Solutions of Nonlinear Differential and Integral Equations via Optimality Results Involving Proximal Mappings

Abstract

This research paper delves into the application of optimality results in orthogonal fuzzy metric spaces to demonstrate the existence and uniqueness of solutions of nonlinear differential equations with boundary conditions and nonlinear integral equations, emphasizing the importance of orthogonal fuzzy metric spaces in extending fixed-point theory. Through introducing this innovative concept, the study provides a theoretical framework for analyzing mappings in diverse scenarios. In this study, we introduce the concept of best proximity point (BPP) within the framework of orthogonal fuzzy metric spaces by employing orthogonal fuzzy proximal contractive mappings. Moreover, this research explores the implications of the established results, considering both self-mappings and non-self mappings that share the same parameter set. Additionally, some examples are provided to illustrate the practical relevance of the proven results and consequences in various mathematical contexts. The findings of this study can open up avenues for further exploration and application in solving real-world problems.

1. Introduction

The utilization of fixed point theory greatly enhances the study of nonlinear analysis, spanning such varied disciplines as biology, chemistry, engineering, and physics. It offers a powerful tool for addressing differential and integral equations, optimization problems, and variational inequalities. With its broad applicability, fixed point theory has become a cornerstone in understanding complex systems and structures. In 1922, Banach’s establishment of the Banach contraction principle and subsequent formulation of the Banach fixed point theorem provided the foundation for numerous advancements in this area. Through his definition of a contraction mapping, Banach introduced a potent method for demonstrating both the existence and uniqueness of fixed points within metric spaces. A significant development in handling ambiguity and uncertainty came with Zadeh’s introduction of fuzzy sets in 1965 [1]. This concept has since been instrumental in resolving numerous real-world problems. The notion of fuzzy metric spaces, first proposed by Kramosil and Michálek [2] in 1975 and later refined by George and Veeramani [3], has provided a framework for dealing with imprecise data and uncertain environments. Over the years, researchers have extensively explored fuzzy metric spaces and their generalizations along with their applications across various domains [4,5,6,7,8,9,10,11,12,13,14,15,16,17], etc.

The foundational work on best proximity points spearheaded by Eldred and Veeramani [18] in 2006 marked a pivotal moment in the field’s development. Their groundbreaking contribution shed light on the existence and uniqueness of such points, primarily through the utilization of proximal contraction mappings. This laid the groundwork for subsequent investigations and advancements in the realm of best proximity points. Subsequent researchers have built upon this foundation, making significant strides in refining and broadening the scope of best proximity point theory. In 2011, S. Basha [19] emphasized optimal results relying on non-self mappings, thereby expanding the theory’s applicability. Later, V.S. Raj [20] introduced weakly contractive non-self mappings, further enriching the field and deepening the understanding of best proximity point results. The field continued to evolve with ongoing modifications and extensions of optimal results, reflecting the dynamic nature of research in this area [21,22,23,24,25,26]. Additionally, the exploration of coupled best proximity results opened up new avenues for applications and theoretical advancements [27,28,29].

In recent years, orthogonal sets and orthogonal metric spaces were introduced by Gordji et al. [30] as a generalization of traditional metric spaces. This led to further advancements in the theory, with subsequent generalizations proposed by various authors [31,32,33,34]. Notably, Hezarjaribi [31] introduced the concept of orthogonal fuzzy metric spaces in 2018, with limited research conducted in this area [35,36,37,38]. However, the absence of the concept of best proximity points within the framework of orthogonal fuzzy metric spaces has created a gap in understanding. Our work addresses a research gap by introducing the conception of best proximity points in the context of orthogonal fuzzy metric spaces and establishing best proximity point results within said generalization of metric space.

This departure from traditional approaches utilizes orthogonal fuzzy contractive mappings and introduces a new inequality involving rational contractions and control functions involving proximal mappings. Through this, we establish some optimality results that deepen our understanding of best proximity points in the context of non-self mappings within orthogonal fuzzy metric spaces. This innovative contribution extends the applicability of best proximity point theory and encourages further exploration in this fertile research area.

2. Preliminaries

The following are some preparatory considerations for the writing of the article:

Definition 1

([2,3]). Consider $\star :[0,1]\times [0,1]⟶[0,1]$ as a binary operator. We claim that ⋆ is a continuous t-norm if it satisfies the following conditions:

(i) ⋆ adheres to commutativity and associativity laws;

(ii) ⋆ is continuous;

(iii) $\mathfrak{c}\star 1=\mathfrak{c}$ for all $\mathfrak{c}\in [0,1]$;

(iv) For any $\mathfrak{c},\mathfrak{g},\mathfrak{p},\mathfrak{v}\in [0,1]$, if $\mathfrak{c}\le \mathfrak{g}$ and $\mathfrak{p}\le \mathfrak{v}$, then $\mathfrak{c}\star \mathfrak{g}\le \mathfrak{p}\star \mathfrak{v}$.

These conditions collectively define the criteria for a function ⋆ to qualify as a continuous t-norm.

Example 1.

Consider ⋆ as a binary operator on the interval $[0,1]$, defined as follows:

$\eta \star \nu =min\{\eta ,\nu \};$

$\eta \star \nu =max\{0,\eta +\nu -1\};\mathrm{or}$

$\eta \star \nu =\eta \nu$.

Note that ⋆ is a continuous t-norm. Also:

(i) Commutativity and associativity laws hold for all cases of ⋆ defined above.

(ii) Continuity: Each case of ⋆ defined above involves basic operations (min, max, multiplication) which are continuous on the interval $[0,1]$.

(iii) For $\mathfrak{c}\in [0,1]$, $\mathfrak{c}\star 1=\mathfrak{c}$, satisfying the condition.

(iv) Suppose $\mathfrak{c}\le \mathfrak{g}$ and $\mathfrak{p}\le \mathfrak{v}$ for $\mathfrak{c},\mathfrak{g},\mathfrak{p},\mathfrak{v}\in [0,1]$.

Because all conditions are met, ⋆ is indeed a continuous t-norm.

Definition 2

([2,3]). Consider $\mathcal{U}$ as a non-empty set, ⋆ as a continuous t-norm, and $\mathfrak{M}:\mathcal{U}\times \mathcal{U}\times (0,+\infty )\to (0,1]$ as a fuzzy set. Then, $(\mathcal{U},\mathfrak{M},\star )$ forms a fuzzy metric space if the following conditions hold:

For any $\mathfrak{z},\mathfrak{g},\mathfrak{b}\in \mathcal{U}$ and $\mathfrak{t},\mathfrak{s}>0$:

(FM1) $\mathfrak{M}(\mathfrak{z},\mathfrak{g},\mathfrak{t})>0$;

(FM2) $\mathfrak{M}(\mathfrak{z},\mathfrak{g},\mathfrak{t})=1$ if and only if $\mathfrak{z}=\mathfrak{w}$;

(FM3) $\mathfrak{M}(\mathfrak{z},\mathfrak{g},\mathfrak{t})=\mathfrak{M}(\mathfrak{g},\mathfrak{z},\mathfrak{t})$;

(FM4) $\mathfrak{M}(\mathfrak{z},\mathfrak{b},\mathfrak{t}+\mathfrak{s})\ge \mathfrak{M}(\mathfrak{z},\mathfrak{g},\mathfrak{t})\star \mathfrak{M}(\mathfrak{g},\mathfrak{b},\mathfrak{s})$;

(FM5) $\mathfrak{M}(\mathfrak{z},\mathfrak{g},·):(0,+\infty )\to (0,1]$ is a continuous function.

These conditions collectively establish the framework for $(\mathcal{U},\mathfrak{M},\star )$ to be regarded as a fuzzy metric space.

Example 2.

Consider $\mathcal{U}={\mathbb{R}}^4$. Let us define a continuous t-norm ⋆ as $\eta \star \nu =\eta \nu$ for all $\eta ,\nu \in [0,1]$. Additionally, define a mapping $\mathfrak{M}:\mathcal{U}\times \mathcal{U}\times (0,+\infty )\to (0,1]$ as follows:

For each $t>0$,

$$\mathfrak{M}(\mathfrak{z},\mathfrak{g},\mathfrak{t})={\displaystyle \frac{\mathfrak{t}}{\mathfrak{t}+{\left({\Sigma }_{j=1}^4{(z_j-w_j)}^3\right)}^{\frac{1}{3}}}},$$

where $\mathfrak{z}=\{{\mathfrak{z}}_1,{\mathfrak{z}}_2,{\mathfrak{z}}_3,{\mathfrak{z}}_4\},\mathfrak{g}=\{{\mathfrak{g}}_1,{\mathfrak{g}}_2,{\mathfrak{g}}_3,{\mathfrak{g}}_4\}\in \mathcal{U}$.

Then, $(\mathcal{U},\mathfrak{M},\star )$ forms a fuzzy metric space.

Example 3.

Consider $\mathcal{U}=\mathcal{C}[0,2]\left(\mathbb{R}\right)$ and ⋆ as a continuous t-norm, defined as $\eta \star \nu =min\{\eta ,\nu \}$ for all $\eta ,\nu \in [0,1]$. Define a mapping $\mathfrak{M}:\mathcal{U}\times \mathcal{U}\times (0,+\infty )\to (0,1]$ by

$$\mathfrak{M}(\mathbf{w}\left(l\right),\mathbf{g}\left(l\right),\mathfrak{t})=\underset{l\in [0,2]}{\sup }{e}^{-{\displaystyle \frac{\left|\mathbf{w}\right(l)-\mathbf{g}(l\left)\right|}{\mathfrak{t}}}},\mathfrak{t}>0.$$

Then, $(\mathcal{U},\mathfrak{M},\star )$ is a fuzzy metric space.

Lemma 1

([3]). $\mathfrak{M}(\mathfrak{z},\mathfrak{w},.)$ is non-decreasing function for any $\mathfrak{z},\mathfrak{w}\in \mathcal{U}$.

Definition 3

([30]). A set $(\mathcal{U},\perp )$ is said to be orthogonal if there exists an element ${\mathfrak{h}}_0\in \mathcal{U}$ such that either ${\mathfrak{h}}_0\perp \mathfrak{h}$ or $\mathfrak{h}\perp {\mathfrak{h}}_0$ holds for every $\mathfrak{z}\in \mathcal{U}$, where ⊥ is a binary relation defined over $\mathcal{U}$.

Definition 4

([31]). In a fuzzy metric space $(\mathcal{U},\mathfrak{M},\star )$ with a binary relation ⊥ defined over $\mathcal{U}$, an ordered quadruple $(\mathcal{U},\mathfrak{M},\star ,\perp )$ is considered to be an orthogonal fuzzy metric space if there exists an element ${\mathfrak{s}}_0\in \mathcal{U}$ such that ${\mathfrak{s}}_0\perp \mathfrak{s}$ holds for every $\mathfrak{s}\in \mathcal{U}$.

Definition 5

([31]). Consider an orthogonal fuzzy metric space $(\mathcal{U},\mathfrak{M},\star ,\perp )$. Then:

(1) A sequence ${\left\{{\mathfrak{h}}_m\right\}}_{m\in \mathbb{N}}$ in $\mathcal{U}$ is termed as an orthogonal sequence (O-sequence) in $\mathcal{U}$ if ${\mathfrak{h}}_m\perp {\mathfrak{h}}_{m+1}\mathrm{for}\mathrm{all}m\in \mathbb{N}$.

(2) An O-sequence ${\left\{{\mathfrak{s}}_m\right\}}_{m\in \mathbb{N}}$ is said to converge to a point $\mathfrak{s}\in \mathfrak{U}$ if, for every $\mathfrak{t}>0,$${lim}_{m\to \infty }\mathfrak{M}({\mathfrak{s}}_m,\mathfrak{s},\mathfrak{t})=1.$

(3) A self-mapping $\mathfrak{T}$ on $\mathcal{U}$ is considered as ⊥-continuous at a point $\mathfrak{h}\in \mathcal{U}$ if, for every O-sequence ${\left\{{\mathfrak{h}}_m\right\}}_{m\in \mathbb{N}}$ of $\mathcal{U}$,

$$\underset{m\to \infty }{lim}\mathfrak{M}({\mathfrak{h}}_m,\mathfrak{h},\mathfrak{t})=1\Rightarrow \underset{m\to \infty }{lim}\mathfrak{M}(\mathfrak{T}{\mathfrak{h}}_m,\mathfrak{T}\mathfrak{h},\mathfrak{t})=1,\mathit{for}\mathit{all}\mathfrak{t}>0.$$

Moreover, $\mathfrak{T}$ is called ⊥-continuous on $\mathcal{U}$ if $\mathfrak{T}$ is ⊥-continuous at each point of $\mathcal{U}$.

(4) A self-mapping $\mathfrak{T}$ on $\mathcal{U}$ is referred to as ⊥-preserving, if for $\mathfrak{z},\mathfrak{g}\in \mathcal{U}$,

$$\mathfrak{z}\perp \mathfrak{g}\Rightarrow \mathfrak{T}\mathfrak{z}\perp \mathfrak{T}\mathfrak{g}.$$

(5) An O-sequence ${\left\{{\mathfrak{s}}_m\right\}}_{m\in \mathbb{N}}$ in $\mathcal{U}$ is described as a Cauchy O-sequence if, for each $0<ϵ<1$ and $\mathfrak{t}>0$, there exists $m_0\in \mathbb{N}$ such that $\mathfrak{M}({\mathfrak{s}}_p,{\mathfrak{s}}_q,\mathfrak{t})>1-ϵ$ $\mathit{for}\mathit{all}p,q\ge m_0$.

(6) $(\mathcal{U},\mathfrak{M},\star ,\perp )$ is purported to be an orthogonal complete (O-complete) fuzzy metric space if every Cauchy O-sequence in $\mathcal{U}$ converges to a point in $\mathcal{U}$.

3. Main Results

In order to demonstrate the existence and uniqueness of solutions of nonlinear differential equations with boundary conditions and nonlinear integral equations, we first introduce the concept of best proximity point and establish optimal results utilizing orthogonal fuzzy proximal contraction mappings.

The following are some essential definitions for the results.

Let us assume that $\mathfrak{K}$ and $\mathfrak{S}$ are non-empty subsets of an orthogonal fuzzy metric space $(\mathcal{U},\mathfrak{M},\star ,\perp )$. We define

$$\begin{array}{cc}\hfill & \mathfrak{M}(\mathfrak{K},\mathfrak{S},\mathfrak{t})=sup\left\{\mathfrak{M}\right(\mathfrak{z},\mathfrak{g},\mathfrak{t}):\mathfrak{z}\in \mathfrak{K}\mathrm{and}\mathfrak{g}\in \mathfrak{S}\},\hfill \\ \hfill & {\mathfrak{K}}_0=\{\mathfrak{z}\in \mathfrak{K}:\mathfrak{M}(\mathfrak{z},\mathfrak{g},\mathfrak{t})=\mathfrak{M}(\mathfrak{K},\mathfrak{S},\mathfrak{t})\mathrm{for}\mathrm{some}\mathfrak{g}\in \mathfrak{S}\},\hfill \\ \hfill & {\mathfrak{S}}_0=\{\mathfrak{g}\in \mathfrak{S}:\mathfrak{M}(\mathfrak{z},\mathfrak{g},\mathfrak{t})=\mathfrak{M}(\mathfrak{K},\mathfrak{S},\mathfrak{t})\mathrm{for}\mathrm{some}\mathfrak{z}\in \mathfrak{K}\}.\hfill \end{array}$$

It is important to note that for every $\mathfrak{z}\in {\mathfrak{K}}_0$, there exists $\mathfrak{g}\in {\mathfrak{S}}_0$ such that $\mathfrak{M}(\mathfrak{z},\mathfrak{g},\mathfrak{t})=\mathfrak{M}(\mathfrak{K},\mathfrak{S},\mathfrak{t})$. Also, for every $\mathfrak{g}\in {\mathfrak{S}}_0$, there exists $\mathfrak{z}\in {\mathfrak{K}}_0$ such that $\mathfrak{M}(\mathfrak{z},\mathfrak{g},\mathfrak{t})=\mathfrak{M}(\mathfrak{K},\mathfrak{S},\mathfrak{t})$.

Definition 6.

(Best Proximity Point in OFMS):  In the context of an orthogonal fuzzy metric space $(\mathcal{U},\mathfrak{M},\star ,\perp )$, let $\mathfrak{K}$ and $\mathfrak{S}$ be two non-empty subsets of $\mathcal{U}$. An element ${\mathfrak{h}}^{*}\in \mathfrak{K}$ is said to be a best proximity point of the mapping $\mathfrak{T}:\mathfrak{K}⟶\mathfrak{S}$ if and only if $\mathfrak{M}({\mathfrak{h}}^{*},\mathfrak{T}{\mathfrak{h}}^{*},\mathfrak{t})=\mathfrak{M}(\mathfrak{K},\mathfrak{S},\mathfrak{t})$.

Definition 7.

($\gamma$-Proximal Admissible Mapping in OFMS): In the context of an orthogonal fuzzy metric space $(\mathcal{U},\mathfrak{M},\star ,\perp )$, let $\mathfrak{K}$ and $\mathfrak{S}$ be two non-empty subsets of $\mathcal{U}$ and let $\gamma :\mathfrak{K}\times \mathfrak{K}\to [0,\infty )$. Then, $\mathfrak{T}:\mathfrak{K}⟶\mathfrak{S}$ is said to be a γ-proximal admissible mapping if $\gamma ({\mathbf{p}}_0,{\mathbf{q}}_0)\ge 1$ and $\mathfrak{M}({\mathbf{p}}_1,\mathfrak{T}{\mathbf{p}}_0,\mathfrak{t})=\mathfrak{M}({\mathbf{q}}_1,\mathfrak{T}{\mathbf{q}}_0,\mathfrak{t})=\mathfrak{M}(\mathfrak{K},\mathfrak{S},\mathfrak{t})$ implies $\gamma ({\mathbf{p}}_1,{\mathbf{q}}_1)\ge 1$ for all ${\mathbf{p}}_1,{\mathbf{p}}_0,{\mathbf{q}}_1,{\mathbf{q}}_0\in \mathfrak{K}$.

Definition 8.

(Proximally ⊥-Preserving Mapping in OFMS): In the context of an orthogonal fuzzy metric space $(\mathcal{U},\mathfrak{M},\star ,\perp )$, let $\mathfrak{K}$ and $\mathfrak{S}$ be a couple of non-empty subsets of $\mathcal{U}$. A mapping $\mathfrak{T}:\mathfrak{K}\to \mathfrak{S}$ is said to be a proximally ⊥-preserving mapping if and only if, for any ${\mathfrak{z}}_1,{\mathfrak{z}}_2,{\mathfrak{w}}_1,{\mathfrak{w}}_2\in \mathfrak{K}$,

$$\left.\begin{array}{cc}\hfill {\mathfrak{w}}_1& \perp {\mathfrak{w}}_2\hfill \\ \hfill \mathfrak{M}({\mathfrak{z}}_1,\mathfrak{T}{\mathfrak{w}}_1,\mathfrak{t})& =\mathfrak{M}(\mathfrak{K},\mathfrak{S},\mathfrak{t})\hfill \\ \hfill \mathfrak{M}({\mathfrak{z}}_2,\mathfrak{T}{\mathfrak{w}}_2,\mathfrak{t})& =\mathfrak{M}(\mathfrak{K},\mathfrak{S},\mathfrak{t})\hfill \end{array}\right\}\Rightarrow {\mathfrak{z}}_1\perp {\mathfrak{z}}_2.$$

Definition 9.

(Fuzzy P-property):  Suppose $\mathfrak{K}$ and $\mathfrak{S}$ are a pair of non-empty subsets of an orthogonal fuzzy metric space $(\mathcal{U},\mathfrak{M},\star ,\perp )$ such that ${\mathfrak{K}}_0\ne \varnothing$. The pair $(\mathfrak{K},\mathfrak{S})$ satisfies the fuzzy P-property if and only if $\mathfrak{M}(a,h,\mathfrak{t})=\mathfrak{M}(\mathfrak{K},\mathfrak{S},\mathfrak{t})=\mathfrak{M}(e,v,\mathfrak{t})$ implies $\mathfrak{M}(a,e,\mathfrak{t})=\mathfrak{M}(h,v,\mathfrak{t})$ for all $a,e\in \mathfrak{K}$ and $h,v\in \mathfrak{S}$.

Definition 10.

Let $\Theta ,\Psi$ be the family of functions $\theta ,\psi :(0,\infty )\to [0,\infty )$, where:

(i) θ is decreasing and ψ is increasing.

(ii) Both must attain continuity.

Definition 11.

Let Φ be the family of functions $\varphi :(0,1]\to [0,\infty )$, where

(i) ϕ is increasing.

(ii) ϕ attains continuity.

(iii) ${\varphi }^n\left(\mathfrak{w}\right)\to 1$ as $n\to \infty$ for each $\mathfrak{w}\in (0,1]$.

Now, we present our results.

Theorem 1.

Consider $\mathfrak{K}$ and $\mathfrak{S}$ as two non-empty closed subsets of an orthogonal complete fuzzy metric space $(\mathcal{U},\mathfrak{M},\star ,\perp )$ where ${\mathfrak{K}}_0\ne \varnothing$. Consider a mapping $\mathfrak{T}:\mathfrak{K}\to \mathfrak{S}$ such that for some $\mathbf{k}\in (0,1)$,

$$\begin{array}{c}\hfill \mathfrak{M}\left(\mathfrak{T}\right(\mathfrak{z}),\mathfrak{T}(\mathfrak{g}),\mathbf{k}\mathfrak{t})\ge \mathfrak{M}(\mathfrak{z},\mathfrak{g},\mathfrak{t})\end{array}$$

for all $\mathfrak{z}\perp \mathfrak{g}$, $\mathfrak{z},\mathfrak{g}\in \mathfrak{K}$ satisfying the following:

(i) $\mathfrak{T}\left({\mathfrak{K}}_0\right)\subseteq {\mathfrak{S}}_0$ and $(\mathfrak{K},\mathfrak{S})$ satisfies the fuzzy P-property.

(ii) There exist ${\mathfrak{z}}_1,{\mathfrak{z}}_0\in \mathcal{A}$ such that

$$\begin{array}{cc}\hfill & \mathfrak{M}({\mathfrak{z}}_1,\mathfrak{T}\left({\mathfrak{z}}_0\right),\mathfrak{t})=\mathfrak{M}(\mathfrak{K},\mathfrak{S},\mathfrak{t})\hfill \\ \hfill & \mathit{and}{\mathfrak{z}}_0\perp {\mathfrak{z}}_1.\hfill \end{array}$$

(iii) $\mathfrak{T}$ is ⊥-continuous and proximally ⊥-preserving.

Then, $\mathfrak{T}$ has a unique best proximity point in $\mathfrak{K}$.

Proof. 

From condition (ii), there exist ${\mathfrak{z}}_0,{\mathfrak{z}}_1\in {\mathfrak{K}}_0$ such that

$$\begin{array}{cc}\hfill & \mathfrak{M}({\mathfrak{z}}_1,\mathfrak{T}\left({\mathfrak{z}}_0\right),\mathfrak{t})=\mathfrak{M}(\mathfrak{K},\mathfrak{S},\mathfrak{t})\hfill \\ & \mathrm{and}{\mathfrak{z}}_0\perp {\mathfrak{z}}_1.\hfill \end{array}$$

Because $\mathfrak{T}\left({\mathfrak{K}}_0\right)\subseteq {\mathfrak{S}}_0$, there exists ${\mathfrak{z}}_2\in {\mathfrak{K}}_0$ such that

$$\begin{array}{c}\hfill \mathfrak{M}({\mathfrak{z}}_2,\mathfrak{T}\left({\mathfrak{z}}_1\right),\mathfrak{t})=\mathfrak{M}(\mathfrak{K},\mathfrak{S},\mathfrak{t}).\end{array}$$

By the proximally ⊥-preserving condition of $\mathfrak{T}$, we obtain

$$\begin{array}{c}\hfill {\mathfrak{z}}_1\perp {\mathfrak{z}}_2.\end{array}$$

Proceeding in this way, we obtain a sequence $\left\{{\mathfrak{z}}_n\right\}\in {\mathfrak{K}}_0$ such that

$$\begin{array}{cc}\hfill \mathfrak{M}({\mathfrak{z}}_{n+1},\mathfrak{T}\left({\mathfrak{z}}_n\right),\mathfrak{t})& =\mathfrak{M}(\mathfrak{K},\mathfrak{S},\mathfrak{t})\mathrm{for}\mathrm{all}n\in \mathbb{N}\hfill \\ \hfill \mathrm{and}{\mathfrak{z}}_n& \perp {\mathfrak{z}}_{n+1}.\hfill \end{array}$$

By the fuzzy P-property,

$$\begin{array}{cc}\hfill \mathfrak{M}({\mathfrak{z}}_n,{\mathfrak{z}}_{n+1},\mathfrak{t})& =\mathfrak{M}(\mathfrak{T}\left({\mathfrak{z}}_{n-1}\right),\mathfrak{T}\left({\mathfrak{z}}_n\right),\mathfrak{t})\hfill \\ & \ge \mathfrak{M}({\mathfrak{z}}_{n-1},{\mathfrak{z}}_n,{\displaystyle \frac{\mathfrak{t}}{\mathbf{k}}})\hfill \\ & .\hfill \\ & .\hfill \\ & .\hfill \\ & \ge \mathfrak{M}({\mathfrak{z}}_0,{\mathfrak{z}}_1,{\displaystyle \frac{\mathfrak{t}}{{\mathbf{k}}^n}}).\hfill \end{array}$$

Therefore, $\underset{n\to \infty }{\lim }\mathfrak{M}({\mathfrak{z}}_n,{\mathfrak{z}}_{n+1},\mathfrak{t})=1$.

Now,

$$\begin{array}{cc}\hfill \mathfrak{M}({\mathfrak{z}}_n,{\mathfrak{z}}_{n+p},\mathfrak{t})& =\mathfrak{M}({\mathfrak{z}}_n,{\mathfrak{z}}_{n+p},{\displaystyle \frac{\mathfrak{t}}{p}}p)\hfill \\ & \ge \mathfrak{M}({\mathfrak{z}}_n,{\mathfrak{z}}_{n+1},{\displaystyle \frac{\mathfrak{t}}{p}})\star \cdots \star \mathfrak{M}({\mathfrak{z}}_{n+p-1},{\mathfrak{z}}_{n+p},{\displaystyle \frac{\mathfrak{t}}{p}})\hfill \\ & \ge \mathfrak{M}({\mathfrak{z}}_0,{\mathfrak{z}}_1,{\displaystyle \frac{\mathfrak{t}}{p{\mathbf{k}}^n}})\star \cdots \star \mathfrak{M}({\mathfrak{z}}_0,{\mathfrak{z}}_1,{\displaystyle \frac{\mathfrak{t}}{p{\mathbf{k}}^{n+p-1}}}).\hfill \end{array}$$

Because $\underset{\mathfrak{t}\to \infty }{\lim }\mathfrak{M}(\mathfrak{z},\mathfrak{g},\mathfrak{t})=1$ for all $\mathfrak{z},\mathfrak{g}\in \mathfrak{K}$, letting $n\to \infty$, we get $\mathfrak{M}({\mathfrak{z}}_n,{\mathfrak{z}}_{n+p},\mathfrak{t})\to 1$.

Thus, $\left\{{\mathfrak{z}}_n\right\}$ is a Cauchy O-sequence.

Because $\mathcal{U}$ is an orthogonal complete fuzzy metric space and $\mathfrak{K}$ is a closed subset in $\mathcal{U}$, there exists ${\mathfrak{z}}^{*}\in \mathfrak{K}$ such that $\underset{n\to \infty }{lim}\mathfrak{M}({\mathfrak{z}}_n,{\mathfrak{z}}^{*},\mathfrak{t})=1$, i.e., ${\mathfrak{z}}_n={\mathfrak{z}}^{*}$.

Now, $\mathfrak{T}$ is ⊥-continuous and $\left\{{\mathfrak{z}}_n\right\}$ is a Cauchy O-sequence.

Because ${\mathfrak{z}}_n\to {\mathfrak{z}}^{*}$, we have $\mathfrak{T}\left({\mathfrak{z}}_n\right)\to \mathfrak{T}\left({\mathfrak{z}}^{*}\right)$ as $n\to \infty$.

Therefore,

$$\begin{array}{c}\hfill \mathfrak{M}(\mathfrak{K},\mathfrak{S},\mathfrak{t})=\mathfrak{M}({\mathfrak{z}}_{n+1},\mathfrak{T}\left({\mathfrak{z}}_n\right),\mathfrak{t})\to \mathfrak{M}({\mathfrak{z}}^{*},\mathfrak{T}\left({\mathfrak{z}}^{*}\right),\mathfrak{t}),\mathrm{as}n\to \infty .\end{array}$$

Thus, ${\mathfrak{z}}^{*}\in \mathfrak{K}$ is a best proximity point of $\mathfrak{T}$.

Assume that ${\mathfrak{g}}^{*}\in \mathcal{A}$ is another best proximity point of $\mathfrak{T}$ such that ${\mathfrak{z}}^{*}\perp {\mathfrak{g}}^{*}$.

Then,

$$\begin{array}{c}\hfill \mathfrak{M}({\mathfrak{z}}^{*},\mathfrak{T}\left({\mathfrak{z}}^{*}\right),\mathfrak{t})=\mathfrak{M}(\mathfrak{K},\mathfrak{S},\mathfrak{t})=\mathfrak{M}({\mathfrak{g}}^{*},\mathfrak{T}\left({\mathfrak{g}}^{*}\right),\mathfrak{t}).\end{array}$$

Using the fuzzy P-property,

$$\begin{array}{cc}\hfill \mathfrak{M}({\mathfrak{z}}^{*},{\mathfrak{g}}^{*},\mathfrak{t})& =\mathfrak{M}(\mathfrak{T}\left({\mathfrak{z}}^{*}\right),\mathfrak{T}\left({\mathfrak{g}}^{*}\right),\mathfrak{t})\hfill \\ & \ge \mathfrak{M}({\mathfrak{z}}^{*},{\mathfrak{g}}^{*},{\displaystyle \frac{\mathfrak{t}}{\mathbf{k}}})\hfill \\ & >\mathfrak{M}({\mathfrak{z}}^{*},{\mathfrak{g}}^{*},\mathfrak{t}),\hfill \end{array}$$

which is a contradiction, i.e., ${\mathfrak{z}}^{*}={\mathfrak{g}}^{*}$.

Hence, $\mathfrak{T}$ has a unique best proximity point. □

Theorem 2.

Consider $(\mathcal{U},\mathfrak{M},\star ,\perp )$ as an O-complete fuzzy metric space and $\mathfrak{K},\mathfrak{S}$ as two non-empty closed subsets of $\mathcal{U}$ such that ${\mathfrak{K}}_0\ne \varnothing$. Consider a γ-proximal admissible mapping $\mathfrak{T}:\mathfrak{K}\to \mathfrak{S}$ such that

$$\begin{array}{c}\hfill \mathfrak{M}\left(\mathfrak{T}\right(\mathfrak{z}),\mathfrak{T}(\mathfrak{g}),\mathfrak{t})\ge \gamma (\mathfrak{z},\mathfrak{g})\varphi \left(\mathfrak{M}\right(\mathfrak{z},\mathfrak{g},\mathfrak{t}\left)\right)\end{array}$$

for all $\mathfrak{z}\perp \mathfrak{g}$, $\mathfrak{z},\mathfrak{g}\in \mathfrak{K}$ satisfying the following:

(i) $\mathfrak{T}\left({\mathfrak{K}}_0\right)\subseteq {\mathfrak{S}}_0$ and $(\mathfrak{K},\mathfrak{S})$ satisfies the fuzzy P-property.

(ii) There exist ${\mathfrak{z}}_1,{\mathfrak{z}}_0\in \mathfrak{K}$ such that

$$\begin{array}{cc}\hfill & \mathfrak{M}({\mathfrak{z}}_1,\mathfrak{T}\left({\mathfrak{z}}_0\right),\mathfrak{t})=\mathfrak{M}(\mathfrak{K},\mathfrak{S},\mathfrak{t})\hfill \\ & \mathit{and}{\mathfrak{z}}_0\perp {\mathfrak{z}}_1.\hfill \end{array}$$

(iii) $\mathfrak{T}$ is ⊥-continuous and proximally ⊥-preserving.

Then, $\mathfrak{T}$ has a unique best proximity point in $\mathfrak{K}$.

Proof. 

From condition (ii), there exist ${\mathfrak{z}}_0,{\mathfrak{z}}_1\in {\mathfrak{K}}_0$ such that

$$\begin{array}{cc}\hfill & \mathfrak{M}({\mathfrak{z}}_1,\mathfrak{T}\left({\mathfrak{z}}_0\right),\mathfrak{t})=\mathfrak{M}(\mathfrak{K},\mathfrak{S},\mathfrak{t})\hfill \\ & \mathrm{and}{\mathfrak{z}}_0\perp {\mathfrak{z}}_1.\hfill \end{array}$$

Because $\mathfrak{T}\left({\mathfrak{K}}_0\right)\subseteq {\mathfrak{S}}_0$, there exists ${\mathfrak{z}}_2\in {\mathfrak{K}}_0$ such that

$$\begin{array}{c}\hfill \mathfrak{M}({\mathfrak{z}}_2,\mathfrak{T}\left({\mathfrak{z}}_1\right),\mathfrak{t})=\mathfrak{M}(\mathfrak{K},\mathfrak{S},\mathfrak{t}).\end{array}$$

Due to the proximally ⊥-preserving condition of $\mathfrak{T}$, we obtain

$$\begin{array}{c}\hfill {\mathfrak{z}}_1\perp {\mathfrak{z}}_2.\end{array}$$

Proceeding in this way, we obtain a sequence $\left\{{\mathfrak{z}}_n\right\}\in {\mathfrak{K}}_0$ such that

$$\begin{array}{cc}\hfill \mathfrak{M}({\mathfrak{z}}_{n+1},\mathfrak{T}\left({\mathfrak{z}}_n\right),\mathfrak{t})& =\mathfrak{M}(\mathfrak{K},\mathfrak{S},\mathfrak{t})\mathrm{for}\mathrm{all}n\in \mathbb{N}\hfill \\ \hfill \mathrm{and}{\mathfrak{z}}_n& \perp {\mathfrak{z}}_{n+1}.\hfill \end{array}$$

By the fuzzy P-property,

$$\begin{array}{cc}\hfill \mathfrak{M}({\mathfrak{z}}_n,{\mathfrak{z}}_{n+1},\mathfrak{t})& =\mathfrak{M}(\mathfrak{T}\left({\mathfrak{z}}_{n-1}\right),\mathfrak{T}\left({\mathfrak{z}}_n\right),\mathfrak{t})\hfill \\ & \ge \gamma ({\mathfrak{z}}_{n-1},{\mathfrak{z}}_n)\varphi \left(\mathfrak{M}({\mathfrak{z}}_{n-1},{\mathfrak{z}}_n,\mathfrak{t})\right)\hfill \\ & \ge \varphi \left(\mathfrak{M}({\mathfrak{z}}_{n-1},{\mathfrak{z}}_n,\mathfrak{t})\right)\hfill \\ & .\hfill \\ & .\hfill \\ & .\hfill \\ & \ge {\varphi }^n\left(\mathfrak{M}({\mathfrak{z}}_0,{\mathfrak{z}}_1,\mathfrak{t})\right).\hfill \end{array}$$

Because ${\varphi }^n\left(\mathfrak{w}\right)\to 1$ as $n\to \infty$, we obtain

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\mathfrak{M}({\mathfrak{z}}_n,{\mathfrak{z}}_{n+1},\mathfrak{t})=1.\end{array}$$

Now,

$$\begin{array}{cc}\hfill \mathfrak{M}({\mathfrak{z}}_n,{\mathfrak{z}}_{n+p},\mathfrak{t})& =\mathfrak{M}({\mathfrak{z}}_n,{\mathfrak{z}}_{n+p},{\displaystyle \frac{\mathfrak{t}}{p}}p)\hfill \\ & \ge \mathfrak{M}({\mathfrak{z}}_n,{\mathfrak{z}}_{n+1},{\displaystyle \frac{\mathfrak{t}}{p}})\star \cdots \star \mathfrak{M}({\mathfrak{z}}_{n+p-1},{\mathfrak{z}}_{n+p},{\displaystyle \frac{\mathfrak{t}}{p}}).\hfill \end{array}$$

Therefore, $\underset{n\to \infty }{\lim }\mathfrak{M}({\mathfrak{z}}_n,{\mathfrak{z}}_{n+p},\mathfrak{t})=1$.

Thus, $\left\{{\mathfrak{z}}_n\right\}$ is a Cauchy O-sequence.

Because $\mathcal{U}$ is an O-complete fuzzy metric space and $\mathfrak{K}$ is a closed subset in $\mathcal{U}$, there exists ${\mathfrak{z}}^{*}\in \mathfrak{K}$ such that $\underset{n\to \infty }{lim}\mathfrak{M}({\mathfrak{z}}_n,{\mathfrak{z}}^{*},\mathfrak{t})=1$, i.e., ${\mathfrak{z}}_n={\mathfrak{z}}^{*}$.

Now, $\mathfrak{T}$ is ⊥-continuous and $\left\{{\mathfrak{z}}_n\right\}$ is a Cauchy O-sequence.

Because ${\mathfrak{z}}_n\to {\mathfrak{z}}^{*}$, we have $\mathfrak{T}\left({\mathfrak{z}}_n\right)\to \mathfrak{T}\left({\mathfrak{z}}^{*}\right)$ as $n\to \infty$.

Hence,

$$\begin{array}{c}\hfill \mathfrak{M}(\mathfrak{K},\mathfrak{S},\mathfrak{t})=\mathfrak{M}({\mathfrak{z}}_{n+1},\mathfrak{T}\left({\mathfrak{z}}_n\right),\mathfrak{t})\to \mathfrak{M}({\mathfrak{z}}^{*},\mathfrak{T}\left({\mathfrak{z}}^{*}\right),\mathfrak{t}),\mathrm{as}n\to \infty .\end{array}$$

Thus, ${\mathfrak{z}}^{*}\in \mathfrak{K}$ is a best proximity point of $\mathfrak{T}$.

Assume that ${\mathfrak{g}}^{*}\in \mathfrak{K}$ has another best proximity point of $\mathfrak{T}$ such that ${\mathfrak{z}}^{*}\perp {\mathfrak{g}}^{*}$.

Then,

$$\begin{array}{c}\hfill \mathfrak{M}({\mathfrak{z}}^{*},\mathfrak{T}\left({\mathfrak{z}}^{*}\right),\mathfrak{t})=\mathfrak{M}(\mathfrak{K},\mathfrak{S},\mathfrak{t})=\mathfrak{M}({\mathfrak{g}}^{*},\mathfrak{T}\left({\mathfrak{g}}^{*}\right),\mathfrak{t}).\end{array}$$

Now, by the fuzzy P-property, we have

$$\begin{array}{cc}\hfill \mathfrak{M}({\mathfrak{z}}^{*},{\mathfrak{g}}^{*},\mathfrak{t})& =\mathfrak{M}(\mathfrak{T}\left({\mathfrak{z}}^{*}\right),\mathfrak{T}\left({\mathfrak{g}}^{*}\right),\mathfrak{t})\hfill \\ & \ge \gamma ({\mathfrak{z}}^{*},{\mathfrak{g}}^{*})\varphi \left(\mathfrak{M}({\mathfrak{z}}^{*},{\mathfrak{g}}^{*},\mathfrak{t})\right)\hfill \\ & \ge \varphi \left(\mathfrak{M}({\mathfrak{z}}^{*},{\mathfrak{g}}^{*},\mathfrak{t})\right)\hfill \\ & >\mathfrak{M}({\mathfrak{z}}^{*},{\mathfrak{g}}^{*},\mathfrak{t}),\hfill \end{array}$$

which is a contradiction, i.e., ${\mathfrak{z}}^{*}={\mathfrak{g}}^{*}$.

Hence, $\mathfrak{T}$ has a unique best proximity point. □

Theorem 3.

Consider $(\mathcal{U},\mathfrak{M},\star ,\perp )$ as an O-complete fuzzy metric space and $\mathfrak{K}$ and $\mathfrak{S}$ as two non-empty closed subsets of $\mathcal{U}$ such that ${\mathfrak{K}}_0\ne \varnothing$. Consider $\mathfrak{T}:\mathfrak{K}\to \mathfrak{S}$ as a γ-proximal admissible mapping such that for some $\wp \ne 0$ and $\mathbf{k}\in (0,1)$,

$$\begin{array}{c}\hfill \gamma (\mathfrak{z},\mathfrak{g})\theta \left(\mathfrak{M}(\mathfrak{T}\left(\mathfrak{z}\right),\mathfrak{T}\left(\mathfrak{g}\right),\mathfrak{t})\right)\le {\displaystyle \frac{1}{cosh(\wp )}}\theta \left({\displaystyle \frac{\mathfrak{M}(\mathfrak{z},\mathfrak{g},\mathfrak{t})}{1-\mathbf{k}\mathfrak{M}(\mathfrak{z},\mathfrak{g},\mathfrak{t})}}\right)-\psi \left(\mathfrak{M}(\mathfrak{T}\left(\mathfrak{z}\right),\mathfrak{T}\left(\mathfrak{g}\right),\mathfrak{t})\right)\end{array}$$

for all $\mathfrak{z}\perp \mathfrak{g}$, $\mathfrak{z},\mathfrak{g}\in \mathfrak{K}$ satisfying the following:

(i) $\mathfrak{T}\left({\mathfrak{K}}_0\right)\subseteq {\mathfrak{S}}_0$ and $(\mathfrak{K},\mathfrak{S})$ satisfies the fuzzy P-property.

(ii) There exist ${\mathfrak{z}}_1,{\mathfrak{z}}_0\in \mathfrak{K}$ such that

$$\begin{array}{cc}\hfill & \mathfrak{M}({\mathfrak{z}}_1,\mathfrak{T}\left({\mathfrak{z}}_0\right),\mathfrak{t})=\mathfrak{M}(\mathfrak{K},\mathfrak{S},\mathfrak{t})\hfill \\ & \mathit{and}{\mathfrak{z}}_0\perp {\mathfrak{z}}_1.\hfill \end{array}$$

(iii) $\mathfrak{T}$ is ⊥-continuous and proximally ⊥-preserving.

(iv) ⊥ is transitive.

Then, $\mathfrak{T}$ has a unique best proximity point in $\mathfrak{K}$.

Proof. 

From condition (ii), there exist ${\mathfrak{z}}_0,{\mathfrak{z}}_1\in {\mathfrak{K}}_0$ such that

$$\begin{array}{cc}\hfill & \mathfrak{M}({\mathfrak{z}}_1,\mathfrak{T}\left({\mathfrak{z}}_0\right),\mathfrak{t})=\mathfrak{M}(\mathfrak{K},\mathfrak{S},\mathfrak{t})\hfill \\ & \mathrm{and}{\mathfrak{z}}_0\perp {\mathfrak{z}}_1.\hfill \end{array}$$

Because $\mathfrak{T}\left({\mathfrak{K}}_0\right)\subseteq {\mathfrak{S}}_0$, there exists ${\mathfrak{z}}_2\in {\mathfrak{K}}_0$ such that

$$\begin{array}{c}\hfill \mathfrak{M}({\mathfrak{z}}_2,\mathfrak{T}\left({\mathfrak{z}}_1\right),\mathfrak{t})=\mathfrak{M}(\mathfrak{K},\mathfrak{S},\mathfrak{t}).\end{array}$$

By the proximally ⊥-preserving condition of $\mathfrak{T}$, we obtain

$$\begin{array}{c}\hfill {\mathfrak{z}}_1\perp {\mathfrak{z}}_2.\end{array}$$

Proceeding in this way, we obtain a sequence $\left\{{\mathfrak{z}}_n\right\}\in {\mathfrak{K}}_0$ such that

$$\begin{array}{cc}\hfill \mathfrak{M}({\mathfrak{z}}_{n+1},\mathfrak{T}\left({\mathfrak{z}}_n\right),\mathfrak{t})& =\mathfrak{M}(\mathfrak{K},\mathfrak{S},\mathfrak{t})\mathrm{for}\mathrm{all}n\in \mathbb{N}\hfill \\ \hfill \mathrm{and}{\mathfrak{z}}_n& \perp {\mathfrak{z}}_{n+1}.\hfill \end{array}$$

Using the fuzzy P-property, we get

$$\begin{array}{cc}\hfill \mathfrak{M}({\mathfrak{z}}_n,{\mathfrak{z}}_{n+1},\mathfrak{t})& =\mathfrak{M}(\mathfrak{T}\left({\mathfrak{z}}_{n-1}\right),\mathfrak{T}\left({\mathfrak{z}}_n\right),\mathfrak{t})\hfill \\ \hfill \Rightarrow \theta \left(\mathfrak{M}({\mathfrak{z}}_n,{\mathfrak{z}}_{n+1},\mathfrak{t})\right)& =\theta \left(\mathfrak{M}(\mathfrak{T}\left({\mathfrak{z}}_{n-1}\right),\mathfrak{T}\left({\mathfrak{z}}_n\right),\mathfrak{t})\right)\hfill \\ & \le \gamma ({\mathfrak{z}}_{n-1},{\mathfrak{z}}_n)\theta \left(\mathfrak{M}(\mathfrak{T}\left({\mathfrak{z}}_{n-1}\right),\mathfrak{T}\left({\mathfrak{z}}_n\right),\mathfrak{t})\right)\hfill \\ & \le {\displaystyle \frac{1}{cosh(\wp )}}\theta \left({\displaystyle \frac{\mathfrak{M}({\mathfrak{z}}_{n-1},{\mathfrak{z}}_n,\mathfrak{t})}{1-\mathbf{k}\mathfrak{M}({\mathfrak{z}}_{n-1},{\mathfrak{z}}_n,\mathfrak{t})}}\right)-\psi \left(\mathfrak{M}(\mathfrak{T}\left({\mathfrak{z}}_{n-1}\right),\mathfrak{T}\left({\mathfrak{z}}_n\right),\mathfrak{t})\right)\hfill \\ & \le {\displaystyle \frac{1}{cosh(\wp )}}\theta \left({\displaystyle \frac{\mathfrak{M}({\mathfrak{z}}_{n-1},{\mathfrak{z}}_n,\mathfrak{t})}{1-\mathbf{k}\mathfrak{M}({\mathfrak{z}}_{n-1},{\mathfrak{z}}_n,\mathfrak{t})}}\right)\hfill \\ & <\theta \left({\displaystyle \frac{\mathfrak{M}({\mathfrak{z}}_{n-1},{\mathfrak{z}}_n,\mathfrak{t})}{1-\mathbf{k}\mathfrak{M}({\mathfrak{z}}_{n-1},{\mathfrak{z}}_n,\mathfrak{t})}}\right)\hfill \\ & <\theta \left(\mathfrak{M}({\mathfrak{z}}_{n-1},{\mathfrak{z}}_n,\mathfrak{t})\right)\left(\mathrm{as}\theta \mathrm{is}\mathrm{decreasing}\right).\hfill \end{array}$$

We define ${\zeta }_n\left(\mathfrak{t}\right)=\mathfrak{M}({\mathfrak{z}}_{n-1},{\mathfrak{z}}_n,\mathfrak{t})$ for all $n\in \mathbb{N}\cup \left\{0\right\}$, meaning that we obtain

$$\begin{array}{c}\hfill \theta \left({\zeta }_n\left(\mathfrak{t}\right)\right)<\theta \left({\displaystyle \frac{{\zeta }_{n-1}\left(\mathfrak{t}\right)}{1-\mathbf{k}{\zeta }_{n-1}\left(\mathfrak{t}\right)}}\right)<\theta \left({\zeta }_{n-1}\left(\mathfrak{t}\right)\right).\end{array}$$

Because $\theta$ is decreasing, ${\zeta }_{n-1}\left(\mathfrak{t}\right)<{\zeta }_n\left(\mathfrak{t}\right)$, i.e., $\left\{{\zeta }_n\left(\mathfrak{t}\right)\right\}$ is an increasing sequence for all $\mathfrak{t}>0$.

Now, we take $\underset{n\to \infty }{lim}{\zeta }_n\left(\mathfrak{t}\right)=\zeta \left(\mathfrak{t}\right)$. We have to show that $\zeta \left(\mathfrak{t}\right)=1$ for all $\mathfrak{t}>0$.

On the contrary, we can assume that $0<\zeta \left({\mathfrak{t}}_0\right)<1$ for some ${\mathfrak{t}}_0>0$.

Letting $n\to \infty$, we get

$$\begin{array}{c}\hfill \theta \left(\zeta \left({\mathfrak{t}}_0\right)\right)\le \theta \left({\displaystyle \frac{\zeta \left({\mathfrak{t}}_0\right)}{1-\mathbf{k}\zeta \left({\mathfrak{t}}_0\right)}}\right)<\theta \left(\zeta \left({\mathfrak{t}}_0\right)\right),\end{array}$$

which is a contradiction; thus, $\zeta \left(\mathfrak{t}\right)=1,$ for all $\mathfrak{t}>0$.

Now, we shall prove that $\left\{{\mathfrak{z}}_n\right\}$ is an O-Cauchy sequence.

If possible, we assume that there exist $\mu \in (0,1),{\mathfrak{t}}_0>0$ and sequences $\left\{n\right(\mathfrak{r}\left)\right\}$ and $\left\{m\right(\mathfrak{r}\left)\right\}$ of natural numbers such that

$$\begin{array}{c}\hfill m\left(\mathfrak{r}\right)>n\left(\mathfrak{r}\right)>\mathfrak{r},\mathfrak{M}({\mathfrak{z}}_{m\left(\mathfrak{r}\right)},{\mathfrak{z}}_{n\left(\mathfrak{r}\right)},{\mathfrak{t}}_0)\le 1-\mu ,\mathfrak{M}({\mathfrak{z}}_{m\left(\mathfrak{r}\right)-1},{\mathfrak{z}}_{n\left(\mathfrak{r}\right)},{\mathfrak{t}}_0)>1-\mu .\end{array}$$

Now, for all $\mathfrak{r}$,

$$\begin{array}{cc}\hfill 1-\mu & \ge \mathfrak{M}({\mathfrak{z}}_{m\left(\mathfrak{r}\right)},{\mathfrak{z}}_{n\left(\mathfrak{r}\right)},{\mathfrak{t}}_0)\hfill \\ & \ge \mathfrak{M}\left({\mathfrak{z}}_{m\left(\mathfrak{r}\right)},{\mathfrak{z}}_{m\left(\mathfrak{r}\right)-1},{\displaystyle \frac{{\mathfrak{t}}_0}{2}}\right)\star \mathfrak{M}\left({\mathfrak{z}}_{m\left(\mathfrak{r}\right)-1},{\mathfrak{z}}_{n\left(\mathfrak{r}\right)},{\displaystyle \frac{{\mathfrak{t}}_0}{2}}\right)\hfill \\ & >{\zeta }_{m\left(\mathfrak{r}\right)-1}\left({\displaystyle \frac{{\mathfrak{t}}_0}{2}}\right)\star (1-\mu ).\hfill \end{array}$$

Letting $n\to \infty$, we obtain

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\mathfrak{M}({\mathfrak{z}}_{m\left(\mathfrak{r}\right)},{\mathfrak{z}}_{n\left(\mathfrak{r}\right)},{\mathfrak{t}}_0)=1-\mu .\end{array}$$

Now,

$$\begin{array}{c}\hfill \mathfrak{M}({\mathfrak{z}}_{m\left(\mathfrak{r}\right)+1},{\mathfrak{z}}_{n\left(\mathfrak{r}\right)+1},{\mathfrak{t}}_0)\ge \mathfrak{M}\left({\mathfrak{z}}_{m\left(\mathfrak{r}\right)+1},{\mathfrak{z}}_{m\left(\mathfrak{r}\right)},{\displaystyle \frac{{\mathfrak{t}}_0}{3}}\right)\star \mathfrak{M}\left({\mathfrak{z}}_{m\left(\mathfrak{r}\right)},{\mathfrak{z}}_{n\left(\mathfrak{r}\right)},{\displaystyle \frac{{\mathfrak{t}}_0}{3}}\right)\star \mathfrak{M}\left({\mathfrak{z}}_{n\left(\mathfrak{r}\right)},{\mathfrak{z}}_{n\left(\mathfrak{r}\right)+1},{\displaystyle \frac{{\mathfrak{t}}_0}{3}}\right).\end{array}$$

Again,

$$\begin{array}{c}\hfill \mathfrak{M}({\mathfrak{z}}_{m\left(\mathfrak{r}\right)},{\mathfrak{z}}_{n\left(\mathfrak{r}\right)},3{\mathfrak{t}}_0)\ge \mathfrak{M}({\mathfrak{z}}_{m\left(\mathfrak{r}\right)},{\mathfrak{z}}_{m\left(\mathfrak{r}\right)+1},{\mathfrak{t}}_0)\star \mathfrak{M}({\mathfrak{z}}_{m\left(\mathfrak{r}\right)+1},{\mathfrak{z}}_{n\left(\mathfrak{r}\right)+1},{\mathfrak{t}}_0)\star \mathfrak{M}({\mathfrak{z}}_{n\left(\mathfrak{r}\right)+1},{\mathfrak{z}}_{n\left(\mathfrak{r}\right)},{\mathfrak{t}}_0).\end{array}$$

Thus,

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\mathfrak{M}({\mathfrak{z}}_{m\left(\mathfrak{r}\right)+1},{\mathfrak{z}}_{n\left(\mathfrak{r}\right)+1},{\mathfrak{t}}_0)=1-\mu .\end{array}$$

Now, we know that

$$\begin{array}{cc}\hfill \mathfrak{M}({\mathfrak{z}}_{m\left(\mathfrak{r}\right)+1},\mathfrak{T}\left({\mathfrak{z}}_{m\left(\mathfrak{r}\right)}\right),{\mathfrak{t}}_0)& =\mathfrak{M}(\mathfrak{K},\mathfrak{S},{\mathfrak{t}}_0),\hfill \\ \hfill \mathfrak{M}({\mathfrak{z}}_{n\left(\mathfrak{r}\right)+1},\mathfrak{T}\left({\mathfrak{z}}_{n\left(\mathfrak{r}\right)}\right),{\mathfrak{t}}_0)& =\mathfrak{M}(\mathfrak{K},\mathfrak{S},{\mathfrak{t}}_0).\hfill \end{array}$$

Because ⊥ is transitive, we obtain ${\mathfrak{z}}_{n\left(\mathfrak{r}\right)}\perp {\mathfrak{z}}_{m\left(\mathfrak{r}\right)}$.

Now, by the fuzzy P-property, we have

$$\begin{array}{cc}\hfill \mathfrak{M}({\mathfrak{z}}_{m\left(\mathfrak{r}\right)+1},{\mathfrak{z}}_{n\left(\mathfrak{r}\right)+1},{\mathfrak{t}}_0)& =\mathfrak{M}(\mathfrak{T}\left({\mathfrak{z}}_{m\left(\mathfrak{r}\right)}\right),\mathfrak{T}\left({\mathfrak{z}}_{n\left(\mathfrak{r}\right)}\right),{\mathfrak{t}}_0)\hfill \\ \hfill \Rightarrow \theta \left(\mathfrak{M}({\mathfrak{z}}_{m\left(\mathfrak{r}\right)+1},{\mathfrak{z}}_{n\left(\mathfrak{r}\right)+1},{\mathfrak{t}}_0)\right)& =\theta \left(\mathfrak{M}(\mathfrak{T}\left({\mathfrak{z}}_{m\left(\mathfrak{r}\right)}\right),\mathfrak{T}\left({\mathfrak{z}}_{n\left(\mathfrak{r}\right)}\right),{\mathfrak{t}}_0)\right)\hfill \\ & \le \gamma ({\mathfrak{z}}_{m\left(\mathfrak{r}\right)},{\mathfrak{z}}_{n\left(\mathfrak{r}\right)})\theta \left(\mathfrak{M}(\mathfrak{T}\left({\mathfrak{z}}_{m\left(\mathfrak{r}\right)}\right),\mathfrak{T}\left({\mathfrak{z}}_{n\left(\mathfrak{r}\right)}\right),{\mathfrak{t}}_0)\right)\hfill \\ & \le {\displaystyle \frac{1}{cosh(\wp )}}\theta \left({\displaystyle \frac{\mathfrak{M}({\mathfrak{z}}_{m\left(\mathfrak{r}\right)},{\mathfrak{z}}_{n\left(\mathfrak{r}\right)},{\mathfrak{t}}_0)}{1-\mathbf{k}\mathfrak{M}({\mathfrak{z}}_{m\left(\mathfrak{r}\right)},{\mathfrak{z}}_{n\left(\mathfrak{r}\right)},{\mathfrak{t}}_0)}}\right)-\psi \left(\mathfrak{M}(\mathfrak{T}\left({\mathfrak{z}}_{m\left(\mathfrak{r}\right)}\right),\mathfrak{T}\left({\mathfrak{z}}_{n\left(\mathfrak{r}\right)}\right),{\mathfrak{t}}_0)\right)\hfill \\ & \le {\displaystyle \frac{1}{cosh(\wp )}}\theta \left({\displaystyle \frac{\mathfrak{M}({\mathfrak{z}}_{m\left(\mathfrak{r}\right)},{\mathfrak{z}}_{n\left(\mathfrak{r}\right)},{\mathfrak{t}}_0)}{1-\mathbf{k}\mathfrak{M}({\mathfrak{z}}_{m\left(\mathfrak{r}\right)},{\mathfrak{z}}_{n\left(\mathfrak{r}\right)},{\mathfrak{t}}_0)}}\right)\hfill \\ & <\theta \left({\displaystyle \frac{\mathfrak{M}({\mathfrak{z}}_{m\left(\mathfrak{r}\right)},{\mathfrak{z}}_{n\left(\mathfrak{r}\right)},{\mathfrak{t}}_0)}{1-\mathbf{k}\mathfrak{M}({\mathfrak{z}}_{m\left(\mathfrak{r}\right)},{\mathfrak{z}}_{n\left(\mathfrak{r}\right)},{\mathfrak{t}}_0)}}\right)\hfill \\ & <\theta \left(\mathfrak{M}({\mathfrak{z}}_{m\left(\mathfrak{r}\right)},{\mathfrak{z}}_{n\left(\mathfrak{r}\right)},{\mathfrak{t}}_0)\right).\hfill \end{array}$$

Letting $n\to \infty$, we get

$$\begin{array}{c}\hfill \theta (1-\mu )\le \theta \left({\displaystyle \frac{1-\mu }{1-\mathbf{k}(1-\mu )}}\right)<\theta (1-\mu ),\end{array}$$

which is a contradiction.

Thus, $\left\{{\mathfrak{z}}_n\right\}$ is an O-Cauchy sequence.

Because $\mathcal{U}$ is an O-complete fuzzy metric space and $\mathfrak{K}$ is a closed subset in $\mathcal{U}$, there exists ${\mathfrak{z}}^{*}\in \mathfrak{K}$ such that $\underset{n\to \infty }{lim}\mathfrak{M}({\mathfrak{z}}_n,{\mathfrak{z}}^{*},\mathfrak{t})=1$, i.e., ${\mathfrak{z}}_n={\mathfrak{z}}^{*}$.

Now, $\mathfrak{T}$ is ⊥-continuous and $\left\{{\mathfrak{z}}_n\right\}$ is a Cauchy O-sequence.

Because ${\mathfrak{z}}_n\to {\mathfrak{z}}^{*}$, we have $\mathfrak{T}\left({\mathfrak{z}}_n\right)\to \mathfrak{T}\left({\mathfrak{z}}^{*}\right)$ as $n\to \infty$.

Hence,

$$\begin{array}{c}\hfill \mathfrak{M}(\mathfrak{K},\mathfrak{S},\mathfrak{t})=\mathfrak{M}({\mathfrak{z}}_{n+1},\mathfrak{T}\left({\mathfrak{z}}_n\right),\mathfrak{t})\to \mathfrak{M}({\mathfrak{z}}^{*},\mathfrak{T}\left({\mathfrak{z}}^{*}\right),\mathfrak{t}),\mathrm{as}n\to \infty .\end{array}$$

Thus, ${\mathfrak{z}}^{*}\in \mathfrak{K}$ is a best proximity point of $\mathfrak{T}$.

Assume ${\mathfrak{g}}^{*}\in \mathfrak{K}$ as another best proximity point of $\mathfrak{T}$ such that ${\mathfrak{z}}^{*}\perp {\mathfrak{g}}^{*}$.

Then,

$$\begin{array}{c}\hfill \mathfrak{M}({\mathfrak{z}}^{*},\mathfrak{T}\left({\mathfrak{z}}^{*}\right),\mathfrak{t})=\mathfrak{M}(\mathfrak{K},\mathfrak{S},\mathfrak{t})=\mathfrak{M}({\mathfrak{g}}^{*},\mathfrak{T}\left({\mathfrak{g}}^{*}\right),\mathfrak{t}).\end{array}$$

Now, by the fuzzy P-property, we obtain

$$\begin{array}{cc}\hfill \mathfrak{M}({\mathfrak{z}}^{*},{\mathfrak{g}}^{*},\mathfrak{t})& =\mathfrak{M}(\mathfrak{T}\left({\mathfrak{z}}^{*}\right),\mathfrak{T}\left({\mathfrak{g}}^{*}\right),\mathfrak{t})\hfill \\ \hfill \Rightarrow \theta \left(\mathfrak{M}({\mathfrak{z}}^{*},{\mathfrak{g}}^{*},\mathfrak{t})\right)& =\theta \left(\mathfrak{M}(\mathfrak{T}\left({\mathfrak{z}}^{*}\right),\mathfrak{T}\left({\mathfrak{g}}^{*}\right),\mathfrak{t})\right)\hfill \\ & \le \gamma ({\mathfrak{z}}^{*},{\mathfrak{g}}^{*})\theta \left(\mathfrak{M}(\mathfrak{T}\left({\mathfrak{z}}^{*}\right),\mathfrak{T}\left({\mathfrak{g}}^{*}\right),\mathfrak{t})\right)\hfill \\ & \le {\displaystyle \frac{1}{cosh(\wp )}}\theta \left({\displaystyle \frac{\mathfrak{M}({\mathfrak{z}}^{*},{\mathfrak{w}}^{*},\mathfrak{t})}{1-\mathbf{k}\mathfrak{M}({\mathfrak{z}}^{*},{\mathfrak{w}}^{*},\mathfrak{t})}}\right)-\psi \left(\mathfrak{M}(\mathfrak{T}\left({\mathfrak{z}}^{*}\right),\mathfrak{T}\left({\mathfrak{w}}^{*}\right),\mathfrak{t})\right)\hfill \\ & \le {\displaystyle \frac{1}{cosh(\wp )}}\theta \left({\displaystyle \frac{\mathfrak{M}({\mathfrak{z}}^{*},{\mathfrak{w}}^{*},\mathfrak{t})}{1-\mathbf{k}\mathfrak{M}({\mathfrak{z}}^{*},{\mathfrak{w}}^{*},\mathfrak{t})}}\right)\hfill \\ & <\theta \left({\displaystyle \frac{\mathfrak{M}({\mathfrak{z}}^{*},{\mathfrak{w}}^{*},\mathfrak{t})}{1-\mathbf{k}\mathfrak{M}({\mathfrak{z}}^{*},{\mathfrak{w}}^{*},\mathfrak{t})}}\right)\hfill \\ & <\theta \left(\mathfrak{M}({\mathfrak{z}}^{*},{\mathfrak{w}}^{*},\mathfrak{t})\right),\hfill \end{array}$$

which is a contradiction. Thus, we have ${\mathfrak{z}}^{*}={\mathfrak{w}}^{*}$.

Hence, $\mathfrak{T}$ has a unique best proximity point. □

Example 4.

Assume $\mathcal{U}=\mathbb{R}\times \mathbb{R}$ and let $\mathfrak{M}:\mathcal{U}\times \mathcal{U}\times (0,\infty )\to (0,1]$ be the standard fuzzy metric, defined as

$$\begin{array}{c}\hfill \mathfrak{M}(({\mathfrak{s}}_1,{\mathfrak{z}}_1),({\mathfrak{s}}_2,{\mathfrak{z}}_2),t)={\displaystyle \frac{\mathfrak{t}}{\mathfrak{t}+|{\mathfrak{s}}_1-{\mathfrak{z}}_1|+|{\mathfrak{s}}_2-{\mathfrak{z}}_2|}}\end{array}$$

for all $\mathfrak{t}>0$. We define $(\mathfrak{u},\mathfrak{q})\perp (\mathfrak{s},\mathfrak{h})\iff \mathfrak{u}\le \mathfrak{s}$ for all $(\mathfrak{u},\mathfrak{q}),(\mathfrak{s},\mathfrak{h})\in \mathcal{U}$ and $\mathfrak{g}\star \mathfrak{k}=min\{\mathfrak{g},\mathfrak{k}\}$ for all $\mathfrak{g},\mathfrak{k}\in (0,1]$.

Now, consider $\mathfrak{K}=\left\{\right(1,-n\left)\right\}\cup \left\{\right(1,0\left)\right\}$ and $\mathfrak{S}=\{(0,{\displaystyle \frac{1}{2n}})\}\cup \left\{(0,0)\right\}$, where $n\in \mathbb{N}$

We define $\mathfrak{T}:\mathfrak{K}\to \mathfrak{S}$ as

$$\begin{array}{cc}\hfill \mathfrak{T}(1,-n)& =\left(0,{\displaystyle \frac{1}{2n}}\right),n\in \mathbb{N},\hfill \\ \hfill \mathit{and}\mathfrak{T}(1,0)& =(0,0).\hfill \end{array}$$

Thus, we have

$$\begin{array}{c}\hfill \mathfrak{M}((1,0),\mathfrak{T}(1,0),\mathfrak{t})=\mathfrak{M}((1,0),(0,0),\mathfrak{t})={\displaystyle \frac{\mathfrak{t}}{\mathfrak{t}+1}}=\mathfrak{M}(\mathfrak{K},\mathfrak{S},\mathfrak{t}).\end{array}$$

Now, taking $\mathbf{k}=0.9$, we can easily verify the following inequality:

$$\begin{array}{c}\hfill \mathfrak{M}(\mathfrak{T}\mathfrak{g},\mathfrak{T}\mathfrak{s},\mathbf{k}\mathfrak{t})\ge \mathfrak{M}(\mathfrak{g},\mathfrak{s},\mathfrak{t}),\forall \mathfrak{g},\mathfrak{s}\in \mathfrak{K}.\end{array}$$

Thus, all the assumptions of Theorem 1 are satisfied and $(1,0)$ is the unique best proximity point of $\mathfrak{T}$.

4. Consequences

Corollary 1.

Suppose $(\mathcal{U},\mathfrak{M},\star ,\perp )$ to be an O-complete fuzzy metric space. Consider $\mathfrak{T}:\mathcal{U}\to \mathcal{U}$ as an ⊥-continuous and ⊥-preserving mapping such that, for some $\mathbf{k}\in (0,1)$,

$$\begin{array}{c}\hfill \mathfrak{M}\left(\mathfrak{T}\right(\mathfrak{z}),\mathfrak{T}(\mathfrak{w}),\mathbf{k}\mathfrak{t})\ge \mathfrak{M}(\mathfrak{z}\mathfrak{w},\mathfrak{t}),\end{array}$$

for all $\mathfrak{z}\perp \mathfrak{w}$, $\mathfrak{z},\mathfrak{w}\in \mathcal{U}$. Then, $\mathfrak{T}$ has a unique fixed point.

Corollary 2.

Suppose $(\mathcal{U},\mathfrak{M},\star ,\perp )$ to be an O-complete fuzzy metric space. Consider $\mathfrak{T}:\mathcal{U}\to \mathcal{U}$ as a γ-admissible, ⊥-continuous, and ⊥-preserving mapping such that, for some $\mathbf{k}\in (0,1)$,

$$\begin{array}{c}\hfill \mathfrak{M}\left(\mathfrak{T}\right(\mathfrak{z}),\mathfrak{T}(\mathfrak{w}),\mathfrak{t})\ge \gamma (\mathfrak{z},\mathfrak{w})\varphi \left(\mathfrak{M}\right(\mathfrak{z},\mathfrak{w},\mathfrak{t}\left)\right)\end{array}$$

for all $\mathfrak{z}\perp \mathfrak{w}$, $\mathfrak{z},\mathfrak{w}\in \mathcal{U}$. Then, $\mathfrak{T}$ has a unique fixed point.

Corollary 3.

Suppose $(\mathcal{U},\mathfrak{M},\star ,\perp )$ to be an O-complete fuzzy metric space. Consider $\mathfrak{T}:\mathcal{U}\to \mathcal{U}$ as a γ-admissible, ⊥-continuous, and ⊥-preserving mapping such that, for some $\mathbf{k}\in (0,1)$ and $\wp \ne 0$,

$$\begin{array}{c}\hfill \gamma (\mathfrak{z},\mathfrak{w})\theta \left(\mathfrak{M}(\mathfrak{T}\left(\mathfrak{z}\right),\mathfrak{T}\left(\mathfrak{w}\right),\mathfrak{t})\right)\le {\displaystyle \frac{1}{cosh(\wp )}}\theta \left({\displaystyle \frac{\mathfrak{M}(\mathfrak{z},\mathfrak{w},\mathfrak{t})}{1-\mathbf{k}\mathfrak{M}(\mathfrak{z},\mathfrak{w},\mathfrak{t})}}\right)-\psi \left(\mathfrak{M}(\mathfrak{T}\left(\mathfrak{z}\right),\mathfrak{T}\left(\mathfrak{w}\right),\mathfrak{t})\right)\end{array}$$

for all $\mathfrak{z}\perp \mathfrak{w}$, $\mathfrak{z},\mathfrak{w}\in \mathcal{U}$. Then, $\mathfrak{T}$ has a unique fixed point.

5. Applications

The findings of this research pave the way for the development of effective techniques in solving nonlinear differential equations, thereby enhancing the capability to analyze and compute solutions in scenarios where traditional methodologies may be inadequate. Moreover, these results have implications for establishing the existence and uniqueness of solutions in nonlinear integral equations. This is particularly significant, as many problems in fractional calculus and nonlinear domains can be reformulated as nonlinear integral equations.

5.1. Application to Differential Equations

In this subsection, we demonstrate how the established findings can be applied to nonlinear differential equations.

Consider $\mathfrak{M}:\mathcal{C}[0,1]\times \mathcal{C}[0,1]\times [0,\infty )\to [0,1]$ as an orthogonal fuzzy metric defined on $\mathcal{C}[0,1]$ as

$$\begin{array}{c}\hfill \mathfrak{M}(\mathfrak{w},\mathfrak{g},\mathfrak{t})=\underset{\mathfrak{s}\in [0,1]}{\sup }{e}^{-{\displaystyle \frac{\left|\mathfrak{w}\right(\mathfrak{s})-\mathfrak{g}(\mathfrak{s}\left)\right|}{\mathfrak{t}}}},\mathfrak{t}>0.\end{array}$$

Now, we evaluate the following problem:

Problem 1.

Consider a differential equation

$$({\mathfrak{D}}^2-\mathfrak{D})\mathbf{z}\left(\mathfrak{s}\right)=\mathfrak{f}(\mathfrak{s},\mathbf{z}\left(\mathfrak{s}\right)),\mathbf{z}\left(0\right)=0\mathit{and}{\mathbf{z}}^{\prime }\left(1\right)=0$$

(1)

for all $\mathbf{z}\in \mathcal{C}[0,1]$ and with $\mathfrak{f}:[0,1]\times \mathbb{R}\to \mathbb{R}$ being continuous.

Now, Green’s function associated with (1) can be obtained as

$$\mathfrak{G}(\mathfrak{s},\xi )=\left\{\begin{array}{c}\hfill -e^{-\xi }+e^{\mathfrak{s}-\xi },0\le \mathfrak{s}\le \xi \le 1,\\ \hfill 1-e^{-\xi },0\le \xi \le \mathfrak{s}\le 1.\end{array}\right.$$

Now, we establish the existence of a unique solution of (1).

Theorem 4.

Along with the above problem, let us consider

$$\begin{array}{c}\hfill 0\le \left|\mathfrak{f}\right(\mathfrak{s},\mathfrak{p}\left(\mathfrak{s}\right))-\mathfrak{f}(\mathfrak{s},\mathfrak{q}\left(\mathfrak{s}\right)\left)\right|\le \left|\mathfrak{p}\right(\mathfrak{s})-\mathfrak{q}(\mathfrak{s}\left)\right|\end{array}$$

for all $\mathfrak{s}\in [0,1]$ and $\mathfrak{p},\mathfrak{q}\in \mathbb{R}$, with the condition satisfying $\underset{\mathfrak{s}\in [0,1]}{\sup }{\int }_0^1\mathfrak{G}(\mathfrak{s},\xi )d\xi \le \mathbf{k}<1$.

In addition, we consider ⋆ as a continuous t-norm defined as $\mathfrak{w}\star \mathfrak{g}=min\{\mathfrak{w},\mathfrak{g}\}$ and ⊥ as the binary relation defined by $\mathfrak{w}\left(\mathfrak{s}\right)\perp \mathfrak{g}\left(\mathfrak{s}\right)\iff \mathfrak{w}\left(\mathfrak{s}\right)\vee \mathfrak{g}\left(\mathfrak{s}\right),\mathfrak{w},\mathfrak{g}\in \mathcal{C}[0,1]$.

Then, the boundary value problem (BVP) (1) attains a unique solution.

Proof. 

The BVP can be expressed as follows:

$$\mathfrak{u}\left(\mathfrak{s}\right)={\int }_0^1\mathfrak{G}(\mathfrak{s},\xi )\mathfrak{f}(\xi ,\mathfrak{u}\left(\xi \right))d\xi ,\forall \mathfrak{s}\in [0,1].$$

We define a metric $\mathfrak{M}$ on the function space $\mathcal{U}:=\mathcal{C}[0,1]$ as

$$\begin{array}{c}\hfill \mathfrak{M}(\mathfrak{w},\mathfrak{g},\mathfrak{t})=\underset{\mathfrak{s}\in [0,1]}{\sup }{e}^{-{\displaystyle \frac{\left|\mathfrak{w}\right(\mathfrak{s})-\mathfrak{g}(\mathfrak{s}\left)\right|}{\mathfrak{t}}}},\mathfrak{t}>0.\end{array}$$

Then, $(\mathcal{U},\mathfrak{M},\star ,\perp )$ is a complete orthogonal fuzzy metric space.

Now, let us define $\mathfrak{T}:\mathcal{U}\to \mathcal{U}$ as

$$\begin{array}{c}\hfill \mathfrak{T}\left(\mathfrak{u}\left(\mathfrak{s}\right)\right)={\int }_0^1\mathfrak{G}(\mathfrak{s},\xi )\mathfrak{f}(\xi ,\mathfrak{u}\left(\xi \right))d\xi .\end{array}$$

Then, for all $\mathfrak{w},\mathfrak{g}\in \mathcal{U}$, we get

$$\begin{array}{cc}\hfill \mathfrak{M}\left(\mathfrak{T}\right(\mathfrak{w}\left(\mathfrak{s}\right)),\mathfrak{T}(\mathfrak{g}\left(\mathfrak{s}\right)),\mathbf{k}\mathfrak{t})& =\underset{\mathfrak{s}\in [0,1]}{\sup }{e}^{-{\displaystyle \frac{\left|\mathfrak{T}\right(\mathfrak{w}\left(\mathfrak{s}\right))-\mathfrak{T}(\mathfrak{g}\left(\mathfrak{s}\right)\left)\right|}{\mathbf{k}\mathfrak{t}}}}\hfill \\ & =\underset{\mathfrak{s}\in [0,1]}{\sup }{e}^{-{\displaystyle \frac{|{\int }_0^1\mathfrak{G}(\mathfrak{s},\xi )\mathfrak{f}(\xi ,\mathfrak{w}\left(\xi \right))d\xi -{\int }_0^1\mathfrak{G}(\mathfrak{s},\xi )\mathfrak{f}(\xi ,\mathfrak{g}\left(\xi \right))d\xi |}{\mathbf{k}\mathfrak{t}}}}\hfill \\ & \ge \underset{\mathfrak{s}\in [0,1]}{\sup }{e}^{-{\displaystyle \frac{{\int }_0^1\mathfrak{G}(\mathfrak{s},\xi )|\mathfrak{f}(\xi ,\mathfrak{w}\left(\xi \right))-\mathfrak{f}(\xi ,\mathfrak{g}\left(\xi \right))|d\xi }{\mathbf{k}\mathfrak{t}}}}\hfill \\ & \ge \underset{\mathfrak{s}\in [0,1]}{\sup }{e}^{-{\displaystyle \frac{|\mathfrak{f}(\mathfrak{s},\mathfrak{w}\left(\mathfrak{s}\right))-\mathfrak{f}(\mathfrak{s},\mathfrak{g}\left(\mathfrak{s}\right))|{\int }_0^1\mathfrak{G}(\mathfrak{s},\xi )d\xi }{\mathbf{k}\mathfrak{t}}}}\hfill \\ & \ge \underset{\mathfrak{s}\in [0,1]}{\sup }{e}^{-{\displaystyle \frac{\mathbf{k}\left|\mathfrak{w}\right(\mathfrak{s})-\mathfrak{g}(\mathfrak{s}\left)\right|}{\mathbf{k}\mathfrak{t}}}}\hfill \\ & =\underset{\mathfrak{s}\in [0,1]}{\sup }{e}^{-{\displaystyle \frac{\left|\mathfrak{w}\right(\mathfrak{s})-\mathfrak{g}(\mathfrak{s}\left)\right|}{\mathfrak{t}}}}\hfill \\ & =\mathfrak{M}\left(\mathfrak{w}\right(\mathfrak{s}),\mathfrak{g}(\mathfrak{s}),\mathfrak{t}).\hfill \end{array}$$

Hence,

$$\begin{array}{c}\hfill \mathfrak{M}\left(\mathfrak{T}\right(\mathfrak{w}),\mathfrak{T}(\mathfrak{g}),\mathbf{k}\mathfrak{t})\ge \mathfrak{M}(\mathfrak{w},\mathfrak{g},\mathfrak{t})\end{array}$$

for some $\mathbf{k}>1$.

Thus, all the hypotheses of Corollary 1 are satisfied, meaning that there exists a unique ${\mathfrak{z}}^{*}\in \mathcal{C}[0,1]$ such that $\mathfrak{T}\left({\mathfrak{z}}^{*}\right)={\mathfrak{z}}^{*}$.

Therefore, the boundary value problem (1) has a unique solution. □

5.2. Application to Integral Equations

In this section, we apply Corollary 1 to demonstrate the existence and uniqueness of a solution to nonlinear integral equations.

Consider $\mathcal{U}=\{\mathfrak{p}\in \mathcal{C}([0,1],\mathbb{R}\left)\right|\mathfrak{p}\left(ϵ\right)\ge 0\}$ and an integral equation of the form

$$\mathfrak{p}\left(ϵ\right)=\mathcal{W}\left(ϵ\right)+\lambda {\int }_0^{ϵ}\mathcal{K}(ϵ,\mu )\mathfrak{p}\left(\mu \right)d\mu ,$$

(2)

where $\lambda >0$, $\mathcal{W}\left(ϵ\right)$ is a fuzzy function of $ϵ$ for $ϵ\in [0,1]$ and $\mathcal{K}:[0,1]\times \mathbb{R}⟶{\mathbb{R}}^{+}$ is an integral kernel. To establish the existence and uniqueness of the solution of (2), we can apply Corollary 1.

Theorem 5.

Define a binary relation ⊥ on $\mathcal{U}$ as $\mathfrak{p}\perp \mathfrak{h}\iff \mathfrak{p}\left(ϵ\right)\mathfrak{h}\left(ϵ\right)\le \mathfrak{p}\left(ϵ\right)\vee \mathfrak{h}\left(ϵ\right)\mathrm{for}\mathrm{all}ϵ\in [0,1]$, $\mathrm{where}\mathfrak{p}\left(ϵ\right)\vee \mathfrak{h}\left(ϵ\right)=\mathfrak{p}\left(ϵ\right)\mathrm{or}\mathfrak{h}\left(ϵ\right)$. Because $\mathfrak{p}\equiv 0\perp \mathfrak{h},$ for all $\mathfrak{h}\in \mathcal{U}$, $(\mathcal{U},\perp )$ is an orthogonal set. Also, consider a function $\mathfrak{M}:\mathcal{U}\times \mathcal{U}\times (0,+\infty )\to (0,1]$ defined by

$$\mathfrak{M}(\mathfrak{p}\left(ϵ\right),\mathfrak{h}\left(ϵ\right),\mathfrak{t})=\underset{ϵ\in [0,1]}{\sup }{e}^{-{\displaystyle \frac{{|\mathfrak{p}\left(ϵ\right)-\mathfrak{h}\left(ϵ\right)|}^{\frac{11}{9}}}{\mathfrak{t}}}},\mathfrak{t}>0$$

and let ⋆ be a continuous t-norm defined as $\alpha \star \beta =min\{\alpha ,\beta \}$. Then, $(\mathcal{U},\mathfrak{M},\star ,\perp )$ is an O-complete fuzzy metric space. This setup guarantees that the integral Equation (2) possesses a unique solution within $\mathcal{U}$.

Proof. 

We define an operator $\mathfrak{T}:\mathcal{U}⟶\mathcal{U}$ as follows:

$$\mathfrak{T}\left(\mathfrak{p}\left(ϵ\right)\right)=\mathcal{W}\left(ϵ\right)+\lambda {\int }_0^{ϵ}\mathcal{K}(ϵ,\mu )\mathfrak{p}\left(\mu \right)d\mu .$$

Observe that for any $\mathfrak{p},\mathfrak{h}\in \mathcal{U}$, $\mathfrak{p}\perp \mathfrak{h}\Rightarrow \mathfrak{T}\mathfrak{p}\perp \mathfrak{T}\mathfrak{h}$. This indicates that $\mathfrak{T}$ is ⊥-preserving.

Moreover, for any $ϵ\in [0,1]$,

$$\begin{array}{cc}\hfill \mathfrak{M}\left(\mathfrak{T}\right(\mathfrak{p}\left(ϵ\right)),\mathfrak{T}(\mathfrak{h}\left(ϵ\right)),\mathbf{k}\mathfrak{t})& =\underset{ϵ\in [0,1]}{\sup }{e}^{-{\displaystyle \frac{|\mathcal{W}\left(ϵ\right)+\lambda {\int }_0^{ϵ}\mathcal{K}(ϵ,\mu )\mathfrak{p}\left(\mu \right)d\mu -\left(\mathcal{W}\left(ϵ\right)+\lambda {\int }_0^{ϵ}\mathcal{K}(ϵ,\mu )\mathfrak{h}\left(\mu \right)d\mu \right){|}^{\frac{11}{9}}}{\mathfrak{t}}}}\hfill \\ \hfill & =\underset{ϵ\in [0,1]}{\sup }{e}^{-{\displaystyle \frac{|(\mathfrak{p}\left(ϵ\right)-\mathfrak{h}\left(ϵ\right))\lambda {\int }_0^{ϵ}{\mathcal{K}(ϵ,\mu )d\mu |}^{\frac{11}{9}}}{\mathfrak{t}}}}\hfill \\ \hfill & \ge \underset{ϵ\in [0,1]}{\sup }{e}^{-{\displaystyle \frac{{|\mathfrak{p}\left(ϵ\right)-\mathfrak{h}\left(ϵ\right)|}^{\frac{11}{9}}}{\mathfrak{t}}}}\hfill \\ \hfill & =\mathfrak{M}\left(\mathfrak{p}\right(ϵ),\mathfrak{h}(ϵ),\mathfrak{t}).\hfill \end{array}$$

Thus, $\mathfrak{T}$ satisfies the condition

$$\mathfrak{M}\left(\mathfrak{T}\right(\mathfrak{p}),\mathfrak{T}(\mathfrak{h}),\mathbf{k}\mathfrak{t})\ge \mathfrak{M}(\mathfrak{p},\mathfrak{h},\mathfrak{t}).$$

(3)

By establishing the fulfillment of condition (3), Corollary 1 guarantees the existence of a unique fixed point in $\mathcal{U}$ for the operator $\mathfrak{T}$. Consequently, integral equations of the form in (2) have a unique solution in $\mathcal{U}$. □

6. Conclusions

There are several techniques for handling nonlinear real-life problems, particularly for nonlinear differential equations and nonlinear integral equations; however, it is essential to check for the existence of solutions to such problems. This research paper has presented a comprehensive exploration showcasing the existence and uniqueness of solutions of nonlinear differential equations and nonlinear integral equations by the application of optimality results within orthogonal fuzzy metric spaces utilizing orthogonal fuzzy proximal contraction mappings, with the introduction of the notion of best proximity point (BPP) within orthogonal fuzzy metric spaces. Furthermore, the implications of these findings have been thoroughly examined, encompassing both self-mappings and non-self mappings sharing the same parameter set. Through the inclusion of illustrative examples, the practical relevance of the established results and their consequences in various mathematical contexts have been demonstrated.

By introducing innovative concepts such as BPP within orthogonal fuzzy metric spaces, this research provides a robust theoretical framework for analyzing mappings in diverse scenarios. The findings presented in this paper not only contribute to advancing the theoretical understanding of extending fuzzy metric spaces but also broaden the applicability of fixed-point theory across different domains. In essence, this research serves as a valuable resource for mathematicians and practitioners alike, offering insights into the intricacies of orthogonal fuzzy metric spaces and their implications in fixed-point theory. It opens up avenues for further exploration and application in solving real-world problems, thereby paving the way for future research endeavors in this field.