# From G-Completeness to M-Completeness

^{1}

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## Abstract

**:**

## 1. Introduction and Preliminaries

**Theorem**

**1**

**Theorem**

**2**

**Definition**

**1.**

- A map $T:X\u27f6X$ is fuzzy contractive if there exists $k\in (0,1)$ such that$$\frac{1}{M(Tx,Ty,t)}}-1\le k\left({\displaystyle \frac{1}{M(x,y,t)}}-1\right),$$
- A sequence ${\left\{{x}_{n}\right\}}_{n}$ in X is fuzzy contractive if there exists $k\in ]0,1[$ such that$$\frac{1}{M({x}_{n+1},{x}_{n+2},t)}}-1\le k\left({\displaystyle \frac{1}{M({x}_{n},{x}_{n+1},t)}}-1\right),$$

**Question 1.**

**Definition**

**2**

- (T1)
- * is commutative and associative;
- (T2)
- * is continuous;
- (T3)
- $a*1=a$ for all $a\in [0,1]$;
- (T4)
- $a*b\le c*d$ when $a\le c$ and $b\le d$, with $a,b,c,d\in [0,1]$.

**Definition**

**3**

- (F1)
- $M(x,y,t)>0$;
- (F2)
- $M(x,y,t)=1$ if and only if $x=y$;
- (F3)
- $M(x,y,t)=M(y,x,t)$;
- (F4)
- $M(x,y,t)*M(y,z,s)\le M(x,z,t+s)$;
- (F5)
- $M(x,y,.):(0,+\infty )\u27f6(0,1]$ is left continuous.

**Definition**

**4**

- (i)
- A sequence ${\left\{{x}_{n}\right\}}_{n}$ converges to $x\in X$ if and only if $M({x}_{n},x,t)\u27f61$ as $n\u27f6+\infty $ for all $t>0$;
- (ii)
- A sequence ${\left\{{x}_{n}\right\}}_{n}$ in X is an M-Cauchy sequence if and only if for all $\epsilon \in (0,1)$ and $t>0$, there exists ${n}_{0}$ such that $M({x}_{n},{x}_{m},t)>1-\epsilon $ for all $m,n\ge {n}_{0}$;
- (iii)
- A sequence ${\left\{{x}_{n}\right\}}_{n}$ in X is an G-Cauchy sequence if and only if for all $p\in \mathbb{N}$, $\epsilon \in (0,1)$ and $t>0$, there exists ${n}_{0}$ such that $M({x}_{n},{x}_{n+p},t)>1-\epsilon $ for all $n\ge {n}_{0}$;
- (iv)
- The fuzzy metric space is G-complete (M-complete resp. ) if every G-Cauchy (M-Cauchy resp.) sequence converges to some $x\in X$.

**Remark**

**1.**

**Definition**

**5.**

- 1.
- We define Ψ as the set of all functions $\phi :(0,1)\times (0,1)\u27f6{\mathbb{R}}_{+}$ satisfying
- (A)
- $\phi (t,t)<1$, for all $t\in (0,1)$;
- (B)
- φ is a continuous function and satisfies $\phi (t,s)\ge 1$ implies that $t\ge s$, for all $t,s\in (0,1)$;

- 2.
- Let $\phi \in \Psi $, a sequence ${\left\{{x}_{n}\right\}}_{n}\subset X$ is said to be a φ-fuzzy contractive sequence if
- (C)
- $\phi \left(M({x}_{n},{x}_{m},t),M({x}_{n+1},{x}_{m+1},t)\right)\ge 1$, for all $n,m\in \mathbb{N}$ and all $t>0$.

**Example**

**1.**

**Example**

**2.**

**Lemma**

**1.**

- *
- ${\left\{M(x,y,{t}_{n})\right\}}_{n}$ is a decreasing and convergent sequence.
- *
- Putting$$\tilde{M}(x,y,t)=\left\{\begin{array}{ccc}\hfill M(x,y,t)& if& t\in (0,\infty );\hfill \\ \hfill \underset{n\to \infty}{lim}M(x,y,{t}_{n})& if& t=0,\hfill \end{array}\right.$$$$\underset{n\to \infty}{lim}\tilde{M}({x}_{n},{x}_{n+1},0)=1.$$

**Proof.**

**Remark**

**2.**

**Example**

**3.**

**Example**

**4.**

## 2. Results

**Theorem**

**3.**

- (I)-
- ${lim}_{n\to \infty}M({x}_{n},{x}_{n+1},t)=1$, for all $t>0$;
- (II)-
- the sequence ${\left\{{x}_{n}\right\}}_{n}$ is a G-Cauchy sequence;
- (III)-
- if ${lim}_{n\to \infty}M({x}_{n},{x}_{n+1},t)=1$ uniformly on $(0,+\infty )$, then the sequence ${\left\{{x}_{n}\right\}}_{n}$ is an M-Cauchy sequence.

**Proof.**

**Corollary**

**1.**

**Proof.**

**Theorem**

**4.**

- (I)-
- ${lim}_{n\to \infty}M({x}_{n},{x}_{n+1},t)=1$ uniformly on $(0,+\infty )$;
- (II)-
- there exists $\phi \phantom{\rule{4pt}{0ex}}in\Psi $ such that ${\left\{{x}_{n}\right\}}_{n}$ is a φ-fuzzy contractive.

**Proof.**

**Theorem**

**5**

**Proof.**

## 3. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Mecheraoui, R.; Mukheimer, A.; Radenović, S.
From G-Completeness to M-Completeness. *Symmetry* **2019**, *11*, 839.
https://doi.org/10.3390/sym11070839

**AMA Style**

Mecheraoui R, Mukheimer A, Radenović S.
From G-Completeness to M-Completeness. *Symmetry*. 2019; 11(7):839.
https://doi.org/10.3390/sym11070839

**Chicago/Turabian Style**

Mecheraoui, Rachid, Aiman Mukheimer, and Stojan Radenović.
2019. "From G-Completeness to M-Completeness" *Symmetry* 11, no. 7: 839.
https://doi.org/10.3390/sym11070839