# w-b-Cone Distance and Its Related Results: A Survey

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## Abstract

**:**

## 1. Introduction and Preliminaries

**Definition**

**1**

**.**Let X be a nonempty set. A real-valued function $D:X\times X\to [0,\infty )$ is said to be a b-metric if the following properties hold:

- $\left({D}_{1}\right)$
- $D(x,y)=0$ iff $x=y$ for all $x,y\in X$;
- $\left({D}_{2}\right)$
- $D(x,y)=D(y,x)$ for all $x,y\in X$;
- $\left({D}_{3}\right)$
- $D(x,z)\le s\left[D(x,y)+D(y,z)\right]$ for all $x,y,z\in X$.

**Definition**

**2**

**.**Consider a metric space $(X,d)$. Let $\rho :X\times X\to [0,+\infty )$ be a function satisfying the following conditions:

- $\left({\rho}_{1}\right)$
- $\rho (x,z)\le \rho (x,y)+\rho (y,z)$ for all $x,y,z\in X$;
- $\left({\rho}_{2}\right)$
- ρ is lower semicontinuous in its second variable; that is, if $x\in X$ and ${y}_{n}\to y$ in X then $\rho (x,y)\le {lim\; inf}_{n}\rho (x,{y}_{n})$;
- $\left({\rho}_{3}\right)$
- for each $\epsilon >0$, there exists $\delta >0$ such that $\rho (z,x)<\delta $ and $\rho (z,y)<\delta $ imply that $d(x,y)<\epsilon $.

**Definition**

**3**

**.**Consider a b-metric space $(X,D)$ with $s\ge 1$. Let $p:X\times X\to [0,+\infty )$ be a function satisfying the following conditions:

- $\left({w}_{1}\right)$
- $p(x,z)\le s\left[p\right(x,y)+p(y,z\left)\right]$ for all $x,y,z\in X$;
- $\left({w}_{2}\right)$
- p is lower b-semicontinuous in its second variable; that is, if $x\in X$ and ${y}_{n}\to y$ in X then $p(x,y)\le s{lim\; inf}_{n}p(x,{y}_{n})$;
- $\left({w}_{3}\right)$
- for each $\epsilon >0$, there exists $\delta >0$ such that $p(z,x)<\delta $ and $p(z,y)<\delta $ imply that $D(x,y)<\epsilon $.

**Theorem**

**1**

**.**Let the underlying cone of an ordered $tvs$ be normal and solid. Then $tvs$ is an ordered normed space.

**Definition**

**4**

**.**Let X be a nonempty set, $s\ge 1$ and $(E,P)$ be an ordered $tvs$. A function $d:X\times X\to P$ is said to be a $tvs$-cone b-metric if the following properties hold:

- $\left({d}_{1}\right)$
- $d(x,y)=\theta $ iff $x=y$ for all $x,y\in X$;
- $\left({d}_{2}\right)$
- $d(x,y)=d(y,x)$ for all $x,y\in X$;
- $\left({d}_{3}\right)$
- $d(x,z)\u2aafs\left[d(x,y)+d(y,z)\right]$ for all $x,y,z\in X$.

**Remark**

**1.**

**Definition**

**5**

**.**Let $\left\{{x}_{n}\right\}$ be a sequence in a $tvs$-cone b-metric space $(X,d)$. Then

- $\left\{{x}_{n}\right\}$ converges to x if for every $c\in E$ with $\theta \ll c$ there exist ${n}_{0}\in \mathbb{N}$ such that $d({x}_{n},x)\ll c$ for all $n>{n}_{0}$, and we write ${x}_{n}\to x$ as $n\to \infty $.
- $\left\{{x}_{n}\right\}$ is called a Cauchy sequence if for every $c\in E$ with $\theta \ll c$ there exist ${n}_{0}\in \mathbb{N}$ such that $d({x}_{n},{x}_{m})\ll c$ for all $m,n>{n}_{0}$.
- X is said to be complete if every Cauchy sequence in X is convergent in X.

- $(*)$
- Let $v\u2aaf\lambda v$ with $v\in P$ and $0\le \lambda <1$. Then $v=\theta $.

## 2. Main Results

**Definition**

**6.**

**Definition**

**7.**

- $\left({q}_{1}\right)$
- $q(x,z)\u2aafs\left[q\right(x,y)+q(y,z\left)\right]$ for all $x,y,z\in X$;
- $\left({q}_{2}\right)$
- $q(x,\xb7):X\to P$ is lower b-semicontinuous for all $x\in X$;
- $\left({q}_{3}\right)$
- for all $\theta \ll c$, there exists $\theta \ll e$ such that $q(z,x)\ll e$ and $q(z,y)\ll e$ implying $d(x,y)\ll c$.

**Remark**

**2.**

- $\left({q}_{2}^{\prime}\right)$
- If ${y}_{n},y\in X$, ${y}_{n}\to y$ as $n\to \infty $ and $g\left(y\right)=q(x,y)$, then $g\left(y\right)-g\left({y}_{n}\right)$ is a c-sequence.

**Remark**

**3.**

**Remark**

**4.**

**Lemma**

**1.**

**Proof.**

**Example**

**1.**

**Example**

**2.**

- $q(x,y)=q(y,x)$ is not presently true;
- $q(x,y)=\theta $ does not certainly equivalent to $x=y$.

**Lemma**

**2.**

- $\left({l}_{1}\right)$
- If $q({\alpha}_{n},\beta )\u2aaf{x}_{n}$ and $q({\alpha}_{n},\gamma )\u2aaf{y}_{n}$ for $n\in \mathbb{N}$, then $\beta =\gamma $. Specially, if $q(\alpha ,\beta )=\theta $ and $q(\alpha ,\gamma )=\theta $, then $\beta =\gamma $.
- $\left({l}_{2}\right)$
- If $q({\alpha}_{n},{\alpha}_{m})\u2aaf{x}_{n}$ for $m>n>{n}_{0}$ (for some ${n}_{0}\in \mathbb{N}$), then $\left\{{\alpha}_{n}\right\}$ is a Cauchy sequence.

**Proof.**

**Theorem**

**2.**

- (i)
- If $fy\ne y$, there exists $c\in \mathit{int}\phantom{\rule{0.166667em}{0ex}}P$ with $c\ne \theta $ such that $c\ll q(x,y)+q(x,fx)$ for each $x\in X$;
- (ii)
- f is continuous.

**Proof.**

**Remark**

**5.**

**Question:**Can the conditions (i) and (ii) be replaced by another condition in Theorem 2?

**Corollary**

**1.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Corollary**

**3.**

**Theorem**

**4.**

**Proof.**

**Remark**

**6.**

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

- (i)
- for all $x\in X$ with $x\subseteq fx$, there exists $c\in \mathrm{int}\phantom{\rule{0.166667em}{0ex}}P$ such that$$\begin{array}{c}\hfill c\ll q(x,y)+q(x,fx)\end{array}$$
- (ii)
- there is ${x}_{0}\in X$ such that ${x}_{0}\subseteq f{x}_{0}$.

**Proof.**

**Remark**

**7.**

## 3. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Babaei, R.; Rahimi, H.; De la Sen, M.; Rad, G.S.
*w*-*b*-Cone Distance and Its Related Results: A Survey. *Symmetry* **2020**, *12*, 171.
https://doi.org/10.3390/sym12010171

**AMA Style**

Babaei R, Rahimi H, De la Sen M, Rad GS.
*w*-*b*-Cone Distance and Its Related Results: A Survey. *Symmetry*. 2020; 12(1):171.
https://doi.org/10.3390/sym12010171

**Chicago/Turabian Style**

Babaei, Reza, Hamidreza Rahimi, Manuel De la Sen, and Ghasem Soleimani Rad.
2020. "*w*-*b*-Cone Distance and Its Related Results: A Survey" *Symmetry* 12, no. 1: 171.
https://doi.org/10.3390/sym12010171