Best Proximity Point for $(\varkappa -\mathsf{ϝ})$-Weak Proximal Contraction in Non-Archimedean Generalized Menger Space with Application to Computer Science

Abstract

This paper introduces a novel framework by merging the concepts of non-Archimedean generalized Menger spaces and ($\varkappa -\mathsf{ϝ}$)-weak proximal contractions. Extending the best proximity point concept to a triple of sets, we establish new existence theorems for these contractions without requiring the probabilistic P-property, representing a meaningful advancement beyond prior findings, which is a significant generalization of existing results. The study leverages two control functions ($\varkappa$ and $\mathsf{ϝ}$) within the contraction condition to derive optimal approximate solutions to fixed-point equations for non-self mappings. Consequently, our core results not only extend but also unify a range of established theorems within classical probabilistic and G-metric spaces. We present a significant application to theoretical computer science by proving that a self-mapping acting on infinite words possesses a unique fixed point.

1. Introduction

Fixed-point theory serves as a foundational pillar of nonlinear analysis, with profound applications across differential equations, optimization, computational mathematics, and integral equations. Among its key generalizations, the concept of best proximity points extends classical fixed point results to non-self mappings, providing solutions to equations of the form $f\left(x\right)=x$ when the mapping does not preserve the domain. This theory has been extensively developed in various generalized metric spaces, particularly in probabilistic frameworks introduced by Menger [1] and formalized by Schweizer and Sklar [2,3]. These probabilistic metric spaces offer a robust structure for modeling uncertainty and approximation behavior, forming the basis for subsequent generalizations such as the generalized Menger (probabilistic G-metric) spaces introduced by Zhou et al. [4], which unify several prior approaches. Within this evolving landscape, best proximity point theory has seen significant advances through the work of researchers such as Saadati [5], Su and Zhang [6], Vetro and Salimi [7], and Basit et al. [8], who typically rely on structural conditions like the probabilistic P-property or metric convexity to guarantee existence and uniqueness, a common thread that our research shares in its core objective.

However, our work introduces several meaningful innovations that distinguish it from earlier studies while building upon this established foundation. To clearly delineate our contributions and explicitly connect them with the aforementioned literature, we propose a novel hybrid framework that merges the concepts of non-Archimedean generalized Menger spaces with $(\varkappa -\mathsf{ϝ})$-weak proximal contractions. Unlike the majority of existing results that are restricted to pairs of sets, we extend the best proximity point concept to a triple of sets $(\mathcal{A},\mathcal{B},\mathcal{C})$. This extension provides a more flexible tool for analyzing cyclic mappings and multi-set optimization problems, a direction relatively unexplored in the current literature. Furthermore, a major theoretical advantage of our approach is the complete elimination of the probabilistic P-property, while this restrictive condition is commonly assumed in earlier works to guarantee convergence, bypassing it offers a significant generalization that directly aligns with recent advances by Zhou et al. [9] in fuzzy metric spaces. Finally, inspired by the latest developments in weak contraction theory [10,11,12,13], we employ two families of auxiliary control functions ($\varkappa$ and $\mathsf{ϝ}$) to govern the contraction condition. This dual-function strategy relaxes traditional continuity and completeness assumptions, enabling substantially weaker contraction hypotheses while rigorously ensuring the existence and uniqueness of the best proximity point.

Our results are presented in a coherent, integrated structure that progresses logically from foundations to applications. We begin by establishing the necessary preliminaries on non-Archimedean generalized Menger spaces, convergence, and topology. We then introduce the best proximity point theory for triples of sets, define $(\varkappa -\mathsf{ϝ})$-weak proximal contractions, and prove our main existence and uniqueness theorems (Theorems 1 and 2). To render our theoretical findings more concrete and to explicitly support these results, we have constructed detailed illustrative examples for both Theorems 1 and 2. These theorems are intrinsically linked: Theorem 2 emerges as a corollary under stronger topological assumptions, demonstrating the adaptability of our framework. Next, we show how these best proximity results naturally specialize to fixed point theorems when $\mathcal{A}=\mathcal{B}=\mathcal{C}$, thereby unifying and extending classical fixed point theory in these spaces, as seen in prior works such as Choudhury and Maity [14]. Finally, we provide a concrete application to the domain of finite and infinite words, building on earlier constructions by Romaguera et al. [15] and Roldán López de Hierro et al. [16], and demonstrating how our abstract theory applies to algorithmic recurrences like that of Quicksort [17]. This progression ensures that each part of the paper builds upon the previous, culminating in a comprehensive theory that connects best proximity points, fixed points, and practical applications.

Scientifically, this research contributes significantly by generalizing and unifying a range of classical theorems in probabilistic metric, G-metric, and fuzzy metric spaces. It opens new avenues for research in fixed point theory, particularly in the study of cyclic mappings and multi-valued contractions in non-Archimedean settings, and invites exploration in other generalized spaces such as strong probabilistic b-metric spaces [18,19]. By bridging pure mathematical analysis with computational theory through rigorous applications, as in the recent work of Laakel et al. [20], we encourage further cross-disciplinary research and advance both the theoretical foundations and practical relevance of best proximity point theory.

2. Materials and Methods

We begin by recalling fundamental definitions, notations, and established results that serve as the foundation for our subsequent analysis.

Let ${\mathbb{R}}^{+}$ denote the interval $[0,+\infty )$, and by $\mathbf{N}$ the set of natural numbers. A central object in probabilistic metric theory is the set ${\Delta }^{+}$, which consists of all functions $\mathcal{P}:[0,+\infty ]\to [0,1]$ satisfying the following three properties:

1.

$\mathcal{P}$ is a monotone non-decreasing function.

2.

$\mathcal{P}$ is left-continuous on $(0,+\infty ]$.

3.

$\mathcal{P}\left(0\right)=0$ and $\mathcal{P}(+\infty )=1$.

Aparticularly important subset of ${\Delta }^{+}$ is ${\mathcal{D}}^{+}$, defined by imposing a stricter limit condition at infinity:

$${\mathcal{D}}^{+}=\left\{\mathcal{P}\in {\Delta }^{+}:\underset{p\to +\infty }{lim}\mathcal{P}\left(p\right)=1\right\}.$$

The elements of ${\Delta }^{+}$ are known as distance distribution functions.

The set ${\Delta }^{+}$ is endowed with a natural partial order, defined by the pointwise comparison of functions. Given two functions $\mathcal{P}$ and $\mathcal{Q}$ in ${\Delta }^{+}$, the relation $\mathcal{P}⩾\mathcal{Q}$ holds precisely when $\mathcal{P}\left(p\right)\ge \mathcal{Q}\left(p\right)$ for all p in the interval $[0,+\infty ]$. Within the ordered subset ${\mathcal{D}}^{+}$, the maximal element with respect to this order is the Heaviside distribution function ${\epsilon }_0$, defined as:

$${\epsilon }_0\left(p\right)=\left\{\begin{array}{cc}0,\hfill & p\le 0,\hfill \\ 1,\hfill & p>0.\hfill \end{array}\right.$$

Definition 1 

([2]). A triangular norm is a binary operation $⊡:[0,1]\times [0,1]\to [0,1]$ that fulfills the following axioms for any $x,y,z,w\in [0,1]$:

1. 

$⊡(x,y)=⊡(y,x)$.

2. 

$⊡\left(x,⊡(y,z)\right)=⊡\left(⊡(x,y),z\right)$.

3. 

If $x\le z$ and $y\le w$, then $⊡(x,y)\le ⊡(z,w)$.

4. 

$⊡(x,1)=x$.

Example 1. 

Two basic examples of continuous triangular norms are:

• ${⊡}_M(x,y)=min\{x,y\}$.
• ${⊡}_P(x,y)=x·y$.

Definition 2 

([16]). A triangular norm ⊡ is called 1-boundary continuous if it continuous at every point of the form $(1,q)$ where $q\in [0,1]$. Formally, this means that for any convergent sequences $\left\{p_n\right\}\to 1$ and $\left\{q_n\right\}\to q$, we have

$$\underset{n\to +\infty }{lim}⊡(p_n,q_n)=⊡(1,q)=q.$$

It follows directly that any continuous triangular norm automatically possesses 1-boundary continuity.

Proposition 1 

([16]). Let ⊡ be a 1-boundary continuous triangular norm, and consider five sequences $\left\{{\alpha }_n\right\}$, $\left\{{\beta }_n\right\}$, $\left\{{\gamma }_n\right\}$, $\left\{u_n\right\}$, $\left\{v_n\right\}$ in $[0,1]$. Assume that:

• ${\alpha }_n\to L$ and $v_n\to L$ for some $L\in [0,1]$,
• ${\beta }_n\to 1$ and $u_n\to 1$,
• the inequalities ${\alpha }_n\ge ⊡({\beta }_n,⊡({\gamma }_n,u_n))\ge v_n$ hold for all $n\in \mathbf{N}$.

Then the sequence $\left\{{\gamma }_n\right\}$ also converges to L.

Definition 3 

([1]). A Menger space is defined as a triple $(\mathcal{U},\mathcal{P},⊡)$ where ⊡ is a triangular norm and $\mathcal{P}:\mathcal{U}\times \mathcal{U}\to {\Delta }^{+}$ is a mapping (with ${\mathcal{P}}_{\alpha ,\beta }$ denoting the value at $(\alpha ,\beta )$) that satisfies the following axioms for all $\alpha ,\beta ,\gamma \in \mathcal{U}$ and $p,q\ge 0$:

1. 

${\mathcal{P}}_{\alpha ,\beta }\left(p\right)=1$ for all $p\ne 0$ is equivalent to $\alpha =\beta$.

2. 

${\mathcal{P}}_{\alpha ,\beta }\left(p\right)={\mathcal{P}}_{\beta ,\alpha }\left(p\right)$.

3. 

${\mathcal{P}}_{\alpha ,\gamma }(p+q)\ge ⊡\left({\mathcal{P}}_{\alpha ,\beta }\left(p\right),{\mathcal{P}}_{\beta ,\gamma }\left(q\right)\right)$.

As a natural generalization of the classical Menger space, the notion of a generalized Menger space was introduced by Zhou and collaborators [4].

Definition 4 

([4]). A generalized Menger space is a triple $(\mathcal{U},\mathcal{F},⊡)$, where $\mathcal{U}$ is a non-empty set, and ⊡ is a triangular norm. Let $\mathcal{F}:\mathcal{U}\times \mathcal{U}\times \mathcal{U}\to {\mathcal{D}}^{+}$ be a mapping (and denote by ${\mathcal{F}}_{\alpha ,\beta ,\gamma }$ its value at the triple $(\alpha ,\beta ,\gamma )$). This mapping is required to satisfy the following properties for all $\alpha ,\beta ,\gamma \in \mathcal{U}$ and all $p>0$:

1. 

${\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right)=1$ is equivalent to $\alpha =\beta =\gamma$.

2. 

${\mathcal{F}}_{\alpha ,\alpha ,\beta }\left(p\right)\ge {\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right)$ with $\beta \ne \gamma$.

3. 

${\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right)={\mathcal{F}}_{\pi (\alpha ,\beta ,\gamma )}\left(p\right)$, where $\pi (\alpha ,\beta ,\gamma )$ is any permutation of $\alpha ,\beta ,\gamma$.

4. 

For all $a\in \mathcal{U}$ and $p,q\ge 0$,

$${\mathcal{F}}_{\alpha ,\beta ,\gamma }(p+q)\ge ⊡\left({\mathcal{F}}_{\alpha ,a,a}\left(p\right),{\mathcal{F}}_{a,\beta ,\gamma }\left(q\right)\right).$$

(1)

Before diving into the formal definitions, it is essential to clarify the intuitive meaning of topology and convergence within the framework of generalized Menger spaces [4]. Unlike classical metric spaces where the topology is induced by a deterministic, real-valued distance between two points, the topology here is probabilistic and ternary. In this context, the “distance” is represented by a distribution function ${\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(t\right)$ that measures the probability that the generalized distance between three points $\alpha ,\beta$, and $\gamma$ is less than t. Consequently, the topology is generated by probabilistic $(\varsigma ,\varrho )$-neighborhood, and a sequence is said to converge to a limit if the probability of the generalized distance between the sequence terms and the limit being less than any $t>0$ approaches 1 as the index goes to infinity. This establishes a robust framework for dealing with approximation and uncertainty.

Consider a generalized Menger space $(\mathcal{U},\mathcal{F},⊡)$ and fix a point $a\in \mathcal{U}$. Given parameters $\varsigma >0$ and $\varrho \in (0,1)$, we define the $(\varsigma ,\varrho )$-neighborhood of a as the collection of all points $b\in \mathcal{U}$ satisfying both inequalities:

$${\mathcal{F}}_{a,b,b}(\varsigma )>1-\varrho \mathrm{and}{\mathcal{F}}_{b,a,a}(\varsigma )>1-\varrho .$$

This set is denoted by:

$$N_a(\varsigma ,\varrho )=\{b\in \mathcal{U}:{\mathcal{F}}_{a,b,b}(\varsigma )>1-\varrho \mathrm{and}{\mathcal{F}}_{b,a,a}(\varsigma )>1-\varrho \}.$$

The family $\mathcal{N}=\{N_a(\varsigma ,\varrho ):a\in \mathcal{U},\varsigma >0,\varrho >0\}$ of all such neighborhoods generates a topology on $\mathcal{U}$.

Building upon this topological framework, we now introduce fundamental convergence concepts for sequences, in generalized Menger spaces.

Definition 5 

([4]). Let $(\mathcal{U},\mathcal{F},⊡)$ be a generalized Menger space and ${\left\{{\alpha }_n\right\}}_{n\in \mathbf{N}}$ a sequence in $\mathcal{U}$.

1. 

The sequence $\left\{{\alpha }_n\right\}$ is said to converge to a point $\gamma \in \mathcal{U}$ if, for every $\varsigma >0$ and $\varrho \in (0,1)$, there exists an integer $M_{\varsigma ,\varrho }>0$ such that

$${\alpha }_n\in N_{\gamma }(\varsigma ,\varrho )\mathit{for}\mathit{all}n>M_{\varsigma ,\varrho }.$$

2. 

We say that $\left\{{\alpha }_n\right\}$ is a Cauchy sequence when, for any given $\varsigma >0$ and $\varrho \in (0,1)$, one can find a positive integer $M_{\varsigma ,\varrho }$ such that

$${\mathcal{F}}_{{\alpha }_n,{\alpha }_m,{\alpha }_l}(\varsigma )>1-\varrho \mathit{holds}\mathit{for}\mathit{all}n,m,l>M_{\varsigma ,\varrho }.$$

3. 

In $(\mathcal{U},\mathcal{F},⊡)$, completeness means that every Cauchy sequence possesses a limit in $\mathcal{U}$.

3. Results

3.1. Non-Archimedean Generalized Menger Space

Definition 6. 

A generalized Menger space $(\mathcal{U},\mathcal{F},⊡)$ is said to be non-Archimedean if the inequality (1) in Definition 4 is replaced by the stronger inequality:

$${\mathcal{F}}_{\alpha ,\gamma ,\beta }(max(p,q))\ge ⊡\left({\mathcal{F}}_{\alpha ,a,a}\left(p\right),{\mathcal{F}}_{a,\gamma ,\beta }\left(q\right)\right)$$

which must hold for all $\alpha ,\gamma ,\beta ,a\in \mathcal{U}$ and $p,q\ge 0$. An equivalent formulation of this condition is:

$${\mathcal{F}}_{\alpha ,\gamma ,\beta }\left(p\right)\ge ⊡\left({\mathcal{F}}_{\alpha ,a,a}\left(p\right),{\mathcal{F}}_{a,\gamma ,\beta }\left(p\right)\right)\mathit{for}\mathit{all}\alpha ,\gamma ,\beta ,a\in \mathcal{U}\mathit{and}p\ge 0.$$

(2)

To rigorously justify why Equation (2) is equivalent to the stronger inequality for all $p,q\ge 0$, notice that the case $p=q$ is trivially given by Equation (2). For the case where $p\ne q$, assume without loss of generality that $p>q$, meaning $max(p,q)=p$. Since the distribution function is non-decreasing, we have ${\mathcal{F}}_{a,\gamma ,\beta }\left(p\right)\ge {\mathcal{F}}_{a,\gamma ,\beta }\left(q\right)$. Furthermore, because the triangular norm ⊡ is non-decreasing in each of its arguments, applying Equation (2) yields:

$${\displaystyle {\mathcal{F}}_{\alpha ,\gamma ,\beta }(max(p,q))={\mathcal{F}}_{\alpha ,\gamma ,\beta }\left(p\right)\ge ⊡\left({\mathcal{F}}_{\alpha ,a,a}\left(p\right),{\mathcal{F}}_{a,\gamma ,\beta }\left(p\right)\right)\ge ⊡\left({\mathcal{F}}_{\alpha ,a,a}\left(p\right),{\mathcal{F}}_{a,\gamma ,\beta }\left(q\right)\right)}$$

which recovers the general non-Archimedean condition.

Example 2. 

Consider the interval $\mathcal{U}=[0,1)$ equipped with the triangular norm ${⊡}_M$. Define a mapping $\mathcal{F}:{\mathcal{U}}^3\times [0,+\infty )\to [0,1]$ by

$${\mathcal{F}}_{\alpha ,\gamma ,\beta }\left(p\right)=\left\{\begin{array}{cc}0,\hfill & p=0,\hfill \\ {\displaystyle \frac{p}{p+max\{\alpha ,\gamma ,\beta \}}},\hfill & p>0,\hfill \end{array}\right.\mathit{for}\mathit{all}\alpha ,\gamma ,\beta \in \mathcal{U}.$$

Then the triple $(\mathcal{U},\mathcal{F},{⊡}_M)$ constitutes a non-Archimedean generalized Menger space.

Remark 1. 

For any non-Archimedean generalized Menger space $(\mathcal{U},\mathcal{F},⊡)$, the following chain of inequalities holds for all $\alpha ,\gamma ,\beta ,a\in \mathcal{U}$ and $p,q\ge 0$:

$${\mathcal{F}}_{\alpha ,\gamma ,\beta }(p+q)\ge {\mathcal{F}}_{\alpha ,\gamma ,\beta }(max(p,q))\ge ⊡\left({\mathcal{F}}_{\alpha ,a,a}\left(p\right),{\mathcal{F}}_{a,\gamma ,\beta }\left(q\right)\right).$$

Consequently, every non-Archimedean generalized Menger space automatically satisfies the axioms of a generalized Menger space. The converse implication; however, does not generally hold.

Since every non-Archimedean generalized Menger space is a generalized Menger space, we adopt the natural topology introduced by Zhou et al. [4] for generalized Menger spaces.

Proposition 2. 

Let $(\mathcal{U},\mathcal{F},⊡)$ be a non-Archimedean generalized Menger space where ⊡ is a 1-boundary continuous triangular norm. Then, the topology defined via the neighborhoods $\left\{N_a(\varsigma ,\varrho )\right\}$ is Hausdorff. Consequently, the limit of any convergent sequence in $\mathcal{U}$ is unique.

Proof. 

We need to show that for any two distinct points $\alpha ,\beta \in \mathcal{U}$, there exist disjoint neighborhoods separating them. Since $\alpha \ne \beta$, by Definition 4 there exists some $\varsigma >0$ such that

$$a_1={\mathcal{F}}_{\alpha ,\beta ,\beta }(\varsigma )<1\mathrm{and}{a}_2={\mathcal{F}}_{\beta ,\alpha ,\alpha }(\varsigma )<1.$$

Set $a=max\{a_1,a_2\}<1$.

We can select numbers $b_1,b_2\in (0,1)$ such that $⊡(b_1,b_2)>a$. This is possible by the 1-boundary continuity of ⊡ and its monotonicity.

Now define two neighborhoods:

$$\begin{array}{cc}\hfill N_{\alpha }& =\{\gamma \in \mathcal{U}:{\mathcal{F}}_{\alpha ,\gamma ,\gamma }(\varsigma )>b_1\mathrm{and}{\mathcal{F}}_{\gamma ,\alpha ,\alpha }(\varsigma )>b_1\},\hfill \\ \hfill N_{\beta }& =\{\gamma \in \mathcal{U}:{\mathcal{F}}_{\beta ,\gamma ,\gamma }(\varsigma )>b_2\mathrm{and}{\mathcal{F}}_{\gamma ,\beta ,\beta }(\varsigma )>b_2\}.\hfill \end{array}$$

Clearly $\alpha \in N_{\alpha }$ and $\beta \in N_{\beta }$ since ${\mathcal{F}}_{\alpha ,\alpha ,\alpha }(\varsigma )={\mathcal{F}}_{\beta ,\beta ,\beta }(\varsigma )=1>b_1,b_2$.

To establish the Hausdorff property, we show that $N_{\alpha }\cap N_{\beta }=\{\varnothing \}$. Suppose, for contradiction, that there exists a point $s\in N_{\alpha }\cap N_{\beta }$. Then by definition:

$${\mathcal{F}}_{\alpha ,s,s}(\varsigma )>b_1,{\mathcal{F}}_{s,\alpha ,\alpha }(\varsigma )>b_1,{\mathcal{F}}_{\beta ,s,s}(\varsigma )>b_2,{\mathcal{F}}_{s,\beta ,\beta }(\varsigma )>b_2.$$

Applying the non-Archimedean inequality (2) with $a=s$, we obtain:

$$\begin{array}{cc}\hfill a_1={\mathcal{F}}_{\alpha ,\beta ,\beta }(\varsigma )& \ge ⊡\left({\mathcal{F}}_{\alpha ,s,s}(\varsigma ),{\mathcal{F}}_{s,\beta ,\beta }(\varsigma )\right)\hfill \\ \hfill & >⊡(b_1,b_2)>a\ge a_1,\hfill \end{array}$$

which is impossible. Similarly,

$$\begin{array}{cc}\hfill a_2={\mathcal{F}}_{\beta ,\alpha ,\alpha }(\varsigma )& \ge ⊡\left({\mathcal{F}}_{\beta ,s,s}(\varsigma ),{\mathcal{F}}_{s,\alpha ,\alpha }(\varsigma )\right)\hfill \\ \hfill & >⊡(b_2,b_1)>a\ge a_2,\hfill \end{array}$$

also a contradiction. Hence, no such s can exist, so $N_{\alpha }$ and $N_{\beta }$ are disjoint neighborhoods of $\alpha$ and $\beta$, respectively. This proves that the $(\varsigma ,\varrho )$-topology is Hausdorff. □

Lemma 1. 

Let $(\mathcal{U},\mathcal{F},⊡)$ be a non-Archimedean generalized Menger space, where ⊡ is a 1-boundary continuous triangular norm. For a sequence $\left\{{\alpha }_n\right\}$ in $\mathcal{U}$, the following conditions are equivalent:

1. 

$\left\{{\alpha }_n\right\}$ is a Cauchy sequence.

2. 

For any $\varsigma >0$ and $\varrho \in (0,1)$, one can find $M\in \mathbf{N}$ such that

$${\mathcal{F}}_{{\alpha }_n,{\alpha }_m,{\alpha }_m}(\varsigma )>1-\varrho \mathit{for}\mathit{every}\mathit{pair}n,m>M.$$

Proof. 

We prove both implications separately.

• 1 ⇒ 2: This can be easily obtained from 2 of Definition 4.
• 2 ⇒ 1: Assume condition 2 holds. Since ⊡ is 1-boundary continuous, for given $\varsigma >0$ and $\varrho >0$, we can choose ${\varrho }_0\in (0,\varrho )$ such that

$$⊡(1-{\varrho }_0,1-{\varrho }_0)>1-\varrho .$$

By Hypothesis 2 applied with ${\varrho }_0$, there exists $M\in \mathbf{N}$ such that

$${\mathcal{F}}_{{\alpha }_n,{\alpha }_m,{\alpha }_m}(\varsigma )>1-{\varrho }_0\mathrm{and}{\mathcal{F}}_{{\alpha }_l,{\alpha }_m,{\alpha }_m}(\varsigma )>1-{\varrho }_0$$

for all $n,m,l\ge M$. Using the non-Archimedean property (2), we have:

$$\begin{array}{cc}\hfill {\mathcal{F}}_{{\alpha }_n,{\alpha }_m,{\alpha }_l}(\varsigma )& \ge ⊡\left({\mathcal{F}}_{{\alpha }_n,{\alpha }_m,{\alpha }_m}(\varsigma ),{\mathcal{F}}_{{\alpha }_l,{\alpha }_m,{\alpha }_m}(\varsigma )\right)\hfill \\ \hfill & \ge ⊡(1-{\varrho }_0,1-{\varrho }_0)\hfill \\ \hfill & >1-\varrho .\hfill \end{array}$$

This verifies the Cauchy criterion. □

Lemma 2. 

Let $(\mathcal{U},\mathcal{F},⊡)$ be a non-Archimedean generalized Menger space, where ⊡ is a 1-boundary continuous triangular norm. Let $\left\{{\alpha }_n\right\},\left\{{\beta }_n\right\},\left\{{\gamma }_n\right\}$ be three sequences in in $\mathcal{U}$ converging, respectively, to points $\alpha ,\beta ,\gamma \in \mathcal{U}$. Then, for every $p>0$:

$$\underset{n\to +\infty }{lim}{\mathcal{F}}_{{\alpha }_n,{\beta }_n,{\gamma }_n}\left(p\right)={\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right).$$

Proof. 

Let $p>0$ be fixed. Since the sequences converge, for any sufficiently small $ϵ>0$ (specifically, choose $ϵ<p$), we have:

$$\underset{n\to +\infty }{lim}{\mathcal{F}}_{{\alpha }_n,\alpha ,\alpha }\left(ϵ\right)=1,\underset{n\to +\infty }{lim}{\mathcal{F}}_{{\beta }_n,\beta ,\beta }\left(ϵ\right)=1,\underset{n\to +\infty }{lim}{\mathcal{F}}_{{\gamma }_n,\gamma ,\gamma }\left(ϵ\right)=1.$$

We first establish the lower bound for the limit. Using the non-Archimedean triangle inequality repeatedly with the pivot points $\alpha ,\beta ,\gamma$:

$$\begin{array}{cc}\hfill {\mathcal{F}}_{{\alpha }_n,{\beta }_n,{\gamma }_n}\left(p\right)& ={\mathcal{F}}_{{\alpha }_n,{\beta }_n,{\gamma }_n}(max(ϵ,p))\hfill \\ \hfill & \ge ⊡\left({\mathcal{F}}_{{\alpha }_n,\alpha ,\alpha }\left(ϵ\right),{\mathcal{F}}_{\alpha ,{\beta }_n,{\gamma }_n}\left(p\right)\right)\hfill \\ \hfill & \ge ⊡\left({\mathcal{F}}_{{\alpha }_n,\alpha ,\alpha }\left(ϵ\right),⊡\left({\mathcal{F}}_{{\beta }_n,\beta ,\beta }\left(ϵ\right),{\mathcal{F}}_{\alpha ,\beta ,{\gamma }_n}\left(p\right)\right)\right)\hfill \\ \hfill & \ge ⊡\left({\mathcal{F}}_{{\alpha }_n,\alpha ,\alpha }\left(ϵ\right),⊡\left({\mathcal{F}}_{{\beta }_n,\beta ,\beta }\left(ϵ\right),⊡({\mathcal{F}}_{{\gamma }_n,\gamma ,\gamma }\left(ϵ\right),{\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right))\right)\right).\hfill \end{array}$$

Taking the limit inferior as $n\to +\infty$, and utilizing the continuity of ⊡ at $(1,1)$ (where $⊡(1,x)=x$):

$$\underset{n\to +\infty }{lim}{\mathcal{F}}_{{\alpha }_n,{\beta }_n,{\gamma }_n}\left(p\right)\ge ⊡(1,⊡(1,⊡(1,{\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right))))={\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right).$$

(3)

Next, we establish the upper bound by reversing the roles. We express the limit term ${\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right)$ using the sequence terms:

$$\begin{array}{cc}\hfill {\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right)& ={\mathcal{F}}_{\alpha ,\beta ,\gamma }(max(ϵ,p))\hfill \\ \hfill & \ge ⊡\left({\mathcal{F}}_{\alpha ,{\alpha }_n,{\alpha }_n}\left(ϵ\right),⊡\left({\mathcal{F}}_{\beta ,{\beta }_n,{\beta }_n}\left(ϵ\right),⊡({\mathcal{F}}_{\gamma ,{\gamma }_n,{\gamma }_n}\left(ϵ\right),{\mathcal{F}}_{{\alpha }_n,{\beta }_n,{\gamma }_n}\left(p\right))\right)\right).\hfill \end{array}$$

Taking the limit superior as $n\to +\infty$, we get:

$${\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right)\ge \underset{n\to +\infty }{lim}{\mathcal{F}}_{{\alpha }_n,{\beta }_n,{\gamma }_n}\left(p\right).$$

Combining this with inequality (3), we obtain:

$${\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right)\ge \underset{n\to +\infty }{lim}{\mathcal{F}}_{{\alpha }_n,{\beta }_n,{\gamma }_n}\left(p\right)\ge \underset{n\to +\infty }{lim}{\mathcal{F}}_{{\alpha }_n,{\beta }_n,{\gamma }_n}\left(p\right)\ge {\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right).$$

This forces equality:

$$\underset{n\to +\infty }{lim}{\mathcal{F}}_{{\alpha }_n,{\beta }_n,{\gamma }_n}\left(p\right)={\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right).$$

□

3.2. Best Proximity Point Theory in Triple Settings

Throughout this section, let $(\mathcal{U},\mathcal{F},⊡)$ denote a non-Archimedean generalized Menger space, and let $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$ be three non-empty subsets of $\mathcal{U}$. We begin by introducing several key constructions.

Definition 7. 

For the triple $(\mathcal{A},\mathcal{B},\mathcal{C})$, we define the optimal proximity function ${\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}:(0,+\infty )\to [0,1]$ by

$${\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right)=sup\{{\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right):\alpha \in \mathcal{A},\beta \in \mathcal{B},\gamma \in \mathcal{C}\},p>0.$$

This value represents the optimal probabilistic proximity achievable among elements of the sets $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$. Indeed, since ${\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right)$ evaluates the probability that the generalized distance among the points is less than p, a higher value inherently indicates a stronger degree of proximity. Consequently, the supremum over all such triplets mathematically quantifies the maximum attainable probability of closeness at the given threshold p.

In the present study, for any nonempty subsets $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$ of a non-Archimedean generalized Menger space $(\mathcal{U},\mathcal{F},⊡)$, the following terminology is applied throughout.

Definition 8. 

We associate to each set a subset of points that can attain this optimal value:

$$\begin{array}{cc}\hfill {\mathcal{A}}_0& =\left\{\alpha \in \mathcal{A}:\exists \beta \in \mathcal{B},\exists \gamma \in \mathcal{C}\mathit{satisfying}{\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right)={\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right)\mathit{for}\mathit{all}p>0\right\},\hfill \\ \hfill {\mathcal{B}}_0& =\left\{\beta \in \mathcal{B}:\exists \alpha \in \mathcal{A},\exists \gamma \in \mathcal{C}\mathit{satisfying}{\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right)={\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right)\mathit{for}\mathit{all}p>0\right\},\hfill \\ \hfill {\mathcal{C}}_0& =\left\{\gamma \in \mathcal{C}:\exists \alpha \in \mathcal{A},\exists \beta \in \mathcal{B}\mathit{satisfying}{\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right)={\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right)\mathit{for}\mathit{all}p>0\right\}.\hfill \end{array}$$

The sets ${\mathcal{A}}_0$, ${\mathcal{B}}_0$, ${\mathcal{C}}_0$ consist of those points that participate in a triple realizing the optimal proximity.

To illustrate Definition 8, let us build upon the non-Archimedean generalized Menger space presented in Example (2).

Example 3. 

Consider $\mathcal{U}=[0,1)$ equipped with the mapping

$${\displaystyle {\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right)=\left\{\begin{array}{cc}0,\hfill & p=0,\hfill \\ {\displaystyle \frac{p}{p+max\{\alpha ,\beta ,\gamma \}}},\hfill & p>0,\hfill \end{array}\right.\forall \alpha ,\gamma ,\beta \in \mathcal{U}.}$$

Let us define three closed subsets of $\mathcal{U}$:

$${\displaystyle \mathcal{A}=\left[\frac{1}{10},\frac{2}{10}\right],\mathcal{B}=\left[\frac{4}{10},\frac{5}{10}\right],\mathcal{C}=\left[\frac{7}{10},\frac{8}{10}\right].}$$

For any triplet $(\alpha ,\beta ,\gamma )\in \mathcal{A}\times \mathcal{B}\times \mathcal{C}$, it is clear from the intervals that $\alpha <\beta <\gamma$, which means $max\{\alpha ,\beta ,\gamma \}=\gamma$. To find the optimal proximity function ${\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right)$, we must take the supremum of ${\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right)$ over all such triplets. Since the function $f\left(x\right)={\displaystyle \frac{p}{p+x}}$ is strictly decreasing for $x\ge 0$, maximizing the probability is equivalent to minimizing $max\{\alpha ,\beta ,\gamma \}=\gamma$.

The minimum possible value for $\gamma \in \mathcal{C}$ is ${\displaystyle \frac{7}{10}}$. Therefore, the optimal proximity function is:

$${\displaystyle {\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right)=\frac{p}{p+0.7}.}$$

Now, we determine the optimal subsets ${\mathcal{A}}_0,{\mathcal{B}}_0$, and ${\mathcal{C}}_0$. A triplet $(\alpha ,\beta ,\gamma )$ achieves this optimal value if and only if $max\{\alpha ,\beta ,\gamma \}=0.7$. Because γ is the only variable that can reach this value (since all elements in $\mathcal{A}$ and $\mathcal{B}$ are strictly less than 0.7), we must strictly choose $\gamma =0.7$, which means ${\mathcal{C}}_0=\left\{0.7\right\}$. However, any choice of $\alpha \in \mathcal{A}$ and $\beta \in \mathcal{B}$, when paired with $\gamma =0.7$, still satisfies $max\{\alpha ,\beta ,0.7\}=0.7$.

Consequently, all elements of $\mathcal{A}$ and $\mathcal{B}$ participate in realizing the optimal proximity, yielding:

$${\displaystyle {\mathcal{A}}_0=\mathcal{A}=\left[\frac{1}{10},\frac{2}{10}\right],{\mathcal{B}}_0=\mathcal{B}=\left[\frac{4}{10},\frac{5}{10}\right],{\mathcal{C}}_0=\left\{\frac{7}{10}\right\}.}$$

In the context of best proximity points, a standard geometric property often required to ensure the uniqueness or existence of solutions is the P-property. We adapt its definition to the context of triplets in non-Archimedean G-Menger spaces as follows:

Definition 9. 

Let $(\mathcal{U},\mathcal{F},⊡)$ be a non-Archimedean generalized Menger space and let $\mathcal{A},\mathcal{B},\mathcal{C}$ be nonempty subsets of $\mathcal{U}$. The triplet $(\mathcal{A},\mathcal{B},\mathcal{C})$) is said to satisfy the probabilistic P-property if, for any elements ${\alpha }_1,{\alpha }_2,{\alpha }_3\in \mathcal{A}$, ${\beta }_1,{\beta }_2,{\beta }_3\in \mathcal{B}$ and ${\gamma }_1,{\gamma }_2,{\gamma }_3\in \mathcal{C}$, the following implication holds:

$$\left.\begin{array}{cc}\hfill {\mathcal{F}}_{{\alpha }_1,{\beta }_1,{\gamma }_1}\left(p\right)& ={\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right)\hfill \\ \hfill {\mathcal{F}}_{{\alpha }_2,{\beta }_2,{\gamma }_2}\left(p\right)& ={\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right)\hfill \\ \hfill {\mathcal{F}}_{{\alpha }_3,{\beta }_3,{\gamma }_3}\left(p\right)& ={\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right)\hfill \end{array}\right\}\Rightarrow {\mathcal{F}}_{{\alpha }_1,{\alpha }_2,{\alpha }_3}\left(p\right)={\mathcal{F}}_{{\beta }_1,{\beta }_2,{\beta }_3}\left(p\right)={\mathcal{F}}_{{\gamma }_1,{\gamma }_2,{\gamma }_3}\left(p\right),$$

for all $p>0$.

To provide a clear, non-trivial geometric illustration of the probabilistic P-property, let us consider the following example involving parallel sets.

Example 4. 

Let $\mathcal{U}={\mathbb{R}}^2$ be equipped with the standard probabilistic G-metric induced by the Euclidean distance d, such that $G(\alpha ,\beta ,\gamma )=d(\alpha ,\beta )+d(\beta ,\gamma )+d(\gamma ,\alpha )$, and the distribution function is given by

$${\displaystyle {\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right)=\left\{\begin{array}{cc}0,\hfill & p=0,\hfill \\ {\displaystyle \frac{p}{p+\mathrm{G}(\alpha ,\beta ,\gamma )}},\hfill & p>0,\hfill \end{array}\right.\forall \alpha ,\gamma ,\beta \in \mathcal{U}.}$$

Consider three parallel horizontal lines in ${\mathbb{R}}^2$:

$${\displaystyle \mathcal{A}=\left\{\right(x,0):x\in \mathbb{R}\},\mathcal{B}=\left\{\right(x,1):x\in \mathbb{R}\},\mathcal{C}=\left\{\right(x,2):x\in \mathbb{R}\}.}$$

Let $\alpha =(x_1,0)\in \mathcal{A}$, $\beta =(x_2,1)\in \mathcal{B}$, and $\gamma =(x_3,2)\in \mathcal{C}$. The sum of the distances between these three points is strictly bounded below by the vertical distances between the lines:

$${\displaystyle G(\alpha ,\beta ,\gamma )\ge |0-1|+|1-2|+|2-0|=1+1+2=4.}$$

This minimal geometric distance $G=4$ is achieved if and only if the three points are perfectly vertically aligned, meaning $x_1=x_2=x_3$. Consequently, the optimal proximity function is ${\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right)={\displaystyle \frac{p}{p+4}}$.

Now, suppose we have three combinations of elements that achieve this optimal distance:

$${\displaystyle {\mathcal{F}}_{{\alpha }_i,{\beta }_i,{\gamma }_i}\left(p\right)={\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right)\mathit{for}i\in \{1,2,3\}.}$$

Based on our previous observation, this implies that for each i, the points must be vertically aligned, so there exist real numbers $x_1,x_2,x_3$ such that ${\alpha }_i=(x_i,0)$, ${\beta }_i=(x_i,1)$, and ${\gamma }_i=(x_i,2)$.

We can now compute the G-distance between the elements of $\mathcal{A}$, between the elements of $\mathcal{B}$, and between the elements of $\mathcal{C}$:

$$\begin{array}{cc}\hfill G({\alpha }_1,{\alpha }_2,{\alpha }_3)& =d((x_1,0),(x_2,0))+d((x_2,0),(x_3,0))+d((x_3,0),(x_1,0))\hfill \\ & =|x_1-x_2|+|x_2-x_3|+|x_3-x_1|.\hfill \end{array}$$

Because shifting the points vertically by 1 or 2 units does not change horizontal Euclidean distances, it trivially follows that:

$${\displaystyle G({\alpha }_1,{\alpha }_2,{\alpha }_3)=G({\beta }_1,{\beta }_2,{\beta }_3)=G({\gamma }_1,{\gamma }_2,{\gamma }_3).}$$

Therefore, substituting these equal G-distances into the probabilistic function yields:

$${\displaystyle \begin{array}{cc}\hfill {\mathcal{F}}_{{\alpha }_1,{\alpha }_2,{\alpha }_3}\left(p\right)& ={\mathcal{F}}_{{\beta }_1,{\beta }_2,{\beta }_3}\left(p\right)\hfill \\ & ={\mathcal{F}}_{{\gamma }_1,{\gamma }_2,{\gamma }_3}\left(p\right),\hfill \end{array}}$$

which proves that the probabilistic P-property is satisfied in this non trivial geometry.

Definition 10. 

Let $(\mathcal{U},\mathcal{F},⊡)$ be a non-Archimedean generalized Menger space and $\mathcal{A},\mathcal{B},\mathcal{C}$ non-empty subsets of $\mathcal{U}$. Consider a mapping $f:\mathcal{A}\cup \mathcal{B}\cup \mathcal{C}\to \mathcal{A}\cup \mathcal{B}\cup \mathcal{C}$ satisfying

$$f\left(\mathcal{A}\right)\subseteq \mathcal{B},f\left(\mathcal{B}\right)\subseteq \mathcal{C},f\left(\mathcal{C}\right)\subseteq \mathcal{A}.$$

A point $\alpha \in \mathcal{A}\cup \mathcal{B}\cup \mathcal{C}$ is called a best proximity point of f if it fulfills the condition

$${\mathcal{F}}_{\alpha ,f\alpha ,f^2\alpha }\left(p\right)={\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right)\mathit{for}\mathit{every}p>0.$$

To illustrate this definition cohesively, let us return to the geometric setting of the three parallel lines introduced in the previous example (4).

Example 5. 

Recall our space $\mathcal{U}={\mathbb{R}}^2$ with the standard probabilistic G-metric, and the three parallel lines:

$${\displaystyle \mathcal{A}=\left\{\right(x,0):x\in \mathbb{R}\},\mathcal{B}=\left\{\right(x,1):x\in \mathbb{R}\},\mathcal{C}=\left\{\right(x,2):x\in \mathbb{R}\}.}$$

We previously established that the optimal proximity function is ${\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right)={\displaystyle \frac{p}{p+4}}$, and this optimal value is achieved if and only if the three points share the same horizontal coordinate.

Now, let us define a cyclic mapping $f:\mathcal{A}\cup \mathcal{B}\cup \mathcal{C}\to \mathcal{A}\cup \mathcal{B}\cup \mathcal{C}$ as follows:

$${\displaystyle f(x,0)=\left(\frac{x}{2},1\right),f(x,1)=\left(\frac{x}{2},2\right),f(x,2)=\left(\frac{x}{2},0\right).}$$

It is trivial to verify that $f\left(\mathcal{A}\right)\subseteq \mathcal{B}$, $f\left(\mathcal{B}\right)\subseteq \mathcal{C}$, and $f\left(\mathcal{C}\right)\subseteq \mathcal{A}$, satisfying the cyclic condition.

We want to find a best proximity point for f. Let $\alpha =(x,0)\in \mathcal{A}$. Applying the mapping twice yields:

$${\displaystyle f\alpha =\left(\frac{x}{2},1\right)\in \mathcal{B}\mathit{and}{f}^2\alpha =f\left(\frac{x}{2},1\right)=\left(\frac{x}{4},2\right)\in \mathcal{C}.}$$

According to Definition 10, α is a best proximity point if and only if ${\mathcal{F}}_{\alpha ,f\alpha ,f^2\alpha }\left(p\right)={\displaystyle \frac{p}{p+4}}$. As established, this requires the three points $\alpha , f\alpha$, and $f^2\alpha$ to be perfectly aligned vertically. Therefore, their horizontal coordinates must be strictly equal:

$${\displaystyle x=\frac{x}{2}=\frac{x}{4}.}$$

The only real number satisfying this equation is $x=0$. Thus, $\alpha =(0,0)$ is a best proximity point in $\mathcal{A}$. Similarly, one can easily verify that $(0,1)\in \mathcal{B}$ and $(0,2)\in \mathcal{C}$ are also best proximity points for this cyclic mapping.

We now introduce two families of auxiliary functions that will govern our contraction conditions.

Denote by $\Xi$ the collection of all functions $\varkappa :(0,1]\to [0,+\infty )$ satisfying:

1.

$\varkappa$ is monotone decreasing: if $p\le q$ then $\varkappa \left(p\right)\ge \varkappa \left(q\right)$ for all $p,q\in (0,1]$;

2.

$\varkappa$ is continuous on $(0,1]$;

3.

$\varkappa \left(p\right)=0$ if and only if $p=1$.

Denote by $\Theta$ the collection of all functions $\mathsf{ϝ}:(0,1]\to [0,+\infty )$ satisfying:

1.

$\mathsf{ϝ}$ is lower semi-continuous on $(0,1]$;

2.

$\mathsf{ϝ}\left(p\right)=0$ if and only if $p=1$.

To illustrate these definitions and avoid redundant verifications in subsequent applications, we present the following standard prototypes of functions belonging to these classes. Consider the mappings $\varkappa ,\mathsf{ϝ}:(0,1]\to [0,+\infty )$ defined by:

$${\displaystyle \varkappa \left(t\right)=\frac{1}{t}-1\mathrm{and}\mathsf{ϝ}\left(t\right)=\frac{1}{2}\left(\frac{1}{t}-1\right).}$$

It is straightforward to verify that $\varkappa \in \Xi$: it maps $(0,1]$ into $[0,+\infty )$, is strictly decreasing, continuous, and satisfies $\varkappa \left(t\right)=0\iff t=1$.

Similarly, $\mathsf{ϝ}\in \Theta$ because it is continuous and $\mathsf{ϝ}\left(t\right)=0\iff t=1$. These specific functions will serve as our primary models throughout the examples in this paper.

Definition 11. 

Let $(\mathcal{U},\mathcal{F},⊡)$ be a non-Archimedean generalized Menger space and $\mathcal{A},\mathcal{B},\mathcal{C}$ non-empty subsets with ${\mathcal{A}}_0,{\mathcal{B}}_0,{\mathcal{C}}_0$ defined as above. A mapping $f:\mathcal{A}\cup \mathcal{B}\cup \mathcal{C}\to \mathcal{A}\cup \mathcal{B}\cup \mathcal{C}$ such that $f\left({\mathcal{A}}_0\right)\subseteq {\mathcal{B}}_0$, $f\left({\mathcal{B}}_0\right)\subseteq {\mathcal{C}}_0$, and $f\left({\mathcal{C}}_0\right)\subseteq {\mathcal{A}}_0$ is called a $(\varkappa -\mathsf{ϝ})$-weak proximal contraction if there exist functions $\varkappa \in \Xi$ and $\mathsf{ϝ}\in \Theta$ with the following property:

For all $\alpha ,u\in {\mathcal{A}}_0$, $\beta ,v\in {\mathcal{B}}_0$, $\gamma ,w\in {\mathcal{C}}_0$, and every $p>0$, whenever

$$\left.\begin{array}{cc}\hfill {\mathcal{F}}_{u,f\alpha ,f^2\alpha }\left(p\right)& ={\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right)\hfill \\ \hfill {\mathcal{F}}_{v,f\beta ,f^2\beta }\left(p\right)& ={\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right)\hfill \\ \hfill {\mathcal{F}}_{w,f\gamma ,f^2\gamma }\left(p\right)& ={\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right)\hfill \end{array}\right\}\mathit{implies}\varkappa \left({\mathcal{F}}_{u,v,w}\left(p\right)\right)\le \varkappa \left({\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right)\right)-\mathsf{ϝ}\left({\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right)\right).$$

Remark 2. 

For the $(\varkappa -\mathsf{ϝ})$ -weak proximal contraction to be mathematically well posed, the left-hand side of the inequality, $\varkappa \left({\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right)\right)$, is non-negative since $\varkappa \left(p\right)\ge 0$. Consequently, this logically forces the right-hand side to be non-negative as well. Therefore, throughout this paper, we inherently assume that any valid pair of control functions $(\varkappa ,\mathsf{ϝ})$ satisfies the condition $\mathsf{ϝ}\left(p\right)\le \varkappa \left(p\right)$ for all $p\in (0,1]$. Our prototype functions clearly satisfy this structural requirement since $\mathsf{ϝ}\left(p\right)={\displaystyle \frac{1}{2}}\varkappa \left(p\right)\le \varkappa \left(p\right)$.

To explicitly demonstrate a mapping satisfying this contraction, let us provide the following example.

Example 6. 

Let $\mathcal{U}=[0,1]$ equipped with the standard probabilistic G-metric where $G(\alpha ,\beta ,\gamma )=d(\alpha ,\beta )+d(\beta ,\gamma )+d(\gamma ,\alpha )$ and the distribution function

$${\displaystyle {\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right)=\left\{\begin{array}{cc}0,\hfill & p=0,\hfill \\ {\displaystyle \frac{p}{p+\mathrm{G}(\alpha ,\beta ,\gamma )}},\hfill & p>0,\hfill \end{array}\right.\forall \alpha ,\gamma ,\beta \in \mathcal{U}.}$$

Let our subsets be $\mathcal{A}=\mathcal{B}=\mathcal{C}=[0,1]$. Since the sets intersect, the optimal proximity function is ${\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right)=1$, which implies ${\mathcal{A}}_0={\mathcal{B}}_0={\mathcal{C}}_0=[0,1]$.

Let us define the control functions $\varkappa \in \Xi$ and $\mathsf{ϝ}\in \Theta$ as follows:

$${\displaystyle \varkappa \left(t\right)=\frac{1}{t}-1\mathit{and}\mathsf{ϝ}\left(t\right)=\frac{1}{2}\left(\frac{1}{t}-1\right).}$$

Define the mapping $f:\mathcal{U}\to \mathcal{U}$ as the constant function $f\left(x\right)=0$.

To verify the $(\varkappa -\mathsf{ϝ})$-weak proximal contraction, assume that for elements $\alpha ,\beta ,\gamma \in \mathcal{U}$ and $u,v,w\in \mathcal{U}$, the antecedent holds:

$${\displaystyle {\mathcal{F}}_{u,f\alpha ,f^2\alpha }\left(p\right)=1,{\mathcal{F}}_{v,f\beta ,f^2\beta }\left(p\right)=1,{\mathcal{F}}_{w,f\gamma ,f^2\gamma }\left(p\right)=1.}$$

Since $f\left(x\right)=0$ for all x, we have $f\alpha =f^2\alpha =0$. The condition ${\mathcal{F}}_{u,0,0}\left(p\right)=1$ implies $G(u,0,0)=0$, which strictly forces $u=0$. Following the same logic for the other sets, we necessarily obtain $v=0$ and $w=0$. Therefore, their proximity is ${\mathcal{F}}_{u,v,w}\left(p\right)={\mathcal{F}}_{0,0,0}\left(p\right)=1$.

Substituting these values into the required contraction inequality yields:

$${\displaystyle \varkappa \left({\mathcal{F}}_{u,v,w}\left(p\right)\right)\le \varkappa \left({\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right)\right)-\mathsf{ϝ}\left({\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right)\right)}$$

$${\displaystyle \varkappa \left(1\right)\le \left(\frac{1}{{\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right)}-1\right)-\frac{1}{2}\left(\frac{1}{{\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right)}-1\right)}$$

$${\displaystyle 0\le \frac{1}{2}\left(\frac{1}{{\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right)}-1\right).}$$

Since probabilities are bounded by ${\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right)\le 1$, the expression on the right side is always non-negative. Thus, the inequality unconditionally holds for any initial choice of $\alpha ,\beta ,\gamma$, perfectly satisfying Definition 11.

Before stating our main result, we introduce a geometric condition necessary to ensure the consistency of the iterative sequence within the proximal sets.

Definition 12. 

Let $(\mathcal{U},\mathcal{F},⊡)$ be a non-Archimedean generalized Menger space and let $\mathcal{A},\mathcal{B},\mathcal{C}$ be nonempty subsets of $\mathcal{U}$. A mapping $f:\mathcal{A}\cup \mathcal{B}\cup \mathcal{C}\to \mathcal{A}\cup \mathcal{B}\cup \mathcal{C}$ such that $f\left({\mathcal{A}}_0\right)\subseteq {\mathcal{B}}_0$, $f\left({\mathcal{B}}_0\right)\subseteq {\mathcal{C}}_0$, and $f\left({\mathcal{C}}_0\right)\subseteq {\mathcal{A}}_0$ is said to be proximal admissible on ${\mathcal{A}}_0$ if for every $\alpha \in {\mathcal{A}}_0$, there exists an element $u\in {\mathcal{A}}_0$ such that:

$${\mathcal{F}}_{u,f\alpha ,f^2\alpha }\left(p\right)={\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right),\forall p>0.$$

To provide a tangible illustration of proximal admissibility, we rely once more on the geometric structure of three parallel lines.

Example 7. 

Consider $\mathcal{U}={\mathbb{R}}^2$ with the standard probabilistic G-metric, and let us reuse the three horizontal parallel lines:

$${\displaystyle {\mathcal{A}}_0=\mathcal{A}=\{(x,0):x\in \mathbb{R}\},{\mathcal{B}}_0=\mathcal{B}=\{(x,1):x\in \mathbb{R}\},{\mathcal{C}}_0=\mathcal{C}=\{(x,2):x\in \mathbb{R}\}.}$$

We know that the optimal proximity is ${\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right)={\displaystyle \frac{p}{p+4}}$, and this optimal value is reached if and only if three points chosen from $\mathcal{A},\mathcal{B}$, and $\mathcal{C}$ are vertically aligned (i.e., they share the same horizontal coordinate).

Let us define a mapping $f:\mathcal{A}\cup \mathcal{B}\cup \mathcal{C}\to \mathcal{A}\cup \mathcal{B}\cup \mathcal{C}$ as follows:

$${\displaystyle f(x,0)=(x^2,1),f(x,1)=(x,2),f(x,2)=(0,0).}$$

Clearly, $f\left({\mathcal{A}}_0\right)\subseteq {\mathcal{B}}_0$, $f\left({\mathcal{B}}_0\right)\subseteq {\mathcal{C}}_0$, and $f\left({\mathcal{C}}_0\right)\subseteq {\mathcal{A}}_0$.

To check if f is proximal admissible on ${\mathcal{A}}_0$, let $\alpha =(x,0)$ be an arbitrary element in ${\mathcal{A}}_0$. Applying the mapping yields:

$${\displaystyle f\alpha =(x^2,1)\in {\mathcal{B}}_0\mathit{and}{f}^2\alpha =f(x^2,1)=(x^2,2)\in {\mathcal{C}}_0.}$$

According to Definition 10, we must find an element $u\in {\mathcal{A}}_0$ such that ${\mathcal{F}}_{u,f\alpha ,f^2\alpha }\left(p\right)={\displaystyle \frac{p}{p+4}}$. Based on the vertical alignment requirement, this equation holds if and only if u shares the same horizontal coordinate as $f\alpha$ and $f^2\alpha$, which is $x^2$.

Since ${\mathcal{A}}_0$ contains all points on the line $y=0$, the element $u=(x^2,0)$ is guaranteed to exist in ${\mathcal{A}}_0$. With this choice of u, the points $u=(x^2,0)$, $f\alpha =(x^2,1)$, and $f^2\alpha =(x^2,2)$ are perfectly aligned, yielding:

$${\displaystyle G(u,f\alpha ,f^2\alpha )=|x^2-x^2|+|1-2|+|2-0|=4\Rightarrow {\mathcal{F}}_{u,f\alpha ,f^2\alpha }\left(p\right)=\frac{p}{p+4}.}$$

Since such a u exists for every possible choice of α, the mapping f is proven to be proximal admissible on ${\mathcal{A}}_0$.

Remark 3. 

It is crucial to distinguish Definition 12 from the probabilistic P-property Definition 9. While the P-property imposes a rigid, global isometric structure on the proximal subsets, proximal admissibility is a much more flexible, local condition.

Geometrically and dynamically, proximal admissibility simply ensures that the mapping f respects the optimal proximity structure along its orbits. When an element $\alpha \in {\mathcal{A}}_0$ is mapped forward to $f\alpha$ and $f^2\alpha$, the admissibility hypothesis guarantees that these projected points do not “drift” into a region where the optimal distance cannot be formed. Instead, it ensures the existence of a base point $u\in {\mathcal{A}}_0$ that perfectly completes the optimal triangular configuration. Thus, admissibility only concerns the preservation of this optimal geometric configuration along the specific sequence generated by f, rather than demanding a global geometric constraint on the entire space.

Theorem 1. 

Let $(\mathcal{U},\mathcal{F},⊡)$ be a complete non-Archimedean generalized Menger space with a 1-boundary continuous triangular norm ⊡. Consider three non-empty closed subsets $\mathcal{A},\mathcal{B},\mathcal{C}$ of $\mathcal{U}$ for which the associated sets ${\mathcal{A}}_0$, ${\mathcal{B}}_0$, and ${\mathcal{C}}_0$ are non-empty. Let

$$f:\mathcal{A}\cup \mathcal{B}\cup \mathcal{C}⟶\mathcal{A}\cup \mathcal{B}\cup \mathcal{C}$$

be a mapping proximal admissible on ${\mathcal{A}}_0$ satisfying the following hypotheses:

(i) 

$f\left({\mathcal{A}}_0\right)\subseteq {\mathcal{B}}_0$, $f\left({\mathcal{B}}_0\right)\subseteq {\mathcal{C}}_0$, and $f\left({\mathcal{C}}_0\right)\subseteq {\mathcal{A}}_0$.

(ii) 

f is a $(\varkappa -\mathsf{ϝ})$-weak proximal contraction for some $\varkappa \in \Xi$ and $\mathsf{ϝ}\in \Theta$.

(iii) 

If $\left\{{\beta }_n\right\}\subseteq {\mathcal{B}}_0$, $\left\{{\gamma }_n\right\}\subseteq {\mathcal{C}}_0$, and $\alpha \in \mathcal{A}$ are such that

$$\underset{n\to +\infty }{lim}{\mathcal{F}}_{\alpha ,{\beta }_n,{\gamma }_n}\left(p\right)={\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right)\mathit{for}\mathit{every}p>0,$$

then necessarily $\alpha \in {\mathcal{A}}_0$. (Analogous statements hold for $\beta \in {\mathcal{B}}_0$ and $\gamma \in {\mathcal{C}}_0$.)

Then f possesses a unique best proximity point ${\alpha }^{\ast }\in {\mathcal{A}}_0$. Consequently, there exist corresponding points ${\beta }^{\ast }=f{\alpha }^{\ast }\in {\mathcal{B}}_0$ and ${\gamma }^{\ast }=f^2{\alpha }^{\ast }\in {\mathcal{C}}_0$ such that $({\alpha }^{\ast },{\beta }^{\ast },{\gamma }^{\ast })$ forms a best proximity triplet.

Proof. 

Consider an arbitrary element ${\alpha }_0$ belonging to ${\mathcal{A}}_0$. Since $f\left({\mathcal{A}}_0\right)\subseteq {\mathcal{B}}_0$ and $f\left({\mathcal{B}}_0\right)\subseteq {\mathcal{C}}_0$, we have $f{\alpha }_0\in {\mathcal{B}}_0$ and $f^2{\alpha }_0\in {\mathcal{C}}_0$. By the hypothesis that f is proximal admissible, there exists ${\alpha }_1\in {\mathcal{A}}_0$ such that

$${\mathcal{F}}_{{\alpha }_1,f{\alpha }_0,f^2{\alpha }_0}\left(p\right)={\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right)\forall p>0.$$

By successive iteration, we produce a sequence $\left\{{\alpha }_n\right\}$ within ${\mathcal{A}}_0$ which fulfills

$${\mathcal{F}}_{{\alpha }_{n+1},f{\alpha }_n,f^2{\alpha }_n}\left(p\right)={\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right)\mathrm{for}\mathrm{all}n\in \mathbf{N},\mathrm{and}\mathrm{for}\mathrm{all}p>0.$$

(4)

Suppose there exists $n_0\in \mathbf{N}$ such that ${\mathcal{F}}_{{\alpha }_{n_0},{\alpha }_{n_0+1},{\alpha }_{n_0+1}}\left(p\right)=1$ for all $p>0$. Then we deduce that

$$\begin{array}{cc}\hfill {\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right)& \ge {\mathcal{F}}_{{\alpha }_{n_0},f{\alpha }_{n_0},f^2{\alpha }_{n_0}}\left(p\right)\hfill \\ & \ge ⊡({\mathcal{F}}_{{\alpha }_{n_0},{\alpha }_{n_0+1},{\alpha }_{n_0+1}}\left(p\right);{\mathcal{F}}_{{\alpha }_{n_0+1},f{\alpha }_{n_0},f^2{\alpha }_{n_0}}\left(p\right))\hfill \\ & =⊡(1;{\mathcal{F}}_{{\alpha }_{n_0+1},f{\alpha }_{n_0},f^2{\alpha }_{n_0}}\left(p\right))\hfill \\ & ={\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right)\forall p>0.\hfill \end{array}$$

Hence, the identity

$${\mathcal{F}}_{{\alpha }_{n_0},f{\alpha }_{n_0},f^2{\alpha }_{n_0}}\left(p\right)={\mathcal{F}}_{A,B,C}\left(p\right)\forall p>0,$$

holds trivially, and ${\alpha }_{n_0}$ constitutes a best proximity point of the mapping f, forming a triple with $f{\alpha }_{n_0}$ and $f^2{\alpha }_{n_0}$. Thus, it is sufficient to assume in the sequel that

$$0<{\mathcal{F}}_{{\alpha }_n,{\alpha }_{n+1},{\alpha }_{n+1}}\left(p\right)<1\mathrm{for}\mathrm{every}n\in \mathbf{N},$$

which is equivalent to the requirement ${\alpha }_n\ne {\alpha }_{n+1}$, for every $n\in \mathbf{N}$. Since f is a $(\varkappa -\mathsf{ϝ})$-weak proximal contraction, the proximal relations at consecutive steps yield

$$\begin{array}{cc}\hfill \varkappa \left({\mathcal{F}}_{{\alpha }_{n+1},{\alpha }_{n+2},{\alpha }_{n+2}}\left(p\right)\right)& \le \varkappa \left({\mathcal{F}}_{{\alpha }_n,{\alpha }_{n+1},{\alpha }_{n+1}}\left(p\right)\right)-\mathsf{ϝ}\left({\mathcal{F}}_{{\alpha }_n,{\alpha }_{n+1},{\alpha }_{n+1}}\left(p\right)\right)\hfill \\ \hfill & \le \varkappa \left({\mathcal{F}}_{{\alpha }_n,{\alpha }_{n+1},{\alpha }_{n+1}}\left(p\right)\right).\hfill \end{array}$$

(5)

Because $\varkappa$ is monotone decreasing, it follows from (5) that

$${\mathcal{F}}_{{\alpha }_{n+1},{\alpha }_{n+2},{\alpha }_{n+2}}\left(p\right)\ge {\mathcal{F}}_{{\alpha }_n,{\alpha }_{n+1},{\alpha }_{n+1}}\left(p\right)\forall p>0.$$

Thus, for any fixed $p>0$, the sequence $\left\{{\mathcal{F}}_{{\alpha }_n,{\alpha }_{n+1},{\alpha }_{n+1}}\left(p\right)\right\}$ is increasing and bounded above by 1, hence it lies within $[0,1]$. Defining

$$r\left(p\right)=\underset{n\to +\infty }{lim}{\mathcal{F}}_{{\alpha }_n,{\alpha }_{n+1},{\alpha }_{n+1}}\left(p\right)\forall p>0.$$

We now show that $r\left(p\right)=1$ for every $p>0$. To see this, take the limit as $n\to +\infty$ in inequality (5) and make use of the continuity of $\varkappa$ along with the lower semi-continuity of $\mathsf{ϝ}$; this yields

$$\varkappa \left(r\right(p\left)\right)\le \varkappa \left(r\right(p\left)\right)-\mathsf{ϝ}\left(r\right(p\left)\right),$$

which necessarily implies $\mathsf{ϝ}\left(r\left(p\right)\right)=0$ and consequently $r\left(p\right)=1$ for all $p>0$.

Next, we establish that $\left\{{\alpha }_n\right\}$ is a Cauchy sequence. Assume, for the sake of contradiction, that $\left\{{\alpha }_n\right\}$ is not a Cauchy. Then one can choose numbers $\varsigma >0$ and $\varrho \in (0,1)$ with the property that for each positive integer k there are indices $n_k,m_k\in \mathbf{N}$ satisfying $n_k>m_k>k$ and

$${\mathcal{F}}_{{\alpha }_{m_k},{\alpha }_{n_k},{\alpha }_{n_k}}(\varsigma )\le 1-\varrho .$$

Let $n_k$ denote the least integer exceeding $m_k$ that satisfies the inequality above,

$${\mathcal{F}}_{{\alpha }_{m_k},{\alpha }_{n_k-1},{\alpha }_{n_k-1}}(\varsigma )>1-\varrho .$$

(6)

So, for all $k\in \mathbf{N}$, we obtain using the rectangle inequality (2):

$$\begin{array}{cc}\hfill 1-\varrho & \ge {\mathcal{F}}_{{\alpha }_{m_k},{\alpha }_{n_k},{\alpha }_{n_k}}(\varsigma )\hfill \\ & \ge ⊡({\mathcal{F}}_{{\alpha }_{m_k},{\alpha }_{n_k-1},{\alpha }_{n_k-1}}(\varsigma );{\mathcal{F}}_{{\alpha }_{n_k-1},{\alpha }_{n_k},{\alpha }_{n_k}}(\varsigma ))\hfill \\ & >⊡(1-\varrho ;{\mathcal{F}}_{{\alpha }_{n_k-1},{\alpha }_{n_k},{\alpha }_{n_k}}(\varsigma )).\hfill \end{array}$$

Since ${lim}_{k\to +\infty }{\mathcal{F}}_{{\alpha }_{n_k-1},{\alpha }_{n_k},{\alpha }_{n_k}}(\varsigma )=1$ (as sequences of consecutive terms converge) and ⊡ is continuous at 1-boundary, we deduce that

$$\underset{k\to +\infty }{lim}{\mathcal{F}}_{{\alpha }_{m_k},{\alpha }_{n_k},{\alpha }_{n_k}}(\varsigma )=1-\varrho .$$

Taking into account that by (6),

$$\begin{array}{cc}\hfill 1-\varrho & \ge ⊡({\mathcal{F}}_{{\alpha }_{m_k},{\alpha }_{n_k-1},{\alpha }_{n_k-1}}(\varsigma );{\mathcal{F}}_{{\alpha }_{n_k-1},{\alpha }_{n_k},{\alpha }_{n_k}}(\varsigma ))\hfill \\ \hfill & \ge ⊡(1-\varrho ;{\mathcal{F}}_{{\alpha }_{n_k-1},{\alpha }_{n_k},{\alpha }_{n_k}}(\varsigma )),\hfill \end{array}$$

for every $k\in \mathbf{N}$. Applying Proposition 1, we deduce that

$$\underset{k\to +\infty }{lim}{\mathcal{F}}_{{\alpha }_{m_k},{\alpha }_{n_k-1},{\alpha }_{n_k-1}}(\varsigma )=1-\varrho ,$$

so that

$$\underset{k\to +\infty }{lim}{\mathcal{F}}_{{\alpha }_{m_k},{\alpha }_{n_k},{\alpha }_{n_k}}(\varsigma )=\underset{k\to +\infty }{lim}{\mathcal{F}}_{{\alpha }_{m_k},{\alpha }_{n_k-1},{\alpha }_{n_k-1}}(\varsigma )=1-\varrho .$$

(7)

Next, for all strictly positive natural number k, using the rectangle inequality of the non-Archimedean generalized Menger (2), we have

$$\begin{array}{cc}\hfill {\mathcal{F}}_{{\alpha }_{m_k},{\alpha }_{n_k},{\alpha }_{n_k}}(\varsigma )& \ge ⊡({\mathcal{F}}_{{\alpha }_{m_k},{\alpha }_{m_k-1},{\alpha }_{m_k-1}}(\varsigma );⊡({\mathcal{F}}_{{\alpha }_{m_k-1},{\alpha }_{n_k-1},{\alpha }_{n_k-1}}(\varsigma );{\mathcal{F}}_{{\alpha }_{n_k-1},{\alpha }_{n_k},{\alpha }_{n_k}}(\varsigma )))\hfill \\ & \ge ⊡({\mathcal{F}}_{{\alpha }_{m_k-1},{\alpha }_{m_k},{\alpha }_{m_k}}(\varsigma );⊡({\mathcal{F}}_{{\alpha }_{m_k-1},{\alpha }_{m_k},{\alpha }_{m_k}}(\varsigma );\hfill \\ & ⊡({\mathcal{F}}_{{\alpha }_{m_k-1},{\alpha }_{n_k-1},{\alpha }_{n_k-1}}(\varsigma );{\mathcal{F}}_{{\alpha }_{n_k-1},{\alpha }_{n_k},{\alpha }_{n_k}}(\varsigma ))\left)\right)\hfill \\ & \ge ⊡({\mathcal{F}}_{{\alpha }_{m_k-1},{\alpha }_{m_k},{\alpha }_{m_k}}(\varsigma );⊡({\mathcal{F}}_{{\alpha }_{m_k-1},{\alpha }_{m_k},{\alpha }_{m_k}}(\varsigma );\hfill \\ & ⊡({\mathcal{F}}_{{\alpha }_{m_k-1},{\alpha }_{m_k},{\alpha }_{m_k}}(\varsigma ),⊡({\mathcal{F}}_{{\alpha }_{m_k},{\alpha }_{n_k-1},{\alpha }_{n_k-1}}(\varsigma );{\mathcal{F}}_{{\alpha }_{n_k-1},{\alpha }_{n_k},{\alpha }_{n_k}}(\varsigma )))\left)\right).\hfill \end{array}$$

Clearly, from the convergence of consecutive terms, ${\mathcal{F}}_{{\alpha }_{m_k-1},{\alpha }_{m_k},{\alpha }_{m_k}}(\varsigma )\to 1$ and ${\mathcal{F}}_{{\alpha }_{n_k-1},{\alpha }_{n_k},{\alpha }_{n_k}}(\varsigma )\to 1$ as $k\to +\infty$. Also, by (7) we established that ${\mathcal{F}}_{{\alpha }_{m_k},{\alpha }_{n_k},{\alpha }_{n_k}}(\varsigma )\to$$1-\varrho$, and ${\mathcal{F}}_{{\alpha }_{m_k},{\alpha }_{n_k-1},{\alpha }_{n_k-1}}(\varsigma )\to 1-\varrho$. Thus, from Proposition 1, we can obtain

$$\underset{k\to +\infty }{lim}{\mathcal{F}}_{{\alpha }_{m_k-1},{\alpha }_{n_k-1},{\alpha }_{n_k-1}}(\varsigma )=1-\varrho .$$

As a consequence,

$$\underset{k\to +\infty }{lim}{\mathcal{F}}_{{\alpha }_{m_k},{\alpha }_{n_k},{\alpha }_{n_k}}(\varsigma )=\underset{k\to +\infty }{lim}{\mathcal{F}}_{{\alpha }_{m_k-1},{\alpha }_{n_k-1},{\alpha }_{n_k-1}}(\varsigma )=1-\varrho .$$

From relation (4) derived earlier, we know that

$${\mathcal{F}}_{{\alpha }_{m_k},f{\alpha }_{m_k-1},f^2{\alpha }_{m_k-1}}(\varsigma )={\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}(\varsigma )$$

and

$${\mathcal{F}}_{{\alpha }_{n_k},f{\alpha }_{n_k-1},f^2{\alpha }_{n_k-1}}(\varsigma )={\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}(\varsigma ).$$

Next, by the $(\varkappa -\mathsf{ϝ})$-weak proximal contraction condition of f, we have

$$\varkappa \left({\mathcal{F}}_{{\alpha }_{m_k},{\alpha }_{n_k},{\alpha }_{n_k}}(\varsigma )\right)\le \varkappa \left({\mathcal{F}}_{{\alpha }_{m_k-1},{\alpha }_{n_k-1},{\alpha }_{n_k-1}}(\varsigma )\right)-\mathsf{ϝ}\left({\mathcal{F}}_{{\alpha }_{m_k-1},{\alpha }_{n_k-1},{\alpha }_{n_k-1}}(\varsigma )\right).$$

Taking the limit as $k\to +\infty$ in the above inequality, and using the continuity of $\varkappa$ and lower semi-continuity of $\mathsf{ϝ}$, we get

$${\displaystyle \varkappa (1-\varrho )\le \varkappa (1-\varrho )-\mathsf{ϝ}(1-\varrho )\Rightarrow \mathsf{ϝ}(1-\varrho )=0,}$$

which contradicts the property of $\mathsf{ϝ}$ (since $1-\varrho <1$). Thus, our assumption was wrong, and $\left\{{\alpha }_n\right\}$ is a Cauchy sequence in ${\mathcal{A}}_0$.

The completeness of $(\mathcal{U},\mathcal{F},⊡)$ ensures that the sequence $\left\{{\alpha }_n\right\}$ converges to some ${\alpha }^{\ast }\in \mathcal{U}$, i.e.,

$$\underset{n\to +\infty }{lim}{\mathcal{F}}_{{\alpha }_n,{\alpha }^{\ast },{\alpha }^{\ast }}\left(p\right)=1\forall p>0.$$

Moreover,

$$\begin{array}{cc}\hfill {\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right)& ={\mathcal{F}}_{{\alpha }_n,f{\alpha }_{n-1},f^2{\alpha }_{n-1}}\left(p\right)\hfill \\ & \ge ⊡({\mathcal{F}}_{{\alpha }_n,{\alpha }^{\ast },{\alpha }^{\ast }}\left(p\right);{\mathcal{F}}_{{\alpha }^{\ast },f{\alpha }_{n-1},f^2{\alpha }_{n-1}}\left(p\right))\hfill \\ & =⊡({\mathcal{F}}_{{\alpha }_n,{\alpha }^{\ast },{\alpha }^{\ast }}\left(p\right);⊡({\mathcal{F}}_{{\alpha }^{\ast },f{\alpha }_{n-1},f^2{\alpha }_{n-1}}\left(p\right);1))\hfill \\ & \ge ⊡({\mathcal{F}}_{{\alpha }_n,{\alpha }^{\ast },{\alpha }^{\ast }}\left(p\right);⊡({\mathcal{F}}_{{\alpha }^{\ast },f{\alpha }_{n-1},f^2{\alpha }_{n-1}}\left(p\right);{\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right))).\hfill \end{array}$$

This implies that

$$\begin{array}{cc}\hfill {\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right)& \ge ⊡({\mathcal{F}}_{{\alpha }_n,{\alpha }^{\ast },{\alpha }^{\ast }}\left(p\right);{\mathcal{F}}_{{\alpha }^{\ast },f{\alpha }_{n-1},f^2{\alpha }_{n-1}}\left(p\right))\hfill \\ & \ge ⊡({\mathcal{F}}_{{\alpha }_n,{\alpha }^{\ast },{\alpha }^{\ast }}\left(p\right);⊡({\mathcal{F}}_{{\alpha }^{\ast },{\alpha }_n,{\alpha }_n}\left(p\right);{\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right))).\hfill \end{array}$$

Passing to the limit as $n\to +\infty$ in the inequality above, we obtain

$${\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right)\ge ⊡(1;\underset{n\to +\infty }{lim}{\mathcal{F}}_{{\alpha }^{\ast },f{\alpha }_{n-1},f^2{\alpha }_{n-1}}\left(p\right))\ge ⊡(1;⊡(1;{\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right))),$$

that is

$$\underset{n\to +\infty }{lim}{\mathcal{F}}_{{\alpha }^{\ast },f{\alpha }_{n-1},f^2{\alpha }_{n-1}}\left(p\right)={\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right),$$

and so by condition (iii) of Theorem 1, ${\alpha }^{\ast }\in {\mathcal{A}}_0$. Since ${\alpha }^{\ast }\in {\mathcal{A}}_0$, $f{\alpha }^{\ast }\in {\mathcal{B}}_0$ and $f^2{\alpha }^{\ast }\in {\mathcal{C}}_0$, then by the hypothesis that f is proximal admissible, there exists ${\alpha }^{\prime }\in {\mathcal{A}}_0$ such that

$${\mathcal{F}}_{{\alpha }^{\prime },f{\alpha }^{\ast },f^2{\alpha }^{\ast }}\left(p\right)={\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right)\mathrm{for}\mathrm{all}p>0.$$

(8)

From (4) and (8), together with the contraction property of f, we deduce:

$$\varkappa \left({\mathcal{F}}_{{\alpha }^{\prime },{\alpha }_n,{\alpha }_n}\left(p\right)\right)\le \varkappa \left({\mathcal{F}}_{{\alpha }^{\ast },{\alpha }_{n-1},{\alpha }_{n-1}}\left(p\right)\right)-\mathsf{ϝ}\left({\mathcal{F}}_{{\alpha }^{\ast },{\alpha }_{n-1},{\alpha }_{n-1}}\left(p\right)\right).$$

Passing to the limit as $n\to +\infty$, using the continuity of $\varkappa$ and $\mathcal{F}$, and the lower semi continuity of $\mathsf{ϝ}$, we obtain

$$\underset{n\to +\infty }{lim}\varkappa \left({\mathcal{F}}_{{\alpha }^{\prime },{\alpha }_n,{\alpha }_n}\left(p\right)\right)\le \varkappa \left(1\right)-\mathsf{ϝ}\left(1\right)=0,$$

which implies that

$$\underset{n\to +\infty }{lim}{\mathcal{F}}_{{\alpha }^{\prime },{\alpha }_n,{\alpha }_n}\left(p\right)=1\forall p>0.$$

Since ${\alpha }_n\to {\alpha }^{\ast }$, by the uniqueness of the limit, we conclude that ${\alpha }^{\prime }={\alpha }^{\ast }$. Thus, substituting ${\alpha }^{\prime }$ with ${\alpha }^{\ast }$ in (8), we get:

$${\mathcal{F}}_{{\alpha }^{\ast },f{\alpha }^{\ast },f^2{\alpha }^{\ast }}\left(p\right)={\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right),\forall p>0.$$

This establishes that ${\alpha }^{\ast }$ is a best proximity point of f.

Finally, to prove the uniqueness of the best proximity point, let us assume there exists another best proximity point $y^{\ast }\in {\mathcal{A}}_0$. This means that ${\mathcal{F}}_{y^{\ast },f{y}^{\ast },f^2{y}^{\ast }}\left(p\right)={\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right)$ for all $p>0$. Since ${\alpha }^{\ast }$ is also a best proximity point, we already know that ${\mathcal{F}}_{{\alpha }^{\ast },f{\alpha }^{\ast },f^2{\alpha }^{\ast }}\left(p\right)={\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right)$.

Applying the $(\varkappa -\mathsf{ϝ})$-weak proximal contraction property of f to these two points, we obtain:

$$\begin{array}{cc}\hfill \varkappa \left({\mathcal{F}}_{{\alpha }^{\ast },y^{\ast },y^{\ast }}\left(p\right)\right)& \le \varkappa \left({\mathcal{F}}_{{\alpha }^{\ast },y^{\ast },y^{\ast }}\left(p\right)\right)-\mathsf{ϝ}\left({\mathcal{F}}_{{\alpha }^{\ast },y^{\ast },y^{\ast }}\left(p\right)\right).\hfill \end{array}$$

This inequality simplifies to $\mathsf{ϝ}\left({\mathcal{F}}_{{\alpha }^{\ast },y^{\ast },y^{\ast }}\left(p\right)\right)\le 0$. Since $\mathsf{ϝ}\in \Theta$, we necessarily have $\mathsf{ϝ}\left({\mathcal{F}}_{{\alpha }^{\ast },y^{\ast },y^{\ast }}\left(p\right)\right)=0$, which in turn implies that ${\mathcal{F}}_{{\alpha }^{\ast },y^{\ast },y^{\ast }}\left(p\right)=1$ for all $p>0$. By the properties of the generalized Menger space, this yields ${\alpha }^{\ast }=y^{\ast }$. Thus, the best proximity point is unique, completing the proof. □

To demonstrate the applicability of Theorem 1, we provide the following concrete example.

Example 8. 

Let $\mathcal{U}={\mathbb{R}}^2$ be endowed with the sum metric

$${\displaystyle G((x_1,y_1),(x_2,y_2),(x_3,y_3))=\sum _{1\le i<j\le 3}\left(\right|x_i-x_j|+|y_i-y_j\left|\right).}$$

We define the non-Archimedean generalized Menger space using the standard probabilistic metric

$${\displaystyle {\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right)=\left\{\begin{array}{cc}0,\hfill & p=0,\hfill \\ {\displaystyle \frac{p}{p+G(\alpha ,\beta ,\gamma )}},\hfill & p>0,\hfill \end{array}\right.\forall \alpha ,\gamma ,\beta \in \mathcal{U}.}$$

equipped with the continuous triangular norm ${⊡}_M$.

Consider the three closed, parallel subsets:

$${\displaystyle \mathcal{A}=\left\{\right(x,0):x\in \mathbb{R}\},\mathcal{B}=\left\{\right(x,1):x\in \mathbb{R}\},\mathcal{C}=\left\{\right(x,2):x\in \mathbb{R}\}.}$$

The optimal proximity is achieved for vertically aligned points, yielding

$${\displaystyle {\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right)=\frac{p}{p+4}.}$$

Note that ${\mathcal{A}}_0=\mathcal{A}$, ${\mathcal{B}}_0=\mathcal{B}$, and ${\mathcal{C}}_0=\mathcal{C}$.

Let us define the mapping $f:\mathcal{A}\cup \mathcal{B}\cup \mathcal{C}\to \mathcal{A}\cup \mathcal{B}\cup \mathcal{C}$ as:

$${\displaystyle f(x,0)=\left(\frac{x}{2},1\right),f(x,1)=(x,2),f(x,2)=\left(\frac{x}{2},0\right).}$$

Condition (i) of Theorem 1 is obviously satisfied.

To verify that f is proximal admissible, let $\alpha =(x,0)\in {\mathcal{A}}_0$. Then $f\alpha =({\displaystyle \frac{x}{2}},1)$ and $f^2\alpha =f({\displaystyle \frac{x}{2}},1)=({\displaystyle \frac{x}{2}},2)$. By choosing $u=({\displaystyle \frac{x}{2}},0)\in {\mathcal{A}}_0$, the triplet $(u,f\alpha ,f^2\alpha )$ is vertically aligned, achieving the optimal distance ${\displaystyle \frac{p}{p+4}}$.

To verify that f is $(\varkappa -\mathsf{ϝ})$-weak proximal contraction: Let ${\alpha }_1=(x,0)$ and ${\alpha }_2=(y,0)$ be in ${\mathcal{A}}_0$. The unique elements $u_1,u_2\in {\mathcal{A}}_0$ that satisfy the optimal distance relations with their respective forward orbits are $u_1=\left({\displaystyle \frac{x}{2}},0\right)$ and $u_2=\left({\displaystyle \frac{y}{2}},0\right)$. First, we evaluate the probabilistic distances:

$${\displaystyle G(u_1,u_2,u_2)=2\left|\frac{x}{2}-\frac{y}{2}\right|=|x-y|\Rightarrow {\mathcal{F}}_{u_1,u_2,u_2}\left(p\right)=\frac{p}{p+|x-y|}.}$$

$${\displaystyle G({\alpha }_1,{\alpha }_2,{\alpha }_2)=2|x-y|\Rightarrow {\mathcal{F}}_{{\alpha }_1,{\alpha }_2,{\alpha }_2}\left(p\right)=\frac{p}{p+2|x-y|}.}$$

Let us define the functions $\varkappa :(0,1]\to [0,+\infty )$ and $\mathsf{ϝ}:(0,1]\to [0,+\infty )$ by

$${\displaystyle \varkappa \left(t\right)=\frac{1}{t}-1\mathit{and}\mathsf{ϝ}\left(t\right)=\frac{1}{2}\left(\frac{1}{t}-1\right).}$$

Substituting our probabilistic distances into these functions, we obtain:

$${\displaystyle \varkappa \left({\mathcal{F}}_{u_1,u_2,u_2}\left(p\right)\right)=\frac{p+|x-y|}{p}-1=\frac{|x-y|}{p}.}$$

$${\displaystyle \varkappa \left({\mathcal{F}}_{{\alpha }_1,{\alpha }_2,{\alpha }_2}\left(p\right)\right)=\frac{p+2|x-y|}{p}-1=\frac{2|x-y|}{p}.}$$

$${\displaystyle \mathsf{ϝ}\left({\mathcal{F}}_{{\alpha }_1,{\alpha }_2,{\alpha }_2}\left(p\right)\right)=\frac{1}{2}\left(\frac{2|x-y|}{p}\right)=\frac{|x-y|}{p}.}$$

Thus, the required inequality is satisfied as an exact identity:

$${\displaystyle \varkappa \left({\mathcal{F}}_{u_1,u_2,u_2}\left(p\right)\right)=\varkappa \left({\mathcal{F}}_{{\alpha }_1,{\alpha }_2,{\alpha }_2}\left(p\right)\right)-\mathsf{ϝ}\left({\mathcal{F}}_{{\alpha }_1,{\alpha }_2,{\alpha }_2}\left(p\right)\right).}$$

Hence, f is rigorously a $(\varkappa -\mathsf{ϝ})$-weak proximal contraction.

Topological property: Let $\left\{{\beta }_n\right\}=\left\{(y_n,1)\right\}\subseteq {\mathcal{B}}_0$, $\left\{{\gamma }_n\right\}=\left\{(z_n,2)\right\}\subseteq {\mathcal{C}}_0$, and $\alpha =(x,0)\in \mathcal{A}$. Assume that for all $p>0$:

$${\displaystyle \underset{n\to +\infty }{lim}{\mathcal{F}}_{\alpha ,{\beta }_n,{\gamma }_n}\left(p\right)={\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right)=\frac{p}{p+4}.}$$

This limit holds if and only if ${lim}_{n\to +\infty }G(\alpha ,{\beta }_n,{\gamma }_n)=4$. By the definition of our metric:

$${\displaystyle G(\alpha ,{\beta }_n,{\gamma }_n)=(|x-y_n|+|y_n-z_n|+|z_n-x|)+(|0-1|+|1-2|+|2-0|).}$$

$${\displaystyle G(\alpha ,{\beta }_n,{\gamma }_n)=|x-y_n|+|y_n-z_n|+|z_n-x|+4.}$$

For the limit to equal 4, we must have ${lim}_{n\to +\infty }\left(\right|x-y_n|+|y_n-z_n|+|z_n-x\left|\right)=0$, which forces $y_n\to x$ and $z_n\to x$. Since $\alpha =(x,0)$ is an element of $\mathcal{A}$, and we have previously established that $\mathcal{A}={\mathcal{A}}_0$, it trivially follows that $\alpha \in {\mathcal{A}}_0$. Therefore, condition (iii) is analytically verified.

Thus, all conditions of Theorem 1 are satisfied, and consequently f admits a best proximity point. Indeed, one easily checks that $(0,0)\in \mathcal{A}$ (and similarly $(0,1)\in \mathcal{B}$ and $(0,2)\in \mathcal{C}$) is such a point, since $f(0,0)=(0,1)$, $f^2(0,0)=(0,2)$ and the triplet $\left(\right(0,0),(0,1),(0,2\left)\right)$ achieves the optimal distance ${\displaystyle \frac{p}{p+4}}$.

As a consequence of Theorem 1, we have the following. Building directly upon the framework established in Theorem 1, we derive a simplified version that imposes stronger topological assumptions on the proximal sets.

Remark 4. 

In Theorem 1, Condition (iii) becomes superfluous if we assume that the proximal sets ${\mathcal{A}}_0,{\mathcal{B}}_0$, and ${\mathcal{C}}_0$ are closed.

To see this, consider the situation that arises in the proof: we have sequences $\left\{{\beta }_n\right\}\subseteq {\mathcal{B}}_0$ and $\left\{{\gamma }_n\right\}\subseteq {\mathcal{C}}_0$ (constructed as $f{\alpha }_{n-1}$ and $f^2{\alpha }_{n-1}$) which are Cauchy and therefore converge in the complete space $\mathcal{U}$. By the closedness of ${\mathcal{B}}_0$ and ${\mathcal{C}}_0$, their limits satisfy ${\beta }^{\ast }\in {\mathcal{B}}_0$ and ${\gamma }^{\ast }\in {\mathcal{C}}_0$. If $\alpha \in \mathcal{A}$ is such that

$$\underset{n\to +\infty }{lim}{\mathcal{F}}_{\alpha ,{\beta }_n,{\gamma }_n}\left(p\right)={\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right)\forall p>0,$$

then the continuity of $\mathcal{F}$ yields

$${\mathcal{F}}_{\alpha ,{\beta }^{\ast },{\gamma }^{\ast }}\left(p\right)={\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right)\forall p>0.$$

Since ${\beta }^{\ast }\in {\mathcal{B}}_0$ and ${\gamma }^{\ast }\in {\mathcal{C}}_0$, the very definition of ${\mathcal{A}}_0$ forces α to belong to ${\mathcal{A}}_0$.

Notice that this argument uses only the closedness of the proximal sets ${\mathcal{B}}_0$ and ${\mathcal{C}}_0$; the closedness of the original sets $\mathcal{A},\mathcal{B},\mathcal{C}$ is not required. Consequently, in any version of Theorem 1 where ${\mathcal{A}}_0,{\mathcal{B}}_0,{\mathcal{C}}_0$ are assumed to be closed, Condition (iii) can be omitted without affecting the validity of the proof.

Theorem 2. 

Let $(\mathcal{U},\mathcal{F},⊡)$ be a complete generalized Menger space whose triangular norm ⊡ is 1-boundary continuous. Let $\mathcal{A},\mathcal{B},\mathcal{C}$ be three non-empty subsets of $\mathcal{U}$ and assume that the proximal sets ${\mathcal{A}}_0,{\mathcal{B}}_0,{\mathcal{C}}_0$ are non-empty and closed. Suppose $f:\mathcal{A}\cup \mathcal{B}\cup \mathcal{C}\to \mathcal{A}\cup \mathcal{B}\cup \mathcal{C}$ is a mapping proximal admissible satisfying:

(i) 

$f\left({\mathcal{A}}_0\right)\subseteq {\mathcal{B}}_0$, $f\left({\mathcal{B}}_0\right)\subseteq {\mathcal{C}}_0$, and $f\left({\mathcal{C}}_0\right)\subseteq {\mathcal{A}}_0$;

(ii) 

f is a $(\varkappa -\mathsf{ϝ})$-weak proximal contraction.

Then f admits a unique best proximity point ${\alpha }^{\ast }\in {\mathcal{A}}_0$.

To highlight the specific utility of Theorem 2 and the preceding Remark, we present an example where the original subsets are not necessarily closed, but their proximal counterparts are closed, non-empty, and perfectly admit a unique best proximity point without requiring Condition (iii).

Example 9. 

Let $\mathcal{U}={\mathbb{R}}^2$ be endowed with the G-metric

$${\displaystyle G((x_1,y_1),(x_2,y_2),(x_3,y_3))=\sum _{1\le i<j\le 3}\left(|x_i-x_j|+|y_i-y_j|\right).}$$

We define the generalized Menger space using the probabilistic metric

$${\displaystyle {\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right)=\left\{\begin{array}{cc}0,\hfill & p=0,\hfill \\ {\displaystyle \frac{p}{p+G(\alpha ,\beta ,\gamma )}},\hfill & p>0,\hfill \end{array}\right.\forall \alpha ,\gamma ,\beta \in \mathcal{U},}$$

and the continuous triangular norm $⊡(a,b)=min(a,b)$.

Consider the three subsets of $\mathcal{U}$:

$$\begin{gathered} {\displaystyle \mathcal{A}=\left([-1,1]\times \left\{0\right\}\right)\cup \left((2,3)\times \left\{0\right\}\right),} \\ {\displaystyle \mathcal{B}=[-1,1]\times \left\{1\right\},} \\ {\displaystyle \mathcal{C}=[-1,1]\times \left\{2\right\}.} \end{gathered}$$

Notice that $\mathcal{A}$ is strictly not closed. The optimal G-metric distance between these sets is achieved exclusively when the x-coordinates are identical. Because elements in $\mathcal{B}$ and $\mathcal{C}$ are restricted to $x\in [-1,1]$, the optimal vertical alignment can only occur if the element in $\mathcal{A}$ also has its x-coordinate in $[-1,1]$. Thus, the minimum distance is $G(\mathcal{A},\mathcal{B},\mathcal{C})=4$, yielding ${\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right)={\displaystyle \frac{p}{p+4}}$.

Consequently, the proximal sets are:

$${\displaystyle {\mathcal{A}}_0=[-1,1]\times \left\{0\right\},{\mathcal{B}}_0=\mathcal{B},{\mathcal{C}}_0=\mathcal{C}.}$$

Remarkably, even though $\mathcal{A}$ is not closed, the proximal set ${\mathcal{A}}_0$ is completely closed and non-empty. This perfectly fits the topological framework of Theorem 2.

Let us define the proximal admissible mapping $f:\mathcal{A}\cup \mathcal{B}\cup \mathcal{C}\to \mathcal{A}\cup \mathcal{B}\cup \mathcal{C}$ such that:

$${\displaystyle f(x,0)=\left\{\begin{array}{cc}\left(\frac{x}{2},1\right)\hfill & \mathit{if}x\in [-1,1]\hfill \\ (0,1)\hfill & \mathit{if}x\in (2,3)\hfill \end{array}\right.,f(x,1)=(x,2),f(x,2)=\left(\frac{x}{2},0\right).}$$

We verify the cyclic behavior on the proximal sets:

$${\displaystyle f\left({\mathcal{A}}_0\right)=[-\frac{1}{2},\frac{1}{2}]\times \left\{1\right\}\subseteq {\mathcal{B}}_0,f\left({\mathcal{B}}_0\right)={\mathcal{C}}_0,\mathit{and}f\left({\mathcal{C}}_0\right)=[-\frac{1}{2},\frac{1}{2}]\times \left\{0\right\}\subseteq {\mathcal{A}}_0.}$$

For Condition (ii), we use the exact same rigorous analytical contraction established in the example of Theorem 1. By defining

$${\displaystyle \varkappa \left(t\right)=\frac{1}{t}-1\in \Xi \mathit{and}\mathsf{ϝ}\left(t\right)=\frac{1}{2}\left(\frac{1}{t}-1\right)\in \Theta .}$$

For any ${\alpha }_1=(x,0)$ and ${\alpha }_2=(y,0)$ in ${\mathcal{A}}_0$ alongside their optimal projection elements $u_1=({\displaystyle \frac{x}{2}},0)$ and $u_2=({\displaystyle \frac{y}{2}},0)$ in ${\mathcal{A}}_0$, we obtain the exact identity:

$${\displaystyle \varkappa \left({\mathcal{F}}_{u_1,u_2,u_2}\left(p\right)\right)=\varkappa \left({\mathcal{F}}_{{\alpha }_1,{\alpha }_2,{\alpha }_2}\left(p\right)\right)-\mathsf{ϝ}\left({\mathcal{F}}_{{\alpha }_1,{\alpha }_2,{\alpha }_2}\left(p\right)\right)=\frac{|x-y|}{p}.}$$

Thus, f is a $(\varkappa -\mathsf{ϝ})$-weak proximal contraction.

Since ${\mathcal{A}}_0,{\mathcal{B}}_0$, and ${\mathcal{C}}_0$ are closed, Theorem 2 guarantees the existence of a unique best proximity point without needing to check any sequence convergence limits (Condition iii of Theorem 1). Indeed, the unique best proximity point is ${\alpha }^{\ast }=(0,0)$.

3.3. Fixed Point Consequences

In this section, we demonstrate that the classical fixed point theorems are natural consequences of our main result. By specializing the geometric configuration to the case where the three subsets coincide with the entire space, i.e., $\mathcal{A}=\mathcal{B}=\mathcal{C}=\mathcal{U}$, the concept of a best proximity point naturally reduces to that of a fixed point.

Indeed, in this setting, the probabilistic distance between the sets becomes maximal, satisfying

$${\mathcal{F}}_{\mathcal{U},\mathcal{U},\mathcal{U}}\left(p\right)=1,\mathrm{for}\mathrm{all}p>0.$$

Consequently, the proximal sets are trivial, with ${\mathcal{A}}_0={\mathcal{B}}_0={\mathcal{C}}_0=\mathcal{U}$. Furthermore, the condition for a point ${\alpha }^{\ast }$ to be a best proximity point,

$${\mathcal{F}}_{{\alpha }^{\ast },f{\alpha }^{\ast },f^2{\alpha }^{\ast }}\left(p\right)={\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right)=1,\mathrm{for}\mathrm{all}p>0,$$

implies that

$${\alpha }^{\ast }=f{\alpha }^{\ast }=f^2{\alpha }^{\ast }.$$

Thus, ${\alpha }^{\ast }$ is a fixed point. Finally, recall that condition (iii) of Theorem 1 requires that if a sequence converges to the optimal distance, the limit point must belong to the proximal set ${\mathcal{A}}_0$. Specifically:

$$\underset{n\to +\infty }{lim}{\mathcal{F}}_{\alpha ,{\beta }_n,{\gamma }_n}\left(p\right)={\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right)\Rightarrow \alpha \in {\mathcal{A}}_0.$$

In our case,

$${\mathcal{F}}_{\mathcal{A},\mathcal{B},\mathcal{C}}\left(p\right)=1,\mathrm{for}\mathrm{all}p>0\mathrm{and}{\mathcal{A}}_0=\mathcal{U}.$$

Since $\alpha$ is an element of the universe $\mathcal{U}$, the conclusion $\alpha \in {\mathcal{A}}_0$ is always true regardless of the sequence’s behavior. Thus, the implication holds as a tautology.

Therefore, the existence and uniqueness of a fixed point follow directly from Theorem 1 without the need for an independent proof.

We formally state the contraction condition and the resulting theorem below.

Definition 13. 

Let $(\mathcal{U},\mathcal{F},⊡)$ be a non-Archimedean generalized Menger space. A self-mapping $f:\mathcal{U}\to \mathcal{U}$ is called a $(\varkappa -\mathsf{ϝ})$-contraction if there exist functions $\varkappa \in \Xi$ and $\mathsf{ϝ}\in \Theta$ such that, for all $\alpha ,\beta ,\gamma \in \mathcal{U}$ and every $p>0$,

$$\varkappa \left({\mathcal{F}}_{f\alpha ,f\beta ,f\gamma }\left(p\right)\right)\le \varkappa \left({\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right)\right)-\mathsf{ϝ}\left({\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right)\right).$$

(9)

Theorem 3. 

Let $(\mathcal{U},\mathcal{F},⊡)$ be a complete non-Archimedean generalized Menger space whose triangular norm ⊡ is 1-boundary continuous. If $f:\mathcal{U}\to \mathcal{U}$ is a $(\varkappa -\mathsf{ϝ})$-contraction, then f possesses a unique fixed point ${\alpha }^{\ast }\in \mathcal{U}$.

To explicitly illustrate the concept of a $(\varkappa -\mathsf{ϝ})$-contraction introduced in Definition 13, and to concurrently demonstrate the practical applicability of Theorem 3, we provide the following concrete example.

Example 10. 

Let $\mathcal{U}=\mathbb{R}$ be equipped with the standard G-metric defined by

$${\displaystyle G(\alpha ,\beta ,\gamma )=|\alpha -\beta |+|\beta -\gamma |+|\gamma -\alpha |.}$$

Consider the non-Archimedean generalized Menger space endowed with the probabilistic metric

$${\displaystyle {\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right)=\left\{\begin{array}{cc}0,\hfill & p=0,\hfill \\ {\displaystyle \frac{p}{p+G(\alpha ,\beta ,\gamma )}},\hfill & p>0,\hfill \end{array}\right.\forall \alpha ,\gamma ,\beta \in \mathcal{U},}$$

and the continuous triangular norm $⊡(a,b)=min(a,b)$.

Let the self-mapping $f:\mathcal{U}\to \mathcal{U}$ be defined by the classic geometric contraction $f\left(\alpha \right)={\displaystyle \frac{\alpha }{2}}$. For any $\alpha ,\beta ,\gamma \in \mathcal{U}$, we compute the G-metric of the mapped points:

$${\displaystyle G(f\alpha ,f\beta ,f\gamma )=\left|\frac{\alpha }{2}-\frac{\beta }{2}\right|+\left|\frac{\beta }{2}-\frac{\gamma }{2}\right|+\left|\frac{\gamma }{2}-\frac{\alpha }{2}\right|=\frac{1}{2}G(\alpha ,\beta ,\gamma ).}$$

Consequently, the probabilistic distance becomes:

$${\displaystyle {\mathcal{F}}_{f\alpha ,f\beta ,f\gamma }\left(p\right)=\frac{p}{p+\frac{1}{2}G(\alpha ,\beta ,\gamma )}=\frac{2p}{2p+G(\alpha ,\beta ,\gamma )}.}$$

Let us define the functions $\varkappa :(0,1]\to [0,+\infty )$ and $\mathsf{ϝ}:(0,1]\to [0,+\infty )$ exactly as in our previous proximity model:

$${\displaystyle \varkappa \left(t\right)=\frac{1}{t}-1\mathit{and}\mathsf{ϝ}\left(t\right)=\frac{1}{2}\left(\frac{1}{t}-1\right).}$$

Evaluating the components of the contraction condition (Equation (9)), we obtain the left-hand side:

$${\displaystyle \varkappa \left({\mathcal{F}}_{f\alpha ,f\beta ,f\gamma }\left(p\right)\right)=\frac{2p+G(\alpha ,\beta ,\gamma )}{2p}-1=\frac{G(\alpha ,\beta ,\gamma )}{2p}.}$$

For the right-hand side, we have:

$${\displaystyle \varkappa \left({\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right)\right)=\frac{p+G(\alpha ,\beta ,\gamma )}{p}-1=\frac{G(\alpha ,\beta ,\gamma )}{p},}$$

$${\displaystyle \mathsf{ϝ}\left({\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right)\right)=\frac{1}{2}\left(\frac{p+G(\alpha ,\beta ,\gamma )}{p}-1\right)=\frac{G(\alpha ,\beta ,\gamma )}{2p}.}$$

Subtracting these yields exactly the left-hand side:

$${\displaystyle \varkappa \left({\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right)\right)-\mathsf{ϝ}\left({\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right)\right)=\frac{G(\alpha ,\beta ,\gamma )}{p}-\frac{G(\alpha ,\beta ,\gamma )}{2p}=\frac{G(\alpha ,\beta ,\gamma )}{2p}.}$$

Therefore, the inequality

$${\displaystyle \varkappa \left({\mathcal{F}}_{f\alpha ,f\beta ,f\gamma }\left(p\right)\right)\le \varkappa \left({\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right)\right)-\mathsf{ϝ}\left({\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right)\right)}$$

is perfectly satisfied as an exact identity for all $\alpha ,\beta ,\gamma \in \mathcal{U}$ and $p>0$. Thus, $f\left(\alpha \right)={\displaystyle \frac{\alpha }{2}}$ is rigorously proven to be a $(\varkappa -\mathsf{ϝ})$-contraction on this space, admitting the unique fixed point ${\alpha }^{\ast }=0$.

3.4. Application to the Computer Science

We now illustrate the applicability of our theoretical framework to theoretical computer science, specifically to the domain of finite and infinite words.

Let $\Sigma$ be a non-empty alphabet and denote by ${\Sigma }_{\infty }$ the set of all finite and infinite words over $\Sigma$. The empty word is denoted by $\epsilon$.

We equip ${\Sigma }_{\infty }$ with the prefix order ⪯ defined by

$$\alpha ⪯\beta \mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}\alpha \mathrm{is}\mathrm{a}\mathit{prefix}\mathrm{of}\beta .$$

For every non-empty word $\alpha \in {\Sigma }_{\infty }$, its length is denoted by $\mathcal{l}\left(\alpha \right)\in [1,+\infty )$; we set $\mathcal{l}\left(\epsilon \right)=0$. If $\alpha$ is finite we write

$$\alpha ={\alpha }_1{\alpha }_2\dots {\alpha }_n\mathrm{with}n=\mathcal{l}\left(\alpha \right);$$

otherwise we write

$$\alpha ={\alpha }_1{\alpha }_2\dots .$$

For $\alpha ,\beta ,\gamma \in {\Sigma }_{\infty }$, the longest common prefix of the three words is written as $\alpha \cap \beta \cap \gamma$. Observe that $\alpha =\beta =\gamma$ holds precisely when they are mutual prefixes of one another and have the same length. We define a generalized Menger structure on ${\Sigma }_{\infty }$ by

$${\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right)=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}p=0,\hfill \\ {\displaystyle exp\left(-\frac{2^{-\mathcal{l}(\alpha \cap \beta \cap \gamma )}}{p}\right),}\hfill & \mathrm{if}p>0\mathrm{and}\alpha ,\beta ,\gamma \mathrm{are}\mathrm{not}\mathrm{all}\mathrm{equal},\hfill \\ 1,\hfill & \mathrm{if}p>0\mathrm{and}\alpha =\beta =\gamma .\hfill \end{array}\right.$$

Equipping this space with the product triangular norm ${⊡}_P$, one verifies that $({\Sigma }_{\infty },\mathcal{F},{⊡}_P)$ becomes a complete non-Archimedean generalized Menger space.

The analysis of the Quicksort algorithm leads to the well-known recurrence (see, e.g., [17])

$${\alpha }_1=0,{\alpha }_m={\displaystyle \frac{2(m-1)}{m}}+{\displaystyle \frac{m+1}{m}}{\alpha }_{m-1},m\ge 2.$$

To connect this recurrence with our fixed point framework, we take the alphabet $\Sigma =[0,+\infty )$ and define a mapping $f:{\Sigma }_{\infty }\to {\Sigma }_{\infty }$ as follows. For a word $\alpha ={\alpha }_1{\alpha }_2\dots$, we set $f\left(\alpha \right)=f{\left(\alpha \right)}_{1}f{\left(\alpha \right)}_2\dots$ where

$$\left\{\begin{array}{c}f{\left(\alpha \right)}_1=0,\hfill \\ {\displaystyle f{\left(\alpha \right)}_m=\frac{2(m-1)}{m}+\frac{m+1}{m}f{\left(\alpha \right)}_{m-1}, m\ge 2.}\hfill \end{array}\right.$$

The mapping f preserves the prefix order: $\alpha ⪯\beta$ if and only if $f\left(\alpha \right)⪯f\left(\beta \right)$. Consequently,

$$f(\alpha \cap \beta \cap \gamma )⪯f\left(\alpha \right)\cap f\left(\beta \right)\cap f\left(\gamma \right)(\mathrm{for}\mathrm{all}\alpha ,\beta ,\gamma \in {\Sigma }_{\infty }),$$

which implies the inequality between lengths:

$$\mathcal{l}\left(f(\alpha \cap \beta \cap \gamma )\right)\le \mathcal{l}\left(f\left(\alpha \right)\cap f\left(\beta \right)\cap f\left(\gamma \right)\right)(\mathrm{for}\mathrm{all}\alpha ,\beta ,\gamma \in {\Sigma }_{\infty }).$$

(10)

We now invoke Theorem 3 to demonstrate that f admits a fixed point. Define auxiliary functions $\varkappa \in \Xi$ and $\mathsf{ϝ}\in \Theta$ by

$$\varkappa \left(p\right)={\displaystyle \frac{1-p}{p}},\mathsf{ϝ}\left(p\right)={\displaystyle \frac{1}{2}}·{\displaystyle \frac{1-p}{p}}(0<p\le 1).$$

Both functions are continuous on $(0,1]$, strictly decreasing, and satisfy $\varkappa \left(1\right)=\mathsf{ϝ}\left(1\right)=0$.

Two cases are considered.

Case 1: $\alpha =\beta =\gamma$. Then ${\mathcal{F}}_{f\left(\alpha \right),f\left(\beta \right),f\left(\gamma \right)}\left(p\right)=1={\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right)$ for every $p>0$, and the contraction inequality holds trivially.

Case 2: $\alpha ,\beta ,\gamma$ are not all equal. Let $L=\mathcal{l}(\alpha \cap \beta \cap \gamma )$ and $L^{\prime }=\mathcal{l}(f\left(\alpha \right)\cap f\left(\beta \right)\cap f\left(\gamma \right))$. From (10) we have $L^{\prime }\ge L+1$. Hence

$${\mathcal{F}}_{f\left(\alpha \right),f\left(\beta \right),f\left(\gamma \right)}\left(p\right)=exp\left(-{\displaystyle \frac{2^{-L^{\prime }}}{p}}\right)\le exp\left(-{\displaystyle \frac{2^{-(L+1)}}{p}}\right)={\left({\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right)\right)}^{1/2}.$$

Set $x={\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right)\in (0,1)$. Then

$$\begin{array}{cc}\hfill \varkappa \left({\mathcal{F}}_{f\left(\alpha \right),f\left(\beta \right),f\left(\gamma \right)}\left(p\right)\right)& ={\displaystyle \frac{1-{\mathcal{F}}_{f\left(\alpha \right),f\left(\beta \right),f\left(\gamma \right)}\left(p\right)}{{\mathcal{F}}_{f\left(\alpha \right),f\left(\beta \right),f\left(\gamma \right)}\left(p\right)}}\hfill \\ \hfill & \le {\displaystyle \frac{1-x^{1/2}}{x^{1/2}}}.\hfill \end{array}$$

A direct computation shows that

$${\displaystyle \frac{1-x^{1/2}}{x^{1/2}}}\le {\displaystyle \frac{1}{2}}·{\displaystyle \frac{1-x}{x}}=\varkappa \left(x\right)-\mathsf{ϝ}\left(x\right).$$

Thus, for every $\alpha ,\beta ,\gamma \in {\Sigma }_{\infty }$ and every $p>0$,

$$\varkappa \left({\mathcal{F}}_{f\left(\alpha \right),f\left(\beta \right),f\left(\gamma \right)}\left(p\right)\right)\le \varkappa \left({\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right)\right)-\mathsf{ϝ}\left({\mathcal{F}}_{\alpha ,\beta ,\gamma }\left(p\right)\right).$$

Since f satisfies the $(\varkappa -\mathsf{ϝ})$-contraction condition (9), all hypotheses of Theorem 3 are satisfied. Consequently, f possesses a unique fixed point ${\alpha }^{\ast }\in {\Sigma }_{\infty }$; translating the equality $f\left({\alpha }^{\ast }\right)={\alpha }^{\ast }$ letter-by-letter yields exactly the Quicksort recurrence. Hence, we obtain

$${\alpha }_1^{\ast }=0,{\alpha }_n^{\ast }={\displaystyle \frac{2(n-1)}{n}}+{\displaystyle \frac{n+1}{n}}{\alpha }_{n-1}^{\ast }(n\ge 2).$$

4. Conclusions

This study is built upon the hypothesis that by merging the frameworks of non-Archimedean generalized Menger spaces with $(\varkappa -\mathsf{ϝ})$-weak proximal contractions, one could establish best proximity point results for cyclic mappings on a triple of sets without relying on the restrictive probabilistic P-property. Our findings confirm this hypothesis, providing new existence and uniqueness theorems that significantly generalize prior work in probabilistic metric spaces. These results matter profoundly to the field as they extend the applicable range of fixed point theory to more irregular, non-convex structures and establish a unified framework that bridges classical results in probabilistic, G-metric, and fuzzy metric spaces. Looking ahead, specific and actionable research directions emerge: extending the triple-set framework to an arbitrary finite number of sets $({\mathcal{A}}_1,\dots ,{\mathcal{A}}_n)$; investigating the stability and data dependence of best proximity points in this setting using quantitative convergence bounds; and applying the established theorems to solve concrete problems in nonlinear programming and distributed algorithm analysis on network spaces. In essence, this work transforms structural limitations into opportunities for exploration, providing a flexible and powerful tool set for solving proximity problems where traditional assumptions fail.