# Several Fixed Point Theorems in Convex b-Metric Spaces and Applications

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1**

- (1)
- ${d}_{b}(u,v)=0$ if and only if $u=v$;
- (2)
- ${d}_{b}(u,v)={d}_{b}(v,u)$;
- (3)
- ${d}_{b}(u,v)\le s[{d}_{b}(u,o)+{d}_{b}(o,v)]$.

**Definition**

**2**

- (1)
- The sequence $\left\{{u}_{n}\right\}$ is said to be convergent in $(H,{d}_{b})$ if there exists ${u}^{*}\in H$ such that $\underset{n\to \infty}{lim}{d}_{b}({u}_{n},{u}^{*})=0$.
- (2)
- The sequence $\left\{{u}_{n}\right\}$ is said to be a Cauchy sequence in $(H,{d}_{b})$, if for every $\epsilon >0$ there exists a positive ${n}_{0}\in \mathbb{N}$ such that ${d}_{b}({u}_{n},{u}_{m})<\epsilon $ for all $n,m>{n}_{0}$ (or, equivalently, $\underset{n,m\to \infty}{lim}{d}_{b}({u}_{n},{u}_{m})=0$).
- (3)
- $(H,{d}_{b})$ is called a complete b-metric space if every Cauchy sequence is convergent in H.

**Definition**

**3**

## 3. Main Results

**Definition**

**4.**

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Theorem**

**1.**

**Proof.**

**Example**

**4.**

**Theorem**

**2.**

**Proof.**

**Example**

**5.**

**Proof.**

**Lemma**

**1**

**Definition**

**5**

**Definition**

**6.**

**Remark**

**1.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

## 4. Applications

**Theorem**

**5.**

**Proof.**

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Chen, L.; Li, C.; Kaczmarek, R.; Zhao, Y.
Several Fixed Point Theorems in Convex *b*-Metric Spaces and Applications. *Mathematics* **2020**, *8*, 242.
https://doi.org/10.3390/math8020242

**AMA Style**

Chen L, Li C, Kaczmarek R, Zhao Y.
Several Fixed Point Theorems in Convex *b*-Metric Spaces and Applications. *Mathematics*. 2020; 8(2):242.
https://doi.org/10.3390/math8020242

**Chicago/Turabian Style**

Chen, Lili, Chaobo Li, Radoslaw Kaczmarek, and Yanfeng Zhao.
2020. "Several Fixed Point Theorems in Convex *b*-Metric Spaces and Applications" *Mathematics* 8, no. 2: 242.
https://doi.org/10.3390/math8020242