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# A new approach to *G*-metric and related fixed point theorems

- Mehdi Asadi
^{1}, - Erdal Karapınar
^{2}Email author and - Peyman Salimi
^{3}

**2013**:454

https://doi.org/10.1186/1029-242X-2013-454

© Asadi et al.; licensee Springer. 2013

**Received:**17 July 2013**Accepted:**16 September 2013**Published:**7 November 2013

## Abstract

Very recently, Samet *et al.* and Jleli and Samet reported that most of fixed point results in the context of *G*-metric space, defined by Sims and Zead, can be derived from the usual fixed point theorems on the usual metric space. In this paper, we state and prove some fixed point theorems in the framework of *G*-metric space that cannot be obtained from the existence results in the context of associated metric space.

## Keywords

- Fixed Point Theorem
- Existence Theorem
- Cauchy Sequence
- Unique Fixed Point
- Expansive Mapping

## 1 Introduction and preliminaries

*G*-metric and investigated the topology of such spaces. The authors also characterized some celebrated fixed point results in the context of

*G*-metric space. Following this initial paper, a number of authors have published many fixed point results on the setting of

*G*-metric space (see,

*e.g.*, [1–33] and the references therein). Samet

*et al.*[24] and Jleli and Samet [25] reported that some published results can be considered as a straight consequence of the existence theorem in the setting of the usual metric space. More precisely, the authors of these two papers noticed that $p(x,y)={p}_{G}(x,y)=G(x,y,y)$ is a quasi-metric whenever $G:X\times X\times X\to [0,\mathrm{\infty})$ is a

*G*-metric. It is evident that each quasi-metric induces a metric. In particular, if the pair $(X,p)$ is a quasi-metric space, then the function defined by

forms a metric on *X*.

The object of this paper is to get some fixed point results in the context of *G*-metric space that cannot be concluded from the existence results. This paper can be considered as a continuation of [27], which was inspired by [26].

First, we recollect some necessary definitions and results in this direction. The notion of *G*-metric spaces is defined as follows.

**Definition 1.1** (See [1])

*X*be a non-empty set, $G:X\times X\times X\to {\mathbb{R}}^{+}$ be a function satisfying the following properties:

- (G1)
$G(x,y,z)=0$ if $x=y=z$,

- (G2)
$0<G(x,x,y)$ for all $x,y?X$ with $x?y$,

- (G3)
$G(x,x,y)=G(x,y,z)$ for all $x,y,z?X$ with $y?z$,

- (G4)
$G(x,y,z)=G(x,z,y)=G(y,z,x)=?$ (symmetry in all three variables),

- (G5)
$G(x,y,z)=G(x,a,a)+G(a,y,z)$ (rectangle inequality) for all $x,y,z,a?X$.

Then function *G* is called a generalized metric or, more specifically, a *G*-metric on *X*, and the pair $(X,G)$ is called a *G*-metric space.

*G*-metric on

*X*induces a metric ${d}_{G}$ on

*X*defined by

For a better understanding of the subject, we give the following examples of *G*-metrics.

**Example 1.1**Let $(X,d)$ be a metric space. Function $G:X\times X\times X\to [0,+\mathrm{\infty})$, defined by

for all $x,y,z\in X$, is a *G*-metric on *X*.

**Example 1.2** (See, *e.g.*, [1])

for all $x,y,z\in X$, is a *G*-metric on *X*.

In their initial paper, Mustafa and Sims [1] also defined the basic topological concepts in *G*-metric spaces as follows.

**Definition 1.2** (See [1])

*G*-metric space, and let $\{{x}_{n}\}$ be a sequence of points of

*X*. We say that $\{{x}_{n}\}$ is

*G*-convergent to $x\in X$ if

that is, for any $\epsilon >0$, there exists $N\in \mathbb{N}$ such that $G(x,{x}_{n},{x}_{m})<\epsilon $ for all $n,m\ge N$. We call *x* the limit of the sequence and write ${x}_{n}\to x$ or ${lim}_{n\to +\mathrm{\infty}}{x}_{n}=x$.

**Proposition 1.1** (See [1])

*Let*$(X,G)$

*be a*

*G*-

*metric space*.

*The following are equivalent*:

- (1)
$\{{x}_{n}\}$

*is**G*-*convergent to**x*, - (2)
$G({x}_{n},{x}_{n},x)\to 0$

*as*$n\to +\mathrm{\infty}$, - (3)
$G({x}_{n},x,x)\to 0$

*as*$n\to +\mathrm{\infty}$, - (4)
$G({x}_{n},{x}_{m},x)\to 0$

*as*$n,m\to +\mathrm{\infty}$.

**Definition 1.3** (See [1])

Let $(X,G)$ be a *G*-metric space. Sequence $\{{x}_{n}\}$ is called a *G*-Cauchy sequence if, for any $\epsilon >0$, there exists $N\in \mathbb{N}$ such that $G({x}_{n},{x}_{m},{x}_{l})<\epsilon $ for all $m,n,l\ge N$, that is, $G({x}_{n},{x}_{m},{x}_{l})\to 0$ as $n,m,l\to +\mathrm{\infty}$.

**Proposition 1.2** (See [1])

*Let*$(X,G)$

*be a*

*G*-

*metric space*.

*Then the following are equivalent*:

- (1)
*sequence*$\{{x}_{n}\}$*is**G*-*Cauchy*, - (2)
*for any*$\epsilon >0$,*there exists*$N\in \mathbb{N}$*such that*$G({x}_{n},{x}_{m},{x}_{m})<\epsilon $*for all*$m,n\ge N$.

**Definition 1.4** (See [1])

A *G*-metric space $(X,G)$ is called *G*-complete if every *G*-Cauchy sequence is *G*-convergent in $(X,G)$.

**Definition 1.5** Let $(X,G)$ be a *G*-metric space. Mapping $F:X\times X\times X\to X$ is said to be continuous if for any three *G*-convergent sequences $\{{x}_{n}\}$, $\{{y}_{n}\}$ and $\{{z}_{n}\}$ converging to *x*, *y* and *z*, respectively, $\{F({x}_{n},{y}_{n},{z}_{n})\}$ is *G*-convergent to $F(x,y,z)$.

Mustafa [4] extended the well-known Banach [34] contraction principle mapping in the framework of *G*-metric spaces as follows.

**Theorem 1.1** (See [4])

*Let*$(X,G)$

*be a complete*

*G*-

*metric space and*$T:X\to X$

*be a mapping satisfying the following condition for all*$x,y,z\in X$:

*where* $k\in [0,1)$. *Then* *T* *has a unique fixed point*.

**Theorem 1.2** (See [4])

*Let*$(X,G)$

*be a complete*

*G*-

*metric space and*$T:X\to X$

*be a mapping satisfying the following condition for all*$x,y\in X$:

*where* $k\in [0,1)$. *Then* *T* *has a unique fixed point*.

**Remark 1.1** We notice that condition (2) implies condition (3). The converse is true only if $k\in [0,\frac{1}{2})$. For details see [4].

**Lemma 1.1** [4]

*By the rectangle inequality*(G5)

*together with the symmetry*(G4),

*we have*

## 2 Main results

**Theorem 2.1**

*Let*$(X,G)$

*be a complete*

*G*-

*metric space and*$T:X\to X$

*be a mapping satisfying the following condition for all*$x,y\in X$:

*where* $k\in [0,1)$. *Then* *T* *has a unique fixed point*.

*Proof*Let ${x}_{0}\in X$ be an arbitrary point, and define the sequence ${x}_{n}$ by ${x}_{n}={T}^{n}({x}_{0})$. By (5), we have

and so, $limG({x}_{n},{x}_{m},{x}_{m})=0$, as $n,m\to \mathrm{\infty}$. Thus, $\{{x}_{n}\}$ is *G*-Cauchy sequence. Due to the completeness of $(X,G)$, there exists $u\in X$ such that $\{{x}_{n}\}$ is *G*-convergent to *u*.

*G*is continuous, then

This contradiction implies that $u=Tu$.

which implies that $u=v$. □

**Example 2.1**Let $X=[0,\mathrm{\infty})$ and

*G*-metric on

*X*. Define $T:X\to X$ by $Tx=\frac{1}{5}x$. Then the condition of Theorem 2.1 holds. In fact,

That is, conditions of Theorem 2.1 hold for this example.

**Corollary 2.1**

*Let*$(X,G)$

*be a complete*

*G*-

*metric space and*$T:X\to X$

*be a mapping satisfying the following condition for all*$x,y,z\in X$:

*where* $0\le a+b<1$. *Then* *T* *has a unique fixed point*.

**Theorem 2.2**

*Let*$(X,G)$

*be a complete*

*G*-

*metric space and*$T:X\to X$

*be a mapping satisfying the following condition for all*$x,y\in X$,

*where*$a+b+c+d<1$

*Then* *T* *has a unique fixed point*.

*Proof*Take ${x}_{0}\in X$. We construct sequence ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$ of points in

*X*in the following way:

*T*has a fixed point. Thus, we suppose that

for all $n\in \mathbb{N}$.

*G*-Cauchy sequence. Due to the completeness of $(X,G)$, there exists $u\in X$ such that $\{{x}_{n}\}$ is

*G*-convergent to

*z*. From (12), with $x={x}_{n}$ and $y=z$, we have

*T*has a fixed point. Hence, we assume that $Tz\ne {T}^{2}z$. Therefore, by (G3), we get

which implies that $G(z,Tz,{T}^{2}z)=0$, *i.e.*, $z=Tz={T}^{2}z$. □

where $\psi (t)=\varphi (t)=0$ if and only if $t=0$.

**Theorem 2.3**

*Let*$(X,G)$

*be a complete*

*G*-

*metric space and*$T:X\to X$

*be a mapping satisfying the following condition for all*$x,y\in X$,

*where*$\psi \in \mathrm{\Psi}$

*and*$\varphi \in \mathrm{\Phi}$

*holds*

*Then* *T* *has a unique fixed point*.

*Proof*Take ${x}_{0}\in X$. We construct sequence ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$ of points in

*X*in the following way:

*T*has a fixed point. Thus, we suppose that

for all $n\in \mathbb{N}$.

*i.e.*,

*G*-Cauchy sequence. Suppose, to the contrary, that there exists $\epsilon >0$, and sequence ${x}_{n(k)}$ of ${x}_{n}$ such that

*G*-complete, then there exist $z\in X$ such that ${x}_{n}\to z$ as $n\to \mathrm{\infty}$. From (15), with $x={x}_{n}$ and $y=z$, we have

*i.e.*, $z=Tz$. To prove uniqueness, suppose that $z\ne u$, such that $Tu=u$. Now, by (15), we get

which implies that $\varphi (G(z,Tz,u))=0$, *i.e.*, $z=u$. □

If we take $\psi (t)=t$ and $\varphi (t)=(1-r)t$ in Theorem 2.3, where $0\le r<1$, then we deduce the following corollary.

**Corollary 2.2**

*Let*$(X,G)$

*be a complete*

*G*-

*metric space and*$T:X\to X$

*be a mapping satisfying the following condition for all*$x,y\in X$,

*where*$0\le r<1$

*holds*

*Then* *T* *has a unique fixed point*.

**Example 2.2**Let $X=[0,\mathrm{\infty})$ and

*G*-metric on

*X*. Define $T:X\to X$ by $Tx=\frac{1}{4}x$. Then all the conditions of Corollary 2.2 (Theorem 2.3) hold. Indeed,

That is, the conditions of Corollary 2.2 (Theorem 2.3) hold for this example.

**Corollary 2.3**

*Let*$(X,G)$

*be a complete*

*G*-

*metric space and*$T:X\to X$

*be a mapping satisfying the following condition for all*$x,y,z\in X$,

*where*$0\le a+b<2$

*holds*

*Then* *T* *has a unique fixed point*.

*Proof*By taking $y=z$, we get

where $0\le \frac{(a+b)}{2}<1$. That is, conditions of Theorem 2.3 hold, and *T* has a unique fixed point. □

## 3 Fixed point results for expansive mappings

In this section, we establish some fixed point results for expansive mappings.

**Theorem 3.1**

*Let*$(X,G)$

*be a complete*

*G*-

*metric space and*$T:X\to X$

*be an onto mapping satisfying the following condition for all*$x,y\in X$,

*where*$\alpha >1$

*holds*

*Then* *T* *has a unique fixed point*.

*Proof*Let ${x}_{0}\in X$, since

*T*is onto, then there exists ${x}_{1}\in X$ such that ${x}_{0}=T{x}_{1}$. By continuing this process, we get ${x}_{n}=T{x}_{n+1}$ for all $n\in \mathbb{N}\cup 0$. In case ${x}_{{n}_{0}}={x}_{{n}_{0}+1}$, for some ${n}_{0}\in \mathbb{N}\cup 0$, then it is clear that ${x}_{{n}_{0}}$ is a fixed point of

*T*. Now, assume that ${x}_{n}\ne {x}_{n+1}$ for all

*n*. From (30), with $x={x}_{n+1}$ and $y={x}_{n}$, we have

*T*is onto, then there exists $w\in X$ such that $z=Tw$. From (30), with $x={x}_{n+1}$ and $y=w$, we have

which is a contradiction. Hence, $u=v$. □

**Theorem 3.2**

*Let*$(X,G)$

*be a complete*

*G*-

*metric space and*$T:X\to X$

*be a mapping satisfying the following condition for all*$x,y\in X$,

*where*$a>1$

*Then* *T* *has a unique fixed point*.

*Proof*Let ${x}_{0}\in X$, since

*T*is onto, then there exists ${x}_{1}\in X$ such that ${x}_{0}=T{x}_{1}$. By continuing this process, we get ${x}_{n}=T{x}_{n+1}$ for all $n\in \mathbb{N}\cup 0$. In case ${x}_{{n}_{0}}={x}_{{n}_{0}+1}$, for some ${n}_{0}\in \mathbb{N}\cup 0$, then it is clear that ${x}_{{n}_{0}}$ is a fixed point of

*T*. Now, assume that ${x}_{n}\ne {x}_{n+1}$ for all

*n*. From (34), with $x={x}_{n+1}$ and $y={x}_{n}$, we have

*G*-metric space, then there exists $z\in X$ such that ${x}_{n}\to z$ as $n\to \mathrm{\infty}$. Consequently, since

*T*is onto, then there exists $w\in X$ such that $z=Tw$. From (34), with $x=w$ and $y={x}_{n+1}$, we have

Taking limit as $n\to \mathrm{\infty}$ in the inequality above, we have $G(w,Tw,{T}^{2}w)=0$. That is, $w=Tw={T}^{2}w$. To prove the uniqueness, suppose that $u\ne v$ such that $Tv=v$ and $Tu=u$. □

## Declarations

### Acknowledgements

The authors thank to anonymous referees for their remarkable comments, suggestions and ideas that helped to improve this paper.

## Authors’ Affiliations

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