Some Common Best Proximity Point Results in Neutrosophic Complete Metric Spaces

Abstract

In this research article, we prove the existence and uniqueness of common best proximity points for a given class of discontinuous mappings. For that, we have introduced the notions of neutrosophic proximally compatible mappings, neutrosophic proximally reciprocal and weak reciprocal mappings, and R-proximally weak reciprocal commuting mappings of type I and type II. We have given examples to validate our findings.

1. Introduction

Best proximity point theory occupies a pivotal position in optimization and nonlinear analysis by providing solutions in scenarios where fixed points are absent. Such situations frequently arise in real-world applications involving mappings between two disjoint sets across fields like engineering, computer science and economics, where self-mapping is impossible. In these cases, best proximity point theory seeks to minimize the distance between two distinct sets by identifying points in one set that are nearest to their corresponding points in the other. This approach generalizes traditional fixed point theory and is essential for addressing problems in which exact solutions are unattainable, but close approximations are both relevant and valuable.

Let $(\Omega ,d)$ be a metric space and let $\Gamma :C\to D$, $\varnothing \ne C,D\subseteq \Omega$. If we can find an element $x\in C$ such that $d(x,\Gamma x)=d(C,D)=\inf \left\{d\right(a,b):a\in C \mathrm{and} b\in D\}$, then x is said to be the proximity point of $\Gamma$ in C. Moreover, x is reduced to a fixed point of $\Gamma$ when $C=D$. Best proximity point theorems deal with the minimum requirements for at least one solution of the optimization problem ${\min }_{x\in C}d(x,\Gamma x)$, which draws the attention of many authors [1,2,3,4,5,6].

Suppose $\Gamma ,{\Gamma }_{\ast }:C\to D$. If we can find an element $x\in C$ such that $d(x,\Gamma x)=d(C,D)=d(x,{\Gamma }_{\ast }x)$, then x is said to be the common best proximity point (CBPP) of mappings $\Gamma$ and ${\Gamma }_{\ast }$ in C. In addition, when $C=D$, the CBPP reduces to a common fixed point. The common best proximity theorem for mappings that satisfy a certain contraction condition has been established by Sadiq Basha et al. [7]. In [8], Shahzad et al. have investigated a common best proximity point theorem for given pair of non-self contraction mappings and also for a pair of non-self contractive mappings. By considering proximally commuting non-self mappings, Sadiq Basha [9] has studied a common best proximity point theorem; and in [10], Mongkolkeha and Kumam discuss an enhanced method compared to the one utilized in [9]. A study on the existence of CBPPs for proximal weak commuting maps and proximally weak reciprocal continuous mappings is given in [11]. In [11], the authors have defined a class of mappings called proximally weak reciprocal continuous mappings which need not be continuous and proved a generalized result on the existence and uniqueness of the CBPP. For more details on studies related to CBPP, one can refer [12,13,14,15,16].

As an extension of regular metric spaces, neutrosophic metric spaces incorporate the idea of indeterminacy, which is crucial for encompassing the ambiguity, uncertainty, and partial information - some traits frequently seen in real-world scenarios. Neutrophic metric spaces allow degrees of truth, falsity, and indeterminacy, in contrast to classical metric spaces that depend on exact numerical distances. This makes them ideal for applications in data analysis, artificial intelligence, and decision-making where information is not always clear or complete. For the analysis of complex systems, this more expansive framework offers a more adaptable and practical mathematical framework, particularly when conventional metrics are unable to adequately represent ambiguity and uncertainty.

Neutrosophic logic and neutrosophic sets were developed by Smarandache [17] as an extension of classical set theory. These sets are characterized by three attributes: the degree of non-membership, the degree of indeterminacy, and the degree of membership. Furthermore, the concept of neutrosophic metric spaces (NMSs) was introduced by Kiriski et al. in [18,19], they established fixed point theorems in complete NMSs. Motivated by the results in [19], Unni and Pragadeeswarar [20] extended the concept of best proximity points to the framework of NMSs. They proved the existence of a best proximity point for a given non-self map in an Archimedean NMS. Notably, in [21], Pragadeeswarar and Unni introduced the notion of CBPP for a pair of non-self maps in complete NMSs.

To date, no research has been done on the existence of common best proximity points for discontinuous non-self maps within neutrosophic complete metric spaces. Inspired by the works of Unni and Pragadeeswarar [11], by proving the existence and uniqueness of the common best proximity points for a particular class of discontinuous non-self maps, we fill this research gap.

The paper is structured as follows. We begin by introducing the notions like neutrosophic proximally compatible mappings, neutrosophic proximally weak reciprocal continuous and neutrosophic R-proximally weak reciprocal commuting mapppings of type I and II. Furthermore, we determine the necessary criteria for a given pair of non-self mappings to share the best proximity point. We also give examples to illustrate our findings.

2. Foundations

This section provides the essential concepts and results needed, in addition to the new definitions we propose. First, let us recall some important fundamentals related to Neutrosophic metric spaces.

Definition 1

([22]). An operation $\ast :[0,1]\times [0,1]\to [0,1]$ is a continuous t-norm (CTN) if ∗ satisfies conditions below:

  (I)

  $a\ast 1=a$

 (II)

 ∗ is commutative and associative

(III) 

∗ is continuous

(IV) 

 $a\ast b\le u\ast v$ whenever $a\le u$, $b\le v$, and $a,b,u,v\in [0,1]$.

Definition 2

([22]). An operation $\diamond :[0,1]\times [0,1]\to [0,1]$ is a continuous t-conorm (CTCN) if ⋄ satisfies conditions below:

  (I)

  $a\diamond 0=a$

 (II)

 ⋄ is commutative and associative

(III) 

⋄ is continuous

(IV) 

 $a\diamond b\le u\diamond v$ whenever $a\le u$ and $b\le v$, and $a,b,u,v\in [0,1]$.

Definition 3

([17]). Let R be a universe of discourse. A neutrosophic set A in R is characterized by three functions:

$$P_{\varphi }\left(x\right), E_{\varphi }\left(x\right), Z_{\varphi }\left(x\right):R\to ]0^{-},1^{+}[$$

where for each element $x\in R$:

• $P_{\varphi }\left(x\right)$ represents the degree of truth-membership of x in A,
• $E_{\varphi }\left(x\right)$ represents the degree of indeterminacy-membership, and
• $Z_{\varphi }\left(x\right)$ represents the degree of falsity-membership.

There is no restriction on the sum of these values; that is,

$$0^{-}\le \inf P_{\varphi }\left(x\right)+\inf E_{\varphi }\left(x\right)+\inf Z_{\varphi }\left(x\right)\le \sup P_{\varphi }\left(x\right)+\sup E_{\varphi }\left(x\right)+\sup Z_{\varphi }\left(x\right)\le 3^{+}.$$

Thus, a neutrosophic set A can be expressed as:

$$A=\{⟨x,P_{\varphi }\left(x\right),E_{\varphi }\left(x\right),Z_{\varphi }\left(x\right)⟩:x\in R\}.$$

Definition 4

([19]). Assume R is a non-empty set. Consider a six-tuple $(R,P_{\varphi },E_{\varphi },Z_{\varphi },\ast ,\diamond )$, where ∗ signifies a continuous t-norm (CTN), ⋄ denotes a continuous t-conorm (CTCN), and $(P_{\varphi }$, $E_{\varphi }$, $Z_{\varphi })$ characterizes the neutrosophic set defined on $R\times R\times (0,\infty )$. If this six-tuple satisfies the subsequent conditions for all $\xi ,\nu ,r\in R$ and for all positive scalars ${\tau }_1,{\tau }_2$:

       (I)

$P_{\varphi }(\xi ,\nu ,{\tau }_1)+E_{\varphi }(\xi ,\nu ,{\tau }_1)+Z_{\varphi }(\xi ,\nu ,{\tau }_1)\le 3$

      (II)

$0\le P_{\varphi }(\xi ,\nu ,{\tau }_1)\le 1$

     (III)

$P_{\varphi }(\xi ,\nu ,{\tau }_1)=1⟺\xi =\nu$

     (IV)

$P_{\varphi }(\xi ,\nu ,{\tau }_1)=P_{\varphi }(\nu ,\xi ,{\tau }_1)$

      (V)

$P_{\varphi }(\xi ,r,{\tau }_1+{\tau }_2)\ge P_{\varphi }(\xi ,\nu ,{\tau }_1)\ast P_{\varphi }(\nu ,r,{\tau }_2)$

     (VI)

The function $P_{\varphi }(\xi ,\nu ,·):[0,\infty )\to [0,1]$ is continuous

    (VII)

$\underset{{\tau }_1\to \infty }{\lim }{P}_{\varphi }(\xi ,\nu ,{\tau }_1)=1$

  (VIII)

 $0\le E_{\varphi }(\xi ,\nu ,{\tau }_1)\le 1$

     (IX)

$E_{\varphi }(\xi ,\nu ,{\tau }_1)=0⟺\xi =\nu$

      (X)

$E_{\varphi }(\xi ,\nu ,{\tau }_1)=E_{\varphi }(\nu ,\xi ,{\tau }_1)$

     (XI)

$E_{\varphi }(\xi ,r,{\tau }_1+{\tau }_2)\le E_{\varphi }(\xi ,\nu ,{\tau }_1)\diamond E_{\varphi }(\nu ,r,{\tau }_2)$

    (XII)

The function $E_{\varphi }(\xi ,\nu ,·):[0,\infty )\to [0,1]$ is continuous

  (XIII)

 $\underset{{\tau }_1\to \infty }{\lim }{E}_{\varphi }(\xi ,\nu ,{\tau }_1)=0$

  (XIV)

 $0\le Z_{\varphi }(\xi ,\nu ,{\tau }_1)\le 1$

    (XV)

$Z_{\varphi }(\xi ,\nu ,{\tau }_1)=0⟺\xi =\nu$

   (XVI)

$Z_{\varphi }(\xi ,\nu ,{\tau }_1)=Z_{\varphi }(\nu ,\xi ,{\tau }_1)$

  (XVII)

$Z_{\varphi }(\xi ,r,{\tau }_1+{\tau }_2)\le Z_{\varphi }(\xi ,\nu ,{\tau }_1)\diamond Z_{\varphi }(\nu ,r,{\tau }_2)$

(XVIII) 

the function $Z_{\varphi }(\xi ,\nu ,·):[0,\infty )\to [0,1]$ is continuous

   (XIX)

 $\underset{{\tau }_1\to \infty }{\lim }{Z}_{\varphi }(\xi ,\nu ,{\tau }_1)=0$

    (XX)

 For ${\tau }_1\le 0$, we have $P_{\varphi }(\xi ,\nu ,{\tau }_1)=0$, $E_{\varphi }(\xi ,\nu ,{\tau }_1)=1$, and $Z_{\varphi }(\xi ,\nu ,{\tau }_1)=1$.

Then, $(P_{\varphi },E_{\varphi },Z_{\varphi })$ is termed a neutrosophic metric on R, and $(R,P_{\varphi },E_{\varphi },Z_{\varphi },\ast ,\diamond )$ is designated a neutrosophic metric space (NMS). The functions $P_{\varphi }$, $E_{\varphi }$, and $Z_{\varphi }$ articulate the degrees of membership for nearness, indeterminacy, and non-nearness, respectively.

Definition 5

([19]). Let $(R,P_{\varphi },E_{\varphi },Z_{\varphi },\ast ,\diamond )$ represent a neutrosophic metric space (NMS).

• A sequence $\left\{{\zeta }_n\right\}$ in R is deemed to converge to a point $\zeta \in R$ if, for every $\tau >0,$ the following limits hold: $\underset{n\to \infty }{\lim }{P}_{\varphi }({\zeta }_n,\zeta ,\tau )=1;$ $\underset{n\to \infty }{\lim }{E}_{\varphi }({\zeta }_n,\zeta ,\tau )=0;$ $\underset{n\to \infty }{\lim }{Z}_{\varphi }({\zeta }_n,\zeta ,\tau )=0.$
• A sequence $\left\{{\zeta }_n\right\}$ in R is declared to be a Cauchy sequence if for every $ϵ>0$ and $\tau >0,$ there exists a natural number $n_0\in \mathbb{N}$ such that for all $n,m\ge n_0,$ we have: $P_{\varphi }({\zeta }_n,{\zeta }_m,\tau )>1-ϵ;$ $E_{\varphi }({\zeta }_n,{\zeta }_m,\tau )<ϵ;$ $Z_{\varphi }({\zeta }_n,{\zeta }_m,\tau )<ϵ$.
• The space is considered complete if every Cauchy sequence within R converges to a point in R.
• The space is considered compact if every sequence $\left\{{\zeta }_n\right\}$ in R admits a convergent subsequence $\left\{{\zeta }_{n_k}\right\}$.

Next, we articulate the concept of a best proximity point for non-self mappings as follows.

Definition 6

([20]). Let $(C,D)$ be a pair of non-empty subsets of a neutrosophic metric space $(R,P_{\varphi },E_{\varphi },Z_{\varphi },\ast ,\diamond ).$ An element $\zeta \in C$ is defined as a best proximity point for the non-self map $\Gamma :C\to D$ if it fulfills the conditions: $P_{\varphi }(\zeta ,\Gamma \zeta ,\tau )=P_{\varphi }(C,D,\tau ),$ $E_{\varphi }(\zeta ,\Gamma \zeta ,\tau )=E_{\varphi }(C,D,\tau ),$ and $Z_{\varphi }(\zeta ,\Gamma \zeta ,\tau )=Z_{\varphi }(C,D,\tau ) \mathit{for} \mathit{all} \tau >0.$

Definition 7

([21]). Let $(C,D)$ represent two non-empty subsets of a neutrosophic metric space $(R,P_{\varphi },E_{\varphi },Z_{\varphi },\ast ,\diamond ).$ An element $\zeta \in C$ is termed a common best proximity point for two non-self mappings ${\Gamma }_1:C\to D$ and ${\Gamma }_2:C\to D$ if it concurrently satisfies the following criteria: $P_{\varphi }(\zeta ,{\Gamma }_1\zeta ,\tau )=P_{\varphi }(\zeta ,{\Gamma }_2\zeta ,\tau )=P_{\varphi }(C,D,\tau ),$ $E_{\varphi }(\zeta ,{\Gamma }_1\zeta ,\tau )=E_{\varphi }(\zeta ,{\Gamma }_2\zeta ,\tau )=E_{\varphi }(C,D,\tau ),$ and $Z_{\varphi }(\zeta ,{\Gamma }_1\zeta ,\tau )=Z_{\varphi }(\zeta ,{\Gamma }_2\zeta ,\tau )=Z_{\varphi }(C,D,\tau ) \mathit{for} \mathit{all} \tau >0.$

Note 1.

(1) In Definition 6, if the sets C and D are identical (i.e., $C=D$), then ζ serves as a fixed point for the self-map Γ.

(2) In Definition 7, if the sets C and D are identical (i.e., $C=D$), then ζ serves as a common fixed point for the self-maps ${\Gamma }_1$ and ${\Gamma }_2$.

Given non-empty subsets C and D of a neutrosophic metric space $(R,P_{\varphi },E_{\varphi },Z_{\varphi },\ast ,\diamond )$, the subsequent definitions defined in [20,21] are employed:

$$\begin{array}{c}\hfill P_{\varphi }(C,D,\tau ):=\sup \{P_{\varphi }(\zeta ,\nu ,\tau ):\zeta \in C \mathrm{and} \nu \in D\},\\ \hfill E_{\varphi }(C,D,\tau ):=\inf \{E_{\varphi }(\zeta ,\nu ,\tau ):\zeta \in C \mathrm{and} \nu \in D\},\\ \hfill Z_{\varphi }(C,D,\tau ):=\inf \{Z_{\varphi }(\zeta ,\nu ,\tau ):\zeta \in C \mathrm{and} \nu \in D\},\forall \tau >0.\end{array}$$

$$C_0=\left\{\zeta \in C|\begin{array}{c}{P}_{\varphi }(\zeta ,\nu ,\tau )=P_{\varphi }(C,D,\tau ),E_{\varphi }(\zeta ,\nu ,\tau )=E_{\varphi }(C,D,\tau ), \mathrm{and} \hfill \\ Z_{\varphi }(\zeta ,\nu ,\tau )=Z_{\varphi }(C,D,\tau )\hspace{1em}\mathrm{for} \mathrm{some} \nu \in D, \forall \tau >0\hfill \end{array}\right\},$$

$$D_0=\left\{\nu \in D|\begin{array}{c}{P}_{\varphi }(\zeta ,\nu ,\tau )=P_{\varphi }(C,D,\tau ),E_{\varphi }(\zeta ,\nu ,\tau )=E_{\varphi }(C,D,\tau ), \mathrm{and} \hfill \\ Z_{\varphi }(\zeta ,\nu ,\tau )=Z_{\varphi }(C,D,\tau )\hspace{1em}\mathrm{for} \mathrm{some} \zeta \in C, \forall \tau >0\hfill \end{array}\right\}.$$

Next, we define the new concepts—like neutrosophic P-property, neutrosophic proximally compatible mappings, neutrosophic proximally weak reciprocal continuous and neutrosophic R-proximally weak reciprocal commuting mappings of types I and II—in order to establish our main result.

Definition 8.

If $C_0$ is non-empty, then the pair $(C,D)$ is said to have the neutrosophic P-property if, for any ${\xi }_1,{\xi }_2\in C_0$ and ${\nu }_1,{\nu }_2\in D_0, \forall \tau >0$

$$\left.\begin{array}{c}{P}_{\varphi }({\xi }_1,{\nu }_1,\tau )=P_{\varphi }(C,D,\tau ),\hfill \\ P_{\varphi }({\xi }_2,{\nu }_2,\tau )=P_{\varphi }(C,D,\tau )\hfill \end{array}\right\}\Rightarrow P_{\varphi }({\xi }_1,{\xi }_2,\tau )=P_{\varphi }({\nu }_1,{\nu }_2,\tau ),$$

$$\left.\begin{array}{c}{E}_{\varphi }({\xi }_1,{\nu }_1,\tau )=E_{\varphi }(C,D,\tau ),\hfill \\ E_{\varphi }({\xi }_2,{\nu }_2,\tau )=E_{\varphi }(C,D,\tau )\hfill \end{array}\right\}\Rightarrow E_{\varphi }({\xi }_1,{\xi }_2,\tau )=E_{\varphi }({\nu }_1,{\nu }_2,\tau ),$$

$$\left.\begin{array}{c}{Z}_{\varphi }({\xi }_1,{\nu }_1,\tau )=Z_{\varphi }(C,D,\tau ),\hfill \\ Z_{\varphi }({\xi }_2,{\nu }_2,\tau )=Z_{\varphi }(C,D,\tau )\hfill \end{array}\right\}\Rightarrow Z_{\varphi }({\xi }_1,{\xi }_2,\tau )=Z_{\varphi }({\nu }_1,{\nu }_2,\tau ).$$

Example 1.

Consider $X={\mathbb{R}}^2$ with metric $d((x_1,y_1),(x_2,y_2)) = |x_1-x_2| + |y_1-y_2|$. Choose $C=\left\{0\right\}\times [3,10]$ and $D=\left\{1\right\}\times [3,10]$. Here, $C_0=C$ and $D_0=D$. Consider the CTN, ∗ as $a\ast b=ab$ and the CCTN ⋄ as $a\diamond b=\max \{a,b\}$. The functions $P_{\varphi },E_{\varphi },Z_{\varphi }:{\mathbb{R}}^2\times {\mathbb{R}}^2\times [0,\infty )$ are defined as follows:

$$\begin{array}{c}{P}_{\varphi }(z_i,z_{i+1},\tau )={\displaystyle \frac{\tau }{\tau +d(z_i,z_{i+1})}},\hfill \\ E_{\varphi }(z_i,z_{i+1},\tau )={\displaystyle \frac{d(z_i,z_{i+1})}{\tau +d(z_i,z_{i+1})}},\hfill \\ Z_{\varphi }(z_i,z_{i+1},\tau )={\displaystyle \frac{d(z_i,z_{i+1})}{\tau }},\hfill \end{array}$$

where $z_i=(x_i,y_i)$ and $z_{i+1}=(x_{i+1},y_{i+1})$, for $x_i,y_i,x_{i+1},y_{i+1}\in \mathbb{R}$, $\forall \tau >0$. And we have,

$$\begin{array}{c}{P}_{\varphi }(C,D,\tau )={\displaystyle \frac{\tau }{\tau +1}},\hfill \\ E_{\varphi }(C,D,\tau )={\displaystyle \frac{1}{\tau +1}},\hfill \\ Z_{\varphi }(C,D,\tau )={\displaystyle \frac{1}{\tau }}, \tau >0.\hfill \end{array}$$

Consider any ${\xi }_1=(0,x_1),{\xi }_2-(0,x_2)\in C_0$ and ${\nu }_1=(1,y_1),{\nu }_2=(1,y_2)\in D_0, \forall \tau >0$, $(x_1,y_1,x_2,y_2\in \mathbb{R})$ such that

$$\left.\begin{array}{c}{P}_{\varphi }({\xi }_1,{\nu }_1,\tau )=P_{\varphi }(C,D,\tau )=P_{\varphi }({\xi }_2,{\nu }_2,\tau )\hfill \\ E_{\varphi }({\xi }_1,{\nu }_1,\tau )=E_{\varphi }(C,D,\tau )=E_{\varphi }({\xi }_2,{\nu }_2,\tau )\hfill \\ Z_{\varphi }({\xi }_1,{\nu }_1,\tau )=Z_{\varphi }(C,D,\tau )=Z_{\varphi }({\xi }_2,{\nu }_2,\tau )\hfill \end{array}\right\}$$

• Claim I: $P_{\varphi }({\xi }_1,{\xi }_2,\tau )=P_{\varphi }({\nu }_1,{\nu }_2,\tau )$
• Given $P_{\varphi }({\xi }_1,{\nu }_1,\tau )=P_{\varphi }(C,D,\tau )=P_{\varphi }({\xi }_2,{\nu }_2,\tau )$. Thus ${\displaystyle \frac{\tau }{\tau +1+|x_1-y_1|}}={\displaystyle \frac{\tau }{\tau +1}}={\displaystyle \frac{\tau }{\tau +1+|x_2-y_2|}}$.
• Hence $x_1=y_1$ and $x_2=y_2$. Now consider $P_{\varphi }({\xi }_1,{\xi }_2,\tau )={\displaystyle \frac{\tau }{\tau +1+|x_1-x_2|}}={\displaystyle \frac{\tau }{\tau +1+|y_1-y_2|}}=P_{\varphi }({\nu }_1,{\nu }_2,\tau )$.
• Claim II: $E_{\varphi }({\xi }_1,{\xi }_2,\tau )=E_{\varphi }({\nu }_1,{\nu }_2,\tau )$
• Given $E_{\varphi }({\xi }_1,{\nu }_1,\tau )=E_{\varphi }(C,D,\tau )=E_{\varphi }({\xi }_2,{\nu }_2,\tau )$. Thus ${\displaystyle \frac{1+|x_1-y_1|}{\tau +1+|x_1-y_1|}}={\displaystyle \frac{1}{\tau +1}}={\displaystyle \frac{1+|x_2-y_2|}{\tau +1+|x_2-y_2|}}$. Hence $x_1=y_1$ and $x_2=y_2$. Now consider $E_{\varphi }({\xi }_1,{\xi }_2,\tau )={\displaystyle \frac{1+|x_1-x_2|}{\tau +1+|x_1-x_2|}}={\displaystyle \frac{1+|y_1-y_2|}{\tau +1+|y_1-y_2|}}=E_{\varphi }({\nu }_1,{\nu }_2,\tau )$.
• Claim III: $Z_{\varphi }({\xi }_1,{\xi }_2,\tau )=Z_{\varphi }({\nu }_1,{\nu }_2,\tau )$

Proof is similar to Claim II.

Definition 9.

Two mappings ${\Gamma }_1,{\Gamma }_2:C\to D$ are said to be neutrosophic proximally compatible if for any sequences $\left\{{\nu }_n\right\},\left\{{\xi }_n\right\}$, and $\left\{{\zeta }_n\right\}$ in C, the following hold:

$$\left.\begin{array}{c}{P}_{\varphi }({\zeta }_n,{\Gamma }_1{\nu }_n,\tau )=P_{\varphi }(C,D,\tau ),\hfill \\ P_{\varphi }({\xi }_n,{\Gamma }_2{\nu }_n,\tau )=P_{\varphi }(C,D,\tau )\hfill \end{array}\right\}\Rightarrow \underset{n\to \infty }{\lim }{P}_{\varphi }({\Gamma }_1{\xi }_n,{\Gamma }_2{\zeta }_n,\tau )=1,$$

$$\left.\begin{array}{c}{E}_{\varphi }({\zeta }_n,{\Gamma }_1{\nu }_n,\tau )=E_{\varphi }(C,D,\tau ),\hfill \\ E_{\varphi }({\xi }_n,{\Gamma }_2{\nu }_n,\tau )=E_{\varphi }(C,D,\tau )\hfill \end{array}\right\}\Rightarrow \underset{n\to \infty }{\lim }{E}_{\varphi }({\Gamma }_1{\xi }_n,{\Gamma }_2{\zeta }_n,\tau )=0,$$

$$\left.\begin{array}{c}{Z}_{\varphi }({\zeta }_n,{\Gamma }_1{\nu }_n,\tau )=Z_{\varphi }(C,D,\tau ),\hfill \\ Z_{\varphi }({\xi }_n,{\Gamma }_2{\nu }_n,\tau )=Z_{\varphi }(C,D,\tau )\hfill \end{array}\right\}\Rightarrow \underset{n\to \infty }{\lim }{Z}_{\varphi }({\Gamma }_1{\xi }_n,{\Gamma }_2{\zeta }_n,\tau )=0,$$

whenever $\underset{n\to \infty }{\lim }{P}_{\varphi }({\xi }_n,t,\tau )=\underset{n\to \infty }{\lim }{P}_{\varphi }({\zeta }_n,t,\tau )=1$, $\underset{n\to \infty }{\lim }{E}_{\varphi }({\xi }_n,t,\tau )=\underset{n\to \infty }{\lim }{E}_{\varphi }({\zeta }_n,t,\tau )=0$, and $\underset{n\to \infty }{\lim }{Z}_{\varphi }({\xi }_n,t,\tau )=\underset{n\to \infty }{\lim }{Z}_{\varphi }({\zeta }_n,t,\tau )=0$, for some $t\in C$ and for all $\tau >0$.

Example 2.

Consider $X={\mathbb{R}}^2$ with metric $d((x_1,y_1),(x_2,y_2))=\sqrt{|x_1-x_2{|}^2+{|y_1-y_2|}^2}$. Choose $C=\left\{0\right\}\times \mathbb{R}$ and $D=\left\{1\right\}\times \mathbb{R}$. Here, $d(C,D)=1$. Define ${\Gamma }_1,{\Gamma }_2:C\to D$ as ${\Gamma }_1(0,y)=(1,{\displaystyle \frac{y}{2}})$ and ${\Gamma }_2(0,y)=(1,{\displaystyle \frac{y}{3}})$. Choose the CTN, ∗ as $a\ast b=ab$ and the CCTN ⋄ as $a\diamond b=\max \{a,b\}$. The functions $P_{\varphi },E_{\varphi },Z_{\varphi }:{\mathbb{R}}^2\times {\mathbb{R}}^2\times [0,\infty )$ are defined as follows:

$$\begin{array}{c}{P}_{\varphi }(z_i,z_{i+1},\tau )={\displaystyle \frac{\tau }{\tau +d(z_i,z_{i+1})}},\hfill \\ E_{\varphi }(z_i,z_{i+1},\tau )={\displaystyle \frac{d(z_i,z_{i+1})}{\tau +d(z_i,z_{i+1})}},\hfill \\ Z_{\varphi }(z_i,z_{i+1},\tau )={\displaystyle \frac{d(z_i,z_{i+1})}{\tau }},\hfill \end{array}$$

where $z_i=(x_i,y_i,z_i)$ and $z_{i+1}=(x_{i+1},y_{i+1},z_{i+1})$, for $x_i,y_i,z_i{x}_{i+1},y_{i+1},z_{i+1}\in \mathbb{R}, \forall \tau >0$. And we have,

$$\begin{array}{c}{P}_{\varphi }(C,D,\tau )={\displaystyle \frac{\tau }{\tau +1}},\hfill \\ E_{\varphi }(C,D,\tau )={\displaystyle \frac{1}{\tau +1}},\hfill \\ Z_{\varphi }(C,D,\tau )={\displaystyle \frac{1}{\tau }}, \tau >0.\hfill \end{array}$$

Consider the sequences $\left(a_n\right)$, $\left(b_n\right)$ and $\left(c_n\right)$ in $\mathbb{R}$ where $\left(a_n\right)$ and $\left(b_n\right)$ converges to some t in $\mathbb{R}$. Thus, $\underset{n\to \infty }{\lim }{P}_{\varphi }((0,b_n),(0,t),\tau )=\underset{n\to \infty }{\lim }{P}_{\varphi }((0,c_n),(0,t),\tau )=1$, $\underset{n\to \infty }{\lim }{E}_{\varphi }((0,b_n),(0,t),\tau )=\underset{n\to \infty }{\lim }{E}_{\varphi }((0,c_n),(0,t),\tau )=0$, and $\underset{n\to \infty }{\lim }{Z}_{\varphi }((0,b_n),(0,t),\tau )=\underset{n\to \infty }{\lim }{Z}_{\varphi }((0,c_n),(0,t),\tau )=0$, for some $t\in \mathbb{R}$ and for all $\tau >0$. Now, for the given sequences $\left(a_n\right)$,$\left(b_n\right)$ and $\left(c_n\right)$ we have

$$\begin{array}{cc}\hfill \hspace{1em}& P_{\varphi }\left((0,b_n),(1,{\displaystyle \frac{a_n}{2}}),\tau \right)={\displaystyle \frac{\tau }{\tau +1}}=P_{\varphi }\left((0,c_n),(1,{\displaystyle \frac{a_n}{3}}),\tau \right),\hfill \\ \hfill \hspace{1em}& E_{\varphi }\left((0,b_n),(1,{\displaystyle \frac{a_n}{2}}),\tau \right)={\displaystyle \frac{1}{\tau +1}}=E_{\varphi }\left((0,c_n),(1,{\displaystyle \frac{a_n}{3}}),\tau \right),\hfill \\ \hfill \hspace{1em}& Z_{\varphi }\left((0,b_n),(1,{\displaystyle \frac{a_n}{2}}),\tau \right)={\displaystyle \frac{1}{\tau }}=Z_{\varphi }\left((0,c_n),(1,{\displaystyle \frac{a_n}{3}}),\tau \right), \forall \tau >0.\hfill \end{array}$$

Thus, the sequences $(b_n)$ and $(c_n)$ are given by $b_n={\displaystyle \frac{a_n}{2}}$ and $c_n={\displaystyle \frac{a_n}{3}}$. Consider,

$$\begin{array}{cc}\hfill \hspace{1em}& \underset{n\to \infty }{\lim }{P}_{\varphi }\left({\Gamma }_1(0,{\displaystyle \frac{a_n}{3}}),{\Gamma }_2(0,{\displaystyle \frac{a_n}{2}}),\tau \right)=\underset{n\to \infty }{\lim }{P}_{\varphi }\left((1,{\displaystyle \frac{a_n}{6}}),(1,{\displaystyle \frac{a_n}{6}}),\tau \right)=1,\hfill \\ \hfill \hspace{1em}& \underset{n\to \infty }{\lim }{E}_{\varphi }\left({\Gamma }_1(0,{\displaystyle \frac{a_n}{3}}),{\Gamma }_2(0,{\displaystyle \frac{a_n}{2}}),\tau \right)=\underset{n\to \infty }{\lim }{E}_{\varphi }\left((1,{\displaystyle \frac{a_n}{6}}),(1,{\displaystyle \frac{a_n}{6}}),\tau \right)=0,\hfill \\ \hfill \hspace{1em}& \underset{n\to \infty }{\lim }{Z}_{\varphi }\left({\Gamma }_1(0,{\displaystyle \frac{a_n}{3}}),{\Gamma }_2(0,{\displaystyle \frac{a_n}{2}}),\tau \right)=\underset{n\to \infty }{\lim }{Z}_{\varphi }\left((1,{\displaystyle \frac{a_n}{6}}),(1,{\displaystyle \frac{a_n}{6}}),\tau \right)=0, \forall \tau >0.\hfill \end{array}$$

Hence, the mappings ${\Gamma }_1$ and ${\Gamma }_2$ are neutrosophic proximally compatible mappings.

Definition 10.

Two mappings ${\Gamma }_1,{\Gamma }_2:C\to D$ are said to be neutrosophic proximally reciprocal continuous if, for any sequences $\left\{{\nu }_n\right\}, \left\{{\xi }_n\right\}$, and $\left\{{\zeta }_n\right\}$ in C with,

$$\left.\begin{array}{c}{P}_{\varphi }({\zeta }_n,{\Gamma }_1{\nu }_n,\tau )=P_{\varphi }(C,D,\tau ),\hfill \\ P_{\varphi }({\xi }_n,{\Gamma }_2{\nu }_n,\tau )=P_{\varphi }(C,D,\tau ),\hfill \\ E_{\varphi }({\zeta }_n,{\Gamma }_1{\nu }_n,\tau )=E_{\varphi }(C,D,\tau ),\hfill \\ E_{\varphi }({\xi }_n,{\Gamma }_2{\nu }_n,\tau )=E_{\varphi }(C,D,\tau ),\hfill \\ Z_{\varphi }({\zeta }_n,{\Gamma }_1{\nu }_n,\tau )=Z_{\varphi }(C,D,\tau ),\hfill \\ Z_{\varphi }({\xi }_n,{\Gamma }_2{\nu }_n,\tau )=Z_{\varphi }(C,D,\tau )\hfill \end{array}\right\}\Rightarrow {\Gamma }_1{\xi }_n\to {\Gamma }_{1}t \mathit{and} {\Gamma }_2{\zeta }_n\to {\Gamma }_{2}t \mathit{as} n\to \infty ,$$

whenever $\underset{n\to \infty }{\lim }{P}_{\varphi }({\xi }_n,t,\tau )=\underset{n\to \infty }{\lim }{P}_{\varphi }({\zeta }_n,t,\tau )=1$, $\underset{n\to \infty }{\lim }{E}_{\varphi }({\xi }_n,t,\tau )=\underset{n\to \infty }{\lim }{E}_{\varphi }({\zeta }_n,t,\tau )=0$, and $\underset{n\to \infty }{\lim }{Z}_{\varphi }({\xi }_n,t,\tau )=\underset{n\to \infty }{\lim }{Z}_{\varphi }({\zeta }_n,t,\tau )=0$, for some $t\in C$ and for all $\tau >0$.

Example 3.

Example 2 is also an example for neutrosophic proximally reciprocal continuous mappings as well.

Definition 11.

Two mappings ${\Gamma }_1,{\Gamma }_2:C\to D$ are said to be neutrosophic proximally weak reciprocal continuous if, for any sequences $\left\{{\nu }_n\right\},\left\{{\xi }_n\right\}$, and $\left\{{\zeta }_n\right\}$ in C, the following holds:

$$\left.\begin{array}{c}{P}_{\varphi }({\zeta }_n,{\Gamma }_1{\nu }_n,\tau )=P_{\varphi }(C,D,\tau ),\hfill \\ P_{\varphi }({\xi }_n,{\Gamma }_2{\nu }_n,\tau )=P_{\varphi }(C,D,\tau ),\hfill \\ E_{\varphi }({\zeta }_n,{\Gamma }_1{\nu }_n,\tau )=E_{\varphi }(C,D,\tau ),\hfill \\ E_{\varphi }({\xi }_n,{\Gamma }_2{\nu }_n,\tau )=E_{\varphi }(C,D,\tau ),\hfill \\ Z_{\varphi }({\zeta }_n,{\Gamma }_1{\nu }_n,\tau )=Z_{\varphi }(C,D,\tau ),\hfill \\ Z_{\varphi }({\xi }_n,{\Gamma }_2{\nu }_n,\tau )=Z_{\varphi }(C,D,\tau )\hfill \end{array}\right\}\Rightarrow {\Gamma }_1{\xi }_n\to {\Gamma }_{1}t \mathit{or} {\Gamma }_2{\zeta }_n\to {\Gamma }_{2}t \mathit{as} n\to \infty ,$$

whenever $\underset{n\to \infty }{\lim }{P}_{\varphi }({\xi }_n,t,\tau )=\underset{n\to \infty }{\lim }{P}_{\varphi }({\zeta }_n,t,\tau )=1$, $\underset{n\to \infty }{\lim }{E}_{\varphi }({\xi }_n,t,\tau )=\underset{n\to \infty }{\lim }{E}_{\varphi }({\zeta }_n,t,\tau )=0$, and $\underset{n\to \infty }{\lim }{Z}_{\varphi }({\xi }_n,t,\tau )=\underset{n\to \infty }{\lim }{Z}_{\varphi }({\zeta }_n,t,\tau )=0$, for some $t\in C$ and for all $\tau >0$.

Example 4.

In Example 2, replace the map ${\Gamma }_2:C\to D$ as ${\Gamma }_2(0,y)=\left\{\begin{array}{cc}(1,{\displaystyle \frac{y}{3}})\hfill & \hspace{1em}y>0\hfill \\ (1,3)\hfill & \hspace{1em}y=0\hfill \end{array}\right.$.

• Consider the sequences $(a_n)$, $(b_n)$ and $(c_n)$ in $\mathbb{R}$ where $(a_n)$ and $(b_n)$ converges to 0. Thus, $\underset{n\to \infty }{\lim }{P}_{\varphi }((0,b_n),(0,0),\tau )=\underset{n\to \infty }{\lim }{P}_{\varphi }((0,c_n),(0,0),\tau )=1$, $\underset{n\to \infty }{\lim }{E}_{\varphi }((0,b_n),(0,0),\tau )=\underset{n\to \infty }{\lim }{E}_{\varphi }((0,c_n),(0,0),\tau )=0$, and $\underset{n\to \infty }{\lim }{Z}_{\varphi }((0,b_n),(0,0),\tau )=\underset{n\to \infty }{\lim }{Z}_{\varphi }((0,c_n),(0,0),\tau )=0$, for all $\tau >0$. Now, for the given sequences $(a_n)$, $(b_n)$ and $(c_n)$ we have,

  $$\begin{array}{c}{P}_{\varphi }\left((0,b_n),(1,{\displaystyle \frac{a_n}{2}}),\tau \right)={\displaystyle \frac{\tau }{\tau +1}}=P_{\varphi }\left((0,c_n),(1,{\displaystyle \frac{a_n}{3}}),\tau \right),\hfill \\ E_{\varphi }\left((0,b_n),(1,{\displaystyle \frac{a_n}{2}}),\tau \right)={\displaystyle \frac{1}{\tau +1}}=E_{\varphi }\left((0,c_n),(1,{\displaystyle \frac{a_n}{3}}),\tau \right),\hfill \\ Z_{\varphi }\left((0,b_n),(1,{\displaystyle \frac{a_n}{2}}),\tau \right)={\displaystyle \frac{1}{\tau }}=Z_{\varphi }\left((0,c_n),(1,{\displaystyle \frac{a_n}{3}}),\tau \right), \forall \tau >0.\hfill \end{array}$$

Thus, the sequences $(b_n)$ and $(c_n)$ are given by $b_n={\displaystyle \frac{a_n}{2}}$ and $c_n={\displaystyle \frac{a_n}{3}}$. Then we have,

$$\begin{array}{cc}\hfill \hspace{1em}& \underset{n\to \infty }{\lim }{P}_{\varphi }\left({\Gamma }_1(0,{\displaystyle \frac{a_n}{3}}),{\Gamma }_1(0,0),\tau \right)=1,\hfill \\ \hfill \hspace{1em}& \underset{n\to \infty }{\lim }{E}_{\varphi }\left({\Gamma }_1(0,{\displaystyle \frac{a_n}{3}}),{\Gamma }_1(0,0),\tau \right)=0,\hfill \\ \hfill \hspace{1em}& \underset{n\to \infty }{\lim }{Z}_{\varphi }\left({\Gamma }_1(0,{\displaystyle \frac{a_n}{3}}),{\Gamma }_1(0,0),\tau \right)=0, \forall \tau >0.\hfill \end{array}$$

Thus ${\Gamma }_1(0,{\displaystyle \frac{a_n}{3}})\to {\Gamma }_1(0,0)$, as $n\to \infty$. But, ${\Gamma }_2(0,{\displaystyle \frac{a_n}{2}})\to (1,0)$, but not to ${\Gamma }_2(0,0)=(1,3)$. Hence, the mappings ${\Gamma }_1$ and ${\Gamma }_2$ are neutrosophic proximally weak reciprocal continuous mappings. In addition, note that ${\Gamma }_1$ and ${\Gamma }_2$ are not neutrosophic proximally reciprocal continuous mappings.

Definition 12.

Two mappings ${\Gamma }_1,{\Gamma }_2:C\to D$ are said to be neutrosophic R-proximally weak reciprocal commuting of type I if there exists $R>0$ such that, for all $\xi ,\zeta ,\nu \in C$, $\tau >0$,

$$\left.\begin{array}{c}{P}_{\varphi }(\zeta ,{\Gamma }_1\nu ,\tau )=P_{\varphi }(C,D,\tau ),\hfill \\ P_{\varphi }(\xi ,{\Gamma }_2\nu ,\tau )=P_{\varphi }(C,D,\tau )\hfill \end{array}\right\}\Rightarrow P_{\varphi }({\Gamma }_1\xi ,{\Gamma }_2\xi ,R\tau )\ge P_{\varphi }(\xi ,\zeta ,\tau ),$$

$$\left.\begin{array}{c}{E}_{\varphi }(\zeta ,{\Gamma }_1\nu ,\tau )=E_{\varphi }(C,D,\tau ),\hfill \\ E_{\varphi }(\xi ,{\Gamma }_2\nu ,\tau )=E_{\varphi }(C,D,\tau )\hfill \end{array}\right\}\Rightarrow E_{\varphi }({\Gamma }_1\xi ,{\Gamma }_2\xi ,R\tau )\le E_{\varphi }(\xi ,\zeta ,\tau ),$$

$$\left.\begin{array}{c}{Z}_{\varphi }(\zeta ,{\Gamma }_1\nu ,\tau )=Z_{\varphi }(C,D,\tau ),\hfill \\ Z_{\varphi }(\xi ,{\Gamma }_2\nu ,\tau )=Z_{\varphi }(C,D,\tau )\hfill \end{array}\right\}\Rightarrow Z_{\varphi }({\Gamma }_1\xi ,{\Gamma }_2\xi ,R\tau )\le Z_{\varphi }(\xi ,\zeta ,\tau ).$$

Definition 13.

Two mappings ${\Gamma }_1,{\Gamma }_2:C\to D$ are said to be neutrosophic R-proximally weak reciprocal commuting of type II if there exists $R>0$ such that, for all $\xi ,\zeta ,\nu \in C$, $\tau >0$,

$$\left.\begin{array}{c}{P}_{\varphi }(\zeta ,{\Gamma }_1\nu ,\tau )=P_{\varphi }(C,D,\tau ),\hfill \\ P_{\varphi }(\xi ,{\Gamma }_2\nu ,\tau )=P_{\varphi }(C,D,\tau )\hfill \end{array}\right\}\Rightarrow P_{\varphi }({\Gamma }_1\zeta ,{\Gamma }_2\zeta ,R\tau )\ge P_{\varphi }(\xi ,\zeta ,\tau ),$$

$$\left.\begin{array}{c}{E}_{\varphi }(\zeta ,{\Gamma }_1\nu ,\tau )=E_{\varphi }(C,D,\tau ),\hfill \\ E_{\varphi }(\xi ,{\Gamma }_2\nu ,\tau )=E_{\varphi }(C,D,\tau )\hfill \end{array}\right\}\Rightarrow E_{\varphi }({\Gamma }_1\zeta ,{\Gamma }_2\zeta ,R\tau )\le E_{\varphi }(\xi ,\zeta ,\tau ),$$

$$\left.\begin{array}{c}{Z}_{\varphi }(\zeta ,{\Gamma }_1\nu ,\tau )=Z_{\varphi }(C,D,\tau ),\hfill \\ Z_{\varphi }(\xi ,{\Gamma }_2\nu ,\tau )=Z_{\varphi }(C,D,\tau )\hfill \end{array}\right\}\Rightarrow Z_{\varphi }({\Gamma }_1\zeta ,{\Gamma }_2\zeta ,R\tau )\le Z_{\varphi }(\xi ,\zeta ,\tau ).$$

The following definitions are derived as a special case of the primary Definitions (Definitions 9–13), when the sets C and D are identical.

Definition 14.

Two mappings ${\Gamma }_1,{\Gamma }_2:C\to C$ are said to be neutrosophic compatible if for every sequences $\left\{{\nu }_n\right\}$ in C, the following hold:

$$\left.\begin{array}{c}\hfill \underset{n\to \infty }{\lim }{P}_{\varphi }({\Gamma }_1{\Gamma }_2{\nu }_n,{\Gamma }_2{\Gamma }_1{\nu }_n,\tau )=1\\ \hfill \underset{n\to \infty }{\lim }{E}_{\varphi }({\Gamma }_1{\Gamma }_2{\nu }_n,{\Gamma }_2{\Gamma }_1{\nu }_n,\tau )=0\\ \hfill \underset{n\to \infty }{\lim }{Z}_{\varphi }({\Gamma }_1{\Gamma }_2{\nu }_n,{\Gamma }_2{\Gamma }_1{\nu }_n,\tau )=0\end{array}\right\},$$

whenever $\underset{n\to \infty }{\lim }{P}_{\varphi }({\Gamma }_1{\nu }_n,t,\tau )=\underset{n\to \infty }{\lim }{P}_{\varphi }({\Gamma }_2{\nu }_n,t,\tau )=1$, $\underset{n\to \infty }{\lim }{E}_{\varphi }({\Gamma }_1{\nu }_n,t,\tau )=\underset{n\to \infty }{\lim }{E}_{\varphi }({\Gamma }_2{\nu }_n,t,\tau )=0$, and $\underset{n\to \infty }{\lim }{Z}_{\varphi }({\Gamma }_1{\nu }_n,t,\tau )=\underset{n\to \infty }{\lim }{Z}_{\varphi }({\Gamma }_2{\nu }_n,t,\tau )=0$, for some $t\in C$ and for all $\tau >0$.

Definition 15.

Two mappings ${\Gamma }_1,{\Gamma }_2:C\to C$ are said to be neutrosophic reciprocal continuous if, for every sequence $\left\{{\nu }_n\right\}$ in C with,

$$\left.\begin{array}{c}\hfill \underset{n\to \infty }{\lim }{P}_{\varphi }({\Gamma }_1{\Gamma }_2{\nu }_n,{\Gamma }_{1}t,\tau )=1\\ \hfill \underset{n\to \infty }{\lim }{E}_{\varphi }({\Gamma }_1{\Gamma }_2{\nu }_n,{\Gamma }_{1}t,\tau )=0\\ \hfill \underset{n\to \infty }{\lim }{Z}_{\varphi }({\Gamma }_1{\Gamma }_2{\nu }_n,{\Gamma }_{1}t,\tau )=0\end{array}\right\},$$

and

$$\left.\begin{array}{c}\hfill \underset{n\to \infty }{\lim }{P}_{\varphi }({\Gamma }_2{\Gamma }_1{\nu }_n,{\Gamma }_{2}t,\tau )=1\\ \hfill \underset{n\to \infty }{\lim }{E}_{\varphi }({\Gamma }_2{\Gamma }_1{\nu }_n,{\Gamma }_{2}t,\tau )=0\\ \hfill \underset{n\to \infty }{\lim }{Z}_{\varphi }({\Gamma }_2{\Gamma }_1{\nu }_n,{\Gamma }_{2}t,\tau )=0\end{array}\right\},$$

whenever $\underset{n\to \infty }{\lim }{P}_{\varphi }({\Gamma }_1{\nu }_n,t,\tau )=\underset{n\to \infty }{\lim }{P}_{\varphi }({\Gamma }_2{\nu }_n,t,\tau )=1$, $\underset{n\to \infty }{\lim }{E}_{\varphi }({\Gamma }_1{\nu }_n,t,\tau )=\underset{n\to \infty }{\lim }{E}_{\varphi }({\Gamma }_2{\nu }_n,t,\tau )=0$, and $\underset{n\to \infty }{\lim }{Z}_{\varphi }({\Gamma }_1{\nu }_n,t,\tau )=\underset{n\to \infty }{\lim }{Z}_{\varphi }({\Gamma }_2{\nu }_n,t,\tau )=0$, for some $t\in C$ and for all $\tau >0$.

Definition 16.

Two mappings ${\Gamma }_1,{\Gamma }_2:C\to C$ are said to be neutrosophic weak reciprocal continuous if, for every sequence $\left\{{\nu }_n\right\}$ in C with,

$$\left.\begin{array}{c}\hfill \underset{n\to \infty }{\lim }{P}_{\varphi }({\Gamma }_1{\Gamma }_2{\nu }_n,{\Gamma }_{1}t,\tau )=1\\ \hfill \underset{n\to \infty }{\lim }{E}_{\varphi }({\Gamma }_1{\Gamma }_2{\nu }_n,{\Gamma }_{1}t,\tau )=0\\ \hfill \underset{n\to \infty }{\lim }{Z}_{\varphi }({\Gamma }_1{\Gamma }_2{\nu }_n,{\Gamma }_{1}t,\tau )=0\end{array}\right\},$$

or

$$\left.\begin{array}{c}\hfill \underset{n\to \infty }{\lim }{P}_{\varphi }({\Gamma }_2{\Gamma }_1{\nu }_n,{\Gamma }_{2}t,\tau )=1\\ \hfill \underset{n\to \infty }{\lim }{E}_{\varphi }({\Gamma }_2{\Gamma }_1{\nu }_n,{\Gamma }_{2}t,\tau )=0\\ \hfill \underset{n\to \infty }{\lim }{Z}_{\varphi }({\Gamma }_2{\Gamma }_1{\nu }_n,{\Gamma }_{2}t,\tau )=0\end{array}\right\},$$

whenever $\underset{n\to \infty }{\lim }{P}_{\varphi }({\Gamma }_1{\nu }_n,t,\tau )=\underset{n\to \infty }{\lim }{P}_{\varphi }({\Gamma }_2{\nu }_n,t,\tau )=1$, $\underset{n\to \infty }{\lim }{E}_{\varphi }({\Gamma }_1{\nu }_n,t,\tau )=\underset{n\to \infty }{\lim }{E}_{\varphi }({\Gamma }_2{\nu }_n,t,\tau )=0$, and $\underset{n\to \infty }{\lim }{Z}_{\varphi }({\Gamma }_1{\nu }_n,t,\tau )=\underset{n\to \infty }{\lim }{Z}_{\varphi }({\Gamma }_2{\nu }_n,t,\tau )=0$, for some $t\in C$ and for all $\tau >0$.

Definition 17.

Two mappings ${\Gamma }_1,{\Gamma }_2:C\to C$ are said to be neutrosophic R-weak reciprocal commuting of type I if there exists $R>0$ such that, for all $\nu \in C$, $\tau >0$,

$$\left.\begin{array}{c}\hfill P_{\varphi }({\Gamma }_1{\Gamma }_2\nu ,{\Gamma }_2{\Gamma }_2\nu ,R\tau )\ge P_{\varphi }({\Gamma }_1\nu ,{\Gamma }_2\nu ,\tau )\\ \hfill E_{\varphi }({\Gamma }_1{\Gamma }_2\nu ,{\Gamma }_2{\Gamma }_2\nu ,R\tau )\le E_{\varphi }({\Gamma }_1\nu ,{\Gamma }_2\nu ,\tau )\\ \hfill Z_{\varphi }({\Gamma }_1{\Gamma }_2\nu ,{\Gamma }_2{\Gamma }_2\nu ,R\tau )\le Z_{\varphi }({\Gamma }_1\nu ,{\Gamma }_2\nu ,\tau )\end{array}\right\}.$$

Definition 18.

Two mappings ${\Gamma }_1,{\Gamma }_2:C\to C$ are said to be neutrosophic R-weak reciprocal commuting of type II if there exists $R>0$ such that, for all $\xi ,\zeta ,\nu \in C$, $\tau >0$,

$$\left.\begin{array}{c}\hfill P_{\varphi }({\Gamma }_1{\Gamma }_1\nu ,{\Gamma }_2{\Gamma }_1\nu ,R\tau )\ge P_{\varphi }({\Gamma }_1\nu ,{\Gamma }_2\nu ,\tau )\\ \hfill E_{\varphi }({\Gamma }_1{\Gamma }_1\nu ,{\Gamma }_2{\Gamma }_1\nu ,R\tau )\le E_{\varphi }({\Gamma }_1\nu ,{\Gamma }_2\nu ,\tau )\\ \hfill Z_{\varphi }({\Gamma }_1{\Gamma }_1\nu ,{\Gamma }_2{\Gamma }_1\nu ,R\tau )\le Z_{\varphi }({\Gamma }_1\nu ,{\Gamma }_2\nu ,\tau )\end{array}\right\}.$$

3. Common Best Proximity Point Result

Theorem 1.

Let $(C,D)$ be a pair of non-empty subsets of a neutrosophic complete metric space $(R,P_{\varphi },E_{\varphi },Z_{\varphi },\ast ,\diamond )$ with $C_0\ne \varnothing$. Let $C_0$ and $D_0$ are closed and $(C,D)$ satisfies neutrosophic P-property. In addition, assume ${\Gamma }_1:C\to D$ and ${\Gamma }_2:C\to D$ are two neutrosophic proximally weak reciprocal continuous non-self mappings satisfying the following conditions:

(a) 

There exists a non-negative real number $k<1$ such that

$$\begin{array}{c}{P}_{\varphi }({\Gamma }_2{\zeta }_1,{\Gamma }_2{\zeta }_2,k\tau )\ge P_{\varphi }({\Gamma }_1{\zeta }_1,{\Gamma }_1{\zeta }_2,\tau )\hfill \\ E_{\varphi }({\Gamma }_2{\zeta }_1,{\Gamma }_2{\zeta }_2,k\tau )\le E_{\varphi }({\Gamma }_1{\zeta }_1,{\Gamma }_1{\zeta }_2,\tau )\hfill \\ Z_{\varphi }({\Gamma }_2{\zeta }_1,{\Gamma }_2{\zeta }_2,k\tau )\le Z_{\varphi }({\Gamma }_1{\zeta }_1,{\Gamma }_1{\zeta }_2,\tau )\hfill \end{array}$$

for all ${\zeta }_1$ and ${\zeta }_2$ in C, for all $\tau >0$.

(b) 

${\Gamma }_2(C_0)\subseteq D_0$ and ${\Gamma }_2(C_0)\subseteq {\Gamma }_1(C_0)$.

If ${\Gamma }_1$ and ${\Gamma }_2$ are neutrosophic proximally compatible or neutrosophic R-proximally weak reciprocal commuting mappings of types I or II. Then, there exists an element $\zeta \in C$ such that

$$\begin{array}{c}{P}_{\varphi }(\zeta ,{\Gamma }_1\zeta ,\tau )=P_{\varphi }(\zeta ,{\Gamma }_2\zeta ,\tau )=P_{\varphi }(C,D,\tau )\hfill \\ E_{\varphi }(\zeta ,{\Gamma }_1\zeta ,\tau )=E_{\varphi }(\zeta ,{\Gamma }_2\zeta ,\tau )=E_{\varphi }(C,D,\tau )\hfill \\ Z_{\varphi }(\zeta ,{\Gamma }_1\zeta ,\tau )=Z_{\varphi }(\zeta ,{\Gamma }_2\zeta ,\tau )=Z_{\varphi }(C,D,\tau ), \forall \tau >0.\hfill \end{array}$$

That is, ζ constitutes a unique common best proximity point for ${\Gamma }_1$ and ${\Gamma }_2$.

Proof. 

Let ${\zeta }_0$ be an element in $C_0$. Given that ${\Gamma }_2(C_0)\subseteq {\Gamma }_1(C_0)$, there exists an element ${\zeta }_1$ in $C_0$ such that ${\Gamma }_2{\zeta }_0={\Gamma }_1{\zeta }_1$. Continuing this process inductively, we establish the existence of a sequence $\left\{{\zeta }_n\right\}$ in $C_0$ such that ${\Gamma }_2{\zeta }_n={\Gamma }_1{\zeta }_{n+1}, \mathrm{for} \mathrm{all} n\in \mathbb{N}.$

• From condition (a), it follows that,

$$\left.\begin{array}{c}{P}_{\varphi }({\Gamma }_2{\zeta }_n,{\Gamma }_2{\zeta }_{n+1},k\tau )\ge P_{\varphi }({\Gamma }_1{\zeta }_n,{\Gamma }_1{\zeta }_{n+1},\tau )=P_{\varphi }({\Gamma }_2{\zeta }_{n-1},{\Gamma }_2{\zeta }_n,\tau ),\hfill \\ E_{\varphi }({\Gamma }_2{\zeta }_n,{\Gamma }_2{\zeta }_{n+1},k\tau )\le E_{\varphi }({\Gamma }_1{\zeta }_n,{\Gamma }_1{\zeta }_{n+1},\tau )=E_{\varphi }({\Gamma }_2{\zeta }_{n-1},{\Gamma }_2{\zeta }_n,\tau ),\hfill \\ Z_{\varphi }({\Gamma }_2{\zeta }_n,{\Gamma }_2{\zeta }_{n+1},k\tau )\le Z_{\varphi }({\Gamma }_1{\zeta }_n,{\Gamma }_1{\zeta }_{n+1},\tau )=Z_{\varphi }({\Gamma }_2{\zeta }_{n-1},{\Gamma }_2{\zeta }_n,\tau ),\hspace{1em}\forall n\in \mathbb{N}, \tau >0 \mathrm{and} k\in (0,1).\hfill \end{array}\right\}$$

(1)

By employing mathematical induction, we derive,

$$\left.\begin{array}{c}{P}_{\varphi }({\Gamma }_2{\zeta }_{n+1},{\Gamma }_2{\zeta }_n,\tau )\ge P_{\varphi }({\Gamma }_2{\zeta }_1,{\Gamma }_2{\zeta }_0,\frac{\tau }{k^n}),\hfill \\ E_{\varphi }({\Gamma }_2{\zeta }_{n+1},{\Gamma }_2{\zeta }_n,\tau )\le E_{\varphi }({\Gamma }_2{\zeta }_1,{\Gamma }_2{\zeta }_0,\frac{\tau }{k^n}),\hfill \\ Z_{\varphi }({\Gamma }_2{\zeta }_{n+1},{\Gamma }_2{\zeta }_n,\tau )\le Z_{\varphi }({\Gamma }_2{\zeta }_1,{\Gamma }_2{\zeta }_0,\frac{\tau }{k^n}),\hspace{1em}\forall n\in \mathbb{N},\tau >0 \mathrm{and} k\in (0,1).\hfill \end{array}\right\}$$

(2)

Hence, for any positive integer $q,$ we obtain,

$$\left.\begin{array}{cc}\hfill P_{\varphi }({\Gamma }_2{\zeta }_{n+q},{\Gamma }_2{\zeta }_n,\tau )& \ge P_{\varphi }({\Gamma }_2{\zeta }_n,{\Gamma }_2{\zeta }_{n+1},\frac{\tau }{q})\ast \cdots (\mathrm{q}\text{-}\mathrm{times})\cdots \ast P_{\varphi }({\Gamma }_2{\zeta }_{n+q-1},{\Gamma }_2{\zeta }_{n+q},\frac{\tau }{q})\hfill \\ \hfill \hspace{1em}& \ge P_{\varphi }({\Gamma }_2{\zeta }_0,{\Gamma }_2{\zeta }_1,\frac{\tau }{q{k}^n})\ast \cdots (\mathrm{q}\text{-}\mathrm{times})\cdots \ast P_{\varphi }({\Gamma }_2{\zeta }_0,{\Gamma }_2{\zeta }_1,\frac{\tau }{q{k}^{n+q-1}}),\hfill \\ \hfill E_{\varphi }({\Gamma }_2{\zeta }_{n+q},{\Gamma }_2{\zeta }_n,\tau )& \le E_{\varphi }({\Gamma }_2{\zeta }_n,{\Gamma }_2{\zeta }_{n+1},\frac{\tau }{q})\diamond \cdots (\mathrm{q}\text{-}\mathrm{times})\cdots \diamond E_{\varphi }({\Gamma }_2{\zeta }_{n+q-1},{\Gamma }_2{\zeta }_{n+q},\frac{\tau }{q})\hfill \\ \hfill \hspace{1em}& \le E_{\varphi }({\Gamma }_2{\zeta }_0,{\Gamma }_2{\zeta }_1,\frac{\tau }{q{k}^n})\diamond \cdots (\mathrm{q}\text{-}\mathrm{times})\cdots \diamond E_{\varphi }({\Gamma }_2{\zeta }_0,{\Gamma }_2{\zeta }_1,\frac{\tau }{q{k}^{n+q-1}}),\hfill \\ \hfill Z_{\varphi }({\Gamma }_2{\zeta }_{n+q},{\Gamma }_2{\zeta }_n,\tau )& \le Z_{\varphi }({\Gamma }_2{\zeta }_n,{\Gamma }_2{\zeta }_{n+1},\frac{\tau }{q})\diamond \cdots (\mathrm{q}\text{-}\mathrm{times})\cdots \diamond Z_{\varphi }({\Gamma }_2{\zeta }_{n+q-1},{\Gamma }_2{\zeta }_{n+q},\frac{\tau }{q})\hfill \\ \hfill \hspace{1em}& \le Z_{\varphi }({\Gamma }_2{\zeta }_0,{\Gamma }_2{\zeta }_1,\frac{\tau }{q{k}^n})\diamond \cdots (\mathrm{q}\text{-}\mathrm{times})\cdots \diamond Z_{\varphi }({\Gamma }_2{\zeta }_0,{\Gamma }_2{\zeta }_1,\frac{\tau }{q{k}^{n+q-1}}), \forall \tau >0.\hfill \end{array}\right\}$$

(3)

Now, from (3) and the properties of a neutrosophic metric space (NMS), we obtain,

$$\begin{array}{c}\underset{n\to \infty }{\lim }{P}_{\varphi }({\Gamma }_2{\zeta }_{n+q},{\Gamma }_2{\zeta }_n,\tau )\ge 1\ast \cdots (\mathrm{q}\text{-}\mathrm{times}) \cdots \ast 1=1,\hfill \\ \underset{n\to \infty }{\lim }{E}_{\varphi }({\Gamma }_2{\zeta }_{n+q},{\Gamma }_2{\zeta }_n,\tau )\le 0\diamond \cdots (\mathrm{q}\text{-}\mathrm{times}) \cdots \diamond 0=0,\hfill \\ \underset{n\to \infty }{\lim }{Z}_{\varphi }({\Gamma }_2{\zeta }_{n+q},{\Gamma }_2{\zeta }_n,\tau )\le 0\diamond \cdots (\mathrm{q}\text{-}\mathrm{times}) \cdots \diamond 0=0, \forall \tau >0.\hfill \end{array}$$

It is thus established that $\left\{{\Gamma }_2{\zeta }_n\right\}$ is a Cauchy sequence in $D_0.$ Let $\left\{{\xi }_n\right\}$ be a sequence of elements in $C_0$ such that,

$$\left.\begin{array}{c}{P}_{\varphi }({\xi }_n,{\Gamma }_2{\zeta }_n,\tau )=P_{\varphi }(C,D,\tau ),\hfill \\ E_{\varphi }({\xi }_n,{\Gamma }_2{\zeta }_n,\tau )=E_{\varphi }(C,D,\tau ),\hfill \\ Z_{\varphi }({\xi }_n,{\Gamma }_2{\zeta }_n,\tau )=Z_{\varphi }(C,D,\tau ),\hspace{1em}\mathrm{for} \mathrm{every} n\in \mathbb{N},\tau >0.\hfill \end{array}\right\}$$

(4)

By the neutrosophic P-property in (4), for $\forall \tau >0$, we get,

$$\left.\begin{array}{c}{P}_{\varphi }({\xi }_n,{\xi }_m,\tau )=P_{\varphi }({\Gamma }_2{\zeta }_n,{\Gamma }_2{\zeta }_m,\tau )=P_{\varphi }(C,D,\tau ),\hfill \\ E_{\varphi }({\xi }_n,{\xi }_m,\tau )=E_{\varphi }({\Gamma }_2{\zeta }_n,{\Gamma }_2{\zeta }_m,\tau )=E_{\varphi }(C,D,\tau ),\hfill \\ Z_{\varphi }({\xi }_n,{\xi }_m,\tau )=Z_{\varphi }({\Gamma }_2{\zeta }_n,{\Gamma }_2{\zeta }_m,\tau )=Z_{\varphi }(C,D,\tau ),\hspace{1em}\forall m,n\in \mathbb{N}.\hfill \end{array}\right\}$$

(5)

From (5) and $\left\{{\Gamma }_2{\zeta }_n\right\}$ is a Cauchy sequence in $D_0,$ clearly, the sequence $\left\{{\xi }_n\right\}$ is a Cauchy sequence. Since $C_0$ is closed, there exists $\xi \in C_0$ such that ${\xi }_n\to \xi$ as $n\to \infty .$

Now, there are three cases to be discussed.

Case 1: Suppose that ${\Gamma }_1$ and ${\Gamma }_2$ are neutrosophic proximally compatible mappings, from (4), we have,

$$\left.\begin{array}{c}{P}_{\varphi }({\xi }_n,{\Gamma }_2{\zeta }_n,\tau )=P_{\varphi }({\xi }_{n-1},{\Gamma }_1{\zeta }_n,\tau )=P_{\varphi }(C,D,\tau ),\hfill \\ E_{\varphi }({\xi }_n,{\Gamma }_2{\zeta }_n,\tau )=E_{\varphi }({\xi }_{n-1},{\Gamma }_1{\zeta }_n,\tau )=E_{\varphi }(C,D,\tau ),\hfill \\ Z_{\varphi }({\xi }_n,{\Gamma }_2{\zeta }_n,\tau )=Z_{\varphi }({\xi }_{n-1},{\Gamma }_1{\zeta }_n,\tau )=Z_{\varphi }(C,D,\tau ),\hspace{1em}\mathrm{for} \mathrm{every} n\in \mathbb{N}, \forall \tau >0.\hfill \end{array}\right\}$$

(6)

From (6) and ${\xi }_n\to \xi$, now by neutrosophic proximally weak reciprocal continuity of ${\Gamma }_1$ and ${\Gamma }_2,$ we get ${\Gamma }_1{\xi }_n\to {\Gamma }_1\xi \mathrm{or} {\Gamma }_2{\xi }_n\to {\Gamma }_2\xi \mathrm{as} n\to \infty .$

Subcase 1: Let ${\Gamma }_1{\xi }_n\to {\Gamma }_1\xi .$ Using neutrosophic proximally compatible mappings in (6) and ${\Gamma }_1{\xi }_n\to {\Gamma }_1\xi$, we get ${\Gamma }_2{\xi }_{n-1}\to {\Gamma }_1\xi .$ By condition (a), we have,

$$\left.\begin{array}{c}{P}_{\varphi }({\Gamma }_2{\xi }_n,{\Gamma }_2\xi ,\tau )\ge P_{\varphi }({\Gamma }_1{\xi }_n,{\Gamma }_1\xi ,{\displaystyle \frac{\tau }{k}})\hfill \\ E_{\varphi }({\Gamma }_2{\xi }_n,{\Gamma }_2\xi ,\tau )\le E_{\varphi }({\Gamma }_1{\xi }_n,{\Gamma }_1\xi ,{\displaystyle \frac{\tau }{k}})\hfill \\ Z_{\varphi }({\Gamma }_2{\xi }_n,{\Gamma }_2\xi ,\tau )\le Z_{\varphi }({\Gamma }_1{\xi }_n,{\Gamma }_1\xi ,{\displaystyle \frac{\tau }{k}})\hfill \end{array}\right\}, \forall \tau >0.$$

(7)

Since ${\Gamma }_2{\xi }_{n-1}\to {\Gamma }_1\xi$ and ${\Gamma }_1{\xi }_n\to {\Gamma }_1\xi ,$ from (7), we get,

$$\begin{array}{cc}\hfill 1\ge P_{\varphi }({\Gamma }_1\xi ,{\Gamma }_2\xi ,\tau )& \ge P_{\varphi }({\Gamma }_1\xi ,{\Gamma }_2{\xi }_n,\frac{\tau }{2})\ast P_{\varphi }({\Gamma }_2{\xi }_n,{\Gamma }_2\xi ,\frac{\tau }{2})\hfill \\ & \ge P_{\varphi }({\Gamma }_1\xi ,{\Gamma }_2{\xi }_n,\frac{\tau }{2})\ast P_{\varphi }({\Gamma }_1{\xi }_n,{\Gamma }_1\xi ,{\displaystyle \frac{\tau }{2k}})\to 1\hspace{1em}\mathrm{as} n\to \infty ,\hfill \\ \hfill 0\le E_{\varphi }({\Gamma }_1\xi ,{\Gamma }_2\xi ,\tau )& \le E_{\varphi }({\Gamma }_1\xi ,{\Gamma }_2{\xi }_n,\frac{\tau }{2})\diamond E_{\varphi }({\Gamma }_2{\xi }_n,{\Gamma }_2\xi ,\frac{\tau }{2})\hfill \\ & \le E_{\varphi }({\Gamma }_1\xi ,{\Gamma }_2{\xi }_n,\frac{\tau }{2})\diamond E_{\varphi }({\Gamma }_1{\xi }_n,{\Gamma }_1\xi ,{\displaystyle \frac{\tau }{2k}})\to 0\hspace{1em}\mathrm{as} n\to \infty ,\hfill \\ \hfill 0\le Z_{\varphi }({\Gamma }_1\xi ,{\Gamma }_2\xi ,\tau )& \le Z_{\varphi }({\Gamma }_1\xi ,{\Gamma }_2{\xi }_n,\frac{\tau }{2})\diamond Z_{\varphi }({\Gamma }_2{\xi }_n,{\Gamma }_2\xi ,\frac{\tau }{2})\hfill \\ & \le Z_{\varphi }({\Gamma }_1\xi ,{\Gamma }_2{\xi }_n,\frac{\tau }{2})\diamond Z_{\varphi }({\Gamma }_1{\xi }_n,{\Gamma }_1\xi ,{\displaystyle \frac{\tau }{2k}})\to 0\hspace{1em}\mathrm{as} n\to \infty , \forall \tau >0.\hfill \end{array}$$

which signifies that ${\Gamma }_2\xi ={\Gamma }_1\xi$. Since ${\Gamma }_2(C_0)\subseteq D_0,$ there exists $\zeta \in C_0$ such that $P_{\varphi }(\zeta ,{\Gamma }_2\xi ,\tau )=P_{\varphi }(C,D,\tau )=P_{\varphi }(\zeta ,{\Gamma }_1\xi ,\tau ),$ $E_{\varphi }(\zeta ,{\Gamma }_2\xi ,\tau )=E_{\varphi }(C,D,\tau )=E_{\varphi }(\zeta ,{\Gamma }_1\xi ,\tau ),$ and $Z_{\varphi }(\zeta ,{\Gamma }_2\xi ,\tau )=Z_{\varphi }(C,D,\tau )=Z_{\varphi }(\zeta ,{\Gamma }_1\xi ,\tau ).$ By neutrosophic proximally compatibility of ${\Gamma }_1$ and ${\Gamma }_2,$ we have $P_{\varphi }({\Gamma }_1\zeta ,{\Gamma }_2\zeta ,\tau )=1,$ and $E_{\varphi }({\Gamma }_1\zeta ,{\Gamma }_2\zeta ,\tau )=Z_{\varphi }({\Gamma }_1\zeta ,{\Gamma }_2\zeta ,\tau )=0.$ That is, ${\Gamma }_1\zeta ={\Gamma }_2\zeta .$ Again, since ${\Gamma }_2(C_0)\subseteq D_0,$ there exists $\eta \in C_0$ such that $P_{\varphi }(\eta ,{\Gamma }_2\zeta ,\tau )=P_{\varphi }(C,D,\tau )=P_{\varphi }(\eta ,{\Gamma }_1\zeta ,\tau ),$ $E_{\varphi }(\eta ,{\Gamma }_2\zeta ,\tau )=E_{\varphi }(C,D,\tau )=E_{\varphi }(\eta ,{\Gamma }_1\zeta ,\tau ),$ and $Z_{\varphi }(\eta ,{\Gamma }_2\zeta ,\tau )=Z_{\varphi }(C,D,\tau )=Z_{\varphi }(\eta ,{\Gamma }_1\zeta ,\tau ).$ Now, by using condition (a), neutrosophic P-property and the definition of neutrosophic metric space, for all $\tau >0$, we get,

$$\left.\begin{array}{cc}\hfill P_{\varphi }(\zeta ,\eta ,\tau )& =P_{\varphi }({\Gamma }_2\xi ,{\Gamma }_2\zeta ,\tau )\hfill \\ & \ge P_{\varphi }({\Gamma }_1\xi ,{\Gamma }_1\zeta ,{\displaystyle \frac{\tau }{k}})=P_{\varphi }(\zeta ,\eta ,{\displaystyle \frac{\tau }{k}})\hfill \\ & \ge P_{\varphi }(\zeta ,\eta ,{\displaystyle \frac{\tau }{k^n}}).\hfill \end{array}\right\}$$

(8)

Analogously,

$$\left.\begin{array}{cc}\hfill E_{\varphi }(\zeta ,\eta ,\tau )& =E_{\varphi }({\Gamma }_2\xi ,{\Gamma }_2\zeta ,\tau )\hfill \\ \hfill \hspace{1em}& \le E_{\varphi }({\Gamma }_1\xi ,{\Gamma }_1\zeta ,{\displaystyle \frac{\tau }{k}})=E_{\varphi }(\zeta ,\eta ,{\displaystyle \frac{\tau }{k}})\hfill \\ \hfill \hspace{1em}& \le E_{\varphi }(\zeta ,\eta ,{\displaystyle \frac{\tau }{k^n}}).\hfill \end{array}\right\}$$

(9)

and

$$\left.\begin{array}{cc}\hfill Z_{\varphi }(\zeta ,\eta ,\tau )& =Z_{\varphi }({\Gamma }_2\xi ,{\Gamma }_2\zeta ,\tau )\hfill \\ \hfill \hspace{1em}& \le Z_{\varphi }({\Gamma }_1\xi ,{\Gamma }_1\zeta ,{\displaystyle \frac{\tau }{k}})=Z_{\varphi }(\zeta ,\eta ,{\displaystyle \frac{\tau }{k}})\hfill \\ \hfill \hspace{1em}& \le Z_{\varphi }(\zeta ,\eta ,{\displaystyle \frac{\tau }{k^n}}).\hfill \end{array}\right\}$$

(10)

As $n\to \infty$ in (8)–(10), we obtain $P_{\varphi }(\zeta ,\eta ,\tau )=1,$ and $E_{\varphi }(\zeta ,\eta ,\tau )=Z_{\varphi }(\zeta ,\eta ,\tau )=0.$ That is, $\zeta =\eta .$ Also, $P_{\varphi }(\zeta ,{\Gamma }_2\zeta ,\tau )=P_{\varphi }(C,D,\tau )=P_{\varphi }(\zeta ,{\Gamma }_1\zeta ,\tau ),$ $E_{\varphi }(\zeta ,{\Gamma }_2\zeta ,\tau )=E_{\varphi }(C,D,\tau )=E_{\varphi }(\zeta ,{\Gamma }_1\zeta ,\tau ),$ and $Z_{\varphi }(\zeta ,{\Gamma }_2\zeta ,\tau )=Z_{\varphi }(C,D,\tau )=Z_{\varphi }(\zeta ,{\Gamma }_1\zeta ,\tau ).$ Hence, $\zeta$ is a common best proximity point of ${\Gamma }_1$ and ${\Gamma }_2.$

Subcase 2: Let ${\Gamma }_2{\xi }_n\to {\Gamma }_2\xi .$ Using neutrosophic proximally compatible mappings in (6) and ${\Gamma }_2{\xi }_{n-1}\to {\Gamma }_2\xi$, we get ${\Gamma }_1{\xi }_n\to {\Gamma }_2\xi .$ Since ${\Gamma }_2(C_0)\subseteq {\Gamma }_1(C_0),$ there exists some $\upsilon \in C_0$ such that ${\Gamma }_2\xi ={\Gamma }_1\upsilon .$ Now, we have ${\Gamma }_1{\xi }_n\to {\Gamma }_1\upsilon$ and ${\Gamma }_2{\xi }_n\to {\Gamma }_1\upsilon .$ Now, by using condition (a), for all $\tau >0$, we get,

$$\begin{array}{cc}\hfill P_{\varphi }({\Gamma }_2\upsilon ,{\Gamma }_2{\xi }_n,\tau )& \ge P_{\varphi }({\Gamma }_1\upsilon ,{\Gamma }_1{\xi }_n,{\displaystyle \frac{\tau }{k}})\hfill \\ \hfill \hspace{1em}& \ge P_{\varphi }({\Gamma }_1\upsilon ,{\Gamma }_1{\xi }_n,{\displaystyle \frac{\tau }{k^n}})\hfill \end{array}$$

In the same manner, for all $\tau >0$, we have,

$$\begin{array}{c}{E}_{\varphi }({\Gamma }_2\upsilon ,{\Gamma }_2{\xi }_n,\tau )\le E_{\varphi }({\Gamma }_1\upsilon ,{\Gamma }_1{\xi }_n,{\displaystyle \frac{\tau }{k^n}}),\hfill \\ Z_{\varphi }({\Gamma }_2\upsilon ,{\Gamma }_2{\xi }_n,\tau )\le Z_{\varphi }({\Gamma }_1\upsilon ,{\Gamma }_1{\xi }_n,{\displaystyle \frac{\tau }{k^n}}).\hfill \end{array}$$

As $n\to \infty$, by using properties of NMS, we obtain, ${\Gamma }_{2}v={\Gamma }_2\xi$. Existence of the BPT can be proved in the same way as in Subcase I.

Case 2: Suppose that ${\Gamma }_1$ and ${\Gamma }_2$ are neutrosophic R-proximally weak reciprocal commuting mappings of type I.

• Since ${\Gamma }_1,{\Gamma }_2$ are R-proximally weak reciprocal commuting mappings of type I, $\forall \tau >0$,

$$\begin{array}{c}{P}_{\varphi }({\xi }_{n+2},{\Gamma }_2{\zeta }_{n+2},\tau )=P_{\varphi }(C,D,\tau )=P_{\varphi }({\xi }_{n+1},{\Gamma }_1{\zeta }_{n+2},\tau ),\hfill \\ E_{\varphi }({\xi }_{n+2},{\Gamma }_2{\zeta }_{n+2},\tau )=E_{\varphi }(C,D,\tau )=E_{\varphi }({\xi }_{n+1},{\Gamma }_1{\zeta }_{n+2},\tau ),\hfill \\ Z_{\varphi }({\xi }_{n+2},{\Gamma }_2{\zeta }_{n+2},\tau )=Z_{\varphi }(C,D,\tau )=Z_{\varphi }({\xi }_{n+1},{\Gamma }_1{\zeta }_{n+2},\tau ),\hfill \end{array}$$

implies

$$\begin{array}{c}{P}_{\varphi }({\Gamma }_1{\xi }_{n+2},{\Gamma }_2{\xi }_{n+2},R\tau )\ge P_{\varphi }({\xi }_{n+2},{\xi }_{n+1},\tau ),\hfill \\ E_{\varphi }({\Gamma }_1{\xi }_{n+2},{\Gamma }_2{\xi }_{n+2},R\tau )\le E_{\varphi }({\xi }_{n+2},{\xi }_{n+1},\tau ),\hfill \\ Z_{\varphi }({\Gamma }_1{\xi }_{n+2},{\Gamma }_2{\xi }_{n+2},R\tau )\le Z_{\varphi }({\xi }_{n+2},{\xi }_{n+1},\tau ),\hfill \end{array}$$

for some $R>0$. As $n\to \infty$, we get,

$$\begin{array}{c}\underset{n\to \infty }{\lim }{P}_{\varphi }({\Gamma }_1{\xi }_{n+2},{\Gamma }_2{\xi }_{n+2},R\tau )\ge \underset{n\to \infty }{\lim }{P}_{\varphi }({\xi }_{n+2},{\xi }_{n+1},\tau )=1,\hfill \\ \underset{n\to \infty }{\lim }{E}_{\varphi }({\Gamma }_1{\xi }_{n+2},{\Gamma }_2{\xi }_{n+2},R\tau )\le \underset{n\to \infty }{\lim }{E}_{\varphi }({\xi }_{n+2},{\xi }_{n+1},\tau )=0,\hfill \\ \underset{n\to \infty }{\lim }{Z}_{\varphi }({\Gamma }_1{\xi }_{n+2},{\Gamma }_2{\xi }_{n+2},R\tau )\le \underset{n\to \infty }{\lim }{Z}_{\varphi }({\xi }_{n+2},{\xi }_{n+1},\tau )=0, \forall \tau >0.\hfill \end{array}$$

Therefore, for all $\tau >0$, we have,

$$\left.\begin{array}{c}\hfill \underset{n\to \infty }{\lim }{P}_{\varphi }({\Gamma }_1{\xi }_{n+2},{\Gamma }_2{\xi }_{n+2},R\tau )=1,\\ \hfill \underset{n\to \infty }{\lim }{E}_{\varphi }({\Gamma }_1{\xi }_{n+2},{\Gamma }_2{\xi }_{n+2},R\tau )=0,\\ \hfill \underset{n\to \infty }{\lim }{Z}_{\varphi }({\Gamma }_1{\xi }_{n+2},{\Gamma }_2{\xi }_{n+2},R\tau )=0.\end{array}\right\}$$

(11)

By neutrosophic proximally weak reciprocal continuity of ${\Gamma }_1$ and ${\Gamma }_2,$ we get ${\Gamma }_1{\xi }_n\to {\Gamma }_1\xi \mathrm{or} {\Gamma }_2{\xi }_n\to {\Gamma }_2\xi \mathrm{as} n\to \infty .$

Subcase 1: Let ${\Gamma }_1{\xi }_n\to {\Gamma }_1\xi$ as $n\to \infty$.

Here, since ${\Gamma }_1{\xi }_n\to {\Gamma }_1\xi$, and by condition (a), we get ${\Gamma }_2{\xi }_n\to {\Gamma }_2\xi$ as $n\to \infty$. Also, by (11), we obtain ${\Gamma }_2{\xi }_n\to {\Gamma }_1\xi$ as $n\to \infty$. Thus ${\Gamma }_1\xi ={\Gamma }_2\xi$.

Existence of the CBPP: Given ${\Gamma }_2(C_0)\subseteq D_0$. Thus, we can find a $w^{\ast }\in C_0$, such that,

$$\begin{array}{c}{P}_{\varphi }(w^{\ast },{\Gamma }_1\xi ,\tau )=P_{\varphi }(C,D,\tau )=P_{\varphi }(w^{\ast },{\Gamma }_2\xi ,\tau ),\hfill \\ E_{\varphi }(w^{\ast },{\Gamma }_1\xi ,\tau )=E_{\varphi }(C,D,\tau )=E_{\varphi }(w^{\ast },{\Gamma }_2\xi ,\tau ),\hfill \\ Z_{\varphi }(w^{\ast },{\Gamma }_1\xi ,\tau )=Z_{\varphi }(C,D,\tau )=Z_{\varphi }(w^{\ast },{\Gamma }_2\xi ,\tau ), \forall \tau >0.\hfill \end{array}$$

Since ${\Gamma }_1$ and ${\Gamma }_2$ are neutrosophic R-proximally weak reciprocal commuting mappings of type I,

$$\begin{array}{c}{P}_{\varphi }({\Gamma }_1{w}^{\ast },{\Gamma }_2{w}^{\ast },R\tau )\ge P_{\varphi }(w^{\ast },w^{\ast },\tau ),\hfill \\ E_{\varphi }({\Gamma }_1{w}^{\ast },{\Gamma }_2{w}^{\ast },R\tau )\le E_{\varphi }(w^{\ast },w^{\ast },\tau ),\hfill \\ Z_{\varphi }({\Gamma }_1{w}^{\ast },{\Gamma }_2{w}^{\ast },R\tau )\le Z_{\varphi }(w^{\ast },w^{\ast },\tau ), \forall \tau >0.\hfill \end{array}$$

Hence ${\Gamma }_1{w}^{\ast }={\Gamma }_2{w}^{\ast }$. In the same manner, since ${\Gamma }_2(C_0)\subseteq D_0$, there exists a $z^{\ast }\in C_0$, such that,

$$\begin{array}{c}{P}_{\varphi }(z^{\ast },{\Gamma }_1{w}^{\ast },\tau )=P_{\varphi }(C,D,\tau )=P_{\varphi }(z^{\ast },{\Gamma }_2{w}^{\ast },\tau ),\hfill \\ E_{\varphi }(z^{\ast },{\Gamma }_1{w}^{\ast },\tau )=E_{\varphi }(C,D,\tau )=E_{\varphi }(z^{\ast },{\Gamma }_2{w}^{\ast },\tau ),\hfill \\ Z_{\varphi }(z^{\ast },{\Gamma }_1{w}^{\ast },\tau )=Z_{\varphi }(C,D,\tau )=Z_{\varphi }(z^{\ast },{\Gamma }_2{w}^{\ast },\tau ) \forall \tau >0.\hfill \end{array}$$

Given that the pair $(C,D)$ satisfies neutrosophic P-property. Thus we obtain, for all $\tau >0$,

$$\begin{array}{cc}\hfill P_{\varphi }(z^{\ast },w^{\ast },\tau )& =P_{\varphi }(z^{\ast },w^{\ast },\tau )\hfill \\ \hfill \hspace{1em}& =P_{\varphi }({\Gamma }_2{w}^{\ast },{\Gamma }_2\xi ,\tau )\hfill \\ \hfill \hspace{1em}& \ge P_{\varphi }({\Gamma }_1{w}^{\ast },{\Gamma }_2\xi ,{\displaystyle \frac{\tau }{k}})\hfill \\ \hfill \hspace{1em}& =P_{\varphi }(z^{\ast },w^{\ast },{\displaystyle \frac{\tau }{k}}).\hfill \end{array}$$

In the same manner, for all $\tau >0$, we get,

$$\begin{array}{c}{E}_{\varphi }(z^{\ast },w^{\ast },\tau )\le E_{\varphi }(z^{\ast },w^{\ast },{\displaystyle \frac{\tau }{k}}),\hfill \\ Z_{\varphi }(z^{\ast },w^{\ast },\tau )\le Z_{\varphi }(z^{\ast },w^{\ast },{\displaystyle \frac{\tau }{k}}).\hfill \end{array}$$

Thus $z^{\ast }=w^{\ast }$. Therefore, $w^{\ast }$ is the CBPP.

Subcase 2: Let ${\Gamma }_2{\xi }_n\to {\Gamma }_2\xi$ as $n\to \infty$.

Since ${\Gamma }_2(C_0)\subseteq {\Gamma }_1(C_0)$, there exists some $r\in C_0$, such that ${\Gamma }_2\xi ={\Gamma }_{1}r$. Thus ${\Gamma }_2{\xi }_n\to {\Gamma }_{1}r$. Now, by (11), for all $\tau >0$, we have,

$$\begin{array}{cc}\hfill & P_{\varphi }({\Gamma }_2{\xi }_{n+1},{\Gamma }_1{\xi }_{n+1},\tau )\to 1,\hfill \\ \hfill & E_{\varphi }({\Gamma }_2{\xi }_{n+1},{\Gamma }_1{\xi }_{n+1},\tau )\to 0,\hfill \\ \hfill & Z_{\varphi }({\Gamma }_2{\xi }_{n+1},{\Gamma }_1{\xi }_{n+1},\tau )\to 0, \mathrm{as} n\to \infty .\hfill \end{array}$$

Therefore, ${\Gamma }_1{\xi }_n\to {\Gamma }_{1}r$ as $n\to \infty$. Finally, consider,

$$\begin{array}{c}{P}_{\varphi }({\Gamma }_{2}r,{\Gamma }_2{\xi }_n,\tau )\ge P_{\varphi }({\Gamma }_{1}r,{\Gamma }_1{\xi }_n,{\displaystyle \frac{\tau }{k}}),\hfill \\ E_{\varphi }({\Gamma }_{2}r,{\Gamma }_2{\xi }_n,\tau )\le E_{\varphi }({\Gamma }_{1}r,{\Gamma }_1{\xi }_n,{\displaystyle \frac{\tau }{k}}),\hfill \\ Z_{\varphi }({\Gamma }_{2}r,{\Gamma }_2{\xi }_n,\tau )\le Z_{\varphi }({\Gamma }_{1}r,{\Gamma }_1{\xi }_n,{\displaystyle \frac{\tau }{k}}), \forall \tau >0.\hfill \end{array}$$

As $n\to \infty$, we obtain ${\Gamma }_{2}r={\Gamma }_{1}r$. Existence can be proved as in the same method used in Subcase 1.

Case 3: Suppose that ${\Gamma }_1$ and ${\Gamma }_2$ are neutrosophic R-proximally weak reciprocal commuting mappings of type II.

• Given ${\Gamma }_1,{\Gamma }_2$ are R-proximally weak reciprocal commuting mappings of type II. Then, $\forall \tau >0$,

$$\begin{array}{c}{P}_{\varphi }({\xi }_{n+2},{\Gamma }_2{\zeta }_{n+2},\tau )=P_{\varphi }(C,D,\tau )=P_{\varphi }({\xi }_{n+1},{\Gamma }_1{\zeta }_{n+2},\tau ),\hfill \\ E_{\varphi }({\xi }_{n+2},{\Gamma }_2{\zeta }_{n+2},\tau )=E_{\varphi }(C,D,\tau )=E_{\varphi }({\xi }_{n+1},{\Gamma }_1{\zeta }_{n+2},\tau ),\hfill \\ Z_{\varphi }({\xi }_{n+2},{\Gamma }_2{\zeta }_{n+2},\tau )=Z_{\varphi }(C,D,\tau )=Z_{\varphi }({\xi }_{n+1},{\Gamma }_1{\zeta }_{n+2},\tau ),\hfill \end{array}$$

implies

$$\begin{array}{c}{P}_{\varphi }({\Gamma }_1{\xi }_{n+1},{\Gamma }_2{\xi }_{n+1},R\tau )\ge P_{\varphi }({\xi }_{n+2},{\xi }_{n+1},\tau ),\hfill \\ E_{\varphi }({\Gamma }_1{\xi }_{n+1},{\Gamma }_2{\xi }_{n+1},R\tau )\le E_{\varphi }({\xi }_{n+2},{\xi }_{n+1},\tau ),\hfill \\ Z_{\varphi }({\Gamma }_1{\xi }_{n+1},{\Gamma }_2{\xi }_{n+1},R\tau )\le Z_{\varphi }({\xi }_{n+2},{\xi }_{n+1},\tau ),\hfill \end{array}$$

for some $R>0$. As $n\to \infty$, we get,

$$\begin{array}{c}\underset{n\to \infty }{\lim }{P}_{\varphi }({\Gamma }_1{\xi }_{n+1},{\Gamma }_2{\xi }_{n+1},R\tau )\ge \underset{n\to \infty }{\lim }{P}_{\varphi }({\xi }_{n+2},{\xi }_{n+1},\tau )=1,\hfill \\ \underset{n\to \infty }{\lim }{E}_{\varphi }({\Gamma }_1{\xi }_{n+1},{\Gamma }_2{\xi }_{n+1},R\tau )\le \underset{n\to \infty }{\lim }{E}_{\varphi }({\xi }_{n+2},{\xi }_{n+1},\tau )=0,\hfill \\ \underset{n\to \infty }{\lim }{Z}_{\varphi }({\Gamma }_1{\xi }_{n+1},{\Gamma }_2{\xi }_{n+1},R\tau )\le \underset{n\to \infty }{\lim }{Z}_{\varphi }({\xi }_{n+2},{\xi }_{n+1},\tau )=0, \forall \tau >0.\hfill \end{array}$$

Therefore, for all $\tau >0$, we get,

$$\left.\begin{array}{c}\hfill \underset{n\to \infty }{\lim }{P}_{\varphi }({\Gamma }_1{\xi }_{n+1},{\Gamma }_2{\xi }_{n+1},R\tau )=1,\\ \hfill \underset{n\to \infty }{\lim }{E}_{\varphi }({\Gamma }_1{\xi }_{n+1},{\Gamma }_2{\xi }_{n+1},R\tau )=0,\\ \hfill \underset{n\to \infty }{\lim }{Z}_{\varphi }({\Gamma }_1{\xi }_{n+1},{\Gamma }_2{\xi }_{n+1},R\tau )=0.\end{array}\right\}$$

(12)

By neutrosophic proximally weak reciprocal continuity of ${\Gamma }_1$ and ${\Gamma }_2,$ we get ${\Gamma }_1{\xi }_n\to {\Gamma }_1\xi \mathrm{or} {\Gamma }_2{\xi }_n\to {\Gamma }_2\xi \mathrm{as} n\to \infty .$

Subcase 1: Let ${\Gamma }_1{\xi }_n\to {\Gamma }_1\xi$ as $n\to \infty$.

Here, since ${\Gamma }_1{\xi }_n\to {\Gamma }_1\xi$, by condition (a), we get ${\Gamma }_2{\xi }_n\to {\Gamma }_2\xi$ as $n\to \infty$. Also, by (12), we have ${\Gamma }_2{\xi }_n\to {\Gamma }_1\xi$ as $n\to \infty$. Thus ${\Gamma }_1\xi ={\Gamma }_2\xi$.

Existence of the CBPP:

Given ${\Gamma }_2(C_0)\subseteq D_0$. Thus, we can find a $w^{\ast }\in C_0$, such that,

$$\begin{array}{c}{P}_{\varphi }(w^{\ast },{\Gamma }_1\xi ,\tau )=P_{\varphi }(C,D,\tau )=P_{\varphi }(w^{\ast },{\Gamma }_2\xi ,\tau ),\hfill \\ E_{\varphi }(w^{\ast },{\Gamma }_1\xi ,\tau )=E_{\varphi }(C,D,\tau )=E_{\varphi }(w^{\ast },{\Gamma }_2\xi ,\tau ),\hfill \\ Z_{\varphi }(w^{\ast },{\Gamma }_1\xi ,\tau )=Z_{\varphi }(C,D,\tau )=Z_{\varphi }(w^{\ast },{\Gamma }_2\xi ,\tau ), \forall \tau >0.\hfill \end{array}$$

Since ${\Gamma }_1$ and ${\Gamma }_2$ are neutrosophic R-proximally weak reciprocal commuting mappings of type II,

$$\begin{array}{c}{P}_{\varphi }({\Gamma }_1{w}^{\ast },{\Gamma }_2{w}^{\ast },R\tau )\ge P_{\varphi }(w^{\ast },w^{\ast },\tau ),\hfill \\ E_{\varphi }({\Gamma }_1{w}^{\ast },{\Gamma }_2{w}^{\ast },R\tau )\le E_{\varphi }(w^{\ast },w^{\ast },\tau ),\hfill \\ Z_{\varphi }({\Gamma }_1{w}^{\ast },{\Gamma }_2{w}^{\ast },R\tau )\le Z_{\varphi }(w^{\ast },w^{\ast },\tau ), \forall \tau >0.\hfill \end{array}$$

Hence ${\Gamma }_1{w}^{\ast }={\Gamma }_2{w}^{\ast }$. Analogously, since ${\Gamma }_2(C_0)\subseteq D_0$, there exists a $z^{\ast }\in C_0$, such that,

$$\begin{array}{c}{P}_{\varphi }(z^{\ast },{\Gamma }_1{w}^{\ast },\tau )=P_{\varphi }(C,D,\tau )=P_{\varphi }(z^{\ast },{\Gamma }_2{w}^{\ast },\tau ),\hfill \\ E_{\varphi }(z^{\ast },{\Gamma }_1{w}^{\ast },\tau )=E_{\varphi }(C,D,\tau )=E_{\varphi }(z^{\ast },{\Gamma }_2{w}^{\ast },\tau ),\hfill \\ Z_{\varphi }(z^{\ast },{\Gamma }_1{w}^{\ast },\tau )=Z_{\varphi }(C,D,\tau )=Z_{\varphi }(z^{\ast },{\Gamma }_2{w}^{\ast },\tau ), \forall \tau >0.\hfill \end{array}$$

Given that the pair $(C,D)$ satisfies neutrosophic P-property. Thus we obtain, for all $\tau >0$,

$$\begin{array}{cc}\hfill P_{\varphi }(z^{\ast },w^{\ast },\tau )& =P_{\varphi }(z^{\ast },w^{\ast },\tau )\hfill \\ \hfill & =P_{\varphi }({\Gamma }_2{w}^{\ast },{\Gamma }_2\xi ,\tau )\hfill \\ \hfill & \ge P_{\varphi }({\Gamma }_1{w}^{\ast },{\Gamma }_2\xi ,{\displaystyle \frac{\tau }{k}})\hfill \\ \hfill & =P_{\varphi }(z^{\ast },w^{\ast },{\displaystyle \frac{\tau }{k}}).\hfill \end{array}$$

In the same manner, for all $\tau >0$, we get,

$$\begin{array}{c}{E}_{\varphi }(z^{\ast },w^{\ast },\tau )\le E_{\varphi }(z^{\ast },w^{\ast },{\displaystyle \frac{\tau }{k}}),\hfill \\ Z_{\varphi }(z^{\ast },w^{\ast },\tau )\le Z_{\varphi }(z^{\ast },w^{\ast },{\displaystyle \frac{\tau }{k}}).\hfill \end{array}$$

Thus $z^{\ast }=w^{\ast }$. Therefore, $w^{\ast }$ is the CBPP.

Subcase 2: Let ${\Gamma }_2{\xi }_n\to {\Gamma }_2\xi$ as $n\to \infty$.

Since ${\Gamma }_2(C_0)\subseteq {\Gamma }_1(C_0)$, there exists some $r\in C_0$, such that ${\Gamma }_2\xi ={\Gamma }_{1}r$. Now, we have, ${\Gamma }_2{\xi }_n\to {\Gamma }_{1}r$. Thus, by (12), for all $\tau >0$, we have,

$$\begin{array}{cc}\hfill & P_{\varphi }({\Gamma }_2{\xi }_{n+1},{\Gamma }_1{\xi }_{n+1},\tau )\to 1,\hfill \\ \hfill & E_{\varphi }({\Gamma }_2{\xi }_{n+1},{\Gamma }_1{\xi }_{n+1},\tau )\to 0,\hfill \\ \hfill & Z_{\varphi }({\Gamma }_2{\xi }_{n+1},{\Gamma }_1{\xi }_{n+1},\tau )\to 0, \mathrm{as} n\to \infty .\hfill \end{array}$$

Therefore, ${\Gamma }_1{\xi }_n\to {\Gamma }_{1}r$ as $n\to \infty$. Next, consider,

$$\begin{array}{c}{P}_{\varphi }({\Gamma }_{2}r,{\Gamma }_2{\xi }_n,\tau )\ge P_{\varphi }({\Gamma }_{1}r,{\Gamma }_1{\xi }_n,{\displaystyle \frac{\tau }{k}}),\hfill \\ E_{\varphi }({\Gamma }_{2}r,{\Gamma }_2{\xi }_n,\tau )\le E_{\varphi }({\Gamma }_{1}r,{\Gamma }_1{\xi }_n,{\displaystyle \frac{\tau }{k}}),\hfill \\ Z_{\varphi }({\Gamma }_{2}r,{\Gamma }_2{\xi }_n,\tau )\le Z_{\varphi }({\Gamma }_{1}r,{\Gamma }_1{\xi }_n,{\displaystyle \frac{\tau }{k}}), \forall \tau >0.\hfill \end{array}$$

As $n\to \infty$, we obtain ${\Gamma }_{2}r={\Gamma }_{1}r$. Existence can be proved as in the same method used in Subcase 1.

Now, we have established the existence of the CBPP.

Claim: CBPP is unique.

Let v and $v^{\ast }$ be two CBPPs of ${\Gamma }_1$ and ${\Gamma }_2$. Then we have,

$$\begin{array}{c}{P}_{\varphi }(v,{\Gamma }_{1}v,\tau )=P_{\varphi }(C,D,\tau )=P_{\varphi }(v,{\Gamma }_{2}v,\tau ),\hfill \\ E_{\varphi }(v,{\Gamma }_{1}v,\tau )=E_{\varphi }(C,D,\tau )=E_{\varphi }(v,{\Gamma }_{2}v,\tau ),\hfill \\ Z_{\varphi }(v,{\Gamma }_{1}v,\tau )=Z_{\varphi }(C,D,\tau )=Z_{\varphi }(v,{\Gamma }_{2}v,\tau ),\hfill \end{array}$$

and

$$\begin{array}{c}{P}_{\varphi }(v^{\ast },{\Gamma }_1{v}^{\ast },\tau )=P_{\varphi }(C,D,\tau )=P_{\varphi }(v^{\ast },{\Gamma }_2{v}^{\ast },\tau ),\hfill \\ E_{\varphi }(v^{\ast },{\Gamma }_1{v}^{\ast },\tau )=E_{\varphi }(C,D,\tau )=E_{\varphi }(v^{\ast },{\Gamma }_2{v}^{\ast },\tau ),\hfill \\ Z_{\varphi }(v^{\ast },{\Gamma }_1{v}^{\ast },\tau )=Z_{\varphi }(C,D,\tau )=Z_{\varphi }(v^{\ast },{\Gamma }_2{v}^{\ast },\tau ),\forall \tau >0.\hfill \end{array}$$

Since the pair $(C,D)$ satisfies neutrosophic P-property, for all $\tau >0$, we get,

$$\begin{array}{cc}\hfill P_{\varphi }(v,v^{\ast },\tau )& =P_{\varphi }({\Gamma }_{2}v,{\Gamma }_2{v}^{\ast },\tau )\hfill \\ \hfill & \ge P_{\varphi }({\Gamma }_{1}v,{\Gamma }_1{v}^{\ast },{\displaystyle \frac{\tau }{k}})\hfill \\ \hfill & =P_{\varphi }(v,v^{\ast },{\displaystyle \frac{\tau }{k}}).\hfill \end{array}$$

In the same manner, for all $\tau >0$, we can obtain,

$$\begin{array}{c}{E}_{\varphi }(v,v^{\ast },\tau )\le E_{\varphi }(v,v^{\ast },{\displaystyle \frac{\tau }{k}}),\hfill \\ Z_{\varphi }(v,v^{\ast },\tau )\le Z_{\varphi }(v,v^{\ast },{\displaystyle \frac{\tau }{k}}).\hfill \end{array}$$

Thus $v=v^{\ast }$. Hence the CBPP is unique. □

Example 5.

Carrying over the assumptions in Example 1, in addition, here we define ${\Gamma }_1,{\Gamma }_2:C\to D$ as ${\Gamma }_1(0,y)=\left\{\begin{array}{cc}(1,3)\hfill & y=3,\hfill \\ (1,7)\hfill & 3<y\le 5,\hfill \\ (1,{\displaystyle \frac{y+1}{2}})\hfill & 5<y\le 10.\hfill \end{array}\right.$ and ${\Gamma }_2(0,y)=\left\{\begin{array}{cc}(1,3)\hfill & y=3, \mathrm{or} 5<y\le 10,\hfill \\ (1,4)\hfill & 3<y\le 5.\hfill \end{array}\right.$. Note that, here the pair $(C,D)$ satisfies neutrosophic P-property, ${\Gamma }_2(C_0)\subseteq D_0,\forall \tau >0$ and ${\Gamma }_2(C_0)\subseteq {\Gamma }_1(C_0)$, $\forall \tau >0$.

Claim 1: ${\Gamma }_1$ and ${\Gamma }_2$ satisfies condition (a). That is, there exists a non-negative real number $k<1$ such that,

$$\begin{array}{c}{P}_{\varphi }({\Gamma }_2{z}_1,{\Gamma }_2{z}_2,k\tau )\ge P_{\varphi }({\Gamma }_1{z}_1,{\Gamma }_1{z}_2,\tau )\hfill \\ E_{\varphi }({\Gamma }_2{z}_1,{\Gamma }_2{z}_2,k\tau )\le E_{\varphi }({\Gamma }_1{z}_1,{\Gamma }_1{z}_2,\tau )\hfill \\ Z_{\varphi }({\Gamma }_2{z}_1,{\Gamma }_2{z}_2,k\tau )\le Z_{\varphi }({\Gamma }_1{z}_1,{\Gamma }_1{z}_2,\tau )\hfill \end{array}$$

for all $z_1=(0,x_1)$ and $z_2=(0,y_2)$ in C, where $x_1,x_2\in [3,10]$.

First, let us find the value of k for which $P_{\varphi }({\Gamma }_2{z}_1,{\Gamma }_2{z}_2,k\tau )\ge P_{\varphi }({\Gamma }_1{z}_1,{\Gamma }_1{z}_2,\tau )$ holds. Thus we obtain,

$$\begin{array}{cc}\hfill & \mathit{for} x_1=3 \mathit{and} y_1=3, \mathit{we} \mathit{get} k\in (0,1),\mathit{for} x_1=3 \mathit{and} 3<y_1\le 5, \mathit{we} \mathit{get} k\in [{\displaystyle \frac{1}{4}},1),\hfill \\ \hfill & \mathit{for} x_1=3 \mathit{and} 5<y_1\le 10, \mathit{we} \mathit{get} k\in (0,1),\mathit{for} 3<x_1\le 5 \mathit{and} y_1=3, \mathit{we} \mathit{get} k\in [{\displaystyle \frac{1}{4}},1),\hfill \\ \hfill & \mathit{for} 3<x_1\le 5 \mathit{and} 3<y_1\le 5, \mathit{we} \mathit{get} k\in (0,1),\mathit{for} 3<x_1\le 5 \mathit{and} 5<y_1\le 10, \mathit{we} \mathit{get} k\in (0,1),\hfill \\ \hfill & \mathit{for} 5<x_1\le 10 \mathit{and} y_1=3, \mathit{we} \mathit{get} k\in (0,1),\mathit{for} 5<x_1\le 10 \mathit{and} 3<y_1\le 5, \mathit{we} \mathit{get} k\in [{\displaystyle \frac{2}{3}},1),\hfill \\ \hfill & \mathit{for} 5<x_1\le 10 \mathit{and} 5<y_1\le 10, \mathit{we} \mathit{get} k\in (0,1).\hfill \end{array}$$

Thus, we can conclude that for $k\in [{\displaystyle \frac{2}{3}},1)$, $P_{\varphi }({\Gamma }_2{z}_1,{\Gamma }_2{z}_2,k\tau )\ge P_{\varphi }({\Gamma }_1{z}_1,{\Gamma }_1{z}_2,\tau )$. In the same pattern, it can be verified that $E_{\varphi }({\Gamma }_2{z}_1,{\Gamma }_2{z}_2,k\tau )\le E_{\varphi }({\Gamma }_1{z}_1,{\Gamma }_1{z}_2,\tau )$ and $Z_{\varphi }({\Gamma }_2{z}_1,{\Gamma }_2{z}_2,k\tau )\le Z_{\varphi }({\Gamma }_1{z}_1,{\Gamma }_1{z}_2,\tau )$ for $k\in [{\displaystyle \frac{2}{3}},1)$. Thus ${\Gamma }_1$ and ${\Gamma }_2$ satisfies condition (a) for $k\in [{\displaystyle \frac{2}{3}},1)$.

Claim 2: ${\Gamma }_1$ and ${\Gamma }_2$ are neutrosophic proximally weak reciprocal continuous.

Consider the sequences $(a_n)$, $(b_n)$ and $(c_n)$ in $[3,10]$ where the sequences $(b_n)$ and $(c_n)$ converges to the same limit, say $t\in [3,10]$. Then, we have, $\underset{n\to \infty }{\lim }{P}_{\varphi }((0,b_n),(0,t),\tau )=\underset{n\to \infty }{\lim }{P}_{\varphi }((0,c_n),(0,t),\tau )=1$, $\underset{n\to \infty }{\lim }{E}_{\varphi }((0,b_n),(0,t),\tau )=\underset{n\to \infty }{\lim }{E}_{\varphi }((0,c_n),(0,t),\tau )=0$, and $\underset{n\to \infty }{\lim }{Z}_{\varphi }((0,b_n),(0,t),\tau )=\underset{n\to \infty }{\lim }{Z}_{\varphi }((0,c_n),(0,t),\tau )=0$, for all $\tau >0$. Now, for the given sequences $(a_n)$,$(b_n)$ and $(c_n)$ we have,

$$\begin{array}{cc}\hfill & P_{\varphi }\left((0,b_n),{\Gamma }_1(0,a_n),\tau \right)={\displaystyle \frac{\tau }{\tau +1}}=P_{\varphi }\left((0,c_n),{\Gamma }_2(0,a_n),\tau \right),\hfill \\ \hfill & E_{\varphi }\left((0,b_n),{\Gamma }_1(0,a_n),\tau \right)={\displaystyle \frac{1}{\tau +1}}=E_{\varphi }\left((0,c_n),{\Gamma }_2(0,a_n),\tau \right),\hfill \\ \hfill & Z_{\varphi }\left((0,b_n),{\Gamma }_1(0,a_n),\tau \right)={\displaystyle \frac{1}{\tau }}=Z_{\varphi }\left((0,c_n),{\Gamma }_2(0,a_n),\tau \right), \forall \tau >0.\hfill \end{array}$$

Given the sequences $(b_n)$ and $(c_n)$ converges to the same limit, we have only 2 cases.

Case I: $b_n=c_n=3$ and $a_n=3, n\in \mathbb{N}$.

Here $\underset{n\to \infty }{\lim }{P}_{\varphi }({\Gamma }_1(0,3),{\Gamma }_1(0,3),\tau )=1=\underset{n\to \infty }{\lim }{P}_{\varphi }({\Gamma }_2(0,3),{\Gamma }_2(0,3),\tau )$, $\underset{n\to \infty }{\lim }{E}_{\varphi }({\Gamma }_1(0,3),$${\Gamma }_1(0,3),\tau )=0=\underset{n\to \infty }{\lim }{E}_{\varphi }({\Gamma }_2(0,3),{\Gamma }_2(0,3),\tau )$ and $\underset{n\to \infty }{\lim }{Z}_{\varphi }({\Gamma }_1(0,3),{\Gamma }_1(0,3),\tau )=0=\underset{n\to \infty }{\lim }{Z}_{\varphi }({\Gamma }_2(0,3),{\Gamma }_2(0,3),\tau )$. Thus, as $n\to \infty$, ${\Gamma }_1(0,c_n)\to {\Gamma }_1(0,3)$ and ${\Gamma }_2(0,b_n)\to {\Gamma }_2(0,3)$.

Case II: $b_n=3,c_n=3+{\displaystyle \frac{x_n}{2}}$ and $a_n=5+x_n$, (where $x_n\in [0,5],\forall n\in \mathbb{N}$ and $x_n\to 0$, as $n\to \infty$).

Here, $\underset{n\to \infty }{\lim }{P}_{\varphi }({\Gamma }_2(0,3+{\displaystyle \frac{x_n}{2}}),{\Gamma }_2(0,3),\tau )=1$, $\underset{n\to \infty }{\lim }{E}_{\varphi }({\Gamma }_2(0,3+{\displaystyle \frac{x_n}{2}}),{\Gamma }_2(0,3),\tau )=0$ and $\underset{n\to \infty }{\lim }{Z}_{\varphi }({\Gamma }_2(0,3+{\displaystyle \frac{x_n}{2}}),{\Gamma }_2(0,3),\tau )=0$. Thus, as $n\to \infty$, ${\Gamma }_2(0,b_n)\to {\Gamma }_2(0,3)$. But, $\underset{n\to \infty }{\lim }{P}_{\varphi }({\Gamma }_1(0,3+{\displaystyle \frac{x_n}{2}}),{\Gamma }_1(0,3),\tau )=\underset{n\to \infty }{\lim }{P}_{\varphi }((1,7),(0,3),\tau )\ne 1$, $\underset{n\to \infty }{\lim }{E}_{\varphi }({\Gamma }_1(0,3+{\displaystyle \frac{x_n}{2}}),{\Gamma }_1(0,3),\tau )=\underset{n\to \infty }{\lim }{E}_{\varphi }((1,7),(0,3),\tau )\ne 0$ and $\underset{n\to \infty }{\lim }{Z}_{\varphi }({\Gamma }_1(0,3+{\displaystyle \frac{x_n}{2}}),{\Gamma }_1(0,3),\tau )=\underset{n\to \infty }{\lim }{Z}_{\varphi }((1,7),(0,3),\tau )\ne 0$. Thus, as $n\to \infty$, ${\Gamma }_2(0,b_n)$ does not converges to ${\Gamma }_2(0,3)$. Hence ${\Gamma }_1$ and ${\Gamma }_2$ are neutrosophic proximally weak reciprocal continuous.

Claim 3: ${\Gamma }_1$ and ${\Gamma }_2$ are neutrosophic proximally compatible.

Consider the sequences $(a_n)$, $(b_n)$ and $(c_n)$ in $[3,10]$ where the sequences $(b_n)$ and $(c_n)$ converges to the same limit, say $t\in [3,10]$. Then, we have, $\underset{n\to \infty }{\lim }{P}_{\varphi }((0,b_n),(0,t),\tau )=\underset{n\to \infty }{\lim }{P}_{\varphi }((0,c_n),(0,t),\tau )=1$, $\underset{n\to \infty }{\lim }{E}_{\varphi }((0,b_n),(0,t),\tau )=\underset{n\to \infty }{\lim }{E}_{\varphi }((0,c_n),(0,t),\tau )=0$, and $\underset{n\to \infty }{\lim }{Z}_{\varphi }((0,b_n),(0,t),\tau )=\underset{n\to \infty }{\lim }{Z}_{\varphi }((0,c_n),(0,t),\tau )=0$, for all $\tau >0$. Now, for the given sequences $(a_n)$,$(b_n)$ and $(c_n)$ we have,

$$\begin{array}{cc}\hfill & P_{\varphi }\left((0,b_n),{\Gamma }_1(0,a_n),\tau \right)={\displaystyle \frac{\tau }{\tau +1}}=P_{\varphi }\left((0,c_n),{\Gamma }_2(0,a_n),\tau \right),\hfill \\ \hfill & E_{\varphi }\left((0,b_n),{\Gamma }_1(0,a_n),\tau \right)={\displaystyle \frac{1}{\tau +1}}=E_{\varphi }\left((0,c_n),{\Gamma }_2(0,a_n),\tau \right),\hfill \\ \hfill & Z_{\varphi }\left((0,b_n),{\Gamma }_1(0,a_n),\tau \right)={\displaystyle \frac{1}{\tau }}=Z_{\varphi }\left((0,c_n),{\Gamma }_2(0,a_n),\tau \right), \forall \tau >0.\hfill \end{array}$$

Given the sequences $(b_n)$ and $(c_n)$ converges to the same limit, we have only 2 cases.

Case I: $b_n=c_n=3$ and $a_n=3, n\in \mathbb{N}$.

Now, we have, $\underset{n\to \infty }{\lim }{P}_{\varphi }({\Gamma }_1(0,3),{\Gamma }_2(0,3),\tau )=1$, $\underset{n\to \infty }{\lim }{E}_{\varphi }({\Gamma }_1(0,3),{\Gamma }_2(0,3),\tau )=0$ and $\underset{n\to \infty }{\lim }{Z}_{\varphi }({\Gamma }_1(0,3),{\Gamma }_2(0,3),\tau )=0$.

Case II: $b_n=3,c_n=3+{\displaystyle \frac{x_n}{2}}$ and $a_n=5+x_n$, (where $x_n\in [0,5],\forall n\in \mathbb{N}$ and $x_n\to 0$, as $n\to \infty$). We obtain $\underset{n\to \infty }{\lim }{P}_{\varphi }({\Gamma }_1(0,3),{\Gamma }_2(0,3+{\displaystyle \frac{x_n}{2}}),\tau )=1$, $\underset{n\to \infty }{\lim }{E}_{\varphi }({\Gamma }_1(0,3),{\Gamma }_2(0,3+{\displaystyle \frac{x_n}{2}}),\tau )=0$ and $\underset{n\to \infty }{\lim }{Z}_{\varphi }({\Gamma }_1(0,3),{\Gamma }_2(0,3+{\displaystyle \frac{x_n}{2}}),\tau )=0$. Hence ${\Gamma }_1$ and ${\Gamma }_2$ are neutrosophic proximally compatible. Thus, ${\Gamma }_1$ and ${\Gamma }_2$ satisfies the conditions in the hypothesis of the Theorem 1. Hence the pair ${\Gamma }_1$ and ${\Gamma }_2$ has a CBPP $(0,3)$ in C.

4. Application

When the sets C and D are identical, we obtain the following results on the existence and uniqueness of common fixed points as an application of Theorem 1.

Corollary 1.

Let C be a non-empty closed subset of a neutrosophic complete metric space $(R,P_{\varphi },E_{\varphi },Z_{\varphi },\ast ,\diamond )$ and consider ${\Gamma }_1,{\Gamma }_2:C\to C$, be two neutrosophic weak reciprocal continuous mappings satisfying the following conditions:

(a) 

There exists a non-negative real number $k<1$ such that

$$\begin{array}{c}{P}_{\varphi }({\Gamma }_2{\zeta }_1,{\Gamma }_2{\zeta }_2,k\tau )\ge P_{\varphi }({\Gamma }_1{\zeta }_1,{\Gamma }_1{\zeta }_2,\tau )\hfill \\ E_{\varphi }({\Gamma }_2{\zeta }_1,{\Gamma }_2{\zeta }_2,k\tau )\le E_{\varphi }({\Gamma }_1{\zeta }_1,{\Gamma }_1{\zeta }_2,\tau )\hfill \\ Z_{\varphi }({\Gamma }_2{\zeta }_1,{\Gamma }_2{\zeta }_2,k\tau )\le Z_{\varphi }({\Gamma }_1{\zeta }_1,{\Gamma }_1{\zeta }_2,\tau )\hfill \end{array}$$

for all ${\zeta }_1$ and ${\zeta }_2$ in C, for all $\tau >0$.

(b) 

${\Gamma }_2(C)\subseteq {\Gamma }_1(C)$.

If ${\Gamma }_1$ and ${\Gamma }_2$ are neutrosophic compatible, then, there exists an element $\zeta \in C$ such that

$$\begin{array}{c}{P}_{\varphi }(\zeta ,{\Gamma }_1\zeta ,\tau )=P_{\varphi }(\zeta ,{\Gamma }_2\zeta ,\tau )\hfill \\ E_{\varphi }(\zeta ,{\Gamma }_1\zeta ,\tau )=E_{\varphi }(\zeta ,{\Gamma }_2\zeta ,\tau )\hfill \\ Z_{\varphi }(\zeta ,{\Gamma }_1\zeta ,\tau )=Z_{\varphi }(\zeta ,{\Gamma }_2\zeta ,\tau ), \forall \tau >0.\hfill \end{array}$$

That is, ζ constitutes a unique common fixed point for ${\Gamma }_1$ and ${\Gamma }_2$.

Corollary 2.

Let C be a non-empty closed subset of a neutrosophic complete metric space $(R,P_{\varphi },E_{\varphi },Z_{\varphi },\ast ,\diamond )$ and consider ${\Gamma }_1,{\Gamma }_2:C\to C$, be two neutrosophic weak reciprocal continuous mappings satisfying the following conditions:

(a) 

There exists a non-negative real number $k<1$ such that

$$\begin{array}{c}{P}_{\varphi }({\Gamma }_2{\zeta }_1,{\Gamma }_2{\zeta }_2,k\tau )\ge P_{\varphi }({\Gamma }_1{\zeta }_1,{\Gamma }_1{\zeta }_2,\tau )\hfill \\ E_{\varphi }({\Gamma }_2{\zeta }_1,{\Gamma }_2{\zeta }_2,k\tau )\le E_{\varphi }({\Gamma }_1{\zeta }_1,{\Gamma }_1{\zeta }_2,\tau )\hfill \\ Z_{\varphi }({\Gamma }_2{\zeta }_1,{\Gamma }_2{\zeta }_2,k\tau )\le Z_{\varphi }({\Gamma }_1{\zeta }_1,{\Gamma }_1{\zeta }_2,\tau )\hfill \end{array}$$

for all ${\zeta }_1$ and ${\zeta }_2$ in C, for all $\tau >0$.

(b) 

${\Gamma }_2(C)\subseteq {\Gamma }_1(C)$.

If ${\Gamma }_1$ and ${\Gamma }_2$ are neutrosophic R-weak reciprocal commuting mappings of type I, then, there exists an element $\zeta \in C$ such that

$$\begin{array}{c}{P}_{\varphi }(\zeta ,{\Gamma }_1\zeta ,\tau )=P_{\varphi }(\zeta ,{\Gamma }_2\zeta ,\tau )\hfill \\ E_{\varphi }(\zeta ,{\Gamma }_1\zeta ,\tau )=E_{\varphi }(\zeta ,{\Gamma }_2\zeta ,\tau )\hfill \\ Z_{\varphi }(\zeta ,{\Gamma }_1\zeta ,\tau )=Z_{\varphi }(\zeta ,{\Gamma }_2\zeta ,\tau ), \forall \tau >0.\hfill \end{array}$$

That is, ζ constitutes a unique common fixed point for ${\Gamma }_1$ and ${\Gamma }_2$.

Corollary 3.

Let C be a non-empty closed subset of a neutrosophic complete metric space $(R,P_{\varphi },E_{\varphi },Z_{\varphi },\ast ,\diamond )$ and consider ${\Gamma }_1,{\Gamma }_2:C\to C$, be two neutrosophic weak reciprocal continuous mappings satisfying the following conditions:

(a) 

There exists a non-negative real number $k<1$ such that

$$\begin{array}{c}{P}_{\varphi }({\Gamma }_2{\zeta }_1,{\Gamma }_2{\zeta }_2,k\tau )\ge P_{\varphi }({\Gamma }_1{\zeta }_1,{\Gamma }_1{\zeta }_2,\tau )\hfill \\ E_{\varphi }({\Gamma }_2{\zeta }_1,{\Gamma }_2{\zeta }_2,k\tau )\le E_{\varphi }({\Gamma }_1{\zeta }_1,{\Gamma }_1{\zeta }_2,\tau )\hfill \\ Z_{\varphi }({\Gamma }_2{\zeta }_1,{\Gamma }_2{\zeta }_2,k\tau )\le Z_{\varphi }({\Gamma }_1{\zeta }_1,{\Gamma }_1{\zeta }_2,\tau )\hfill \end{array}$$

for all ${\zeta }_1$ and ${\zeta }_2$ in C, for all $\tau >0$.

(b) 

${\Gamma }_2(C)\subseteq {\Gamma }_1(C)$.

If ${\Gamma }_1$ and ${\Gamma }_2$ are neutrosophic R-weak reciprocal commuting mappings of type II, then, there exists an element $\zeta \in C$ such that

$$\begin{array}{c}{P}_{\varphi }(\zeta ,{\Gamma }_1\zeta ,\tau )=P_{\varphi }(\zeta ,{\Gamma }_2\zeta ,\tau )\hfill \\ E_{\varphi }(\zeta ,{\Gamma }_1\zeta ,\tau )=E_{\varphi }(\zeta ,{\Gamma }_2\zeta ,\tau )\hfill \\ Z_{\varphi }(\zeta ,{\Gamma }_1\zeta ,\tau )=Z_{\varphi }(\zeta ,{\Gamma }_2\zeta ,\tau ), \forall \tau >0.\hfill \end{array}$$

That is, ζ constitutes a unique common fixed point for ${\Gamma }_1$ and ${\Gamma }_2$.

5. Conclusions

The importance of the best proximity point arises when we have to deal with mappings where finding classical fixed points is not possible. In [11], Unni et al. introduced a new class of discontinuous mappings and established the existence and uniqueness of CBPP for this new class of mappings in a complete metric space. Motivated by this, in this work, we extended the aforementioned result to the framework of complete NMSs. Our result extends many existing results in the literature, as NMSs are extensions of traditional metric spaces.