Advancements in Best Proximity Points: A Study in $\mathcal{F}$-Metric Spaces with Applications

Abstract

The purpose of this study is to explore the existence and uniqueness of the best proximity points for $\left(\alpha ,\Theta \right)$-proximal contractions, a novel concept introduced in the context of $\mathcal{F}$-metric spaces. Moreover, we provide an example to show the usability of the obtained results. To broaden the scope of this research area, we leverage our best proximity point results to demonstrate the existence and uniqueness of solutions for differential equations (equation of motion) and also for fractional differential equations.

1. Introduction

The fundamental concept of a metric space, independently introduced by Maurice Fréchet [1], lays the groundwork for fixed point theory. A metric space provides the mathematical setting in which fixed point theorems are often formulated and proven. The concept of a metric space captures the idea of distance between points, and fixed point theorems establish conditions under which certain mappings in metric spaces have points that map to themselves. In 1922, Stefan Banach established the first result in this theory, which is well known as the Banach contraction principle [2]. Jleli et al. [3] introduced a novel form of contraction called $\Theta$-contraction and formulated a fresh fixed point theorem to extend the Banach contraction principle.

Czerwik [4] introduced the innovative idea of b-metric spaces and demonstrated a fixed point theorem for mappings of a contractive nature. The b-metric space serves as an extension of metric spaces, relaxing the triangle inequality and incorporating the b-metric inequality as a more lenient condition. Consequently, b-metric spaces play a crucial role in mathematics by offering a versatile framework for investigating structures based on distance that may not adhere strictly to the triangle inequality, enabling the exploration of a diverse range of phenomena not fully captured by traditional metric spaces. Subsequently, Jleli et al. [5] introduced an innovative metric space referred to as the $\mathcal{F}$-metric space, expanding beyond the traditional realms of both classical metric space and b-metric space. The concept of $\mathcal{F}$-metric space provides a broader context for studying fixed point theorems and has implications for various areas of mathematics, including functional analysis and topology.

On the other hand, Basha [6] gave the concept of a best proximity point in the 1960s, extending the idea of a fixed point in a more generalized form. The concept of a best proximity point is a powerful tool for analyzing the behavior of functions and sets in metric spaces. It has been extensively studied and has found applications in a wide range of fields. Later on, Eldred et al. [7] studied the properties of best proximity points and provided several conditions that guarantee the existence and convergence of best proximity points for different types of mappings. Best proximity points are pivotal in diverse mathematical fields, including optimization, approximation theory, ordinary differential equations and fractional differential equations. Best proximity points are particularly useful in ordinary differential equations and fractional differential equations, as they provide a powerful tool for analyzing the behavior of solutions and establishing existence and uniqueness results. Gabeleh et al. [8] discussed the existence of a solution for a system of differential equations by using the best proximity point methods under suitable assumptions. Patle et al. [9] obtained Sadovskii-type best proximity point results with an application to fractional differential equations. Very recently, Lateef [10] obtained best proximity point results for ( $\alpha$ − $\psi$ )-contraction in the context of $\mathcal{F}$-metric spaces. Some coupled best proximity points on $\mathcal{F}$-metric spaces endowed with an arbitrary binary relation are also established by Lateef [10]. For further details from this standpoint, we direct readers to [11,12,13,14,15,16,17,18,19,20]. In this manuscript, we introduce the notion of $\left(\alpha ,\Theta \right)$-proximal contraction within the framework of $\mathcal{F}$-metric space, establishing the existence and uniqueness of best proximity points for these contractions. To showcase the practical implications, we utilize our findings on best proximity points to illustrate solutions for differential equations, including the equation of motion, as well as fractional differential equations.

2. Preliminaries

As a foundation for this section, we present the precise definition of a metric space.

Definition 1

([1]). Consider a non-empty set $\mathcal{M}$. A function $d:\mathcal{M}\times \mathcal{M}\to$ $[0,\infty )$ is deemed a metric if the following conditions are satisfied:

$(m1):$$d(\mathfrak{o},\ell )=0$ if and only if $\mathfrak{o}=\ell ;$

$(m2):d(\mathfrak{o},\ell )=d(\ell ,\mathfrak{o}),$

$(m3):$   $d(\mathfrak{o},\nu )\le d(\mathfrak{o},\ell )+d(\ell ,\nu ),$

for all $\mathfrak{o},\ell ,\nu \in \mathcal{M}.$

The pair $(\mathcal{M},d)$ is consequently referred to as a metric space.

Theorem 1

([2]). Consider a self-mapping $\mathcal{L}$ on a complete metric space $(\mathcal{M},d).$ If there exists a constant $\lambda \in [0,1)$ such that

$$d(\mathcal{L}\mathfrak{o},\mathcal{L}\ell )\le \lambda d(\mathfrak{o},\ell ),$$

for all $\mathfrak{o},\ell \in \mathcal{M}$, then $\mathcal{L}$ possesses a unique fixed point.

Jleli et al. [3] presented a novel form of contraction termed $\Theta$-contraction and derived several fresh fixed point theorems applicable to this type of contraction within the realm of generalized metric spaces.

Let $\Psi$ be a set of mappings $\Theta :(0,\infty )\to (1,\infty )$ satisfying

( ${\Theta }_1$ )

$\Theta$ ( $\mathfrak{o}$ ) $\le \Theta$ (ℓ) for all $\mathfrak{o}\le \ell$,

( ${\Theta }_2$ )

For $\left\{{\mathfrak{o}}_n\right\}\subseteq {\mathbb{R}}^{+}$, ${lim}_{n\to \infty }\Theta ({\mathfrak{o}}_n)=1$ ⟺ ${lim}_{n\to \infty }({\mathfrak{o}}_n)=0;$

( ${\Theta }_3$ )

There exists $0<r<1$ and $l\in (0,\infty ]$ such that ${lim}_{\mathfrak{o}\to 0^{+}}\frac{\Theta (\mathfrak{o})-1}{{\mathfrak{o}}^r}=l.$

Definition 2

([3]). A function $\mathcal{L}:\mathcal{M}\to \mathcal{M}$ is characterized as a Θ-contraction when there exists a function Θ adhering to conditions ( ${\Theta }_1$ )–( ${\Theta }_3$ ) and a constant $\lambda \in (0,1)$ such that for every $\mathfrak{o},\ell \in \mathcal{M}$,

$$\begin{array}{c}d(\mathcal{L}\mathfrak{o},\mathcal{L}\ell )>0⟹\Theta (d(\mathcal{L}\mathfrak{o},\mathcal{L}\ell ))\le {[\Theta (d(\mathfrak{o},\ell ))]}^{\lambda }.\hfill \end{array}$$

(1)

Theorem 2

([3]). In the scenario where $(\mathcal{M},d)$ forms a complete metric space and $\mathcal{L}:\mathcal{M}\to \mathcal{M}$ is a Θ-contraction, it follows that $\mathcal{L}$ possesses a unique fixed point.

Czerwik [4] extended the concept of the classical metric space in this manner.

Definition 3

([4]). Consider a non-empty set $\mathcal{M}$ and $s\ge 1$. A function A function $d:\mathcal{M}\times \mathcal{M}\to$ $[0,\infty )$ is termed a b-metric if the following conditions are met:

$(b1):$$d(\mathfrak{o},\ell )=0$ if and only if $\mathfrak{o}=\ell ;$

$(b2):d(\mathfrak{o},\ell )=d(\ell ,\mathfrak{o}),$

$(b3):$   $d(\mathfrak{o},\nu )\le s[d(\mathfrak{o},\ell )+d(\ell ,\nu )],$

for all $\mathfrak{o},\ell ,\nu \in \mathcal{M}.$

The combination $(\mathcal{M},d)$ is consequently denoted as a b-metric space.

In recent times, Jleli et al. [5] presented an intriguing expansion of a metric space using this approach.

Consider $\mathcal{F}$ as the collection of continuous functions $f:(0,+\infty )\to \mathbb{R}$ that adhere to the following criteria:

( ${\mathcal{F}}_1$ )

$0<s<t$ implies $f(s)\le f(t);$

( ${\mathcal{F}}_2$ )

For all $\left\{t_n\right\}\subseteq {\mathbb{R}}^{+}$, ${lim}_{n\to \infty }{t}_n=0$ ⟺ ${lim}_{n\to \infty }f(t_n)=-\infty .$

Definition 4

([5]). Let $\mathcal{M}\ne \varnothing$ and let $d:\mathcal{M}\times \mathcal{M}\to [0,+\infty )$ be a continuous function. Assume that there exist $(f,\hslash )\in \mathcal{F}\times [0,+\infty )$ such that

(D${}_1$)

$(\mathfrak{o},\ell )\in \mathcal{M}\times \mathcal{M}$, $d(\mathfrak{o},\ell )=0$ if and only if $\mathfrak{o}=\ell$.

(D${}_2$)

$d(\mathfrak{o},\ell )=d(\ell ,\mathfrak{o})$, for all $(\mathfrak{o},\ell )\in \mathcal{M}\times \mathcal{M}.$

(D${}_3$)

For every $(\mathfrak{o},\ell )\in \mathcal{M}\times \mathcal{M}$, for every $N\in \mathbb{N}$, $N\ge 2$, and for every ${(u_i)}_{i=1}^N\subset \mathcal{M},$ with $(u_1,u_N)=(\mathfrak{o},\ell )$, we have

$$d(\mathfrak{o},\ell )>0\Rightarrow f(d(\mathfrak{o},\ell ))\le f({\displaystyle \sum _{i=1}^{N-1}}d(u_i,u_{i+1}))+\hslash .$$

Then $(\mathcal{M},d)$ is called an $\mathcal{F}$-metric space.

Example 1

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

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

with $f(ı)=ln(ı)$ and $\hslash =ln(3)$, is an $\mathcal{F}$-metric.

Definition 5

([5]). Consider $(\mathcal{M},d)$ as an $\mathcal{F}$-metric space.

(i) Consider a sequence $\left\{{\kappa }_n\right\}$ $\subseteq \mathcal{M}$. Then $\left\{{\mathfrak{o}}_n\right\}$ is characterized as $\mathcal{F}$-convergent to $\mathfrak{o}\in \mathcal{M}$ if $\left\{{\mathfrak{o}}_n\right\}$ converges to $\mathfrak{o}$ with respect to the $\mathcal{F}$-metric d.

(ii) A sequence $\left\{{\mathfrak{o}}_n\right\}$ in $\mathcal{F}$-metric space $(\mathcal{M},d)$ is denoted as $\mathcal{F}$-Cauchy, if and only if

$$\underset{n,m\to \infty }{lim}d({\mathfrak{o}}_n,{\mathfrak{o}}_m)=0.$$

(iii) If each $\mathcal{F}$-Cauchy sequence within $\mathcal{M}$ is $\mathcal{F}$-convergent to a point in $\mathcal{M}$, then $(\mathcal{M},d)$ is considered $\mathcal{F}$-complete.

Theorem 3

([5]). Consider $(\mathcal{M},d)$ to be an $\mathcal{F}$-metric space with a mapping $\mathcal{L}:\mathcal{M}\to \mathcal{M}$. Suppose that the following conditions hold:

(i) $(\mathcal{M},d)$ is $\mathcal{F}$-complete;

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

$$d(\mathcal{L}(\mathfrak{o}),\mathcal{L}(\ell ))\le \lambda d(\mathfrak{o},\ell ).$$

Then, $\mathcal{L}$ possesses a unique fixed point ${\mathfrak{o}}^{*}\in \mathcal{M}$. Furthermore, for any ${\mathfrak{o}}_0\in \mathcal{M}$, the sequence $\left\{{\mathfrak{o}}_n\right\}\subset \mathcal{M}$ defined by

$${\mathfrak{o}}_{n+1}=\mathcal{L}({\mathfrak{o}}_n),n\in \mathbb{N},$$

is $\mathcal{F}$-convergent to ${\mathfrak{o}}^{*}$.

Inspired by the work of Lateef [10], we give the concept of best proximity point within the framework of an $\mathcal{F}$-metric space in the following manner.

Definition 6

([10]). Let $(\mathcal{M},d)$ be an $\mathcal{F}$-metric space and $\mathcal{E}$, $\mathcal{H}$ $\in N(\mathcal{M})$. A point ${\mathfrak{o}}^{*}$ $\in \mathcal{E}$ is characterized as the best proximity point of $\mathcal{L}:\mathcal{E}\to \mathcal{H}$ when it meets the condition that

$$d({\mathfrak{o}}^{*},\mathcal{L}{\mathfrak{o}}^{*})\le d(\mathcal{E},\mathcal{H}).$$

Consistent with Lateef et al. [10], we give the $\mathcal{F}$-distance among the two nonempty sets $\mathcal{E}$ and $\mathcal{H}$ that fulfill the property P.

Definition 7

([10]). Let $(\mathcal{M},d)$ be an $\mathcal{F}$-metric space and $\mathcal{E}$, $\mathcal{H}$ $\in N(\mathcal{M})$, then $d(\mathcal{E},\mathcal{H})$ is $\mathcal{F}$-distance between two nonempty sets $\mathcal{E}$ and $\mathcal{H}.$ Now define ${\mathcal{E}}_0$ and ${\mathcal{H}}_0$ by

$$\begin{array}{ccc}\hfill {\mathcal{E}}_0& =& \{\mathfrak{o}\in \mathcal{E}:\mathit{there}\mathit{exists}u\in \mathcal{H}\mathit{such}\mathit{that}d(\mathfrak{o},u)=d(\mathcal{E},\mathcal{H})\}\hfill \\ \hfill {\mathcal{H}}_0& =& \{u\in \mathcal{H}:\mathit{there}\mathit{exists}\mathfrak{o}\in \mathcal{E}\mathit{such}\mathit{that}d(\mathfrak{o},u)=d(\mathcal{E},\mathcal{H})\}.\hfill \end{array}$$

The pair $(\mathcal{E},\mathcal{H})$ is deemed to possess the property P if ${\mathcal{E}}_0\ne \varnothing$ and

$$\mathfrak{o},\ell \in {\mathcal{E}}_0,u,v\in {\mathcal{H}}_0,d(\mathfrak{o},u)=d(\ell ,v)=d(\mathcal{E},\mathcal{H})⟹d(\mathfrak{o},\ell )=d(u,v).$$

Definition 8

([10]). Let $(\mathcal{M},d)$ be an $\mathcal{F}$-metric space and $\mathcal{E}$, $\mathcal{H}$ $\in N(\mathcal{M})$. A mapping $\mathcal{L}:\mathcal{E}\to \mathcal{H}$ is called α-proximal admissible if there exists $\alpha :\mathcal{E}\times \mathcal{E}\to [0,\infty )$ such that

$$\left.\begin{array}{ccc}\alpha (\mathfrak{o},\ell )& \ge & 1\\ d(u,\mathcal{L}\mathfrak{o})& =& d(\mathcal{E},\mathcal{H})\\ d(v,\mathcal{L}\ell )& =& d(\mathcal{E},\mathcal{H})\end{array}\right\}\Rightarrow \alpha (u,v)\ge 1,$$

where $\mathfrak{o},\ell ,u,v\in \mathcal{E}.$

Lateef [10] proved the following best proximity result.

Theorem 4

([10]). Let $(\mathcal{M},d)$ be a complete $\mathcal{F}$-metric space and $\mathcal{E}$, $\mathcal{H}$ $\in Cl(\mathcal{M})$ such that ${\mathcal{E}}_0\ne ⌀$. Assume that there exist the mapping $\mathcal{L}:\mathcal{E}\to \mathcal{H}$ and the comparison functions $\alpha :\mathcal{E}\times \mathcal{E}\to [0,\infty )$ and $\psi :[0,\infty )\to [0,\infty )$ such that the following assertions hold:

(i)

$$\alpha (\mathfrak{o},\ell )d(\mathcal{L}\mathfrak{o},\mathcal{L}\ell )\le \psi \left(d(\mathfrak{o},\ell )\right),$$

(ii) The mapping $\mathcal{L}:\mathcal{E}\to \mathcal{H}$ is α-proximal admissible mapping;

(iii) $\mathcal{L}({\mathcal{E}}_0)\subseteq {\mathcal{H}}_0$ and $(\mathcal{E},\mathcal{H})$ satisfies the property P;

(iv) There exists ${\mathfrak{o}}_0,{\mathfrak{o}}_1\in {\mathcal{E}}_0$ such that

$$d({\mathfrak{o}}_1,\mathcal{L}{\mathfrak{o}}_0)=d(\mathcal{E},\mathcal{H}),\mathit{and}\alpha ({\mathfrak{o}}_0,{\mathfrak{o}}_1)\ge 1,$$

(v) $\mathcal{L}$ is continuous or for a sequence $\left\{{\mathfrak{o}}_n\right\}\subseteq$ $\mathcal{E}$ such that $\alpha ({\mathfrak{o}}_n,{\mathfrak{o}}_{n+1})$ $\ge 1,$ for all n and ${\mathfrak{o}}_n\to \mathfrak{o}$   $\in \mathcal{E}$ as $n\to \infty$, then there exists $\left\{{\mathfrak{o}}_{n(k)}\right\}$ of $\left\{{\mathfrak{o}}_n\right\}$ such that $\alpha ({\mathfrak{o}}_{n(k)},\mathfrak{o})\ge 1,$ for all k.

Then there exists ${\mathfrak{o}}^{*}\in \mathcal{E}$ such that $d({\mathfrak{o}}^{*},\mathcal{L}{\mathfrak{o}}^{*})\le d(\mathcal{E},\mathcal{H}).$

3. Results and Discussion

We shall use $N(\mathcal{M})$ and $Cl(\mathcal{M})$ to refer to the sets of all non-empty and closed subsets of $\mathcal{M}$, respectively.

Definition 9.

Let $(\mathcal{M},d)$ is $\mathcal{F}$-metric space and $\mathcal{E}$, $\mathcal{H}$ $\in N(\mathcal{M}).$ A mapping $\mathcal{L}:\mathcal{E}\to \mathcal{H}$ is characterized as an $\left(\alpha ,\Theta \right)$-proximal contraction if there exist the functions $\alpha :\mathcal{E}\times \mathcal{E}\to [0,\infty ),$ $\Theta \in \Psi$ and the constants $\lambda \in (0,1)$ such that

$$\alpha (\mathfrak{o},\ell )\Theta \left(d(\mathcal{L}\mathfrak{o},\mathcal{L}\ell )\right)\le {\left[\Theta (d(\mathfrak{o},\ell ))\right]}^{\lambda }$$

(2)

for all $\mathfrak{o},\ell \in \mathcal{E}$.

Theorem 5.

Let $(\mathcal{M},d)$ be a complete $\mathcal{F}$-metric space and $\mathcal{E}$, $\mathcal{H}$ $\in Cl(\mathcal{M})$ such that ${\mathcal{E}}_0\ne ⌀$. Let $\alpha :\mathcal{E}\times \mathcal{E}\to [0,\infty )$ and $\Theta \in \Psi .$ Assume that $\mathcal{L}:\mathcal{E}\to \mathcal{H}$ is an $\left(\alpha ,\Theta \right)$-proximal contraction that fulfills the following conditions:

(i) $\mathcal{L}$ is α-proximal admissible mapping;

(ii) $\mathcal{L}({\mathcal{E}}_0)\subseteq {\mathcal{H}}_0$ and $(\mathcal{E},\mathcal{H})$ satisfies the property P;

(iii) There exists ${\mathfrak{o}}_0,{\mathfrak{o}}_1\in {\mathcal{E}}_0$ such that

$$d({\mathfrak{o}}_1,\mathcal{L}{\mathfrak{o}}_0)=d(\mathcal{E},\mathcal{H}),\mathit{and}\alpha ({\mathfrak{o}}_0,{\mathfrak{o}}_1)\ge 1.$$

(iv) $\mathcal{L}$ is continuous.

Then there exists ${\mathfrak{o}}^{*}\in \mathcal{E}$ such that $d({\mathfrak{o}}^{*},\mathcal{L}{\mathfrak{o}}^{*})\le d(\mathcal{E},\mathcal{H}).$

Proof. 

By the hypothesis (iii), there exists ${\mathfrak{o}}_0,{\mathfrak{o}}_1\in {\mathcal{E}}_0$ such that

$$d({\mathfrak{o}}_1,\mathcal{L}{\mathfrak{o}}_0)=d(\mathcal{E},\mathcal{H}),\alpha ({\mathfrak{o}}_0,{\mathfrak{o}}_1)\ge 1.$$

(3)

Since $\mathcal{L}({\mathcal{E}}_0)\subseteq {\mathcal{H}}_0,$ there exists ${\mathfrak{o}}_2\in {\mathcal{E}}_0$ such that

$$d({\mathfrak{o}}_2,\mathcal{L}{\mathfrak{o}}_1)=d(\mathcal{E},\mathcal{H}).$$

Now, we have $\alpha ({\mathfrak{o}}_0,{\mathfrak{o}}_1)\ge 1,d({\mathfrak{o}}_1,\mathcal{L}{\mathfrak{o}}_0)=d(\mathcal{E},\mathcal{H})$ and $d({\mathfrak{o}}_2,\mathcal{L}{\mathfrak{o}}_1)=d(\mathcal{E},\mathcal{H}).$ As the mapping $\mathcal{L}$ is $\alpha$-proximal admissible, we obtain $\alpha ({\mathfrak{o}}_1,{\mathfrak{o}}_2)\ge 1.$ Hence,

$$d({\mathfrak{o}}_2,\mathcal{L}{\mathfrak{o}}_1)=d(\mathcal{E},\mathcal{H}),\alpha ({\mathfrak{o}}_1,{\mathfrak{o}}_2)\ge 1$$

(4)

Again, since $\mathcal{L}({\mathcal{E}}_0)\subseteq {\mathcal{H}}_0,$ there exists ${\mathfrak{o}}_3\in {\mathcal{E}}_0$ such that

$$d({\mathfrak{o}}_3,\mathcal{L}{\mathfrak{o}}_2)=d(\mathcal{E},\mathcal{H}).$$

Now, we have $\alpha ({\mathfrak{o}}_1,{\mathfrak{o}}_2)\ge 1,d({\mathfrak{o}}_2,\mathcal{L}{\mathfrak{o}}_1)=d(\mathcal{E},\mathcal{H})$ and $d({\mathfrak{o}}_3,\mathcal{L}{\mathfrak{o}}_2)=d(\mathcal{E},\mathcal{H}).$ As the mapping $\mathcal{L}$ is $\alpha$-proximal admissible, we obtain $\alpha ({\mathfrak{o}}_2,{\mathfrak{o}}_3)\ge 1.$ Hence

$$d({\mathfrak{o}}_3,\mathcal{L}{\mathfrak{o}}_2)=d(\mathcal{E},\mathcal{H}),\alpha ({\mathfrak{o}}_2,{\mathfrak{o}}_3)\ge 1.$$

(5)

Utilizing the inductive approach, we can systematically construct $\left\{{\mathfrak{o}}_n\right\}\subset {\mathcal{E}}_0$ such that

$$d({\mathfrak{o}}_{n+1},\mathcal{L}{\mathfrak{o}}_n)=d(\mathcal{E},\mathcal{H}),\alpha ({\mathfrak{o}}_n,{\mathfrak{o}}_{n+1})\ge 1$$

(6)

for all $n\in$ $\mathbb{N}\cup \left\{0\right\}.$ Assume that ${\mathfrak{o}}_k={\mathfrak{o}}_{k+1}$ for some k. From (6), we have

$$d({\mathfrak{o}}_k,\mathcal{L}{\mathfrak{o}}_k)=d({\mathfrak{o}}_{k+1},\mathcal{L}{\mathfrak{o}}_k)=d(\mathcal{E},\mathcal{H})$$

i.e., ${\mathfrak{o}}_k$ is a best proximity point of $\mathcal{L}$. Therefore, we posit that $d({\mathfrak{o}}_{n-1},{\mathfrak{o}}_n)>0$ for all $n\in$ $\mathbb{N}\cup \left\{0\right\}.$ As ( $\mathcal{E},\mathcal{H}$ ) satisfies the property P, we summarize from (6) that

$$d({\mathfrak{o}}_n,{\mathfrak{o}}_{n+1})=d(\mathcal{L}{\mathfrak{o}}_{n-1},\mathcal{L}{\mathfrak{o}}_n),$$

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

$$\begin{array}{ccc}\hfill 1& <& \Theta \left(d({\mathfrak{o}}_n,{\mathfrak{o}}_{n+1})\right)\le \alpha ({\mathfrak{o}}_n,{\mathfrak{o}}_{n+1})\Theta \left(d({\mathfrak{o}}_n,{\mathfrak{o}}_{n+1})\right)\hfill \\ & =& \alpha ({\mathfrak{o}}_n,{\mathfrak{o}}_{n+1})\Theta \left(d(\mathcal{L}{\mathfrak{o}}_{n-1},\mathcal{L}{\mathfrak{o}}_n)\right)\hfill \\ & \le & {\left[\Theta (d({\mathfrak{o}}_{n-1},{\mathfrak{o}}_n))\right]}^{\lambda }\hfill \end{array}$$

(7)

for all $n\ge 0.$ which further implies that

$$1<\Theta \left(d({\mathfrak{o}}_n,{\mathfrak{o}}_{n+1})\right)\le {\left[\Theta (d({\mathfrak{o}}_{n-1},{\mathfrak{o}}_n))\right]}^{\lambda }\le {\left[\Theta (d({\mathfrak{o}}_{n-2},{\mathfrak{o}}_{n-1}))\right]}^{{\lambda }^2}\le \dots \le {\left[\Theta (d({\mathfrak{o}}_0,{\mathfrak{o}}_1))\right]}^{{\lambda }^n}$$

(8)

for all $n\in \mathbb{N}$. Given that $\Theta$ belongs to the set $\Psi ,$ letting n approach infinity in the equation yields

$$\underset{n\to \infty }{lim}\Theta \left(d({\mathfrak{o}}_n,{\mathfrak{o}}_{n+1})\right)=1$$

(9)

which implies that

$$\underset{n\to \infty }{lim}d({\mathfrak{o}}_n,{\mathfrak{o}}_{n+1})=0$$

(10)

by the condition ( ${\Theta }_2$ ). Now condition ( ${\Theta }_3$ ) guarantees the existence of $0<r<1$ and $l\in (0,\infty ]$ such that

$$\underset{n\to \infty }{lim}\frac{\Theta (d({\mathfrak{o}}_n,{\mathfrak{o}}_{n+1}))-1}{d{({\mathfrak{o}}_n,{\mathfrak{o}}_{n+1})}^r}=l.$$

(11)

Assuming $l<\infty ,$ let $\rho =\frac{l}{2}>0.$ Based on the definition of the limit, there is an $n_0\in \mathbb{N}$ such that

$$|\frac{\Theta (d({\mathfrak{o}}_n,{\mathfrak{o}}_{n+1}))-1}{d{({\mathfrak{o}}_n,{\mathfrak{o}}_{n+1})}^r}-l|\le \rho$$

for all $n>n_0.$ This compels that

$$\frac{\Theta (d({\mathfrak{o}}_n,{\mathfrak{o}}_{n+1}))-1}{d{({\mathfrak{o}}_n,{\mathfrak{o}}_{n+1})}^r}\ge l-\rho =\frac{l}{2}=\rho$$

for all $n>n_0.$ Then

$$nd{({\mathfrak{o}}_n,{\mathfrak{o}}_{n+1})}^r\le \mu n[\Theta (d({\mathfrak{o}}_n,{\mathfrak{o}}_{n+1}))-1]$$

(12)

for all $n>n_0,$ where $\mu =\frac{1}{\rho }.$ Considering the case where $l=\infty ,$ let $\rho >0$ be an arbitrary positive number. Invoking the definition of the limit, there exists $n_0\in \mathbb{N}$ such that

$$\rho \le \frac{\Theta (d({\mathfrak{o}}_n,{\mathfrak{o}}_{n+1}))-1}{d{({\mathfrak{o}}_n,{\mathfrak{o}}_{n+1})}^r}$$

for all $n>n_0.$ This leads to the conclusion that

$$nd{({\mathfrak{o}}_n,{\mathfrak{o}}_{n+1})}^r\le \mu n[\Theta (d({\mathfrak{o}}_n,{\mathfrak{o}}_{n+1}))-1]$$

for all $n>n_0,$ where $\mu =\frac{1}{\rho }.$ Hence, across all possibilities, there exist $\mu >0$ and $n_0\in \mathbb{N}$ such that

$$nd{({\mathfrak{o}}_n,{\mathfrak{o}}_{n+1})}^r\le \mu n[\Theta (d({\mathfrak{o}}_n,{\mathfrak{o}}_{n+1}))-1]$$

(13)

for all $n>n_0.$ Hence, by (8) and (13), we obtain

$$nd{({\mathfrak{o}}_n,{\mathfrak{o}}_{n+1})}^r\le \mu n({[\Theta (d({\mathfrak{o}}_n,{\mathfrak{o}}_{n+1}))]}^{r^n}-1).$$

(14)

By setting n as infinity in the preceding inequality, we establish that

$$\underset{n\to \infty }{lim}nd{({\mathfrak{o}}_n,{\mathfrak{o}}_{n+1})}^r=0.$$

Hence, there exists $n_1\in \mathbb{N}$ such that

$$d{({\mathfrak{o}}_n,{\mathfrak{o}}_{n+1})}^r\le \frac{1}{n^{1/r}}$$

(15)

for all $n>n_1.$ Now by (15) for $m>n>n_1,$ we have

$$d({\mathfrak{o}}_n,{\mathfrak{o}}_m)\le {\displaystyle \sum _{i=n}^{m-1}}d({\mathfrak{o}}_i,{\mathfrak{o}}_{i+1})\le {\displaystyle \sum _{i=n}^{m-1}}\frac{1}{i^{1/r}}\le {\displaystyle \sum _{i=1}^{\infty }}\frac{1}{i^{1/r}}.$$

(16)

Let $ϵ>0$ be fixed and $(f,\hslash )\in \mathcal{F}\times [0,+\infty )$ be such that ( $D_3$ ) is satisfied. By ( ${\mathcal{F}}_2$ ), there exists $\delta >0$ such that

$$0<ı<\delta ⟹f(ı)<f(\epsilon )-\hslash .$$

(17)

for $m>n>n_1.$ Using ( $D_3$ ), (15) and (17), we obtain $d({\mathfrak{o}}_n,{\mathfrak{o}}_m)>0,$ $m>n>n_1$ implies

$$\begin{array}{ccc}\hfill f(d({\mathfrak{o}}_m,{\mathfrak{o}}_n))& \le & f\left({\displaystyle \sum _{i=n}^{m-1}}d({\mathfrak{o}}_i,{\mathfrak{o}}_{i+1})\right)+\hslash \hfill \\ & \le & f\left({\displaystyle \sum _{i=1}^{\infty }}\frac{1}{i^{1/r}}\right)+\hslash <f(ϵ).\hfill \end{array}$$

By virtue of ( ${\mathcal{F}}_1$ ), this implies that $d({\mathfrak{o}}_m,{\mathfrak{o}}_n)<ϵ,$ $m>n>n_1.$ It establishes that { ${\mathfrak{o}}_n$ } is $\mathcal{F}$-Cauchy. Since $(\mathcal{M},d)$ is $\mathcal{F}$-complete and $\mathcal{E}$ is closed, there exists ${\mathfrak{o}}^{*}\in \mathcal{E}$ such that $\left\{{\mathfrak{o}}_n\right\}$ is $\mathcal{F}$-convergent to ${\mathfrak{o}}^{*}$, i.e.,

$$\underset{n\to \infty }{lim}d({\mathfrak{o}}_n,{\mathfrak{o}}^{*})=0.$$

(18)

Otherwise, $\mathcal{L}$ is continuous. Then, we obtain $\mathcal{L}{\mathfrak{o}}_n\to \mathcal{L}{\mathfrak{o}}^{*}$ as $n\to \infty .$ Using the continuity of d, we obtain

$$d(\mathcal{E},\mathcal{H})=d({\mathfrak{o}}_{n+1},\mathcal{L}{\mathfrak{o}}_n)\to d({\mathfrak{o}}^{*},\mathcal{L}{\mathfrak{o}}^{*})$$

as $n\to \infty .$ Therefore, $d({\mathfrak{o}}^{*},\mathcal{L}{\mathfrak{o}}^{*})$ $=d(\mathcal{E},\mathcal{H})$. □

Theorem 6.

Let $(\mathcal{M},d)$ be a complete $\mathcal{F}$-metric space and $\mathcal{E}$, $\mathcal{H}$ $\in Cl(\mathcal{M})$ such that ${\mathcal{E}}_0\ne ⌀$. Let $\alpha :\mathcal{E}\times \mathcal{E}\to [0,\infty )$ and $\Theta \in \Psi .$ Suppose that $\mathcal{L}:\mathcal{E}\to \mathcal{H}$ is an $\left(\alpha ,\Theta \right)$-proximal contraction. Let it satisfy the following conditions:

(i) $\mathcal{L}$ is α-proximal admissible mapping;

(ii) $\mathcal{L}({\mathcal{E}}_0)\subseteq {\mathcal{H}}_0$ and $(\mathcal{E},\mathcal{H})$ satisfies the property P;

(iii) There exists ${\mathfrak{o}}_0,{\mathfrak{o}}_1\in {\mathcal{E}}_0$ such that

$$d({\mathfrak{o}}_1,\mathcal{L}{\mathfrak{o}}_0)=d(\mathcal{E},\mathcal{H}),\mathit{and}\alpha ({\mathfrak{o}}_0,{\mathfrak{o}}_1)\ge 1.$$

(iv) If $\left\{{\mathfrak{o}}_n\right\}\subseteq$ $\mathcal{E}$ such that $\alpha ({\mathfrak{o}}_n,{\mathfrak{o}}_{n+1})$ $\ge 1,$ for all n and ${\mathfrak{o}}_n\to \mathfrak{o}$ $\in \mathcal{E}$ as $n\to \infty$, then there exists $\left\{{\mathfrak{o}}_{n(k)}\right\}$ of $\left\{{\mathfrak{o}}_n\right\}$ such that $\alpha ({\mathfrak{o}}_{n(k)},\mathfrak{o})\ge 1,$ for all k.

Then there exists ${\mathfrak{o}}^{*}\in \mathcal{M}$ such that $d({\mathfrak{o}}^{*},\mathcal{L}{\mathfrak{o}}^{*})\le d(\mathcal{E},\mathcal{H}).$

Proof. 

In support of the findings in Theorem 5, there exists a sequence $\left\{{\mathfrak{o}}_n\right\}\subset \mathcal{E}$ such that the inequality (2) is satisfied and ${\mathfrak{o}}_n$ $\to {\mathfrak{o}}^{*}$ as $n\to \infty ,$ i.e.,

$$\underset{n\to \infty }{lim}d({\mathfrak{o}}_n,{\mathfrak{o}}^{*})=0.$$

From condition (iii), there exists $\left\{{\mathfrak{o}}_{n(k)}\right\}$ of $\left\{{\mathfrak{o}}_n\right\}$ such that $\alpha ({\mathfrak{o}}_{n(k)},{\mathfrak{o}}^{*})\ge 1,$ for all $k.$ We affirm that $\mathcal{L}{\mathfrak{o}}_{n(k)}\to \mathcal{L}{\mathfrak{o}}^{*}$ as $k\to \infty .$ So by (2), we obtain

$$\begin{array}{ccc}\hfill \Theta \left(d(\mathcal{L}{\mathfrak{o}}_{n(k)},\mathcal{L}{\mathfrak{o}}^{*})\right)& \le & \alpha ({\mathfrak{o}}_{n(k)},{\mathfrak{o}}^{*})\Theta \left(d(\mathcal{L}{\mathfrak{o}}_{n(k)},\mathcal{L}{\mathfrak{o}}^{*})\right)\hfill \\ & \le & {\left[\Theta \left(d({\mathfrak{o}}_{n(k)},{\mathfrak{o}}^{*})\right)\right]}^{\lambda }\hfill \\ & <& \Theta \left(d({\mathfrak{o}}_{n(k)},{\mathfrak{o}}^{*})\right)\hfill \end{array}$$

since $\lambda <1,$ which implies by ( ${\Theta }_1$ ), we have

$$d(\mathcal{L}{\mathfrak{o}}_{n(k)},\mathcal{L}{\mathfrak{o}}^{*})<d({\mathfrak{o}}_{n(k)},{\mathfrak{o}}^{*}).$$

Taking $k\to \infty$ and using the continuity of d, we have

$${\sigma }_F(\mathcal{E},\mathcal{H})=d({\mathfrak{o}}_{n(k)+1},\mathcal{L}{\mathfrak{o}}_{n(k)})\to d({\mathfrak{o}}^{*},\mathcal{L}{\mathfrak{o}}^{*})$$

as $n\to \infty .$ Therefore,

$$d({\mathfrak{o}}^{*},\mathcal{L}{\mathfrak{o}}^{*})={\sigma }_F(\mathcal{E},\mathcal{H})$$

which completes the proof of the theorem. □

Definition 10.

Let $\mathcal{L}:\mathcal{E}\to \mathcal{H}$ and $\alpha :\mathcal{E}\times \mathcal{E}\to [0,\infty )$. The mapping $\mathcal{L}$ is said to be $(\alpha ,{\sigma }_F)$-regular if for all $(\mathfrak{o},\ell )\in {\alpha }^{-1}[0,1)$, there exists $\varrho \in {\mathcal{E}}_0$ such that

$$\alpha (\mathfrak{o},\ell )\ge 1\mathit{and}\alpha (\ell ,\varrho )\ge 1.$$

Theorem 7.

Assuming that $\mathcal{L}$ is $(\alpha ,\sigma )$-regular along with the conditions of Theorem 5 (respectively, Theorem 6), we can deduce the existence of a unique element ${\mathfrak{o}}^{*}\in \mathcal{E}$ satisfying the inequality $d({\mathfrak{o}}^{*},\mathcal{L}{\mathfrak{o}}^{*})\le d(\mathcal{E},\mathcal{H}).$

Proof. 

As established by Theorem 5, the set of best proximity points of $\mathcal{L}$ is nonempty, indicating the presence of a best proximity point ${\mathfrak{o}}^{*}\in {\mathcal{E}}_0$. Assuming another best proximity point ${\ell }^{*}\in {\mathcal{E}}_0$ of $\mathcal{L}$, i.e.,

$$d(\mathcal{L}{\mathfrak{o}}^{*},{\mathfrak{o}}^{*})=d(\mathcal{L}{\ell }^{*},{\ell }^{*})=d(\mathcal{E},\mathcal{H}).$$

(19)

Using the property P and (19), we obtain that

$$d(\mathcal{L}{\mathfrak{o}}^{*},\mathcal{L}{\ell }^{*})=d({\mathfrak{o}}^{*},{\ell }^{*}).$$

(20)

We explore two potential cases.

Case 1. Assuming $\alpha ({\mathfrak{o}}^{*},{\ell }^{*})\ge 1$ and utilizing the Equation (19), we deduce that

$$\begin{array}{ccc}\hfill \Theta \left(d({\mathfrak{o}}^{*},{\ell }^{*})\right)& =& \Theta \left(d(\mathcal{L}{\mathfrak{o}}^{*},\mathcal{L}{\ell }^{*})\right)\le \alpha ({\mathfrak{o}}^{*},{\ell }^{*})\Theta \left(d(\mathcal{L}{\mathfrak{o}}^{*},\mathcal{L}{\ell }^{*})\right)\hfill \\ & \le & {\left[\Theta (d({\mathfrak{o}}^{*},{\ell }^{*}))\right]}^{\lambda }<\Theta (d({\mathfrak{o}}^{*},{\ell }^{*}))\hfill \end{array}$$

since $\lambda <1,$ which implies by ( ${\Theta }_1$ ) that $d({\mathfrak{o}}^{*},{\ell }^{*})<d({\mathfrak{o}}^{*},{\ell }^{*}),$ a contradiction. Thus, ${\mathfrak{o}}^{*}={\ell }^{*}.$

Case 2. If $\alpha ({\mathfrak{o}}^{*},{\ell }^{*})<1.$

By supposition, there exists ${\varrho }_0\in {\mathcal{E}}_0$ such that $\alpha ({\mathfrak{o}}^{*},{\varrho }_0)\ge 1$ and $\alpha ({\ell }^{*},{\varrho }_0)\ge 1$. Since $\mathcal{L}({\mathcal{E}}_0)\subseteq {\mathcal{H}}_0$, there exists ${\varrho }_1\in {\mathcal{E}}_0$ such that

$$d({\varrho }_1,\mathcal{L}{\varrho }_0)=d(\mathcal{E},\mathcal{H}).$$

Now, we have

$$\alpha ({\mathfrak{o}}^{*},{\varrho }_0)\ge 1$$

$$d({\mathfrak{o}}^{*},\mathcal{L}{\mathfrak{o}}^{*})=d(\mathcal{E},\mathcal{H}),$$

$$d({\varrho }_1,\mathcal{L}{\varrho }_0)=d(\mathcal{E},\mathcal{H}).$$

As $\mathcal{L}$ is $\alpha$-proximal admissible, we have $\alpha ({\mathfrak{o}}^{*},{\varrho }_1)$ $\ge 1$. Hence

$$d({\varrho }_1,\mathcal{L}{\varrho }_0)=d(\mathcal{E},\mathcal{H})\mathrm{and}\alpha ({\mathfrak{o}}^{*},{\varrho }_1)\ge 1.$$

Following this approach, we can iteratively construct a sequence  $\left\{{\varrho }_n\right\}$ in ${\mathcal{E}}_0$ such that

$$d({\varrho }_{n+1},\mathcal{L}{\varrho }_n)=d(\mathcal{E},\mathcal{H})\mathrm{and}\alpha ({\mathfrak{o}}^{*},{\varrho }_n)\ge 1$$

(21)

for all $n\ge 0.$ As a consequence of property P and (21), it can be inferred that

$$d({\varrho }_{n+1},{\mathfrak{o}}^{*})=d(\mathcal{L}{\varrho }_n,\mathcal{L}{\mathfrak{o}}^{*})$$

(22)

for all $n\in \mathbb{N}\cup \left\{0\right\}.$ Since $\mathcal{L}$ is $\left(\alpha ,\Theta \right)$-proximal contraction, we have

$$\begin{array}{ccc}\hfill 1& <& \Theta \left(d({\varrho }_{n+1},{\mathfrak{o}}^{*})\right)=\Theta \left(d(\mathcal{L}{\varrho }_n,\mathcal{L}{\mathfrak{o}}^{*})\right)\hfill \\ & \le & \alpha ({\varrho }_n,{\mathfrak{o}}^{*})\Theta \left(d(\mathcal{L}{\varrho }_n,\mathcal{L}{\mathfrak{o}}^{*})\right)\hfill \\ & \le & {\left[\Theta (d({\varrho }_n,{\mathfrak{o}}^{*}))\right]}^{\lambda }\hfill \end{array}$$

for all $n\ge 0.$ Thus, we have

$$1<\Theta \left(d({\varrho }_{n+1},{\mathfrak{o}}^{*})\right)\le {\left[\Theta (d({\varrho }_n,{\mathfrak{o}}^{*}))\right]}^{\lambda }\le \dots \le {\left[\Theta (d({\varrho }_0,{\mathfrak{o}}^{*}))\right]}^{{\lambda }^n}$$

(23)

Taking the limit as $n\to \infty$ in (23), we have

$$\underset{n\to \infty }{lim}\Theta \left(d({\varrho }_{n+1},{\mathfrak{o}}^{*})\right)=1$$

then by ( ${\Theta }_2$ ), we have

$$\underset{n\to \infty }{lim}d({\varrho }_{n+1},{\mathfrak{o}}^{*})=0$$

which implies that ${\varrho }_{n+1}\to {\mathfrak{o}}^{*}$ whenever $n\to \infty$. Thus { ${\varrho }_n\}\to {\mathfrak{o}}^{*}.$ Therefore, in each of the analyzed scenarios, the sequence { ${\varrho }_n\}\to {\mathfrak{o}}^{*}$ as $n\to \infty$. Similarly, we can demonstrate that { ${\varrho }_n\}\to {\ell }^{*}$ as $n\to \infty$. Due to the uniqueness of the limit, we conclude that ${\mathfrak{o}}^{*}={\ell }^{*}$. □

We now present an example that serves to demonstrate the applicability and validity of our findings.

Example 2.

Let $\mathcal{M}$ = $\mathbb{R}$ and $d:\mathcal{M}\times \mathcal{M}\to [0,+\infty )$ be defined by $d(\mathfrak{o},\ell )=\left|\mathfrak{o}-\ell \right|.$ Then, ( $\mathcal{M}$,d) is an $\mathcal{F}$-complete $\mathcal{F}$-metric space. Consider the two closed subsets of $\mathcal{M}$ as $\mathcal{E}=\left[-1,1\right]\cup \left\{4,8\right\}$ and $\mathcal{H}=\left[-\frac{1}{2},\frac{1}{2}\right]\cup \left\{4,8\right\}.$ The compactness of $\mathcal{H}$ is established, implying its approximate compactness relative to $\mathcal{E}$. Define $\mathcal{L}:\mathcal{E}\to \mathcal{H}$ as

$$\mathcal{L}\mathfrak{o}=\left\{\begin{array}{c}\frac{1}{2}\mathfrak{o},\mathit{if}\mathfrak{o}\in \left[-1,1\right],\\ 0,\mathit{if}\mathfrak{o}=4,\\ 0,\mathit{if}\mathfrak{o}=8.\end{array}\right.$$

Evidently, $d(\mathcal{E},\mathcal{H})=0$ and

$$\begin{array}{ccc}\hfill {\mathcal{E}}_0& =& \left\{\mathfrak{o}\in \mathcal{E}:d(\mathfrak{o},\ell )=d(\mathcal{E},\mathcal{H})=0\mathit{for}\mathit{some}\ell \in \mathcal{H}\right\}=\mathcal{H},\hfill \\ \hfill {\mathcal{H}}_0& =& \left\{\ell \in \mathcal{H}:d(\mathfrak{o},\ell )=d(\mathcal{E},\mathcal{H})=0\mathit{for}\mathit{some}\ell \in \mathcal{E}\right\}=\mathcal{H}.\hfill \end{array}$$

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

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

Suppose that $u,v,\mathfrak{o},\ell$ belong to $\mathcal{E}$, where

$$\left\{\begin{array}{c}\alpha (\mathfrak{o},\ell )\ge 1,\\ d(u,\mathcal{L}\mathfrak{o})=d(\mathcal{E},\mathcal{H}),\\ d(v,\mathcal{L}\ell )=d(\mathcal{E},\mathcal{H}).\end{array}\right.$$

Then

$$\left\{\begin{array}{c}\mathfrak{o},\ell \in \left[-1,1\right]\\ d(u,\mathcal{L}\mathfrak{o})=0\\ d(v,\mathcal{L}\ell )=0.\end{array}\right.$$

Thus, $u=\mathcal{L}\mathfrak{o}=\frac{1}{2}\mathfrak{o}$ and $v=\mathcal{L}\ell =\frac{1}{2}\ell .$ Define $\Theta :(0,\infty )\to (1,\infty )$ by $\Theta (t)=e^t.$ Then $\Theta \in \Psi .$ Now

$$\alpha (\mathfrak{o},\ell )\Theta \left(d(\mathcal{L}\mathfrak{o},\mathcal{L}\ell )\right)=e^{\frac{1}{2}\left|\mathfrak{o}-\ell \right|}={\left[e^{\left|\mathfrak{o}-\ell \right|}\right]}^{\frac{1}{2}}={\left[\Theta \left(d(\mathfrak{o},\ell )\right)\right]}^{\lambda }$$

for $\lambda =\frac{1}{2}.$ Hence, $\mathcal{L}:\mathcal{E}\to \mathcal{H}$ is an $\left(\alpha ,\Theta \right)$-proximal contraction. Now, we show that $\mathcal{L}$ is α-proximal admissible. Suppose that $u,v,\mathfrak{o},\ell$ belong to $\mathcal{E}$ such that

$$\left\{\begin{array}{c}\alpha (\mathfrak{o},\ell )\ge 1,\\ d(u,\mathcal{L}\mathfrak{o})=d(\mathcal{E},\mathcal{H}),\\ d(v,\mathcal{L}\ell )=d(\mathcal{E},\mathcal{H}).\end{array}\right.$$

Then, we have $\mathfrak{o},\ell \in \left[-1,1\right].$ Hence

$$u=\mathcal{L}\mathfrak{o}\in \left[-\frac{1}{2},\frac{1}{2}\right]\subseteq \left[-1,1\right]$$

and

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

Hence, $\alpha (u,v)=\alpha (\mathcal{L}\mathfrak{o},\mathcal{L}\ell$ $)\ge 1,$ which shows that $\mathcal{L}$ is α-proximal admissible. It is evident that there exists ${\mathfrak{o}}_0,{\mathfrak{o}}_1\in \mathcal{E}$. In this way, $d({\mathfrak{o}}_1,\mathcal{L}{\mathfrak{o}}_0)=d(\mathcal{E},\mathcal{H}),$ and  $\alpha ({\mathfrak{o}}_0,{\mathfrak{o}}_1)\ge 1.$ Assume that $\left\{{\mathfrak{o}}_n\right\}\subseteq$ $\mathcal{E}$ such that $\alpha ({\mathfrak{o}}_n,{\mathfrak{o}}_{n+1})$ $\ge 1,$ for all n and ${\mathfrak{o}}_n\to \mathfrak{o}$ $\in \mathcal{E}$ as $n\to \infty$. Hence,

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

Since $\left[-1,1\right]$ is closed, thus, we obtain $\mathfrak{o}\in \left[-1,1\right]$ and hence, $\alpha ({\mathfrak{o}}_n,\mathfrak{o})$ $\ge 1,$ for all $n\in \mathbb{N}.$ Having established that all conditions of Theorems 5 and 6 are met, it follows that $\mathcal{L}$ possesses at least one best proximity point. This point, denoted by 0, satisfies

$$d(0,\mathcal{L}0)=d(\mathcal{E},\mathcal{H}).$$

Remark 1.

In Theorems 5 and 6.

(i) If we define $\alpha :\mathcal{E}\times \mathcal{E}\to [0,\infty )$ by $\alpha (\mathfrak{o},\ell )=1,$ for all $\mathfrak{o},\ell \in \mathcal{E}$ and the function $\Theta :(0,\infty )\to (1,\infty )$ by $\Theta (t)=e^t,$ then we arrive at the same key conclusions as Basha et al. [6] for $\mathcal{F}$-metric spaces.

(ii) If we consider $\mathcal{E}=\mathcal{H}=\mathcal{M}$ and $\alpha :\mathcal{E}\times \mathcal{E}\to [0,\infty )$ by $\alpha (\mathfrak{o},\ell )=1,$ we obtain the main outcome of Jleli et al. [5].

(iii) If we take $f(t)=lnt,$ for $t>0$ and $\hslash =0$ in Definition 4, our analysis replicates a result established by Hussain et al. [21].

4. Consequences

Corollary 1.

Let $(\mathcal{M},d)$ be an $\mathcal{F}$-complete $\mathcal{F}$-metric space, $\mathcal{E}$, $\mathcal{H}$ $\in Cl(\mathcal{M})$ such that ${\mathcal{E}}_0\ne ⌀$. Let $\Theta \in \Psi .$ Suppose that $\mathcal{L}:\mathcal{E}\to \mathcal{H}$, meeting the following requirements:

(i) $\mathcal{L}({\mathcal{E}}_0)\subseteq {\mathcal{H}}_0$ and $(\mathcal{E},\mathcal{H})$ satisfies the property P;

(ii) $d(\mathcal{L}\mathfrak{o},\mathcal{L}\ell )>0$ implies

$$\Theta \left(d(\mathcal{L}\mathfrak{o},\mathcal{L}\ell )\right)\le {\left[\Theta (d(\mathfrak{o},\ell ))\right]}^{\lambda },$$

for all $\mathfrak{o},\ell \in \mathcal{E}.$

Then, there exists ${\mathfrak{o}}^{*}\in \mathcal{M}$ such that $d({\mathfrak{o}}^{*},\mathcal{L}{\mathfrak{o}}^{*})\le d(\mathcal{E},\mathcal{H}).$

Proof. 

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

$$\alpha (\mathfrak{o},\ell )=1$$

for all $\mathfrak{o},\ell \in \mathcal{E}.$ Evidently $\mathcal{L}$ is $\alpha$-proximal admissible by the definition of $\alpha$, and additionally, it must be an $\left(\alpha ,\Theta \right)$-proximal contraction. On the other hand, for any $\mathfrak{o}\in {\mathcal{E}}_0$, since $\mathcal{L}({\mathcal{E}}_0)\subseteq {\mathcal{H}}_0$, there exists $\ell \in {\mathcal{E}}_0$ such that $\sigma (\mathcal{L}\mathfrak{o},\ell )=\sigma (\mathcal{E},\mathcal{H})$. Furthermore, from the hypothesis (ii), we obtain

$$\Theta \left(d(\mathcal{L}\mathfrak{o},\mathcal{L}\ell )\right)\le {\left[\Theta (d(\mathfrak{o},\ell ))\right]}^{\lambda }<\Theta (d(\mathfrak{o},\ell ))$$

which implies by ( ${\Theta }_1$ ) that

$$d(\mathcal{L}\mathfrak{o},\mathcal{L}\ell )<d(\mathfrak{o},\ell ).$$

The preceding inequality implies the continuity of $\mathcal{L}$. Consequently, all the prerequisites of Theorem 5 are met, guaranteeing the existence of the best proximity point of $\mathcal{L}$. Furthermore, based on Theorem 4 and the definition of the function $\alpha :\mathcal{E}\times \mathcal{E}\to [0,\infty )$, we can establish the uniqueness of this best proximity point. □

If we take $\Theta (t)=e^t$, for $t>0$ in Theorem 1, we establish this result.

Corollary 2.

Let $(\mathcal{M},d)$ be a complete $\mathcal{F}$-metric space, $\mathcal{E}$, $\mathcal{H}$ $\in Cl(\mathcal{M})$ such that ${\mathcal{E}}_0\ne ⌀$. Let $\Theta \in \Psi .$ Assume that $\mathcal{L}:\mathcal{E}\to \mathcal{H}$, satisfying these assertions:

(i) $\mathcal{L}({\mathcal{E}}_0)\subseteq {\mathcal{H}}_0$ and $(\mathcal{E},\mathcal{H})$ satisfies the property P;

(ii) There exists $\lambda \in (0,1)$ such that $d(\mathcal{L}\mathfrak{o},\mathcal{L}\ell )\le \lambda d(\mathfrak{o},\ell ),$ for all $\mathfrak{o},\ell \in \mathcal{E}.$

Then there exists ${\mathfrak{o}}^{*}\in \mathcal{M}$ such that $d({\mathfrak{o}}^{*},\mathcal{L}{\mathfrak{o}}^{*})\le d(\mathcal{E},\mathcal{H}).$

Now we establish proximity results in $\mathcal{F}$-metric spaces endowed with binary relation.

Given the $\mathcal{F}$-metric space $(\mathcal{M},d)$ and the binary relation R over $\mathcal{M}$, let

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

Evidently,

$$\mathfrak{o},\ell \in \mathcal{M},\mathfrak{o}S\ell ⟺\mathfrak{o}R\ell \mathrm{or}\ell R\mathfrak{o}.$$

Definition 11.

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

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

for all ${\mathfrak{o}}_1,{\mathfrak{o}}_2,u_1,u_2\in \mathcal{E}.$

Corollary 3.

Let $(\mathcal{M},d)$ be a complete $\mathcal{F}$-metric space, $\mathcal{E}$, $\mathcal{H}$ $\in Cl(\mathcal{M})$ such that ${\mathcal{E}}_0\ne ⌀$. Let R be a binary relation over $\mathcal{M}$. Assume that $\mathcal{L}:\mathcal{E}\to \mathcal{H}$ is continuous, satisfying these assertions:

(i) $\mathcal{L}({\mathcal{E}}_0)\subseteq {\mathcal{H}}_0$ and $(\mathcal{E},\mathcal{H})$ satisfies the P property;

(ii) $\mathcal{L}$ is a proximal comparative mapping;

(iii) There exists ${\mathfrak{o}}_0,{\mathfrak{o}}_1\in {\mathcal{E}}_0$ such that

$$d({\mathfrak{o}}_1,\mathcal{L}{\mathfrak{o}}_0)=d(\mathcal{E},\mathcal{H}),\mathit{and}{\mathfrak{o}}_{0}S{\mathfrak{o}}_1,$$

(iv) There exists $\Theta \in$ Ψ such that

$$\mathfrak{o},\ell \in \mathcal{E},\mathfrak{o}S\ell \mathit{implies}\Theta \left(d(\mathcal{L}\mathfrak{o},\mathcal{L}\ell )\right)\le {\left[\Theta (d(\mathfrak{o},\ell ))\right]}^{\lambda }$$

(24)

Then there exists ${\mathfrak{o}}^{*}\in \mathcal{M}$ such that $d({\mathfrak{o}}^{*},\mathcal{L}{\mathfrak{o}}^{*})\le d(\mathcal{E},\mathcal{H}).$

Proof. 

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

$$\alpha (\mathfrak{o},\ell )=\left\{\begin{array}{c}1\mathrm{if}\mathfrak{o}S\ell \\ 0\mathrm{otherwise}.\end{array}\right.$$

Suppose that

$$\left\{\begin{array}{c}\alpha ({\mathfrak{o}}_1,{\mathfrak{o}}_2)\ge 1\\ d(u_1,\mathcal{L}{\mathfrak{o}}_1)=d(\mathcal{E},\mathcal{H})\\ d(u_2,\mathcal{L}{\mathfrak{o}}_2)=d(\mathcal{E},\mathcal{H})\end{array}\right.$$

for some ${\mathfrak{o}}_1,{\mathfrak{o}}_2,u_1,u_2\in \mathcal{E}$. By the definition of $\alpha$, we obtain that

$$\left\{\begin{array}{c}{\mathfrak{o}}_{1}S{\mathfrak{o}}_2,\\ d(u_1,\mathcal{L}{\mathfrak{o}}_1)=d(\mathcal{E},\mathcal{H})\\ d(u_2,\mathcal{L}{\mathfrak{o}}_2)=d(\mathcal{E},\mathcal{H})\end{array}\right.$$

Invoking supposition (ii), we deduce that $u_{1}S{u}_2$. Applying the definition of $\alpha$, we find that $\alpha (u_1,u_2)\ge 1.$ Therefore, we demonstrated that $\mathcal{L}$ is $\alpha$-proximal admissible. Supposition (iii) leads to the conclusion

$$d({\mathfrak{o}}_1,\mathcal{L}{\mathfrak{o}}_0)=d(\mathcal{E},\mathcal{H})$$

and $\alpha ({\mathfrak{o}}_0,{\mathfrak{o}}_1)\ge 1$. In conclusion, condition (iv) entails that

$$\alpha (\mathfrak{o},\ell )\Theta \left(d(\mathcal{L}\mathfrak{o},\mathcal{L}\ell )\right)\le {\left[\Theta (d(\mathfrak{o},\ell ))\right]}^{\lambda }.$$

Being an ( $\alpha ,\Theta$ )-proximal contraction, $\mathcal{L}$ fulfills all the requirements of Theorem 5, and hence, the required result is directly obtainable from the theorem. □

To forego the assumption of $\mathcal{L}$ ’s continuity, we employ another assumption.

Corollary 4.

Let $\mathcal{E}$, $\mathcal{H}\in Cl(\mathcal{M}),$ where $(\mathcal{M},d)$ is a complete $\mathcal{F}$-metric space such that ${\mathcal{E}}_0\ne \varnothing$. Given a binary relation $\mathcal{R}$ over a set $\mathcal{M}$, let $\mathcal{L}:\mathcal{E}\to \mathcal{H}$ be a mapping that fulfills the following conditions:

(i) $\mathcal{L}({\mathcal{E}}_0)\subseteq {\mathcal{H}}_0$ and $(\mathcal{E},\mathcal{H})$ satisfies the property P;

(ii) $\mathcal{L}$ is a proximal comparative mapping,

(iii) There exists ${\mathfrak{o}}_0,{\mathfrak{o}}_1\in {\mathcal{E}}_0$ such that

$$d({\mathfrak{o}}_1,\mathcal{L}{\mathfrak{o}}_0)=d(\mathcal{E},\mathcal{H}),\mathit{and}{\mathfrak{o}}_{0}S{\mathfrak{o}}_1,$$

(iv) There exist the function $\Theta \in$ Ψ and the constant $\lambda \in (0,1)$ such that

$$\mathfrak{o},\ell \in \mathcal{E},\mathfrak{o}S\ell \mathit{implies}\Theta \left(d(\mathcal{L}\mathfrak{o},\mathcal{L}\ell )\right)\le {\left[\Theta (d(\mathfrak{o},\ell ))\right]}^{\lambda }$$

(v) If $\left\{{\mathfrak{o}}_n\right\}$ in $\mathcal{M}$ and $\mathfrak{o}\in \mathcal{M}$ are such that ${\mathfrak{o}}_{n}S{\mathfrak{o}}_{n+1},$ for all $n\ge 0$ and ${lim}_{n\to \infty }d({\mathfrak{o}}_n,\mathfrak{o})=0,$ then there exists  $\left\{{\mathfrak{o}}_{n(k)}\right\}$ of $\left\{{\mathfrak{o}}_n\right\}$ such that ${\mathfrak{o}}_{n(k)}S\mathfrak{o}$ for all k.

Then there exists ${\mathfrak{o}}^{*}\in \mathcal{M}$ such that $d({\mathfrak{o}}^{*},\mathcal{L}{\mathfrak{o}}^{*})\le d(\mathcal{E},\mathcal{H}).$

Proof. 

If we consider $\alpha :\mathcal{E}\times \mathcal{E}\to [0,\infty )$ given by

$$\alpha (\mathfrak{o},\ell )=\left\{\begin{array}{c}1\mathrm{if}\mathfrak{o}S\ell \\ 0\mathrm{otherwise}.\end{array}\right.$$

Furthermore, noting that assertion (v) entails condition (J), we can apply Theorem 6 to arrive at the desired conclusion. □

Theorem 8.

In addition to the hypotheses of Corollary 3 (resp. Corollary 4), suppose that the following conditions hold: for any pair $(\mathfrak{o},\ell )\in \mathcal{E}\times \mathcal{E}$ such that $(\mathfrak{o},\ell )\notin S$, there exists an element $\varrho \in {\mathcal{E}}_0$ satisfying $\mathfrak{o}S\varrho$ and $\ell S\varrho$. Under these conditions, $\mathcal{L}$ possesses a unique best proximity point.

5. Application

Let $C([0,1])$ be the set of all continuous functions defined on the closed interval $[0,1]$ and $d:C([0,1])\times C([0,1])\to \mathbb{R}$ be an $\mathcal{F}$-metric defined by

$$d\left(\mathfrak{o},\ell \right)={∥\mathfrak{o}-\ell ∥}_{\infty }=\underset{t\in [0,1]}{max}\left|\mathfrak{o}\left(t\right)-\ell \left(t\right)\right|,$$

(25)

then the pair ( $C([0,1]),d$ ) embodies an $\mathcal{F}$-complete $\mathcal{F}$-metric space (see [13]).

Problem 1.

A particle with mass m is initially stationary at $\mathfrak{o}=0$ and $t=0.$ A force g begins acting on it in the X direction, causing its velocity to instantaneously increase from 0 to 1 immediately after $t=0$. Determine the particle’s position at time t:

$$\left\{\begin{array}{c}m\frac{d^2\mathfrak{o}}{d{t}^2}=g\left(t,\mathfrak{o}\left(t\right)\right),t\in \left[0,1\right]\\ \mathfrak{o}\left(0\right)={\mathfrak{o}}^{/}(0)=1,\end{array}\right.$$

(26)

where g is a continuous function from the rectangle $\left[0,1\right]\times \mathbb{R}$ to $\mathbb{R}$. The green function connected with (26) is defined by

$$G(t,s)=\left\{\begin{array}{c}t\mathit{if}0\le t\le s\le 1,\\ 2t-s\mathit{if}0\le s\le t\le 1.\end{array}\right.$$

Let $\phi :\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ be a function. We will proceed under the assumption that the following are true:

(i) $\left|g\left(t,\mathfrak{o}\right)-g\left(t,\ell \right)\right|\le {max}_{\mathfrak{o},\ell \in \mathbb{R}}\left|\mathfrak{o}-\ell \right|$ for all $t\in \left[0,1\right]$ with $\phi \left(\mathfrak{o},\ell \right)\ge 0,$

(ii) There exists ${\mathfrak{o}}_0\in C([0,1])$ such that $\phi \left({\mathfrak{o}}_0(t),\mathcal{L}{\mathfrak{o}}_0(t)\right)\ge 0$ for all $t\in \left[0,1\right],$ where $\mathcal{L}:C([0,1])\to C([0,1]),$

(iii) $\phi \left(\mathfrak{o}(t),\ell (t)\right)\ge 0$ implies $\phi \left(\mathcal{L}\mathfrak{o}(t),\mathcal{L}\ell (t)\right)\ge 0$, for all $\mathfrak{o},\ell \in C([0,1])$ and $t\in [0,1],$

(iv) If $\left\{{\mathfrak{o}}_n\right\}$ is a sequence in $\mathcal{M}$ such that ${\mathfrak{o}}_n\to \mathfrak{o}$ in $C([0,1])$ and $\phi \left({\mathfrak{o}}_n(t),{\mathfrak{o}}_{n+1}(t)\right)\ge 0,$ for all $n\in \mathbb{N},$ then $\phi \left({\mathfrak{o}}_n(t),\mathfrak{o}(t)\right)\ge 0,$ for all $n\in \mathbb{N}.$

Theorem 9.

Assume that the conditions (i)–(iv) hold, then (26) has a solution in $C{([0,1])}^2.$

Proof. 

It is a well-established fact that the existence of a solution $\mathfrak{o}\in C{([0,1])}^2$ to (26) is equivalent to the existence of a solution $\mathfrak{o}\in C([0,1])$ to the integral equation

$$\mathfrak{o}(t)={\displaystyle {\int }_0^1}G\left(t,s\right)g\left(s,\mathfrak{o}(s)\right)ds$$

for all $t\in [0,1].$ Consider $\mathcal{L}:C([0,1])\to C([0,1])$ to be a function defined by

$$\mathcal{L}\mathfrak{o}(t)={\displaystyle {\int }_0^1}G\left(t,s\right)g\left(s,\mathfrak{o}(s)\right)ds.$$

Let $\mathfrak{o},\ell \in C([0,1])$ in a manner that $\phi \left(\mathfrak{o}(t),\ell (t)\right)\ge 0,$ for all $t\in [0,1].$ In accordance with (i), we obtain

$$\begin{array}{ccc}\hfill \left|\mathcal{L}\mathfrak{o}(t)-\mathcal{L}\ell (t)\right|& =& \left|{\displaystyle {\int }_0^1}G\left(t,s\right)g\left(s,\mathfrak{o}(s)\right)ds-{\displaystyle {\int }_0^1}G\left(t,s\right)g\left(s,\ell (s)\right)ds\right|\hfill \\ & =& \left|{\displaystyle {\int }_0^1}\left(G\left(t,s\right)g\left(s,\mathfrak{o}(s)\right)-G\left(t,s\right)g\left(s,\ell (s)\right)\right)ds\right|\hfill \\ & \le & {\displaystyle {\int }_0^1}G\left(t,s\right)\left|g\left(s,\mathfrak{o}(s)\right)-g\left(s,\ell (s)\right)\right|ds\hfill \\ & \le & {\displaystyle {\int }_0^1}G\left(t,s\right)\left|max\left|\mathfrak{o}(s)-\ell (s)\right|\right|ds\hfill \\ & \le & {∥\mathfrak{o}-\ell ∥}_{\infty }·\underset{t\in [0,1]}{sup}\left({\displaystyle {\int }_0^1}G\left(t,s\right)ds\right).\hfill \end{array}$$

Since ${\displaystyle {\int }_0^1}G\left(t,s\right)ds=-\frac{t^2}{2}+\frac{1}{2},$ for all $t\in [0,1],$ then ${sup}_{t\in [0,1]}\left({\displaystyle {\int }_0^1}G\left(t,s\right)ds\right)=\frac{1}{2}.$ Then from the above inequality, we have

$${∥\mathcal{L}\mathfrak{o}(t)-\mathcal{L}\ell (t)∥}_{\infty }\le \frac{1}{2}{∥\mathfrak{o}-\ell ∥}_{\infty }.$$

Taking the exponential on both sides, we have

$$\begin{array}{ccc}\hfill e^{{∥\mathcal{L}\mathfrak{o}(t)-\mathcal{L}\ell (t)∥}_{\infty }}& \le & e^{\frac{1}{2}{∥\mathfrak{o}-\ell ∥}_{\infty }}\hfill \\ & =& {\left[e^{{∥\mathfrak{o}-\ell ∥}_{\infty }}\right]}^{\frac{1}{2}}\hfill \end{array}$$

(27)

for all $\mathfrak{o},\ell \in C([0,1]).$ Now we consider a function $\Theta :\left(0,+\infty \right)\to \left(1,+\infty \right)$ by

$$\Theta (t)=e^t.$$

Also, define $\alpha :C([0,1])\times C([0,1])\to [0,\infty )$ by

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

Then, from (27) and the definition of the $\mathcal{F}$-metric d, we have

$$\alpha (\mathfrak{o},\ell )\Theta \left(d(\mathcal{L}\mathfrak{o},\mathcal{L}\ell )\right)\le {\left[\Theta (d(\mathfrak{o},\ell ))\right]}^{\lambda }$$

with $\lambda =\frac{1}{2}.$ Now from (ii), there exists ${\mathfrak{o}}_0\in C([0,1])$ to ensure that $\phi \left({\mathfrak{o}}_0(t),\mathcal{L}{\mathfrak{o}}_0(t)\right)\ge 0$ implies that $\alpha \left({\mathfrak{o}}_0,\mathcal{L}{\mathfrak{o}}_0\right)\ge 1$ for $t\in [0,1].$ Subsequently, for the functions $\mathfrak{o},\ell \in C([0,1])$ with $\alpha \left(\mathfrak{o},\ell \right)\ge 1,$ we have

$$\phi \left(\mathfrak{o}(t),\ell (t)\right)\ge 0,\mathrm{for}\mathrm{all}t\in [0,1]$$

which implies by (iii) that

$$\phi \left(\mathcal{L}\mathfrak{o}(t),\mathcal{L}\ell (t)\right)\ge 0\mathrm{for}\mathrm{all}t\in [0,1].$$

It yields

$$\alpha (\mathcal{L}\mathfrak{o},\mathcal{L}\ell )\ge 1,$$

and therefore, $\mathcal{L}$ is deemed $\alpha$-admissible. According to Theorem 5, that $\mathcal{L}$ admits a fixed point $\mathfrak{o}$ within the space $C([0,1])$ which consequently serves as the solution to (26). □

Application to Fractional Calculus

Consider the set $C([0,1]),$ comprising all continuous functions defined on the closed interval $[0,1].$ Then, $(C([0,1]),d)$ constitutes an $\mathcal{F}$-metric space with an $\mathcal{F}$-metric defined by (25). To demonstrate the applicability of our fixed point theorem, we shall consider a nonlinear differential equation of fractional order

$${}^C{D}^{\eta }(\mathfrak{o}(t))+g(t,\mathfrak{o}(t)),\left(0\le t\le 1,\eta <1\right)$$

(28)

along with the integral boundary conditions

$$\mathfrak{o}(0)=0,\mathfrak{o}(1)=0,$$

where ${}^C{D}^{\eta }$ represents the Caputo fractional derivative of order $\eta$ for a continuous function $h:\left[0,+\infty \right)\to \mathbb{R}$ given as

$${}^C{D}^{\eta }h(t)=\frac{1}{\Gamma (j-\eta )}{\displaystyle {\int }_0^t}{\left(t-s\right)}^{j-\eta -1}{h}^j(s)ds,$$

$\left(j-1<\eta <j,j=\left[\eta \right]+1\right)$ and $g:\left[0,1\right]\times \mathbb{R}\to \mathbb{R}$ is a continuous function, and Green’s function linked with Problem (28) is defined by

$$G(t,s)=\left\{\begin{array}{c}{\left(t(1-s)\right)}^{a-1}-{(t-s)}^{a-1},\mathrm{if}0\le t\le s\le 1,\\ \frac{{\left(t(1-s)\right)}^{a-1}}{\Gamma (a)},\mathrm{if}0\le s\le t\le 1.\end{array}\right.$$

Let us assume that the following conditions are satisfied:

(i) $\left|g\left(t,\mathfrak{o}\right)-g\left(t,\ell \right)\right|\le \lambda \left|\mathfrak{o}-\ell \right|$ for all $t\in \left[0,1\right]$ with $\phi \left(\mathfrak{o},\ell \right)\ge 0;$

(ii) There exists ${\mathfrak{o}}_0\in \mathcal{M}$ such that $\phi \left({\mathfrak{o}}_0(t),\mathcal{L}{\mathfrak{o}}_0(t)\right)\ge 0$ for all $t\in \left[0,1\right],$ where $\mathcal{L}:C([0,1])\to C([0,1])$ is defined by

$$\mathcal{L}\mathfrak{o}(t)={\displaystyle {\int }_0^1}G\left(t,s\right)g\left(s,\mathfrak{o}(s)\right)ds.$$

(iii) $\phi \left(\mathfrak{o}(t),\ell (t)\right)\ge 0$ implies $\phi \left(\mathcal{L}\mathfrak{o}(t),\mathcal{L}\ell (t)\right)\ge 0$, for all $\mathfrak{o},\ell \in \mathcal{M}$ and $t\in [0,1],$

(iv) If $\left\{{\mathfrak{o}}_n\right\}$ is a sequence in $\mathcal{M}$ such that ${\mathfrak{o}}_n\to \mathfrak{o}$ in $\mathcal{M}$ and $\phi \left({\mathfrak{o}}_n(t),{\mathfrak{o}}_{n+1}(t)\right)\ge 0$ for all $n\in \mathbb{N},$ then $\phi \left({\mathfrak{o}}_n(t),\mathfrak{o}(t)\right)\ge 0$ for all $n\in \mathbb{N}.$

Theorem 10.

Subject to conditions (i)–(iv), the differential Equation (28) possesses a solution within the function space $C^2([0,1]).$

Proof. 

The equivalence between the existence of a solution $\mathfrak{o}\in {\mathcal{M}}^2$ to (28) and the existence of a solution $\mathfrak{o}\in \mathcal{M}$ to the integral equation is a cornerstone of functional analysis

$$\mathfrak{o}(t)={\displaystyle {\int }_0^1}G\left(t,s\right)g\left(s,\mathfrak{o}(s)\right)ds$$

for all $t\in [0,1].$ Let $\mathfrak{o},\ell \in \mathcal{M}$ such that $\phi \left(\mathfrak{o}(t),\ell (t)\right)\ge 0$ for all $t\in [0,1].$ By (i), we have

$$\begin{array}{ccc}\hfill \left|\mathcal{L}\mathfrak{o}(t)-\mathcal{L}\ell (t)\right|& =& \left|{\displaystyle {\int }_0^1}G\left(t,s\right)f\left(s,\mathfrak{o}(s)\right)ds-{\displaystyle {\int }_0^1}G\left(t,s\right)f\left(s,\ell (s)\right)ds\right|\hfill \\ & \le & {\displaystyle {\int }_0^1}\left|G\left(t,s\right)\right|\left|f\left(s,\mathfrak{o}(s)\right)-f\left(s,\ell (s)\right)\right|ds\hfill \\ & \le & {\displaystyle {\int }_0^1}\left|G\left(t,s\right)\right|\lambda \left|\mathfrak{o}(s)-\ell (s)\right|ds\hfill \\ & \le & {\displaystyle {\int }_0^1}\left|G\left(t,s\right)\right|\lambda max\left|\mathfrak{o}(s)-\ell (s)\right|ds\hfill \\ & \le & \lambda {∥\mathfrak{o}-\ell ∥}_{\infty }·\underset{t\in [0,1]}{sup}\left({\displaystyle {\int }_0^1}G\left(t,s\right)ds\right).\hfill \end{array}$$

Since ${\displaystyle {\int }_0^1}G\left(t,s\right)ds=\frac{t^{a-1}{(1-t)}^{a-1}}{-a\Gamma (a)}+\frac{t^{a-1}}{a\Gamma (a)}+\frac{t^{a-1}}{a}+\frac{t^{a-1}{(1-t)}^{a-1}}{-a\Gamma (a)}$ for all $t\in [0,1],$ then

$$\underset{t\in [0,1]}{sup}\left({\displaystyle {\int }_0^1}G\left(t,s\right)ds\right)\le 1.$$

Then from the above inequality, we have

$${∥\mathcal{L}\mathfrak{o}(t)-\mathcal{L}\ell (t)∥}_{\infty }\le \lambda {∥\mathfrak{o}-\ell ∥}_{\infty }.$$

Taking the exponential on both sides, we have

$$\begin{array}{ccc}\hfill e^{{∥\mathcal{L}\mathfrak{o}(t)-\mathcal{L}\ell (t)∥}_{\infty }}& \le & e^{\lambda {∥\mathfrak{o}-\ell ∥}_{\infty }}\hfill \\ & =& {\left[e^{{∥\mathfrak{o}-\ell ∥}_{\infty }}\right]}^{\lambda }\hfill \end{array}$$

(29)

for all $\mathfrak{o},\ell \in \mathcal{M}.$ Now we consider a function $\Theta :\left(0,+\infty \right)\to \left(1,+\infty \right)$ by

$$\Theta (t)=e^t.$$

Also define $\alpha :\mathcal{E}\times \mathcal{E}\to [0,\infty )$ by

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

Then from (29) and the definition of the $\mathcal{F}$-metric d, we have

$$\alpha (\mathfrak{o},\ell )\Theta \left(d(\mathcal{L}\mathfrak{o},\mathcal{L}\ell )\right)\le {\left[\Theta (d(\mathfrak{o},\ell ))\right]}^{\lambda }$$

Now from (ii), there exists ${\mathfrak{o}}_0\in \mathcal{M}$ such that $\phi \left({\mathfrak{o}}_0(t),\mathcal{L}{\mathfrak{o}}_0(t)\right)\ge 0$ implies that $\alpha \left({\mathfrak{o}}_0,\mathcal{L}{\mathfrak{o}}_0\right)\ge 1$ for $t\in [0,1].$ Next for any $\mathfrak{o},\ell \in \mathcal{M}$ with $\alpha \left(\mathfrak{o},\ell \right)\ge 1,$ we have

$$\phi \left(\mathfrak{o}(t),\ell (t)\right)\ge 0\mathrm{for}\mathrm{all}t\in [0,1]$$

which implies by (iii) that

$$\phi \left(\mathcal{L}\mathfrak{o}(t),\mathcal{L}\ell (t)\right)\ge 0\mathrm{for}\mathrm{all}t\in [0,1].$$

It yields

$$\alpha (\mathcal{L}\mathfrak{o},\mathcal{L}\ell )\ge 1.$$

Since $\mathcal{L}$ satisfies the conditions for $\alpha$-admissibility (Theorem 5), it has a fixed point $\mathfrak{o}$ in $C([0,1])$, which is also the solution of (28). □

6. Conclusions

In this paper, we proposed the notion of an $\left(\alpha ,\Theta \right)$-proximal contraction within the framework of $\mathcal{F}$-metric space and obtained the existence and uniqueness of the best proximity points for such contractions. To demonstrate the practicality of our results, we presented a non-trivial example. With the goal of further enhancing the appeal of this research direction, we extended our results to novel applications of fractional calculus and also to the equation of motion modeling differential equations.

The outcomes of this investigation can be generalized to encompass the best proximity points of multivalued mappings in the framework of $\mathcal{F}$-metric spaces. Moreover, the existence of common and coupled best proximity points for both self-mappings and non-self-mappings can be demonstrated within this framework. The application of our established results to solving fractional differential equations and ordinary differential equations provides an avenue for future exploration. Additionally, the applicability of our results in the context of $\mathcal{F}$-bipolar metric spaces constitutes another potential direction for future development in this field.