# Some Globally Stable Fixed Points in b-Metric Spaces

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

- 1.
**Reflexive**if $\phantom{\rule{3.33333pt}{0ex}}x\u2aafx,\phantom{\rule{3.33333pt}{0ex}}\forall x\in X$.- 2.
**Antisymmetric**if $x\u2aafy$ and $y\u2aafx\u27f9x=y,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\forall x,y\in X.$- 3.
**Transitive**if $x\u2aafy$ and $y\u2aafz\u27f9x\u2aafz,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\forall x,y,z\in X.$

**partial order**if it satisfies all of the above conditions (1–3), we call the pair (X,$\u2aaf)$ a partial ordered set.

**Definition**

**2.**

**Regular**if whenever $x\u2aafy\u2aafz$, then $max\left\{\mathsf{\Psi}\right(x,y),\mathsf{\Psi}(y,z\left)\right\}\le \mathsf{\Psi}(x,z),\phantom{\rule{3.33333pt}{0ex}}\forall x,y,z\in X$, where $(X,\u2aaf)$ is an ordered space, max function is from ${\mathbb{R}}_{+}\times {\mathbb{R}}_{+}$ to ${\mathbb{R}}_{+}$.

**Definition**

**3.**

**Definition**

**4.**

**metric**on X is a function $d:X\times X\to {\mathbb{R}}_{+}$ such that,

**Definition**

**5.**

**b-metric**on X is a function ${d}_{b}:X\times X\to {\mathbb{R}}_{+}$ such that,

**Example**

**1.**

**Lemma**

**1.**

**Example**

**2.**

**Proof.**

**Definition**

**6.**

**asymptotically contractive in a b-metric space $(X,{d}_{b})$**if

**Definition**

**7.**

**Example**

**3.**

**Definition**

**8.**

- 1.
- ψ is continuous and nondecreasing.
- 2.
- $\psi \left(t\right)=0$ iff $t=0.$

**Definition**

**9.**

## 3. Main Results

**Assumption**

**A1.**

**Assumption**

**A2.**

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1.**

**Theorem**

**2.**

**Proof**

**of**

**Theorem**

**2.**

**The forward case:**Let ${d}_{b}({x}_{n},\widehat{x})\to 0$, we have

**The backward case:**Let ${d}_{b}({x}_{n},T{x}_{n})\to 0$, we have

**Theorem**

**3.**

**Proof**

**of**

**Theorem**

**3.**

**Corollary**

**1.**

**Corollary**

**2.**

**Theorem**

**4.**

**Proof**

**of**

**Theorem**

**4.**

**Theorem**

**5.**

**Proof.**

**Corollary**

**3.**

**Corollary**

**4.**

**Corollary**

**5.**

**Corollary**

**6.**

- 1.
- $\psi :[0,+\infty )\to [0,+\infty )$ is an altering distance function.
- 2.
- $\varphi :[0,+\infty )\times [0,+\infty )\to [0,+\infty )$ is a continuous function with $\varphi (x,y)=0$ if and only if $x=y=0$.

**Corollary**

**7.**

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Batsari, U.Y.; Kumam, P.; Sitthithakerngkiet, K.
Some Globally Stable Fixed Points in *b*-Metric Spaces. *Symmetry* **2018**, *10*, 555.
https://doi.org/10.3390/sym10110555

**AMA Style**

Batsari UY, Kumam P, Sitthithakerngkiet K.
Some Globally Stable Fixed Points in *b*-Metric Spaces. *Symmetry*. 2018; 10(11):555.
https://doi.org/10.3390/sym10110555

**Chicago/Turabian Style**

Batsari, Umar Yusuf, Poom Kumam, and Kanokwan Sitthithakerngkiet.
2018. "Some Globally Stable Fixed Points in *b*-Metric Spaces" *Symmetry* 10, no. 11: 555.
https://doi.org/10.3390/sym10110555