Integral Equations: New Solutions via Generalized Best Proximity Methods

Abstract

This paper introduces the concept of proximal $(\alpha ,F)$-contractions in $\mathcal{F}$-metric spaces. We establish novel results concerning the existence and uniqueness of best proximity points for such mappings. The validity of our findings is corroborated through a non-trivial example. Furthermore, we demonstrate the applicability of these results by proving the existence of solutions for Volterra integral equations related to population growth models. This approach not only extends best proximity theory, but also paves the way for further research in applied mathematics and beyond.

1. Introduction

The renowned Banach contraction principle [1], established by Stefan Banach in 1922, is a pioneer result in fixed point theory. Building upon this foundation, Wardowski [2] provided an innovative form of contraction known as the F-contraction and concurrently developed a fresh fixed point theorem, effectively broadening the scope beyond the constraints of the Banach fixed point theorem. This innovative approach has significantly contributed to the understanding and application of contraction mappings in various mathematical contexts. Ali et al. [3] utilized the concept of F-contractions and proved fixed point results for set-valued mappings. Subsequently, Sgroi et al. [4] established some new fixed point results for F-contractions and explored their applications in solving certain functional and integral equations.

In a ground-breaking move, Czerwik [5] introduced the concept of b-metric spaces (b-MSs), extending the reach of fixed point theorems beyond classical metric spaces (MSs). His relaxed “b-metric inequality” replaced the familiar triangle inequality, opening a door to analyze previously excluded diverse distance structures. Building upon this innovation, Jleli et al. [6] further pushed the boundaries with the novel $\mathcal{F}$-metric spaces ($\mathcal{F}$-MSs), a flexible framework encompassing both classical MSs and b-MSs. This broader canvas promises rich results in fixed point theory across numerous mathematical fields, including functional analysis and topology.

Moving beyond fixed points, Basha [7] pioneered the concept of best proximity points in the 1960s, offering a broader lens for analyzing sets in metric spaces and the functions. This powerful tool has found applications in diverse fields, from optimization and approximation theory to differential equations. Building on this foundation, Eldred et al. [8] delved deeper, establishing conditions for the existence and uniqueness of best proximity points for various mappings. This framework proved pivotal in fields like optimization, differential equations (both ordinary and fractional), and, more recently, homotopy theory through the work of Şahina [9]. Jain et al. [10] further cemented the connection by applying best proximity results to differential equations. Khan et al. [11] established some best proximity point results for new generalized proximal contractions in the background of metric spaces. Recently, Lateef [12] obtained best proximity point results for ($\alpha$-$\psi$)-contractions in the framework of $\mathcal{F}$-MSs and generalized some well-known results given in classical metric spaces. A deeper exploration of these applications can be found in [13,14,15,16,17].

In spite of this, best proximity point theory stands as a powerful mathematical framework with wide-ranging applications in various scientific disciplines. In the context of integral equations governing population growth models, best proximity point theory offers a versatile tool to explore and understand the dynamics of evolving populations. Furthermore, understanding the fixed points of integral equations allows for predictions about the long-term behavior of populations and the effects of perturbations, contributing to the field of ecological modeling and conservation biology. For more details on the applications of integral equations to population growth models, we encourage the readers to refer to [18,19].

In the present research article, we introduce the notion of proximal $(\alpha ,F)$-contraction in the context of an $\mathcal{F}$-MS and prove best proximity point theorems for the aforementioned contractions. Moreover, we furnish a non-trivial example to show the validity of the obtained results. To demonstrate the practical applications, we investigate the solution for Volterra integral equations related to population growth models.

2. Preliminaries

In fixed point theory, the first and pioneer theorem is the following Banach contraction principle.

Theorem 1

([1]). Let $\mathcal{H}:(\mathcal{Q},d)\to (\mathcal{Q},d)$ be a mapping defined on a complete MS $(\mathcal{Q},d).$ If there exists $\lambda \in [0,1)$, such that

$$d(\mathcal{H}x,\mathcal{H}y)\le \lambda d(x,y),$$

for all $x,y\in \mathcal{Q}$, then $\mathcal{H}$ possesses a unique fixed point.

Wardowski [2] introduced an innovative form of contraction known as F-contraction, unveiling numerous novel fixed point theorems tailored to this contraction type in the domain of generalized metric spaces.

Consider $\Psi$ as a collection of functions $F:{\mathbb{R}}^{+}\to \mathbb{R}$ that satisfy.

($F_1$) 

F(x) $\le F$(y) for $x\le y$,

($F_2$) 

for $\left\{x_n\right\}\subseteq {\mathbb{R}}^{+}$, ${lim}_{n\to \infty }F\left(x_n\right)=-\infty$⟺ ${lim}_{n\to \infty }\left(x_n\right)=0;$

($F_3$) 

there exists $r\in \left(0,1\right)$, such that $\underset{x\to 0^{+}}{lim}{x}^{r}F\left(x\right)=0$.

Definition 1

([2]). A function $\mathcal{H}:\mathcal{Q}\to \mathcal{Q}$ is defined as an F-contraction when there exist a function F satisfying the conditions ($F_1$)–($F_3$) and some constant $\tau >0$, such that for every $x,y\in \mathcal{Q}$,

$$\begin{array}{c}d(\mathcal{H}x,\mathcal{H}y)>0⟹\tau +F\left(d\right(\mathcal{H}x,\mathcal{H}y\left)\right)\le F\left(d\right(x,y\left)\right).\hfill \end{array}$$

(1)

Theorem 2

([2]). Given a complete MS $(\mathcal{Q},d)$ and an F-contraction mapping $\mathcal{H}:\mathcal{Q}\to \mathcal{Q}$, a unique fixed point for $\mathcal{H}$ is guaranteed.

Czerwik [5] expanded the notion of the conventional MS in this fashion.

Definition 2

([5]). Let $\mathcal{Q}\ne Ø$ and $s\ge 1$. A mapping $d:\mathcal{Q}\times \mathcal{Q}\to$ $[0,\infty )$ is defined as a b-metric if it satisfies these assertions:

$\left(b1\right):d(x,y)=0$⟺$x=y;$

$\left(b2\right):d(x,y)=d(y,x),$

$\left(b3\right):d(x,)\le s\left[d\right(x,y)+d(y,\left)\right],$

For all $x,y,\in \mathcal{Q}.$

The pair $(\mathcal{Q},d)$ is thereby designated as a b-MS.

Recently, Jleli et al. [6] introduced a compelling extension of an MS utilizing this technique.

Let $\mathcal{F}$ represent the ensemble of continuous mappings $⋏:(0,+\infty )\to \mathbb{R}$ that only fulfill the conditions ($F_1$) and ($F_2$).

Definition 3

([6]). Consider $\mathcal{Q}\ne Ø$ and let $d:\mathcal{Q}\times \mathcal{Q}\to [0,+\infty )$ be a continuous mapping. Suppose that there exists $(⋏,\beta )\in \mathcal{F}\times [0,+\infty )$, such that

(D₁)

$(x,y)\in \mathcal{Q}\times \mathcal{Q}$, $d(x,y)=0$ if and only if $x=y$.

(D₂)

$d(x,y)=d(y,x)$, for all $(x,y)\in \mathcal{Q}\times \mathcal{Q}.$

(D₃)

For every $(x,y)\in \mathcal{Q}\times \mathcal{Q}$, for every $N\in \mathbb{N}$, $N\ge 2$, and for every ${\left(u_i\right)}_{i=1}^N\subset \mathcal{Q},$ with $(u_1,u_N)=(x,y)$, we have

$$d(x,y)>0\Rightarrow ⋏\left(d(x,y)\right)\le ⋏\left(\sum _{i=1}^{N-1}d(u_i,u_{i+1})\right)+\beta .$$

Under these conditions, $(\mathcal{Q},d)$ is designated as an $\mathcal{F}$-MS.

Example 1

([6]). The mapping $d:\mathbb{R}\times \mathbb{R}\to [0,+\infty )$

$$d(x,y)=\left\{\begin{array}{c}{(x-y)}^{2}if(x,y)\in [0,3]\times [0,3]\\ |x-y|if(x,y)\notin [0,3]\times [0,3]\end{array}\right.$$

with $⋏\left(\mathsf{ı}\right)=ln\left(\mathsf{ı}\right)$ and $\beta =ln\left(3\right)$, is an $\mathcal{F}$-metric.

Definition 4

([6]). Consider $(\mathcal{Q},d)$ as an $\mathcal{F}$-MS.

(i)

A sequence $\left\{x_n\right\}\subseteq$ $\mathcal{Q}$ is said to be convergent to $x\in \mathcal{Q}$ if $x_n\to x$ regarding d.

(ii)

A sequence $\left\{x_n\right\}\subseteq$ $\mathcal{Q}$ is represented as Cauchy, if

$$\underset{n,m\to \infty }{lim}d(x_n,x_m)=0.$$

(iii)

If every Cauchy sequence in $\mathcal{Q}$ converges to an element in $\mathcal{Q}$, then $(\mathcal{Q},d)$ is recognized as complete.

Theorem 3

([6]). Let $(\mathcal{Q},d)$ be an $\mathcal{F}$-MS and $\mathcal{H}:\mathcal{Q}\to \mathcal{Q}$ be a self-mapping. Assume that the subsequent conditions are satisfied:

(i)

$(\mathcal{Q},d)$ is complete,

(ii)

there exists $\lambda \in (0,1)$, such that

$$d\left(\mathcal{H}\right(x),\mathcal{H}(y\left)\right)\le \lambda d(x,y).$$

Then, $\mathcal{H}$ possesses a unique fixed point $x*\in \mathcal{Q}$. Furthermore, for any $x_0\in \mathcal{Q}$, the sequence $\left\{x_n\right\}\subset \mathcal{Q}$ defined by

$$x_{n+1}=\mathcal{H}\left(x_n\right),n\in \mathbb{N},$$

is convergent to $x*$.

Drawing inspiration from the contributions of Lateef [12], we denote non-empty subsets of $\mathcal{Q}$ by $N\left(\mathcal{Q}\right)$ and closed subsets of $\mathcal{Q}$ by $Cl\left(\mathcal{Q}\right)$.

Definition 5

([12]). Let $(\mathcal{Q},d)$ is $\mathcal{F}$-MS and $\mathcal{Y}$,$\mathcal{Z}$ $\in N\left(\mathcal{Q}\right)$. A point $x*$ $\in \mathcal{Y}$ is said to be the best proximity point of $\mathcal{H}:\mathcal{Y}\to \mathcal{Z}$ if the following inequality holds

$$d(x*,\mathcal{H}x*)\le d(\mathcal{Y},\mathcal{Z}),$$

where $d(\mathcal{Y},\mathcal{Z})$ is $\mathcal{F}$-distance between the sets $\mathcal{Y}$ and $\mathcal{Z}$ which is defined as follows

$$d(\mathcal{Y},\mathcal{Z})=\underset{\begin{array}{c}\omega \in \mathcal{Y}\\ \varrho \in \mathcal{Z}\end{array}}{inf}d\left(\omega ,\varrho \right).$$

Definition 6

([12]). Let $(\mathcal{Q},d)$ be $\mathcal{F}$-MS, and $\mathcal{Y}$,$\mathcal{Z}$ $\in N\left(\mathcal{Q}\right)$, and $d(\mathcal{Y},\mathcal{Z})$ is $\mathcal{F}$-distance between $\mathcal{Y}$ and $\mathcal{Z}.$ Now define ${\mathcal{Y}}_0$ and ${\mathcal{Z}}_0$ by

$$\begin{array}{ccc}\hfill {\mathcal{Y}}_0& =& \{x\in \mathcal{Y}:\mathit{there}\mathit{exists}u\in \mathcal{Z}\mathit{such}\mathit{that}d(x,u)=d(\mathcal{Y},\mathcal{Z}\left)\right\}\hfill \\ \hfill {\mathcal{Z}}_0& =& \{u\in \mathcal{Z}:\mathit{there}\mathit{exists}x\in \mathcal{Y}\mathit{such}\mathit{that}d(x,u)=d(\mathcal{Y},\mathcal{Z}\left)\right\}.\hfill \end{array}$$

The couple $(\mathcal{Y},\mathcal{Z})$ is considered to exhibit the property P if ${\mathcal{Y}}_0\ne Ø$ and

$$x,y\in {\mathcal{Y}}_0,u,v\in {\mathcal{Z}}_0,d(x,u)=d(y,v)=d(\mathcal{Y},\mathcal{Z})⟹d(x,y)=d(u,v).$$

Definition 7

([12]). Let $(\mathcal{Q},d)$ be $\mathcal{F}$-MS and $\mathcal{Y}$,$\mathcal{Z}$ $\in N\left(\mathcal{Q}\right)$. A mapping $\mathcal{H}:\mathcal{Y}\to \mathcal{Z}$ is said to be α-proximal admissible if there exists $\alpha :\mathcal{Y}\times \mathcal{Y}\to [0,\infty )$, such that

$$\left.\begin{array}{ccc}\alpha (x,y)& \ge & 1\\ d(u,\mathcal{H}x)& =& d(\mathcal{Y},\mathcal{Z})\\ d(v,\mathcal{H}y)& =& d(\mathcal{Y},\mathcal{Z})\end{array}\right\}\Rightarrow \alpha (u,v)\ge 1,$$

where $x,y,u,v\in \mathcal{Y}.$

Lateef [12] established the subsequent best proximity theorem.

Theorem 4

([12]). Consider $(\mathcal{Q},d)$ as a complete $\mathcal{F}$-MS and $\mathcal{Y}$,$\mathcal{Z}$ $\in Cl\left(\mathcal{Q}\right)$, such that ${\mathcal{Y}}_0\ne ⌀$. Assume the existence of the mapping $\mathcal{H}:\mathcal{Y}\to \mathcal{Z}$ and the comparison functions $\alpha :\mathcal{Y}\times \mathcal{Y}\to [0,\infty )$ and $\psi :[0,\infty )\to [0,\infty ),$ satisfying the following conditions:

(i)

$$\alpha (x,y)d(\mathcal{H}x,\mathcal{H}y)\le \psi \left(d(x,y)\right),$$

(ii)

the mapping $\mathcal{H}:\mathcal{Y}\to \mathcal{Z}$ is α-proximal admissible mapping,

(iii)

$\mathcal{H}\left({\mathcal{Y}}_0\right)\subseteq {\mathcal{Z}}_0$ and $(\mathcal{Y},\mathcal{Z})$ satisfies property P;

(iv)

there exists $x_0,x_1\in {\mathcal{Y}}_0$, such that

$$d(x_1,\mathcal{H}{x}_0)=d(\mathcal{Y},\mathcal{Z}),\mathit{and}\alpha (x_0,x_1)\ge 1,$$

(v)

either $\mathcal{H}$ is continuous or for $\left\{x_n\right\}\subseteq$ $\mathcal{Y}$ satisfying $\alpha (x_n,x_{n+1})$ $\ge 1$ and $x_n\to x$ $\in \mathcal{Y}$ as $n\to \infty$, then there exists a subsequence $\left\{x_{n\left(k\right)}\right\}$ of $\left\{x_n\right\}$, such that $\alpha (x_{n\left(k\right)},x)\ge 1,$ for all k. Then, $\mathcal{H}$ has a best proximity point.

3. Results and Discussion

Throughout the research article, we consistently denote the complete $\mathcal{F}$-MS as $(\mathcal{Q},d)$.

Definition 8.

Let $\mathcal{Y}$,$\mathcal{Z}$ $\in N\left(\mathcal{Q}\right).$ A mapping $\mathcal{H}:\mathcal{Y}\to \mathcal{Z}$ is defined as a proximal $\left(\alpha ,F\right)$-contraction if the functions $\alpha :\mathcal{Y}\times \mathcal{Y}\to [0,\infty ),$ $F\in \Psi$ and the constants $\tau >0$ exist, such that

$$\tau +\alpha (x,y)F\left(d(\mathcal{H}x,\mathcal{H}y)\right)\le F\left(d(x,y)\right)$$

(2)

for all $x,y\in \mathcal{Y}$.

Theorem 5.

Let $\mathcal{Y}$,$\mathcal{Z}$ $\in Cl\left(\mathcal{Q}\right)$, such that ${\mathcal{Y}}_0\ne ⌀$. Let $\alpha :\mathcal{Y}\times \mathcal{Y}\to [0,\infty )$ and $F\in \Psi .$ Consider $\mathcal{H}:\mathcal{Y}\to \mathcal{Z}$ as a proximal $\left(\alpha ,F\right)$-contraction satisfying the subsequent conditions:

(i)

$\mathcal{H}$ is α-proximal admissible mapping,

(ii)

$\mathcal{H}\left({\mathcal{Y}}_0\right)\subseteq {\mathcal{Z}}_0$ and $(\mathcal{Y},\mathcal{Z})$ satisfies property P;

(iii)

there exists $x_0,x_1\in {\mathcal{Y}}_0$, such that

$$d(x_1,\mathcal{H}{x}_0)=d(\mathcal{Y},\mathcal{Z}),\mathit{and}\alpha (x_0,x_1)\ge 1.$$

(iv)

$\mathcal{H}$ is continuous.

Then, $\mathcal{H}$ has a best proximity point.

Proof. 

According to assumption (iii), there exists $x_0,x_1\in {\mathcal{Y}}_0$ such that

$$d(x_1,\mathcal{H}{x}_0)=d(\mathcal{Y},\mathcal{Z}),\alpha (x_0,x_1)\ge 1.$$

(3)

As $\mathcal{H}\left({\mathcal{Y}}_0\right)\subseteq {\mathcal{Z}}_0,$ there exists $x_2\in {\mathcal{Y}}_0$ such that

$$d(x_2,\mathcal{H}{x}_1)=d(\mathcal{Y},\mathcal{Z}).$$

Now, we have $\alpha (x_0,x_1)\ge 1,d(x_1,\mathcal{H}{x}_0)=d(\mathcal{Y},\mathcal{Z})$ and $d(x_2,\mathcal{H}{x}_1)=d(\mathcal{Y},\mathcal{Z}).$ Since $\mathcal{H}$ is $\alpha$-proximal admissible, so we get $\alpha (x_1,x_2)\ge 1.$ Hence,

$$d(x_2,\mathcal{H}{x}_1)=d(\mathcal{Y},\mathcal{Z}),\alpha (x_1,x_2)\ge 1$$

(4)

Again, as $\mathcal{H}\left({\mathcal{Y}}_0\right)\subseteq {\mathcal{Z}}_0,$ there exists $x_3\in {\mathcal{Y}}_0$, such that

$$d(x_3,\mathcal{H}{x}_2)=d(\mathcal{Y},\mathcal{Z}).$$

Now, we have $\alpha (x_1,x_2)\ge 1,d(x_2,\mathcal{H}{x}_1)=d(\mathcal{Y},\mathcal{Z})$ and $d(x_3,\mathcal{H}{x}_2)=d(\mathcal{Y},\mathcal{Z}).$ As $\mathcal{H}$ is $\alpha$-proximal admissible, we obtain $\alpha (x_2,x_3)\ge 1.$ Hence,

$$d(x_3,\mathcal{H}{x}_2)=d(\mathcal{Y},\mathcal{Z}),\alpha (x_2,x_3)\ge 1.$$

(5)

Applying the inductive method, we can generate a sequence $\left\{x_n\right\}\subset {\mathcal{Y}}_0$, such that

$$d(x_{n+1},\mathcal{H}{x}_n)=d(\mathcal{Y},\mathcal{Z}),\alpha (x_n,x_{n+1})\ge 1$$

(6)

for all $n\in$ $\mathbb{N}\cup \left\{0\right\}.$ Suppose $x_k=x_{k+1}$ for some k. Referring to (6), we obtain

$$d(x_k,\mathcal{H}{x}_k)=d(x_{k+1},\mathcal{H}{x}_k)=d(\mathcal{Y},\mathcal{Z})$$

i.e., $x_k$ achieves the best proximity point to $\mathcal{H}$. Hence, we assert that $d(x_{n-1},x_n)>0,$ for all $n\in$ $\mathbb{N}\cup \left\{0\right\}.$ From hypothesis (ii) and (6), we infer that

$$d(x_n,x_{n+1})=d(\mathcal{H}{x}_{n-1},\mathcal{H}{x}_n),$$

for all $n\in$ $\mathbb{N}\cup \left\{0\right\}.$ So, by (2), we have

$$\begin{array}{ccc}\hfill \tau +F\left(d(x_n,x_{n+1})\right)& \le & \tau +\alpha (x_n,x_{n+1})F\left(d(x_n,x_{n+1})\right)\hfill \\ & =& \tau +\alpha (x_n,x_{n+1})F\left(d(\mathcal{H}{x}_{n-1},\mathcal{H}{x}_n)\right)\hfill \\ & \le & F\left(d(x_{n-1},x_n)\right)\hfill \end{array}$$

(7)

for all $n\ge 0.$ This additionally implies that

$$\begin{array}{ccc}\hfill F\left(d(x_n,x_{n+1})\right)& \le & F\left(d(x_{n-1},x_n)\right)-\tau \hfill \\ & \le & F\left(d(x_{n-2},x_{n-1})\right)-2\tau \hfill \\ \hfill & \le & \dots \le F\left(d(x_0,x_1)\right)-n\tau \hfill \end{array}$$

(8)

for all $n\in \mathbb{N}$. Considering F as a member of the set, allowing n to approach infinity in the equation results in

$$\underset{n\to \infty }{lim}F\left(d(x_n,x_{n+1})\right)=-\infty$$

(9)

which yields that

$$\underset{n\to \infty }{lim}d(x_n,x_{n+1})=0$$

(10)

by hypothesis ($F_2$). Now, by ($F_3$), there exists $r\in \left(0,1\right)$, such that

$$\underset{n\to \infty }{lim}{\left[d(x_n,x_{n+1})\right]}^{r}F\left(d(x_n,x_{n+1})\right)=0.$$

By the inequality (8), we have

$$\begin{array}{ccc}& & {\left[d(x_n,x_{n+1})\right]}^{r}F\left(d(x_n,x_{n+1})\right)-{\left[d(x_n,x_{n+1})\right]}^{r}F\left(d(x_0,x_1)\right)\hfill \\ & \le & {\left[d(x_n,x_{n+1})\right]}^r\left(F\left(d(x_0,x_1)\right)-n\tau \right)-{\left[d(x_n,x_{n+1})\right]}^{r}F\left(d(x_0,x_1)\right)\hfill \\ & \le & -n\tau {\left[d(x_n,x_{n+1})\right]}^r\le 0.\hfill \end{array}$$

(11)

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

$$\underset{n\to \infty }{lim}n{\left[d(x_n,x_{n+1})\right]}^r=0.$$

(12)

Hence, $\underset{n\to \infty }{lim}{n}^{\frac{1}{r}}d(x_n,x_{n+1})=0$ and there exists $n_1\in \mathbb{N}$ such that $n^{\frac{1}{r}}d(x_n,x_{n+1})\le 1$ for all $n\ge n_1.$ So, we have

$$d(x_n,x_{n+1})\le \frac{1}{n^{1/r}}$$

(13)

Now, by (13) for $n_1<n<m,$ we have

$$d(x_n,x_m)\le \sum _{i=n}^{m-1}d(x_i,x_{i+1})\le \sum _{i=n}^{m-1}\frac{1}{i^{1/r}}\le \sum _{i=1}^{\infty }\frac{1}{i^{1/r}}.$$

(14)

Let $ϵ>0$ be fixed and $(⋏,\beta )\in \mathcal{F}\times [0,+\infty )$ be such that (D₃) is satisfied. By ($F_2$), $\delta >0$ exists, such that

$$0<\mathsf{ı}<\delta ⟹⋏\left(\mathsf{ı}\right)<⋏\left(\epsilon \right)-y.$$

(15)

for $m>n>n_1.$ Employing ($D_3$), along with references (14) and (15), we deduce that $d(x_n,x_m)>0,$ where $m>n>n_1$, thereby implying

$$\begin{array}{ccc}\hfill ⋏\left(d(x_m,x_n)\right)& \le & ⋏\left(\sum _{i=n}^{m-1}d(x_i,x_{i+1})\right)+\beta \hfill \\ & \le & ⋏\left(\sum _{i=1}^{\infty }\frac{1}{i^{1/r}}\right)+\beta <⋏\left(ϵ\right).\hfill \end{array}$$

From condition ($F_1$), it follows that the distance between successive terms is $d(x_m,x_n)<ϵ,$ $m>n>n_1.$ This confirms that the sequence {$x_n$} is Cauchy. As $(\mathcal{Q},d)$ is complete and $\mathcal{Y}$ is closed, there exists $x*\in \mathcal{Y}$, such that $\left\{x_n\right\}$ is convergent to $x*$, i.e.,

$$\underset{n\to \infty }{lim}d(x_n,x*)=0.$$

(16)

Next, as $\mathcal{H}$ is a continuous, we can conclude that $\mathcal{H}{x}_n\to \mathcal{H}x*$ as $n\to \infty .$ By leveraging the fact that d is continuous, we obtain

$$d(\mathcal{Y},\mathcal{Z})=d(x_{n+1},\mathcal{H}{x}_n)\to d(x*,\mathcal{H}x*)$$

as $n\to \infty .$ Hence, $d(x^{*},\mathcal{H}x*)$ $=d(\mathcal{Y},\mathcal{Z})$. □

Theorem 6.

Let $\mathcal{Y}$, $\mathcal{Z}$ $\in Cl\left(\mathcal{Q}\right)$, such that ${\mathcal{Y}}_0\ne ⌀$. Let $\alpha :\mathcal{Y}\times \mathcal{Y}\to [0,\infty )$ and $F\in \Psi .$ Suppose that $\mathcal{H}:\mathcal{Y}\to \mathcal{Z}$ is proximal ($\alpha ,F)$-contraction, let it fulfill the subsequent criteria:

(i)

$\mathcal{H}$ is α-proximal admissible mapping,

(ii)

$\mathcal{H}\left({\mathcal{Y}}_0\right)\subseteq {\mathcal{Z}}_0$ and $(\mathcal{Y},\mathcal{Z})$ satisfies the property P;

(iii)

there exists $x_0,x_1\in {\mathcal{Y}}_0$ such that

$$d(x_1,\mathcal{H}{x}_0)=d(\mathcal{Y},\mathcal{Z}),and\alpha (x_0,x_1)\ge 1.$$

(iv)

If $\left\{x_n\right\}\subseteq$ $\mathcal{Y}$ is a sequence satisfying $\alpha (x_n,x_{n+1})$ $\ge 1$ and $x_n\to x$ $\in \mathcal{Y}$ as $n\to \infty$, then there exists a subsequence $\left\{x_{n\left(k\right)}\right\}$ of $\left\{x_n\right\}$ such that $\alpha (x_{n\left(k\right)},x)\ge 1,$ for all k.

Then, $\mathcal{H}$ has a best proximity point.

Proof. 

To corroborate the outcomes presented in Theorem 5, there is a sequence $\left\{x_n\right\}\subset \mathcal{Y}$ for which the inequality (2) holds true, and $x_n$ $\to x*$ as $n\to \infty ,$ i.e.,

$$\underset{n\to \infty }{lim}d(x_n,x*)=0.$$

From assumption (iii), there is $\left\{x_{n\left(k\right)}\right\}$ of $\left\{x_n\right\}$ with $\alpha (x_{n\left(k\right)},x*)\ge 1,$ for all $k.$ We affirm that $\mathcal{H}{x}_{n\left(k\right)}\to \mathcal{H}x*$ as $k\to \infty .$ Using (2), we obtain

$$\begin{array}{ccc}\hfill \tau +F\left(d(\mathcal{H}{x}_{n\left(k\right)},\mathcal{H}x*)\right)& \le & \tau +\alpha (x_{n\left(k\right)},x*)F\left(d(\mathcal{H}{x}_{n\left(k\right)},\mathcal{H}x*)\right)\hfill \\ & \le & F\left(d(x_{n\left(k\right)},x*)\right)\hfill \end{array}$$

which implies

$$\begin{array}{ccc}\hfill F\left(d(\mathcal{H}{x}_{n\left(k\right)},\mathcal{H}x*)\right)& \le & F\left(d(x_{n\left(k\right)},x*)\right)-\tau \hfill \\ & <& F\left(d(x_{n\left(k\right)},x*)\right)\hfill \end{array}$$

as $\tau >0.$ Consequently, in accordance with ($F_1$), we obtain

$$d(\mathcal{H}{x}_{n\left(k\right)},\mathcal{H}x*)<d(x_{n\left(k\right)},x*).$$

Letting k approaches to infinity and leveraging the continuity of d, we obtain

$$d(\mathcal{Y},\mathcal{Z})=d(x_{n\left(k\right)+1},\mathcal{H}{x}_{n\left(k\right)})\to d(x^{*},\mathcal{H}x*)$$

as $n\to \infty .$ Therefore,

$$d(x*,\mathcal{H}x*)=d(\mathcal{Y},\mathcal{Z})$$

which completes the proof of the theorem. □

Definition 9.

Let $\mathcal{H}:\mathcal{Y}\to \mathcal{Z}$ and $\alpha :\mathcal{Y}\times \mathcal{Y}\to [0,\infty )$. The mapping $\mathcal{H}$ is said to be $(\alpha ,d)$-regular if for all $(x,y)\in {\alpha }^{-1}[0,1)$, there exists $\varkappa \in {\mathcal{Y}}_0$, such that

$$\alpha (x,y)\ge 1\mathit{and}\alpha (y,\varkappa )\ge 1.$$

Theorem 7.

Suppose $\mathcal{H}$ is $(\alpha ,d)$-regular, in conjunction with the assertions outlined in Theorem 5 (resp. Theorem 6). In such a scenario, we can infer the existence of a unique point $x*\in \mathcal{Y}$ that satisfies the inequality $d(x*,\mathcal{H}x*)\le d(\mathcal{Y},\mathcal{Z}).$

Proof. 

As proven by the Theorem 5, the collection of points exhibiting the best proximity to $\mathcal{H}$ is guaranteed to contain at least one element, indicating the presence of a best proximity point $x*\in {\mathcal{Y}}_0$. Assuming another best proximity point $y*\in {\mathcal{Y}}_0$ of $\mathcal{H}$, i.e.,

$$d(\mathcal{H}x*,x*)=d(\mathcal{H}y*,y*)=d(\mathcal{Y},\mathcal{Z}).$$

(17)

Employing the hypothesis (ii) of Theorem 5 and (17), we obtain that

$$d(\mathcal{H}x*,\mathcal{H}y*)=d(x*,y*).$$

(18)

We examine two possible cases:

Case 1. Assuming $\alpha (x*,y*)\ge 1$ and utilizing (17), we deduce that

$$\begin{array}{ccc}\hfill \tau +F\left(d(x*,y*)\right)& =& \tau +F\left(d(\mathcal{H}x*,\mathcal{H}y*)\right)\hfill \\ & \le & \tau +\alpha (x*,y*)F\left(d(\mathcal{H}x*,\mathcal{H}y*)\right)\hfill \\ & \le & F\left(d(x*,y*)\right)\hfill \end{array}$$

which implies

$$\begin{array}{ccc}\hfill F\left(d(x*,y*)\right)& \le & F\left(d(x*,y*)\right)-\tau \hfill \\ & <& F\left(d(x*,y*)\right)\hfill \end{array}$$

as $\tau >0,$ which implies by ($F_1$) that $d(x*,y*)<d(x*,y*),$ is a contradiction. Thus, $x*=y*.$

Case 2. If $\alpha (x*,y*)<1.$

By supposition, ${\varkappa }_0\in {\mathcal{Y}}_0$ exists, such that $\alpha (x*,{\varkappa }_0)\ge 1$ and $\alpha (y*,{\varkappa }_0)\ge 1$. As $\mathcal{H}\left({\mathcal{Y}}_0\right)\subseteq {\mathcal{Z}}_0$, ${\varkappa }_1\in {\mathcal{Y}}_0$ such that

$$d({\varkappa }_1,\mathcal{H}{\varkappa }_0)=d(\mathcal{Y},\mathcal{Z}).$$

Now, we have

$$\alpha (x*,{\varkappa }_0)\ge 1$$

$$d(x*,\mathcal{H}x*)=d(\mathcal{Y},\mathcal{Z}),$$

$$d({\varkappa }_1,\mathcal{H}{\varkappa }_0)=d(\mathcal{Y},\mathcal{Z}).$$

As $\mathcal{H}$ is $\alpha$-proximal admissible, so we have $\alpha (x*,{\varkappa }_1)$ $\ge 1$. Hence,

$$d({\varkappa }_1,\mathcal{H}{\varkappa }_0)=d(\mathcal{Y},\mathcal{Z})\mathrm{and}\alpha (x*,{\varkappa }_1)\ge 1.$$

Adopting this methodology, we can systematically generate a sequence $\left\{{\varkappa }_n\right\}$ in ${\mathcal{Y}}_0$, such that

$$d({\varkappa }_{n+1},\mathcal{H}{\varkappa }_n)=d(\mathcal{Y},\mathcal{Z})\mathrm{and}\alpha (x*,{\varkappa }_n)\ge 1$$

(19)

for all $n\ge 0.$ Due to the implications of hypothesis (ii) of Theorem 5 and (19), this can be deduced that

$$d({\varkappa }_{n+1},x*)=d(\mathcal{H}{\varkappa }_n,\mathcal{H}x*)$$

(20)

for all $n\in \mathbb{N}\cup \left\{0\right\}.$ As $\mathcal{H}$ is proximal $\left(\alpha ,F\right)$-contraction, we obtain

$$\begin{array}{ccc}\hfill \tau +F\left(d({\varkappa }_{n+1},x*)\right)& =& \tau +F\left(d(\mathcal{H}{\varkappa }_n,\mathcal{H}x*)\right)\hfill \\ & \le & \tau +\alpha ({\varkappa }_n,x*)F\left(d(\mathcal{H}{\varkappa }_n,\mathcal{H}x*)\right)\hfill \\ & \le & F\left(d({\varkappa }_n,x*)\right)\hfill \end{array}$$

for all $n\ge 0.$ Hence, we obtain

$$F\left(d({\varkappa }_{n+1},x*)\right)\le F\left(d({\varkappa }_n,x*)\right)-\tau \le \dots \le F\left(d({\varkappa }_0,x*)\right)-n\tau$$

(21)

Taking limit as $n\to \infty$ in (21), we have

$$\underset{n\to \infty }{lim}F\left(d({\varkappa }_{n+1},x*)\right)=-\infty ,$$

then by ($F_2$), we have

$$\underset{n\to \infty }{lim}d({\varkappa }_{n+1},x*)=0,$$

which yields that ${\varkappa }_{n+1}$ converges to $x*$ as n approaches infinity, establishing {${\varkappa }_n\}\to x*.$ Consequently, in both of the analyzed scenarios, the sequence {${\varkappa }_n\}$ converges to $x*$ as $n\to \infty$. Analogously, we can show that {${\varkappa }_n\}\to y*$ as $n\to \infty$. As the limit is unique, we deduce that $x*=y*$.

We now provide an example to illustrate the relevance and soundness of our findings. □

Example 2.

Let $\mathcal{Q}$ = $\mathbb{R}$ and $d:\mathcal{Q}\times \mathcal{Q}\to [0,+\infty )$ be defined by $d(x,y)=\left|x-y\right|.$ In thia case, ($\mathcal{Q}$,d) constitutes a complete $\mathcal{F}$-MS. Let us take two closed subsets of $\mathcal{Q}$, denoted as $\mathcal{Y}=\left[-1,1\right]\cup \left\{5,10\right\}$ and $\mathcal{Z}=\left[-\frac{1}{2},\frac{1}{2}\right]\cup \left\{5,10\right\}.$ The compactness of $\mathcal{Z}$ is verified, indicating its approximate compactness concerning $\mathcal{Y}$. Define the mapping $\mathcal{H}:\mathcal{Y}\to \mathcal{Z}$ as

$$\mathcal{H}x=\left\{\begin{array}{c}\frac{1}{2}x,\mathit{if}x\in \left[-1,1\right],\\ 0,\mathit{if}x=5,\\ 0,\mathit{if}x=10.\end{array}\right.$$

Evidently, $d(\mathcal{Y},\mathcal{Z})=0$ and

$$\begin{array}{ccc}\hfill {\mathcal{Y}}_0& =& \left\{x\in \mathcal{Y}:d(x,y)=d(\mathcal{Y},\mathcal{Z})=0\mathit{for}\mathit{some}y\in \mathcal{Z}\right\}=\mathcal{Z},\hfill \\ \hfill {\mathcal{Z}}_0& =& \left\{y\in \mathcal{Z}:d(x,y)=d(\mathcal{Y},\mathcal{Z})=0\mathit{for}\mathit{some}y\in \mathcal{Y}\right\}=\mathcal{Z}.\hfill \end{array}$$

Clearly, we have that $\mathcal{H}\left({\mathcal{Y}}_0\right)\subseteq {\mathcal{Z}}_0.$ Now, we define $\alpha :\mathcal{Y}\times \mathcal{Y}\to [0,\infty )$ by

$$\alpha (x,y)=\left\{\begin{array}{c}1,\mathit{if}x,y\in \left[-1,1\right],\\ 0,\mathit{otherwise}.\end{array}\right.$$

Assume that $u,v,x,y\in$ $\mathcal{Y}$, where

$$\left\{\begin{array}{c}\hspace{1em} \alpha (x,y)\ge 1,\\ d(u,\mathcal{H}x)=d(\mathcal{Y},\mathcal{Z}),\\ d(v,\mathcal{H}y)=d(\mathcal{Y},\mathcal{Z}).\end{array}\right.$$

Then

$$\left\{\begin{array}{c} x,y\in \left[-1,1\right]\\ d(u,\mathcal{H}x)=0\\ d(v,\mathcal{H}y)=0.\end{array}\right.$$

Hence, $u=\mathcal{H}x=\frac{1}{2}x$ and $v=\mathcal{H}y=\frac{1}{2}y.$ Define $F:(0,\infty )\to \mathbb{R}$ by $F\left(t\right)=lnt,$ for $t>0.$ Then, $F\in \Psi .$ Now

$$\tau +\alpha (x,y)F\left(d(\mathcal{H}x,\mathcal{H}y)\right)=ln2+ln\left(\frac{1}{2}\left|x-y\right|\right)=ln\left(\left|x-y\right|\right)=F\left(d(x,y)\right)$$

for $\tau =ln2>0.$ Hence $\mathcal{H}:\mathcal{Y}\to \mathcal{Z}$ is a proximal$\left(\alpha ,F\right)$-contraction. Now, let us demonstrate that $\mathcal{H}$ is α-proximal admissible. Assume that $u,v,x,y\in \mathcal{Y}$, such that

$$\left\{\begin{array}{c}\alpha (x,y)\ge 1,\\ d(u,\mathcal{H}x)=d(\mathcal{Y},\mathcal{Z}),\\ d(v,\mathcal{H}y)=d(\mathcal{Y},\mathcal{Z}).\end{array}\right.$$

Then, we have $x,y\in \left[-1,1\right].$ Hence,

$$u=\mathcal{H}x\in \left[-\frac{1}{2},\frac{1}{2}\right]\subseteq \left[-1,1\right]$$

and

$$v=\mathcal{H}y\in \left[-\frac{1}{2},\frac{1}{2}\right]\subseteq \left[-1,1\right].$$

Hence, $\alpha (u,v)=\alpha (\mathcal{H}x,\mathcal{H}y$ $) \ge 1,$ demonstrating that $\mathcal{H}$ is α-proximal admissible. It is clear that $x_0,x_1\in \mathcal{Y}$ exists in this way $d(x_1,\mathcal{H}{x}_0)=d(\mathcal{Y},\mathcal{Z}),$ and $\alpha (x_0,x_1)\ge 1.$ Suppose that $\left\{x_n\right\}\subseteq$ $\mathcal{Y}$, such that $\alpha (x_n,x_{n+1})$ $\ge 1,$ for all n and $x_n\to x$ $\in \mathcal{Y}$ as $n\to \infty$. Hence,

$$x_n\in \left[-1,1\right],\mathit{for}\mathit{all}n\in \mathbb{N}.$$

As $\left[-1,1\right]$ is a closed set, we conclude that $x\in \left[-1,1\right]$ and consequently $\alpha (x_n,x)$ $\ge 1,$ for all $n\in \mathbb{N}.$ After verifying that all of the assertions of Theorems 5 and 6 are fulfilled, it yields that $\mathcal{H}$ has at least one best proximity point 0 satisfying

$$d(0,\mathcal{H}0)=d(\mathcal{Y},\mathcal{Z}).$$

Remark 1.

In Theorems 5 and 6.

(i)

If we define $\alpha :\mathcal{Y}\times \mathcal{Y}\to [0,\infty )$ by $\alpha (x,y)=1,$ for all $x,y\in \mathcal{Y}$ and the mapping $F:(0,\infty )\to \mathbb{R}$ by $F\left(t\right)=lnt,$ for $t>0,$ In such a scenario, we reach identical pivotal conclusions as presented by Basha [7] in the context of $\mathcal{F}$-metric spaces.

(ii)

When considering $\mathcal{Y}=\mathcal{Z}=\mathcal{Q}$ and defining $\alpha :\mathcal{Y}\times \mathcal{Y}\to [0,\infty )$ by $\alpha (x,y)=1,$ we obtain the key result of Wardowski et al. [2].

(iii)

If we choose $⋏\left(t\right)=lnt,$ for $t>0$, $\beta =0$ in Definition 3 and $\alpha :\mathcal{Y}\times \mathcal{Y}\to [0,\infty )$ by $\alpha (x,y)=1$, our scrutiny reproduces a finding obtained by Omidvari et al. [20].

4. Generalizations and Extensions

Corollary 1.

Let $\mathcal{Y}$, $\mathcal{Z}$ $\in Cl\left(\mathcal{Q}\right)$, such that ${\mathcal{Y}}_0\ne ⌀$. Let $F\in \Psi$ and $\tau >0.$ Suppose that $\mathcal{H}:\mathcal{Y}\to \mathcal{Z}$, meeting the following requirements:

(i)

$\mathcal{H}\left({\mathcal{Y}}_0\right)\subseteq {\mathcal{Z}}_0$ and $(\mathcal{Y},\mathcal{Z})$ satisfies property P;

(ii)

$d(\mathcal{H}x,\mathcal{H}y)>0$⟹

$$\tau +F\left(d(\mathcal{H}x,\mathcal{H}y)\right)\le F\left(d(x,y)\right),$$

for all $x,y\in \mathcal{Y}.$

Then, there exists $x*\in \mathcal{Q}$, such that $d(x*,\mathcal{H}x*)\le d(\mathcal{Y},\mathcal{Z}).$

Proof. 

Define $\alpha :\mathcal{Y}\times \mathcal{Y}\to [0,\infty )$ by

$$\alpha (x,y)=1$$

for all $x,y\in \mathcal{Y}.$ Clearly, $\mathcal{H}$ is $\alpha$-proximal by the definition of $\alpha$, and, moreover, it is required to be a proximal $\left(\alpha ,F\right)$-contraction. Conversely, for any $x\in {\mathcal{Y}}_0$, since $\mathcal{H}\left({\mathcal{Y}}_0\right)\subseteq {\mathcal{Z}}_0$, there exists $y\in {\mathcal{Y}}_0$, such that $\sigma (\mathcal{H}x,y)=\sigma (\mathcal{Y},\mathcal{Z})$. Additionally, based on hypothesis (ii), we obtain

$$\tau +F\left(d(\mathcal{H}x,\mathcal{H}y)\right)\le F\left(d(x,y)\right)$$

which yields

$$\begin{array}{ccc}\hfill F\left(d(\mathcal{H}x,\mathcal{H}y)\right)& \le & F\left(d\right(x,y\left)\right)-\tau \hfill \\ & <& F\left(d\right(x,y\left)\right)\hfill \end{array}$$

which implies by ($F_1$) that

$$d(\mathcal{H}x,\mathcal{H}y)<d(x,y).$$

The above inequality establishes the continuity of $\mathcal{H}$. Thus, all of the conditions outlined in Theorem 5 are satisfied, ensuring the existence of the best proximity point for $\mathcal{H}$. Moreover, referring to Theorem 4 and the definition of the function $\alpha :\mathcal{Y}\times \mathcal{Y}\to [0,\infty )$, we can prove that this point is unique. □

By choosing $F\left(t\right)=lnt$ in Theorem 1, we validate this outcome.

Corollary 2.

Let $\mathcal{Y}$,$\mathcal{Z}$ $\in Cl\left(\mathcal{Q}\right)$, such that ${\mathcal{Y}}_0\ne ⌀$. Let $F\in \Psi .$ Assume that $\mathcal{H}:\mathcal{Y}\to \mathcal{Z}$, satisfying these assertions:

(i)

$\mathcal{H}\left({\mathcal{Y}}_0\right)\subseteq {\mathcal{Z}}_0$ and $(\mathcal{Y},\mathcal{Z})$ satisfies property P;

(ii)

there exists $k\in (0,1)$, such that $d(\mathcal{H}x,\mathcal{H}y)\le \lambda d(x,y),$ for all $x,y\in \mathcal{Y}.$

Then, there exists $x*\in \mathcal{Q}$, such that $d(x*,\mathcal{H}x*)\le d(\mathcal{Y},\mathcal{Z}).$

We now derive proximity theorems in $\mathcal{F}$-MS equipped with a binary relation.

In the context of the $\mathcal{F}$-MS $(\mathcal{Q},d)$ and the binary relation R on $\mathcal{Q}$, consider the following:

$$S=R\cup R^{-1}.$$

Clearly,

$$x,y\in \mathcal{Q},xSy⟺xRy\mathrm{or}yRx.$$

Definition 10.

A mapping $\mathcal{H}:\mathcal{Y}\to \mathcal{Z}$ is called a proximal comparative mapping if

$$\left.\begin{array}{c}{x}_{1}S{x}_2\\ d(u_1,\mathcal{H}{u}_1)=d(\mathcal{Y},\mathcal{Z})\\ d(u_2,\mathcal{H}{u}_2)=d(\mathcal{Y},\mathcal{Z})\end{array}\right\}\Rightarrow u_{1}S{u}_2$$

for all $x_1,x_2,u_1,u_2\in \mathcal{Y}.$

Corollary 3.

Let $\mathcal{Y}$, $\mathcal{Z}$ $\in Cl\left(\mathcal{Q}\right)$, such that ${\mathcal{Y}}_0\ne ⌀$. Let R denote a binary relation on the set $\mathcal{Q}$. Let us suppose that $\mathcal{H}:\mathcal{Y}\to \mathcal{Z}$ is continuous and proximal comparative, and it fulfills the following assertions:

(i)

$\mathcal{H}\left({\mathcal{Y}}_0\right)\subseteq {\mathcal{Z}}_0$ and $(\mathcal{Y},\mathcal{Z})$ satisfies property P;

(ii)

there exists $x_0,x_1\in {\mathcal{Y}}_0$, such that

$$d(x_1,\mathcal{H}{x}_0)=d(\mathcal{Y},\mathcal{Z}),\mathit{and}{x}_{0}S{x}_1,$$

(iii)

there exists $F \in$ Ψ and $\tau >0$, such that

$$x,y\in \mathcal{Y},xSy\mathit{implies}\tau +F\left(d(\mathcal{H}x,\mathcal{H}y)\right)\le F\left(d(x,y)\right)$$

(22)

Then, there exists $x*\in \mathcal{Q}$, such that $d(x*,\mathcal{H}x*)\le d(\mathcal{Y},\mathcal{Z}).$

Proof. 

Define $\alpha :\mathcal{Y}\times \mathcal{Y}\to [0,\infty )$ by:

$$\alpha (x,y)=\left\{\begin{array}{c}1\mathrm{if}xSy\\ 0\mathrm{otherwise}.\end{array}\right.$$

Suppose that

$$\left\{\begin{array}{c}\alpha (x_1,x_2)\ge 1\\ d(u_1,\mathcal{H}{x}_1)=d(\mathcal{Y},\mathcal{Z})\\ d(u_2,\mathcal{H}{x}_2)=d(\mathcal{Y},\mathcal{Z})\end{array}\right.$$

for some $x_1,x_2,u_1,u_2\in \mathcal{Y}$. By the definition of $\alpha$, we get that

$$\left\{\begin{array}{c}{x}_{1}S{x}_2,\\ d(u_1,\mathcal{H}{x}_1)=d(\mathcal{Y},\mathcal{Z})\\ d(u_2,\mathcal{H}{x}_2)=d(\mathcal{Y},\mathcal{Z})\end{array}\right..$$

Invoking the supposition that $\mathcal{H}:\mathcal{Y}\to \mathcal{Z}$ is a proximal comparative, we deduce that $u_{1}S{u}_2$. Applying the definition of $\alpha$, we find that $\alpha (u_1,u_2)\ge 1.$ Hence, we have established that $\mathcal{H}$ is $\alpha$-proximal admissible. The assumption (ii) consequently results in the conclusion

$$d(x_1,\mathcal{H}{x}_0)=d(\mathcal{Y},\mathcal{Z})$$

and $\alpha (x_0,x_1)\ge 1$. To conclude, condition (iii) implies that

$$\tau +\alpha (x,y)F\left(d(\mathcal{H}x,\mathcal{H}y)\right)\le F\left(d(x,y)\right).$$

Being a proximal($\alpha ,F$)-contraction, $\mathcal{H}$ satisfies all of the conditions outlined in Theorem 5, and, consequently, the desired result can be directly derived from the theorem. □

To avoid relying on the continuity assumption of $\mathcal{H}$, we introduce an alternative assumption.

Corollary 4.

Let $\mathcal{Y}$, $\mathcal{Z}\in Cl\left(\mathcal{Q}\right)$, such that ${\mathcal{Y}}_0\ne Ø$ and $\mathcal{R}$ is a binary relation on $\mathcal{Q}$, let $\mathcal{H}:\mathcal{Y}\to \mathcal{Z}$ be a proximal comparative mapping that fulfills the these assumptions:

(i)

$\mathcal{H}\left({\mathcal{Y}}_0\right)\subseteq {\mathcal{Z}}_0$ and $(\mathcal{Y},\mathcal{Z})$ satisfies property P;

(ii)

there exists $x_0,x_1\in {\mathcal{Y}}_0$, such that

$$d(x_1,\mathcal{H}{x}_0)=d(\mathcal{Y},\mathcal{Z}),\mathit{and}{x}_{0}S{x}_1,$$

(iii)

there exists $F\in$ Ψ and some constant $\tau >0$, such that

$$x,y\in \mathcal{Y},xSy\mathit{implies}\tau +F\left(d(\mathcal{H}x,\mathcal{H}y)\right)\le F\left(d(x,y)\right)$$

(iv)

if $\left\{x_n\right\}$ in $\mathcal{Q}$ and $x\in \mathcal{Q}$ are such that $x_{n}S{x}_{n+1},$ for all $n\in \mathbb{N}$ and ${lim}_{n\to \infty }d(x_n,x)=0$, then there exists $\left\{x_{n\left(k\right)}\right\}$ of $\left\{x_n\right\}$ in such a manner $x_{n\left(k\right)}Sx,$ for all k.

Then, $\mathcal{H}$ has a best proximity point.

Proof. 

Define $\alpha :\mathcal{Y}\times \mathcal{Y}\to [0,\infty )$ by

$$\alpha (x,y)=\left\{\begin{array}{c}1\mathrm{if}xSy\\ 0\mathrm{otherwise}.\end{array}\right.$$

Then, we can employ Theorem 6 to reach the intended conclusion. □

Theorem 8.

Besides the assumptions of Corollary 3 (resp. Corollary 4), assume that the subsequent conditions are met: for all $(x,y)\in \mathcal{Y}\times \mathcal{Y}$, such that $(x,y)\notin S$, there exists an element $\varkappa \in {\mathcal{Y}}_0$ satisfying $xS\varkappa$ and $yS\varkappa$. In light of these specified conditions, $\mathcal{H}$ has a unique best proximity point.

5. Application

Best proximity theory, rooted in functional analysis and metric space theory, provides a framework for identifying optimal points that minimize distances or discrepancies. In the context of integral equations for population growth, this theory allows researchers to discern optimal states that represent equilibrium or stability, shedding light on critical aspects such as carrying capacity and sustainable population sizes. To develop a robust and realistic population growth model that accurately captures the dynamic interactions within a population over time, we examined the subsequent Volterra integral equation of the second kind

$$x\left(t\right)=g\left(t\right)+\underset{a}{\overset{b}{\int }}K\left(t,s,x\left(s\right)\right)ds.$$

(23)

In this equation

• $x\left(t\right)$ represents the unknown function to be determined, often related to the population density at time t in the context of population growth models.
• $g\left(t\right)$ is a given function representing the initial condition or a known part of the solution.
• $K\left(t,s,x\left(s\right)\right)$ is the kernel of the integral equation, characterizing the rate or intensity of the interaction or the influence of the population at time s on the population at time t.

We are interested in finding the solution to Equation (23), which tells us that how the population at time t depends not only on the initial conditions $\left(g\left(t\right)\right)$, but also on the interactions with its own past $\left(x\left(s\right)\right)$.

Let $C\left(\right[a,b\left]\right)$ be the set of all continuous functions defined on the closed interval $[a,b]$ and let $d:C\left(\right[a,b\left]\right)\times C\left(\right[a,b\left]\right)\to \mathbb{R}$ be an $\mathcal{F}$-metric defined by

$$d\left(x,y\right)={∥x-y∥}_{\infty }=\underset{t\in [a,b]}{max}\left|x\left(t\right)-y\left(t\right)\right|,$$

(24)

then, the pair ($C\left(\right[a,b\left]\right),d$) incorporates a complete $\mathcal{F}$-MS (see [13]).

Theorem 9.

Suppose that the following statements hold:

where $g:[a,b]\to \mathbb{R}$ is continuous and $K:[a,b]\times [a,b]\times C\left(\right[a,b]\to \mathbb{R}$ is integrable with respect to x on $[a,b].$

(i)

for all $t,s\in [a,b]$ and $x\left(s\right),y\left(s\right)\in C\left(\right[a,b],$ we have

$$\left|K\left(t,s,x\left(s\right)\right)-K\left(t,s,y\left(s\right)\right)\right|\le \frac{e^{-\tau }}{(b-a)}\left(max\left|x\left(s\right)-y\left(s\right)\right|\right),$$

(ii)

there exists $x_0\in C([a,b]$, such that $\phi \left(x_0\left(t\right),\mathcal{H}{x}_0\left(t\right)\right)\ge 0,$ for a given function $\phi :\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ and a mapping $\mathcal{H}:C\left(\right[a,b]\to C([a,b]$ defined by

$$\mathcal{H}x(t)=g\left(t\right)+{\int }_a^{b}K\left(t,s,x\left(s\right)\right)ds$$

(iii)

for each $t\in [a,b]$ and $x,y\in C\left(\right[a,b],$ $\phi \left(x\left(t\right),y\left(t\right)\right)\ge 0$ implies $\phi \left(\mathcal{H}x\left(t\right),\mathcal{H}y\left(t\right)\right)\ge 0,$

(iv)

if $\left\{x_n\right\}\subseteq$ $C\left(\right[a,b]$, such that $x_n\to x$ in $C\left(\right[a,b]$ and $\phi \left(x_n\left(t\right),x_{n+1}\left(t\right)\right)\ge 0$ for all $n\in \mathbb{N},$ then, $\phi \left(x_n\left(t\right),x\left(t\right)\right)\ge 0$ for all $n\in \mathbb{N}.$

Then, the nonlinear integral Equation (23) has a solution in $C\left(\right[a,b].$

Proof. 

The mapping $\mathcal{H}:C\left(\right[a,b]\to C([a,b]$ which is defined as

$$\mathcal{H}x\left(t\right)=g\left(t\right)+\underset{a}{\overset{b}{\int }}K\left(t,s,x\left(s\right)\right)ds$$

for all $x\in C\left(\right[a,b],$ is continuous. Consider

$$\begin{array}{ccc}\hfill \left|\mathcal{H}x\left(t\right)-\mathcal{H}y\left(t\right)\right|& =& \left|{\int }_a^{b}K\left(t,s,x\left(s\right)\right)ds-{\int }_a^{b}K\left(t,s,y\left(s\right)\right)ds\right|\hfill \\ & \le & \underset{a}{\overset{b}{\int }}\left|K\left(t,s,x\left(s\right)\right)-K\left(t,s,y\left(s\right)\right)\right|ds\hfill \\ & \le & \frac{e^{-\tau }}{(b-a)}\left(max\left|x\left(s\right)-y\left(s\right)\right|\right){\int }_a^{b}ds\hfill \\ & \le & \frac{e^{-\tau }}{(b-a)}(b-a)\left(max\left|x\left(s\right)-y\left(s\right)\right|\right)\hfill \\ & \le & e^{-\tau }\left(max\left|x\left(s\right)-y\left(s\right)\right|\right)\hfill \\ & =& e^{-\tau }{∥x-y∥}_{\infty }.\hfill \end{array}$$

From the above inequality, we obtain that

$$d(\mathcal{H}x,\mathcal{H}y)\le e^{-\tau }d(x,y).$$

Logarithmizing both sides, we have

$$ln\left(d(\mathcal{H}x,\mathcal{H}y)\right)\le ln\left(e^{-\tau }d(x,y)\right),$$

that is,

$$ln\left(d(\mathcal{H}x,\mathcal{H}y)\right)\le ln\left(e^{-\tau }\right)+ln\left(d(x,y)\right),$$

that is,

$$ln\left(d(\mathcal{H}x,\mathcal{H}y)\right)\le -\tau +ln\left(d(x,y)\right).$$

Thus

$$\tau +ln\left(d(\mathcal{H}x,\mathcal{H}y)\right)\le ln\left(d(x,y)\right).$$

(25)

Now, consider $F:{\mathbb{R}}^{+}\to \mathbb{R}$ by $F\left(t\right)=ln\left(t\right)$ for $t>0.$ Then, $F\in \Psi .$ Also define $\alpha :\mathcal{Y}\times \mathcal{Y}\to [0,\infty )$ by

$$\alpha (x,y)=\left\{\begin{array}{c}1\mathrm{if}\phi \left(x\left(t\right),y\left(t\right)\right)\ge 0,\mathrm{for}t\in [0,1],\\ 0,\mathrm{otherwise}.\end{array}\right..$$

Then, from (25) and the definition of the $F:(0,+\infty )\to \mathbb{R}$, we have

$$\tau +\alpha (x,y)F\left(d(\mathcal{H}x,\mathcal{H}y)\right)\le F\left(d(x,y)\right).$$

Now, from (ii) there exists $x_0\in \mathcal{Q}$, such that $\phi \left(x_0\left(t\right),\mathcal{H}{x}_0\left(t\right)\right)\ge 0$ yields that $\alpha \left(x_0,\mathcal{H}{x}_0\right)\ge 1$ for $t\in [a,b].$ Next for any $x,y\in \mathcal{Q}$ with $\alpha \left(x,y\right)\ge 1,$ we have

$$\phi \left(x\left(t\right),y\left(t\right)\right)\ge 0,\mathrm{for}\mathrm{all}t\in [a,b]$$

which implies by (iii) that

$$\phi \left(\mathcal{H}x\left(t\right),\mathcal{H}y\left(t\right)\right)\ge 0,\mathrm{for}\mathrm{all}t\in [a,b].$$

It yields

$$\alpha (\mathcal{H}x,\mathcal{H}y)\ge 1.$$

As $\mathcal{H}$ satisfies hypothesis (i) of Theorem 5, it possesses a fixed point x in $C\left(\right[a,b\left]\right)$, which is also the solution of (23). □

6. Conclusions

In this research article, we introduce the concept of proximal ($\alpha$,F)-contractions in the context of $\mathcal{F}$-metric spaces and established the best proximity point results for the aforementioned contractions. To illustrate the real-world relevance of these findings, a non-trivial example is provided. As an application of our leading result, we investigated the solutions of Volterra integral equations.

This research findings unlock a promising path for generalizing the best proximity point concept to multivalued mappings within the $\mathcal{F}$-metric framework. Further exploration of the applicability of these results in solving fractional and ordinary differential equations presents an exciting avenue for future research. Furthermore, investigating the potential of applying our findings to the setting of orthogonal $\mathcal{F}$-metric spaces opens a supplementary promising approach for future advancement in this field.