# A Discussion on the Existence of Best Proximity Points That Belong to the Zero Set

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## Abstract

**:**

_{p}-admissible; α

_{p}-admissible weak (F,φ)-proximal contraction; G-proximal graphic contraction; φ-best proximity point

## 1. Introduction and Preliminaries

**Definition**

**1**

**(p1)**- $p(\xi ,\xi )=p(\eta ,\eta )=p(\xi ,\eta )\iff \xi =\eta ,$
**(p2)**- $p(\xi ,\xi )\le p(\xi ,\eta ),$
**(p3)**- $p(\xi ,\eta )=p(\eta ,\xi ),$
**(p4)**- $p(\xi ,\eta )\le p(\xi ,\zeta )+p(\zeta ,\eta )-p(\zeta ,\zeta )$

**(1)**- ${m}_{\xi \eta}=min\left(\right)open="\{"\; close="\}">\rho (\xi ,\xi ),\rho (\eta ,\eta )$
**(2)**- ${M}_{\xi \eta}=max\left(\right)open="\{"\; close="\}">\rho (\xi ,\xi ),\rho (\eta ,\eta )$

**Definition**

**2**

**(m1)**- $\rho (\xi ,\xi )=\rho (\eta ,\eta )=\rho (\xi ,\eta )\iff \xi =\eta ,$
**(m2)**- ${m}_{\xi \eta}\le \rho (\xi ,\eta )$
**(m3)**- $\rho (\xi ,\eta )=\rho (\eta ,\xi ),$
**(m4)**- $\rho (\xi ,\eta )-{m}_{\xi \eta}\le \rho (\xi ,\zeta )-{m}_{\xi \zeta}+\rho (\zeta ,\eta )-{m}_{\zeta \eta}$

**Example**

**1.**

**Definition**

**3**

**.**Let $(S,\rho )$ be an M-metric space and $\xi \in S.$ A sequence $\left(\right)$ in S is called:

**(1)**- $M-$ convergent to $\xi \in S$ if and only if$$\underset{n\to \infty}{lim}(\rho ({\xi}_{n},\xi )-{m}_{{\xi}_{n},\xi})=0,$$
**(2)**- $M-$ Cauchy sequence if and only if$$\underset{n,m\to \infty}{lim}(\rho ({\xi}_{n},{\xi}_{m})-{m}_{{\xi}_{n},{\xi}_{m}})\phantom{\rule{4.pt}{0ex}}and\phantom{\rule{4.pt}{0ex}}\underset{n,m\to \infty}{lim}({M}_{{\xi}_{n},{\xi}_{m}}-{m}_{{\xi}_{n},{\xi}_{m}}),$$exist (and are finite).

**Definition**

**4**

**Remark**

**1**

- (r1)
- $0\le {M}_{\xi \eta}+{m}_{\xi \eta}=\rho (\xi ,\xi )+\rho (\eta ,\eta ),$
- (r2)
- $0\le {M}_{\xi \eta}-{m}_{\xi \eta}=[\rho (\xi ,\xi )-\rho (\eta ,\eta )],$
- (r3)
- ${M}_{\xi \eta}-{m}_{\xi \eta}\le ({M}_{\xi \zeta}-{m}_{\xi \zeta})+({M}_{\zeta \eta}-{m}_{\zeta \eta}).$

- (F1)
- $max\left(\right)open="\{"\; close="\}">s,t$ for all $s,t,r\in [0,\infty ),$
- (F2)
- F is continuous,
- (F3)
- $F(0,0,0)=0$.

**Example**

**2**

**.**Let $i=\left(\right)open="\{"\; close="\}">1,2,3$ Define ${F}_{i}:{[0,\infty )}^{3}\to [0,\infty )$ as follows:

**Definition**

**5**

**Remark**

**2**

**.**Note that ${\sum}_{n=1}^{+\infty}{\psi}^{n}\left(t\right)<\infty $ implies ${lim}_{n\to \infty}{\psi}^{n}\left(t\right)=0,$ for all $t\in (0,\infty ).$

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

**Definition**

**9.**

## 2. Main Results

**Definition**

**10.**

**Definition**

**11.**

**Theorem**

**1.**

**Proof.**

**(F1)**that

**(m2)**, we have

**(m4)**, we have

**(m4)**, we have

**Uniqueness:**Let $\alpha (\xi ,\eta )\ge 0,$ for all $\xi ,\eta \in {\phi}_{T}\left(A\right).$ Suppose that ${\xi}^{*}$ and w are two $\phi $ -best proximity points of T with ${\xi}^{*}\ne w$. Hence

**Corollary**

**1.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**Corollary**

**3.**

**Proof.**

**Example**

**3.**

**Example**

**4.**

## 3. Application to Fixed Point Theory

**Remark**

**3.**

**Definition**

**12.**

**Corollary**

**4.**

**Proof.**

**(m4)**, we have

**Uniqueness:**Let $\alpha (\xi ,\eta )\ge 0$ for all $\xi ,\eta \in {\phi}_{F}\left(S\right).$ Suppose that ${\xi}^{*}$ and w are two $\phi -$ fixed point of T with ${\xi}^{*}\ne w$. Hence

## 4. Application to Graph Theory

**Definition**

**13**

**.**1. A graph’s subgraph is a graph whose vertex set is a subset of $V\left(G\right)$ and whose edge set is a subset of $E\left(G\right).$

- 2
- Let ξ and η be two vertices of a graph $G.$ A path from ξ to η of length n(where $n\in \mathbb{N}\cup \left\{0\right\})$ in a graph G is a sequence $\{{\xi}_{n}:n=0,1,2,\dots ,n\}$ of $n+1$ distinct vertices such that ${\xi}_{0}=\xi ,$ ${\xi}_{n}=\eta $ and $({\xi}_{i},{\xi}_{i+1})\in E\left(G\right)$ for $i=1,2,\dots ,n.$
- 3
- A graph G is called connected graph if there exist a path between any two vertices of graph G and if $\tilde{G}$ is connected then G is said to be weakly connected graph.
- 4
- A path is called elementary if no vertices appear more than once in it.

**Definition**

**14.**

**Theorem**

**2.**

**Proof.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Karapınar, E.; Abbas, M.; Farooq, S.
A Discussion on the Existence of Best Proximity Points That Belong to the Zero Set. *Axioms* **2020**, *9*, 19.
https://doi.org/10.3390/axioms9010019

**AMA Style**

Karapınar E, Abbas M, Farooq S.
A Discussion on the Existence of Best Proximity Points That Belong to the Zero Set. *Axioms*. 2020; 9(1):19.
https://doi.org/10.3390/axioms9010019

**Chicago/Turabian Style**

Karapınar, Erdal, Mujahid Abbas, and Sadia Farooq.
2020. "A Discussion on the Existence of Best Proximity Points That Belong to the Zero Set" *Axioms* 9, no. 1: 19.
https://doi.org/10.3390/axioms9010019