A Novel Approach to Some Proximal Contractions with Examples of Its Application

Abstract

In this article, we will introduce a new generalized proximal $\theta$-contraction for multivalued and single-valued mappings named ${(\mathcal{f}-{\theta }_{\kappa })}_{C_P}$-proximal contraction and ${(\mathcal{f}-{\theta }_{\kappa })}_{B_P}$-proximal contraction. Using these newly constructed proximal contractions, we will establish new results for the coincidence best proximity point, best proximity point, and fixed point for multivalued mappings in the context of rectangular metric space. Also, we will reduce these contractions for single-valued mappings, named ${\left({\theta }_{\kappa }\right)}_{C_P}$-proximal contraction and ${\left({\theta }_{\kappa }\right)}_{B_P}$-proximal contraction, to establish results for the coincidence proximity point, best proximity point, and fixed point results. We will give some illustrated examples for our newly generated results with graphical representations. In the last section, we will also find the solution to the equation of motion by using our defined results.

1. Introduction

The term $\mathcal{T}\left(\mathcal{x}\right)=\mathcal{x}$ denotes a fixed point for a self-mapping $\mathcal{T}$ on a metric space. However, it is evident that not every mapping exhibits such a property. We present some examples of such mappings specified on other sets; we call them non-self-mappings and translation mappings. When this happens, our objective shifts towards identifying those elements that are very close to their images, and we call them best approximations; in fact, these approximations are actually fixed points. Some new developments in fixed point theory are discussed in [1,2].

Let $\mathcal{T}:₱\to \mathcal{Q}$, a non-self-mapping, having no fixed point. However, when we approach the image of a point under this mapping, we continuously observe progress. This highlights an important idea called best proximity. Obtaining an optimal global solution, denoted as $\mathcal{x}\to đ(\mathcal{x},\mathcal{T}\mathcal{x})$, is the key to the best proximity theorem. When we have the mapping $\mathcal{T}:₱\to \mathcal{Q}$, for all $\mathcal{x}\in ₱$, the distance $đ(\mathcal{T}\mathcal{x},\mathcal{x})$ is at a minimum of $đ(₱,\mathcal{Q})$. Constructing a close approximation $\mathcal{x}$ for the relation $\mathcal{T}\mathcal{x}=\mathcal{x}$ ensures that $đ(\mathcal{x},\mathcal{T}\mathcal{x})=đ(₱,\mathcal{Q})$ is satisfied.

The main concept of best proximity point was first introduced in 1997 by Sadiq Basha et al. [3,4]. The method to obtain the best proximity point was given by Eldred [5]. Umer I. et al. [6] presented fuzzy adaptations of several well-known iterative mappings. Additionally, they established various concrete conditions on real-valued functions $\mathcal{J},\mathcal{S}:(0,1]\to \mathbb{R}$ to ensure the existence of best proximity points for generalized fuzzy $(\mathcal{J},\mathcal{S})$-iterative mappings in the framework of fuzzy metric spaces.

Youns et al. [7] established coincidence point, best proximity point, and fixed point results for multivalued proximal contractions within the framework of b-metric spaces, utilizing an alternating distance function. Komal et al. [8] obtained theorems for the best coincidence proximity points via Geraghty-type proximal cyclic contraction in metric space. Moreover, Jleli et al.’s paper [9] proposed the idea of $\theta$-contraction. The proximity theorem has undergone various modifications and adaptations, as evidenced by works such as [10]. Sametric et al. [11] introduce a new concept of $\alpha -\psi$-contractive type mappings and establish fixed point theorems for such mappings in complete metric spaces, and gave applications to ordinary differential equations.

Here, we explore applications of fixed point theory in nonlinear systems. It is used to analyze the light-clock model [12], which describes mass as an emergent property of entangled photons, and to study battery capacity gaps and multipartite entanglement [13] under LOCC transformations. Fixed point theory also aids in the identification of nonlinear state-space models (NSSM) [14] and ensures the stability of solutions in the expectation maximization (EM) algorithm. Additionally, it was applied to analyze nonlinear mappings in a bionic bimanual robot teleoperation system [15] and to study the stability and convergence of hand detection and pose estimation algorithms, ensuring accurate motion transfer between human and robot.

The nonlinear nature of the diffusive wave model (DWM) [16] aligns with fixed point theory, enabling the exploration of forward and inverse problems regarding solution existence and stability. Fixed point theory is also applied to study the complex dynamics of Sun-perturbed Earth–Moon triangular libration points [17] and to analyze the Sobolev norm growth in the Hartree equation [18]. Furthermore, in the design and control of bioinspired soft robotic hands [19], fixed point theory helps analyze complex nonlinear dynamics and ensures stability and convergence, particularly in material compliance, actuator dynamics, and controller performance in uncertain environments.

After conducting a thorough analysis of the literature, we have discovered important information that has led to remarkable findings about the best coincidence proximity points in the rectangular metric space. We will introduce new $\theta$-type proximal contractions for multivalued mappings with an alternating distance function $\mathcal{f}$. We reduce our findings to the result of fixed points that are mentioned as corollaries of the notion of best proximity points. Significantly, we give descriptive examples to demonstrate the best coincidence points to fixed points with graphical representations. These examples support the validity and importance of the conclusions reported in this manuscript in addition to clarifying our findings. In the last section, we offer a concise approach to solving the equation of motion using our findings. This application not only validates our theoretical results, but also demonstrates their strength in practical, real-world scenarios.

2. Preliminaries

To set the stage for our investigation, we first gather some foundational definitions, key concepts, and well-established results. These serve as the building blocks that guide us toward formulating and achieving our new findings.

Definition 1

([20]). If the following given conditions are satisfied by the function $đ:\mathrm{ʘ}\times \mathrm{ʘ}\to \mathbb{R}$, where ʘ is a non-empty set, then đ is considered as a metric, such that

1.

$đ(\mathcal{x},\mathcal{y})\ge 0,$

2.

$đ(\mathcal{x},\mathcal{y})=0$ iff $\mathcal{x}=\mathcal{y},$

3.

$đ(\mathcal{x},\mathcal{y})=đ(\mathcal{y},\mathcal{x}),$

4.

$đ(\mathcal{x},\zeta )+đ(\zeta ,\mathcal{y})\ge đ(\mathcal{x},\mathcal{y})$, for each $\mathcal{x},\mathcal{y},\zeta \in \mathrm{ʘ}.$

Then, we call $(\mathrm{ʘ},đ)$ a metric space.

From [20], here, we have few examples of the distance.

Example 1.

The functions ${đ}_1,{đ}_2,{đ}_3:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ are given as

$$\begin{array}{ccc}\hfill {đ}_1(\mathcal{x},\mathcal{y})& =& \sqrt{\left|\mathcal{x}-\mathcal{y}\right|,}\hfill \\ \hfill {đ}_2(\mathcal{x},\mathcal{y})& =& \left|{\displaystyle \frac{1}{\mathcal{x}}}-{\displaystyle \frac{1}{\mathcal{y}}}\right|,\hfill \\ \hfill {đ}_3(\mathcal{x},\mathcal{y})& =& \left|\mathcal{x}-\mathcal{y}\right|,\hfill \end{array}$$

Then, ${đ}_1$ and ${đ}_3$ are metrics defined on $\mathbb{R}$. Furthermore, ${đ}_2$ defines a metric on non-zero real numbers $\mathbb{R}\setminus \left\{0\right\}$.

Example 2.

For the real numbers ${\mathcal{x}}_1,{\mathcal{y}}_1,{\mathcal{x}}_2,{\mathcal{y}}_2$, suppose $({\mathcal{x}}_1,{\mathcal{y}}_1)={\mathrm{ʐ}}_1$ and $({\mathcal{x}}_2,{\mathcal{y}}_2)={\mathrm{ʐ}}_2$; then, $đ({\mathrm{ʐ}}_1,{\mathrm{ʐ}}_2)$ is given as

$$đ({\mathrm{ʐ}}_1,{\mathrm{ʐ}}_2)=\left|{\mathcal{x}}_1-{\mathcal{x}}_2,\right|+\left|{\mathcal{y}}_1-{\mathcal{y}}_2\right|,$$

where đ is a metric defined on ${\mathbb{R}}^2=\mathrm{ʘ}.$

Example 3.

Let $\mathrm{ʘ}\ne \varphi$, with the metric ${đ}_0$ and the metric space denoted as $(\mathrm{ʘ},{đ}_0)$. The function ${đ}_0:\mathrm{ʘ}\times \mathrm{ʘ}\to \mathbb{R}$ is defined as follows:

$${đ}_0(\mathcal{x},\mathcal{y})=\left\{\begin{array}{c}0,\mathrm{if}\mathcal{x}=\mathcal{y},\\ 1,\mathrm{if}\mathcal{x}\ne \mathcal{y}.\end{array}\right.$$

The metric ${đ}_0$ is termed a discrete metric, and $(\mathrm{ʘ},{đ}_0)$ is termed a discrete metric space.

In 2000, Branciari generalized the metric spaces concept in [21] as follows:

Definition 2.

If the conditions given below are fulfilled by the function $đ:\mathrm{ʘ}\times \mathrm{ʘ}\to {\mathbb{R}}^{+}$, where ʘ is a non-empty set, then đ is considered to be a rectangular metric, such that

1.

$đ(\mathcal{x},\mathcal{y})\ge 0,$

2.

$đ(\mathcal{x},\mathcal{y})=0,\iff \mathcal{x}=\mathcal{y},$

3.

$đ(\mathcal{x},\mathcal{y})=đ(\mathcal{y},\mathcal{x}),$

4.

$đ(\mathcal{x},\zeta )+đ(\zeta ,\mathfrak{u})+đ(\mathfrak{u},\mathcal{y})\ge đ(\mathcal{x},\mathcal{y})$ for all $\mathcal{x},\mathcal{y}\in \mathrm{ʘ}\setminus \{\zeta ,\mathfrak{u}\}.$

Then, $(\mathrm{ʘ},đ)$ is termed a rectangular metric space.

Definition 3

([21]). If a sequence $\left\{{\mathrm{ʐ}}_n\right\}\subseteq (\mathrm{ʘ},đ)$ (a rectangular metric space), then the following holds:

(i)

We can write ${lim}_{n\to \infty }{\mathrm{ʐ}}_n=\mathrm{ʐ}$ if and only if the sequence $\left\{{\mathrm{ʐ}}_n\right\}$ converges to a point $\mathrm{ʐ}\in \mathrm{ʘ}$.

(ii)

If the sequence $\left\{{\mathrm{ʐ}}_n\right\}$ satisfies the condition $\underset{n,m\to \infty }{lim}đ({\mathrm{ʐ}}_n,{\mathrm{ʐ}}_m)=0$, then it is known to be Cauchy.

(iii)

If, in the rectangular metric space ʘ, every Cauchy sequence is convergent, then we call it a complete metric space, and vice versa.

(iv)

The uniqueness of limits is ensured for every convergent sequence in the rectangular metric space ʘ.

Definition 4.

Let $(\mathrm{ʘ},đ)$ be a rectangular metric space with subsets $₱\ne \varphi$ and $\mathcal{Q}\ne \varphi$. Then, the following is how we define ${₱}_0$ and ${\mathcal{Q}}_0$:

$$\begin{array}{cc}\hfill {₱}_0& =\{\mathcal{x}\in ₱:đ(₱,\mathcal{Q})=đ(\mathcal{x},\mathcal{y})\mathit{for}\mathit{some}\mathcal{y}\in \mathcal{Q}\},\hfill \\ \hfill {\mathcal{Q}}_0& =\{\mathcal{y}\in \mathcal{Q}:đ(₱,\mathcal{Q})=đ(\mathcal{x},\mathcal{y})\mathit{for}\mathit{some}\mathcal{x}\in ₱\},\hfill \end{array}$$

where

$$inf\left\{đ\right(\mathcal{x},\mathcal{y}):\mathcal{x}\in ₱,\mathcal{y}\in \mathcal{Q}\}=đ(₱,\mathcal{Q}),$$

which is the distance between sets $₱$ and $\mathcal{Q}$.

Definition 5

([22]). Let $(\mathrm{ʘ},đ)$, a rectangular metric space, with subsets $₱\ne \varphi \ne \mathcal{Q}$; then, $(₱,\mathcal{Q})$, known to satisfy the $\mathcal{P}$-property if

$$đ({\mathrm{ʐ}}_1,{\mathcal{x}}_1)=đ({\mathrm{ʐ}}_2,{\mathcal{x}}_2)=đ(₱,\mathcal{Q}),$$

implies that

$$đ({\mathrm{ʐ}}_1,{\mathrm{ʐ}}_2)=đ({\mathcal{x}}_1,{\mathcal{x}}_2)$$

where ${\mathcal{x}}_1,{\mathcal{x}}_2\in \mathcal{Q}$ and ${\mathrm{ʐ}}_1,{\mathrm{ʐ}}_2\in ₱$.

Definition 6

([23]). Let $\mathcal{f}:{\mathbb{R}}^{+}⟶{\mathbb{R}}^{+}$. Upon meeting the axioms given below, it is termed an alternating distance function:

(i)

$\mathcal{f}$ is continuous.

(ii)

If $\mathcal{x}\le \mathcal{y}$, then $\mathcal{f}\left(\mathcal{x}\right)\le \mathcal{f}\left(\mathcal{y}\right).$

(iii)

For all $\mathcal{x}>0$, $\mathcal{f}\left(\mathcal{x}\right)>0$.

Definition 7

([9]). Let the function $\theta :(0,\infty )\to (1,\infty )$. Upon meeting the axioms given below, the

function is termed a $\theta -$contraction, and collectively, the family of these functions is known as $\mathrm{\Theta }.$:

(${\theta }_1$)

θ is non-decreasing.

(${\theta }_2$)

For every $\left\{{\mathrm{ʐ}}_n\right\}\subseteq \mathbb{R},\underset{n\to \infty }{lim}$${\mathrm{ʐ}}_n=0$, if and only if $\underset{n\to \infty }{lim}$$\theta \left({\mathrm{ʐ}}_n\right)=1$.

(${\theta }_3$)

For $\underset{\mathrm{ʐ}\to 0^{+}}{lim}$${\displaystyle \frac{\theta \left(\mathrm{ʐ}\right)-1}{{\mathrm{ʐ}}^k}}=\mathcal{l},\exists k\in (0,1)$ and $\mathcal{l}\in (0,\infty ]$.

A self-map $\mathcal{T}:\mathrm{ʘ}\to \mathrm{ʘ}$ defined on $(\mathrm{ʘ},đ)$, a rectangular metric space, is identified as a $\theta$-contraction, where $\theta$ is from the family $\mathrm{\Theta }$, and $\kappa \in (0,1)$ serves as a constant.

$$đ(\mathcal{T}\left(\mathcal{x}\right),\mathcal{T}\left(\mathcal{y}\right))\ne 0,\Rightarrow \theta \left(đ(\mathcal{T}\left(\mathcal{x}\right),\mathcal{T}\left(\mathcal{y}\right))\right)\le {\left[\theta \left(đ\right(\mathcal{x},\mathcal{y}\left)\right)\right]}^{\kappa }$$

where $\mathcal{x},\mathcal{y}$ are elements of $\mathrm{ʘ}.$

Example 4.

Define some functions as follows: for all $t>0$,

1.

${\theta }_1\left(\mathcal{x}\right)=e^{\sqrt{\mathcal{x}}}$;

2.

${\theta }_2\left(\mathcal{x}\right)=e^{\sqrt{\mathcal{x}{e}^{\mathcal{x}}}}$;

3.

${\theta }_3\left(\mathcal{x}\right)=e^{\mathcal{x}}$;

4.

${\theta }_4\left(\mathcal{x}\right)=cosh\left(\mathcal{x}\right)$;

5.

${\theta }_5\left(\mathcal{x}\right)=1+ln(1+\mathcal{x})$;

6.

${\theta }_6\left(\mathcal{x}\right)=e^{\mathcal{x}{e}^{\mathcal{x}}}$.

Then, ${\theta }_1,{\theta }_2,{\theta }_3,{\theta }_4,{\theta }_5,{\theta }_6\in \mathrm{\Theta }$.

Definition 8

([24]). Let $\mathfrak{CB}\left(\mathrm{ʘ}\right)$ be the family of all closed, bounded subsets of ʘ. The symbol $\mathcal{H}$ denotes the Hausdorff distance associated with metric đ, given by

$$\mathcal{H}(₱,\mathcal{Q})=max\left\{\underset{\zeta \in ₱}{sup}\mathfrak{D}(\zeta ,\mathcal{Q}),\underset{\mathcal{y}\in \mathcal{Q}}{sup}\mathfrak{D}(\mathcal{y},₱)\right\},$$

for $\mathcal{Q},₱\in \mathfrak{CB}\left(\mathrm{ʘ}\right),$ with

$$\mathfrak{D}(\zeta ,\mathcal{Q})=inf\left\{đ\right(\zeta ,\mathcal{y}):\mathcal{y}\in \mathcal{Q}\}.$$

Here, we introduce a new notion as follows:

$${\mathfrak{D}}^{*}(\zeta ,\mathcal{y})=\mathfrak{D}(\zeta ,\mathcal{y})-đ(₱,\mathcal{Q}),$$

for $\zeta \in ₱$ and $\mathcal{y}\in \mathcal{Q}.$

Throughout this study, we consider the rectangular metric function đ to be a continuously defined mapping.

3. Multivalued Mapping Results

Herein, we explore theorems concerning multivalued coincidence points, fixed points, and best proximity points within (ʘ, đ), a complete rectangular metric space.

Definition 9.

In $(\mathrm{ʘ},đ)$, a rectangular metric space, let $₱\ne \varphi \ne \mathcal{Q}$ be two closed subsets of ʘ. Consider two mappings $(\mathfrak{g},\mathcal{T})$: mapping $\mathcal{T}:₱\to \mathfrak{CB}\left(\mathcal{Q}\right)$ and the self-mapping $\mathfrak{g}:₱\to ₱$. These mappings satisfy

$$\mathfrak{D}(\mathfrak{g}\mathrm{ʐ},\mathcal{T}\mathrm{ʐ})=đ(₱,\mathcal{Q}),$$

Then, the point $\mathrm{ʐ}\in ₱$ is termed a best coincidence proximity point of the couple $(\mathfrak{g},\mathcal{T})$.

Remark 1.

The best coincidence point theorems provide a unified framework encompassing both best proximity and fixed points. In particular, when $I_{₱}=\mathfrak{g}$, each coincidental best proximity point inherently serves as the best proximity point of $\mathcal{T}$. Furthermore, if $\mathcal{T}$ simplifies to a self-map, the concept of the best proximity point seamlessly transforms into the fixed point.

Definition 10.

Consider a multivalued map $\mathcal{T}:₱\to \mathfrak{CB}\left(\mathcal{Q}\right)$ and a self-map $\mathfrak{g}:₱\to ₱$. The couple $(\mathfrak{g},\mathcal{T})$ is known to satisfy the ${(\mathcal{f}-{\theta }_{\kappa })}_{C_P}$-proximal contraction condition if for $0<\kappa <1$ and $\forall \mathfrak{u},\mathcal{x},\mathfrak{v},\mathrm{ʐ}\in ₱$, the following is satisfied:

$$\begin{array}{cc}\hfill đ(₱,\mathcal{Q})& =\mathfrak{D}\left(\mathfrak{g}\mathfrak{u},\mathcal{T}\mathrm{ʐ}\right)\hfill \\ \hfill đ(₱,\mathcal{Q})& =\mathfrak{D}\left(\mathfrak{g}\mathfrak{v},\mathcal{T}\mathcal{x}\right)\hfill \end{array}$$

This implies that

$$\theta \left(\mathcal{f}\left(\mathcal{H}(\mathcal{T}\mathrm{ʐ},\mathcal{T}\mathcal{x})\right)\right)\le {\left[\theta \left(\mathcal{f}\right(\mathcal{S}(\mathfrak{u},\mathfrak{v},\mathrm{ʐ},\mathcal{x})\left)\right)\right]}^{\kappa },$$

where

$$\mathcal{S}(\mathfrak{u},\mathfrak{v},\mathrm{ʐ},\mathcal{x})=max\left[\mathfrak{D}(\mathfrak{g}\mathfrak{u},\mathcal{T}\mathfrak{u})-đ(\mathfrak{g}\mathfrak{u},\mathfrak{g}\mathfrak{v})-\mathfrak{D}(\mathfrak{g}\mathcal{x},\mathcal{T}\mathfrak{u}),đ(\mathfrak{g}\mathfrak{u},\mathfrak{g}\mathfrak{v}),đ(\mathfrak{g}\mathrm{ʐ},\mathfrak{g}\mathcal{x}),{\mathfrak{D}}^{*}(\mathfrak{g}\mathfrak{v},\mathcal{T}\mathfrak{u})\right].$$

Definition 11.

Consider $\mathcal{T}:₱\to \mathfrak{CB}\left(\mathcal{Q}\right)$, a multivalued mapping, that is termed a ${(\mathcal{f}-{\theta }_{\kappa })}_{B_P}$-proximal contraction if for $\kappa \in (0,1)$ and $\forall \mathfrak{u},\mathcal{x},\mathfrak{v},\mathrm{ʐ}\in ₱$, the following holds:

$$\begin{array}{cc}\hfill \mathfrak{D}(\mathfrak{u},\mathcal{T}\mathrm{ʐ})& =đ(₱,\mathcal{Q})\hfill \\ \hfill \mathfrak{D}(\mathfrak{v},\mathcal{T}\mathcal{x})& =đ(₱,\mathcal{Q})\hfill \end{array}$$

This implies that

$$\theta \left(\mathcal{f}\left(\mathcal{H}(\mathcal{T}\mathrm{ʐ},\mathcal{T}\mathcal{x})\right)\right)\le {\left[\theta \left(\mathcal{f}\right(\mathcal{S}(\mathfrak{u},\mathfrak{v},\mathrm{ʐ},\mathcal{x})\left)\right)\right]}^{\kappa },$$

with

$$\mathcal{S}(\mathfrak{u},\mathfrak{v},\mathrm{ʐ},\mathcal{x})=max\left[\mathfrak{D}(\mathfrak{u},\mathcal{T}\mathfrak{u})-đ(\mathfrak{u},\mathfrak{v})-\mathfrak{D}(\mathcal{x},\mathcal{T}\mathfrak{u}),đ(\mathrm{ʐ},\mathcal{x}),đ(\mathfrak{u},\mathfrak{v}),{\mathfrak{D}}^{*}(\mathfrak{v},\mathcal{T}\mathfrak{u})\right].$$

When the mapping $\mathfrak{g}={\mathcal{I}}_{₱}$ is considered, every ${(\mathcal{f}-{\theta }_{\kappa })}_{C_P}$-proximal contraction naturally transitions into a ${(\mathcal{f}-{\theta }_{\kappa })}_{B_P}$-proximal contraction. Here, ${\mathcal{I}}_{₱}$ represents the identity mapping.

Theorem 1.

Let $(\mathrm{ʘ},đ)$ be a complete rectangular metric space with closed subsets $₱\ne \varphi \ne \mathcal{Q}$, fulfilling the $\mathcal{P}$-property. Assume continuous mappings $\mathcal{T}:₱\to \mathfrak{CB}\left(\mathcal{Q}\right)$ and $\mathfrak{g}:₱\to ₱$, such that ${₱}_0\ne \varphi$, ${₱}_0\subseteq \mathfrak{g}\left({₱}_0\right)$, and $\mathcal{T}\left({₱}_0\right)\subseteq {\mathcal{Q}}_0$. Assume further that $\mathfrak{g}$ is a continuous and injective mapping, and that the couple $(\mathfrak{g},\mathcal{T})$ satisfies ${(\mathcal{f}-{\theta }_{\kappa })}_{C_P}$-proximal contraction, along with an alternating distance function $\mathcal{f}$. Based on such premises, there is a point termed the coincidence best proximity point for the couple $(\mathfrak{g},\mathcal{T})$ in $₱$.

Proof. 

Let us take ${\mathrm{ʐ}}_0\in {₱}_0$, an arbitrary element. Since $\mathcal{T}\left({₱}_0\right)$ is a subset of ${\mathcal{Q}}_0$, and ${₱}_0$ is contained in $\mathfrak{g}\left({₱}_0\right)$, there is a point ${\mathrm{ʐ}}_1\in {₱}_0$ such that

$$\mathfrak{D}(\mathfrak{g}{\mathrm{ʐ}}_1,\mathcal{T}{\mathrm{ʐ}}_0)=đ(₱,\mathcal{Q}).$$

Furthermore, given that $\mathcal{T}{\mathrm{ʐ}}_1\in \mathcal{T}\left({₱}_0\right)$, which is a subset of ${\mathcal{Q}}_0$, and that ${₱}_0$ is within $\mathfrak{g}\left({₱}_0\right),$ this suggests that there is a point ${\mathrm{ʐ}}_2\in {₱}_0$ such that

$$\mathfrak{D}(\mathfrak{g}{\mathrm{ʐ}}_2,\mathcal{T}{\mathrm{ʐ}}_1)=đ(₱,\mathcal{Q}).$$

Utilizing the $\mathcal{P}$-property, we have

$$đ(\mathfrak{g}{\mathrm{ʐ}}_1,\mathfrak{g}{\mathrm{ʐ}}_2)=\mathcal{H}(\mathcal{T}{\mathrm{ʐ}}_0,\mathcal{T}{\mathrm{ʐ}}_1).$$

Moreover, since the couple $(\mathfrak{g},\mathcal{T})$ meets the condition of ${(\mathcal{f}-{\theta }_{\kappa })}_{C_p}$-proximal contraction, it follows that

$$\theta \left(\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_1,\mathfrak{g}{\mathrm{ʐ}}_2)\right)\right)\le {\left[\theta \left(\mathcal{f}\left(\mathcal{S}({\mathrm{ʐ}}_1,{\mathrm{ʐ}}_2,{\mathrm{ʐ}}_0,{\mathrm{ʐ}}_1)\right)\right)\right]}^{\kappa },$$

(1)

where

$$\begin{array}{ccc}\hfill \mathcal{S}({\mathrm{ʐ}}_1,{\mathrm{ʐ}}_2,{\mathrm{ʐ}}_0,{\mathrm{ʐ}}_1)& =& max\left\{\begin{array}{c}đ(\mathfrak{g}{\mathrm{ʐ}}_0,\mathfrak{g}{\mathrm{ʐ}}_1),đ(\mathfrak{g}{\mathrm{ʐ}}_1,\mathfrak{g}{\mathrm{ʐ}}_2),{\mathfrak{D}}^{*}(\mathfrak{g}{\mathrm{ʐ}}_1,\mathcal{T}{\mathrm{ʐ}}_0),\\ \mathfrak{D}(\mathfrak{g}{\mathrm{ʐ}}_0,\mathcal{T}{\mathrm{ʐ}}_0)-đ(\mathfrak{g}{\mathrm{ʐ}}_0,\mathfrak{g}{\mathrm{ʐ}}_1)-\mathfrak{D}(\mathfrak{g}{\mathrm{ʐ}}_2,\mathcal{T}{\mathrm{ʐ}}_0),\end{array}\right\},\hfill \\ & \le & max\left\{\begin{array}{c}đ(\mathfrak{g}{\mathrm{ʐ}}_0,\mathfrak{g}{\mathrm{ʐ}}_1),đ(\mathfrak{g}{\mathrm{ʐ}}_1,\mathfrak{g}{\mathrm{ʐ}}_2),\mathfrak{D}(\mathfrak{g}{\mathrm{ʐ}}_1,\mathcal{T}{\mathrm{ʐ}}_0)-\mathfrak{D}(₱,\mathcal{Q}),\\ đ(\mathfrak{g}{\mathrm{ʐ}}_0,\mathfrak{g}{\mathrm{ʐ}}_1)+đ(\mathfrak{g}{\mathrm{ʐ}}_1,\mathfrak{g}{\mathrm{ʐ}}_2)+\mathfrak{D}(\mathfrak{g}{\mathrm{ʐ}}_1,\mathcal{T}{\mathrm{ʐ}}_0)\\ -đ(\mathfrak{g}{\mathrm{ʐ}}_0,\mathfrak{g}{\mathrm{ʐ}}_1)-\mathfrak{D}(\mathfrak{g}{\mathrm{ʐ}}_2,\mathcal{T}{\mathrm{ʐ}}_0),\end{array}\right\}\hfill \\ & \le & max\left\{\begin{array}{c}đ(\mathfrak{g}{\mathrm{ʐ}}_0,\mathfrak{g}{\mathrm{ʐ}}_1),đ(\mathfrak{g}{\mathrm{ʐ}}_1,\mathfrak{g}{\mathrm{ʐ}}_2),0,đ(\mathfrak{g}{\mathrm{ʐ}}_1,\mathfrak{g}{\mathrm{ʐ}}_2)\end{array}\right\}\hfill \\ & \le & max\left\{đ(\mathfrak{g}{\mathrm{ʐ}}_0,\mathfrak{g}{\mathrm{ʐ}}_1),đ(\mathfrak{g}{\mathrm{ʐ}}_1,\mathfrak{g}{\mathrm{ʐ}}_2)\right\},\hfill \end{array}$$

so we can say that

$$\mathcal{S}({\mathrm{ʐ}}_1,{\mathrm{ʐ}}_2,{\mathrm{ʐ}}_0,{\mathrm{ʐ}}_1)\le max\left\{đ(\mathfrak{g}{\mathrm{ʐ}}_0,\mathfrak{g}{\mathrm{ʐ}}_1),đ(\mathfrak{g}{\mathrm{ʐ}}_1,\mathfrak{g}{\mathrm{ʐ}}_2)\right\}.$$

If we choose $max\left\{đ(\mathfrak{g}{\mathrm{ʐ}}_0,\mathfrak{g}{\mathrm{ʐ}}_1),đ(\mathfrak{g}{\mathrm{ʐ}}_1,\mathfrak{g}{\mathrm{ʐ}}_2)\right\}=đ(\mathfrak{g}{\mathrm{ʐ}}_1,\mathfrak{g}{\mathrm{ʐ}}_2),$ then inequality (1) becomes

$$\theta \left(\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_1,\mathfrak{g}{\mathrm{ʐ}}_2)\right)\right)\le {\left[\theta \left(\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_1,\mathfrak{g}{\mathrm{ʐ}}_2)\right)\right)\right]}^{\kappa }$$

which is impossible.

If $max\left\{đ(\mathfrak{g}{\mathrm{ʐ}}_0,\mathfrak{g}{\mathrm{ʐ}}_1),đ(\mathfrak{g}{\mathrm{ʐ}}_1,\mathfrak{g}{\mathrm{ʐ}}_2)\right\}=đ(\mathfrak{g}{\mathrm{ʐ}}_0,\mathfrak{g}{\mathrm{ʐ}}_1),$ then inequality (1) will be

$$\theta \left(\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_1,\mathfrak{g}{\mathrm{ʐ}}_2)\right)\right)\le {\left[\theta \left(\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_0,\mathfrak{g}{\mathrm{ʐ}}_1)\right)\right)\right]}^{\kappa },$$

since we obtain

$$đ(\mathfrak{g}{\mathrm{ʐ}}_1,\mathfrak{g}{\mathrm{ʐ}}_2)\le đ(\mathfrak{g}{\mathrm{ʐ}}_0,\mathfrak{g}{\mathrm{ʐ}}_1).$$

Now, let ${\mathrm{ʐ}}_1$ be an arbitrary element in ${₱}_0$. Since $\mathcal{T}\left({₱}_0\right)\subset {\mathcal{Q}}_0$ and ${₱}_0$ is contained within $\mathfrak{g}\left({₱}_0\right)$, there exists an element ${\mathrm{ʐ}}_2\in {₱}_0$ such that

$$\mathfrak{D}(\mathfrak{g}{\mathrm{ʐ}}_2,\mathcal{T}{\mathrm{ʐ}}_1)=đ(₱,\mathcal{Q}).$$

Furthermore, given that $\mathcal{T}{\mathrm{ʐ}}_2\in \mathcal{T}\left({₱}_0\right)$, which is a subset of ${\mathcal{Q}}_0$, and that ${₱}_0$ is contained within $\mathfrak{g}\left({₱}_0\right),$ there is an element ${\mathrm{ʐ}}_3$ in ${₱}_0$ such that

$$\mathfrak{D}(\mathfrak{g}{\mathrm{ʐ}}_3,\mathcal{T}{\mathrm{ʐ}}_2)=đ(₱,\mathcal{Q}).$$

Utilizing the $\mathcal{P}$-property, we have

$$đ(\mathfrak{g}{\mathrm{ʐ}}_2,\mathfrak{g}{\mathrm{ʐ}}_3)=\mathcal{H}(\mathcal{T}{\mathrm{ʐ}}_1,\mathcal{T}{\mathrm{ʐ}}_2).$$

Furthermore, since the couple $(\mathfrak{g},\mathcal{T})$ meets the condition of ${(\mathcal{f}-{\theta }_{\kappa })}_{C_p}$-proximal contraction, it follows that

$$\theta \left(\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_2,\mathfrak{g}{\mathrm{ʐ}}_3)\right)\right)\le {\left[\theta \left(\mathcal{f}\left(\mathcal{S}({\mathrm{ʐ}}_2,{\mathrm{ʐ}}_3,{\mathrm{ʐ}}_1,{\mathrm{ʐ}}_2)\right)\right)\right]}^{\kappa },$$

(2)

where

$$\begin{array}{ccc}\hfill \mathcal{S}({\mathrm{ʐ}}_2,{\mathrm{ʐ}}_3,{\mathrm{ʐ}}_1,{\mathrm{ʐ}}_2)& =& max\left\{\begin{array}{c}đ(\mathfrak{g}{\mathrm{ʐ}}_1,\mathfrak{g}{\mathrm{ʐ}}_1),đ(\mathfrak{g}{\mathrm{ʐ}}_2,\mathfrak{g}{\mathrm{ʐ}}_3),{\mathfrak{D}}^{*}(\mathfrak{g}{\mathrm{ʐ}}_2,\mathcal{T}{\mathrm{ʐ}}_1),\\ \mathfrak{D}(\mathfrak{g}{\mathrm{ʐ}}_1,\mathcal{T}{\mathrm{ʐ}}_1)-đ(\mathfrak{g}{\mathrm{ʐ}}_1,\mathfrak{g}{\mathrm{ʐ}}_2)-\mathfrak{D}(\mathfrak{g}{\mathrm{ʐ}}_3,\mathcal{T}{\mathrm{ʐ}}_1),\end{array}\right\},\hfill \\ & \le & max\left\{\begin{array}{c}đ(\mathfrak{g}{\mathrm{ʐ}}_1,\mathfrak{g}{\mathrm{ʐ}}_2),đ(\mathfrak{g}{\mathrm{ʐ}}_2,\mathfrak{g}{\mathrm{ʐ}}_3),\mathfrak{D}(\mathfrak{g}{\mathrm{ʐ}}_2,\mathcal{T}{\mathrm{ʐ}}_1)-\mathfrak{D}(₱,\mathcal{Q}),\\ đ(\mathfrak{g}{\mathrm{ʐ}}_1,\mathfrak{g}{\mathrm{ʐ}}_2)+đ(\mathfrak{g}{\mathrm{ʐ}}_2,\mathfrak{g}{\mathrm{ʐ}}_3)+\mathfrak{D}(\mathfrak{g}{\mathrm{ʐ}}_2,\mathcal{T}{\mathrm{ʐ}}_1)\\ -đ(\mathfrak{g}{\mathrm{ʐ}}_1,\mathfrak{g}{\mathrm{ʐ}}_2)-\mathfrak{D}(\mathfrak{g}{\mathrm{ʐ}}_3,\mathcal{T}{\mathrm{ʐ}}_1),\end{array}\right\}\hfill \\ & \le & max\left\{\begin{array}{c}đ(\mathfrak{g}{\mathrm{ʐ}}_1,\mathfrak{g}{\mathrm{ʐ}}_2),đ(\mathfrak{g}{\mathrm{ʐ}}_2,\mathfrak{g}{\mathrm{ʐ}}_3),0,đ(\mathfrak{g}{\mathrm{ʐ}}_2,\mathfrak{g}{\mathrm{ʐ}}_3)\end{array}\right\}\hfill \\ & \le & max\left\{đ(\mathfrak{g}{\mathrm{ʐ}}_1,\mathfrak{g}{\mathrm{ʐ}}_2),đ(\mathfrak{g}{\mathrm{ʐ}}_2,\mathfrak{g}{\mathrm{ʐ}}_3)\right\},\hfill \end{array}$$

so we can say that

$$\mathcal{S}({\mathrm{ʐ}}_2,{\mathrm{ʐ}}_3,{\mathrm{ʐ}}_1,{\mathrm{ʐ}}_2)\le max\left\{đ(\mathfrak{g}{\mathrm{ʐ}}_1,\mathfrak{g}{\mathrm{ʐ}}_2),đ(\mathfrak{g}{\mathrm{ʐ}}_2,\mathfrak{g}{\mathrm{ʐ}}_3)\right\}.$$

If we choose $max\left\{đ(\mathfrak{g}{\mathrm{ʐ}}_1,\mathfrak{g}{\mathfrak{x}}_2),đ(\mathfrak{g}{\mathrm{ʐ}}_2,\mathfrak{g}{\mathrm{ʐ}}_3)\right\}=$$đ(\mathfrak{g}{\mathrm{ʐ}}_2,\mathfrak{g}{\mathrm{ʐ}}_3),$ then inequality (2) becomes

$$\theta \left(\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_2,\mathfrak{g}{\mathrm{ʐ}}_3)\right)\right)\le {\left[\theta \left(\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_2,\mathfrak{g}{\mathrm{ʐ}}_3)\right)\right)\right]}^{\kappa }$$

which is impossible. If $max\left\{đ(\mathfrak{g}{\mathrm{ʐ}}_1,\mathfrak{g}{\mathrm{ʐ}}_2),đ(\mathfrak{g}{\mathrm{ʐ}}_2,\mathfrak{g}{\mathrm{ʐ}}_3)\right\}=đ(\mathfrak{g}{\mathrm{ʐ}}_1,\mathfrak{g}{\mathrm{ʐ}}_2),$ then inequality (2) will be

$$\theta \left(\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_2,\mathfrak{g}{\mathrm{ʐ}}_3)\right)\right)\le {\left[\theta \left(\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_1,\mathfrak{g}{\mathrm{ʐ}}_2)\right)\right)\right]}^{\kappa },$$

since we obtain

$$đ(\mathfrak{g}{\mathrm{ʐ}}_2,\mathfrak{g}{\mathrm{ʐ}}_3)\le đ(\mathfrak{g}{\mathrm{ʐ}}_1,\mathfrak{g}{\mathrm{ʐ}}_2).$$

Similarly, since ${\mathrm{ʐ}}_n\in {₱}_0$ and $\mathcal{T}\left({₱}_0\right)\subseteq {\mathcal{Q}}_0,$ there is ${\mathrm{ʐ}}_{n+1}\in {₱}_0$, which gives

$$\mathfrak{D}(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathcal{T}{\mathrm{ʐ}}_n)=đ(₱,\mathcal{Q}).$$

(3)

After selecting ${\mathrm{ʐ}}_{n+1}$, which satisfies the given condition, there exists an element ${\mathrm{ʐ}}_{n+2}\in {₱}_0$ that meets the condition

$$đ(₱,\mathcal{Q})=\mathfrak{D}(\mathfrak{g}\left({\mathrm{ʐ}}_{n+2}\right),\mathcal{T}\left({\mathrm{ʐ}}_{n+1}\right))$$

(4)

for all positive integers n, and utilizing the $\mathcal{P}$-property, we obtain

$$đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2})=\mathcal{H}(\mathcal{T}{\mathrm{ʐ}}_n,\mathcal{T}{\mathrm{ʐ}}_{n+1}).$$

Moreover, since the couple $(\mathfrak{g},\mathcal{T})$ meets the condition of ${(\mathcal{f}-{\theta }_{\kappa })}_{C_p}$-proximal contraction, and by applying Equations (3) and (4), we obtain

$$\theta \left(\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2})\right)\right)\le {\left[\theta \left(\mathcal{f}\left(\mathcal{S}({\mathrm{ʐ}}_{n+1},{\mathrm{ʐ}}_{n+2},{\mathrm{ʐ}}_n,{\mathrm{ʐ}}_{n+1})\right)\right)\right]}^{\kappa },$$

(5)

where

$$\begin{array}{ccc}\hfill \mathcal{S}({\mathrm{ʐ}}_{n+1},{\mathrm{ʐ}}_{n+2},{\mathrm{ʐ}}_n,{\mathrm{ʐ}}_{n+1})& =& max\left\{\begin{array}{c}đ(\mathfrak{g}{\mathrm{ʐ}}_n,\mathfrak{g}{\mathrm{ʐ}}_{n+1}),đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2}),{\mathfrak{D}}^{*}(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathcal{T}{\mathrm{ʐ}}_n),\\ \mathfrak{D}(\mathfrak{g}{\mathrm{ʐ}}_n,\mathcal{T}{\mathrm{ʐ}}_n)-đ(\mathfrak{g}{\mathrm{ʐ}}_n,\mathfrak{g}{\mathrm{ʐ}}_{n+1})-\mathfrak{D}(\mathfrak{g}{\mathrm{ʐ}}_{n+2},\mathcal{T}{\mathrm{ʐ}}_n)\end{array}\right\}\hfill \end{array}$$

$$\begin{array}{ccc}& \le & max\left\{\begin{array}{c}\mathfrak{D}(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathcal{T}{\mathrm{ʐ}}_n)-\mathfrak{D}(₱,\mathcal{Q}),đ(\mathfrak{g}{\mathrm{ʐ}}_n,\mathfrak{g}{\mathrm{ʐ}}_{n+1}),đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2}),\\ đ(\mathfrak{g}{\mathrm{ʐ}}_n,\mathfrak{g}{\mathrm{ʐ}}_{n+1})+đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2})+\mathfrak{D}(\mathfrak{g}{\mathrm{ʐ}}_{n+2},\mathcal{T}{\mathrm{ʐ}}_n)\\ -đ(\mathfrak{g}{\mathrm{ʐ}}_n,\mathfrak{g}{\mathrm{ʐ}}_{n+1})-\mathfrak{D}(\mathfrak{g}{\mathrm{ʐ}}_{n+2},\mathcal{T}{\mathrm{ʐ}}_n)\end{array}\right\}\hfill \\ & \le & max\left\{đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2}),đ(\mathfrak{g}{\mathrm{ʐ}}_n,\mathfrak{g}{\mathrm{ʐ}}_{n+1}),đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2}),0\right\}\hfill \\ & \le & max\left\{đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2}),đ(\mathfrak{g}{\mathrm{ʐ}}_n,\mathfrak{g}{\mathrm{ʐ}}_{n+1})\right\}.\hfill \end{array}$$

So, we can say that

$$\mathcal{S}({\mathrm{ʐ}}_{n+1},{\mathrm{ʐ}}_{n+2},{\mathrm{ʐ}}_n,{\mathrm{ʐ}}_{n+1})\le max\left\{đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2}),đ(\mathfrak{g}{\mathrm{ʐ}}_n,\mathfrak{g}{\mathrm{ʐ}}_{n+1})\right\}.$$

If we choose $max\left\{đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2}),đ(\mathfrak{g}{\mathrm{ʐ}}_n,\mathfrak{g}{\mathfrak{x}}_{n+1})\right\}=đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2}),$ then inequality (5) transforms into

$$\theta \left(\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2})\right)\right)\le {\left[\theta \left(\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2})\right)\right)\right]}^{\kappa },$$

which is not impossible. If $max\left\{đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2}),đ(\mathfrak{g}{\mathrm{ʐ}}_n,\mathfrak{g}{\mathrm{ʐ}}_{n+1})\right\}=đ(\mathfrak{g}{\mathrm{ʐ}}_n,\mathfrak{g}{\mathrm{ʐ}}_{n+1}),$ then Equation (5) becomes

$${\left[\theta \left(\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_n,\mathfrak{g}{\mathrm{ʐ}}_{n+1})\right)\right)\right]}^{\kappa }\ge \theta \left(\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2})\right)\right),$$

(6)

since we obtain

$$đ(\mathfrak{g}{\mathrm{ʐ}}_n,\mathfrak{g}{\mathrm{ʐ}}_{n+1})\ge đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2}).$$

Therefore, the sequence $\left\{(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2})\right\}$ is monotonically nonincreasing and bounded from below. As a result, there exists $\lambda \ge 0$ such that

$$0\le \lambda =\underset{n\to \infty }{lim}đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2}).$$

(7)

Assuming that ${lim}_{n\to \infty }đ(\mathfrak{g}{\mathfrak{x}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2})=\lambda >0$, with Equations (6) and (7), we have:

$$\theta \left(\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2})\right)\right)\le {\left[\theta \left(\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_{n-1},\mathfrak{g}{\mathrm{ʐ}}_n)\right)\right)\right]}^{{\kappa }^2}.$$

Proceeding with the process, we arrive at

$$\theta \left(\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2})\right)\right)\le {\left[\theta \left(\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_0,\mathfrak{g}{\mathrm{ʐ}}_1)\right)\right)\right]}^{{\kappa }^n}.$$

(8)

As a consequence of the prior equation, we obtain

$$\underset{n\to \infty }{lim}\theta \left(\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2})\right)\right)=1.$$

Using equation (${\theta }_2$) of Definition 7 gives

$$\underset{n\to \infty }{lim}\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2})\right)=0.$$

(9)

Again, using equation $\left({\theta }_3\right)$ of Definition 7, there is a $\mathrm{ъ}\in (0,1)$, such that

$$\begin{array}{ccc}\hfill \underset{đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2}\to 0)}{lim}\left({\displaystyle \frac{\theta \left[\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2})\right)\right]-1}{{\left[\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2})\right)\right]}^{\mathrm{ъ}}}}\right)& =& l\hfill \\ \hfill \underset{n\to \infty }{lim}\left({\displaystyle \frac{\theta \left[\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2})\right)\right]-1}{{\left[\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2})\right)\right]}^{\mathrm{ъ}}}}\right)& =& l\hfill \end{array}$$

(10)

So, we can write it for $l<\infty$ and $C={\displaystyle \frac{l}{2}}$ as follows:

$$\left|{\displaystyle \frac{\theta \left[\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2})\right)\right]-1}{{\left[\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2})\right)\right]}^{\mathrm{ъ}}}}\right|\le C,\forall n\ge n_0$$

Hence, we obtain

$$\begin{array}{ccc}\hfill -C& \le & {\displaystyle \frac{\theta \left[\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2})\right)\right]-1}{{\left[\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2})\right)\right]}^{\mathrm{ъ}}}}-l\le C,\forall n\ge n_0,\forall n\ge n_0\hfill \end{array}$$

From this, we have

$$l-C\le {\displaystyle \frac{\theta \left[\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2})\right)\right]-1}{{\left[\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2})\right)\right]}^{\mathrm{ъ}}}},\forall n\ge n_0$$

Since $C={\displaystyle \frac{l}{2}}$, we have

$$C\le {\displaystyle \frac{\theta \left[\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2})\right)\right]-1}{{\left[\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2})\right)\right]}^{\mathrm{ъ}}}},\forall n\ge n_0$$

From these, we obtain

$${\left[\mathcal{f}\left\{đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2})\right\}\right]}^{\mathrm{ъ}}\le {\displaystyle \frac{1}{C}}\left[\theta \left[\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2})\right)\right]-1\right]$$

By using (8), we obtain

$${\left[\mathcal{f}\left\{đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2})\right\}\right]}^{\mathrm{ъ}}\le {\displaystyle \frac{1}{C}}\left[\theta {\left[\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_0,\mathfrak{g}{\mathrm{ʐ}}_1)\right)\right]}^{{\kappa }^n}-1\right]$$

Now, by taking the limit as $n\to \infty$, we obtain

$$\underset{n\to \infty }{lim}{\left[\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2})\right)\right]}^{\mathrm{ъ}}=0$$

Assume that ${\mathcal{y}}_n=\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2})\right)$; from this, the above inequality is given as

$$\underset{n\to \infty }{lim}{\left({\mathcal{y}}_n\right)}^{\mathrm{ъ}}=0.$$

(11)

From Equation (11), it is evident that there exists an integer $n_1\in \mathbb{N}$ such that, for any given $\epsilon >0$,

$$\begin{array}{ccc}\hfill \left|{\left({\mathcal{y}}_n\right)}^{\mathrm{ъ}}-0\right|& <& \epsilon ,\mathrm{for}\mathrm{all}n\ge n_1\hfill \\ \hfill \left|{\left({\mathcal{y}}_n\right)}^{\mathrm{ъ}}\right|& <& \epsilon \hfill \\ \hfill \left({\mathcal{y}}_n\right)& <& \epsilon .\hfill \end{array}$$

Next, we aim to demonstrate that the sequence $\left\{\mathfrak{g}{\mathrm{ʐ}}_n\right\}$ is Cauchy within the rectangular metric space $(\mathrm{ʘ},đ)$, which is complete too. Therefore, $\forall n,m\in \mathbb{N}$, we have

$$\begin{array}{ccc}\hfill \mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_m,\mathfrak{g}{\mathrm{ʐ}}_n)\right)& \le & \mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_m,\mathfrak{g}{\mathrm{ʐ}}_{n+2})\right)+{\mathcal{y}}_{n+1}+{\mathcal{y}}_n\hfill \\ & \le & \mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_m,\mathfrak{g}{\mathrm{ʐ}}_{n+4})\right)+{\mathcal{y}}_{n+3}+{\mathcal{y}}_{n+2}+{\mathcal{y}}_{n+1}+{\mathcal{y}}_n\hfill \\ & & ⋮\hfill \\ & \le & {\mathcal{y}}_{m-1}+{\mathcal{y}}_{m-2}+\dots +{\mathcal{y}}_{n+1}+{\mathcal{y}}_n\hfill \\ & <& \epsilon .\hfill \end{array}$$

Thus, for all positive integers $n\ge n_1,$$p\in \mathbb{N}$, the above inequality will be as follows:

$$\underset{m,n\to \infty }{lim}\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_m,\mathfrak{g}{\mathrm{ʐ}}_n)\right)<\epsilon .$$

Thus, the Cauchy sequence $\left\{\mathfrak{g}{\mathrm{ʐ}}_n\right\}$ converges within the space $(\mathrm{ʘ},đ)$. Suppose it converges to ${\mathrm{ʐ}}^{*}$ in ${₱}_0\subseteq ₱$, where $₱$ is a closed set. This implies that the sequence $\left\{{\mathrm{ʐ}}_n\right\}\subseteq {₱}_0$ satisfies ${\mathrm{ʐ}}_n\to {\mathrm{ʐ}}^{*}$. Given that $(\mathfrak{g},\mathcal{T})$ are continuous mappings, we conclude that

$$\mathfrak{D}(\mathfrak{g}{\mathrm{ʐ}}^{*},\mathcal{T}{\mathrm{ʐ}}^{*})=đ(₱,\mathcal{Q}).$$

Thus, ${\mathrm{ʐ}}^{*}$ is a coincidence proximity point for the couple $(\mathfrak{g},\mathcal{T})$. □

Example 5.

Take a set $\mathrm{ʘ}=\{20,21,22,23,24,25\}$ with the metric $đ:\mathrm{ʘ}\times \mathrm{ʘ}\to [0,\infty )$ defined as

$$\begin{array}{|ccccccc|}\hline đ& 20& 21& 22& 23& 24& 25\\ 20& 0& 3& 4& 2& 6& 7\\ 21& 3& 0& 5& 8& 2& 6\\ 22& 4& 5& 0& 7& 8& 2\\ 23& 2& 8& 7& 0& 3& 4\\ 24& 6& 2& 8& 3& 0& 5\\ 25& 7& 6& 2& 4& 5& 0\\ \hline\end{array}$$

We see that $(\mathrm{ʘ},đ)$ becomes a complete rectangular metric space. Given that $₱=\{20,21,22\}$ and $\mathcal{Q}=\{23,24,25\}$ are nonempty closed subsets of $(\mathrm{ʘ},đ)$, meeting the $\mathcal{P}$-property condition, here, ${₱}_0=₱$, ${\mathcal{Q}}_0=\mathcal{Q}$. Also, we obtain $đ(₱,\mathcal{Q})=2$. The map $\mathcal{T}:₱\to \mathfrak{CB}\left(\mathcal{Q}\right)$, defined as

$$\mathcal{T}\mathrm{ʐ}=\left\{\begin{array}{cc}24,\hfill & \mathit{when}\mathrm{ʐ}\in \{20,21\}\hfill \\ \{23,25\},\hfill & \mathit{when}\mathrm{ʐ}\in \left\{22\right\},\hfill \end{array}\right.$$

and $\mathfrak{g}:₱\to ₱$, given as

$$\mathfrak{g}\mathrm{ʐ}=\left\{\begin{array}{c}20,\mathit{when}\mathrm{ʐ}=22\\ 21,\mathit{when}\mathrm{ʐ}=21\\ 22,\mathit{when}\mathrm{ʐ}=20.\end{array}\right.$$

We see $\mathcal{T}\left({₱}_0\right)$, a subset of ${\mathcal{Q}}_0$, and $\mathfrak{g}\left({₱}_0\right)$ a subset of ${₱}_0.$ Then, $(\mathfrak{g},\mathcal{T})$ meets the condition of ${(\mathcal{f}-{\theta }_{\kappa })}_{C_p}$-proximal contraction,

$$\theta \left(\mathcal{f}\left(\mathcal{H}(\mathcal{T}\mathrm{ʐ},\mathcal{T}\mathcal{x})\right)\right)\le {\left[\theta \left(\mathcal{f}\right(\mathcal{S}(\mathfrak{u},\mathfrak{v},\mathrm{ʐ},\mathcal{x})\left)\right)\right]}^{\kappa },$$

(12)

for all $\mathfrak{v},\mathcal{x},\mathfrak{u},\mathrm{ʐ}\in ₱$. Therefore,

$$\begin{array}{ccc}\hfill đ(₱,\mathcal{Q})& =& \mathfrak{D}(\mathfrak{g}21,\mathcal{T}20),\hfill \\ \hfill đ(₱,\mathcal{Q})& =& \mathfrak{D}(\mathfrak{g}20,\mathcal{T}22),\hfill \end{array}$$

with $\mathcal{x}=22$, $\mathfrak{u}=21,$ $\mathrm{ʐ}=20$, and $\mathfrak{v}=20$. After few simple steps, we obtain

$$\mathcal{H}(\mathcal{T}20,\mathcal{T}22)=\mathcal{H}(24,\{23,25\left\}\right)=3,$$

and

$$\begin{array}{ccc}\hfill \mathcal{S}(21,20,20,22)& =& max\left(\begin{array}{c}đ(\mathfrak{g}20,\mathfrak{g}22),đ(\mathfrak{g}22,\mathfrak{g}20),{\mathfrak{D}}^{*}(\mathfrak{g}21,\mathcal{T}20),\\ \mathfrak{D}(\mathfrak{g}20,\mathcal{T}20)-đ(\mathfrak{g}20,\mathfrak{g}22)-\mathfrak{D}(\mathfrak{g}20,\mathcal{T}20)\end{array}\right)\hfill \\ & =& max\left(\begin{array}{c}đ(22,20),đ(20,22),\mathfrak{D}(21,24)-đ(₱,\mathcal{Q}),\\ \mathfrak{D}(22,24)-đ(22,20)-\mathfrak{D}(22,24)\end{array}\right)\hfill \\ & =& max\left(4,20,4,-4\right)=4.\hfill \end{array}$$

Take function θ as

$$\begin{array}{c}\hfill \theta \left(\mathcal{y}\right)=e^{\mathcal{y}}.\end{array}$$

For $\mathcal{f}=2\mathcal{T}$, we obtain $\mathcal{f}\left(\mathcal{H}\right(\mathcal{T}\mathrm{ʐ},\mathcal{T}\mathcal{x}\left)\right)=6$, $\mathcal{f}\left(\mathcal{S}\right(u,v,\mathrm{ʐ},\mathcal{x}\left)\right)=8$; then, the inequality (12) will be

$$\theta \left(6\right)\le {\left[\theta \left(8\right)\right]}^{\kappa }.$$

for all $\kappa >0.75$. As a result, all of the Theorem 1’s circumstances are met; here, point 21 is the coincidence proximity point for the couple $(\mathfrak{g},\mathcal{T}).$

Corollary 1.

Let $(\mathrm{ʘ},đ)$ (a complete rectangular metric space) with closed subsets $₱\ne \varphi \ne \mathcal{Q}$ meet the $\mathcal{P}$-property and $\varphi \ne {₱}_0$. Let a continuous map $\mathcal{T}:₱\to \mathfrak{CB}\left(\mathcal{Q}\right)$ with $\mathcal{T}\left({₱}_0\right)\subseteq {\mathcal{Q}}_0$. It follows that if $\mathcal{T}$ accomplishes the ${(\mathcal{f}-{\theta }_{\kappa })}_{B_P}$-proximal contraction condition, then it admits a best proximity point.

Proof. 

Under the assumption that $I_{₱}=\mathfrak{g}$, where $I_{₱}$ denotes the identity map, the remainder of the proof unfolds in accordance with Theorem 1. □

Observe that in Theorem 1, by setting $₱=\mathrm{ʘ}=\mathcal{Q}$, we derive the corresponding results presented below.

Corollary 2.

Assume that $\mathcal{T}:\mathrm{ʘ}\to \mathfrak{CB}\left(\mathrm{ʘ}\right)$ is a multivalued mapping on a complete rectangular metric space, and that $\mathcal{T}$ meets the condition of ${\left({\theta }_{\kappa }\right)}_{Bp}$-proximal contraction with the function $\mathcal{f}$. Under the mentioned assumptions, a fixed point for $\mathcal{T}$ is confirmed to exist.

Proof. 

Let $₱=\mathrm{ʘ}=\mathcal{Q}$. Under this assumption, the remaining part directly corresponds to Theorem 1. □

Example 6.

Consider $\mathrm{ʘ}=\{21,22,23,24\}$; then, we can conclude that $(\mathrm{ʘ},đ)$ constitutes a complete rectangular metric space, where the metric function $đ$ is given as follows:

$$đ(\mathrm{ʐ},\mathcal{x})=đ(\mathcal{x},\mathrm{ʐ})=\left\{\begin{array}{cc}0,\hfill & \mathit{if}\mathrm{ʐ}=\mathcal{x},\hfill \\ 1,\hfill & \mathit{if}(\mathrm{ʐ},\mathcal{x})\in \left\{\right(21,23),(22,23\left)\right\},\hfill \\ 2,\hfill & \mathit{if}(\mathrm{ʐ},\mathcal{x})\in \left\{\right(21,24),(22,24),(23,24\left)\right\},\hfill \\ 3,\hfill & \mathit{if}(\mathrm{ʐ},\mathcal{x})=(21,22).\hfill \end{array}\right.$$

Also, multivalued mapping $\mathcal{T}:\mathrm{ʘ}\to \mathfrak{CB}\left(\mathrm{ʘ}\right)$ is given as

$$\mathcal{T}\mathrm{ʐ}=\left\{\begin{array}{cc}21,\hfill & \mathit{if}\mathrm{ʐ}=\{21,22,23\}\hfill \\ \{22,23\},\hfill & \mathit{if}\mathrm{ʐ}=24.\hfill \end{array}\right.$$

Now, $\mathcal{T}$ meets the ${(\mathcal{f}-{\theta }_{\kappa })}_{B_P}-$proximal contraction condition

$$\theta \left(\mathcal{f}\left(\mathcal{H}(\mathcal{T}\mathrm{ʐ},\mathcal{T}\mathcal{x})\right)\right)\le \left[\theta \left(\mathcal{f}\right(\mathcal{S}(\mathrm{ʐ},\mathcal{x})\left)\right)\right],$$

(13)

for all $\mathrm{ʐ},\mathcal{x}\in \mathrm{ʘ}$. Let the function θ be $\theta \left(\mathcal{y}\right)=e^{\mathcal{y}}$ for $\mathcal{f}=2\mathcal{t}$.

Let $\mathrm{ʐ}\in \{21,22,23\}$ with $\mathcal{x}=24$; then, $\mathcal{T}\mathrm{ʐ}=21$, and $\mathcal{T}\mathcal{x}=\{22,23\}$. Therefore, $\mathcal{H}(\mathcal{T}\mathrm{ʐ},\mathcal{T}\mathcal{x})=1$, and we also obtain

$$\begin{array}{cc}\hfill \mathcal{S}(\mathrm{ʐ},\mathcal{x})& =max\left\{đ(\mathcal{x},\mathcal{T}(\mathcal{x}\left)\right),đ(\mathrm{ʐ},\mathcal{x}),đ(\mathrm{ʐ},\mathcal{T}y)-đ(\mathrm{ʐ},\mathcal{x})\right\}\hfill \\ \hfill & =max\left\{đ\right(2,1)-đ(2,4),2,đ(4,1\left)\right\}\hfill \\ \hfill & =max\{2,2,3-2\}=2.\hfill \end{array}$$

Therefore,

$$\mathcal{f}\left(\mathcal{H}\right(\mathcal{T}\mathrm{ʐ},\mathcal{T}\mathcal{x}\left)\right)=\mathcal{f}\left(1\right)=2\left(1\right)=2,\mathrm{and}\mathcal{f}\left(\mathcal{S}\right(\mathrm{ʐ},\mathcal{x}\left)\right)=\mathcal{f}\left(2\right)=4.$$

Now, by applying θ, we obtain

$$\theta \left(\mathcal{f}\left(\mathcal{H}(\mathcal{T}\mathrm{ʐ},\mathcal{T}\mathcal{x})\right)\right)=\theta \left(2\right)=e^2=7.3891,$$

and

$$\theta \left(\mathcal{f}\left(\mathcal{S}(\mathrm{ʐ},\mathcal{x})\right)\right)=\theta \left(4\right)=e^4=54.5982$$

All of the conditions stated in Corollary 2 are fulfilled, which confirms that 1 is a fixed point for $\mathcal{T}$.

4. Single-Valued Mappings Results

This part of the manuscript explores the coincidence proximity point results within the complete rectangular metric spaces, with a particular emphasis on single-valued mappings.

Definition 12.

Let us take two mappings $(\mathfrak{g},\mathcal{T}),$ such that the scenario whereby $\mathfrak{g}:₱\to ₱$ and $\mathcal{T}:₱\to \mathcal{Q}$ is termed a ${\left({\theta }_{\kappa }\right)}_{C_p}-$proximal contraction if for $\kappa \in (0,1)$,

$$\begin{array}{c}đ(₱,\mathcal{Q})=đ\left(\mathfrak{g}\mathfrak{u},\mathcal{T}\mathrm{ʐ}\right)\\ đ(₱,\mathcal{Q})=đ\left(\mathfrak{g}\mathfrak{v},\mathcal{T}\mathcal{x}\right)\end{array}$$

implying that

$$\theta \left(\mathcal{f}\left(đ(\mathcal{T}\mathrm{ʐ},\mathcal{T}\mathcal{x})\right)\right)\le {\left[\theta \left(\mathcal{f}\right(\mathcal{S}(\mathfrak{u},\mathfrak{v},\mathrm{ʐ},\mathcal{x})\left)\right)\right]}^{\kappa },$$

where

$$\mathcal{S}(\mathfrak{u},\mathfrak{v},\mathrm{ʐ},\mathcal{x})=max\left\{đ(\mathfrak{g}\mathrm{ʐ},\mathfrak{g}\mathcal{x}),đ(\mathfrak{g}\mathfrak{u},\mathfrak{g}\mathfrak{v}),đ(\mathfrak{g}\mathfrak{u},\mathcal{T}\mathfrak{u})-đ(\mathfrak{g}\mathfrak{u},\mathfrak{g}\mathfrak{v})-đ(\mathfrak{g}\mathcal{x},\mathcal{T}\mathfrak{u}),{đ}^{*}(\mathfrak{g}\mathfrak{v},\mathcal{T}\mathfrak{u})\right\}$$

for all $\mathfrak{u},\mathfrak{v},\mathrm{ʐ},\mathcal{x}\in ₱.$

Definition 13.

Consider a map $\mathcal{T}:₱\to \mathcal{Q}$ that is termed to satisfy the ${\left({\theta }_{\kappa }\right)}_{B_P}-$proximal contraction condition if for $\kappa \in (0,1)$,

$$\begin{array}{c}đ(₱,\mathcal{Q})=đ\left(\mathfrak{u},\mathcal{T}\mathrm{ʐ}\right)\\ đ(₱,\mathcal{Q})=đ\left(\mathfrak{v},\mathcal{T}\mathcal{x}\right)\end{array}$$

which implies that

$$\theta \left(\mathcal{f}\left(đ(\mathcal{T}\mathrm{ʐ},\mathcal{T}\mathcal{x})\right)\right)\le {\left[\theta \left(\mathcal{f}\right(\mathcal{S}(\mathfrak{u},\mathfrak{v},\mathrm{ʐ},\mathcal{x})\left)\right)\right]}^{\kappa },$$

where

$$\begin{array}{c}\hfill \mathcal{S}(\mathfrak{u},\mathfrak{v},\mathrm{ʐ},\mathcal{x})=max\left\{đ(\mathrm{ʐ},\mathcal{x}),đ(\mathfrak{u},\mathfrak{v}),đ(\mathfrak{u},\mathcal{T}\mathfrak{u})-đ(\mathfrak{u},\mathfrak{v})-đ(\mathcal{x},\mathcal{T}\mathfrak{u}),{đ}^{*}(\mathfrak{v},\mathcal{T}\mathfrak{u})\right\}.\end{array}$$

for all $\mathfrak{u},\mathfrak{v},\mathrm{ʐ},\mathcal{x}$ in $₱.$

While choosing $\mathfrak{g}$ as $I_{₱}$, ${\left({\theta }_{\kappa }\right)}_{C_p}$-proximal contractions reduce to the standardized form of ${\left({\theta }_{\kappa }\right)}_{B_p}$-proximal contractions.

Theorem 2.

Consider two mappings, $\mathfrak{g}:₱\to ₱$ and $\mathcal{T}:₱\to \mathcal{Q}$, with $₱\ne \varphi \ne \mathcal{Q}$, closed subsets of $(\mathrm{ʘ},đ)$, with the $\mathcal{P}$-property, where $\mathcal{T}\left({₱}_0\right)\subseteq {\mathcal{Q}}_0$ and ${₱}_0\subseteq \mathfrak{g}\left({₱}_0\right)$. If the couple $(\mathfrak{g},\mathcal{T})$ of continuous maps contains injective mapping $\mathfrak{g}$, these mappings fulfill the condition of ${\left({\theta }_{\kappa }\right)}_{C_p}$-proximal contraction. Under the mentioned assumptions, a best coincidence proximity point for the couple $(\mathfrak{g},\mathcal{T})$ is confirmed to exist.

Proof. 

As single-valued mappings are a special case of multivalued mappings, for ${\mathrm{ʐ}}_n\in {₱}_0$ and $\mathcal{T}\left({₱}_0\right)\subseteq {\mathcal{Q}}_0,$ there is ${\mathrm{ʐ}}_{n+1}\in {₱}_0$, which gives

$$đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathcal{T}{\mathrm{ʐ}}_n)=đ(₱,\mathcal{Q}).$$

(14)

After selecting ${\mathrm{ʐ}}_{n+1}$, which satisfies the given condition, there exists an element ${\mathrm{ʐ}}_{n+2}\in {₱}_0$ that meets the condition

$$đ(₱,\mathcal{Q})=đ(\mathfrak{g}\left({\mathrm{ʐ}}_{n+2}\right),\mathcal{T}\left({\mathrm{ʐ}}_{n+1}\right))$$

(15)

For all positive integers n, utilizing the $\mathcal{P}$-property, we obtain

$$đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2})=đ(\mathcal{T}{\mathrm{ʐ}}_n,\mathcal{T}{\mathrm{ʐ}}_{n+1}).$$

Moreover, since the couple $(\mathfrak{g},\mathcal{T})$ meets the condition of ${\left({\theta }_{\kappa }\right)}_{C_p}$-proximal contraction, and by applying Equations (14) and (15), we obtain

$$\theta \left(\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2})\right)\right)\le {\left[\theta \left(\mathcal{f}\left(\mathcal{S}({\mathrm{ʐ}}_{n+1},{\mathrm{ʐ}}_{n+2},{\mathrm{ʐ}}_n,{\mathrm{ʐ}}_{n+1})\right)\right)\right]}^{\kappa },$$

(16)

where

$$\begin{array}{ccc}\hfill \mathcal{S}({\mathrm{ʐ}}_{n+1},{\mathrm{ʐ}}_{n+2},{\mathrm{ʐ}}_n,{\mathrm{ʐ}}_{n+1})& =& max\left\{\begin{array}{c}đ(\mathfrak{g}{\mathrm{ʐ}}_n,\mathfrak{g}{\mathrm{ʐ}}_{n+1}),đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2}),{đ}^{*}(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathcal{T}{\mathrm{ʐ}}_n),\\ đ(\mathfrak{g}{\mathrm{ʐ}}_n,\mathcal{T}{\mathrm{ʐ}}_n)-đ(\mathfrak{g}{\mathrm{ʐ}}_n,\mathfrak{g}{\mathrm{ʐ}}_{n+1})-đ(\mathfrak{g}{\mathrm{ʐ}}_{n+2},\mathcal{T}{\mathrm{ʐ}}_n)\end{array}\right\}\hfill \\ & \le & max\left\{\begin{array}{c}đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathcal{T}{\mathrm{ʐ}}_n)-đ(₱,\mathcal{Q}),đ(\mathfrak{g}{\mathrm{ʐ}}_n,\mathfrak{g}{\mathrm{ʐ}}_{n+1}),đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2}),\\ đ(\mathfrak{g}{\mathrm{ʐ}}_n,\mathfrak{g}{\mathrm{ʐ}}_{n+1})+đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2})+đ(\mathfrak{g}{\mathrm{ʐ}}_{n+2},\mathcal{T}{\mathrm{ʐ}}_n)\\ -đ(\mathfrak{g}{\mathrm{ʐ}}_n,\mathfrak{g}{\mathrm{ʐ}}_{n+1})-đ(\mathfrak{g}{\mathrm{ʐ}}_{n+2},\mathcal{T}{\mathrm{ʐ}}_n)\end{array}\right\}\hfill \\ & \le & max\left\{đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2}),đ(\mathfrak{g}{\mathrm{ʐ}}_n,\mathfrak{g}{\mathrm{ʐ}}_{n+1}),đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2}),0\right\}\hfill \\ & \le & max\left\{đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2}),đ(\mathfrak{g}{\mathrm{ʐ}}_n,\mathfrak{g}{\mathrm{ʐ}}_{n+1})\right\}.\hfill \end{array}$$

So, we can say that

$$\mathcal{S}({\mathrm{ʐ}}_{n+1},{\mathrm{ʐ}}_{n+2},{\mathrm{ʐ}}_n,{\mathrm{ʐ}}_{n+1})\le max\left\{đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2}),đ(\mathfrak{g}{\mathrm{ʐ}}_n,\mathfrak{g}{\mathrm{ʐ}}_{n+1})\right\}.$$

If we choose $max\left\{đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2}),đ(\mathfrak{g}{\mathrm{ʐ}}_n,\mathfrak{g}{\mathfrak{x}}_{n+1})\right\}=đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2}),$ then inequality (16) transforms into

$$\theta \left(\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2})\right)\right)\le {\left[\theta \left(\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2})\right)\right)\right]}^{\kappa },$$

which is not impossible. If $max\left\{đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2}),đ(\mathfrak{g}{\mathrm{ʐ}}_n,\mathfrak{g}{\mathrm{ʐ}}_{n+1})\right\}=đ(\mathfrak{g}{\mathrm{ʐ}}_n,\mathfrak{g}{\mathrm{ʐ}}_{n+1}),$ then Equation (16) becomes

$${\left[\theta \left(\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_n,\mathfrak{g}{\mathrm{ʐ}}_{n+1})\right)\right)\right]}^{\kappa }\ge \theta \left(\mathcal{f}\left(đ(\mathfrak{g}{\mathrm{ʐ}}_{n+1},\mathfrak{g}{\mathrm{ʐ}}_{n+2})\right)\right),$$

(17)

Here, our contraction holds, and the remaining part directly corresponds to Theorem 1. □

Example 7.

Consider the set $\mathrm{ʘ}=\{17,18,19,20,21,22\}$; then, $(\mathrm{ʘ},đ)$ becomes a complete rectangular metric space. Here, the metric function $đ:\mathrm{ʘ}\times \mathrm{ʘ}\to [0,\infty )$ is

$$đ(\mathrm{ʐ},\mathcal{x})=\left\{\begin{array}{cc}0,\hfill & \mathit{if}\mathrm{ʐ}=\mathcal{x},\hfill \\ \sqrt{|\mathrm{ʐ}-\mathcal{x}|}\hfill & \mathit{if}\mathrm{ʐ}\ne \mathcal{x}\hfill \end{array}\right.$$

Suppose that $\mathcal{Q}=\{18,20,22\}$ and $₱=\{17,19,21\}$ are two closed subsets of $(\mathrm{ʘ},đ).$ We see that $đ(₱,\mathcal{Q})=1$ and meets the $\mathcal{P}-$property, where ${₱}_0=₱$ and ${\mathcal{Q}}_0=\mathcal{Q}$. Also, map $\mathcal{T}:₱\to \mathcal{Q}$ is

$$\mathcal{T}\mathrm{ʐ}=\left\{\begin{array}{cc}20,\hfill & \mathit{if}\mathrm{ʐ}=21\hfill \\ 18,\hfill & \mathit{if}\mathrm{ʐ}=\{17,19\},\hfill \end{array}\right.$$

and $\mathfrak{g}:₱\to ₱$ is

$$\mathfrak{g}\mathrm{ʐ}=\left\{\begin{array}{c}21,\mathit{if}\mathrm{ʐ}=19\\ 19,\mathit{if}\mathrm{ʐ}=21\\ 17,\mathit{if}\mathrm{ʐ}=17\end{array}\right.$$

We clearly see that $\mathcal{T}\left({₱}_0\right)\subseteq {\mathcal{Q}}_0$ and $\mathfrak{g}\left({₱}_0\right)\subseteq {₱}_0.$ Now, our aims to prove the couple $(\mathfrak{g},\mathcal{T})$ fulfill the condition of ${\left({\theta }_{\kappa }\right)}_{C_p}$-proximal contraction,

$$\theta \left(\mathcal{f}\left(đ(\mathcal{T}\mathrm{ʐ},\mathcal{T}\mathcal{x})\right)\right)\le {\left[\theta \left(\mathcal{f}\right(\mathcal{S}(\mathfrak{u},\mathfrak{v},\mathrm{ʐ},\mathcal{x})\left)\right)\right]}^{\kappa },$$

(18)

for all $\mathfrak{u},\mathcal{x},\mathfrak{v},\mathrm{ʐ}\in ₱$. Since

$$\begin{array}{ccc}\hfill đ(₱,\mathcal{Q})& =& đ(\mathfrak{g}17,\mathcal{T}19),\hfill \\ \hfill đ(₱,\mathcal{Q})& =& đ(\mathfrak{g}19,\mathcal{T}21),\hfill \end{array}$$

here, $\mathfrak{v}=19,$ $\mathrm{ʐ}=19,$ $\mathfrak{u}=17,$ and $\mathcal{x}=21$. After routine calculation, we have

$$đ(\mathcal{T}19,\mathcal{T}21)=đ(18,20)=\sqrt{2},$$

and

$$\begin{array}{ccc}\hfill \mathcal{S}(17,19,19,21)& =& max\left(\begin{array}{c}đ(\mathfrak{g}17,\mathfrak{g}19),đ(\mathfrak{g}19,\mathfrak{g}21),{đ}^{*}(\mathfrak{g}19,\mathcal{T}17),\\ đ(\mathfrak{g}17,\mathcal{T}17)-đ(\mathfrak{g}17,\mathfrak{g}19)-đ(\mathfrak{g}21,\mathcal{T}17)\end{array}\right)\hfill \\ & =& max\left(\begin{array}{c}đ(\mathfrak{g}17,\mathfrak{g}19),đ(\mathfrak{g}19,\mathfrak{g}21),đ(\mathfrak{g}19,\mathcal{T}17)-đ(₱,\mathcal{Q}),\\ đ(\mathfrak{g}17,\mathcal{T}17)-đ(\mathfrak{g}17,\mathfrak{g}19)-đ(\mathfrak{g}21,\mathcal{T}17)\end{array}\right)\hfill \\ & =& max\left(\begin{array}{c}đ(17,21),đ(21,19),đ(21,18)-đ(₱,\mathcal{Q}),\\ đ(17,18)-đ(17,21)-đ(19,18)\end{array}\right)\hfill \\ & =& max\left(\sqrt{2},2,1-2-\sqrt{2},\sqrt{3}-1\right)\hfill \\ & =& 2.\hfill \end{array}$$

Here, θ functions as $\theta \left(\mathcal{y}\right)=e^{\mathcal{y}}$. For $\mathcal{f}=2\mathcal{t}$ and $\kappa >0.26$, we calculate $\mathcal{f}\left(đ(\mathcal{T}\mathrm{ʐ},\mathcal{T}\mathcal{x})\right)=2\sqrt{2}$, $\mathcal{f}\left(\mathcal{S}\right(\mathfrak{u},\mathfrak{v},\mathrm{ʐ},\mathcal{x}\left)\right)=4$ and then inequality (18), given as

$$\theta \left(2\sqrt{2}\right)\le {\left[\theta \left(4\right)\right]}^{\kappa }$$

Thus, the requirements as per Theorem 2 are met, confirming that 17 serves as the coincidence best proximity point for the couple $(\mathfrak{g},\mathcal{T})$.

Corollary 3.

Let $\mathcal{T}:₱\to \mathcal{Q}$, with $₱\ne Ø\ne \mathcal{Q}$, closed subsets of $(\mathrm{ʘ},đ)$, satisfying the $\mathcal{P}$-property, and the given metric is complete. Additionally, assume that $\mathcal{T}\left({₱}_0\right)\subseteq {\mathcal{Q}}_0$. The mapping $\mathcal{T}$ admits a proximity point if it is continuous and satisfies the ${\left({\theta }_{\kappa }\right)}_{B_p}$-proximal contraction condition.

Proof. 

If we take $\mathfrak{g}=I_{₱}$ (here, $I_{₱}$ denotes the identity mapping on $₱$), and ${\mathrm{ʐ}}_n\in {₱}_0$, and $\mathcal{T}\left({₱}_0\right)\subseteq {\mathcal{Q}}_0,$ then there is ${\mathrm{ʐ}}_{n+1}\in {₱}_0$, which gives

$$đ({\mathrm{ʐ}}_{n+1},\mathcal{T}{\mathrm{ʐ}}_n)=đ(₱,\mathcal{Q}).$$

(19)

After selecting ${\mathrm{ʐ}}_{n+1}$, which satisfies the given condition, there exists an element ${\mathrm{ʐ}}_{n+2}\in {₱}_0$ that meets the condition

$$đ(₱,\mathcal{Q})=đ(\left({\mathrm{ʐ}}_{n+2}\right),\mathcal{T}\left({\mathrm{ʐ}}_{n+1}\right))$$

(20)

For all positive integers n, utilizing the $\mathcal{P}$-property, we obtain

$$đ({\mathrm{ʐ}}_{n+1},{\mathrm{ʐ}}_{n+2})=đ(\mathcal{T}{\mathrm{ʐ}}_n,\mathcal{T}{\mathrm{ʐ}}_{n+1}).$$

Moreover, since $\mathcal{T}$ meets the condition of ${\left({\theta }_{\kappa }\right)}_{B_p}$-proximal contraction, and by applying Equations (19) and (20), we obtain

$$\theta \left(\mathcal{f}\left(đ({\mathrm{ʐ}}_{n+1},{\mathrm{ʐ}}_{n+2})\right)\right)\le {\left[\theta \left(\mathcal{f}\left(\mathcal{S}({\mathrm{ʐ}}_{n+1},{\mathrm{ʐ}}_{n+2},{\mathrm{ʐ}}_n,{\mathrm{ʐ}}_{n+1})\right)\right)\right]}^{\kappa },$$

(21)

where

$$\begin{array}{ccc}\hfill \mathcal{S}({\mathrm{ʐ}}_{n+1},{\mathrm{ʐ}}_{n+2},{\mathrm{ʐ}}_n,{\mathrm{ʐ}}_{n+1})& =& max\left\{\begin{array}{c}đ({\mathrm{ʐ}}_n,{\mathrm{ʐ}}_{n+1}),đ({\mathrm{ʐ}}_{n+1},{\mathrm{ʐ}}_{n+2}),{đ}^{*}({\mathrm{ʐ}}_{n+1},\mathcal{T}{\mathrm{ʐ}}_n),\\ đ({\mathrm{ʐ}}_n,\mathcal{T}{\mathrm{ʐ}}_n)-đ({\mathrm{ʐ}}_n,{\mathrm{ʐ}}_{n+1})-đ({\mathrm{ʐ}}_{n+2},\mathcal{T}{\mathrm{ʐ}}_n)\end{array}\right\}\hfill \\ & \le & max\left\{\begin{array}{c}đ({\mathrm{ʐ}}_{n+1},\mathcal{T}{\mathrm{ʐ}}_n)-đ(₱,\mathcal{Q}),đ({\mathrm{ʐ}}_n,{\mathrm{ʐ}}_{n+1}),đ({\mathrm{ʐ}}_{n+1},{\mathrm{ʐ}}_{n+2}),\\ đ({\mathrm{ʐ}}_n,{\mathrm{ʐ}}_{n+1})+đ({\mathrm{ʐ}}_{n+1},{\mathrm{ʐ}}_{n+2})+đ({\mathrm{ʐ}}_{n+2},\mathcal{T}{\mathrm{ʐ}}_n)\\ -đ({\mathrm{ʐ}}_n,{\mathrm{ʐ}}_{n+1})-đ({\mathrm{ʐ}}_{n+2},\mathcal{T}{\mathrm{ʐ}}_n)\end{array}\right\}\hfill \\ & \le & max\left\{đ({\mathrm{ʐ}}_{n+1},{\mathrm{ʐ}}_{n+2}),đ({\mathrm{ʐ}}_n,{\mathrm{ʐ}}_{n+1}),đ({\mathrm{ʐ}}_{n+1},{\mathrm{ʐ}}_{n+2}),0\right\}\hfill \\ & \le & max\left\{đ({\mathrm{ʐ}}_{n+1},{\mathrm{ʐ}}_{n+2}),đ({\mathrm{ʐ}}_n,{\mathrm{ʐ}}_{n+1})\right\}.\hfill \end{array}$$

So, we can say that

$$\mathcal{S}({\mathrm{ʐ}}_{n+1},{\mathrm{ʐ}}_{n+2},{\mathrm{ʐ}}_n,{\mathrm{ʐ}}_{n+1})\le max\left\{đ({\mathrm{ʐ}}_{n+1},{\mathrm{ʐ}}_{n+2}),đ({\mathrm{ʐ}}_n,{\mathrm{ʐ}}_{n+1})\right\}.$$

If we choose $max\left\{đ({\mathrm{ʐ}}_{n+1},{\mathrm{ʐ}}_{n+2}),đ({\mathrm{ʐ}}_n,{\mathfrak{x}}_{n+1})\right\}=đ({\mathrm{ʐ}}_{n+1},{\mathrm{ʐ}}_{n+2}),$ then inequality (21) transforms into

$$\theta \left(\mathcal{f}\left(đ({\mathrm{ʐ}}_{n+1},{\mathrm{ʐ}}_{n+2})\right)\right)\le {\left[\theta \left(\mathcal{f}\left(đ({\mathrm{ʐ}}_{n+1},{\mathrm{ʐ}}_{n+2})\right)\right)\right]}^{\kappa },$$

which is not impossible. If $max\left\{đ({\mathrm{ʐ}}_{n+1},{\mathrm{ʐ}}_{n+2}),đ({\mathrm{ʐ}}_n,{\mathrm{ʐ}}_{n+1})\right\}=đ({\mathrm{ʐ}}_n,{\mathrm{ʐ}}_{n+1}),$ then Equation (21) becomes

$${\left[\theta \left(\mathcal{f}\left(đ({\mathrm{ʐ}}_n,{\mathrm{ʐ}}_{n+1})\right)\right)\right]}^{\kappa }\ge \theta \left(\mathcal{f}\left(đ({\mathrm{ʐ}}_{n+1},{\mathrm{ʐ}}_{n+2})\right)\right),$$

(22)

Here, our contraction holds, and then, the remaining part directly corresponds to Theorem 1, and we obtain a proximity point. □

The following example shows that in the best proximity study, we are dealing with two sequences at a time, which is an advantage of best proximity. We show this phenomenon numerically and graphically.

Example 8.

Let us have $\mathrm{ʘ}=C[0,3]$, and the metric $đ(\mathcal{a},♭)={sup}_{\mathrm{\omega }\in [0,1]}|{\mathcal{a}}^8\left(\mathrm{\omega }\right)-{♭}^8\left(\mathrm{\omega }\right)|$. Let

$$₱=\left\{\alpha \right(\mathrm{\omega }):\alpha \in [0,1\left]\right\}$$

and

$$\mathcal{Q}=\left\{\alpha \right(\mathrm{\omega }-2):\alpha \in [0,1\left]\right\}$$

Here, $đ(₱,\mathcal{Q})=2$, with ${₱}_0=₱$ and ${\mathcal{Q}}_0=\mathcal{Q}$. Now, non-self-mapping $\mathcal{T}:₱\to \mathcal{Q}$ is defined as

$$\mathcal{T}\left(\mathcal{a}\left(\mathrm{\omega }\right)\right)={\displaystyle \frac{8}{9}}(\mathrm{\omega }-2)+(\mathrm{\omega }-2){\int }_0^1\left({\mathcal{a}}^8\left(s\right)\right)ds$$

So, $\mathcal{T}\left({₱}_0\right)\subset {\mathcal{Q}}_0.$ Take $\mathcal{a}\left(\mathrm{\omega }\right),♭\left(\mathrm{\omega }\right),\mathfrak{w}\left(\mathrm{\omega }\right),\mathfrak{u}\left(\mathrm{\omega }\right)\in ₱$, which gives

$$đ(₱,\mathcal{Q})=đ\left(\mathfrak{w}\right(\mathrm{\omega }),\mathcal{T}(\mathcal{a}\left(\mathrm{\omega }\right)\left)\right)$$

$$đ(₱,\mathcal{Q})=đ\left(\mathfrak{u}\right(\mathrm{\omega }),\mathcal{T}(♭\left(\mathrm{\omega }\right)\left)\right)$$

Now,

$$\begin{array}{cc}\hfill đ\left(\mathcal{T}\right(\mathcal{a}\left(\mathrm{\omega }\right)),\mathcal{T}(♭\left(\mathrm{\omega }\right)\left)\right)& =\underset{\mathrm{\omega }\in [0,1]}{sup}\left|\begin{array}{cc}& {\displaystyle \frac{8}{9}}(\mathrm{\omega }-2)+(\mathrm{\omega }-2){\int }_0^1\left({\mathcal{a}}^8\left(s\right)\right)ds\hfill \\ & -{\displaystyle \frac{8}{9}}(\mathrm{\omega }-2)-(\mathrm{\omega }-2){\int }_0^1\left({♭}^8\left(s\right)\right)ds\hfill \end{array}\right|\hfill \\ \hfill & =\underset{\mathrm{\omega }\in [0,1]}{sup}\left|(\mathrm{\omega }-2){\int }_0^1\left({\mathcal{a}}^8\left(s\right)\right)ds-(\mathrm{\omega }-2){\int }_0^1\left({♭}^8\left(s\right)\right)ds\right|\hfill \\ \hfill & \le \underset{\mathrm{\omega }\in [0,1]}{sup}(\mathrm{\omega }-2){\int }_0^1\left|{\mathcal{a}}^8\left(s\right)-{♭}^8\left(s\right)\right|ds\hfill \\ \hfill & =(\mathrm{\omega }-2)\underset{\mathrm{\omega }\in [0,1]}{sup}\left|{\mathcal{a}}^8\left(\mathrm{\omega }\right)-{♭}^8\left(\mathrm{\omega }\right)\right|{\int }_0^{1}ds\hfill \\ \hfill & =(\mathrm{\omega }-2)đ\left(\mathcal{a}\right(\mathrm{\omega }),♭(\mathrm{\omega }\left)\right)\hfill \\ \hfill & \le (\mathrm{\omega }-2)max\left\{\begin{array}{c}đ\left(\mathcal{a}\right(\mathrm{\omega }),♭(\mathrm{\omega }\left)\right),đ\left(\mathfrak{w}\right(\mathrm{\omega }),\mathfrak{u}(\mathrm{\omega }\left)\right),\hfill \\ đ\left(\mathfrak{w}\right(\mathrm{\omega }),\mathcal{T}(\mathfrak{w}\left(\mathrm{\omega }\right)\left)\right)-đ\left(\mathfrak{w}\right(\mathrm{\omega }),\mathfrak{u}(\mathrm{\omega }\left)\right)\hfill \\ -đ(♭(\mathrm{\omega }),\mathcal{T}(\mathfrak{w}\left(\mathrm{\omega }\right)\left)\right),\hfill \\ đ\left(\mathcal{a}\right(\omega ),\mathcal{T}(\mathcal{a}\left(\mathrm{\omega }\right)\left)\right)-đ\left(\mathcal{a}\right(\mathrm{\omega }),♭(\mathrm{\omega }\left)\right)\hfill \end{array}\right\}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le (\mathrm{\omega }-2)\mathcal{S}\left(\mathcal{a}\left(\mathrm{\omega }\right),♭\left(\mathrm{\omega }\right),\mathfrak{w}\left(\mathrm{\omega }\right),\mathfrak{u}\left(\mathrm{\omega }\right)\right)\hfill \\ \hfill & \le \left(\mathrm{\omega }\right)\mathcal{S}\left(\mathcal{a}\left(\mathrm{\omega }\right),♭\left(\mathrm{\omega }\right),\mathfrak{w}\left(\mathrm{\omega }\right),\mathfrak{u}\left(\mathrm{\omega }\right)\right).\hfill \end{array}$$

Now, taking $\mathcal{f}\left(\mathfrak{t}\right)=2\mathfrak{t}$ and $\theta \left(\zeta \right)=e^{\zeta }$ implies that

$$\theta \left(\mathcal{f}\left(đ\left(\mathcal{T}\right(\mathcal{a}\left(\mathrm{\omega }\right)),\mathcal{T}(♭\left(\mathrm{\omega }\right))\right)\right)\le {\left[\theta \left(\mathcal{f}\left(\mathcal{S}\left(\mathcal{a}\right(\mathrm{\omega }),♭(\mathrm{\omega }),\mathfrak{w}(\mathrm{\omega }),\mathfrak{u}(\mathrm{\omega }\left)\right)\right)\right)\right]}^{\left(\mathrm{\omega }\right)}$$

Hence, ${\left({\theta }_{\kappa }\right)}_{B_P}$ holds. Now, we show the numerical convergence of both sequences ${\mathcal{a}}_n$ and $\mathcal{T}\left({\mathcal{a}}_n\right)$ in Table 1 given below, taking the initial guess ${\mathcal{a}}_0=0$.

Table 1. Convergence of sequences ${\mathcal{a}}_n$ and $\mathcal{T}\left({\mathcal{a}}_n\right)$.

Sr. No.	Sequence ${\mathcal{a}}_{\mathit{n}}$	Sequence $\mathcal{T}\left({\mathcal{a}}_{\mathit{n}}\right)$
00	0	${\displaystyle \frac{8}{9}}(\mathrm{\omega }-2)=0.8889(\mathrm{\omega }-2)$
01	$0.8889\mathrm{\omega }$	$0.9322(\mathrm{\omega }-2)$
02	$0.9322\mathrm{\omega }$	$0.9522(\mathrm{\omega }-2)$
03	$0.9522\mathrm{\omega }$	$0.9640(\mathrm{\omega }-2)$
04	$0.9640\mathrm{\omega }$	$0.9717(\mathrm{\omega }-2)$
05	$0.9717\mathrm{\omega }$	$0.9772(\mathrm{\omega }-2)$
06	$0.9772\mathrm{\omega }$	$0.9813(\mathrm{\omega }-2)$
07	$0.9813\mathrm{\omega }$	$0.9869(\mathrm{\omega }-2)$
08	$0.9869\mathrm{\omega }$	$0.9888(\mathrm{\omega }-2)$
09	$0.9888\mathrm{\omega }$	$0.9904(\mathrm{\omega }-2)$
10	$0.9904\mathrm{\omega }$	$0.9918(\mathrm{\omega }-2)$
⋮	⋮	⋮
31	$0.9997\mathrm{\omega }$	$0.9998(\mathrm{\omega }-2)$
32	$0.9998\mathrm{\omega }$	$0.9999(\mathrm{\omega }-2)$
33	$0.9999\mathrm{\omega }$	$(\mathrm{\omega }-2)$
34	ω	$(\mathrm{\omega }-2)$

Now, we have the graphical convergence of both sequences, which are generated at the same time and also converge at same time in different places. In Figure 1, we show the convergence of sequence ${\mathcal{a}}_n$, and in Figure 2, the convergence of $\mathcal{T}\left({\mathcal{a}}_n\right)$.

Figure 1. Convergence behavior of sequence ${\mathcal{a}}_n$.

Figure 2. Convergence behavior of sequence $\mathcal{T}\left({\mathcal{a}}_n\right)$.

5. Fixed Point Results

If we set $\mathcal{Q}=₱=\mathrm{ʘ}$ in Theorem 2 from the earlier section, then we obtain a fixed point and simultaneously fulfill the preceding considerations.

Definition 14.

A map $\mathcal{T}:\mathrm{ʘ}\to \mathrm{ʘ}$ is known to be ${\left({\theta }_{\kappa }\right)}_{F_P}$-contraction, if for $\kappa \in (0,1)$, the following holds:

$$\theta \left(\mathcal{f}\left(đ(\mathcal{T}\mathfrak{v},\mathcal{T}\mathfrak{u})\right)\right)\le {\left[\theta \left(\mathcal{f}\right(\mathcal{S}(\mathfrak{v},\mathfrak{u})\left)\right)\right]}^{\kappa },$$

where $\forall \mathfrak{v},\mathfrak{u}\in \mathrm{ʘ}$

$$\begin{array}{ccc}\hfill \mathcal{S}(\mathfrak{v},\mathfrak{u})& =& max\left\{đ\right(\mathfrak{u},\mathcal{T}\mathfrak{v}),đ(\mathfrak{v},\mathfrak{u}),đ(\mathfrak{v},\mathcal{T}\mathfrak{v})-đ(\mathfrak{v},\mathfrak{u}\left)\right\}.\hfill \end{array}$$

Theorem 3.

Let $\mathcal{T}:\mathrm{ʘ}\to \mathrm{ʘ}$ be a self-mapping on $\left(\mathrm{ʘ},đ\right)$ (a complete rectangular metric space). If $\mathcal{T}$ meets the condition of ${\left({\theta }_{\kappa }\right)}_{F_P}$-contraction with $\mathcal{f}$, then $\mathcal{T}$ possesses a fixed point. Here $\mathcal{f}$ is an alternating distance function.

Proof. 

When $₱=\mathrm{ʘ}=\mathcal{Q}$ is considered, and for ${\mathrm{ʐ}}_n\in \mathrm{ʘ}$, there is ${\mathrm{ʐ}}_{n+1}\in \mathrm{ʘ}$. Moreover, since $\mathcal{T}$ meets the condition of $({\theta }_{\kappa }{)}_{F_p}$-contraction, and by applying equation, we obtain

$$\theta \left(\mathcal{f}\left(đ({\mathrm{ʐ}}_{n+1},{\mathrm{ʐ}}_{n+2})\right)\right)\le {\left[\theta \left(\mathcal{f}\left(\mathcal{S}({\mathrm{ʐ}}_n,{\mathrm{ʐ}}_{n+1})\right)\right)\right]}^{\kappa },$$

(23)

where

$$\begin{array}{ccc}\hfill \mathcal{S}({\mathrm{ʐ}}_n,{\mathrm{ʐ}}_{n+1})& =& max\left\{đ({\mathrm{ʐ}}_{n+1},\mathcal{T}{\mathrm{ʐ}}_n),đ({\mathrm{ʐ}}_n,{\mathrm{ʐ}}_{n+1}),đ({\mathrm{ʐ}}_n,\mathcal{T}{\mathrm{ʐ}}_n)-đ({\mathrm{ʐ}}_n,{\mathrm{ʐ}}_{n+1})\right\}\hfill \\ & \le & max\left\{đ({\mathrm{ʐ}}_{n+1},{\mathrm{ʐ}}_{n+1}),đ({\mathrm{ʐ}}_n,{\mathrm{ʐ}}_{n+1}),đ({\mathrm{ʐ}}_n,{\mathrm{ʐ}}_{n+1})-đ({\mathrm{ʐ}}_n,{\mathrm{ʐ}}_{n+1})\right\}\hfill \\ & \le & max\left\{đ({\mathrm{ʐ}}_n,{\mathrm{ʐ}}_{n+1})\right\}.\hfill \end{array}$$

So, Equation (23) becomes

$${\left[\theta \left(\mathcal{f}\left(đ({\mathrm{ʐ}}_n,{\mathrm{ʐ}}_{n+1})\right)\right)\right]}^{\kappa }\ge \theta \left(\mathcal{f}\left(đ({\mathrm{ʐ}}_{n+1},{\mathrm{ʐ}}_{n+2})\right)\right),$$

(24)

Here, our contraction holds, and then, the remaining part directly corresponds to Theorem 1.

□

Example 9.

: If we take $₱=\left\{{\displaystyle \frac{1}{n}}:n=\{6,7,8,9,10\}\right\}$ and $\mathcal{Q}=[1,2]$, the metric $đ:\mathrm{ʘ}\times \mathrm{ʘ}\to \mathbb{R}$ is as follows:

$$đ\left({\displaystyle \frac{1}{7}},{\displaystyle \frac{1}{8}}\right)=0.03=đ\left({\displaystyle \frac{1}{9}},{\displaystyle \frac{1}{10}}\right),$$

$$đ\left({\displaystyle \frac{1}{7}},{\displaystyle \frac{1}{10}}\right)=0.02=đ\left({\displaystyle \frac{1}{8}},{\displaystyle \frac{1}{9}}\right),$$

$$đ\left({\displaystyle \frac{1}{7}},{\displaystyle \frac{1}{9}}\right)=0.06=đ\left({\displaystyle \frac{1}{10}},{\displaystyle \frac{1}{8}}\right),$$

and

$$đ(\mathrm{ʐ},\mathcal{x})=|\mathrm{ʐ}-\mathcal{x}|,\mathit{all}\mathit{other}\mathrm{ʐ},\mathcal{x}\in \mathrm{ʘ}.$$

Here, $\mathrm{ʘ}=₱\cup \mathcal{Q}$. We see that $(\mathrm{ʘ},đ)$ meets the rectangular metric conditions but does not fulfill the condition of a metric space. If we take $\mathrm{ʐ}={\displaystyle \frac{1}{7}},\mathcal{x}={\displaystyle \frac{1}{9}}$ and $\mathfrak{z}={\displaystyle \frac{1}{8}}$, then

$$đ\left({\displaystyle \frac{1}{7}},{\displaystyle \frac{1}{9}}\right)=0.06,$$

On the other hand,

$$đ\left({\displaystyle \frac{1}{7}},{\displaystyle \frac{1}{8}}\right)+đ\left({\displaystyle \frac{1}{8}},{\displaystyle \frac{1}{9}}\right)=0.03+0.02=0.05,$$

which gives

$$đ\left({\displaystyle \frac{1}{7}},{\displaystyle \frac{1}{8}}\right)+đ\left({\displaystyle \frac{1}{8}},{\displaystyle \frac{1}{9}}\right)\le đ\left({\displaystyle \frac{1}{7}},{\displaystyle \frac{1}{9}}\right),$$

which shows that $(\mathrm{ʘ},đ)$ does not fulfill the metric definition but it meets conditions of the rectangular metric.

Now, define $\mathcal{T}:\mathrm{ʘ}\to \mathrm{ʘ}$, where $\mathrm{ʘ}=₱\cup \mathcal{Q}$, such that

$$\mathcal{T}\mathrm{ʐ}=\left\{\begin{array}{cc}1,\hfill & \mathit{if}\mathrm{ʐ}\in ₱\hfill \\ {\displaystyle \frac{1}{2}}\mathrm{ʐ}+{\displaystyle \frac{1}{2}},\hfill & \mathit{if}\mathrm{ʐ}\in \mathcal{Q}.\hfill \end{array}\right.$$

This map $\mathcal{T}$ meets the ${\left({\theta }_{\kappa }\right)}_{F_p}-$contraction condition

$$\theta \left(\mathcal{f}\left(đ(\mathcal{T}\mathrm{ʐ},\mathcal{T}\mathcal{x})\right)\right)\le {\left[\theta \left(\mathcal{f}\right(\mathcal{S}(\mathrm{ʐ},\mathcal{x})\left)\right)\right]}^{\kappa },$$

(25)

for all $\mathrm{ʐ},\mathcal{x}\in \mathrm{ʘ}$.

• Case (i): If $\mathrm{ʐ}\in ₱$ and $\mathcal{x}\in \mathcal{Q}$, we take $\mathrm{ʐ}={\displaystyle \frac{1}{7}}$ and $\mathcal{x}=1.5$; then, the distance $đ(\mathcal{T}x,\mathcal{T}y)=0.25$, with $\mathcal{S}\left({\displaystyle \frac{1}{7}},1.5\right)=1.357.$
• Case (ii): If both points are taken from set $\mathcal{Q}$, then we have two cases. In the first one, we select closely situated elements, whereas, in the second case, the chosen points are more distant from each other.

(a)

If $\mathrm{ʐ}=1.1$ and $\mathcal{x}=1$, then đ$(\mathcal{T}\mathrm{ʐ},\mathcal{T}\mathcal{x})=0.05$ and $\mathcal{S}(\mathrm{ʐ},\mathcal{x})=0.1.$

(b)

If $\mathrm{ʐ}=1$ and $\mathcal{x}=2$, then $\mathrm{đ}(\mathcal{T}\mathrm{ʐ},\mathcal{T}\mathcal{x})=0.5$ and $\mathcal{S}(\mathrm{ʐ},\mathcal{x})=1.$

Now, θ is defined as

$$\theta \left(\mathcal{x}\right)=e^{\mathcal{x}}.$$

For $\mathcal{f}\left(\mathcal{t}\right)=2\mathcal{t}$, with $\kappa >0.5$, we obtain inequality (25) in all cases.

Thus, all requirements mentioned in Theorem 3 are met. So, 1 is clearly a fixed point of $\mathcal{T}$.

The following analysis investigates the convergence behavior of the sequence by considering different initial guesses. The results presented in Table 2 clearly highlight the distinct convergence behaviors observed in each scenario.

Table 2. Convergence behavior of sequences with various initial guesses.

Sr. No.	${\mathrm{ʐ}}_0=2$	$1.8$	$1.6$	$1.4$	$1.2$
1	1.5	1.4	1.3	1.2	1.1
2	1.25	1.2	1.15	1.1	1.05
3	1.125	1.1	1.075	1.05	1.025
4	1.0625	1.05	1.0375	1.025	1.0125
5	1.03125	1.025	1.01875	1.00125	1.00625
6	1.01563	1.0125	1.00938	1.00625	1.00313
7	1.00781	1.00625	1.00468	1.00313	1.00157
8	1.00391	1.00313	1.00234	1.00157	1.00079
9	1.00196	1.00157	1.00117	1.00079	1.00040
⋮	⋮	⋮	⋮	⋮	⋮
16	1.00002	1.00001	1.00001	1	1
17	1.00001	1	1	1	1
18	1	1	1	1	1

The graph below illustrates the behavior of sequences generated from various initial guesses. Figure 3 illustrates the convergence trends of these sequences, offering valuable analytical insights.

Figure 3. Convergence behavior with various initial points.

6. Application to Equation of Motion

Fixed point theory has several applications, such as in the hybrid pneumatic–electric-driven climbing robot [25], which features bionic flexible feet and involves nonlinear control systems. Fixed point theory ensures stability and convergence within the neural control framework, facilitating proper limb-to-foot coordination and enhancing the robot’s adaptability and performance in climbing. Additionally, the proposed 3D reconstruction algorithm for bubble mesostructures [26] employs nonlinear imaging techniques, where fixed point theory ensures the stability and convergence of the reconstruction process. To demonstrate the practicality of our results, we apply the proposed generalized contraction to the equation of motion, a classical second-order differential equation in mechanics.

Consider the set $C[0,1]$, a family of continuous functions defined on $[0,1]$. Let $đ:C[0,1]\times C[0,1]\to \mathbb{R}$, defined by $đ(\mathfrak{u},\mathfrak{v})={sup}_{\mathcal{t}\in [0,1]}|\mathfrak{u}\left(\mathcal{t}\right)-\mathfrak{v}\left(\mathcal{t}\right)|$. It is widely acknowledged that $\left(C\right[0,1],đ)$ constitutes a complete metric space.

Now, let us turn our attention to the following problem.

Problem: Let us consider a scenario where a particle with unit mass m, at time zero, is at rest, such that $\mathrm{ʐ}=0$ and $\mathrm{𝓉}=0$. At $\mathrm{𝓉}=0$, a force f begins to act on the particle in the direction of the x-axis, causing its velocity to abruptly change from 0 to 1. Our objective is to determine the position of the particle at time $\mathrm{𝓉}$. This can be described by the equation of motion

$$m{\displaystyle \frac{d^2\mathrm{ʐ}}{d{\mathrm{𝓉}}^2}}=f(\mathrm{𝓉},\mathrm{ʐ}\left(\mathrm{𝓉}\right))$$

(26)

with the initial conditions $\mathrm{ʐ}\left(0\right)=0$ and ${\mathrm{ʐ}}^{\prime }\left(0\right)=1$, where $f:[0,1]\times \mathbb{R}\to \mathbb{R}$ is a continuous function.

We can define the Green’s function associated with this equation as

$$\mathcal{G}(\mathrm{𝓉},s)=\left\{\begin{array}{cc}\mathrm{𝓉}\hfill & \mathrm{if}0\le \mathrm{𝓉}\le s\le 1,\hfill \\ 2\mathrm{𝓉}-s\hfill & \mathrm{if}0\le s\le \mathrm{𝓉}\le 1.\hfill \end{array}\right.$$

Consider a function, $f:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$, satisfying the following conditions:

• $\left|f\right(\mathrm{𝓉},\mathcal{a})-f(\mathrm{𝓉},♭\left)\right|\le |\mathcal{a}-♭|$ for all $\mathrm{𝓉}\in [0,1]$ and $\mathcal{a},♭\in \mathbb{R}$ with $f(\mathcal{a},♭)\ge 0$.
• There is ${\mathrm{ʐ}}_0\in C[0,1]$, which gives $f({\mathrm{ʐ}}_0\left(\mathrm{𝓉}\right),\mathcal{T}{\mathrm{ʐ}}_0\left(\mathrm{𝓉}\right))\ge 0,\forall \mathrm{𝓉}\in [0,1]$, where $\mathcal{T}:C[0,1]\to C[0,1]$.

Now, we aim to establish the existence of a solution to the second-order differential equation.

Theorem 4.

Under the given assumptions (1)–(2), the equation of motion possesses a solution within the space $C^2\left([0,1]\right)$.

Proof. 

It is a well-established fact that finding a solution to Equation (26) is equivalent to finding $\mathrm{ʐ}\in C\left(\right[0,1\left]\right)$ that satisfies the associated integral equation.

$$\mathrm{ʐ}\left(\mathrm{𝓉}\right)={\int }_0^1\mathcal{G}(\mathrm{𝓉},s)f(s,\mathrm{ʐ}\left(s\right))ds,\mathrm{𝓉}\in [0,1]$$

Let $\mathcal{T}:C[0,1]\to C[0,1]$, given by

$$\mathcal{T}\left(\mathrm{ʐ}\left(\mathrm{𝓉}\right)\right)={\int }_0^1\mathcal{G}(\mathrm{𝓉},s)f(s,\mathrm{ʐ}\left(s\right))ds.$$

Now, suppose that $\mathrm{ʐ},\mathcal{x}\in C[0,1]$ such that $\mathcal{f}\left(\mathrm{ʐ}\right(\mathrm{𝓉}),\mathcal{x}(\mathrm{𝓉}\left)\right)\ge 0,\forall \mathrm{𝓉}\in [0,1]$. So,

$$\begin{array}{cc}\hfill đ\left(\mathcal{T}\mathrm{ʐ}\right(\mathrm{𝓉}),\mathcal{T}\mathcal{x}(\mathrm{𝓉}\left)\right)& =\left|\mathcal{T}\mathrm{ʐ}\left(\mathrm{𝓉}\right)-\mathcal{T}\mathcal{x}\left(\mathrm{𝓉}\right)\right|\hfill \\ \hfill & =\left|{\int }_0^1\mathcal{G}(\mathrm{𝓉},s)f(s,\mathrm{ʐ}\left(s\right))ds-{\int }_0^1\mathcal{G}(\mathrm{𝓉},s)f(s,\mathcal{x}\left(s\right))ds\right|\hfill \\ \hfill & \le {\int }_0^1\mathcal{G}(\mathrm{𝓉},s)ds\left|f(s,\mathrm{ʐ}(s\left)\right)-f(s,\mathcal{x}(s\left)\right)\right|\hfill \\ \hfill & \le {\int }_0^1\mathcal{G}(\mathrm{𝓉},s)ds\left|\mathrm{ʐ}\left(s\right)-\mathcal{x}\left(s\right)\right|\hfill \\ \hfill & \le sup\left\{{\int }_0^1\mathcal{G}(s,\mathrm{𝓉})ds\right\}\mathcal{S}(\mathrm{ʐ},\mathcal{x})\hfill \end{array}$$

Since ${\int }_0^1\mathcal{G}(\mathrm{𝓉},s)=-{\displaystyle \frac{{\mathrm{𝓉}}^2}{2}}+{\displaystyle \frac{1}{2}}\forall \mathrm{𝓉}\in [0,1]$, $sup\left\{{\int }_0^1\mathcal{G}(s,\mathrm{𝓉})ds\right\}={\displaystyle \frac{1}{2}}$. Hence, we obtain

$$đ(\mathcal{T}\mathrm{ʐ}\left(\mathrm{𝓉}\right),\mathcal{x}\left(\mathrm{𝓉}\right))\le {\displaystyle \frac{1}{2}}\mathcal{S}(\mathrm{ʐ},\mathcal{x})$$

Since $\mathcal{f}\left(\mathrm{𝓉}\right)=2\mathrm{𝓉}$, we obtain

$$\mathcal{f}\left(đ(\mathcal{T}\mathrm{ʐ}\left(\mathrm{𝓉}\right),\mathcal{T}\mathcal{x}\left(\mathrm{𝓉}\right))\right)\le \mathcal{f}\left({\displaystyle \frac{1}{2}}\mathcal{S}(\mathrm{ʐ},\mathcal{x})\right)$$

and $\theta \left(\mathcal{x}\right)=e^{\mathcal{x}}$; then, by performing a simple calculation, we obtain

$$\theta \left[\mathcal{f}\left(đ\left(\mathcal{T}\mathrm{ʐ}\right(\mathrm{𝓉}),\mathcal{T}\mathcal{x}(\mathrm{𝓉}\left)\right)\right)\right]\le {\left[\theta \left(\mathcal{f}\right(\mathcal{S}(\mathrm{ʐ},\mathcal{x})\left)\right)\right]}^{{\displaystyle \frac{1}{2}}}$$

So, we fulfill the ${\left({\theta }_{\kappa }\right)}_{F_P}$-contraction condition; then, according to Theorem 3, $\mathcal{T}$ has a unique fixed point, which is the solution to the equation of motion. □

7. Conclusions

We investigate the best coincidence proximity points in the context of rectangular metric space for multivalued maps. We employ the alternating distance function $\mathcal{f}$ and develop $\theta$-type generalized contractions for both single-valued and multi-valued non-self-maps to examine the existence of these best coincidence, best proximity, and fixed point results. To extend the applicability of our core findings, we give examples of single-valued and multi-valued mappings. Furthermore, we showcase the practical importance of our research by applying our findings to solve problems related to the equation of motion. These results have significant implications in both the theoretical and practical realms, assisting researchers in the study of fixed point theory and addressing difficulties in mechanical and engineering disciplines.

Future directions:

One can extend this concept towards the framework of $(\varphi -\psi )$-contractions [27], which represents a further generalization of several types of contraction mappings. Future research in fixed point theory can greatly enhance stability and performance across various fields. In human-exoskeleton control, fixed point methods can stabilize mode switching in multimodal systems [28]. For electrohydrodynamic flows, they can ensure the convergence of lattice Boltzmann simulations [29]. In structural mechanics, fixed point theory can improve the stability of finite element models for reinforced concrete under torsional load [30,31].