# Best Proximity Point Results for Geraghty Type Ƶ-Proximal Contractions with an Application

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## Abstract

**:**

## 1. Introduction

**Definition**

**1**

**.**A simulation function is a mapping $\zeta :[0,\infty )\times [0,\infty )\to \mathbb{R}$ so that:

- $\left({\zeta}_{1}\right)$
- $\zeta (0,0)=0$;
- $\left({\zeta}_{2}\right)$
- $\zeta (\mu ,\eta )<\eta -\mu $ for all $\mu ,\eta >0$;
- $\left({\zeta}_{3}\right)$
- If $\left({\mu}_{n}\right),\left({\eta}_{n}\right)$ are sequences in $(0,\infty )$ so that $\underset{n\to \infty}{lim}{\mu}_{n}=\underset{n\to \infty}{lim}{\eta}_{n}>0,$ then$$\underset{n\to \infty}{lim\; sup}\zeta ({\mu}_{n},{\eta}_{n})<0.$$

**Theorem**

**1**

**.**Let $(M,d)$ be a complete metric space and $\mathcal{T}:M\to M$ be a $\mathcal{Z}$-contraction with respect to $\zeta \in \mathcal{Z}$—that is,

## 2. Preliminaries

**Definition**

**2**

**.**Let $P,Q$ be two non-empty subsets of a metric space, $(M,d).$ A mapping $\mathcal{T}:P\to Q$ is called a Geraghty contraction if there is $\beta \in \mathsf{\Sigma}$, so that for all $\xi ,\omega \in P$

**Definition**

**3**

**.**Let $\mathcal{T}:P\to Q$ and $\alpha :P\times P\to \left[0,\infty \right)$ be given mappings. Then, $\mathcal{T}$ is called α-proximal admissible if

**Definition**

**4**

**.**Let $\mathcal{T}:P\to Q$ and $\alpha ,\eta :P\times P\to \left[0,\infty \right)$ be given mappings. Such $\mathcal{T}$ is said to be $(\alpha ,\eta )$-proximal admissible if

**Definition**

**5**

**.**Let P and Q be two non-empty subsets of a metric space, $(M,d).$ A non-self-mapping $\mathcal{T}:P\to Q$ is called a $\mathcal{Z}$-proximal contraction if there is a simulation function ζ so that

**Definition**

**6.**

- (1)
- $\mathcal{T}$ is $(\alpha ,\eta )$-proximal admissible;
- (2)
- $\alpha (u,v)\ge \eta (u,v)\phantom{\rule{4pt}{0ex}}and\phantom{\rule{4pt}{0ex}}\alpha (v,z)\ge \eta (v,z)\phantom{\rule{4pt}{0ex}}implies\phantom{\rule{4.pt}{0ex}}that\phantom{\rule{4pt}{0ex}}\alpha (u,z)\ge \eta (u,z),$ for all $u,v,z\in P.$

**Definition**

**7.**

**Remark**

**1.**

## 3. Main Results

**Theorem**

**2.**

- (i)
- P is closed and $\mathcal{T}\left({P}_{0}\right)\subseteq {Q}_{0}$;
- (ii)
- $\mathcal{T}$ is triangular $(\alpha ,\eta )$-proximal admissible;
- (iii)
- There are ${u}_{0},{u}_{1}\in {P}_{0}$ so that $d({u}_{1},\mathcal{T}{u}_{0})=d(P,Q)$ and $\alpha \left({u}_{0},{u}_{1}\right)\ge \eta \left({u}_{0},{u}_{1}\right)$;
- (iv)
- $\mathcal{T}$ is a continuous Geraghty type $\mathcal{Z}$-proximal contraction.

**Proof.**

- (C)
- If a sequence $\left\{{u}_{n}\right\}$ in P is convergent to $u\in P$ so that $\alpha \left({u}_{n},{u}_{n+1}\right)\ge \eta \left({u}_{n},{u}_{n+1}\right)$, then $\alpha \left({u}_{n},u\right)\ge \eta \left({u}_{n},u\right)$ for all $n\in \mathbb{N}.$

**Theorem**

**3.**

- (i)
- P is closed and $\mathcal{T}\left({P}_{0}\right)\subseteq {Q}_{0}$;
- (ii)
- $\mathcal{T}$ is triangular $(\alpha ,\eta )$-proximal admissible;
- (iii)
- there are ${u}_{0},{u}_{1}\in {P}_{0}$ so that $d({u}_{1},\mathcal{T}{u}_{0})=d(P,Q)$ and $\alpha \left({u}_{0},{u}_{1}\right)\ge \eta \left({u}_{0},{u}_{1}\right)$;
- (iv)
- the condition $\left(C\right)$ holds and $\mathcal{T}$ is a Geraghty type $\mathcal{Z}$-proximal contraction.

**Proof.**

**Example**

**1.**

**Corollary**

**1.**

**Proof.**

## 4. Some Consequences

**Definition**

**8.**

**Theorem**

**4.**

- (i)
- P is closed and $\mathcal{T}\left({P}_{0}\right)\subseteq {Q}_{0}$;
- (ii)
- $\mathcal{T}$ is ⪯-proximal increasing;
- (iii)
- There are ${u}_{0},{u}_{1}\in {P}_{0}$ so that $d({u}_{1},\mathcal{T}{u}_{0})=d(P,Q)$ and ${u}_{0}\u2aaf{u}_{1}$;
- (iv)
- $\mathcal{T}$ is continuous or, for every sequence $\left\{{u}_{n}\right\}$ in P is convergent to $u\in P$ so that ${u}_{n}\u2aaf{u}_{n+1}$, we have ${u}_{n}\u2aafu$ for all $n\in \mathbb{N};$
- (v)
- There exist $\zeta \in \mathcal{Z}$ and $\beta \in \mathsf{\Sigma}$, such that for all $u,v,\rho ,\nu \in P,$$$\left.\begin{array}{c}\hfill u\u2aafv\\ \hfill d(\rho ,\mathcal{T}u)=d(P,Q)\\ \hfill d(\nu ,\mathcal{T}v)=d(P,Q)\end{array}\right\}\u27f9\zeta (d(\rho ,\nu ),\beta \left(d(u,v)\right)d(u,v))\ge 0.$$

**Definition**

**9.**

- (1)
- for all ${u}_{1},{u}_{2},{p}_{1},{p}_{2}\in P$,$$\left.\begin{array}{c}\hfill ({u}_{1},{u}_{2})\in E\left(G\right)\\ \hfill d({p}_{1},\mathcal{T}{u}_{1})=d(P,Q)\\ \hfill d({p}_{2},\mathcal{T}{u}_{2})=d(P,Q)\end{array}\right\}\u27f9({p}_{1},{p}_{2})\in E\left(G\right);$$
- (2)
- $(u,v)\in E\left(G\right)\phantom{\rule{4pt}{0ex}}and\phantom{\rule{4pt}{0ex}}(v,z)\in E\left(G\right)\phantom{\rule{4pt}{0ex}}implies\phantom{\rule{4.pt}{0ex}}that\phantom{\rule{4pt}{0ex}}(u,z)\in E\left(G\right),$ for all $u,v,z\in P.$

**Theorem**

**5.**

- (i)
- P is closed and $\mathcal{T}\left({P}_{0}\right)\subseteq {Q}_{0}$;
- (ii)
- $\mathcal{T}$ is triangular G-proximal;
- (iii)
- There are ${u}_{0},{u}_{1}\in {P}_{0}$ so that $d({u}_{1},\mathcal{T}{u}_{0})=d(P,Q)$ and $({u}_{0},{u}_{1})\in E\left(G\right)$;
- (iv)
- $\mathcal{T}$ is continuous or, for every sequence $\left\{{u}_{n}\right\}$ in P is convergent to $u\in P$ so that $({u}_{n},{u}_{n+1})\in E\left(G\right)$, we have $({u}_{n},u)\in E\left(G\right)$ for all $n\in \mathbb{N};$
- (v)
- There exist $\zeta \in \mathcal{Z}$ and $\beta \in \mathsf{\Sigma}$ such that for all $u,v,\rho ,\nu \in P,$$$\left.\begin{array}{c}\hfill (u,v)\in E\left(G\right)\\ \hfill d(\rho ,\mathcal{T}u)=d(P,Q)\\ \hfill d(\nu ,\mathcal{T}v)=d(P,Q)\end{array}\right\}\u27f9\zeta (d(\rho ,\nu ),\beta \left(d(u,v)\right)d(u,v))\ge 0.$$

## 5. A Variational Inequality Problem

**Lemma**

**1**

**.**Let $z\in H$. Then, $u\in C$ is such that $\langle u-z,y-u\rangle \ge 0$, for all $y\in C$ iff $u={P}_{C}z$.

**Lemma**

**2**

**.**Let $S:H\to H$. Then, $u\in C$ is a solution of $\langle Su,v-u\rangle \ge 0$, for all $v\in C$, if $u={P}_{C}(u-\lambda Su)$, with $\lambda >0$.

**Theorem**

**6.**

**Proof.**

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Işık, H.; Aydi, H.; Mlaiki, N.; Radenović, S.
Best Proximity Point Results for Geraghty Type *Ƶ*-Proximal Contractions with an Application. *Axioms* **2019**, *8*, 81.
https://doi.org/10.3390/axioms8030081

**AMA Style**

Işık H, Aydi H, Mlaiki N, Radenović S.
Best Proximity Point Results for Geraghty Type *Ƶ*-Proximal Contractions with an Application. *Axioms*. 2019; 8(3):81.
https://doi.org/10.3390/axioms8030081

**Chicago/Turabian Style**

Işık, Hüseyin, Hassen Aydi, Nabil Mlaiki, and Stojan Radenović.
2019. "Best Proximity Point Results for Geraghty Type *Ƶ*-Proximal Contractions with an Application" *Axioms* 8, no. 3: 81.
https://doi.org/10.3390/axioms8030081