Common Fixed Point Theorems in Complex-Valued Controlled Metric Spaces with Application

Abstract

The objective of this article is to establish common fixed point results in the background of complex-valued controlled metric spaces for generalized rational contractions. Our findings generalize a number of well-established results in the literature. To highlight the uniqueness of our key finding, we present an example. As a demonstration of the applicability of our principal theorem, we solve the Volterra integral equation.

1. Introduction

Fixed point theory is a well-established and prominent area of mathematics, known for its extensive applications across numerous disciplines. The foundational concept of a metric spaces (MSs), essential to fixed point (FP) theory, was originally introduced by M. Fréchet [1] in 1906. The pioneering work of Banach [2], with his contraction principle, marked the beginning of this field of study. This principle offers a robust method for establishing the existence and uniqueness of FPs for specific classes of mappings. Later on, Fisher [3] initiated the concept of a rational contraction in the purview of MSs and established a FP theorem that generalizes the Banach contraction principle. The inherent symmetry of MSs, especially in preserving distances and relationships between points, contributes to their greater utility. Over time, the significance of this idea has inspired many researchers to explore various extensions and generalizations of metric spaces, leading to important advancements in the field in recent years. Azam et al. [4] defined the idea of a complex-valued metric space (CVMS) in 2011 and presented corresponding common fixed-point (CFP) results for two single-valued functions. The motivation behind the introduction of CVMSs was to address certain rational expressions that cannot be effectively defined within the framework of cone metric spaces (CMSs). As a result, several FP results that are unprovable in CMSs can be handled within CVMSs, making them a distinct subclass of CMSs. The concept of a CMS is rooted in the underlying Banach space structure, which lacks the properties of a division ring. In contrast, CVMSs allow for the study of generalized FP results that involve division. Subsequently, Rouzkard et al. [5] introduced a rational expression into Azam’s contraction framework, thereby generalizing the primary result of Azam. Klin-eam et al. [6] added two new terms in the contractive condition of Rouzkard et al. [5] and generalized the above outcomes. In 2013, Rao [7] further extended the notion of CVMSs by developing the conception of complex-valued b-metric spaces (CVbMS). Mukheimer [8] strengthened the concept of CVbMS and established CFP results for generalized contractive mappings. Dubey [9] extended the result of Rouzkard et al. [5] to complex valued b-metric spaces and proved some common fixed point results. Naimatullah et al. [10,11] advanced this area by proposing the idea of complex-valued extended b-metric spaces (CVEbMS) as a generalization of CVbMSs, and they established fixed-point results for generalized contractions within this broader framework. Afterwards, Ahmad et al. [12] utilized the thought of CVEbMS and proved CFP theorems for rational contractions, incorporating some incontestable control functions. More recently, Aslam et al. [13] introduced a new extension of CVbMS by defining the notion of complex-valued controlled metric space (CVCMS) and established FP theorems. For further information on this topic, we recommend consulting [14,15,16,17,18,19].

In this research article, we obtain CFP results on CVCMSs for generalized rational contractions. By applying our main result, we derive the prominent findings of Fisher [3], Azam et al. [4], Rouzkard et al. [5], Mukheimer [8], and Aslam et al. [13] for self-mappings in CVCMSs. To underscore the innovative nature of our primary theorem, we address the Volterra integral equation.

2. Preliminaries

In this section, we discuss essential definitions from the literature that pertain to our research. The notion of distance was formalized axiomatically by Frechet in the early nineteenth century, introducing the concept of a metric. Since then, numerous scholars have explored this concept, leading to a vast body of research.

Definition 1

([1]). Let $\Xi \ne \varnothing$ and $\mathfrak{e}:\Xi \times \Xi \to {\mathbb{R}}^{+}$ be a function gratifying

(i) 

$0\le \mathfrak{e}(\mathfrak{a},\hslash )$ and $\mathfrak{e}(\mathfrak{a},\hslash )=0$⟺$\mathfrak{a}=\hslash$,

(ii) 

$\mathfrak{e}(\mathfrak{a},\hslash )=\mathfrak{e}(\hslash ,\mathfrak{a}),$

(iii) 

$\mathfrak{e}(\mathfrak{a},\hslash )\le \mathfrak{e}(\mathfrak{a},\nu )+\mathfrak{e}(\nu ,\hslash ),$

for all $\mathfrak{a},\hslash ,\nu \in \Xi ;$ then, $(\Xi ,\mathfrak{e})$ is said to be an MS.

Fisher ([3]) defined the notion of a rational contraction in the framework of MS in such a manner.

Definition 2

([3]). Let $\left(\Xi ,\mathfrak{e}\right)$ be an MS. A mapping $R:\Xi$$\to \Xi$ is defined as a rational contraction if there exist the constants $\kappa \in [0,1)$ with $\rho +\kappa <1$ such that

$$\mathfrak{e}\left(R\mathfrak{a},R\hslash \right)\le \rho e\left(\mathfrak{a},\hslash \right)+\kappa {\displaystyle \frac{\mathfrak{e}\left(\mathfrak{a},R\mathfrak{a}\right)\mathfrak{e}\left(\hslash ,R\hslash \right)}{1+\mathfrak{e}\left(\mathfrak{a},\hslash \right)}},$$

for all $\mathfrak{a},\hslash \in \Xi .$

Theorem 1

([3]). Let $\left(\Xi ,\mathfrak{e}\right)$ be a complete MS and $R:\Xi$$\to \Xi$ be a rational contraction; then, R has a unique FP.

Azam et al. [4] gave the notion of CVMS as follows.

Definition 3

([4]). Let ${\varsigma }_1,{\varsigma }_2\in \mathbb{C}$ (set of complex numbers) and ≾ be a partial order on $\mathbb{C}$ that is characterized as follows:

$${\varsigma }_1\precsim {\varsigma }_2\iff Re\left({\varsigma }_1\right)⩽Re\left({\varsigma }_2\right),Im\left({\varsigma }_1\right)⩽Im\left({\varsigma }_2\right).$$

It follows that

$${\varsigma }_1\precsim {\varsigma }_2,$$

if one of these assertions is satisfied:

$$\begin{array}{ccc}\hfill \left(\mathrm{a}\right)\mathrm{Re}\left({\varsigma }_1\right)& =& \mathrm{Re}\left({\varsigma }_2\right),\mathrm{Im}\left({\varsigma }_1\right)<\mathrm{Im}\left({\varsigma }_2\right),\hfill \\ \hfill \left(\mathrm{b}\right)\mathrm{Re}\left({\varsigma }_1\right)& <& \mathrm{Re}\left({\varsigma }_2\right),\mathrm{Im}\left({\varsigma }_1\right)=\mathrm{Im}\left({\varsigma }_2\right),\hfill \\ \hfill \left(\mathrm{c}\right)\mathrm{Re}\left({\varsigma }_1\right)& <& \mathrm{Re}\left({\varsigma }_2\right),\mathrm{Im}\left({\varsigma }_1\right)<\mathrm{Im}\left({\varsigma }_2\right),\hfill \\ \hfill \left(\mathrm{d}\right)\mathrm{Re}\left({\varsigma }_1\right)& =& \mathrm{Re}\left({\varsigma }_2\right),\mathrm{Im}\left({\varsigma }_1\right)=\mathrm{Im}\left({\varsigma }_2\right).\hfill \end{array}$$

Definition 4

([4]). Let $\Xi \ne \varnothing$ and $\mathfrak{e}:\Xi \times \Xi \to \mathbb{C}$ be a function gratifying

(i) 

$0\precsim \mathfrak{e}(\mathfrak{a},\hslash )$ and $\mathfrak{e}(\mathfrak{a},\hslash )=0$⟺$\mathfrak{a}=\hslash$,

(ii) 

$\mathfrak{e}(\mathfrak{a},\hslash )=\mathfrak{e}(\hslash ,\mathfrak{a}),$

(iii) 

$\mathfrak{e}(\mathfrak{a},\hslash )\precsim \mathfrak{e}(\mathfrak{a},\nu )+\mathfrak{e}(\nu ,\hslash ),$

for all $\mathfrak{a},\hslash ,\nu \in \Xi ;$ then, $(\Xi ,\mathfrak{e})$ is said to be CVMS.

Example 1

([4]). Let $\Xi =[0,1]$ and $\mathfrak{a},\hslash \in \Xi .$ Define $\mathfrak{e}:\Xi \times \Xi \to \mathbb{C}$ by

$$\mathfrak{e}(\mathfrak{a},\hslash )=\left\{\begin{array}{c}0,\mathit{if}\mathfrak{a}=\hslash ,\\ {\displaystyle \frac{i}{2}},\mathit{if}\mathfrak{a}\ne \hslash .\end{array}\right.$$

Then, $(\Xi ,\mathfrak{e})$ is CVMS.

Rao [7] gave the concept of CVbMS as follows.

Definition 5

([7]). Consider a mapping $\mathfrak{e}$ from the Cartesian product $\Xi \times \Xi$ to the complex numbers, where Ξ is the empty set and $s\ge 1$ such that

(i) 

$0\precsim \mathfrak{e}(\mathfrak{a},\hslash )$ and $\mathfrak{e}(\mathfrak{a},\hslash )=0$⟺$\mathfrak{a}=\hslash$,

(ii) 

$\mathfrak{e}(\mathfrak{a},\hslash )=\mathfrak{e}(\hslash ,\mathfrak{a}),$

(iii) 

$\mathfrak{e}(\mathfrak{a},\hslash )\precsim s\left[\mathfrak{e}(\mathfrak{a},\nu )+\mathfrak{e}(\nu ,\hslash )\right],$

for all $\mathfrak{a},\hslash ,\nu \in \Xi ;$ then, $(\Xi ,\mathfrak{e})$ is called a CVbMS.

Example 2

([7]). Let $\Xi =[0,1].$ Define $\mathfrak{e}:\Xi \times \Xi \to \mathbb{C}$ by

$$\mathfrak{e}(\mathfrak{a},\hslash )={|\mathfrak{a}-\hslash |}^2+i{|\mathfrak{a}-\hslash |}^2$$

for all $\mathfrak{a},\hslash \in \Xi ;$ then, $(\Xi ,\mathfrak{e})$ is CVbMS with $s=2$.

Naimatullah et al. [10] defined the concept of CVEbMS in this way.

Definition 6

([10]). Let $\Xi \ne \varnothing$, $\varphi :\Xi \times \Xi \to [1,\infty )$ and $\mathfrak{e}:\Xi \times \Xi \to \mathbb{C}$ be a function such that

(i) 

$0\precsim \mathfrak{e}(\mathfrak{a},\hslash )$ and $\mathfrak{e}(\mathfrak{a},\hslash )=0$⟺$\mathfrak{a}=\hslash$;

(ii) 

$\mathfrak{e}(\mathfrak{a},\hslash )=\mathfrak{e}(\hslash ,\mathfrak{a}),$

(iii) 

$\mathfrak{e}(\mathfrak{a},\hslash )\precsim \varphi (\mathfrak{a},\hslash )\left[\mathfrak{e}(\mathfrak{a},\nu )+\mathfrak{e}(\nu ,\hslash )\right],$

for all $\mathfrak{a},\hslash ,\nu \in \Xi ;$ then, $(\Xi ,\mathfrak{e})$ is called a CVEbMS.

Example 3

([10]). Let $\Xi \ne ⌀$ and $\varphi :\Xi \times \Xi \to [1,\infty )$ be defined by

$$\varphi (\mathfrak{a},\hslash )={\displaystyle \frac{1+\mathfrak{a}+\hslash }{\mathfrak{a}+\hslash }}$$

and $\mathfrak{e}:\Xi \times \Xi \to \mathbb{C}$ by

(i) $\mathfrak{e}(\mathfrak{a},\hslash )={\displaystyle \frac{i}{\mathfrak{a}\hslash }},$$\forall 0<\mathfrak{a},\hslash \le 1;$

(ii) $\mathfrak{e}(\mathfrak{a},\hslash )=0\iff \mathfrak{a}=\hslash ,$$\forall 0\le \mathfrak{a},\hslash \le 1;$

(iii) $\mathfrak{e}(\mathfrak{a},0)=\mathfrak{e}(0,\mathfrak{a})={\displaystyle \frac{i}{\mathfrak{a}}},\forall 0<\mathfrak{a}\le 1.$

Then, $(\Xi ,\mathfrak{e})$ is a CVEbMS.

Recently, Aslam et al. [13] defined the notion of CVCMS in the following way.

Definition 7

([13]). Let $\Xi \ne \varnothing$, $\phi :\Xi \times \Xi \to [1,\infty )$ and $\mathfrak{e}$ is a complex-valued function on $\Xi \times \Xi$ satisfying

(i) 

$0\precsim \mathfrak{e}(\mathfrak{a},\hslash )$ and $\mathfrak{e}(\mathfrak{a},\hslash )=0$⟺$\mathfrak{a}=\hslash$;

(ii) 

$\mathfrak{e}(\mathfrak{a},\hslash )=\mathfrak{e}(\hslash ,\mathfrak{a}),$

(iii) 

$\mathfrak{e}(\mathfrak{a},\hslash )\precsim \phi (\mathfrak{a},\nu )\mathfrak{e}(\mathfrak{a},\nu )+\phi (\nu ,\hslash )\mathfrak{e}(\nu ,\hslash ),$ for all $\mathfrak{a},\hslash ,\nu \in \Xi .$

Then, $(\Xi ,\mathfrak{e})$ is called a CVCMS.

Remark 1.

If $\phi (\mathfrak{a},\hslash )=s\ge 1,$ for all $\mathfrak{a},\hslash \in \Xi$, it follows that $(\Xi ,\mathfrak{e})$ is a CVbMS. Consequently, every CVbMS is a CVCMS. Moreover, under the condition $\varphi =\phi ,$ a CVCMS does not generally qualify as a CVEbMS when considering the same function.

Example 4

([13]). Let $\Xi =[0,\infty )$ and $\phi :\Xi \times \Xi \to [1,\infty )$ be a function defined by

$$\phi (\mathfrak{a},\hslash )=\left\{\begin{array}{c}1,\mathit{if}\mathfrak{a},\hslash \in [0,1]\\ 1+\mathfrak{a}+\hslash ,\mathit{otherwise}\end{array}\right.$$

and $\mathfrak{e}:\Xi \times \Xi \to \mathbb{C}$ by

$$\mathfrak{e}(\mathfrak{a},\hslash )=\left\{\begin{array}{c}0,\mathit{if}\mathfrak{a}=\hslash \\ i,\mathit{if}\mathfrak{a}\ne \hslash .\end{array}\right.$$

Then, $(\Xi ,\mathfrak{e})$ is a CVCMS.

Example 5.

Let $\Xi =U\cup V$ with $U=\left\{\left({\displaystyle \frac{1}{n}}\right):n\in \mathbb{N}\right\}$ and V is set of positive integers. Define $\phi :\Xi \times \Xi \to [1,\infty )$ by

$$\phi (\mathfrak{a},\hslash )=5r$$

where $r>0$ and $\mathfrak{e}:\Xi \times \Xi \to \mathbb{C}$ by

$$\mathfrak{e}(\mathfrak{a},\hslash )=\left\{\begin{array}{c}0,\mathit{if}\mathfrak{a}=\hslash \\ 2ri,\mathit{if}\mathfrak{a},\hslash \in U\\ {\displaystyle \frac{ri}{2}},\mathit{otherwise}.\end{array}\right.$$

Then, $(\Xi ,\phi ,\mathfrak{e})$ is a CVCMS.

Lemma 1

([13]). Let $(\Xi ,\mathfrak{e})$ be a CVCMS and let $\left\{{\mathfrak{a}}_n\right\}$$\subseteq \Xi$. Then, $\left\{{\mathfrak{a}}_n\right\}$ converges to $\mathfrak{a}$ iff $\left|\mathfrak{e}({\mathfrak{a}}_n,\mathfrak{a})\right|\to 0$ as $n\to \infty .$

Lemma 2

([13]). Let $(\Xi ,\mathfrak{e})$ be a CVCMS and let $\left\{{\mathfrak{a}}_n\right\}$$\subseteq \Xi$. Then, $\left\{{\mathfrak{a}}_n\right\}$ is a Cauchy sequence iff $\left|\mathfrak{e}({\mathfrak{a}}_n,{\mathfrak{a}}_{n+m})\right|\to 0$ as $n\to \infty ,$ where $m\in \mathbb{N}.$

3. Main Results

This section is dedicated to presenting novel CFP theorems for self-mappings within the framework of complete CVCMSs.

Theorem 2.

Let $\left(\Xi ,\mathfrak{e}\right)$ be a complete CVCMS, $\phi :\Xi \times \Xi \to [1,\infty )$ and let $R,T:\Xi$$\to \Xi$. Assume that there exist the constants $\rho ,\kappa ,\mu \in [0,1)$ with $\rho +\kappa +\mu <1$ such that

$$\mathfrak{e}\left(R\mathfrak{a},T\hslash \right)⪯\rho \mathfrak{e}\left(\mathfrak{a},\hslash \right)+\kappa {\displaystyle \frac{\mathfrak{e}\left(\mathfrak{a},R\mathfrak{a}\right)\mathfrak{e}\left(\hslash ,T\hslash \right)}{1+\mathfrak{e}\left(\mathfrak{a},\hslash \right)}}+\mu {\displaystyle \frac{\mathfrak{e}\left(\hslash ,R\mathfrak{a}\right)\mathfrak{e}\left(\mathfrak{a},T\hslash \right)}{1+\mathfrak{e}\left(\mathfrak{a},\hslash \right)}}.$$

(1)

For a point ${\mathfrak{a}}_0\in \Xi$ and a sequence {${\mathfrak{a}}_n$} defined by

$${\mathfrak{a}}_{2n+1}=T{\mathfrak{a}}_{2n}\mathit{and}{\mathfrak{a}}_{2n+2}=R{\mathfrak{a}}_{2n+1},$$

(2)

suppose that ${lim}_{n\to \infty }\phi \left({\mathfrak{a}}_n,\mathfrak{a}\right)$ and ${lim}_{n\to \infty }\phi \left(\mathfrak{a},{\mathfrak{a}}_n\right)$ exist and are finite and

$$\underset{m\ge 1}{max}\underset{i\to \infty }{lim}{\displaystyle \frac{\phi \left({\mathfrak{a}}_{i+1},{\mathfrak{a}}_{i+2}\right)}{\phi \left({\mathfrak{a}}_i,{\mathfrak{a}}_{i+1}\right)}}\phi \left({\mathfrak{a}}_{i+1},{\mathfrak{a}}_m\right)<{\displaystyle \frac{1}{\pi }},$$

(3)

where $\pi ={\displaystyle \frac{\rho }{1-\kappa }}<1$. Then, R and T have a unique CFP.

Proof. 

Let ${\mathfrak{a}}_0$ be any point in $\Xi$ and the sequence {${\mathfrak{a}}_n$} be defined by (2). By (1), we have

$$\begin{array}{ccc}\hfill \mathfrak{e}\left({\mathfrak{a}}_{2n},{\mathfrak{a}}_{2n+1}\right)& =& \mathfrak{e}\left(R{\mathfrak{a}}_{2n-1},T{\mathfrak{a}}_{2n}\right)⪯\rho \mathfrak{e}\left({\mathfrak{a}}_{2n-1},{\mathfrak{a}}_{2n}\right)+\kappa {\displaystyle \frac{\mathfrak{e}\left({\mathfrak{a}}_{2n-1},R{\mathfrak{a}}_{2n-1}\right)\mathfrak{e}\left({\mathfrak{a}}_{2n},T{\mathfrak{a}}_{2n}\right)}{1+\mathfrak{e}\left({\mathfrak{a}}_{2n-1},{\mathfrak{a}}_{2n}\right)}}\hfill \\ & & +\mu {\displaystyle \frac{\mathfrak{e}\left({\mathfrak{a}}_{2n},R{\mathfrak{a}}_{2n-1}\right)\mathfrak{e}\left({\mathfrak{a}}_{2n-1},T{\mathfrak{a}}_{2n}\right)}{1+\mathfrak{e}\left({\mathfrak{a}}_{2n-1},{\mathfrak{a}}_{2n}\right)}}\hfill \\ & =& \rho \mathfrak{e}\left({\mathfrak{a}}_{2n-1},{\mathfrak{a}}_{2n}\right)+\kappa {\displaystyle \frac{\mathfrak{e}\left({\mathfrak{a}}_{2n-1},{\mathfrak{a}}_{2n}\right)\mathfrak{e}\left({\mathfrak{a}}_{2n},{\mathfrak{a}}_{2n+1}\right)}{1+\mathfrak{e}\left({\mathfrak{a}}_{2n-1},{\mathfrak{a}}_{2n}\right)}}\hfill \end{array}$$

which implies that

$$\begin{array}{ccc}\hfill \left|\mathfrak{e}\left({\mathfrak{a}}_{2n},{\mathfrak{a}}_{2n+1}\right)\right|& \le & \rho \left|\mathfrak{e}\left({\mathfrak{a}}_{2n-1},{\mathfrak{a}}_{2n}\right)\right|+\kappa {\displaystyle \frac{\left|\mathfrak{e}\left({\mathfrak{a}}_{2n-1},{\mathfrak{a}}_{2n}\right)\right|}{\left|1+\mathfrak{e}\left({\mathfrak{a}}_{2n-1},{\mathfrak{a}}_{2n}\right)\right|}}\left|\mathfrak{e}\left({\mathfrak{a}}_{2n},{\mathfrak{a}}_{2n+1}\right)\right|\hfill \\ & \le & \rho \left|\mathfrak{e}\left({\mathfrak{a}}_{2n-1},{\mathfrak{a}}_{2n}\right)\right|+\kappa \left|\mathfrak{e}\left({\mathfrak{a}}_{2n},{\mathfrak{a}}_{2n+1}\right)\right|.\hfill \end{array}$$

It follows that

$$\left(1-\kappa \right)\left|\mathfrak{e}\left({\mathfrak{a}}_{2n},{\mathfrak{a}}_{2n+1}\right)\right|\le \rho \left|\mathfrak{e}\left({\mathfrak{a}}_{2n-1},{\mathfrak{a}}_{2n}\right)\right|$$

that is,

$$\left|\mathfrak{e}\left({\mathfrak{a}}_{2n},{\mathfrak{a}}_{2n+1}\right)\right|\le {\displaystyle \frac{\rho }{1-\kappa }}\left|\mathfrak{e}\left({\mathfrak{a}}_{2n-1},{\mathfrak{a}}_{2n}\right)\right|.$$

(4)

Similarly, we have

$$\begin{array}{ccc}\hfill \mathfrak{e}\left({\mathfrak{a}}_{2n+1},{\mathfrak{a}}_{2n+2}\right)& =& \mathfrak{e}\left(T{\mathfrak{a}}_{2n},R{\mathfrak{a}}_{2n+1}\right)\hfill \\ & =& \mathfrak{e}\left(R{\mathfrak{a}}_{2n+1},T{\mathfrak{a}}_{2n}\right)⪯\rho \mathfrak{e}\left({\mathfrak{a}}_{2n+1},{\mathfrak{a}}_{2n}\right)+\kappa {\displaystyle \frac{\mathfrak{e}\left({\mathfrak{a}}_{2n+1},R{\mathfrak{a}}_{2n+1}\right)\mathfrak{e}\left({\mathfrak{a}}_{2n},T{\mathfrak{a}}_{2n}\right)}{1+\mathfrak{e}\left({\mathfrak{a}}_{2n+1},{\mathfrak{a}}_{2n}\right)}}\hfill \\ & & +\mu {\displaystyle \frac{\mathfrak{e}\left({\mathfrak{a}}_{2n},R{\mathfrak{a}}_{2n+1}\right)\mathfrak{e}\left({\mathfrak{a}}_{2n+1},T{\mathfrak{a}}_{2n}\right)}{1+\mathfrak{e}\left({\mathfrak{a}}_{2n+1},{\mathfrak{a}}_{2n}\right)}}\hfill \\ & =& \rho \mathfrak{e}\left({\mathfrak{a}}_{2n+1},{\mathfrak{a}}_{2n}\right)+\kappa {\displaystyle \frac{\mathfrak{e}\left({\mathfrak{a}}_{2n+1},{\mathfrak{a}}_{2n+2}\right)\mathfrak{e}\left({\mathfrak{a}}_{2n},{\mathfrak{a}}_{2n+1}\right)}{1+\mathfrak{e}\left({\mathfrak{a}}_{2n+1},{\mathfrak{a}}_{2n}\right)}},\hfill \end{array}$$

which implies that

$$\begin{array}{ccc}\hfill \left|\mathfrak{e}\left({\mathfrak{a}}_{2n+1},{\mathfrak{a}}_{2n+2}\right)\right|& \le & \rho \left|\mathfrak{e}\left({\mathfrak{a}}_{2n+1},{\mathfrak{a}}_{2n}\right)\right|+\kappa \left|\mathfrak{e}\left({\mathfrak{a}}_{2n+1},{\mathfrak{a}}_{2n+2}\right)\right|{\displaystyle \frac{\left|\mathfrak{e}\left({\mathfrak{a}}_{2n},{\mathfrak{a}}_{2n+1}\right)\right|}{\left|1+\mathfrak{e}\left({\mathfrak{a}}_{2n+1},{\mathfrak{a}}_{2n}\right)\right|}}\hfill \\ & \le & \rho \left|\mathfrak{e}\left({\mathfrak{a}}_{2n+1},{\mathfrak{a}}_{2n}\right)\right|+\kappa \left|\mathfrak{e}\left({\mathfrak{a}}_{2n+1},{\mathfrak{a}}_{2n+2}\right)\right|\hfill \\ & =& \rho \left|\mathfrak{e}\left({\mathfrak{a}}_{2n},{\mathfrak{a}}_{2n+1}\right)\right|+\kappa \left|\mathfrak{e}\left({\mathfrak{a}}_{2n+1},{\mathfrak{a}}_{2n+2}\right)\right|.\hfill \end{array}$$

This entails that

$$\left(1-\kappa \right)\left|\mathfrak{e}\left({\mathfrak{a}}_{2n+1},{\mathfrak{a}}_{2n+2}\right)\right|\le \rho \left|\mathfrak{e}\left({\mathfrak{a}}_{2n},{\mathfrak{a}}_{2n+1}\right)\right|,$$

which yields that

$$\left|\mathfrak{e}\left({\mathfrak{a}}_{2n+1},{\mathfrak{a}}_{2n+2}\right)\right|\le {\displaystyle \frac{\rho }{1-\kappa }}\left|\mathfrak{e}\left({\mathfrak{a}}_{2n},{\mathfrak{a}}_{2n+1}\right)\right|.$$

(5)

Let $\pi =$${\displaystyle \frac{\rho }{1-\kappa }}<1.$ Then, from (4) and (5), we have

$$\left|\mathfrak{e}\left({\mathfrak{a}}_n,{\mathfrak{a}}_{n+1}\right)\right|\le \pi \left|\mathfrak{e}\left({\mathfrak{a}}_{n-1},{\mathfrak{a}}_n\right)\right|$$

for all $n\in \mathbb{N}.$ Inductively, we can construct a sequence $\left\{{\mathfrak{a}}_n\right\}$ in $\Xi$ such that

$$\left|\mathfrak{e}\left({\mathfrak{a}}_n,{\mathfrak{a}}_{n+1}\right)\right|\le \pi \left|\mathfrak{e}\left({\mathfrak{a}}_{n-1},{\mathfrak{a}}_n\right)\right|\le {\pi }^2\left|\mathfrak{e}\left({\mathfrak{a}}_{n-2},{\mathfrak{a}}_{n-1}\right)\right|\le ···\le {\pi }^n\left|\mathfrak{e}\left({\mathfrak{a}}_0,{\mathfrak{a}}_1\right)\right|$$

for all $n\in \mathbb{N}.$ Now, for $m>n$, we obtain

$$\begin{array}{ccc}\hfill \left|\mathfrak{e}\left({\mathfrak{a}}_n,{\mathfrak{a}}_m\right)\right|& \le & \phi \left({\mathfrak{a}}_n,{\mathfrak{a}}_{n+1}\right)\left|\mathfrak{e}\left({\mathfrak{a}}_n,{\mathfrak{a}}_{n+1}\right)\right|+\phi \left({\mathfrak{a}}_{n+1},{\mathfrak{a}}_m\right)\left|\mathfrak{e}\left({\mathfrak{a}}_{n+1},{\mathfrak{a}}_m\right)\right|\hfill \\ & \le & \phi \left({\mathfrak{a}}_n,{\mathfrak{a}}_{n+1}\right)\left|\mathfrak{e}\left({\mathfrak{a}}_n,{\mathfrak{a}}_{n+1}\right)\right|+\phi \left({\mathfrak{a}}_{n+1},{\mathfrak{a}}_m\right)\phi \left({\mathfrak{a}}_{n+1},{\mathfrak{a}}_{n+2}\right)\left|\mathfrak{e}\left({\mathfrak{a}}_{n+1},{\mathfrak{a}}_{n+2}\right)\right|\hfill \\ & & +\phi \left({\mathfrak{a}}_{n+1},{\mathfrak{a}}_m\right)\phi \left({\mathfrak{a}}_{n+2},{\mathfrak{a}}_m\right)\left|\mathfrak{e}\left({\mathfrak{a}}_{n+2},{\mathfrak{a}}_m\right)\right|\hfill \\ & \le & \phi \left({\mathfrak{a}}_n,{\mathfrak{a}}_{n+1}\right)\left|\mathfrak{e}\left({\mathfrak{a}}_n,{\mathfrak{a}}_{n+1}\right)\right|+\phi \left({\mathfrak{a}}_{n+1},{\mathfrak{a}}_m\right)\phi \left({\mathfrak{a}}_{n+1},{\mathfrak{a}}_{n+2}\right)\left|\mathfrak{e}\left({\mathfrak{a}}_{n+1},{\mathfrak{a}}_{n+2}\right)\right|\hfill \\ & & +\phi \left({\mathfrak{a}}_{n+1},{\mathfrak{a}}_m\right)\phi \left({\mathfrak{a}}_{n+2},{\mathfrak{a}}_m\right)\phi \left({\mathfrak{a}}_{n+2},{\mathfrak{a}}_{n+3}\right)\left|\mathfrak{e}\left({\mathfrak{a}}_{n+2},{\mathfrak{a}}_{n+3}\right)\right|\hfill \\ & & +\phi \left({\mathfrak{a}}_{n+1},{\mathfrak{a}}_m\right)\phi \left({\mathfrak{a}}_{n+2},{\mathfrak{a}}_m\right)\phi \left({\mathfrak{a}}_{n+3},{\mathfrak{a}}_m\right)\left|\mathfrak{e}\left({\mathfrak{a}}_{n+3},{\mathfrak{a}}_m\right)\right|\hfill \\ & \le & ···\le \phi \left({\mathfrak{a}}_n,{\mathfrak{a}}_{n+1}\right)\left|\mathfrak{e}\left({\mathfrak{a}}_n,{\mathfrak{a}}_{n+1}\right)\right|+\sum _{l=n+1}^{m-2}\left(\prod _{j=n+1}^l\phi \left({\mathfrak{a}}_j,{\mathfrak{a}}_m\right)\right)\phi \left({\mathfrak{a}}_l,{\mathfrak{a}}_{l+1}\right)\left|\mathfrak{e}\left({\mathfrak{a}}_l,{\mathfrak{a}}_{l+1}\right)\right|\hfill \\ & & +\prod _{k=n+1}^{m-1}\phi \left({\mathfrak{a}}_k,{\mathfrak{a}}_m\right)\left|\mathfrak{e}\left({\mathfrak{a}}_{m-1},{\mathfrak{a}}_m\right)\right|\hfill \\ & \le & \phi \left({\mathfrak{a}}_n,{\mathfrak{a}}_{n+1}\right){\pi }^n\left|\mathfrak{e}\left({\mathfrak{a}}_0,{\mathfrak{a}}_1\right)\right|+\sum _{l=n+1}^{m-2}\left(\prod _{j=n+1}^l\phi \left({\mathfrak{a}}_j,{\mathfrak{a}}_m\right)\right)\phi \left({\mathfrak{a}}_l,{\mathfrak{a}}_{l+1}\right){\pi }^l\left|\mathfrak{e}\left({\mathfrak{a}}_0,{\mathfrak{a}}_1\right)\right|\hfill \\ & & +\prod _{k=n+1}^{m-1}\phi \left({\mathfrak{a}}_k,{\mathfrak{a}}_m\right){\pi }^{m-1}\left|\mathfrak{e}\left({\mathfrak{a}}_0,{\mathfrak{a}}_1\right)\right|\hfill \\ & \le & \phi \left({\mathfrak{a}}_n,{\mathfrak{a}}_{n+1}\right){\pi }^n\left|\mathfrak{e}\left({\mathfrak{a}}_0,{\mathfrak{a}}_1\right)\right|+\sum _{l=n+1}^{m-2}\left(\prod _{j=n+1}^l\phi \left({\mathfrak{a}}_j,{\mathfrak{a}}_m\right)\right)\phi \left({\mathfrak{a}}_l,{\mathfrak{a}}_{l+1}\right){\pi }^l\left|\mathfrak{e}\left({\mathfrak{a}}_0,{\mathfrak{a}}_1\right)\right|\hfill \\ & & +\left(\prod _{k=n+1}^{m-1}\phi \left({\mathfrak{a}}_k,{\mathfrak{a}}_m\right)\right){\pi }^{m-1}\phi \left({\mathfrak{a}}_{m-1},{\mathfrak{a}}_m\right)\left|\mathfrak{e}\left({\mathfrak{a}}_0,{\mathfrak{a}}_1\right)\right|\hfill \\ & =& \phi \left({\mathfrak{a}}_n,{\mathfrak{a}}_{n+1}\right){\pi }^n\left|\mathfrak{e}\left({\mathfrak{a}}_0,{\mathfrak{a}}_1\right)\right|+\sum _{l=n+1}^{m-1}\left(\prod _{j=n+1}^l\phi \left({\mathfrak{a}}_j,{\mathfrak{a}}_m\right)\right)\phi \left({\mathfrak{a}}_l,{\mathfrak{a}}_{l+1}\right){\pi }^l\left|\mathfrak{e}\left({\mathfrak{a}}_0,{\mathfrak{a}}_1\right)\right|\hfill \\ & \le & \phi \left({\mathfrak{a}}_n,{\mathfrak{a}}_{n+1}\right){\pi }^n\left|\mathfrak{e}\left({\mathfrak{a}}_0,{\mathfrak{a}}_1\right)\right|+\sum _{l=n+1}^{m-1}\left(\prod _{j=0}^l\phi \left({\mathfrak{a}}_j,{\mathfrak{a}}_m\right)\right)\phi \left({\mathfrak{a}}_l,{\mathfrak{a}}_{l+1}\right){\pi }^l\left|\mathfrak{e}\left({\mathfrak{a}}_0,{\mathfrak{a}}_1\right)\right|.\hfill \end{array}$$

Let

$$S_u=\sum _{l=0}^u\left(\prod _{j=0}^l\phi \left({\mathfrak{a}}_j,{\mathfrak{a}}_m\right)\right)\phi \left({\mathfrak{a}}_l,{\mathfrak{a}}_{l+1}\right){\pi }^l.$$

Hence, we have

$$\left|\mathfrak{e}\left({\mathfrak{a}}_n,{\mathfrak{a}}_m\right)\right|\le \left|\mathfrak{e}\left({\mathfrak{a}}_0,{\mathfrak{a}}_1\right)\right|\left[{\pi }^n\phi \left({\mathfrak{a}}_n,{\mathfrak{a}}_{n+1}\right)+\left(S_{m-1}-S_n\right)\right].$$

(6)

Using the inequality in (3) and the ratio test, we determine that ${lim}_{n,m\to \infty }{S}_n$ exists, indicating that {$S_n$} is Cauchy. If we let $n,m\to \infty$ in the inequality in (6), we arrive at

$$\left|\mathfrak{e}\left({\mathfrak{a}}_n,{\mathfrak{a}}_m\right)\right|\to 0.$$

By the conclusion of Lemma (2), $\left\{{\mathfrak{a}}_n\right\}$ is a Cauchy sequence. Due to the completeness of $\Xi$, there then exists a point ${\mathfrak{a}}^{*}$ such that ${\mathfrak{a}}_n\to {\mathfrak{a}}^{*}\in \Xi$ as $n\to \infty .$ Now, we show that ${\mathfrak{a}}^{*}$ is an FP of $R.$ From (1), we have

$$\begin{array}{ccc}\hfill \mathfrak{e}\left({\mathfrak{a}}^{*},R{\mathfrak{a}}^{*}\right)& ⪯& \phi \left({\mathfrak{a}}^{*},{\mathfrak{a}}_{2n+2}\right)\mathfrak{e}\left({\mathfrak{a}}^{*},{\mathfrak{a}}_{2n+2}\right)+\phi \left({\mathfrak{a}}_{2n+2},R{\mathfrak{a}}^{*}\right)\mathfrak{e}\left({\mathfrak{a}}_{2n+2},R{\mathfrak{a}}^{*}\right)\hfill \\ & ⪯& \phi \left({\mathfrak{a}}^{*},{\mathfrak{a}}_{2n+2}\right)\mathfrak{e}\left({\mathfrak{a}}^{*},{\mathfrak{a}}_{2n+2}\right)+\phi \left({\mathfrak{a}}_{2n+2},R{\mathfrak{a}}^{*}\right)\mathfrak{e}\left(T{\mathfrak{a}}_{2n+1},R{\mathfrak{a}}^{*}\right)\hfill \\ & =& \phi \left({\mathfrak{a}}^{*},{\mathfrak{a}}_{2n+2}\right)\mathfrak{e}\left({\mathfrak{a}}^{*},{\mathfrak{a}}_{2n+2}\right)+\phi \left({\mathfrak{a}}_{2n+2},R{\mathfrak{a}}^{*}\right)\mathfrak{e}\left(R{\mathfrak{a}}^{*},T{\mathfrak{a}}_{2n+1}\right)\hfill \\ & ⪯& \phi \left({\mathfrak{a}}^{*},{\mathfrak{a}}_{2n+2}\right)\mathfrak{e}\left({\mathfrak{a}}^{*},{\mathfrak{a}}_{2n+2}\right)+\phi \left({\mathfrak{a}}_{2n+2},R{\mathfrak{a}}^{*}\right)\left(\begin{array}{c}\rho \mathfrak{e}\left({\mathfrak{a}}^{*},{\mathfrak{a}}_{2n+1}\right)\\ +\kappa {\displaystyle \frac{\mathfrak{e}\left({\mathfrak{a}}^{*},R{\mathfrak{a}}^{*}\right)\mathfrak{e}\left({\mathfrak{a}}_{2n+1},T{\mathfrak{a}}_{2n+1}\right)}{1+\mathfrak{e}\left({\mathfrak{a}}^{*},{\mathfrak{a}}_{2n+1}\right)}}\\ +\mu {\displaystyle \frac{\mathfrak{e}\left({\mathfrak{a}}_{2n+1},R{\mathfrak{a}}^{*}\right)\mathfrak{e}\left({\mathfrak{a}}^{*},T{\mathfrak{a}}_{2n+1}\right)}{1+\mathfrak{e}\left({\mathfrak{a}}^{*},{\mathfrak{a}}_{2n+1}\right)}}\end{array}\right)\hfill \\ & =& \phi \left({\mathfrak{a}}^{*},{\mathfrak{a}}_{2n+2}\right)\mathfrak{e}\left({\mathfrak{a}}^{*},{\mathfrak{a}}_{2n+2}\right)+\phi \left({\mathfrak{a}}_{2n+2},R{\mathfrak{a}}^{*}\right)\left(\begin{array}{c}\rho \mathfrak{e}\left({\mathfrak{a}}^{*},{\mathfrak{a}}_{2n+1}\right)\\ +\kappa {\displaystyle \frac{\mathfrak{e}\left({\mathfrak{a}}^{*},R{\mathfrak{a}}^{*}\right)\mathfrak{e}\left({\mathfrak{a}}_{2n+1},{\mathfrak{a}}_{2n+2}\right)}{1+\mathfrak{e}\left({\mathfrak{a}}^{*},{\mathfrak{a}}_{2n+1}\right)}}\\ +\mu {\displaystyle \frac{\mathfrak{e}\left({\mathfrak{a}}_{2n+1},R{\mathfrak{a}}^{*}\right)\mathfrak{e}\left({\mathfrak{a}}^{*},{\mathfrak{a}}_{2n+2}\right)}{1+\mathfrak{e}\left({\mathfrak{a}}^{*},{\mathfrak{a}}_{2n+1}\right)}}\end{array}\right)\hfill \end{array}$$

This implies that

$$\left|\mathfrak{e}\left({\mathfrak{a}}^{*},R{\mathfrak{a}}^{*}\right)\right|\le \phi \left({\mathfrak{a}}^{*},{\mathfrak{a}}_{2n+2}\right)\left|\mathfrak{e}\left({\mathfrak{a}}^{*},{\mathfrak{a}}_{2n+2}\right)\right|+\phi \left({\mathfrak{a}}_{2n+2},R{\mathfrak{a}}^{*}\right)\left(\begin{array}{c}\rho \left|\mathfrak{e}\left({\mathfrak{a}}^{*},{\mathfrak{a}}_{2n+1}\right)\right|\\ +\kappa \left|\mathfrak{e}\left({\mathfrak{a}}^{*},R{\mathfrak{a}}^{*}\right)\right|{\displaystyle \frac{\left|\mathfrak{e}\left({\mathfrak{a}}_{2n+1},{\mathfrak{a}}_{2n+2}\right)\right|}{\left|1+\mathfrak{e}\left({\mathfrak{a}}^{*},{\mathfrak{a}}_{2n+1}\right)\right|}}\\ +\mu \left|\mathfrak{e}\left({\mathfrak{a}}_{2n+1},R{\mathfrak{a}}^{*}\right)\right|{\displaystyle \frac{\left|\mathfrak{e}\left({\mathfrak{a}}^{*},{\mathfrak{a}}_{2n+2}\right)\right|}{\left|1+\mathfrak{e}\left({\mathfrak{a}}^{*},{\mathfrak{a}}_{2n+1}\right)\right|}}\end{array}\right).$$

Letting $n\to \infty ,$ we have $\left|\mathfrak{e}\left({\mathfrak{a}}^{*},R{\mathfrak{a}}^{*}\right)\right|=0.$ Thus, ${\mathfrak{a}}^{*}=R{\mathfrak{a}}^{*}.$ Now, we prove that ${\mathfrak{a}}^{*}$ is an FP of $T.$ By (1), we have

$$\begin{array}{ccc}\hfill \mathfrak{e}\left({\mathfrak{a}}^{*},T{\mathfrak{a}}^{*}\right)& ⪯& \phi \left({\mathfrak{a}}^{*},{\mathfrak{a}}_{2n+1}\right)\mathfrak{e}\left({\mathfrak{a}}^{*},{\mathfrak{a}}_{2n+1}\right)+\phi \left({\mathfrak{a}}_{2n+1},T{\mathfrak{a}}^{*}\right)\mathfrak{e}\left({\mathfrak{a}}_{2n+1},T{\mathfrak{a}}^{*}\right)\hfill \\ & ⪯& \phi \left({\mathfrak{a}}^{*},{\mathfrak{a}}_{2n+1}\right)\mathfrak{e}\left({\mathfrak{a}}^{*},{\mathfrak{a}}_{2n+1}\right)+\phi \left({\mathfrak{a}}_{2n+1},T{\mathfrak{a}}^{*}\right)\mathfrak{e}\left(R{\mathfrak{a}}_{2n},T{\mathfrak{a}}^{*}\right)\hfill \\ & ⪯& \phi \left({\mathfrak{a}}^{*},{\mathfrak{a}}_{2n+1}\right)\mathfrak{e}\left({\mathfrak{a}}^{*},{\mathfrak{a}}_{2n+1}\right)+\phi \left({\mathfrak{a}}_{2n+1},T{\mathfrak{a}}^{*}\right)\left(\begin{array}{c}\rho \mathfrak{e}\left({\mathfrak{a}}_{2n},{\mathfrak{a}}^{*}\right)\\ +\kappa {\displaystyle \frac{\mathfrak{e}\left({\mathfrak{a}}_{2n},R{\mathfrak{a}}_{2n}\right)\mathfrak{e}\left({\mathfrak{a}}^{*},T{\mathfrak{a}}^{*}\right)}{1+\mathfrak{e}\left({\mathfrak{a}}_{2n},{\mathfrak{a}}^{*}\right)}}\\ +\mu {\displaystyle \frac{\mathfrak{e}\left({\mathfrak{a}}^{*},R{\mathfrak{a}}_{2n}\right)\mathfrak{e}\left({\mathfrak{a}}_{2n},T{\mathfrak{a}}^{*}\right)}{1+\mathfrak{e}\left({\mathfrak{a}}_{2n},{\mathfrak{a}}^{*}\right)}}\end{array}\right)\hfill \\ & ⪯& \phi \left({\mathfrak{a}}^{*},{\mathfrak{a}}_{2n+1}\right)\mathfrak{e}\left({\mathfrak{a}}^{*},{\mathfrak{a}}_{2n+1}\right)+\phi \left({\mathfrak{a}}_{2n+1},T{\mathfrak{a}}^{*}\right)\left(\begin{array}{c}\rho \mathfrak{e}\left({\mathfrak{a}}_{2n},{\mathfrak{a}}^{*}\right)\\ +\kappa {\displaystyle \frac{\mathfrak{e}\left({\mathfrak{a}}_{2n},{\mathfrak{a}}_{2n+1}\right)\mathfrak{e}\left({\mathfrak{a}}^{*},T{\mathfrak{a}}^{*}\right)}{1+\mathfrak{e}\left({\mathfrak{a}}_{2n},{\mathfrak{a}}^{*}\right)}}\\ +\mu {\displaystyle \frac{\mathfrak{e}\left({\mathfrak{a}}^{*},{\mathfrak{a}}_{2n+1}\right)\mathfrak{e}\left({\mathfrak{a}}_{2n},T{\mathfrak{a}}^{*}\right)}{1+\mathfrak{e}\left({\mathfrak{a}}_{2n},{\mathfrak{a}}^{*}\right)}}\end{array}\right)\hfill \end{array}$$

This implies that

$$\begin{array}{ccc}\hfill \left|\mathfrak{e}\left({\mathfrak{a}}^{*},T{\mathfrak{a}}^{*}\right)\right|& \le & \phi \left({\mathfrak{a}}^{*},{\mathfrak{a}}_{2n+1}\right)\left|\mathfrak{e}\left({\mathfrak{a}}^{*},{\mathfrak{a}}_{2n+1}\right)\right|\hfill \\ & & +\phi \left({\mathfrak{a}}_{2n+1},T{\mathfrak{a}}^{*}\right)\left(\begin{array}{c}\rho \left|\mathfrak{e}\left({\mathfrak{a}}_{2n},{\mathfrak{a}}^{*}\right)\right|\\ +\kappa {\displaystyle \frac{\left|\mathfrak{e}\left({\mathfrak{a}}_{2n},{\mathfrak{a}}_{2n+1}\right)\right|}{\left|1+\mathfrak{e}\left({\mathfrak{a}}_{2n},{\mathfrak{a}}^{*}\right)\right|}}\left|\mathfrak{e}\left({\mathfrak{a}}^{*},T{\mathfrak{a}}^{*}\right)\right|\\ +\mu {\displaystyle \frac{\left|\mathfrak{e}\left({\mathfrak{a}}^{*},{\mathfrak{a}}_{2n+1}\right)\right|}{\left|1+\mathfrak{e}\left({\mathfrak{a}}_{2n},{\mathfrak{a}}^{*}\right)\right|}}\left|\mathfrak{e}\left({\mathfrak{a}}_{2n},T{\mathfrak{a}}^{*}\right)\right|\end{array}\right).\hfill \end{array}$$

Letting $n\to \infty ,$ we have $\left|\mathfrak{e}\left({\mathfrak{a}}^{*},T{\mathfrak{a}}^{*}\right)\right|=0.$ Hence, ${\mathfrak{a}}^{*}=T{\mathfrak{a}}^{*}.$ Therefore ${\mathfrak{a}}^{*}$ is a CFP of R and $T.$ We shall now demonstrate the uniqueness of ${\mathfrak{a}}^{*}$. We assume that there exists another CFP of ${\mathfrak{a}}^{/}$ of R and T, that is,

$${\mathfrak{a}}^{/}=R{\mathfrak{a}}^{/}=T{\mathfrak{a}}^{/},$$

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

$$\begin{array}{ccc}\hfill \mathfrak{e}\left({\mathfrak{a}}^{*},{\mathfrak{a}}^{/}\right)& =& \mathfrak{e}\left(R{\mathfrak{a}}^{*},T{\mathfrak{a}}^{/}\right)\hfill \\ & ⪯& \rho \mathfrak{e}\left({\mathfrak{a}}^{*},{\mathfrak{a}}^{/}\right)+\kappa {\displaystyle \frac{\mathfrak{e}\left({\mathfrak{a}}^{*},R{\mathfrak{a}}^{*}\right)\mathfrak{e}\left({\mathfrak{a}}^{/},T{\mathfrak{a}}^{/}\right)}{1+\mathfrak{e}\left({\mathfrak{a}}^{*},{\mathfrak{a}}^{/}\right)}}\hfill \\ & & +\mu {\displaystyle \frac{\mathfrak{e}\left({\mathfrak{a}}^{/},R{\mathfrak{a}}^{*}\right)\mathfrak{e}\left({\mathfrak{a}}^{*},T{\mathfrak{a}}^{/}\right)}{1+\mathfrak{e}\left({\mathfrak{a}}^{*},{\mathfrak{a}}^{/}\right)}}\hfill \\ & =& \rho \mathfrak{e}\left({\mathfrak{a}}^{*},{\mathfrak{a}}^{/}\right)+\kappa {\displaystyle \frac{\mathfrak{e}\left({\mathfrak{a}}^{*},{\mathfrak{a}}^{*}\right)\mathfrak{e}\left({\mathfrak{a}}^{/},{\mathfrak{a}}^{/}\right)}{1+\mathfrak{e}\left({\mathfrak{a}}^{*},{\mathfrak{a}}^{/}\right)}}\hfill \\ & & +\mu {\displaystyle \frac{\mathfrak{e}\left({\mathfrak{a}}^{/},{\mathfrak{a}}^{*}\right)\mathfrak{e}\left({\mathfrak{a}}^{*},{\mathfrak{a}}^{/}\right)}{1+\mathfrak{e}\left({\mathfrak{a}}^{*},{\mathfrak{a}}^{/}\right)}}.\hfill \end{array}$$

This yields that we have

$$\begin{array}{ccc}\hfill \left|\mathfrak{e}\left({\mathfrak{a}}^{*},{\mathfrak{a}}^{/}\right)\right|& \le & \rho \left|\mathfrak{e}\left({\mathfrak{a}}^{*},{\mathfrak{a}}^{/}\right)\right|\hfill \\ & & +\mu \left|\mathfrak{e}\left({\mathfrak{a}}^{*},{\mathfrak{a}}^{/}\right)\right|\left|{\displaystyle \frac{\mathfrak{e}\left({\mathfrak{a}}^{*},{\mathfrak{a}}^{/}\right)}{1+\mathfrak{e}\left({\mathfrak{a}}^{*},{\mathfrak{a}}^{/}\right)}}\right|\hfill \\ & \le & \rho \left|\mathfrak{e}\left({\mathfrak{a}}^{*},{\mathfrak{a}}^{/}\right)\right|+\mu \left|\mathfrak{e}\left({\mathfrak{a}}^{*},{\mathfrak{a}}^{/}\right)\right|\hfill \\ & =& \left(\rho +\mu \right)\left|\mathfrak{e}\left({\mathfrak{a}}^{*},{\mathfrak{a}}^{/}\right)\right|.\hfill \end{array}$$

As $\rho +\mu <1,$ we have

$$\left|\mathfrak{e}\left({\mathfrak{a}}^{*},{\mathfrak{a}}^{/}\right)\right|=0.$$

Thus, ${\mathfrak{a}}^{*}={\mathfrak{a}}^{/}.$ □

Corollary 1.

Let $\left(\Xi ,\mathfrak{e}\right)$ be a complete CVCMS, $\phi :\Xi \times \Xi \to [1,\infty )$ and let $R:\Xi$$\to \Xi$. Assume that there exist the constants $\rho ,\kappa ,\mu \in [0,1)$ with $\rho +\kappa +\mu <1$ such that

$$\mathfrak{e}\left(R\mathfrak{a},R\hslash \right)⪯\rho \mathfrak{e}\left(\mathfrak{a},\hslash \right)+\kappa {\displaystyle \frac{\mathfrak{e}\left(\mathfrak{a},R\mathfrak{a}\right)\mathfrak{e}\left(\hslash ,R\hslash \right)}{1+\mathfrak{e}\left(\mathfrak{a},\hslash \right)}}+\mu {\displaystyle \frac{\mathfrak{e}\left(\hslash ,R\mathfrak{a}\right)\mathfrak{e}\left(\mathfrak{a},R\hslash \right)}{1+\mathfrak{e}\left(\mathfrak{a},\hslash \right)}},$$

For a point ${\mathfrak{a}}_0\in \Xi$ and a sequence {${\mathfrak{a}}_n$} defined by

$${\mathfrak{a}}_{n+1}=R{\mathfrak{a}}_n$$

suppose that ${lim}_{n\to \infty }\phi \left({\mathfrak{a}}_n,\mathfrak{a}\right)$ and ${lim}_{n\to \infty }\phi \left(\mathfrak{a},{\mathfrak{a}}_n\right)$ exist and are finite and

$$\underset{m\ge 1}{max}\underset{i\to \infty }{lim}{\displaystyle \frac{\phi \left({\mathfrak{a}}_{i+1},{\mathfrak{a}}_{i+2}\right)}{\phi \left({\mathfrak{a}}_i,{\mathfrak{a}}_{i+1}\right)}}\phi \left({\mathfrak{a}}_{i+1},{\mathfrak{a}}_m\right)<{\displaystyle \frac{1}{\pi }}$$

where $\pi ={\displaystyle \frac{\rho }{1-\kappa }}<1$. Then, R has a unique FP.

Proof. 

Set $T=R$ in Theorem 2. □

Corollary 2.

Let $\left(\Xi ,\mathfrak{e}\right)$ be a complete CVCMS, $\phi :\Xi \times \Xi \to [1,\infty )$, and let $R,T:\Xi$$\to \Xi$. Assume that there exist the constants $\rho ,\kappa \in [0,1)$ with $\rho +\kappa <1$ such that

$$\mathfrak{e}\left(R\mathfrak{a},T\hslash \right)⪯\rho \mathfrak{e}\left(\mathfrak{a},\hslash \right)+\kappa {\displaystyle \frac{\mathfrak{e}\left(\mathfrak{a},R\mathfrak{a}\right)\mathfrak{e}\left(\hslash ,T\hslash \right)}{1+\mathfrak{e}\left(\mathfrak{a},\hslash \right)}}.$$

For a point ${\mathfrak{a}}_0\in \Xi$ and a sequence {${\mathfrak{a}}_n$} defined by

$${\mathfrak{a}}_{2n+1}=T{\mathfrak{a}}_{2n}\mathrm{and}{\mathfrak{a}}_{2n+2}=R{\mathfrak{a}}_{2n+1}$$

suppose that ${lim}_{n\to \infty }\phi \left({\mathfrak{a}}_n,\mathfrak{a}\right)$ and ${lim}_{n\to \infty }\phi \left(\mathfrak{a},{\mathfrak{a}}_n\right)$ exist and are finite and

$$\underset{m\ge 1}{max}\underset{i\to \infty }{lim}{\displaystyle \frac{\phi \left({\mathfrak{a}}_{i+1},{\mathfrak{a}}_{i+2}\right)}{\phi \left({\mathfrak{a}}_i,{\mathfrak{a}}_{i+1}\right)}}\phi \left({\mathfrak{a}}_{i+1},{\mathfrak{a}}_m\right)<{\displaystyle \frac{1}{\pi }}$$

where $\pi ={\displaystyle \frac{\rho }{1-\kappa }}<1$. Then, R and T have a unique CFP.

Proof. 

Take $\mu =0$ in Theorem 2. □

Corollary 3.

Let $\left(\Xi ,\mathfrak{e}\right)$ be a complete CVCMS, $\phi :\Xi \times \Xi \to [1,\infty )$, and let $R,T:\Xi$$\to \Xi$. Assume that there exist the constants $\rho ,\mu \in [0,1)$ with $\rho +\mu <1$ such that

$$\mathfrak{e}\left(R\mathfrak{a},T\hslash \right)⪯\rho \mathfrak{e}\left(\mathfrak{a},\hslash \right)+\mu {\displaystyle \frac{\mathfrak{e}\left(\hslash ,R\mathfrak{a}\right)\mathfrak{e}\left(\mathfrak{a},T\hslash \right)}{1+\mathfrak{e}\left(\mathfrak{a},\hslash \right)}},$$

For a point ${\mathfrak{a}}_0\in \Xi$ and a sequence {${\mathfrak{a}}_n$} defined by

$${\mathfrak{a}}_{2n+1}=T{\mathfrak{a}}_{2n}\mathrm{and}{\mathfrak{a}}_{2n+2}=R{\mathfrak{a}}_{2n+1}$$

suppose that ${lim}_{n\to \infty }\phi \left({\mathfrak{a}}_n,\mathfrak{a}\right)$ and ${lim}_{n\to \infty }\phi \left(\mathfrak{a},{\mathfrak{a}}_n\right)$ exist and are finite and

$$\underset{m\ge 1}{max}\underset{i\to \infty }{lim}{\displaystyle \frac{\phi \left({\mathfrak{a}}_{i+1},{\mathfrak{a}}_{i+2}\right)}{\phi \left({\mathfrak{a}}_i,{\mathfrak{a}}_{i+1}\right)}}\phi \left({\mathfrak{a}}_{i+1},{\mathfrak{a}}_m\right)<{\displaystyle \frac{1}{\rho }},$$

then, R and T have a unique CFP.

Proof. 

Take $\kappa =0$ in Theorem 2. □

Corollary 4.

Let $\left(\Xi ,\mathfrak{e}\right)$ be a complete CVCMS, $\phi :\Xi \times \Xi \to [1,\infty )$, and let $R,T:\Xi$$\to \Xi$. Assume that there exists a constant $\rho \in [0,1)$ such that

$$\mathfrak{e}\left(R\mathfrak{a},T\hslash \right)⪯\rho \mathfrak{e}\left(\mathfrak{a},\hslash \right).$$

For a point ${\mathfrak{a}}_0\in \Xi$ and a sequence {${\mathfrak{a}}_n$} defined by

$${\mathfrak{a}}_{2n+1}=T{\mathfrak{a}}_{2n}\mathit{and}{\mathfrak{a}}_{2n+2}=R{\mathfrak{a}}_{2n+1},$$

suppose that ${lim}_{n\to \infty }\phi \left({\mathfrak{a}}_n,\mathfrak{a}\right)$ and ${lim}_{n\to \infty }\phi \left(\mathfrak{a},{\mathfrak{a}}_n\right)$ exist and are finite and

$$\underset{m\ge 1}{max}\underset{i\to \infty }{lim}{\displaystyle \frac{\phi \left({\mathfrak{a}}_{i+1},{\mathfrak{a}}_{i+2}\right)}{\phi \left({\mathfrak{a}}_i,{\mathfrak{a}}_{i+1}\right)}}\phi \left({\mathfrak{a}}_{i+1},{\mathfrak{a}}_m\right)<{\displaystyle \frac{1}{\rho }},$$

then, R and T have a unique CFP.

Proof. 

Take $\mu =\kappa =0$ in Theorem 2. □

Corollary 5.

Let $\left(\Xi ,\mathfrak{e}\right)$ be a complete CVCMS, $\phi :\Xi \times \Xi \to [1,\infty )$, and let $R:\Xi$$\to \Xi$. Suppose that there exists a constant $\rho \in [0,1)$ such that

$$\mathfrak{e}\left(R\mathfrak{a},R\hslash \right)⪯\rho \mathfrak{e}\left(\mathfrak{a},\hslash \right).$$

For a point ${\mathfrak{a}}_0\in \Xi$ and a sequence {${\mathfrak{a}}_n$} defined by

$${\mathfrak{a}}_{n+1}=R{\mathfrak{a}}_n,$$

suppose that ${lim}_{n\to \infty }\phi \left({\mathfrak{a}}_n,\mathfrak{a}\right)$ and ${lim}_{n\to \infty }\phi \left(\mathfrak{a},{\mathfrak{a}}_n\right)$ exist and are finite and

$$\underset{m\ge 1}{max}\underset{i\to \infty }{lim}{\displaystyle \frac{\phi \left({\mathfrak{a}}_{i+1},{\mathfrak{a}}_{i+2}\right)}{\phi \left({\mathfrak{a}}_i,{\mathfrak{a}}_{i+1}\right)}}\phi \left({\mathfrak{a}}_{i+1},{\mathfrak{a}}_m\right)<{\displaystyle \frac{1}{\rho }},$$

then, R has a unique FP.

Proof. 

Take $R=T$ in the above corollary. □

Example 6.

Let $\Xi =\left\{0,1,2\right\}$ and $\phi :\Xi \times \Xi \to [1,\infty )$ be a function defined by

$$\phi (0,0)=4,\phi (1,1)={\displaystyle \frac{5}{2}},\phi (2,2)={\displaystyle \frac{7}{4}}$$

and

$$\begin{array}{ccc}\hfill \phi (0,1)& =& 2=\phi (1,0),\hfill \\ \hfill \phi (0,2)& =& {\displaystyle \frac{5}{2}}=\phi (2,0),\hfill \\ \hfill \phi (1,2)& =& 2=\phi (2,1)\hfill \end{array}$$

and $\mathfrak{e}:\Xi \times \Xi \to \mathbb{C}$ by

$$\mathfrak{e}(\mathfrak{a},\mathfrak{a})=0+0i$$

for all $\mathfrak{a}\in \Xi$ and

$$\mathfrak{e}(0,1)=1+i=\mathfrak{e}(1,0)$$

$$\mathfrak{e}(0,2)=3+3i=\mathfrak{e}(2,0)$$

$$\mathfrak{e}(1,2)=2+2i=\mathfrak{e}(2,1)$$

for all $\mathfrak{a},\hslash \in$$\Xi .$ The first two conditions are straightforward to verify. It suffices to demonstrate the triangle inequality of CVCMS. Thus, we consider the following three cases.

• Case 1. If $\mathfrak{a}=0$ and $\hslash =1,$ then

  $$\begin{array}{ccc}\hfill \mathfrak{e}(\mathfrak{a},\hslash )& =& \mathfrak{e}(0,1)=1+i\precsim {\displaystyle \frac{23}{2}}+{\displaystyle \frac{23}{2}}i={\displaystyle \frac{5}{2}}\left(3+3i\right)+2\left(2+2i\right)\hfill \\ & =& \phi (0,2)\mathfrak{e}(0,2)+\phi (2,1)\mathfrak{e}(2,1)\hfill \\ & =& \phi (\mathfrak{a},\nu )\mathfrak{e}(\mathfrak{a},\nu )+\phi (\nu ,\hslash )\mathfrak{e}(\nu ,\hslash ).\hfill \end{array}$$
• Case 2. If $\mathfrak{a}=0$ and $\hslash =2,$ then

  $$\begin{array}{ccc}\hfill \mathfrak{e}(\mathfrak{a},\hslash )& =& \mathfrak{e}(0,2)=3+3i\precsim 5+5i=2\left(1+i\right)+2\left(2+2i\right)\hfill \\ & =& \phi (0,1)\mathfrak{e}(0,1)+\phi (1,2)\mathfrak{e}(1,2)\hfill \\ & =& \phi (\mathfrak{a},\nu )\mathfrak{e}(\mathfrak{a},\nu )+\phi (\nu ,\hslash )\mathfrak{e}(\nu ,\hslash ).\hfill \end{array}$$
• Case 3. If $\mathfrak{a}=1$ and $\hslash =2,$ then

  $$\begin{array}{ccc}\hfill \mathfrak{e}(\mathfrak{a},\hslash )& =& \mathfrak{e}(1,2)=2+2i\precsim {\displaystyle \frac{19}{2}}+{\displaystyle \frac{19}{2}}i=2\left(1+i\right)+{\displaystyle \frac{5}{2}}\left(3+3i\right)\hfill \\ & =& \phi (1,0)\mathfrak{e}(1,0)+\phi (0,2)\mathfrak{e}(0,2)\hfill \\ & =& \phi (\mathfrak{a},\nu )\mathfrak{e}(\mathfrak{a},\nu )+\phi (\nu ,\hslash )\mathfrak{e}(\nu ,\hslash ).\hfill \end{array}$$

  Therefore, $(\Xi ,\mathfrak{e})$ is CVCMS. Moreover, the set Ξ is finite, so $(\Xi ,\mathfrak{e})$ is automatically complete. Define $R,T:\Xi$ $\to \Xi$ by

  $$R\mathfrak{a}=1,\mathit{if}\mathfrak{a}\in \Xi ,$$

  and

  $$T\mathfrak{a}=\left\{\begin{array}{c}1,\mathit{if}\mathfrak{a}\in \{0,1\},\\ 0,\mathit{if}\mathfrak{a}=2.\end{array}\right.$$

  If $\mathfrak{a}\in \Xi$ and $\hslash \in \{0,1\},$ then $\mathfrak{e}\left(R\mathfrak{a},T\hslash \right)=0+0i,$ so the inequality in (1) holds trivially. But, if $\mathfrak{a}\in \Xi$ and $\hslash =2$ (fixed), then Inequality (1) becomes

  $$\mathfrak{e}\left(1,0\right)⪯\rho \mathfrak{e}\left(\mathfrak{a},2\right)+\kappa {\displaystyle \frac{\mathfrak{e}\left(\mathfrak{a},1\right)\mathfrak{e}\left(2,0\right)}{1+\mathfrak{e}\left(\mathfrak{a},2\right)}}+\mu {\displaystyle \frac{\mathfrak{e}\left(2,1\right)\mathfrak{e}\left(\mathfrak{a},2\right)}{1+\mathfrak{e}\left(\mathfrak{a},2\right)}}.$$

  Now, consider the different cases for different values of $\mathfrak{a}.$

• Case 1. If $\mathfrak{a}=0,$ then

  $$\begin{array}{ccc}\hfill \mathfrak{e}\left(\mathfrak{a},2\right)& =& 3+3i\hfill \\ \hfill \mathfrak{e}\left(\mathfrak{a},1\right)& =& 1+i.\hfill \end{array}$$

  Substituting these values and simplifying, we can verify that the inequality in (1) holds for $\rho ={\displaystyle \frac{1}{3}},\kappa ={\displaystyle \frac{1}{3}}$, and $\mu ={\displaystyle \frac{3}{10}}$.
• Case 2. If $\mathfrak{a}=1,$ then

  $$\begin{array}{ccc}\hfill \mathfrak{e}\left(\mathfrak{a},2\right)& =& 2+2i\hfill \\ \hfill \mathfrak{e}\left(\mathfrak{a},1\right)& =& 0+0i.\hfill \end{array}$$

  $$\begin{array}{ccc}\hfill \mathfrak{e}\left(1,0\right)& =& 1+i⪯{\displaystyle \frac{202}{195}}+{\displaystyle \frac{238}{195}}i={\displaystyle \frac{1}{3}}\left(2+2i\right)+{\displaystyle \frac{1}{3}}\left(0+0i\right)+{\displaystyle \frac{3}{10}}\left({\displaystyle \frac{16}{13}}+{\displaystyle \frac{24}{13}}i\right)\hfill \\ & =& \rho \mathfrak{e}\left(\mathfrak{a},2\right)+\kappa {\displaystyle \frac{\mathfrak{e}\left(\mathfrak{a},1\right)\mathfrak{e}\left(2,0\right)}{1+\mathfrak{e}\left(\mathfrak{a},2\right)}}+\mu {\displaystyle \frac{\mathfrak{e}\left(2,1\right)\mathfrak{e}\left(\mathfrak{a},2\right)}{1+\mathfrak{e}\left(\mathfrak{a},2\right)}}.\hfill \end{array}$$

  Hence, the inequality in (1) holds for $\rho ={\displaystyle \frac{1}{3}},\kappa ={\displaystyle \frac{1}{3}}$, and $\mu ={\displaystyle \frac{3}{10}}.$
• Case 3. If $\mathfrak{a}=2,$ then

  $$\begin{array}{ccc}\hfill \mathfrak{e}\left(\mathfrak{a},2\right)& =& 0+0i\hfill \\ \hfill \mathfrak{e}\left(\mathfrak{a},1\right)& =& 2+2i.\hfill \end{array}$$

  $$\begin{array}{ccc}\hfill \mathfrak{e}\left(1,0\right)& =& 1+i⪯4i={\displaystyle \frac{1}{3}}\left(0+0i\right)+{\displaystyle \frac{1}{3}}\left(0+12i\right)+{\displaystyle \frac{3}{10}}\left(0+0i\right)\hfill \\ & =& \rho \mathfrak{e}\left(\mathfrak{a},2\right)+\kappa {\displaystyle \frac{\mathfrak{e}\left(\mathfrak{a},1\right)\mathfrak{e}\left(2,0\right)}{1+\mathfrak{e}\left(\mathfrak{a},2\right)}}+\mu {\displaystyle \frac{\mathfrak{e}\left(2,1\right)\mathfrak{e}\left(\mathfrak{a},2\right)}{1+\mathfrak{e}\left(\mathfrak{a},2\right)}}.\hfill \end{array}$$

  Now, since 4 (the magnitude of $4i$) is greater than $\sqrt{2}$ (the magnitude of $1+i$), the inequality in (1) holds for $\rho ={\displaystyle \frac{1}{3}},\kappa ={\displaystyle \frac{1}{3}}$, and $\mu ={\displaystyle \frac{3}{10}}.$ Moreover, $\rho +\kappa +\mu ={\displaystyle \frac{29}{30}}<1.$ Thus, all hypotheses in Theorem 2 are fulfilled and 1 is the unique CFP of R of $T.$

Corollary 6.

Let $\left(\Xi ,\mathfrak{e}\right)$ be a complete CVCMS, $\phi :\Xi \times \Xi \to [1,\infty )$, and let $R:\Xi$$\to \Xi$. Assume that there exist the constants $\rho ,\kappa ,\mu \in [0,1)$ with $\rho +\kappa +\mu <1$ such that

$$\mathfrak{e}\left(R^n\mathfrak{a},R^n\hslash \right)⪯\rho \mathfrak{e}\left(\mathfrak{a},\hslash \right)+\kappa {\displaystyle \frac{\mathfrak{e}\left(\mathfrak{a},R^n\mathfrak{a}\right)\mathfrak{e}\left(\hslash ,R^n\hslash \right)}{1+\mathfrak{e}\left(\mathfrak{a},\hslash \right)}}+\mu {\displaystyle \frac{\mathfrak{e}\left(\hslash ,R^n\mathfrak{a}\right)\mathfrak{e}\left(\mathfrak{a},R^n\hslash \right)}{1+\mathfrak{e}\left(\mathfrak{a},\hslash \right)}},$$

For a point ${\mathfrak{a}}_0\in \Xi$ and a sequence {${\mathfrak{a}}_n$} defined by

$${\mathfrak{a}}_{n+1}=R{\mathfrak{a}}_n$$

suppose that ${lim}_{n\to \infty }\phi \left({\mathfrak{a}}_n,\mathfrak{a}\right)$ and ${lim}_{n\to \infty }\phi \left(\mathfrak{a},{\mathfrak{a}}_n\right)$ exist and are finite and

$$\underset{m\ge 1}{max}\underset{i\to \infty }{lim}{\displaystyle \frac{\phi \left({\mathfrak{a}}_{i+1},{\mathfrak{a}}_{i+2}\right)}{\phi \left({\mathfrak{a}}_i,{\mathfrak{a}}_{i+1}\right)}}\phi \left({\mathfrak{a}}_{i+1},{\mathfrak{a}}_m\right)<{\displaystyle \frac{1}{\pi }}$$

where $\pi ={\displaystyle \frac{\rho }{1-\kappa }}<1$. Then, R has a unique FP.

Proof. 

From Corollary (1), we have $\mathfrak{a}\in \Xi$ such that $R^n\mathfrak{a}=\mathfrak{a}.$ Now, from

$$\begin{array}{ccc}\hfill \mathfrak{e}\left(R\mathfrak{a},\mathfrak{a}\right)& =& \mathfrak{e}\left(R{R}^n\mathfrak{a},R^n\mathfrak{a}\right)\hfill \\ & =& \mathfrak{e}\left(R^{n}R\mathfrak{a},R^n\mathfrak{a}\right)⪯\rho \mathfrak{e}\left(R\mathfrak{a},\mathfrak{a}\right)+\kappa {\displaystyle \frac{\mathfrak{e}\left(R\mathfrak{a},R^{n}R\mathfrak{a}\right)\mathfrak{e}\left(\mathfrak{a},R^n\mathfrak{a}\right)}{1+\mathfrak{e}\left(R\mathfrak{a},\mathfrak{a}\right)}}\hfill \\ & & +\mu {\displaystyle \frac{\mathfrak{e}\left(\mathfrak{a},R^{n}R\mathfrak{a}\right)\mathfrak{e}\left(R\mathfrak{a},R^n\mathfrak{a}\right)}{1+\mathfrak{e}\left(R\mathfrak{a},\mathfrak{a}\right)}}\hfill \\ & ⪯& \rho \mathfrak{e}\left(R\mathfrak{a},\mathfrak{a}\right)+\kappa {\displaystyle \frac{\mathfrak{e}\left(R\mathfrak{a},R\mathfrak{a}\right)\mathfrak{e}\left(\mathfrak{a},\mathfrak{a}\right)}{1+\mathfrak{e}\left(R\mathfrak{a},\mathfrak{a}\right)}}+\mu {\displaystyle \frac{\mathfrak{e}\left(\mathfrak{a},R\mathfrak{a}\right)\mathfrak{e}\left(R\mathfrak{a},\mathfrak{a}\right)}{1+\mathfrak{e}\left(R\mathfrak{a},\mathfrak{a}\right)}}\hfill \\ & =& \rho \mathfrak{e}\left(R\mathfrak{a},\mathfrak{a}\right)+\mu {\displaystyle \frac{\mathfrak{e}\left(\mathfrak{a},R\mathfrak{a}\right)\mathfrak{e}\left(R\mathfrak{a},\mathfrak{a}\right)}{1+\mathfrak{e}\left(R\mathfrak{a},\mathfrak{a}\right)}}\hfill \end{array}$$

which implies that

$$\begin{array}{ccc}\hfill \left|\mathfrak{e}\left(R\mathfrak{a},\mathfrak{a}\right)\right|& \le & \rho \left|\mathfrak{e}\left(R\mathfrak{a},\mathfrak{a}\right)\right|+\mu \left|\mathfrak{e}\left(\mathfrak{a},R\mathfrak{a}\right)\right|\left|{\displaystyle \frac{\mathfrak{e}\left(R\mathfrak{a},\mathfrak{a}\right)}{1+\mathfrak{e}\left(R\mathfrak{a},\mathfrak{a}\right)}}\right|\hfill \\ & \le & \rho \left|\mathfrak{e}\left(R\mathfrak{a},\mathfrak{a}\right)\right|+\mu \left|\mathfrak{e}\left(\mathfrak{a},R\mathfrak{a}\right)\right|\hfill \\ & =& \left(\rho +\mu \right)\left|\mathfrak{e}\left(\mathfrak{a},R\mathfrak{a}\right)\right|\hfill \end{array}$$

which is possible only whenever $\left|\mathfrak{e}\left(\mathfrak{a},R\mathfrak{a}\right)\right|=0.$ Thus, $R\mathfrak{a}=\mathfrak{a}.$ □

3.1. Fixed-Point Results in Complex-Valued b-Metric Spaces

Corollary 7

([9]). Let $\left(\Xi ,\mathfrak{e}\right)$ be a complete CVbMS and let $R,T:\Xi$$\to \Xi$. Assume that there exist the constants $\rho ,\kappa ,\mu \in [0,1)$ with $\rho +\kappa +\mu <1$ such that

$$\mathfrak{e}\left(R\mathfrak{a},T\hslash \right)⪯\rho \mathfrak{e}\left(\mathfrak{a},\hslash \right)+\kappa {\displaystyle \frac{\mathfrak{e}\left(\mathfrak{a},R\mathfrak{a}\right)\mathfrak{e}\left(\hslash ,T\hslash \right)}{1+\mathfrak{e}\left(\mathfrak{a},\hslash \right)}}+\mu {\displaystyle \frac{\mathfrak{e}\left(\hslash ,R\mathfrak{a}\right)\mathfrak{e}\left(\mathfrak{a},T\hslash \right)}{1+\mathfrak{e}\left(\mathfrak{a},\hslash \right)}},$$

for all $\mathfrak{a},\hslash \in \Xi ,$ then R and T have a unique CFP.

Proof. 

Define $\phi :\Xi \times \Xi \to [1,\infty )$; by $\phi (\mathfrak{a},\hslash )=s\ge 1$ in Theorem 2, we obtain the conclusion. □

Corollary 8

([9]). Let $\left(\Xi ,\mathfrak{e}\right)$ be a complete CVbMS and let $R:\Xi$$\to \Xi$. Assume that there exist the constants $\rho ,\kappa ,\mu \in [0,1)$ with $\rho +\kappa +\mu <1$ such that

$$\mathfrak{e}\left(R\mathfrak{a},R\hslash \right)⪯\rho \mathfrak{e}\left(\mathfrak{a},\hslash \right)+\kappa {\displaystyle \frac{\mathfrak{e}\left(\mathfrak{a},R\mathfrak{a}\right)\mathfrak{e}\left(\hslash ,R\hslash \right)}{1+\mathfrak{e}\left(\mathfrak{a},\hslash \right)}}+\mu {\displaystyle \frac{\mathfrak{e}\left(\hslash ,R\mathfrak{a}\right)\mathfrak{e}\left(\mathfrak{a},R\hslash \right)}{1+\mathfrak{e}\left(\mathfrak{a},\hslash \right)}},$$

for all $\mathfrak{a},\hslash \in \Xi ,$ then R has a unique FP.

Proof. 

Take $T=R$ in Corollary 7. □

Corollary 9

([8]). Let $\left(\Xi ,\mathfrak{e}\right)$ be a complete CVbMS, $s\ge 1$, and let $R,T:\Xi$$\to \Xi$. Assume that there exist the constants $\rho ,\kappa \in [0,1)$ with $\rho +\kappa <1$ such that

$$\mathfrak{e}\left(R\mathfrak{a},T\hslash \right)⪯\rho \mathfrak{e}\left(\mathfrak{a},\hslash \right)+\kappa {\displaystyle \frac{\mathfrak{e}\left(\mathfrak{a},R\mathfrak{a}\right)\mathfrak{e}\left(\hslash ,T\hslash \right)}{1+\mathfrak{e}\left(\mathfrak{a},\hslash \right)}},$$

for all $\mathfrak{a},\hslash \in \Xi ,$ then R and T have a unique CFP.

Proof. 

Take $\mu =0$ in Corollary 7. □

Corollary 10

([8]). Let $\left(\Xi ,\mathfrak{e}\right)$ be a complete CVbMS, $s\ge 1$, and let $R:\Xi$$\to \Xi$. Assume that there exist the constants $\rho ,\kappa \in [0,1)$ with $\rho +\kappa <1$ such that

$$\mathfrak{e}\left(R\mathfrak{a},R\hslash \right)⪯\rho \mathfrak{e}\left(\mathfrak{a},\hslash \right)+\kappa {\displaystyle \frac{\mathfrak{e}\left(\mathfrak{a},R\mathfrak{a}\right)\mathfrak{e}\left(\hslash ,R\hslash \right)}{1+\mathfrak{e}\left(\mathfrak{a},\hslash \right)}},$$

for all $\mathfrak{a},\hslash \in \Xi ;$ then, R has a unique FP.

Proof. 

Take R=T in the aforementioned corollary. □

Corollary 11.

Let $\left(\Xi ,\mathfrak{e}\right)$ be a complete CVbMS and let $R,T:\Xi$$\to \Xi$. Assume that there exist the constants $\rho ,\kappa ,\mu \in [0,1)$ with $\rho +\kappa +\mu <1$ such that

$$\mathfrak{e}\left(R\mathfrak{a},T\hslash \right)⪯\rho \mathfrak{e}\left(\mathfrak{a},\hslash \right)+\mu {\displaystyle \frac{\mathfrak{e}\left(\hslash ,R\mathfrak{a}\right)\mathfrak{e}\left(\mathfrak{a},T\hslash \right)}{1+\mathfrak{e}\left(\mathfrak{a},\hslash \right)}},$$

for all $\mathfrak{a},\hslash \in \Xi ;$ then, R and T have a unique CFP.

Proof. 

Take $\kappa =0$ in Corollary 7. □

Corollary 12.

Let $\left(\Xi ,\mathfrak{e}\right)$ be a complete CVbMS and let $R:\Xi$ $\to \Xi$. Assume that there exist the constants $\rho ,\kappa ,\mu \in [0,1)$ with $\rho +\kappa +\mu <1$ such that

$$\mathfrak{e}\left(R\mathfrak{a},R\hslash \right)⪯\rho \mathfrak{e}\left(\mathfrak{a},\hslash \right)+\mu {\displaystyle \frac{\mathfrak{e}\left(\hslash ,R\mathfrak{a}\right)\mathfrak{e}\left(\mathfrak{a},R\hslash \right)}{1+\mathfrak{e}\left(\mathfrak{a},\hslash \right)}},$$

for all $\mathfrak{a},\hslash \in \Xi ;$ then, R has a unique FP.

Proof. 

Consider the case where R equals T in the preceding corollary. □

3.2. Fixed-Point Results in Complex-Valued Metric Spaces

Corollary 13

([5]). Let $\left(\Xi ,\mathfrak{e}\right)$ be a complete CVMS and let $R,T:\Xi$$\to \Xi$. Assume that there exist the constants $\rho ,\kappa ,\mu \in [0,1)$ with $\rho +\kappa +\mu <1$ such that

$$\mathfrak{e}\left(R\mathfrak{a},T\hslash \right)⪯\rho \mathfrak{e}\left(\mathfrak{a},\hslash \right)+\kappa {\displaystyle \frac{\mathfrak{e}\left(\mathfrak{a},R\mathfrak{a}\right)\mathfrak{e}\left(\hslash ,T\hslash \right)}{1+\mathfrak{e}\left(\mathfrak{a},\hslash \right)}}+\mu {\displaystyle \frac{\mathfrak{e}\left(\hslash ,R\mathfrak{a}\right)\mathfrak{e}\left(\mathfrak{a},T\hslash \right)}{1+\mathfrak{e}\left(\mathfrak{a},\hslash \right)}},$$

for all $\mathfrak{a},\hslash \in \Xi ;$ then, R and T have a unique CFP.

Proof. 

Define $\phi :\Xi \times \Xi \to [1,\infty )$; by $\phi (\mathfrak{a},v)=\phi (v,\hslash )=1$ in Theorem 2, we obtain the conclusion. □

Corollary 14

([5]). Let $\left(\Xi ,\mathfrak{e}\right)$ be a complete CVMS and let $R:\Xi$$\to \Xi$. Assume that there exist the constants $\rho ,\kappa ,\mu \in [0,1)$ with $\rho +\kappa +\mu <1$ such that

$$\mathfrak{e}\left(R\mathfrak{a},R\hslash \right)⪯\rho \mathfrak{e}\left(\mathfrak{a},\hslash \right)+\kappa {\displaystyle \frac{\mathfrak{e}\left(\mathfrak{a},R\mathfrak{a}\right)\mathfrak{e}\left(\hslash ,R\hslash \right)}{1+\mathfrak{e}\left(\mathfrak{a},\hslash \right)}}+\mu {\displaystyle \frac{\mathfrak{e}\left(\hslash ,R\mathfrak{a}\right)\mathfrak{e}\left(\mathfrak{a},R\hslash \right)}{1+\mathfrak{e}\left(\mathfrak{a},\hslash \right)}},$$

for all $\mathfrak{a},\hslash \in \Xi ;$ then, R has a unique FP.

Proof. 

Take $T=R$ in Corollary 13. □

Corollary 15

([4]). Let $\left(\Xi ,\mathfrak{e}\right)$ be a complete CVMS and let $R,T:\Xi$$\to \Xi$. Assume that there exist the constants $\rho ,\kappa \in [0,1)$ with $\rho +\kappa <1$ such that

$$\mathfrak{e}\left(R\mathfrak{a},T\hslash \right)⪯\rho \mathfrak{e}\left(\mathfrak{a},\hslash \right)+\kappa {\displaystyle \frac{\mathfrak{e}\left(\mathfrak{a},R\mathfrak{a}\right)\mathfrak{e}\left(\hslash ,T\hslash \right)}{1+\mathfrak{e}\left(\mathfrak{a},\hslash \right)}},$$

for all $\mathfrak{a},\hslash \in \Xi ;$ then, R and T have a unique CFP.

Proof. 

Take $\mu =0$ in Corollary 13. □

Corollary 16

([4]). Let $\left(\Xi ,\mathfrak{e}\right)$ be a complete CVMS and let $R:\Xi$$\to \Xi$. Assume that there exist the constants $\rho ,\kappa \in [0,1)$ with $\rho +\kappa <1$ such that

$$\mathfrak{e}\left(R\mathfrak{a},R\hslash \right)⪯\rho \mathfrak{e}\left(\mathfrak{a},\hslash \right)+\kappa {\displaystyle \frac{\mathfrak{e}\left(\mathfrak{a},R\mathfrak{a}\right)\mathfrak{e}\left(\hslash ,R\hslash \right)}{1+\mathfrak{e}\left(\mathfrak{a},\hslash \right)}},$$

for all $\mathfrak{a},\hslash \in \Xi ,$ then R has a unique FP.

Proof. 

Take $T=R$ in Corollary 15. □

Remark 2.

If we restrict the set of complex numbers to only real numbers, the concept of a CVMS reduces to an MS. As a result, Fisher’s main theorem [3] becomes a direct consequence of the aforementioned corollary.

4. Applications

Integral equations are mathematical equations that involve an unknown function appearing as an integral. They arise in various fields, including physics, engineering, and economics. One significant application of integral equations is in the study of integral-type equations.

In the present section, we consider the following Volterra integral equation:

$$\mathfrak{a}\left(t\right)=g\left(t\right)+{\int }_0^{t}K(t,s)l(s,\mathfrak{a}\left(s\right))\mathfrak{e}s,$$

(7)

for $t\in [0,1],$ where $l(·,·):\left[0,1\right]\times \mathbb{R}\to \mathbb{R}$, $g:\left[0,1\right]\to \mathbb{R}$ are continuous functions that are bounded. Furthermore, $K(·,·):\left[0,1\right]\times \left[0,1\right]\to {\mathbb{R}}^{+}$ is a function such that $K(t,·)\in L^1\left(\left[0,1\right]\right)$ (the space of Lebesgue integrable functions on $\left[0,1\right]$).

Let $\Xi =C\left(\right[0,1],\mathbb{R})$ and $\mathfrak{e}:\Xi \times \Xi \to \mathbb{C}$ be a function given in this way:

$$\mathfrak{e}(\mathfrak{a},\hslash )=\underset{t\in \left[0,1\right]}{max}∥\mathfrak{a}\left(t\right)-\hslash \left(t\right)∥e^{-i\omega t}$$

with $\omega \in \left(1,{\displaystyle \frac{1}{\lambda }}\right]$ and $\lambda \in \left(0,1\right).$ Define $\phi :\Xi \times \Xi \to [1,\infty )$ by

$$\phi (\mathfrak{a},\hslash )=\left\{\begin{array}{c}1,\mathrm{if}\mathfrak{a},\hslash \in [0,1]\\ max\left\{\mathfrak{a}\left(t\right),\hslash \left(t\right)\right\}+\omega ,\mathrm{otherwise}.\end{array}\right.$$

One can infer that $\left(\Xi ,\mathfrak{e}\right)$ is a complete CVCMS (see [13]).

Theorem 3.

Let $\Xi =C\left(\right[0,1],\mathbb{R})$ be the set of all continuous and real-valued functions that are defined on $[0,1].$ Assume that the operator $R:\Xi \to \Xi$ is defined as

$$R\mathfrak{a}\left(t\right)=g\left(t\right)+{\int }_0^{t}K(t,s)l(s,\mathfrak{a}\left(s\right))\mathfrak{e}s,$$

and assume that these axioms hold:

(i) there exists the function $K(·,·):\left[0,1\right]\times \left[0,1\right]\to {\mathbb{R}}^{+}$ from the space of Lebesgue integrable functions on $\left[0,1\right]$ such that

$$∥{\int }_0^{t}K(t,s)\mathfrak{e}s∥<1,$$

(ii) $\left|l(t,\mathfrak{a}(t\left)\right)-l(t,\hslash (t\left)\right)\right|\le {\displaystyle \frac{1}{\omega e^{i\omega t}}}\left|\mathfrak{a}\left(t\right)-\hslash \left(t\right)\right|,$ for all $\mathfrak{a},\hslash \in \Xi$ and $\omega \in \left(1,{\displaystyle \frac{1}{\lambda }}\right]$ with $\lambda \in \left(0,1\right).$

It follows that the integral in Equation (7) admits a unique solution.

Proof. 

It is evident that to find the solution of integral Equation (7) is equivalent to finding a point $\mathfrak{a}\in \Xi ,$ the FP of the operator R.

Consider

$$\begin{array}{ccc}\hfill \left|R\mathfrak{a}\left(t\right)-R\hslash \left(t\right)\right|& =& \left|{\int }_0^{t}K(t,s)l(s,\mathfrak{a}\left(s\right))\mathfrak{e}s-{\int }_0^{t}K(t,s)l(s,\hslash \left(s\right))\mathfrak{e}s\right|\hfill \\ & \le & \left|{\int }_0^{t}K(t,s)\left[l(s,\mathfrak{a}(s\left)\right)-l(s,\hslash (s\left)\right)\right]\mathfrak{e}s\right|\hfill \\ & \le & \left|{\int }_0^{t}K(t,s)\mathfrak{e}s\right|\left({\int }_0^t\left|l(s,\mathfrak{a}(s\left)\right)-l(s,\hslash (s\left)\right)\right|\mathfrak{e}s\right)\hfill \\ & \le & \left|K(t,s)\mathfrak{e}s\right|\left({\int }_0^t\left|l(s,\mathfrak{a}(s\left)\right)-l(s,\hslash (s\left)\right)\right|\mathfrak{e}s\right)\hfill \\ & \le & {\int }_0^{t}K(t,s)\mathfrak{e}s·{\displaystyle \frac{1}{\omega e^{i\omega t}}}\left({\int }_0^t\left|\mathfrak{a}\left(s\right)-\hslash \left(s\right)\right|\mathfrak{e}s\right)\hfill \\ & \le & e^{i\omega t}\left({\int }_0^{t}K(t,s)e^{-i\omega t}\mathfrak{e}s\right)·{\displaystyle \frac{1}{\omega }}\left({\int }_0^t\left|\mathfrak{a}\left(s\right)-\hslash \left(s\right)\right|e^{-i\omega t}\mathfrak{e}s\right)\hfill \end{array}$$

By taking the supremum over the variable t in the preceding inequality, we arrive at

$$\underset{t\in \left[0,1\right]}{sup}\left|R\mathfrak{a}\left(t\right)-R\hslash \left(t\right)\right|e^{-i\omega t}⪯\left({\int }_0^t\underset{t\in \left[0,1\right]}{sup}K(t,s)e^{-i\omega t}\mathfrak{e}s\right)·{\displaystyle \frac{1}{\omega }}\left(\underset{t\in \left[0,1\right]}{sup}\left|\mathfrak{a}\left(t\right)-\hslash \left(t\right)\right|e^{-i\omega t}\mathfrak{e}s\right).$$

Given hypothesis (i), it follows that

$$\mathfrak{e}(R\mathfrak{a},R\hslash )={∥R\mathfrak{a}-R\hslash ∥}_{\infty }⪯{\displaystyle \frac{1}{\omega }}{∥\mathfrak{a}-\hslash ∥}_{\infty }={\displaystyle \frac{1}{\omega }}\mathfrak{e}(\mathfrak{a},\hslash ).$$

Then, all the conditions in Corollary 5 hold for $\rho ={\displaystyle \frac{1}{\omega }}<1$, and $\mathfrak{a}$ is the FP of the mapping R; thus, Equation (7) has a unique solution. □

5. Conclusions

In this article, we have utilized the notion of complete CVCMSs and obtained CFP results for generalized rational contractions. Our theorems generalized some famous results from the literature, including the central theorems of Azam et al. [4], Rouzkard et al. [5], Mukheimer [8], and Aslam et al. [13]. We provided a concrete example of the innovative nature of our leading result. As a demonstration of the applicability of our principal theorem, we solved the Volterra integral equation. It is anticipated that the findings presented in this article will contribute to future advancements in the field of CVCMS.

The future work in this way will target studying the CFPs of self-mappings in the setting of CVCMS. Differential and integral equations can be solved by applying the new results.