A fixed point approach to predator-prey dynamics via nonlinear mixed Volterra–Fredholm integral equations in complex-valued suprametric spaces

1.   Introduction

Fixed point (FP) theory is an extensive and multifaceted area of mathematics, primarily divided into three major branches: metric, topological, and discrete FP theory. Among these, metric FP theory plays a crucial role as a foundational framework, focusing on establishing the existence and uniqueness of fixed points for self-mappings defined on metric spaces (MSs). This theory is inherently linked to the concepts of distance and convergence, which are the core features of MSs.

The notion of an MS was first introduced by Maurice Fréchet ^[1] in 1906, describing a set equipped with a metric or distance function that measures the distance between any two points in the space. Over the years, this concept has been generalized and extended by various mathematicians, resulting in the development of advanced structures. Key generalizations include the partial MSs by Matthews ^[2], the $b$-MSs introduced by Czerwik ^[3], the rectangular MSs by Branciari ^[4], the cone MSs developed by Huang ^[5], and the supra-metric spaces (SMS) introduced by Berzig ^[6].

These generalizations have expanded the scope of metric FP theory, facilitating the analysis of more intricate mathematical systems and their applications. The SMS framework, proposed by Berzig ^[6], employs a relaxed version of the triangle inequality to prove FP results, such as the Banach contraction theorem, within this setting. Additional research in ordered vector spaces and nonlinear integral and matrix equations has also been conducted within this framework. Building upon SMSs, Berzig ^[7,8,9] later extended the concept by generalizing the triangle inequality, introducing two new spaces: generalized SMSs and $b$-SMSs.

On the other hand, complex numbers, introduced by the Italian mathematician Gerolamo Cardano in the 16th century while solving cubic equations, have become an integral part of mathematics and its applications. Represented in the form $z = a+bi,$ where $a$ and $b$ are real numbers and $i$ is the imaginary unit satisfying $i^{2} = -1,$ complex numbers extend the concept of real numbers and enable the solution of equations that lack real solutions, such as $x^{2}+1 = 0.$ The importance of complex numbers spans various fields: in mathematics, they underpin the fundamental theorem of algebra and play a central role in analysis, geometry, and number theory; in physics, they are essential in quantum mechanics, electromagnetism, and wave theory; in engineering, they simplify computations in signal processing, electrical circuit analysis, and control systems; and in computer science, they are utilized in graphics, fractals, and algorithms for modeling and simulations. Building upon the foundation of complex numbers, the idea of complex-valued metric spaces (CVMS) was given by Azam et al. ^[10] to extend classical MSs by defining the distance between points as a complex number rather than a real number. A complex-valued MS consists of a set $\Psi$ and a mapping $\mathfrak{c}:\Psi \times \Psi \rightarrow \mathbb{C}$ satisfying properties analogous to those of real-valued MSs, including non-negativity, symmetry, and a generalized triangle inequality. These spaces have proven valuable in FP theory, enabling the exploration of more general contraction mappings and providing a framework for solving advanced mathematical problems, including nonlinear equations and integral equations. Rouzkard et al. ^[11] extended Azam et al.'s results ^[10] by including a rational expression to the contractive condition. Following this, Sintunavarat et al. ^[12] further extended these results using control functions with one variable. Sitthikul et al. ^[13] advanced the theory by employing control functions with two variables in the context of CVMSs. Very recently, Panda et al. ^[14] merged the concepts of SMSs and CVMSs and pioneered the notion of complex-valued suprametric spaces (CVSMSs). In this way, they subsequently proved several common FP theorems for rational contractions within this framework. They applied their results to solve complex nonlinear integral equations via contractive mappings. For a more in-depth discussion, the readers may consult References ^{[15,16,17,18]}.

FP theorems serve as a cornerstone in functional analysis, offering a systematic approach to validate the existence and uniqueness of solutions across a wide spectrum of mathematical equations, including integral equations. These theorems are particularly valuable in resolving complex nonlinear problems, such as mixed Volterra–Fredholm integral equations, where they furnish a rigorous methodology to demonstrate solvability and ensure a unique solution under certain conditions. The nonlinear mixed Volterra–Fredholm integral equation, in turn, acts as a flexible mathematical tool for simulating dynamic ecological systems, enabling researchers to analyze species' interactions, including predator-prey dynamics and competition. For further information, the readers are encouraged to consult References ^[19,20].

In this research article, we prove common FP (CFP) theorems for rational contractions involving control functions dependent on multiple variables within the framework of CVSMSs. The results are further extended to the case of single-variable control functions and constants. Through this approach, we derive some important FP results in SMSs, CVMSs, and CVSMSs, which emerge as special cases of our main theorem. To underscore the novelty of the main result, an illustrative example is provided. The results are practically demonstrated by solving a nonlinear mixed Volterra–Fredholm integral equation that models predator-prey dynamics.

2.   Preliminaries

The concept of MS was introduced by Fréchet ^[1] in 1906, defined as follows:

Definition 1. (^[6]) Let $\Psi \not = \emptyset$ and $\mathfrak{c}:\Psi \times \Psi \rightarrow \mathbb{ \mathbb{R} }^{+}$ be a function such that

(ⅰ) $0\leq \mathfrak{c}(\mathfrak{v}, \mathfrak{o})\ $and $\mathfrak{c} (\mathfrak{v}, \mathfrak{o}) = 0\Longleftrightarrow$ $\mathfrak{v} = \mathfrak{o}$,

(ⅱ) $\mathfrak{c}(\mathfrak{v}, \mathfrak{o}) = \mathfrak{c}(\mathfrak{o }, \mathfrak{v}),$

(ⅲ) $\mathfrak{c}(\mathfrak{v}, \mathfrak{o})\leq \mathfrak{c}(\mathfrak{v}, \varpi)+\mathfrak{c}(\varpi, \mathfrak{o}),$

for all $\mathfrak{v}, \mathfrak{o}, \varpi \in \Psi;$ then $(\Psi, \mathfrak{ c})$ is said to be an MS.

Berzig ^[6] introduced the concept of SMS in this way.

Definition 2. (^[6]) Let $\Psi \not = \emptyset$ and $\curlywedge \geq 0$. Consider a function $\mathfrak{c}:\Psi \times \Psi \rightarrow \mathbb{ \mathbb{R} }^{+}$ such that

(ⅰ) $0\leq \mathfrak{c}(\mathfrak{v}, \mathfrak{o})\ $and $\mathfrak{c} (\mathfrak{v}, \mathfrak{o}) = 0$ $\Longleftrightarrow$ $\mathfrak{v} = \mathfrak{o}$,

(ⅱ) $\mathfrak{c}(\mathfrak{v}, \mathfrak{o}) = \mathfrak{c}(\mathfrak{o }, \mathfrak{v}),$

(ⅲ) $\mathfrak{c}(\mathfrak{v}, \mathfrak{o})\leq \mathfrak{c}(\mathfrak{v}, \varpi)+\mathfrak{c}(\varpi, \mathfrak{o})+\curlywedge \mathfrak{c}(\mathfrak{v}, \varpi)\mathfrak{c}(\varpi, \mathfrak{o}),$

for all $\mathfrak{v}, \mathfrak{o}, \varpi \in \Psi;$ then $(\Psi, \mathfrak{ c})$ is called an SMS.

Example 1. Let $\Psi = \left \{ 0, 1, 2\right \}$ and define the distance function $\mathfrak{c}:\Psi \times \Psi \rightarrow \mathbb{ \mathbb{R} }^{+}$ as follows:

and

Let's choose $\curlywedge = 1.5.$ Then, $\left(\Psi, \mathfrak{c}\right)$ is an SMS but not an MS because the triangle of an MS is not satisfied, that is,

In the context of the newly introduced SMS, Berzig ^[6] proved the Banach contraction principle.

Theorem 1. (^[6]) Let $\left(\Psi, \mathfrak{c}\right)$ be a complete SMS and $\mathfrak{A}:\Psi$ $\rightarrow \Psi$. If $\Bbbk \in \lbrack 0, 1)$ exists such that

for all $\mathfrak{v}, \mathfrak{o}\in$ $\Psi$, then $\mathfrak{A}$ has a unique FP.

On the other hand, Azam et al. ^[10] introduced the concept of a CVMS by replacing the real numbers with complex numbers in the codomain, thereby extending the classical definition of an MS to the complex number system.

Let$\ z_{1}$ and $z_{2}$ be complex numbers. A partial order $\precsim$ on $\mathbb{C}$ is defined by:

It follows that

if one of these conditions is met:

Azam et al. ^[10] defined the concept of CVMS as follows:

Definition 3. (^[10]) Let $\Psi \not = \emptyset.$ Let $\mathfrak{c}:\Psi \times \Psi \rightarrow \mathbb{C}$ be a function such that

(ⅰ) $0\precsim \mathfrak{c}(\mathfrak{v}, \mathfrak{o})\ $and $\mathfrak{c}(\mathfrak{v}, \mathfrak{o}) = 0$ $\Longleftrightarrow$ $\mathfrak{ v} = \mathfrak{o}$,

(ⅱ) $\mathfrak{c}(\mathfrak{v}, \mathfrak{o}) = \mathfrak{c}(\mathfrak{o }, \mathfrak{v}),$

(ⅲ) $\mathfrak{c}(\mathfrak{v}, \mathfrak{o})\precsim \mathfrak{c}(\mathfrak{v}, \varpi)+\mathfrak{c}(\varpi, \mathfrak{o}),$

for all $\mathfrak{v}, \mathfrak{o}, \varpi \in \Psi;$ then $(\Psi, \mathfrak{ c})$ is said to be a CVMS.

Example 2. (^[10]) Let $\Psi = [0, 1]$ and $\mathfrak{v}, \mathfrak{o}\in \Psi.$ Define $\mathfrak{c}:\Psi \times \Psi \rightarrow \mathbb{C}$ by

Then, $(\Psi, \mathfrak{c})$ is a CVMS.

Azam et al. ^[10] established the following FP theorem in the context of CVMSs.

Theorem 2. (^[10]) Let $\left(\Psi, \mathfrak{c}\right)$ be a complete CVMS and $\mathfrak{A}, \mathfrak{B}:\Psi$ $\rightarrow \Psi$. If the non-negative real numbers $\Bbbk _{1}$ and $\Bbbk _{2}$ with $\Bbbk _{1}+\Bbbk _{2} < 1$ exist such that

for all $\mathfrak{v}, \mathfrak{o}\in$ $\Psi$, then $\mathfrak{A}$ and $\mathfrak{B}$ possess a unique CFP.

Rouzkard et al. ^[11] refined the contractive condition of Azam et al. ^[10] through the inclusion of a rational expression, subsequently demonstrating the following result.

Theorem 3. (^[11]) Let $\left(\Psi, \mathfrak{c}\right)$ be a complete CVMS and $\mathfrak{A}, \mathfrak{B}:\Psi$ $\rightarrow \Psi$. If $\Bbbk _{1}, \Bbbk _{2}, \Bbbk _{3}\in \lbrack 0, 1)$ with $\Bbbk _{1}+\Bbbk _{2}+\Bbbk _{3} < 1$ exist such that

for all $\mathfrak{v}, \mathfrak{o}\in$ $\Psi$, then $\mathfrak{A}$ and $\mathfrak{B}$ possess a unique CFP.

Sintunavarat et al. ^[12] extended the primary finding of Azam et al. ^[10] by substituting the constants with single-variable control functions.

Theorem 4. (^[12]) Let $\left(\Psi, \mathfrak{c}\right)$ be a complete CVMS and $\mathfrak{A}, \mathfrak{B}:$ $\Psi$ $\rightarrow \Psi$. Assuming the existence of the functions $\Bbbk _{1}, \Bbbk _{2}:\Psi \mathfrak{\rightarrow \lbrack }0, 1\mathfrak{)}$ such that

(a) $\Bbbk _{1}\left(\mathfrak{BAv}\right) \leq \Bbbk _{1}\left(\mathfrak{v }\right)$ $\Bbbk _{2}\left(\mathfrak{BAv}\right) \leq \Bbbk _{2}\left(\mathfrak{v} \right),$

(b) $\Bbbk _{1}\left(\mathfrak{v}\right) +\Bbbk _{2}\left(\mathfrak{v} \right) < 1,$

(c)

for all $\mathfrak{v}, \mathfrak{o}\in$ $\Psi \mathfrak{, }$ then $\mathfrak{A }$ and $\mathfrak{B}$ possess a unique CFP.

Sitthikul et al. ^[13] presented the following result for the functions of two variables in the framework of CVMSs.

Theorem 5. (^[13]) Let $\left(\Psi, \mathfrak{c}\right)$ be a complete CVMS and $\mathfrak{A}, \mathfrak{B}:$ $\Psi$ $\rightarrow \Psi$. Assuming the existence of the functions $\Bbbk _{1}, \Bbbk _{2}, \Bbbk _{3}:\Psi \mathfrak{ \times }\Psi \mathfrak{\rightarrow }[0, 1)$ such that for all $\mathfrak{v}, \mathfrak{o}\in$ $\Psi \mathfrak{, }$

(a) $\Bbbk _{1}\left(\mathfrak{BAv}, \mathfrak{o}\right) \leq \Bbbk _{1}\left(\mathfrak{v}, \mathfrak{o}\right)$ and $\Bbbk _{1}\left(\mathfrak{v}, \mathfrak{ABo}\right) \leq \Bbbk _{1}\left(\mathfrak{v}, \mathfrak{o}\right),$

$\Bbbk _{2}\left(\mathfrak{BAv}, \mathfrak{o}\right) \leq \Bbbk _{2}\left(\mathfrak{v}, \mathfrak{o}\right)$ and $\Bbbk _{2}\left(\mathfrak{v}, \mathfrak{ABo}\right) \leq \Bbbk _{2}\left(\mathfrak{v}, \mathfrak{o}\right),$

$\Bbbk _{3}\left(\mathfrak{BAv}, \mathfrak{o}\right) \leq \Bbbk _{3}\left(\mathfrak{v}, \mathfrak{o}\right)$ and $\Bbbk _{3}\left(\mathfrak{v}, \mathfrak{ABo}\right) \leq \Bbbk _{3}\left(\mathfrak{v}, \mathfrak{o}\right),$

(b) $\Bbbk _{1}\left(\mathfrak{v}, \mathfrak{o}\right) +\Bbbk _{2}\left(\mathfrak{v}, \mathfrak{o}\right) +\Bbbk _{3}\left(\mathfrak{v}, \mathfrak{o} \right) < 1,$

(c)

for all $\mathfrak{v}, \mathfrak{o}\in \Psi,$ then $\mathfrak{A}$ and $\mathfrak{B}$ possess a unique CFP.

Recently, Panda et al. ^[14] merged the concepts of SMSs and CVMSs and introduced the notion of CVSMS in this manner.

Definition 4. (^[14]) Let $\Psi \not = \emptyset$ and $\curlywedge \geq 0.$ Consider a function $\mathfrak{c}:\Psi \times \Psi \rightarrow \mathbb{C}$ such that

(ⅰ) $0\precsim \mathfrak{c}(\mathfrak{v}, \mathfrak{o})\ $and $\mathfrak{c}(\mathfrak{v}, \mathfrak{o}) = 0$ $\Longleftrightarrow$ $\mathfrak{ v} = \mathfrak{o}$,

(ⅱ) $\mathfrak{c}(\mathfrak{v}, \mathfrak{o}) = \overline{\mathfrak{c}(\mathfrak{o}, \mathfrak{v})}, \ $

(ⅲ) $\mathfrak{c}(\mathfrak{v}, \mathfrak{o})\precsim \mathfrak{c}(\mathfrak{v}, \varpi)+\mathfrak{c}(\varpi, \mathfrak{o})+\curlywedge \mathfrak{c}(\mathfrak{v}, \varpi)\mathfrak{c}(\varpi, \mathfrak{o}),$

for all $\mathfrak{v}, \mathfrak{o}, \varpi \in \Psi;$ then $(\Psi, \mathfrak{ c})$ is called a CVSMS.

Example 3. Let $\Psi = \mathbb{C}$ (the set of complex numbers) and $\curlywedge = 1.$ Define the function $\mathfrak{c}:\Psi \times \Psi \rightarrow \mathbb{C}$ by

where $\left \vert \mathfrak{v}-\mathfrak{o}\right \vert$ is the modulus of the difference between $\mathfrak{v}$ and $\mathfrak{o, }$ and $i$ is the imaginary unit. Then, $\left(\Psi, \mathfrak{c}\right)$ is a CVSMS.

Panda et al. ^[14] proved the following result in the setting of CVSMSs.

Theorem 6. (^[14]) Let $\left(\Psi, \mathfrak{c}\right)$ be a complete CVSMS and $\mathfrak{A}, \mathfrak{B}:\Psi$ $\rightarrow \Psi$. If the non-negative real numbers $\Bbbk _{1}, \Bbbk _{2}$ and $\Bbbk _{3}$ with $\Bbbk _{1}+\Bbbk _{2}+\Bbbk _{3} < 1$ exist such that

for all $\mathfrak{v}, \mathfrak{o}\in$ $\Psi$, then $\mathfrak{A}$ and $\mathfrak{B}$ possess a unique CFP.

The following lemma plays a key role in the proof of our main result.

Lemma 1. (^[14]) Let $(\Psi, \mathfrak{c})\ $be a CVSMS, and let $\left \{ \mathfrak{v}_{r}\right \}$ $\subseteq \Psi$ be a sequence. Then, the following hold:

(ⅰ). Convergence: The sequence $\left \{ \mathfrak{v}_{r}\right \}$ converges to $\mathfrak{v}$ if and only if $\left \vert \mathfrak{c}(\mathfrak{v}_{r}, \mathfrak{v})\right \vert \rightarrow 0\ $as$\ r\rightarrow \infty.$

(ⅱ). Cauchy Condition: The sequence $\left \{ \mathfrak{v}_{r}\right \}$ is a Cauchy sequence if and only if$\ \left \vert \mathfrak{c}(\mathfrak{v}_{r}, \mathfrak{v}_{r+m})\right \vert \rightarrow 0\ $as$\ r\rightarrow \infty, \ $ for any $m\in \mathbb{N}.$

In this research work, we prove common FP theorems for rational contractions involving control functions of multiple variables in the framework of CVSMSs. In this way, we derive some important FP results, including the fundamental theorem of Panda et al. ^[14] in the context of CVSMSs, along with the significant findings of Azam et al. ^[10], Rouzkard et al. ^[11], Sintunavarat et al. ^[12], and Sitthikul et al. ^[13] within the scope of CVMSs, and the core theorem of Berzig ^[6] in the setting of SMSs. To emphasize the novelty of the main result, we also provide an illustrative example. Furthermore, the results are practically demonstrated by solving a nonlinear mixed Volterra–Fredholm integral equation, which models predator-prey dynamics.

3.   Main results

We establish the following proposition, which is necessary for the proof of our prime result:

Proposition 1. Let $\mathfrak{A}, \mathfrak{B}:$ $\Psi$ $\rightarrow \Psi$ be self-mappings and $\mathfrak{v}_{0}\in$ $\Psi.$ Define the sequence {$\mathfrak{v}_{r}$} by

for all $r = 0, 1, 2, ...$

Suppose a function $\Bbbk _{1}:\Psi \mathfrak{\times }\Psi \mathfrak{\rightarrow }[0, 1)$ exist such that

for all $\mathfrak{v}, \mathfrak{o}\in \Psi.$ Then

for all $\mathfrak{v}, \mathfrak{o}\in \Psi$ and $r = 0, 1, 2, ...$

We now state and prove the following lemma, which plays a crucial role in establishing our main result.

Lemma 2. Let $\Bbbk _{1}, \Bbbk _{2}:\Psi \mathfrak{\times }\Psi \mathfrak{ \rightarrow }[0, 1)$ and $\mathfrak{v}, \mathfrak{o}\in$ $\Psi.$ If $\mathfrak{A}, \mathfrak{B}:$ $\Psi$ $\rightarrow \Psi$ satisfy

in which case

Proof. We can write

Similarly, we have

We now present the leading theorem of this section. □

Theorem 7. Let $\left(\Psi, \mathfrak{c}\right)$ be a complete CVSMS and $\mathfrak{A}, \mathfrak{B}:$ $\Psi$ $\rightarrow \Psi$. Assuming the existence of the functions $\Bbbk _{1}, \Bbbk _{2}, \Bbbk _{3}:\Psi \mathfrak{ \times }\Psi \mathfrak{\rightarrow }[0, 1)$ such that for all $\mathfrak{v}, \mathfrak{o}\in$ $\Psi \mathfrak{, }$

(a) $\Bbbk _{1}\left(\mathfrak{BAv}, \mathfrak{o}\right) \leq \Bbbk _{1}\left(\mathfrak{v}, \mathfrak{o}\right)$ and $\Bbbk _{1}\left(\mathfrak{v}, \mathfrak{ABo}\right) \leq \Bbbk _{1}\left(\mathfrak{v}, \mathfrak{o}\right),$

$\Bbbk _{2}\left(\mathfrak{BAv}, \mathfrak{o}\right) \leq \Bbbk _{2}\left(\mathfrak{v}, \mathfrak{o}\right)$ and $\Bbbk _{2}\left(\mathfrak{v}, \mathfrak{ABo}\right) \leq \Bbbk _{2}\left(\mathfrak{v}, \mathfrak{o}\right),$

$\Bbbk _{3}\left(\mathfrak{BAv}, \mathfrak{o}\right) \leq \Bbbk _{3}\left(\mathfrak{v}, \mathfrak{o}\right)$ and $\Bbbk _{3}\left(\mathfrak{v}, \mathfrak{ABo}\right) \leq \Bbbk _{3}\left(\mathfrak{v}, \mathfrak{o}\right),$

(b) $\Bbbk _{1}\left(\mathfrak{v}, \mathfrak{o}\right) +\Bbbk _{2}\left(\mathfrak{v}, \mathfrak{o}\right) +\Bbbk _{3}\left(\mathfrak{v}, \mathfrak{o} \right) < 1,$

(c)

for all $\mathfrak{v}, \mathfrak{o}\in \Psi,$ then $\mathfrak{A}$ and $\mathfrak{B}$ admit a unique CFP.

Proof. Let $\mathfrak{v}, \mathfrak{o}\in \Psi \mathfrak{.}$ From Eq (3.1), we have

By Lemma 2, we get

Similarly, we have

By Lemma 2, we get

Let $\mathfrak{v}_{0}\in \Psi$ be given. Define the sequence {$\mathfrak{v} _{r}$} inductively by

From Proposition 1, Eqs (3.2), and (3.3), for all $r = 0, 1, 2, ..$.

which implies that

Similarly, we have

which implies that

Let $\mu =$ $\frac{\Bbbk _{1}\left(\mathfrak{v}_{0}, \mathfrak{v}_{1}\right) }{1-\Bbbk _{2}\left(\mathfrak{v}_{0}, \mathfrak{v}_{1}\right) } < 1.$ Then from Eqs (3.4) and (3.5), we have

for all $r\in \mathbb{N}.$ Recursively, we can generate a sequence $\{ \mathfrak{v}_{r}\}$ in $\Psi$ such that

That is,

for all $r\in \mathbb{N}.$ Now for $m > r$, we get

We now apply the triangle inequality again to the second term $\left \vert \mathfrak{c}\left(\mathfrak{v}_{r+1}, \mathfrak{v}_{m}\right) \right \vert,$ and we have

We now apply the triangle inequality again to the second term $\left \vert \mathfrak{c}\left(\mathfrak{v}_{r+2}, \mathfrak{v}_{m}\right) \right \vert,$ and we have

and so on

Recursively substituting each inequality into the preceding one (3.7) and simplifying, we have

By the inequality (3.6), we have

From the inequality (3.8), we get

which implies

Now, observe that $\left(1+\curlywedge \mu ^{i}\left \vert \mathfrak{c}\left(\mathfrak{v}_{0}, \mathfrak{v}_{1}\right) \right \vert \right) \geq 1,$ for all $i.$ Therefore,

With the product term bounded below by $1$, we can simplify the summation

Since $\sum \limits_{k = r}^{m-1}\mu ^{k}$ is a finite geometric series with the first term $\mu ^{r}$ and the common ratio $\mu,$ the sum of a finite geometric series is given by

As $m\rightarrow \infty,$ $\mu ^{m-r}\rightarrow 0.$ Therefore, the sum converges to $\frac{\mu ^{r}}{1-\mu };$ that is,

Since $\mu < 1,$ the expression $\frac{\mu ^{r}}{1-\mu }$ approaches $0$ as $r\rightarrow \infty.$ Letting $r\rightarrow \infty$ in inequality (3.9) and employing the established facts, we arrive at

Therefore, by Lemma 1 (ⅱ), $\left \{ \mathfrak{v}_{r}\right \}$ is Cauchy. As $\Psi$ is complete, so $\mathfrak{v}^{\ast }\in \Psi$ exist such that $\mathfrak{v}_{r}\rightarrow \mathfrak{v}^{\ast }$ as $r\rightarrow \infty.$ Thus

Now, we show that $\mathfrak{v}^{\ast }$ is a FP of $\mathfrak{A}.$ From (3.1), we have

This implies that

Letting $r\rightarrow \infty,$ we have $\left \vert \mathfrak{c}\left(\mathfrak{v}^{\ast }, \mathfrak{Av}^{\ast }\right) \right \vert = 0.$ Thus $\mathfrak{v}^{\ast } = \mathfrak{Av}^{\ast }.$ To demonstrate that $\mathfrak{v }^{\ast }$ is a FP of $\mathfrak{B},$ we observe from (3.1) that

This implies that

□

Letting $r\rightarrow \infty,$ we have $\left \vert \mathfrak{c}\left(\mathfrak{v}^{\ast }, \mathfrak{Bv}^{\ast }\right) \right \vert = 0,$ a therefore, $\mathfrak{v}^{\ast } = \mathfrak{Bv}^{\ast }.$ Thus, $\mathfrak{v}^{\ast }$ is a CFP of $\mathfrak{A}$ and $\mathfrak{B}.$ To establish the uniqueness of $\mathfrak{v}^{\ast },$ suppose, to the contrary, that another CFP $\mathfrak{v}^{/}$ of $\mathfrak{A}$ and $\mathfrak{B}$ exist. This implies that

but $\mathfrak{v}^{\ast }\not = \mathfrak{v}^{/}.$ From (3.1), we have

This implies that we have

As $\Bbbk _{1}\left(\mathfrak{v}^{\ast }, \mathfrak{v}^{/}\right) +\Bbbk _{3}\left(\mathfrak{v}^{\ast }, \mathfrak{v}^{/}\right) < 1,$ we have

Thus $\mathfrak{v}^{\ast } = \mathfrak{v}^{/}.$

Corollary 1. Let $\left(\Psi, \mathfrak{c}\right)$ be a complete CVSMS and $\mathfrak{A}, \mathfrak{B}:$ $\Psi$ $\rightarrow \Psi$. Assuming the existence of the functions $\Bbbk _{1}, \Bbbk _{2}:\Psi \mathfrak{\times }\Psi \mathfrak{ \rightarrow \lbrack }0, 1\mathfrak{)}$ such that

(a) $\Bbbk _{1}\left(\mathfrak{BAv}, \mathfrak{o}\right) \leq \Bbbk _{1}\left(\mathfrak{v}, \mathfrak{o}\right)$ and $\Bbbk _{1}\left(\mathfrak{v}, \mathfrak{ABo}\right) \leq \Bbbk _{1}\left(\mathfrak{v}, \mathfrak{o}\right),$

$\Bbbk _{2}\left(\mathfrak{BAv}, \mathfrak{o}\right) \leq \Bbbk _{2}\left(\mathfrak{v}, \mathfrak{o}\right)$ and $\Bbbk _{2}\left(\mathfrak{v}, \mathfrak{ABo}\right) \leq \Bbbk _{2}\left(\mathfrak{v}, \mathfrak{o}\right),$

(b) $\Bbbk _{1}\left(\mathfrak{v}, \mathfrak{o}\right) +\Bbbk _{2}\left(\mathfrak{v}, \mathfrak{o}\right) < 1,$

(c)

for all $\mathfrak{v}, \mathfrak{o}\in$ $\Psi \mathfrak{, }$ then $\mathfrak{A}$ and $\mathfrak{B}$ admit a unique CFP.

Proof. Define $\Bbbk _{3}:\Psi \mathfrak{\times }\Psi \mathfrak{\rightarrow \lbrack }0, 1\mathfrak{)}$ by $\Bbbk _{3}\left(\mathfrak{v}, \mathfrak{o}\right) = 0$ in Theorem 7. □

Corollary 2. Let $\left(\Psi, \mathfrak{c}\right)$ be a complete CVSMS and $\mathfrak{A}, \mathfrak{B}:$ $\Psi$ $\rightarrow \Psi$. Assuming the existence of the functions $\Bbbk _{1}, \Bbbk _{3}:\Psi \mathfrak{\times }\Psi \mathfrak{ \rightarrow }[0, 1)$ such that

(a) $\Bbbk _{1}\left(\mathfrak{BAv}, \mathfrak{o}\right) \leq \Bbbk _{1}\left(\mathfrak{v}, \mathfrak{o}\right)$ and $\Bbbk _{1}\left(\mathfrak{v}, \mathfrak{ABo}\right) \leq \Bbbk _{1}\left(\mathfrak{v}, \mathfrak{o}\right),$

$\Bbbk _{3}\left(\mathfrak{BAv}, \mathfrak{o}\right) \leq \Bbbk _{3}\left(\mathfrak{v}, \mathfrak{o}\right)$ and $\Bbbk _{3}\left(\mathfrak{v}, \mathfrak{ABo}\right) \leq \Bbbk _{3}\left(\mathfrak{v}, \mathfrak{o}\right),$

(b) $\Bbbk _{1}\left(\mathfrak{v}, \mathfrak{o}\right) +\Bbbk _{3}\left(\mathfrak{v}, \mathfrak{o}\right) < 1,$

(c)

for all $\mathfrak{v}, \mathfrak{o}\in$ $\Psi \mathfrak{, }$ then $\mathfrak{A}$ and $\mathfrak{B}$ admit a unique CFP.

Proof. Define $\Bbbk _{2}:\Psi \mathfrak{\times }\Psi \mathfrak{\rightarrow \lbrack }0, 1\mathfrak{)}$ by $\Bbbk _{2}\left(\mathfrak{v}, \mathfrak{o}\right) = 0$ in Theorem 7. □

Corollary 3. Let $\left(\Psi, \mathfrak{c}\right)$ be a complete CVSMS and $\mathfrak{A}, \mathfrak{B}:$ $\Psi$ $\rightarrow \Psi$. Assuming the existence of the function $\Bbbk _{1}:\Psi \mathfrak{\times }\Psi \mathfrak{\rightarrow \lbrack }0, 1\mathfrak{)}$ such that

(a) $\Bbbk _{1}\left(\mathfrak{BAv}, \mathfrak{o}\right) \leq \Bbbk _{1}\left(\mathfrak{v}, \mathfrak{o}\right)$ and $\Bbbk _{1}\left(\mathfrak{v}, \mathfrak{ABo}\right) \leq \Bbbk _{1}\left(\mathfrak{v}, \mathfrak{o}\right),$

(b)

for all $\mathfrak{v}, \mathfrak{o}\in$ $\Psi$, then $\mathfrak{A}$ and $\mathfrak{B}$ admit a unique CFP.

Proof. Define $\Bbbk _{2}, \Bbbk _{3}:\Psi \mathfrak{\times }\Psi \mathfrak{ \rightarrow \lbrack }0, 1\mathfrak{)}$ by $\Bbbk _{2}\left(\mathfrak{v}, \mathfrak{o}\right) = \Bbbk _{3}\left(\mathfrak{v}, \mathfrak{o}\right) = 0$ in Theorem 7. □

Corollary 4. Let $\left(\Psi, \mathfrak{c}\right)$ be a complete CVSMS and $\mathfrak{A}:\Psi$ $\rightarrow \Psi$. Assuming the existence of the functions $\Bbbk _{1}, \Bbbk _{2}, \Bbbk _{3}:\Psi \mathfrak{\times }\Psi \mathfrak{ \rightarrow \lbrack }0, 1\mathfrak{)}$ such that

(a) $\Bbbk _{1}\left(\mathfrak{Av}, \mathfrak{o}\right) \leq \Bbbk _{1}\left(\mathfrak{v}, \mathfrak{o}\right)$ and $\Bbbk _{1}\left(\mathfrak{v}, \mathfrak{Ao}\right) \leq \Bbbk _{1}\left(\mathfrak{v}, \mathfrak{o}\right),$

$\Bbbk _{2}\left(\mathfrak{Av}, \mathfrak{o}\right) \leq \Bbbk _{2}\left(\mathfrak{v}, \mathfrak{o}\right)$ and $\Bbbk _{2}\left(\mathfrak{v}, \mathfrak{Ao}\right) \leq \Bbbk _{2}\left(\mathfrak{v}, \mathfrak{o}\right),$

$\Bbbk _{3}\left(\mathfrak{Av}, \mathfrak{o}\right) \leq \Bbbk _{3}\left(\mathfrak{v}, \mathfrak{o}\right)$ and $\Bbbk _{3}\left(\mathfrak{v}, \mathfrak{Ao}\right) \leq \Bbbk _{3}\left(\mathfrak{v}, \mathfrak{o}\right),$

(b) $\Bbbk _{1}\left(\mathfrak{v}, \mathfrak{o}\right) +\Bbbk _{2}\left(\mathfrak{v}, \mathfrak{o}\right) +\Bbbk _{3}\left(\mathfrak{v}, \mathfrak{o} \right) < 1,$

(c)

for all $\mathfrak{v}, \mathfrak{o}\in$ $\Psi$, then $\mathfrak{A}$ has a unique FP.

Proof. Take $\mathfrak{A} = \mathfrak{B}$ in Theorem 7. □

Corollary 5. Let $\left(\Psi, \mathfrak{c}\right)$ be a complete CVSMS and $\mathfrak{A} :\Psi$ $\rightarrow \Psi$. Assuming the existence of the functions $\Bbbk _{1}, \Bbbk _{2}:\Psi \mathfrak{\times }\Psi \mathfrak{\rightarrow \lbrack } 0, 1\mathfrak{)}$ such that

(a) $\Bbbk _{1}\left(\mathfrak{Av}, \mathfrak{o}\right) \leq \Bbbk _{1}\left(\mathfrak{v}, \mathfrak{o}\right)$ and $\Bbbk _{1}\left(\mathfrak{v}, \mathfrak{Ao}\right) \leq \Bbbk _{1}\left(\mathfrak{v}, \mathfrak{o}\right),$

$\Bbbk _{2}\left(\mathfrak{Av}, \mathfrak{o}\right) \leq \Bbbk _{2}\left(\mathfrak{v}, \mathfrak{o}\right)$ and $\Bbbk _{2}\left(\mathfrak{v}, \mathfrak{Ao}\right) \leq \Bbbk _{2}\left(\mathfrak{v}, \mathfrak{o}\right),$

(b) $\Bbbk _{1}\left(\mathfrak{v}, \mathfrak{o}\right) +\Bbbk _{2}\left(\mathfrak{v}, \mathfrak{o}\right) < 1,$

(c)

for all $\mathfrak{v}, \mathfrak{o}\in$ $\Psi$, then $\mathfrak{A}$ has a unique FP.

Proof. Define $\Bbbk _{3}:\Psi \mathfrak{\times }\Psi \mathfrak{\rightarrow \lbrack }0, 1\mathfrak{)}$ by $\Bbbk _{3}\left(\mathfrak{v}, \mathfrak{o}\right) = 0$ in Corollary 4.

□

Corollary 6. Let $\left(\Psi, \mathfrak{c}\right)$ be a complete CVSMS and $\mathfrak{A}:\Psi$ $\rightarrow \Psi$. Assuming the existence of the function $\Bbbk _{1}:\Psi \mathfrak{\times }\Psi \mathfrak{\rightarrow \lbrack }0, 1 \mathfrak{)}$ such that

(a) $\Bbbk _{1}\left(\mathfrak{Av}, \mathfrak{o}\right) \leq \Bbbk _{1}\left(\mathfrak{v}, \mathfrak{o}\right)$ and $\Bbbk _{1}\left(\mathfrak{v}, \mathfrak{Ao}\right) \leq \Bbbk _{1}\left(\mathfrak{v}, \mathfrak{o}\right),$

(b)

for all $\mathfrak{v}, \mathfrak{o}\in$ $\Psi$, then $\mathfrak{A}$ has a unique FP.

Proof. Define $\Bbbk _{2}, \Bbbk _{3}:\Psi \mathfrak{\times }\Psi \mathfrak{ \rightarrow \lbrack }0, 1\mathfrak{)}$ by $\Bbbk _{2}\left(\mathfrak{v}, \mathfrak{o}\right) = \Bbbk _{3}\left(\mathfrak{v}, \mathfrak{o}\right) = 0$ in Corollary 4. □

Example 4. Let $\Psi = [0, 1],$ define $\mathfrak{c}:\Psi \times \Psi \rightarrow \mathbb{ C}$ by

for all $\mathfrak{v}, \mathfrak{o}\in$ $\Psi.$ Now, we prove that $(\Psi, \mathfrak{c})$ is complete CVSMS.

Condition (ⅰ).

● Non-negativity: For $\mathfrak{c}(\mathfrak{v}, \mathfrak{o}) = \left \vert \mathfrak{v}-\mathfrak{o}\right \vert +i\left \vert \mathfrak{v}-\mathfrak{o}\right \vert :$

Thus, $0+0i\preceq \mathfrak{c}(\mathfrak{v}, \mathfrak{o})$ is satisfied.

● Zero iff equality:

$\mathfrak{c}(\mathfrak{v}, \mathfrak{o}) = 0+0i$ implies $\left \vert \mathfrak{v}-\mathfrak{o}\right \vert = 0,$ which means $\mathfrak{v} = \mathfrak{o}.$ Conversely, if $\mathfrak{v} = \mathfrak{o},$ then $\left \vert \mathfrak{v}-\mathfrak{o}\right \vert = 0$ implies $\mathfrak{c}(\mathfrak{v}, \mathfrak{o}) = 0+0i.$

Condition (ⅱ).

Since

symmetry is satisfied.

Condition (ⅲ).

We now satisfy the triangle inequality, that is,

for all $\mathfrak{v}, \varpi, \mathfrak{o}\in \Psi$ and for some $\curlywedge \geq 0.$

(1). $\mathfrak{c}(\mathfrak{v}, \varpi)+\mathfrak{c}(\varpi, \mathfrak{o}) = \left(\left \vert \mathfrak{v}-\varpi \right \vert +i\left \vert \mathfrak{v} -\varpi \right \vert \right) +\left(\left \vert \varpi -\mathfrak{o} \right \vert +i\left \vert \varpi -\mathfrak{o}\right \vert \right).$ Simplifying, we get

and

(2). $\mathfrak{c}(\mathfrak{v}, \varpi)\mathfrak{c}(\varpi, \mathfrak{o}) = \left(\left \vert \mathfrak{v}-\varpi \right \vert +i\left \vert \mathfrak{v}-\varpi \right \vert \right) \left(\left \vert \varpi - \mathfrak{o}\right \vert +i\left \vert \varpi -\mathfrak{o}\right \vert \right).$ Simplifying, we get

Now,

Compare $\mathfrak{c}(\mathfrak{v}, \mathfrak{o}) = \left \vert \mathfrak{v}- \mathfrak{o}\right \vert +i\left \vert \mathfrak{v}-\mathfrak{o}\right \vert.$

Real part:

by the triangle inequality in $\mathbb{R}.$

Imaginary part:

which follows from non-negativity of $\curlywedge$ and the terms. Thus, Condition (ⅲ) is satisfied. Hence, $(\Psi, \mathfrak{c})$ is a complete CVSMS.

Define a self mapping $\mathfrak{A}:\Psi$ $\rightarrow \Psi$ by

Consider $\Bbbk _{1}:\Psi \times \Psi \mathfrak{\rightarrow \lbrack }0, 1 \mathfrak{)}$, where

In this case

Thus,

and

Now assume that

Since $1+\mathfrak{v}+\mathfrak{o} > 1$ for all $\mathfrak{v}, \mathfrak{o} \in \lbrack 0, 1],$ we have

Therefore,

Thus,

Hence, all the conditions of Corollary 6 are fulfilled, establishing that $\frac{1}{2}$ is the unique FP of the mapping $\mathfrak{A}.$

Corollary 7. Let $\left(\Psi, \mathfrak{c}\right)$ be a complete CVSMS and $\mathfrak{A} :\Psi$ $\rightarrow \Psi$. Assuming the existence of the functions $\Bbbk _{1}, \Bbbk _{2}, \Bbbk _{3}:\Psi \mathfrak{\times }\Psi \mathfrak{\rightarrow \lbrack }0, 1\mathfrak{)}$ such that

(a) $\Bbbk _{1}\left(\mathfrak{Av}, \mathfrak{o}\right) \leq \Bbbk _{1}\left(\mathfrak{v}, \mathfrak{o}\right)$ and $\Bbbk _{1}\left(\mathfrak{v}, \mathfrak{Ao}\right) \leq \Bbbk _{1}\left(\mathfrak{v}, \mathfrak{o}\right),$

$\Bbbk _{2}\left(\mathfrak{Av}, \mathfrak{o}\right) \leq \Bbbk _{2}\left(\mathfrak{v}, \mathfrak{o}\right)$ and $\Bbbk _{2}\left(\mathfrak{v}, \mathfrak{Ao}\right) \leq \Bbbk _{2}\left(\mathfrak{v}, \mathfrak{o}\right),$

$\Bbbk _{3}\left(\mathfrak{Av}, \mathfrak{o}\right) \leq \Bbbk _{3}\left(\mathfrak{v}, \mathfrak{o}\right)$ and $\Bbbk _{3}\left(\mathfrak{v}, \mathfrak{Ao}\right) \leq \Bbbk _{3}\left(\mathfrak{v}, \mathfrak{o}\right),$

(b) $\Bbbk _{1}\left(\mathfrak{v}, \mathfrak{o}\right) +\Bbbk _{2}\left(\mathfrak{v}, \mathfrak{o}\right) +\Bbbk _{3}\left(\mathfrak{v}, \mathfrak{o} \right) < 1,$

(c)

for all $\mathfrak{v}, \mathfrak{o}\in$ $\Psi$, then $\mathfrak{A}$ has a unique FP.

Proof. By Corollary (4), $\mathfrak{v}\in \Psi$ exist such that $\mathfrak{A}^{r}\mathfrak{v} = \mathfrak{v}.$ In this case,

which implies that

which is possible only whenever $\left \vert \mathfrak{c}\left(\mathfrak{Av}, \mathfrak{v}\right) \right \vert = 0.$ Thus $\mathfrak{Av} = \mathfrak{v}.$ □

We now turn our attention to deriving FP results for contractive conditions encompassing control functions of a single variable.

Corollary 8. Let $\left(\Psi, \mathfrak{c}\right)$ be a complete CVSMS and $\mathfrak{A}, \mathfrak{B}:$ $\Psi$ $\rightarrow \Psi$. Assuming the existence of the functions $\Bbbk _{1}, \Bbbk _{2}, \Bbbk _{3}:\Psi \mathfrak{ \rightarrow \lbrack }0, 1\mathfrak{)}$ such that

(a) $\Bbbk _{1}\left(\mathfrak{BAv}\right) \leq \Bbbk _{1}\left(\mathfrak{v }\right),$

$\Bbbk _{2}\left(\mathfrak{BAv}\right) \leq \Bbbk _{2}\left(\mathfrak{v} \right),$

$\Bbbk _{3}\left(\mathfrak{BAv}\right) \leq \Bbbk _{3}\left(\mathfrak{v} \right),$

(b) $\Bbbk _{1}\left(\mathfrak{v}\right) +\Bbbk _{2}\left(\mathfrak{v} \right) +\Bbbk _{3}\left(\mathfrak{v}\right) < 1,$

(c)

for all $\mathfrak{v}, \mathfrak{o}\in$ $\Psi \mathfrak{, }$ then $\mathfrak{A }$ and $\mathfrak{B}$ possess a unique CFP.

Proof. Define $\Bbbk _{1}, \Bbbk _{2}, \Bbbk _{3}:\Psi \mathfrak{\times }\Psi \mathfrak{\rightarrow \lbrack }0, 1\mathfrak{)}$ by

for all $\mathfrak{v}, \mathfrak{o}\in \Psi \mathfrak{.}$ For all $\mathfrak{v}, \mathfrak{o}\in \Psi \mathfrak{, }$ we then have □

(a) $\Bbbk _{1}\left(\mathfrak{BAv}, \mathfrak{o}\right) = \Bbbk _{1}\left(\mathfrak{BAv}\right) \leq \Bbbk _{1}\left(\mathfrak{v}\right) = \Bbbk _{1}\left(\mathfrak{v}, \mathfrak{o}\right)$ and $\Bbbk _{1}\left(\mathfrak{v}, \mathfrak{ABo}\right) = \Bbbk _{1}\left(\mathfrak{v}\right) = \Bbbk _{1}\left(\mathfrak{v}, \mathfrak{o}\right),$

$\Bbbk _{2}\left(\mathfrak{BAv}, \mathfrak{o}\right) = \Bbbk _{2}\left(\mathfrak{BAv}\right) \leq \Bbbk _{2}\left(\mathfrak{v}\right) = \Bbbk _{2}\left(\mathfrak{v}, \mathfrak{o}\right)$ and $\Bbbk _{2}\left(\mathfrak{v}, \mathfrak{ABo}\right) = \Bbbk _{2}\left(\mathfrak{v}\right) = \Bbbk _{2}\left(\mathfrak{v}, \mathfrak{o}\right),$

$\Bbbk _{3}\left(\mathfrak{BAv}, \mathfrak{o}\right) = \Bbbk _{3}\left(\mathfrak{BAv}\right) \leq \Bbbk _{3}\left(\mathfrak{v}\right) = \Bbbk _{3}\left(\mathfrak{v}, \mathfrak{o}\right)$ and $\Bbbk _{3}\left(\mathfrak{v}, \mathfrak{ABo}\right) = \Bbbk _{3}\left(\mathfrak{v}\right) = \Bbbk _{3}\left(\mathfrak{v}, \mathfrak{o}\right),$

(b) $\Bbbk _{1}\left(\mathfrak{v}, \mathfrak{o}\right) +\Bbbk _{2}\left(\mathfrak{v}, \mathfrak{o}\right) +\Bbbk _{3}\left(\mathfrak{v}, \mathfrak{o} \right) = \Bbbk _{1}\left(\mathfrak{v}\right) +\Bbbk _{2}\left(\mathfrak{v} \right) +\Bbbk _{3}\left(\mathfrak{v}\right) < 1,$

(c)

By Theorem 7, $\mathfrak{A}$ and $\mathfrak{B}$ possess a unique CFP.

Corollary 9. Let $\left(\Psi, \mathfrak{c}\right)$ be a complete CVSMS and $\mathfrak{A}, \mathfrak{B}:$ $\Psi$ $\rightarrow \Psi$. Assuming the existence of the functions $\Bbbk _{1}, \Bbbk _{2}:\Psi \mathfrak{\rightarrow \lbrack }0, 1\mathfrak{)}$ such that

(a) $\Bbbk _{1}\left(\mathfrak{BAv}\right) \leq \Bbbk _{1}\left(\mathfrak{v }\right),$

$\Bbbk _{2}\left(\mathfrak{BAv}\right) \leq \Bbbk _{2}\left(\mathfrak{v} \right),$

(b) $\Bbbk _{1}\left(\mathfrak{v}\right) +\Bbbk _{2}\left(\mathfrak{v} \right) < 1,$

(c)

for all $\mathfrak{v}, \mathfrak{o}\in$ $\Psi \mathfrak{, }$ then $\mathfrak{A }$ and $\mathfrak{B}$ possess a unique CFP.

Proof. Define $\Bbbk _{3}:\Psi \mathfrak{\rightarrow \lbrack }0, 1\mathfrak{)}$ by $\Bbbk _{3}\left(\mathfrak{v}\right) = 0$ in Corollary 8. □

Corollary 10. Let $\left(\Psi, \mathfrak{c}\right)$ be a complete CVSMS and $\mathfrak{A}, \mathfrak{B}:$ $\Psi$ $\rightarrow \Psi$. Assuming the existence of the function $\Bbbk _{1}:\Psi \mathfrak{\rightarrow \lbrack }0, 1\mathfrak{)}$ such that

(a) $\Bbbk _{1}\left(\mathfrak{BAv}\right) \leq \Bbbk _{1}\left(\mathfrak{v }\right),$

(b)

for all $\mathfrak{v}, \mathfrak{o}\in$ $\Psi \mathfrak{, }$ then $\mathfrak{A }$ and $\mathfrak{B}$ possess a unique CFP.

Proof. Define $\Bbbk _{2}, \Bbbk _{3}:\Psi \mathfrak{\rightarrow \lbrack }0, 1 \mathfrak{)}$ by $\Bbbk _{2}\left(\mathfrak{v}\right) = \Bbbk _{3}\left(\mathfrak{v}\right) = 0$ in Corollary 8. □

Remark 1. The main result of Panda et al. ^[14] follows directly from Corollary 8 by defining the functions $\Bbbk _{1}, \Bbbk _{2}$ and $\Bbbk _{3}$ as $\Bbbk _{1}, \Bbbk _{2}, \Bbbk _{3}:\Psi \mathfrak{\rightarrow \lbrack }0, 1\mathfrak{)}$ with $\Bbbk _{1}\left(\cdot \right) = \Bbbk _{1},$ $\Bbbk _{2}\left(\cdot \right) = \Bbbk _{2}$ and $\Bbbk _{3}\left(\cdot \right) = \Bbbk _{3}$.

Corollary 11. Let $\left(\Psi, \mathfrak{c}\right)$ be a complete CVSMS and let $\mathfrak{A}:\Psi$ $\rightarrow \Psi$. If the non-negative real numbers $\Bbbk _{1}, \Bbbk _{2}$ and $\Bbbk _{3}$ with $\Bbbk _{1}+\Bbbk _{2}+\Bbbk _{3} < 1$ exist such that

for all $\mathfrak{v}, \mathfrak{o}\in$ $\Psi$, then $\mathfrak{A}$ has a unique FP.

Proof. The proof follows directly from Remark 1 by setting $\mathfrak{A} = \mathfrak{B}$, which ensures the existence of a unique FP for $\mathfrak{A}$. □

Corollary 12. Let $\left(\Psi, \mathfrak{c}\right)$ be a complete CVSMS and $\mathfrak{A}, \mathfrak{B}:\Psi$ $\rightarrow \Psi$. If $\Bbbk _{1}, \Bbbk _{2}\in \lbrack 0, 1)$ with $\Bbbk _{1}+\Bbbk _{2} < 1$ exist such that

for all $\mathfrak{v}, \mathfrak{o}\in$ $\Psi$, then $\mathfrak{A}$ and $\mathfrak{B}$ possess a unique CFP.

Proof. Setting $\Bbbk _{3} = 0$ in Remark 1 yields the desired result. □

Corollary 13. Let $\left(\Psi, \mathfrak{c}\right)$ be a complete CVSMS and $\mathfrak{A}, \mathfrak{B}:\Psi$ $\rightarrow \Psi$. If $\Bbbk _{1}\in \lbrack 0, 1)$ exists such that

for all $\mathfrak{v}, \mathfrak{o}\in$ $\Psi$, then $\mathfrak{A}$ and $\mathfrak{B}$ possess a unique CFP.

Proof. Take $\Bbbk _{2} = 0$ in Corollary 12. □

Remark 2. By choosing $\mathfrak{A} = \mathfrak{B}$ and restricting the complex numbers to real numbers by setting the imaginary part to $0$ in Corollary 13, the principal result of Berzig ^[6] follows.

Remark 3. The primary result presented by Sitthikul et al. ^[13] is a direct consequence of Theorem 7 when the parameter $\curlywedge = 0.$

Remark 4. The principal finding of Sintunavarat et al. ^[12] emerges as a special case of Corollary 9 by taking $\curlywedge = 0.$

Remark 5. By applying Remark 1 and choosing $\curlywedge = 0,$ the main result of Rouzkard et al. ^[11] follows naturally.

Remark 6. Taking $\curlywedge = 0$ in Corollary 12, we obtain the leading result of Azam et al. ^[10].

4.   Applications

FP theorems are a powerful tool in functional analysis, providing a framework to prove the existence and uniqueness of solutions to various mathematical equations, including integral equations. These theorems are particularly valuable in addressing complex problems such as nonlinear mixed Volterra–Fredholm integral equations, where their application provides a systematic approach to establish solvability and ensure uniqueness under specific conditions.

The ensuing section is concerned with the investigation of the following nonlinear mixed Volterra–Fredholm integral equation:

where $\mathfrak{v}(t)$ is the unknown function to be determined; $f$ is a given function from $[a, b]$ to $\mathbb{C}$; $K_{1}(t, \mathfrak{o})$ is a specific form of the kernel representing the Fredholm-type integral term, with fixed limits of integration over $[a, b];$ $K_{2}(t, \mathfrak{o})$ is a specific form of the kernel representing the Volterra-type integral term, with the upper limit of integration dependent on $t.$ The function $\mathfrak{o}$ is a transformation function that maps the integration variable $t$ to a new variable. $\Phi _{1},$ $\Phi _{2}:\mathbb{R\rightarrow C}$ are nonlinear functions of the unknown function $\mathfrak{v}.$

Nonlinear mixed Volterra–Fredholm integral equations, such as (4.1), play a crucial role in modeling biological and ecological systems, particularly in the study of species interactions. Predator-prey dynamics, competition, and other ecological processes often exhibit both temporal and spatial dependencies, which can be effectively captured using this integral framework. In a predator-prey system, Eq (4.1) accounts for both local (time-dependent) and global (space-dependent) effects as follows:

● $\mathfrak{v}(t)$ $-:$ Predator population at time $t;$

● $f(t)-:$ External influences (e.g., seasonal effects on predator reproduction);

● $K_{1}(t, \mathfrak{o})-:$ Kernel representing spatial interactions (e.g., prey's availability across regions);

● $\Phi _{1}(\mathfrak{v}(\mathfrak{o}))-:$ Nonlinear effects of resource competition or habitat constraints;

● $K_{2}(t, \mathfrak{o})-:$ Kernel for temporal interactions, representing predator-prey encounter rates;

● $\Phi _{2}(\mathfrak{v}(\mathfrak{o})-:$ Nonlinear predator response to prey density (e.g., functional response).

This integral formulation provides a robust mathematical foundation for analyzing predator-prey interactions and can be extended to other ecological and biological systems exhibiting similar spatial-temporal dependencies. By employing FP theory, we establish the conditions under which (4.1) admits a unique solution, ensuring its theoretical and practical relevance in ecological modeling.

We now present and prove a theorem that establishes the solution of a mixed Volterra–Fredholm integral equation using the FP result obtained in this study.

Theorem 8. Consider $\Psi = C\left([a, b]\right)$. Define $\mathfrak{c}:C\left([a, b]\right) \times C\left([a, b]\right) \rightarrow \mathbb{C}$ by

in which case, $\left(\Psi, \mathfrak{c}\right)$ is a CVSMS. Assume the following:

(ⅰ) $M > 0$ exists such that

for all $t\in \lbrack a, b].$

(ⅱ) The nonlinear functions $\Phi _{1}$ and $\Phi _{2}$ satisfy the contractive condition

for some constants $0 < \lambda, \beta < 1$ and for all $\mathfrak{v}_{1}, \mathfrak{v}_{2}\in C\left([a, b]\right).$

(ⅲ) There is a function $\Bbbk _{1}\mathfrak{:}C\left([a, b]\right) \times C\left([a, b]\right) \rightarrow \lbrack 0, 1)$ defined by

where $M, \lambda, \beta, a, b$ are such that $M\left(\lambda +\beta \right) \left(b-a\right) \mathfrak{ < }1.$

Then the integral equation

possesses a unique solution in $C\left([a, b]\right).$

Proof. Define the mapping $\mathfrak{A}:C\left([a, b]\right) \rightarrow C\left([a, b]\right)$ by

Consider $\mathfrak{v}_{1}, \mathfrak{v}_{2}\in C\left([a, b]\right).$ In this case,

Now, for each $t\in \lbrack a, b],$ we have

This simplifies to

Using the contractive property of $\Phi _{1}$ and $\Phi _{2},$ we obtain

Since $\left \Vert \mathfrak{v}_{1}(\mathfrak{o}(s))-\mathfrak{v}_{2}(\mathfrak{o}(s))\right \Vert \leq \max_{t\in \lbrack a, b]}\left \Vert \mathfrak{v}_{1}(\mathfrak{o}(t))-\mathfrak{v}_{2}(\mathfrak{o} (t))\right \Vert = \frac{\mathfrak{c}(\mathfrak{v}_{1}, \mathfrak{v}_{2})}{ e^{i\tan ^{-1}(a)}},$

Therefore,

and

Substituting these into the inequality (4.2), we have

Taking the maximum for $t\in \lbrack a, b],$ the term $\lambda \left(b-a\right) +\beta \left(t-a\right)$ achieves its maximum at $t = b,$ giving

Since $\mathfrak{c}$ is defined as the maximum over $t\in \lbrack a, b],$

Including the exponential factor $e^{i\tan ^{-1}(a)}$ in the inequality (4.3), we have

Moreover, since $M\left(\lambda +\beta \right) \left(b-a\right) < 1$ and $\frac{\left \Vert \mathfrak{v}_{1}-\mathfrak{v}_{2}\right \Vert }{\mathfrak{c}(\mathfrak{v}_{1}, \mathfrak{v}_{2})}\leq 1,$ it follows by the definition of $\Bbbk _{1}\mathfrak{:}C\left([a, b]\right) \times C\left([a, b]\right) \rightarrow \lbrack 0, 1)$ that $\Bbbk _{1}\mathfrak{(v}_{1}, \mathfrak{v}_{2} \mathfrak{) < }1.$ Hence, all the conditions of Corollary 6 are satisfied. Therefore, the mixed Volterra–Fredholm integral equation admits a unique solution, identical to its unique FP. □

Example 5. Consider the nonlinear mixed Volterra–Fredholm integral equation

where $f(t) = 1+t+e^{t}(1-e),$ $K_{1}(t, \mathfrak{o}) = e^{t-\mathfrak{o} }, K_{2}(t, \mathfrak{o}) = \left(t-\mathfrak{o}\right),$ $\Phi _{1}(\mathfrak{ v}) = \mathfrak{v}^{2}$, and $\Phi _{2}(\mathfrak{v}) = \mathfrak{v}$. This integral equation has a unique solution $\mathfrak{v}(t) = e^{t}$.

5.   Conclusion and future work

In this research work, we have explored the concept of CVSMSs and have established CFP theorems for rational contractions incorporating two variables control functions. Additionally, the study has derived CFP theorems for rational contractions with control functions of a single variable in the same setting. Our results directly imply the main findings of Berzig ^[6], Azam et al. ^[10], Rouzkard et al. ^[11], Sintunavarat et al. ^[12], Sitthikul et al. ^[13], and Panda et al. ^[14]. A demonstrative example has been provided to showcase the novelty of the primary results. The results are practically demonstrated by solving a nonlinear mixed Volterra–Fredholm integral equation that models predator-prey dynamics.

Future research directions include extending CFP theorems to multi-valued, fuzzy, and $L$-fuzzy mappings within the framework of CVSMSs. Additionally, exploring differential and integral inclusions in this setting presents a promising avenue for further study. It is expected that the findings of this research will inspire continued investigations and refinements, potentially broadening the scope of applications for these results.

Use of AI tools declaration

The author confirms that no Artificial Intelligence (AI) tools were utilized in the creation of this article.

Acknowledgments

This work was funded by the University of Jeddah, Jeddah, Saudi Arabia, under grant (No. UJ-24-DR-3126–1). The author acknowledges with thanks the University of Jeddah technical and financial support.

Conflicts of interest

The author declares no conflicts of interest.