

$$\begin{array}{ccc}\hfill {\omega }_1\left(\mathcal{O}\mathfrak{I}\varkappa \right)& \le & {\omega }_1\left(\varkappa \right)\hfill \\ \hfill {\omega }_2\left(\mathcal{O}\mathfrak{I}\varkappa \right)& \le & {\omega }_2\left(\varkappa \right)\hfill \\ \hfill {\omega }_3\left(\mathcal{O}\mathfrak{I}\varkappa \right)& \le & {\omega }_3\left(\varkappa \right),\hfill \end{array}$$

(b) 

$${\omega }_1\left(\varkappa \right)+2{\omega }_2\left(\varkappa \right)+{\omega }_3\left(\varkappa \right)<1,$$

(c) 

$$\begin{array}{ccc}\hfill d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{l}\right)& ⪯& {\omega }_1\left(\varkappa \right)d\left(\varkappa ,\mathfrak{l}\right)+{\omega }_2\left(\varkappa \right)\left(d\left(\varkappa ,\mathfrak{I}\varkappa \right)+d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)\right)\hfill \\ & & +{\omega }_3\left(\varkappa \right){\displaystyle \frac{d\left(\varkappa ,\mathfrak{I}\varkappa \right)d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}},\hfill \end{array}$$

for all $\varkappa ,\mathfrak{l}\in \Omega .$

Then $\mathfrak{I}$ and $\mathcal{O}$ posses a unique CFP.

Proof. 

Take ${\omega }_4:\Omega \to [0,1)$ by ${\omega }_4\left(\varkappa \right)=0$ in Corollary 11. □

Corollary 13.

Let $\mathfrak{I},\mathcal{O}:\Omega$$\to \Omega$. Assume that there exist functions ${\omega }_1,{\omega }_2,{\omega }_4:\Omega \to [0,1)$ satisfying the following conditions:

(a) 

$$\begin{array}{ccc}\hfill {\omega }_1\left(\mathcal{O}\mathfrak{I}\varkappa \right)& \le & {\omega }_1\left(\varkappa \right)\hfill \\ \hfill {\omega }_2\left(\mathcal{O}\mathfrak{I}\varkappa \right)& \le & {\omega }_2\left(\varkappa \right)\hfill \\ \hfill {\omega }_4\left(\mathcal{O}\mathfrak{I}\varkappa \right)& \le & {\omega }_4\left(\varkappa \right),\hfill \end{array}$$

(b) 

$${\omega }_1\left(\varkappa \right)+2{\omega }_2\left(\varkappa \right)+{\omega }_4\left(\varkappa \right)<1,$$

(c) 

$$\begin{array}{ccc}\hfill d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{l}\right)& ⪯& {\omega }_1\left(\varkappa \right)d\left(\varkappa ,\mathfrak{l}\right)+{\omega }_2\left(\varkappa \right)\left(d\left(\varkappa ,\mathfrak{I}\varkappa \right)+d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)\right)\hfill \\ & & +{\omega }_4\left(\varkappa \right){\displaystyle \frac{d\left(\mathfrak{l},\mathfrak{I}\varkappa \right)d\left(\varkappa ,\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}},\hfill \end{array}$$

for all $\varkappa ,\mathfrak{l}\in \Omega .$

Then $\mathfrak{I}$ and $\mathcal{O}$ have a unique CFP.

Proof. 

Assigning ${\omega }_3:\Omega \to [0,1)$ by ${\omega }_3\left(\varkappa \right)=0$ in Corollary 11. □

Corollary 14.

Let $\mathfrak{I},\mathcal{O}:\Omega$$\to \Omega$. Assume that there exist functions ${\omega }_1,{\omega }_3,{\omega }_4:\Omega \to [0,1)$ satisfying the following conditions:

(a) 

$$\begin{array}{ccc}\hfill {\omega }_1\left(\mathcal{O}\mathfrak{I}\varkappa \right)& \le & {\omega }_1\left(\varkappa \right)\hfill \\ \hfill {\omega }_3\left(\mathcal{O}\mathfrak{I}\varkappa \right)& \le & {\omega }_3\left(\varkappa \right)\hfill \\ \hfill {\omega }_4\left(\mathcal{O}\mathfrak{I}\varkappa \right)& \le & {\omega }_4\left(\varkappa \right),\hfill \end{array}$$

(b) 

$${\omega }_1\left(\varkappa \right)+{\omega }_3\left(\varkappa \right)+{\omega }_4\left(\varkappa \right)<1,$$

(c) 

$$\begin{array}{ccc}\hfill d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{l}\right)& ⪯& {\omega }_1\left(\varkappa \right)d\left(\varkappa ,\mathfrak{l}\right)\hfill \\ & & +{\omega }_3\left(\varkappa \right){\displaystyle \frac{d\left(\varkappa ,\mathfrak{I}\varkappa \right)d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}}+{\omega }_4\left(\varkappa \right){\displaystyle \frac{d\left(\mathfrak{l},\mathfrak{I}\varkappa \right)d\left(\varkappa ,\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}},\hfill \end{array}$$

for all $\varkappa ,\mathfrak{l}\in \Omega .$

Then $\mathfrak{I}$ and $\mathcal{O}$ posses a unique CFP.

Proof. 

Assigning ${\omega }_2:\Omega \to [0,1)$ by ${\omega }_2\left(\varkappa \right)=0$ in Corollary 11. □

Corollary 15.

Let $\mathfrak{I},\mathcal{O}:\Omega$$\to \Omega$. Assume that there exist functions ${\omega }_1,{\omega }_2:\Omega \to [0,1)$ satisfying the following conditions:

(a) 

$$\begin{array}{ccc}\hfill {\omega }_1\left(\mathcal{O}\mathfrak{I}\varkappa \right)& \le & {\omega }_1\left(\varkappa \right)\hfill \\ \hfill {\omega }_2\left(\mathcal{O}\mathfrak{I}\varkappa \right)& \le & {\omega }_2\left(\varkappa \right),\hfill \end{array}$$

(b) 

$${\omega }_1\left(\varkappa \right)+2{\omega }_2\left(\varkappa \right)<1,$$

(c) 

$$d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{l}\right)⪯{\omega }_1\left(\varkappa \right)d\left(\varkappa ,\mathfrak{l}\right)+{\omega }_2\left(\varkappa \right)\left(d\left(\varkappa ,\mathfrak{I}\varkappa \right)+d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)\right),$$

for all $\varkappa ,\mathfrak{l}\in \Omega .$

Then $\mathfrak{I}$ and $\mathcal{O}$ posses a unique CFP.

Proof. 

Assigning ${\omega }_3,{\omega }_4:\Omega \to [0,1)$ by ${\omega }_3\left(\varkappa \right)={\omega }_4\left(\varkappa \right)=0$ in Corollary 11. □

Corollary 16.

Let $\mathfrak{I},\mathcal{O}:\Omega$$\to \Omega$. Assume that there exist functions ${\omega }_1,{\omega }_3:\Omega \to [0,1)$ satisfying the following conditions:

(a) 

$$\begin{array}{ccc}\hfill {\omega }_1\left(\mathcal{O}\mathfrak{I}\varkappa \right)& \le & {\omega }_1\left(\varkappa \right)\hfill \\ \hfill {\omega }_3\left(\mathcal{O}\mathfrak{I}\varkappa \right)& \le & {\omega }_3\left(\varkappa \right),\hfill \end{array}$$

(b) 

$${\omega }_1\left(\varkappa \right)+{\omega }_3\left(\varkappa \right)<1,$$

(c) 

$$\begin{array}{ccc}\hfill d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{l}\right)& ⪯& {\omega }_1\left(\varkappa \right)d\left(\varkappa ,\mathfrak{l}\right)\hfill \\ & & +{\omega }_3\left(\varkappa \right){\displaystyle \frac{d\left(\varkappa ,\mathfrak{I}\varkappa \right)d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}},\hfill \end{array}$$

for all $\varkappa ,\mathfrak{l}\in \Omega .$

Then $\mathfrak{I}$ and $\mathcal{O}$ posses a unique CFP.

Proof. 

Assigning ${\omega }_2,{\omega }_4:\Omega \to [0,1)$ by ${\omega }_2\left(\varkappa \right)={\omega }_4\left(\varkappa \right)=0$ in Corollary 11. □

Remark 2.

If we specialize the above control functions of one variable to constant values, that is, set ${\omega }_1(·)={\omega }_1,$ ${\omega }_2(·)={\omega }_2,$ ${\omega }_3(·)={\omega }_3$ and ${\omega }_4(·)={\omega }_4$ in Corollary 11 yields the following results.

Corollary 17.

Let $\mathfrak{I},\mathcal{O}:\Omega$$\to \Omega$. Suppose there are constants ${\omega }_1,{\omega }_2,{\omega }_3,{\omega }_4\in [0,1)$ such that ${\omega }_1+2{\omega }_2+{\omega }_3+{\omega }_4<1$ and

$$\begin{array}{ccc}\hfill d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{l}\right)& ⪯& {\omega }_{1}d\left(\varkappa ,\mathfrak{l}\right)+{\omega }_2\left(d\left(\varkappa ,\mathfrak{I}\varkappa \right)+d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)\right)\hfill \\ & & +{\omega }_3{\displaystyle \frac{d\left(\varkappa ,\mathfrak{I}\varkappa \right)d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}}+{\omega }_4{\displaystyle \frac{d\left(\mathfrak{l},\mathfrak{I}\varkappa \right)d\left(\varkappa ,\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}},\hfill \end{array}$$

for all $\varkappa ,\mathfrak{l}\in \Omega .$ Then $\mathfrak{I}$ and $\mathcal{O}$ admit a unique CFP.

Proof. 

Assigning ${\omega }_1(·)={\omega }_1,$ ${\omega }_2(·)={\omega }_2,$ ${\omega }_3(·)={\omega }_3$ and ${\omega }_4(·)={\omega }_4$ in Corollary 11. □

Corollary 18.

Let $\mathfrak{I},\mathcal{O}:\Omega$$\to \Omega$. Suppose there are constants ${\omega }_1,{\omega }_2,{\omega }_3\in [0,1)$ with ${\omega }_1+2{\omega }_2+{\omega }_3<1$ such that

$$\begin{array}{ccc}\hfill d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{l}\right)& ⪯& {\omega }_{1}d\left(\varkappa ,\mathfrak{l}\right)+{\omega }_2\left(d\left(\varkappa ,\mathfrak{I}\varkappa \right)+d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)\right)\hfill \\ & & +{\omega }_3{\displaystyle \frac{d\left(\varkappa ,\mathfrak{I}\varkappa \right)d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}},\hfill \end{array}$$

for all $\varkappa ,\mathfrak{l}\in \Omega .$ Then $\mathfrak{I}$ and $\mathcal{O}$ posses a unique CFP.

Proof. 

Assigning ${\omega }_4=0$ in Corollary 17. □

Corollary 19.

Let $\mathfrak{I},\mathcal{O}:\Omega$$\to \Omega$. Suppose there are constants ${\omega }_1,{\omega }_2,{\omega }_4\in [0,1)$ with ${\omega }_1+2{\omega }_2+{\omega }_4<1$ such that

$$\begin{array}{ccc}\hfill d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{l}\right)& ⪯& {\omega }_{1}d\left(\varkappa ,\mathfrak{l}\right)+{\omega }_2\left(d\left(\varkappa ,\mathfrak{I}\varkappa \right)+d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)\right)\hfill \\ & & +{\omega }_4{\displaystyle \frac{d\left(\mathfrak{l},\mathfrak{I}\varkappa \right)d\left(\varkappa ,\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}},\hfill \end{array}$$

for all $\varkappa ,\mathfrak{l}\in \Omega .$ Then $\mathfrak{I}$ and $\mathcal{O}$ posses a unique CFP.

Proof. 

Assigning ${\omega }_3=0$ in Corollary 17. □

The next result, due to Panda et al. [16], follows directly from Corollary 17.

Corollary 20.

Let $\mathfrak{I},\mathcal{O}:\Omega$$\to \Omega$. Suppose there are constants ${\omega }_1,{\omega }_3,{\omega }_4\in [0,1)$ with ${\omega }_1+{\omega }_3+{\omega }_4<1$ such that

$$\begin{array}{ccc}\hfill d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{l}\right)& ⪯& {\omega }_{1}d\left(\varkappa ,\mathfrak{l}\right)\hfill \\ & & +{\omega }_3{\displaystyle \frac{d\left(\varkappa ,\mathfrak{I}\varkappa \right)d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}}+{\omega }_4{\displaystyle \frac{d\left(\mathfrak{l},\mathfrak{I}\varkappa \right)d\left(\varkappa ,\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}},\hfill \end{array}$$

for all $\varkappa ,\mathfrak{l}\in \Omega .$ Then $\mathfrak{I}$ and $\mathcal{O}$ have a unique CFP.

Proof. 

Assigning ${\omega }_2=0$ in Corollary 17. □

Corollary 21.

Let $\mathfrak{I},\mathcal{O}:\Omega$$\to \Omega$. Suppose there are constants ${\omega }_1,{\omega }_2,{\omega }_4\in [0,1)$ with ${\omega }_1+2{\omega }_2<1$ such that

$$d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{l}\right)⪯{\omega }_{1}d\left(\varkappa ,\mathfrak{l}\right)+{\omega }_2\left(d\left(\varkappa ,\mathfrak{I}\varkappa \right)+d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)\right),$$

for all $\varkappa ,\mathfrak{l}\in \Omega .$ Then $\mathfrak{I}$ and $\mathcal{O}$ posses a unique CFP.

Proof. 

Assigning ${\omega }_3={\omega }_4=0$ in Corollary 17. □

Corollary 22.

Let $\mathfrak{I},\mathcal{O}:\Omega$$\to \Omega$. Suppose there are constants ${\omega }_1,{\omega }_3\in [0,1)$ with ${\omega }_1+{\omega }_3<1$ such that

$$\begin{array}{ccc}\hfill d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{l}\right)& ⪯& {\omega }_{1}d\left(\varkappa ,\mathfrak{l}\right)\hfill \\ & & +{\omega }_3{\displaystyle \frac{d\left(\varkappa ,\mathfrak{I}\varkappa \right)d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}},\hfill \end{array}$$

for all $\varkappa ,\mathfrak{l}\in \Omega .$ Then $\mathfrak{I}$ and $\mathcal{O}$ posses a unique CFP.

Proof. 

Assigning ${\omega }_2={\omega }_4=0$ in Corollary 17. □

5. Core Contributions in the Framework of Complex-Valued Metric Spaces

Accordingly, we obtain several FP results within the framework of CVMSs by setting $\aleph =0$ in Definition 4 and employing Theorem 1. For the scope of this section, $\left(\Omega ,d\right)$ is assumed to be a CVMS.

Corollary 23.

Let $\mathfrak{I},\mathcal{O}:\Omega$$\to \Omega$. Assume that there exist the functions ${\varpi }_1,{\varpi }_2,{\varpi }_3,{\varpi }_4:\Omega \times \Omega \to [0,1)$ such that

(a) 

$$\begin{array}{ccc}\hfill {\varpi }_1\left(\mathcal{O}\mathfrak{I}\varkappa ,\mathfrak{l}\right)& \le & {\varpi }_1\left(\varkappa ,\mathfrak{l}\right),{\varpi }_1\left(\varkappa ,\mathfrak{I}\mathcal{O}\mathfrak{l}\right)\le {\varpi }_1\left(\varkappa ,\mathfrak{l}\right)\hfill \\ \hfill {\varpi }_2\left(\mathcal{O}\mathfrak{I}\varkappa ,\mathfrak{l}\right)& \le & {\varpi }_2\left(\varkappa ,\mathfrak{l}\right),{\varpi }_2\left(\varkappa ,\mathfrak{I}\mathcal{O}\mathfrak{l}\right)\le {\varpi }_2\left(\varkappa ,\mathfrak{l}\right)\hfill \\ \hfill {\varpi }_3\left(\mathcal{O}\mathfrak{I}\varkappa ,\mathfrak{l}\right)& \le & {\varpi }_3\left(\varkappa ,\mathfrak{l}\right),{\varpi }_3\left(\varkappa ,\mathfrak{I}\mathcal{O}\mathfrak{l}\right)\le {\varpi }_3\left(\varkappa ,\mathfrak{l}\right)\hfill \\ \hfill {\varpi }_4\left(\mathcal{O}\mathfrak{I}\varkappa ,\mathfrak{l}\right)& \le & {\varpi }_4\left(\varkappa ,\mathfrak{l}\right),{\varpi }_4\left(\varkappa ,\mathfrak{I}\mathcal{O}\mathfrak{l}\right)\le {\varpi }_4\left(\varkappa ,\mathfrak{l}\right),\hfill \end{array}$$

(b) 

$${\varpi }_1\left(\varkappa ,\mathfrak{l}\right)+2{\varpi }_2\left(\varkappa ,\mathfrak{l}\right)+{\varpi }_3\left(\varkappa ,\mathfrak{l}\right)+{\varpi }_4\left(\varkappa ,\mathfrak{l}\right)<1,$$

(c) 

$$\begin{array}{ccc}\hfill d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{l}\right)& ⪯& {\varpi }_1\left(\varkappa ,\mathfrak{l}\right)d\left(\varkappa ,\mathfrak{l}\right)+{\varpi }_2\left(\varkappa ,\mathfrak{l}\right)\left(d\left(\varkappa ,\mathfrak{I}\varkappa \right)+d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)\right)\hfill \\ & & +{\varpi }_3\left(\varkappa ,\mathfrak{l}\right){\displaystyle \frac{d\left(\varkappa ,\mathfrak{I}\varkappa \right)d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}}+{\varpi }_4\left(\varkappa ,\mathfrak{l}\right){\displaystyle \frac{d\left(\mathfrak{l},\mathfrak{I}\varkappa \right)d\left(\varkappa ,\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}},\hfill \end{array}$$

for all $\varkappa ,\mathfrak{l}\in \Omega .$ Then $\mathfrak{I}$ and $\mathcal{O}$ have a unique CFP.

Proof. 

Take $\aleph =0$ in Definition 4 and employing Theorem 1. □

We proceed to derive the principal theorem of Sitthikul et al. [15], which can be obtained as a direct outcome of Corollary 23.

Corollary 24.

Let $\mathfrak{I},\mathcal{O}:\Omega$$\to \Omega$. Assume that there exist the functions ${\varpi }_1,{\varpi }_3,{\varpi }_4:\Omega \times \Omega \to [0,1)$ such that

(a) 

$$\begin{array}{ccc}\hfill {\varpi }_1\left(\mathcal{O}\mathfrak{I}\varkappa ,\mathfrak{l}\right)& \le & {\varpi }_1\left(\varkappa ,\mathfrak{l}\right),{\varpi }_1\left(\varkappa ,\mathfrak{I}\mathcal{O}\mathfrak{l}\right)\le {\varpi }_1\left(\varkappa ,\mathfrak{l}\right)\hfill \\ \hfill {\varpi }_3\left(\mathcal{O}\mathfrak{I}\varkappa ,\mathfrak{l}\right)& \le & {\varpi }_3\left(\varkappa ,\mathfrak{l}\right),{\varpi }_3\left(\varkappa ,\mathfrak{I}\mathcal{O}\mathfrak{l}\right)\le {\varpi }_3\left(\varkappa ,\mathfrak{l}\right)\hfill \\ \hfill {\varpi }_4\left(\mathcal{O}\mathfrak{I}\varkappa ,\mathfrak{l}\right)& \le & {\varpi }_4\left(\varkappa ,\mathfrak{l}\right),{\varpi }_4\left(\varkappa ,\mathfrak{I}\mathcal{O}\mathfrak{l}\right)\le {\varpi }_4\left(\varkappa ,\mathfrak{l}\right),\hfill \end{array}$$

(b) 

$${\varpi }_1\left(\varkappa ,\mathfrak{l}\right)+{\varpi }_3\left(\varkappa ,\mathfrak{l}\right)+{\varpi }_4\left(\varkappa ,\mathfrak{l}\right)<1,$$

(c) 

$$\begin{array}{ccc}\hfill d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{l}\right)& ⪯& {\varpi }_1\left(\varkappa ,\mathfrak{l}\right)d\left(\varkappa ,\mathfrak{l}\right)\hfill \\ & & +{\varpi }_3\left(\varkappa ,\mathfrak{l}\right){\displaystyle \frac{d\left(\varkappa ,\mathfrak{I}\varkappa \right)d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}}+{\varpi }_4\left(\varkappa ,\mathfrak{l}\right){\displaystyle \frac{d\left(\mathfrak{l},\mathfrak{I}\varkappa \right)d\left(\varkappa ,\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}},\hfill \end{array}$$

for all $\varkappa ,\mathfrak{l}\in \Omega .$ Then $\mathfrak{I}$ and $\mathcal{O}$ have a unique CFP.

Proof. 

Take ${\varpi }_2:\Omega \times \Omega \to [0,1)$ by ${\varpi }_2\left(\varkappa ,\mathfrak{l}\right)=0$ in Corollary 23. □

Remark 3.

By assigning ${\varpi }_1,{\varpi }_2,{\varpi }_3,{\varpi }_4:\Omega \times \Omega \to [0,1)$ to zero in various possible combinations in Corollary 23, several corollaries can be obtained for the control functions involving two variables.

Next, by setting $\aleph =0$ in Definition 4 and applying it to Corollary 5, we derive CFP results under contractive conditions involving control functions of a single variable.

Corollary 25.

Let $\mathfrak{I},\mathcal{O}:\Omega$$\to \Omega$. Assume that there are functions ${\omega }_1,{\omega }_2,{\omega }_3,{\omega }_4:\Omega \to [0,1)$ satisfying the following conditions:

(a) 

$$\begin{array}{ccc}\hfill {\omega }_1\left(\mathfrak{I}\varkappa \right)& \le & {\omega }_1\left(\varkappa \right),{\omega }_1\left(\mathcal{O}\varkappa \right)\le {\omega }_1\left(\varkappa \right)\hfill \\ \hfill {\omega }_2\left(\mathfrak{I}\varkappa \right)& \le & {\omega }_2\left(\varkappa \right),{\omega }_2\left(\mathcal{O}\varkappa \right)\le {\omega }_2\left(\varkappa \right)\hfill \\ \hfill {\omega }_3\left(\mathfrak{I}\varkappa \right)& \le & {\omega }_3\left(\varkappa \right),{\omega }_3\left(\mathcal{O}\varkappa \right)\le {\omega }_3\left(\varkappa \right)\hfill \\ \hfill {\omega }_4\left(\mathfrak{I}\varkappa \right)& \le & {\omega }_4\left(\varkappa \right),{\omega }_4\left(\mathcal{O}\varkappa \right)\le {\omega }_4\left(\varkappa \right),\hfill \end{array}$$

(b) 

$${\omega }_1\left(\varkappa \right)+2{\omega }_2\left(\varkappa \right)+{\omega }_3\left(\varkappa \right)+{\omega }_4\left(\varkappa \right)<1,$$

(c) 

$$\begin{array}{ccc}\hfill d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{l}\right)& ⪯& {\omega }_1\left(\varkappa \right)d\left(\varkappa ,\mathfrak{l}\right)+{\omega }_2\left(\varkappa \right)\left(d\left(\varkappa ,\mathfrak{I}\varkappa \right)+d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)\right)\hfill \\ & & +{\omega }_3\left(\varkappa \right){\displaystyle \frac{d\left(\varkappa ,\mathfrak{I}\varkappa \right)d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}}+{\omega }_4\left(\varkappa \right){\displaystyle \frac{d\left(\mathfrak{l},\mathfrak{I}\varkappa \right)d\left(\varkappa ,\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}},\hfill \end{array}$$

for all $\varkappa ,\mathfrak{l}\in \Omega .$

Then $\mathfrak{I}$ and $\mathcal{O}$ admit a unique CFP.

Proof. 

Take $\aleph =0$ in Corollary 5. □

Building on Corollary 25, we recover the central theorem of Sintunavarat et al. [14] as a natural consequence.

Corollary 2.

([14]). Let $\mathfrak{I},\mathcal{O}:\Omega$$\to \Omega$. Assume that there are functions ${\omega }_1,{\omega }_3:\Omega \to [0,1)$ satisfying the following conditions:

(a) 

$$\begin{array}{ccc}\hfill {\omega }_1\left(\mathfrak{I}\varkappa \right)& \le & {\omega }_1\left(\varkappa \right),{\omega }_1\left(\mathcal{O}\varkappa \right)\le {\omega }_1\left(\varkappa \right)\hfill \\ \hfill {\omega }_3\left(\mathfrak{I}\varkappa \right)& \le & {\omega }_3\left(\varkappa \right),{\omega }_3\left(\mathcal{O}\varkappa \right)\le {\omega }_3\left(\varkappa \right),\hfill \end{array}$$

(b) 

$${\omega }_1\left(\varkappa \right)+{\omega }_3\left(\varkappa \right)<1,$$

(c) 

$$\begin{array}{ccc}\hfill d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{l}\right)& ⪯& {\omega }_1\left(\varkappa \right)d\left(\varkappa ,\mathfrak{l}\right)\hfill \\ & & +{\omega }_3\left(\varkappa \right){\displaystyle \frac{d\left(\varkappa ,\mathfrak{I}\varkappa \right)d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}},\hfill \end{array}$$

for all $\varkappa ,\mathfrak{l}\in \Omega .$ Then $\mathfrak{I}$ and $\mathcal{O}$ admit a unique CFP.

Proof. 

Take ${\omega }_2,{\omega }_4:\Omega \to [0,1)$ with ${\omega }_2\left(\varkappa \right)={\omega }_4\left(\varkappa \right)=0$ in Corollary 25. □

Subsequently, considering $\aleph =0$ in Definition 4 and invoking Corollary 17, we derive FP results under contractive mappings characterized by constant parameters.

Corollary 27.

Let $\mathfrak{I},\mathcal{O}:\Omega$$\to \Omega$. Assume that there are constants ${\omega }_1,{\omega }_2,{\omega }_3,{\omega }_4\in [0,1)$ such that ${\omega }_1+2{\omega }_2+{\omega }_3+{\omega }_4<1$ and

$$\begin{array}{ccc}\hfill d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{l}\right)& ⪯& {\omega }_{1}d\left(\varkappa ,\mathfrak{l}\right)+{\omega }_2\left(d\left(\varkappa ,\mathfrak{I}\varkappa \right)+d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)\right)\hfill \\ & & +{\omega }_3{\displaystyle \frac{d\left(\varkappa ,\mathfrak{I}\varkappa \right)d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}}+{\omega }_4{\displaystyle \frac{d\left(\mathfrak{l},\mathfrak{I}\varkappa \right)d\left(\varkappa ,\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}},\hfill \end{array}$$

for all $\varkappa ,\mathfrak{l}\in \Omega .$ Then $\mathfrak{I}$ and $\mathcal{O}$ possess a unique CFP.

Proof. 

Take $\aleph =0$ in Corollary 17. □

The fundamental result of Rouzkard et al. [13] can be derived directly as a special case of Corollary 27.

Corollary 28.

([13]). Let $\mathfrak{I},\mathcal{O}:\Omega$$\to \Omega$. Assume that there are constants ${\omega }_1,{\omega }_3,{\omega }_4\in [0,1)$ with ${\omega }_1+{\omega }_3+{\omega }_4<1$ such that

$$\begin{array}{ccc}\hfill d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{l}\right)& ⪯& {\omega }_{1}d\left(\varkappa ,\mathfrak{l}\right)\hfill \\ & & +{\omega }_3{\displaystyle \frac{d\left(\varkappa ,\mathfrak{I}\varkappa \right)d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}}+{\omega }_4{\displaystyle \frac{d\left(\mathfrak{l},\mathfrak{I}\varkappa \right)d\left(\varkappa ,\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}},\hfill \end{array}$$

for all $\varkappa ,\mathfrak{l}\in \Omega .$ Then $\mathfrak{I}$ and $\mathcal{O}$ possess a unique CFP.

Proof. 

Take ${\omega }_2=0$ in Corollary 27. □

Corollary 27 encompasses the main theorem of Azam et al. [12] as a particular case.

Corollary 29.

([12]). Let $\mathfrak{I},\mathcal{O}:\Omega$$\to \Omega$. Assume that there are constants ${\omega }_1,{\omega }_3\in [0,1)$ with ${\omega }_1+{\omega }_3<1$ such that

$$\begin{array}{ccc}\hfill d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{l}\right)& ⪯& {\omega }_{1}d\left(\varkappa ,\mathfrak{l}\right)\hfill \\ & & +{\omega }_3{\displaystyle \frac{d\left(\varkappa ,\mathfrak{I}\varkappa \right)d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}},\hfill \end{array}$$

for all $\varkappa ,\mathfrak{l}\in \Omega .$ Then $\mathfrak{I}$ and $\mathcal{O}$ possess a unique CFP.

Proof. 

Take ${\omega }_2={\omega }_4=0$ in Corollary 27. □

6. Fundamental Developments in the Study of Suprametric Spaces

When $\mathbb{C}$ is restricted to $\mathbb{R}$ in Definition 4 (i.e., the imaginary component is zero), the notion of CVSMS reduces to SMS, and Theorem 1 yields several results as its special cases. This section examines $\left(\Omega ,d\right)$ under the assumption that it forms a SMS. As an immediate consequence of Theorem 1, we present the following result.

Corollary 30.

Let $\mathfrak{I},\mathcal{O}:\Omega$$\to \Omega$. Assume that there are functions ${\varpi }_1,{\varpi }_2,{\varpi }_3,{\varpi }_4:\Omega \times \Omega \to [0,1)$ satisfying the following conditions:

(a) 

$$\begin{array}{ccc}\hfill {\varpi }_1\left(\mathcal{O}\mathfrak{I}\varkappa ,\mathfrak{l}\right)& \le & {\varpi }_1\left(\varkappa ,\mathfrak{l}\right),{\varpi }_1\left(\varkappa ,\mathfrak{I}\mathcal{O}\mathfrak{l}\right)\le {\varpi }_1\left(\varkappa ,\mathfrak{l}\right)\hfill \\ \hfill {\varpi }_2\left(\mathcal{O}\mathfrak{I}\varkappa ,\mathfrak{l}\right)& \le & {\varpi }_2\left(\varkappa ,\mathfrak{l}\right),{\varpi }_2\left(\varkappa ,\mathfrak{I}\mathcal{O}\mathfrak{l}\right)\le {\varpi }_2\left(\varkappa ,\mathfrak{l}\right)\hfill \\ \hfill {\varpi }_3\left(\mathcal{O}\mathfrak{I}\varkappa ,\mathfrak{l}\right)& \le & {\varpi }_3\left(\varkappa ,\mathfrak{l}\right),{\varpi }_3\left(\varkappa ,\mathfrak{I}\mathcal{O}\mathfrak{l}\right)\le {\varpi }_3\left(\varkappa ,\mathfrak{l}\right)\hfill \\ \hfill {\varpi }_4\left(\mathcal{O}\mathfrak{I}\varkappa ,\mathfrak{l}\right)& \le & {\varpi }_4\left(\varkappa ,\mathfrak{l}\right),{\varpi }_4\left(\varkappa ,\mathfrak{I}\mathcal{O}\mathfrak{l}\right)\le {\varpi }_4\left(\varkappa ,\mathfrak{l}\right),\hfill \end{array}$$

(b) 

$${\varpi }_1\left(\varkappa ,\mathfrak{l}\right)+2{\varpi }_2\left(\varkappa ,\mathfrak{l}\right)+{\varpi }_3\left(\varkappa ,\mathfrak{l}\right)+{\varpi }_4\left(\varkappa ,\mathfrak{l}\right)<1,$$

(c) 

$$\begin{array}{ccc}\hfill d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{l}\right)& \le & {\varpi }_1\left(\varkappa ,\mathfrak{l}\right)d\left(\varkappa ,\mathfrak{l}\right)+{\varpi }_2\left(\varkappa ,\mathfrak{l}\right)\left(d\left(\varkappa ,\mathfrak{I}\varkappa \right)+d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)\right)\hfill \\ & & +{\varpi }_3\left(\varkappa ,\mathfrak{l}\right){\displaystyle \frac{d\left(\varkappa ,\mathfrak{I}\varkappa \right)d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}}+{\varpi }_4\left(\varkappa ,\mathfrak{l}\right){\displaystyle \frac{d\left(\mathfrak{l},\mathfrak{I}\varkappa \right)d\left(\varkappa ,\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}},\hfill \end{array}$$

for all $\varkappa ,\mathfrak{l}\in \Omega .$

Then $\mathfrak{I}$ and $\mathcal{O}$ possess a unique CFP.

Remark 4.

By setting the control functions ${\varpi }_1,{\varpi }_2,{\varpi }_3,{\varpi }_4:\Omega \times \Omega \to [0,1)$ to zero and taking the mappings $\mathfrak{I}$ and $\mathcal{O}$ to be identical, several results can be derived from the above statement.

Corollary 31.

Let $\mathfrak{I},\mathcal{O}:\Omega$$\to \Omega$. Assume that there are functions ${\omega }_1,{\omega }_2,{\omega }_3,{\omega }_4:\Omega \to [0,1)$ satisfying the following conditions:

(a) 

$$\begin{array}{ccc}\hfill {\omega }_1\left(\mathfrak{I}\varkappa \right)& \le & {\omega }_1\left(\varkappa \right),{\omega }_1\left(\mathcal{O}\varkappa \right)\le {\omega }_1\left(\varkappa \right)\hfill \\ \hfill {\omega }_2\left(\mathfrak{I}\varkappa \right)& \le & {\omega }_2\left(\varkappa \right),{\omega }_2\left(\mathcal{O}\varkappa \right)\le {\omega }_2\left(\varkappa \right)\hfill \\ \hfill {\omega }_3\left(\mathfrak{I}\varkappa \right)& \le & {\omega }_3\left(\varkappa \right),{\omega }_3\left(\mathcal{O}\varkappa \right)\le {\omega }_3\left(\varkappa \right)\hfill \\ \hfill {\omega }_4\left(\mathfrak{I}\varkappa \right)& \le & {\omega }_4\left(\varkappa \right),{\omega }_4\left(\mathcal{O}\varkappa \right)\le {\omega }_4\left(\varkappa \right),\hfill \end{array}$$

(b) 

$${\omega }_1\left(\varkappa \right)+2{\omega }_2\left(\varkappa \right)+{\omega }_3\left(\varkappa \right)+{\omega }_4\left(\varkappa \right)<1,$$

(c) 

$$\begin{array}{ccc}\hfill d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{l}\right)& \le & {\omega }_1\left(\varkappa \right)d\left(\varkappa ,\mathfrak{l}\right)+{\omega }_2\left(\varkappa \right)\left(d\left(\varkappa ,\mathfrak{I}\varkappa \right)+d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)\right)\hfill \\ & & +{\omega }_3\left(\varkappa \right){\displaystyle \frac{d\left(\varkappa ,\mathfrak{I}\varkappa \right)d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}}+{\omega }_4\left(\varkappa \right){\displaystyle \frac{d\left(\mathfrak{l},\mathfrak{I}\varkappa \right)d\left(\varkappa ,\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}},\hfill \end{array}$$

for all $\varkappa ,\mathfrak{l}\in \Omega .$

Then $\mathfrak{I}$ and $\mathcal{O}$ admit a unique CFP.

Corollary 32.

Let $\mathfrak{I},\mathcal{O}:\Omega$$\to \Omega$. Assume that there exist functions ${\omega }_1,{\omega }_2,{\omega }_3,{\omega }_4:\Omega \to [0,1)$ satisfying the following conditions:

(a) 

$$\begin{array}{ccc}\hfill {\omega }_1\left(\mathcal{O}\mathfrak{I}\varkappa \right)& \le & {\omega }_1\left(\varkappa \right)\hfill \\ \hfill {\omega }_2\left(\mathcal{O}\mathfrak{I}\varkappa \right)& \le & {\omega }_2\left(\varkappa \right)\hfill \\ \hfill {\omega }_3\left(\mathcal{O}\mathfrak{I}\varkappa \right)& \le & {\omega }_3\left(\varkappa \right)\hfill \\ \hfill {\omega }_4\left(\mathcal{O}\mathfrak{I}\varkappa \right)& \le & {\omega }_4\left(\varkappa \right),\hfill \end{array}$$

(b) 

$${\omega }_1\left(\varkappa \right)+2{\omega }_2\left(\varkappa \right)+{\omega }_3\left(\varkappa \right)+{\omega }_4\left(\varkappa \right)<1,$$

(c) 

$$\begin{array}{ccc}\hfill d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{l}\right)& \le & {\omega }_1\left(\varkappa \right)d\left(\varkappa ,\mathfrak{l}\right)+{\omega }_2\left(\varkappa \right)\left(d\left(\varkappa ,\mathfrak{I}\varkappa \right)+d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)\right)\hfill \\ & & +{\omega }_3\left(\varkappa \right){\displaystyle \frac{d\left(\varkappa ,\mathfrak{I}\varkappa \right)d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}}+{\omega }_4\left(\varkappa \right){\displaystyle \frac{d\left(\mathfrak{l},\mathfrak{I}\varkappa \right)d\left(\varkappa ,\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}},\hfill \end{array}$$

for all $\varkappa ,\mathfrak{l}\in \Omega .$

Then $\mathfrak{I}$ and $\mathcal{O}$ possess a unique CFP.

Remark 5.

By assigning zero to the control functions ${\omega }_1,{\omega }_2,{\omega }_3,{\omega }_4:\Omega \to [0,1)$ and considering the mappings $\mathfrak{I}$ and $\mathcal{O}$ as identical, multiple results follow directly from the Corollaries 31 and 32.

Corollary 33.

Let $\mathfrak{I},\mathcal{O}:\Omega$$\to \Omega$. Assume that there exist the constants ${\omega }_1,{\omega }_2,{\omega }_3,{\omega }_4\in [0,1)$ such that ${\omega }_1+2{\omega }_2+{\omega }_3+{\omega }_4<1$ and

$$\begin{array}{ccc}\hfill d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{l}\right)& \le & {\omega }_{1}d\left(\varkappa ,\mathfrak{l}\right)+{\omega }_2\left(d\left(\varkappa ,\mathfrak{I}\varkappa \right)+d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)\right)\hfill \\ & & +{\omega }_3{\displaystyle \frac{d\left(\varkappa ,\mathfrak{I}\varkappa \right)d\left(\mathfrak{l},\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}}+{\omega }_4{\displaystyle \frac{d\left(\mathfrak{l},\mathfrak{I}\varkappa \right)d\left(\varkappa ,\mathcal{O}\mathfrak{l}\right)}{1+d\left(\varkappa ,\mathfrak{l}\right)}},\hfill \end{array}$$

for all $\varkappa ,\mathfrak{l}\in \Omega .$ Then $\mathfrak{I}$ and $\mathcal{O}$ possess a unique CFP.

Corollary 34.

Let $\left(\Omega ,d\right)$ be a complete SMS and $\mathfrak{I},\mathcal{O}:\Omega$$\to \Omega$. Assume that there exist the constants ${\omega }_1\in [0,1)$ such that

$$d\left(\mathfrak{I}\varkappa ,\mathcal{O}\mathfrak{l}\right)\le {\omega }_{1}d\left(\varkappa ,\mathfrak{l}\right),$$

for all $\varkappa ,\mathfrak{l}\in \Omega .$ Then $\mathfrak{I}$ and $\mathcal{O}$ possess a unique CFP.

Proof. 

Take ${\omega }_2={\omega }_3={\omega }_4=0$ in Corollary 33. □

Using this approach, we obtain the principal result of Berzig [8] in the context of SMS.

Corollary 35.

([8]). Let $\mathcal{O}:\Omega$$\to \Omega$. Assume that there exist the constants ${\omega }_1\in [0,1)$ such that

$$d\left(\mathcal{O}\varkappa ,\mathcal{O}\mathfrak{l}\right)\le {\omega }_{1}d\left(\varkappa ,\mathfrak{l}\right),$$

for all $\varkappa ,\mathfrak{l}\in \Omega .$ Then, $\mathcal{O}$ possesses a unique FP.

Remark 6.

Setting the constants ${\omega }_1,{\omega }_2,{\omega }_3,{\omega }_4\in [0,1)$ to zero and taking the mappings $\mathfrak{I}$ and $\mathcal{O}$ to be identical, various results including Kannan-type [3] and Fisher-type FP theorems [4] can be directly obtained from Corollary 33 in the framework of SMS.

7. Existence and Unique Solution of Reimann-Lioville Fractional Integrals

The application of FP theorems to solve differential and integral equations has become a cornerstone in modern mathematical research. This method offers a solid framework for proving the solutions across a broad range of equations., particularly those involving nonlocal operators such as fractional integrals and derivatives. The utility of FP theory in this context is underscored by its ability to handle the complexities introduced by nonlinearity and memory effects inherent in fractional calculus.

In recent years, there has been a significant surge in the application of FP results to fractional integral equations. These equations, which generalize classical integral equations by incorporating fractional-order integrals, are pivotal in modeling phenomena with memory and hereditary properties across various disciplines, including physics, engineering, and biology. The Riemann–Liouville fractional integral, in particular, serves as a fundamental operator in this domain. Recent works have applied FP theory to fractional integral equations, including Zhou et al. [27], Ramaswamy et al. [28] and Deb et al. [29]

Building upon these advancements, this section establishes a FP framework within CVSMS to prove the continuous solutions to a general Riemann–Liouville fractional integral equation. By leveraging the properties of CVSMS and introducing appropriate control functions, we demonstrate that under suitable conditions, the associated operator has a unique FP, which corresponds to the unique continuous solution of the integral equation.

Consider the Riemann–Liouville fractional integral equation of order $\alpha$

$$\varkappa \left(t\right)=f\left(t\right)+{\displaystyle \frac{\lambda }{\Gamma \left(\alpha \right)}}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{\alpha -1}g\left(s,\varkappa \left(s\right)\right)ds,t\in [0,1],$$

(13)

where

• $\varkappa :[0,1]\to \mathbb{R}$ is the unknown continuous function (solution) we aim to find.
• $f:[0,1]\to \mathbb{R}$ is a given continuous function representing the forcing term or initial input.
• $\lambda \in \mathbb{C}$ is a complex constant parameter that can scale the integral term.
• $\alpha \in \mathbb{C}$ (with $R\left(\alpha \right)>0)$ denotes the order of the Riemann–Liouville fractional integral, with $\Gamma \left(\alpha \right)$ being the Gamma function.
• $g:[0,1]\times \mathbb{R}\to \mathbb{R}$ is a continuous function, often assumed to satisfy a Lipschitz condition in the second variable to ensure uniqueness of the solution.
• The integral ${\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{\alpha -1}g\left(s,\varkappa \left(s\right)\right)ds$ is the Riemann–Liouville fractional integral of order $\alpha .$

Let $\Omega =\mathcal{C}\left([0,1],\mathbb{R}\right)$ denote the Banach space of real-valued continuous functions on $[0,1].$ We endow $d:\Omega \times \Omega \to \mathbb{C}$ with the complex-valued suprametric

$$d\left(\varkappa ,\mathfrak{l}\right)=\left(1+i\right)\underset{t\in [0,1]}{sup}\left(\left|\varkappa \left(t\right)-\mathfrak{l}\left(t\right)\right|\right),\forall \varkappa ,\mathfrak{l}\in \Omega$$

where $i=\sqrt{-1}.$ It is easy to verify that $(\Omega ,d)$ is a complete CVSMS with $\aleph \ge 0$, which provides a suitable setting for applying generalized FP results.

Let $g:[0,1]\times \mathbb{R}\to \mathbb{R}$ satisfy a Lipschitz condition in its second argument with Lipschitz constant $L>0$, that is,

$$\left|g\left(s,\varkappa \right)-g\left(s,\mathfrak{l}\right)\right|\le L\left|\varkappa -\mathfrak{l}\right|,$$

for all $s\in$$[0,1],$$\varkappa ,\mathfrak{l}\in \mathbb{R}.$ We define the Riemann–Liouville fractional integral operator $\mathcal{O}:$$\Omega$$\to \Omega$ by

$$\left(\mathcal{O}\varkappa \right)\left(t\right)=f\left(t\right)+{\displaystyle \frac{\lambda }{\Gamma \left(\alpha \right)}}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{\alpha -1}g\left(s,\varkappa \left(s\right)\right)ds,$$

where $f\in \mathcal{C}\left([0,1],\mathbb{R}\right)$ is a given continuous function. To apply a FP theorem in CVSMS, we introduce a control function ${\varpi }_1:\Omega \times \Omega \to [0,1)$ by

$${\varpi }_1\left(\varkappa ,\mathfrak{l}\right)={\displaystyle \frac{\left|\lambda \right|L}{\left|\Gamma \left(\alpha \right)\right|R\left(\alpha \right)}}+{\displaystyle \frac{{∥\varkappa -\mathfrak{l}∥}_{\infty }}{1+{∥\varkappa -\mathfrak{l}∥}_{\infty }}}·\left(1-{\displaystyle \frac{\left|\lambda \right|L}{\left|\Gamma \left(\alpha \right)\right|R\left(\alpha \right)}}\right),$$

where ${∥\varkappa -\mathfrak{l}∥}_{\infty }={sup}_{t\in [0,1]}\left(\left|\varkappa \left(t\right)-\mathfrak{l}\left(t\right)\right|\right).$ The condition $\left|\lambda \right|L<\left|\Gamma \left(\alpha \right)\right|R\left(\alpha \right)$ ensures that ${\varpi }_1\left(\varkappa ,\mathfrak{l}\right)\in [0,1)$ for all $\varkappa \ne \mathfrak{l}.$

With this setup, we can state the main result of this section.

Theorem 2.

Let all the above assumptions hold. Then the operator $\mathcal{O}:\Omega$$\to \Omega$ has a unique FP ${\varkappa }^{*}\in \Omega$. Moreover, ${\varkappa }^{*}$ is the unique continuous solution of the Riemann–Liouville fractional integral Equation (13).

Proof. 

Since g and f are continuous and the Riemann–Liouville integral preserves continuity, $\left(\mathcal{O}\varkappa \right)\left(t\right)\in \mathcal{C}\left([0,1],\mathbb{R}\right),$ so $\mathcal{O}:$$\Omega$$\to \Omega$ is well-defined.

For $\varkappa ,\mathfrak{l}\in \Omega$, we have

$$\begin{array}{ccc}\hfill d\left(\mathcal{O}\varkappa ,\mathcal{O}\mathfrak{l}\right)& =& \left(1+i\right)\underset{t\in [0,1]}{sup}\left(\left|\left(\mathcal{O}\varkappa \right)\left(t\right)-\left(\mathcal{O}\mathfrak{l}\right)\left(t\right)\right|\right)\hfill \\ & =& \left(1+i\right)\underset{t\in [0,1]}{sup}\left|{\displaystyle \frac{\lambda }{\Gamma \left(\alpha \right)}}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{\alpha -1}\left(g\left(s,\varkappa \left(s\right)\right)-g\left(s,\mathfrak{l}\left(s\right)\right)\right)ds\right|\hfill \\ & ⪯& \left(1+i\right)\underset{t\in [0,1]}{sup}{\displaystyle \frac{\left|\lambda \right|}{\left|\Gamma \left(\alpha \right)\right|}}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{Re\left(\alpha \right)-1}\left|g\left(s,\varkappa \left(s\right)\right)-g\left(s,\mathfrak{l}\left(s\right)\right)\right|ds.\hfill \end{array}$$

Using the Lipschitz condition for g, that is,

$$\left|g\left(s,\varkappa \left(s\right)\right)-g\left(s,\mathfrak{l}\left(s\right)\right)\right|\le L\left|\varkappa \left(s\right)-\mathfrak{l}\left(s\right)\right|\le L{∥\varkappa -\mathfrak{l}∥}_{\infty }.$$

So

$$d\left(\mathcal{O}\varkappa ,\mathcal{O}\mathfrak{l}\right)⪯\left(1+i\right)\underset{t\in [0,1]}{sup}{\displaystyle \frac{\left|\lambda \right|L}{\left|\Gamma \left(\alpha \right)\right|}}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{Re\left(\alpha \right)-1}ds·{∥\varkappa -\mathfrak{l}∥}_{\infty }.$$

Since

$$\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{R\left(\alpha \right)-1}ds={\displaystyle \frac{t^{R\left(\alpha \right)}}{R\left(\alpha \right)}}\le {\displaystyle \frac{1}{R\left(\alpha \right)}},t\in [0,1].$$

Therefore

$$\begin{array}{ccc}\hfill d\left(\mathcal{O}\varkappa ,\mathcal{O}\mathfrak{l}\right)& ⪯& \left(1+i\right)\underset{t\in [0,1]}{sup}{\displaystyle \frac{\left|\lambda \right|L}{\left|\Gamma \left(\alpha \right)\right|}}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{R\left(\alpha \right)-1}ds·{∥\varkappa -\mathfrak{l}∥}_{\infty }\hfill \\ & ⪯& {\displaystyle \frac{\left|\lambda \right|L}{\left|\Gamma \left(\alpha \right)\right|R\left(\alpha \right)}}d\left(\varkappa ,\mathfrak{l}\right)\hfill \\ & ⪯& {\varpi }_1\left(\varkappa ,\mathfrak{l}\right)d\left(\varkappa ,\mathfrak{l}\right).\hfill \end{array}$$

Since $\left|\lambda \right|L<\left|\Gamma \left(\alpha \right)\right|R\left(\alpha \right),$ so by the definition of ${\varpi }_1\left(\varkappa ,\mathfrak{l}\right),$ we have

$${\varpi }_1\left(\varkappa ,\mathfrak{l}\right)={\displaystyle \frac{\left|\lambda \right|L}{\left|\Gamma \left(\alpha \right)\right|R\left(\alpha \right)}}+{\displaystyle \frac{{∥\varkappa -\mathfrak{l}∥}_{\infty }}{1+{∥\varkappa -\mathfrak{l}∥}_{\infty }}}·\left(1-{\displaystyle \frac{\left|\lambda \right|L}{\left|\Gamma \left(\alpha \right)\right|R\left(\alpha \right)}}\right)\in [0,1).$$

Therefore, all the assumptions of Corollary 1 are satisfied, with ${\varpi }_2\left(\varkappa ,\mathfrak{l}\right)={\varpi }_3\left(\varkappa ,\mathfrak{l}\right)={\varpi }_4\left(\varkappa ,\mathfrak{l}\right)=0.$ Hence $\mathcal{O}$ has a unique FP ${\varkappa }^{*}\in \Omega$, which is also the unique continuous solution of the integral equation

$$\varkappa \left(t\right)=f\left(t\right)+{\displaystyle \frac{\lambda }{\Gamma \left(\alpha \right)}}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{\alpha -1}g\left(s,\varkappa \left(s\right)\right)ds,t\in [0,1].$$

 □

Example 6.

Consider the Riemann–Liouville fractional integral equation of order $\alpha =0.5.$

$$\varkappa \left(t\right)=1+{\displaystyle \frac{0.4}{\Gamma \left(0.5\right)}}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{-0.5}sin\left(\varkappa \left(s\right)\right)ds,t\in [0,1].$$

Here, the components are:

• $\varkappa :[0,1]\to \mathbb{R}$ is the unknown continuous function (solution) we aim to find.
• $f\left(t\right)=1$ is a continuous forcing term, and a non-constant kernel
• $\lambda =0.4$ is a complex constant,
• $\alpha =0.5$ is the order of the Riemann–Liouville fractional integral, with $\Gamma \left(\alpha \right)$ being the Gamma function.
• $g(s,\varkappa \left(s\right))=sin\left(\varkappa \left(s\right)\right)$ is a continuous function satisfying a Lipschitz condition in the second argument with Lipschitz constant $L=1.$

With these parameters, the condition

$$\left|\lambda \right|L=0.4<\left|\Gamma \left(\alpha \right)\right|R\left(\alpha \right)\approx 0.886$$

is satisfied. Therefore, by the theorem given above, there exists a unique continuous solution ${\varkappa }^{*}\in \mathcal{C}\left([0,1],\mathbb{R}\right)$ of this integral equation.

Remark 7. (Behavior Near the Threshold).

As $\left|\lambda \right|L\to \left|\Gamma \left(\alpha \right)\right|R{\left(\alpha \right)}^{-},$ the constant part

$${\displaystyle \frac{\left|\lambda \right|L}{\left|\Gamma \left(\alpha \right)\right|R\left(\alpha \right)}}$$

tends to 1 from below. Crucially, the control function

$${\varpi }_1\left(\varkappa ,\mathfrak{l}\right)={\displaystyle \frac{\left|\lambda \right|L}{\left|\Gamma \left(\alpha \right)\right|R\left(\alpha \right)}}+{\displaystyle \frac{{∥\varkappa -\mathfrak{l}∥}_{\infty }}{1+{∥\varkappa -\mathfrak{l}∥}_{\infty }}}·\left(1-{\displaystyle \frac{\left|\lambda \right|L}{\left|\Gamma \left(\alpha \right)\right|R\left(\alpha \right)}}\right),$$

satisfies ${\varpi }_1\left(\varkappa ,\mathfrak{l}\right)<1$, for all $\varkappa ,\mathfrak{l}\in \Omega ,$ even arbitrarily close to the threshold. Thus, unlike classical Banach-type inequalities, our condition remains admissible and stable near the boundary, showing that the method is robust.

Remark 8. (Beyond Lipschitz Nonlinearities).

The two-variable control structure ${\varpi }_1\left(\varkappa ,\mathfrak{l}\right)$ naturally accommodates nonlinearities that are not Lipschitz continuous. For instance, suppose g satisfies a Hölder condition

$$\left|g\left(s,\varkappa \right)-g\left(s,\mathfrak{l}\right)\right|\le \varpi {\left|\varkappa -\mathfrak{l}\right|}^{\beta },0<\beta <1.$$

Following the same estimation steps, one obtains

$$d\left(\mathcal{O}\varkappa ,\mathcal{O}\mathfrak{l}\right)⪯{\displaystyle \frac{\left|\lambda \right|L}{\left|\Gamma \left(\alpha \right)\right|R\left(\alpha \right)}}d{\left(\varkappa ,\mathfrak{l}\right)}^{\beta }.$$

Since $d{\left(\varkappa ,\mathfrak{l}\right)}^{\beta }$ is not linear in $d\left(\varkappa ,\mathfrak{l}\right),$ the classical Banach FP theorem does not apply. However, we can incorporate this behavior into a suitably modified control function, e.g.,

$${\tilde{\varpi }}_1\left(\varkappa ,\mathfrak{l}\right)={\displaystyle \frac{\left|\lambda \right|L}{\left|\Gamma \left(\alpha \right)\right|R\left(\alpha \right)}}{∥\varkappa -\mathfrak{l}∥}_{\infty }^{\beta -1}+{\displaystyle \frac{{∥\varkappa -\mathfrak{l}∥}_{\infty }}{1+{∥\varkappa -\mathfrak{l}∥}_{\infty }}}·\left(1-{\displaystyle \frac{\left|\lambda \right|L}{\left|\Gamma \left(\alpha \right)\right|R\left(\alpha \right)}}{∥\varkappa -\mathfrak{l}∥}_{\infty }^{\beta -1}\right),$$

which remains in $[0,1)$ for $\varkappa \ne \mathfrak{l}$ under appropriate parameter restrictions. This illustrates the flexibility of the framework to handle a wider class of non-linearities than standard Banach-type contractions.

Remark 9. (Robustness With Respect to ${∥\varkappa -\mathfrak{l}∥}_{\infty }$). The term

$${\displaystyle \frac{{∥\varkappa -\mathfrak{l}∥}_{\infty }}{1+{∥\varkappa -\mathfrak{l}∥}_{\infty }}}$$

ensures that the control function ${\varpi }_1\left(\varkappa ,\mathfrak{l}\right)$ behaves smoothly and remains bounded away from 1 for all pairs $\varkappa ,\mathfrak{l}\in \Omega .$ When ${∥\varkappa -\mathfrak{l}∥}_{\infty }$ is small, the contraction coefficient is close to the constant ${\displaystyle \frac{\left|\lambda \right|L}{\left|\Gamma \left(\alpha \right)\right|R\left(\alpha \right)}}$ giving a strong contraction similar to the classical case. For large deviations, the term approaches $1,$ but the factor $1-{\displaystyle \frac{\left|\lambda \right|L}{\left|\Gamma \left(\alpha \right)\right|R\left(\alpha \right)}}$ attenuates its contribution, preventing the overall control function from becoming ineffective. This built-in-self adjustment makes the contraction condition more stable across different scales of the distance between functions, which is advantageous for proving convergence of successive approximations even when the initial guess is far from the FP. It provides stronger contraction for small differences and prevents blow-up for large differences.

This makes the proposed contraction more stable and more general than constant-coefficient or one-variable control conditions used in earlier works.

8. Conclusions and Future Work

In this study, the concept of CVSMSs was explored, and common FP theorems for generalized contractions with control functions of two variables were established. As a result, several known results were recovered as special cases: the leading theorems of Abdou [21], Shammaky et al. [22] and Panda et al. [16] within the framework of complex-valued suprametric spaces, the main results of Azam et al. [12], Rouzkard et al. [13], Sintunavarat et al. [14] and Sitthikul et al. [15] in CVMSs, and the prime result of Berzig [8] in SMSs. To demonstrate the novelty and applicability of the proposed results, an illustrative example was provided, and the developed theorems were successfully applied to solve a Riemann–Liouville fractional integral equation.

Future studies will aim to generalize the common FP theorems to multi-valued mappings within the framework of CVSMSs. Furthermore, the investigation of differential and integral inclusions in this setting will be pursued. It is anticipated that these developments will stimulate further research and enhancements, potentially expanding the applicability of the current results.