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Remarks on some coupled fixed point theorems in Gmetric spaces
Fixed Point Theory and Applicationsvolume 2013, Article number: 2 (2013)
Abstract
In this paper, we show that, unexpectedly, most of the coupled fixed point theorems in the context of (ordered) Gmetric spaces are in fact immediate consequences of usual fixed point theorems that are either well known in the literature or can be obtained easily.
MSC:47H10, 54H25.
1 Introduction
Investigation of the existence and uniqueness of fixed points of certain mappings in the framework of metric spaces is one of the centers of interests in nonlinear functional analysis. The Banach contraction mapping principle [1] is the limelight result in this direction: A self mapping T on a complete metric space $(X,d)$ has a unique fixed point if there exists $0\le k<1$ such that $d(Tx,Ty)=kd(x,y)$ for all $x,y\in X$. Fixed point theory has a wide application in almost all fields of quantitative sciences such as economics, biology, physics, chemistry, computer science and many branches of engineering. It is quite natural to consider various generalizations of metric spaces in order to address the needs of these quantitative sciences. In this respect quasimetric spaces, ultrametric spaces, uniform spaces, fuzzy metric spaces, partial metric spaces, cone metric spaces and bmetric spaces can be listed as wellknown examples (see e.g. [2–6]). Consequently, the concept of a Gmetric space was introduced by Mustafa and Sims [7] in 2004. The authors discussed the topological properties of this space and proved the analog of the Banach contraction mapping principle in the context of Gmetric spaces (see e.g. [8–15]).
On the other hand, Ran and Reuring [16] proved the existence and uniqueness of a fixed point of a contraction mapping in partially ordered complete metric spaces. Following this initial work, a number of authors have investigated fixed points of various mappings and their applications in the theory of differential equations (see, e.g., [17–35]). Afterwords, GnanaBhaskar and Lakshmikantham [22] proved the existence and uniqueness of a coupled fixed point (defined by Guo and Laksmikantham [36]) in the context of partially ordered metric spaces by introducing the notion of mixed monotone property. In this remarkable paper, GnanaBhaskar and Lakshmikantham [22] also gave some applications related to the existence and uniqueness of a solution of periodic boundary value problems. Following this trend, many authors have studied the (common) coupled fixed points (see, e.g., [17–28, 31, 37–47]).
In this paper, we show that, unexpectedly, most of the coupled fixed point theorems in the context of (ordered) Gmetric spaces are in fact immediate consequences of wellknown fixed point theorems in the literature.
2 Preliminaries
We start with basic definitions and a detailed overview of the essential results developed in the interesting works mentioned above. Throughout this paper, ℕ is the set of nonnegative integers, and ${\mathbb{N}}^{\ast}$ is the set of positive integers.
Definition 2.1 (See [7])
Let X be a nonempty set and $G:X\times X\times X\to {\mathbb{R}}^{+}$ 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 $x,y\in X$ with $x\ne y$,

(G3)
$G(x,x,y)\le G(x,y,z)$ for all $x,y,z\in X$ with $y\ne z$,

(G4)
$G(x,y,z)=G(x,z,y)=G(y,z,x)=\cdots $ (symmetry in all three variables),

(G5)
$G(x,y,z)\le G(x,a,a)+G(a,y,z)$ for all $x,y,z,a\in X$ (rectangle inequality).
Then the function G is called a generalized metric or, more specially, a Gmetric on X, and the pair $(X,G)$ is called a Gmetric space.
Every Gmetric on X defines a metric ${d}_{G}$ on X by
Example 2.1 Let $(X,d)$ be a metric space. The function $G:X\times X\times X\to [0,+\mathrm{\infty})$, defined as
or
for all $x,y,z\in X$, is a Gmetric on X.
Definition 2.2 (See [7])
Let $(X,G)$ be a Gmetric space, and let $\{{x}_{n}\}$ be a sequence of points of X. We say that $\{{x}_{n}\}$ is Gconvergent 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 2.1 (See [7])
Let $(X,G)$ be a Gmetric space. The following are equivalent:

(1)
$\{{x}_{n}\}$ is Gconvergent 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 2.3 (See [7])
Let $(X,G)$ be a Gmetric space. A sequence $\{{x}_{n}\}$ is called a GCauchy sequence if, for any $\epsilon >0$, there is $N\in \mathbb{N}$ 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 2.2 (See [7])
Let $(X,G)$ be a Gmetric space. Then the following are equivalent:

(1)
the sequence $\{{x}_{n}\}$ is GCauchy,

(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 2.4 (See [7])
A Gmetric space $(X,G)$ is called Gcomplete if every GCauchy sequence is Gconvergent in $(X,G)$.
We will use the following result which can be easily derived from the definition of Gmetric space (see, e.g., [7]).
Lemma 2.1 Let $(X,G)$ be a Gmetric space. Then
Definition 2.5 (See [7])
Let $(X,G)$ be a Gmetric space. A mapping $T:X\to X$ is said to be Gcontinuous if $\{T({x}_{n})\}$ is Gconvergent to $T(x)$ where $\{{x}_{n}\}$ is any Gconvergent sequence converging to x.
We characterize this definition for a mapping $F:X\times X\to X$. A mapping $F:X\times X\to X$ is said to be continuous if $\{F({x}_{n},{y}_{n})\}$ is Gconvergent to $F(x,y)$ where $\{{x}_{n}\}$ and $\{{y}_{n}\}$ are any two Gconvergent sequences converging to x and y, respectively.
Definition 2.6 Let $(X,\u2aaf)$ be a partially ordered set, $(X,G)$ be a Gmetric space and $g:X\to X$ be a mapping. A partially ordered Gmetric space, $(X,G,\u2aaf)$, is called gordered complete if for each convergent sequence ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}\subset X$, the following conditions hold:

(OC_{1}) if $\{{x}_{n}\}$ is a nonincreasing sequence in X such that ${x}_{n}\to {x}^{\ast}$, then $g{x}^{\ast}\u2aafg{x}_{n}$ $\mathrm{\forall}n\in \mathbb{N}$,

(OC_{2}) if $\{{y}_{n}\}$ is a nondecreasing sequence in X such that ${y}_{n}\to {y}^{\ast}$, then $g{y}^{\ast}\u2ab0g{y}_{n}$ $\mathrm{\forall}n\in \mathbb{N}$.
In particular, a partially ordered Gmetric space, $(X,G,\u2aaf)$, is called ordered complete when g is equal to an identity mapping in (OC_{1}) and (OC_{2}).
In [48], Mustafa characterized the wellknown Banach contraction principle mapping in the context of Gmetric spaces in the following ways.
Theorem 2.1 (See [48])
Let $(X,G)$ be a complete Gmetric 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 2.2 (See [48])
Let $(X,G)$ be a complete Gmetric 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 2.1 The condition (2) implies the condition (3). The converse is true only if $k\in [0,\frac{1}{2})$. For details, see [48].
In 1987, Guo and Lakshmikantham [36] introduced the notion of coupled fixed point. The concept of coupled fixed point was reconsidered by GnanaBhaskar and Lakshmikantham [22] in 2006. In this paper, they proved the existence and uniqueness of a coupled fixed point of an operator $F:X\times X\to X$ on a partially ordered metric space under a condition called the mixed monotone property.
Definition 2.7 ([22])
Let $(X,\u2aaf)$ be a partially ordered set and $F:X\times X\to X$. The mapping F is said to have the mixed monotone property if $F(x,y)$ is monotone nondecreasing in x and monotone nonincreasing in y; that is, for any $x,y\in X$,
and
Definition 2.8 ([22])
An element $(x,y)\in X\times X$ is called a coupled fixed point of the mapping $F:X\times X\to X$ if
The results in [22] were extended by Ćirić and Lakshmikantham in [23] by defining the mixed gmonotone property.
Definition 2.9 Let $(X,\u2aaf)$ be a partially ordered set, $F:X\times X\to X$ and $g:X\to X$. The function F is said to have the mixed gmonotone property if $F(x,y)$ is monotone gnondecreasing in x and is monotone gnonincreasing in y; that is, for any $x,y\in X$,
and
It is clear that Definition 2.9 reduces to Definition 2.7 when g is the identity.
Definition 2.10 An element $(x,y)\in X\times X$ is called a coupled coincidence point of mappings $F:X\times X\to X$ and $g:X\to X$ if
and is called a coupled common fixed point of F and g if
The mappings F and g are said to commute if
for all $x,y\in X$.
3 Auxiliary results
We first state the following theorem about the existence and uniqueness of a common fixed point which can be considered as a generalization of Theorem 2.1.
Theorem 3.1 Let $(X,G)$ be a Gmetric space. Let $T:X\to X$ and $g:X\to X$ be two mappings such that
for all x, y, z. Assume that T and g satisfy the following conditions:

(A1)
$T(X)\subset g(X)$,

(A2)
$g(X)$ is Gcomplete,

(A3)
g is Gcontinuous and commutes with T.
If $k\in [0,1)$, then there is a unique $x\in X$ such that $gx=Tx=x$.
Proof Let ${x}_{0}\in X$. By assumption (A1), there exists ${x}_{1}\in X$ such that $T{x}_{0}=g{x}_{1}$. By the same arguments, there exists ${x}_{2}\in X$ such that $T{x}_{1}=g{x}_{2}$. Inductively, we define a sequence $\{{x}_{n}\}$ in the following way:
Due to (6), we have
by taking $x=y={x}_{n+1}$ and $z={x}_{n}$. Thus, for each natural number n, we have
We will show that $\{g{x}_{n}\}$ is a Cauchy sequence. By the rectangle inequality, we have for $m>n$
Letting $n,m\to \mathrm{\infty}$ in (9), we get that $G(g{x}_{m},g{x}_{m},g{x}_{n})\to 0$. Hence, $\{g{x}_{n}\}$ is a GCauchy sequence in $g(X)$. Since $(g(X),G)$ is Gcomplete, then there exists $z\in X$ such that $\{g{x}_{n}\}\to z$. Since g is Gcontinuous, we have $\{gg{x}_{n}\}$ is Gconvergent to gz. On the other hand, we have $gg{x}_{n+1}=gT{x}_{n}=Tg{x}_{n}$ since g and T commute. Thus,
Letting $n\to \mathrm{\infty}$ and using the fact that the metric G is continuous, we get that
Hence $gz=Tz$. The sequence $\{g{x}_{n+1}\}$ is Gconvergent to z since $\{g{x}_{n+1}\}$ is a subsequence of $\{g{x}_{n}\}$. So, we have
Letting $n\to \mathrm{\infty}$ and using the fact that G is continuous, we obtain that
Hence we have $z=gz=Tz$. We will show that z is the unique common fixed point of T and g. Suppose that, contrary to our claim, there exists another common fixed point $w\in X$ with $w\ne z$. From (6) we have
which is a contradiction since $k<1$. Hence, the common fixed point of T and g is unique. □
Theorem 3.2 Let $(X,G)$ be a Gmetric space. Let $T:X\to X$ and $g:X\to X$ be two mappings such that
for all x, y. Assume that T and g satisfy the following conditions:

(A1)
$T(X)\subset g(X)$,

(A2)
$g(X)$ is Gcomplete,

(A3)
g is Gcontinuous and commutes with T.
If $k\in [0,1)$, then there is a unique $x\in X$ such that $gx=Tx=x$.
Proof Following the lines of the proof of Theorem 3.1 by taking $y=z$, one can easily get the result. □
In [16], Ran and Reurings established the following fixed point theorem that extends the Banach contraction principle to the setting of ordered metric spaces.
Theorem 3.3 (Ran and Reurings [16])
Let $(X,\u2aaf)$ be an ordered set endowed with a metric d and $T:X\to X$ be a given mapping. Suppose that the following conditions hold:

(i)
$(X,d)$ is complete;

(ii)
T is continuous and nondecreasing (with respect to ⪯);

(iii)
there exists ${x}_{0}\in X$ such that ${x}_{0}\u2aafT{x}_{0}$;

(iv)
there exists a constant $k\in (0,1)$ such that for all $x,y\in X$ with $x\u2ab0y$,
$$d(Tx,Ty)\le kd(x,y).$$
Then T has a fixed point. Moreover, if for all $(x,y)\in X\times X$ there exists $z\in X$ such that $x\u2aafz$ and $y\u2aafz$, we obtain uniqueness of the fixed point.
The result of Ran and Reurings [16] can be also proved in the framework of a Gmetric space.
Theorem 3.4 Let $(X,\u2aaf)$ be an ordered set endowed with a Gmetric and $T:X\to X$ be a given mapping. Suppose that the following conditions hold:

(i)
$(X,G)$ is Gcomplete;

(ii)
T is Gcontinuous and nondecreasing (with respect to ⪯);

(iii)
there exists ${x}_{0}\in X$ such that ${x}_{0}\u2aafT{x}_{0}$;

(iv)
there exists a constant $k\in (0,1)$ such that for all $x,y,z\in X$ with $x\u2ab0y\u2ab0z$,
$$G(Tx,Ty,Tz)\le kG(x,y,z).$$(11)
Then T has a fixed point. Moreover, if for all $(x,y)\in X\times X$ there exists $w\in X$ such that $x\u2aafw$ and $y\u2aafw$, we obtain uniqueness of the fixed point.
Proof Let ${x}_{0}\in X$ be a point satisfying (iii), that is, ${x}_{0}\u2aafT{x}_{0}$. We define a sequence $\{{x}_{n}\}$ in X as follows:
Regarding that T is a nondecreasing mapping together with (12), we have ${x}_{0}\u2aafT{x}_{0}={x}_{1}$ implies ${x}_{1}=T{x}_{0}\u2aafT{x}_{1}={x}_{2}$. Inductively, we obtain
Assume that there exists ${n}_{0}$ such that ${x}_{{n}_{0}}={x}_{{n}_{0}+1}$. Since ${x}_{{n}_{0}}={x}_{{n}_{0}+1}=T{x}_{{n}_{0}}$, then ${x}_{{n}_{0}}$ is the fixed point of T, which completes the existence part of the proof. Suppose that ${x}_{n}\ne {x}_{n+1}$ for all $n\in \mathbb{N}$. Thus, by (13) we have
Put $x=y={x}_{n}$ and $z={x}_{n1}$ in (11). Then
Then we have
which, upon letting $n\to \mathrm{\infty}$, implies
On the other hand, by Lemma 2.1 we have
The inequality (17) with $x={x}_{n}$ and $y={x}_{n1}$ becomes
Letting $n\to \mathrm{\infty}$ in (18), we get
We will show that the sequence $\{{x}_{n}\}$ is a Cauchy sequence in the metric space $(X,{d}_{G})$ where ${d}_{G}$ is given in (1). For $n\ge l$ we have
and making use of (15) and (18), we obtain
Hence,
that is, the sequence $\{{x}_{n}\}$ is Cauchy in $(X,{d}_{G})$ and hence $\{{x}_{n}\}$ is GCauchy in $(X,G)$ (see Proposition 9 in [7]). Since the space $(X,G)$ is Gcomplete, then $(X,{d}_{G})$ is complete (see Proposition 10 in [7]). Thus, $\{{x}_{n}\}$ is Gconvergent to a number, say $u\in X$, that is,
We show now that $u\in X$ is a fixed point of T, that is, $u=Tu$. By the Gcontinuity of T, the sequence $\{T{x}_{n}\}=\{{x}_{n+1}\}$ converges to Tu, that is,
The rectangle inequality on the other hand gives
Passing to limit as $n\to \mathrm{\infty}$ in (25), we conclude that $G(u,Tu,Tu)=0$. Hence, $u=Tu$, that is, u∈ is a fixed point of T.
To prove the uniqueness, we assume that $v\in X$ is another fixed point of T such that $v\ne u$. We examine two cases. For the first case, assume that either $v\u2aafu$ or $u\u2aafv$. Then we substitute $x=u$ and $y=z=v$ in (11) which yields $G(Tv,Tu,Tu)\le kG(u,v,v)$. This is true only for $k=1$, but $k\in (0,1)$ by definition. Thus, the fixed point of T is unique.
For the second case, we suppose that neither $v\u2aafu$ nor $u\u2aafv$ holds. Then by assumption (iv), there exists $w\in X$ such that $u\u2aafw$ and $v\u2aafw$. Substituting $x=y=w$ and $z=u$ in (11), we get that $G(Tw,Tw,Tu)=G(Tw,Tw,u)\le kG(w,w,u)$. Since T is nondecreasing, $Tu\u2aafTw$. Substitute now $x=y=Tw$ and $z=Tu$, which implies $G({T}^{2}w,{T}^{2}w,{T}^{2}u)\le kG(Tw,Tw,Tu)\le {k}^{2}G(w,w,u)$. Continuing in this way, we conclude $G({T}^{n}w,{T}^{n}w,u)\le {k}^{n}G(w,w,u)$. Passing to limit as $n\to \mathrm{\infty}$, we get
Similarly, if we take $x=y=w$ and $z=v$ in (11), then we obtain
From (26) and (27), we deduce $\{{T}^{n}w\}\to u$ and $\{{T}^{n}w\}\to v$. The uniqueness of the limit implies that $u=v$. Hence, the fixed point of T is unique. □
Corollary 3.1 Let $(X,\u2aaf)$ be an ordered set endowed with a Gmetric and $T:X\to X$ be a given mapping. Suppose that the following conditions hold:

(i)
$(X,G)$ is Gcomplete;

(ii)
T is Gcontinuous and nondecreasing (with respect to ⪯);

(iii)
there exists ${x}_{0}\in X$ such that ${x}_{0}\u2aafT{x}_{0}$;

(iv)
there exists a constant $k\in (0,1)$ such that for all $x,y\in X$ with $x\u2ab0y$,
$$G(Tx,Ty,Ty)\le kG(x,y,y).$$(28)
Then T has a fixed point. Moreover, if for all $(x,y)\in X\times X$ there exists $w\in X$ such that $x\u2aafw$ and $y\u2aafw$, we obtain uniqueness of the fixed point.
Proof It is sufficient to take $z=y$ in the proof of Theorem 3.4. □
Nieto and López [49] extended the result of Ran and Reurings [16] for a mapping T not necessarily continuous by assuming an additional hypothesis on $(X,\u2aaf,d)$.
Theorem 3.5 (Nieto and López [49])
Let $(X,\u2aaf)$ be an ordered set endowed with a metric d and $T:X\to X$ be a given mapping. Suppose that the following conditions hold:

(i)
$(X,d)$ is complete;

(ii)
X is ordered complete;

(iii)
T is nondecreasing;

(iv)
there exists ${x}_{0}\in X$ such that ${x}_{0}\u2aafT{x}_{0}$;

(v)
there exists a constant $k\in (0,1)$ such that for all $x,y\in X$ with $x\u2ab0y$,
$$d(Tx,Ty)\le kd(x,y).$$
Then T has a fixed point. Moreover, if for all $(x,y)\in X\times X$ there exists $w\in X$ such that $x\u2aafw$ and $y\u2aafw$, we obtain uniqueness of the fixed point.
The result of Nieto and López [49] can also be proved in the framework of Gmetric space.
Theorem 3.6 Let $(X,\u2aaf)$ be an ordered set endowed with a Gmetric and $T:X\to X$ be a given mapping. Suppose that the following conditions hold:

(i)
$(X,G)$ is Gcomplete;

(ii)
X is ordered complete;

(iii)
T is nondecreasing;

(iv)
there exists ${x}_{0}\in X$ such that ${x}_{0}\u2aafT{x}_{0}$;

(v)
there exists a constant $k\in (0,1)$ such that for all $x,y,z\in X$ with $x\u2ab0y\u2ab0z$,
$$G(Tx,Ty,Tz)\le kG(x,y,z).$$(29)
Then T has a fixed point. Moreover, if for all $(x,y)\in X\times X$ there exists $w\in X$ such that $x\u2aafw$ and $y\u2aafw$, we obtain uniqueness of the fixed point.
Proof Following the lines in the proof of Theorem 3.4, we have a sequence $\{{x}_{n}\}$ which is Gconvergent to $u\in X$. Due to (ii), we have ${x}_{n}\u2aafu$ for all n. We will show that u is a fixed point of T. Suppose on the contrary that $u\ne Tu$, that is, ${d}_{G}(u,Tu)>0$. Regarding (1) and (29) with $x={x}_{n}$, $y=z=Tu$, we have
Passing to limit as $n\to \mathrm{\infty}$, we get ${d}_{G}(u,Tu)=0$, which is a contradiction. Hence, $Tu=u$. Uniqueness of u can be observed as in the proof of Theorem 3.4. □
Corollary 3.2 Let $(X,\u2aaf)$ be an ordered set endowed with a Gmetric and $T:X\to X$ be a given mapping. Suppose that the following conditions hold:

(i)
$(X,G)$ is Gcomplete;

(ii)
X is ordered complete;

(iii)
T is nondecreasing;

(iv)
there exists ${x}_{0}\in X$ such that ${x}_{0}\u2aafT{x}_{0}$;

(v)
there exists a constant $k\in (0,1)$ such that for all $x,y\in X$ with $x\u2ab0y$,
$$G(Tx,Ty,Ty)\le kG(x,y,y).$$(31)
Then T has a fixed point. Moreover, if for all $(x,y)\in X\times X$ there exists $w\in X$ such that $x\u2aafw$ and $y\u2aafw$, we obtain uniqueness of the fixed point.
Proof It is sufficient to take $z=y$ in the proof of Theorem 3.6. □
Denote by Ψ the set of functions $\psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ satisfying the following conditions:
(${\mathrm{\Psi}}_{0}$) ${\psi}^{1}(\{0\})=0$,
(${\mathrm{\Psi}}_{1}$) $\psi (t)<t$ for all $t>0$;
(${\mathrm{\Psi}}_{2}$) ${lim}_{r\to {t}^{+}}\psi (r)<t$.
Following the work of Ćirić et al. [50], we generalize the abovementioned results by means of introducing a function g. More specifically, we modify the definitions and theorems according to the presence of the function g.
Definition 3.1 (See [50])
Let $(X,\u2aaf)$ be an ordered set and $T:X\to X$ and $g:X\to X$ be given mappings. The mapping T is called gnondecreasing if for every $x,y\in X$,
Theorem 3.7 Let $(X,\u2aaf)$ be an ordered set endowed with a Gmetric and $T:X\to X$ and $g:X\to X$ be given mappings. Suppose that the following conditions hold:

(i)
$(X,G)$ is Gcomplete;

(ii)
T is Gcontinuous;

(iii)
T is gnondecreasing;

(iv)
there exists ${x}_{0}\in X$ such that $g{x}_{0}\u2aafT{x}_{0}$;

(v)
$T(X)\subset g(X)$ and g is Gcontinuous and commutes with T;

(vi)
there exists a function $\phi \in \mathrm{\Psi}$ such that for all $x,y,z\in X$ with $gx\u2ab0gy\u2ab0gz$,
$$G(Tx,Ty,Tz)\le \phi (G(gx,gy,gz)).$$(32)
Then T and g have a coincidence point, that is, there exists $w\in X$ such that $gw=Tw$.
Proof Let ${x}_{0}\in X$ such that $g{x}_{0}\u2aafT{x}_{0}$. Since $T(X)\subset g(X)$, we can choose ${x}_{1}$ such that $g{x}_{1}=T{x}_{0}$. Again, by $T(X)\subset g(X)$, we can choose ${x}_{2}$ such that $g{x}_{2}=T{x}_{1}$. By repeating the same argument, we construct the sequence $\{g{x}_{n}\}$ in the following way:
Regarding that T is a gnondecreasing mapping together with (33), we observe that
Inductively, we obtain
If there exists ${n}_{0}$ such that $g{x}_{{n}_{0}}=g{x}_{{n}_{0}+1}$, then $g{x}_{{n}_{0}}=g{x}_{{n}_{0}+1}=T{x}_{{n}_{0}}$, that is, T and g have a coincidence point which completes the proof. Assume that $g{x}_{n}\ne g{x}_{n+1}$ for all $n\in \mathbb{N}$.
Regarding (34), we set $x=y={x}_{n+1}$ and $z={x}_{n}$ in (32). Then we get
which is equivalent to
since $\phi (t)<t$ for all $t>0$. Let ${t}_{n}=G(g{x}_{n+1},g{x}_{n+1},g{x}_{n})$. Then $\{{t}_{n}\}$ is a positive nonincreasing sequence. Thus, there exists $L\ge 0$ such that
We will show that $L=0$. Suppose that contrary to our claim, $L>0$. Letting $n\to \mathrm{\infty}$ in (35), we get
which is a contradiction. Hence, we have
We will show that $\{g{x}_{n}\}$ is a GCauchy sequence. Suppose on the contrary that the sequence $\{g{x}_{n}\}$ is not GCauchy. Then there exists $\epsilon >0$ and sequences of natural numbers $\{m(k)\}$, $\{l(k)\}$ such that for each natural number k,
and we have
Corresponding to $l(k)$, the number $m(k)$ is chosen to be the smallest number for which (38) holds. Hence, we have
By using (G5), we obtain that
Regarding (37) and letting $n\to \mathrm{\infty}$ in the previous inequality, we deduce
Again by the rectangle inequality (G5), together with (G4) and Lemma 2.1, we get that
Setting $x=y={x}_{m(k)}$ and $z={x}_{l(k)}$, the inequality (32) implies
Combining the inequalities (41) and (42), we find
Taking (37) and (40) into account and letting $k\to \mathrm{\infty}$ in (43), we obtain that
which is a contradiction. Hence, $\{g{x}_{n}\}$ is a GCauchy sequence in the Gmetric space $(X,G)$. Since $(X,G)$ is Gcomplete, there exists $w\in X$ such that $\{g{x}_{n}\}$ is Gconvergent to w. By Proposition 2.1, we have
The Gcontinuity of g implies that the sequence $\{gg{x}_{n}\}$ is Gconvergent to gw, that is,
On the other hand, due to the commutativity of T and g, we can write
and the Gcontinuity of T implies that the sequence $\{Tg{x}_{n}\}=\{gg{x}_{n+1}\}$ Gconverges to Tw so that
By the uniqueness of the limit, the expressions (45) and (47) yield that $gw=Tw$. Indeed, from the rectangle inequality, we get
which implies $G(gw,Tw,Tw)=0$ upon letting $n\to \mathrm{\infty}$. Hence, $gw=Tw$. □
In the next theorem, Gcontinuity of T is no longer required. However, we require the gordered completeness of X.
Theorem 3.8 Let $(X,\u2aaf)$ be an ordered set endowed with a Gmetric and $T:X\to X$ and $g:X\to X$ be given mappings. Suppose that the following conditions hold:

(i)
$(X,G)$ is Gcomplete;

(ii)
X is gordered complete;

(iii)
T is gnondecreasing (with respect to ⪯);

(iv)
there exists ${x}_{0}\in X$ such that $g{x}_{0}\u2aafT{x}_{0}$;

(v)
$T(X)\subset g(X)$ and g is Gcontinuous and commutes with T;

(vi)
there exists a function $\phi \in \mathrm{\Psi}$ such that for all $x,y,z\in X$ with $gx\u2ab0gy\u2ab0gz$,
$$G(Tx,Ty,Tz)\le \phi (G(gx,gy,gz)).$$(49)
Then T and g have a coincidence point, that is, there exists $w\in X$ such that $gw=Tw$.
Proof Following the lines of the proof of Theorem 3.7, we define a sequence $\{g{x}_{n}\}$ and conclude that it is a GCauchy sequence in the Gcomplete, Gmetric space $(X,G)$. Thus, there exists $w\in X$ such that $g{x}_{n}$ is Gconvergent to gw. Since $\{g{x}_{n}\}$ is nondecreasing and X is gordered complete, we have $g{x}_{n}\u2aafgw$. If $gw=g{x}_{n}$ for some natural number n, then T and g have a coincidence point. Indeed, $gw=g{x}_{n}\u2aafg{x}_{n+1}=T{x}_{n}\u2aafgw$ and hence $g{x}_{n}=T{x}_{n}$. Suppose that $gw\ne g{x}_{n}$. By the rectangle inequality together with the inequality (49) and the property (${\mathrm{\Psi}}_{1}$), we have
Letting $n\to \mathrm{\infty}$ in the inequality above, we get that $G(Tw,gw,gw)=0$. Hence, $Tw=gw$. □
If we take $\phi (t)=kt$, where $k\in [0,1)$ in Theorem 3.7 and Theorem 3.8, we deduce the following corollaries, respectively.
Corollary 3.3 Let $(X,\u2aaf)$ be an ordered set endowed with a Gmetric and $T:X\to X$ and $g:X\to X$ be given mappings. Suppose that the following conditions hold:

(i)
$(X,G)$ is Gcomplete;

(ii)
T is Gcontinuous;

(iii)
T is gnondecreasing (with respect to ⪯);

(iv)
there exists ${x}_{0}\in X$ such that $g{x}_{0}\u2aafT{x}_{0}$;

(v)
$T(X)\subset g(X)$ and g is Gcontinuous and commutes with T;

(vi)
there exists $k\in [0,1)$ such that for all $x,y,z\in X$ with $gx\u2ab0gy\u2ab0gz$,
$$G(Tx,Ty,Tz)\le kG(gx,gy,gz).$$(50)
Then T and g have a coincidence point, that is, there exists $w\in X$ such that $gw=Tw$.
Corollary 3.4 Let $(X,\u2aaf)$ be an ordered set endowed with a Gmetric and $T:X\to X$ and $g:X\to X$ be given mappings. Suppose that the following conditions hold:

(i)
$(X,G)$ is Gcomplete;

(ii)
X is gordered complete;

(iii)
T is gnondecreasing (with respect to ⪯);

(iv)
there exists ${x}_{0}\in X$ such that $g{x}_{0}\u2aafT{x}_{0}$;

(v)
$T(X)\subset g(X)$ and g is Gcontinuous and commutes with T;

(vi)
there exists $k\in [0,1)$ such that for all $x,y,z\in X$ with $gx\u2ab0gy\u2ab0gz$,
$$G(Tx,Ty,Tz)\le kG(gx,gy,gz).$$(51)
Then T and g have a coincidence point, that is, there exists $w\in X$ such that $gw=Tw$.
If we take $z=y$ in Theorem 3.7 and Theorem 3.8, we obtain the following particular cases.
Corollary 3.5 Let $(X,\u2aaf)$ be an ordered set endowed with a Gmetric and $T:X\to X$ and $g:X\to X$ be given mappings. Suppose that the following conditions hold:

(i)
$(X,G)$ is Gcomplete;

(ii)
T is Gcontinuous;

(iii)
T is gnondecreasing;

(iv)
there exists ${x}_{0}\in X$ such that $g{x}_{0}\u2aafT{x}_{0}$;

(v)
$T(X)\subset g(X)$ and g is Gcontinuous and commutes with T;

(vi)
there exists a function $\phi \in \mathrm{\Psi}$ such that for all $x,y\in X$ with $gx\u2ab0gy$,
$$G(Tx,Ty,Ty)\le \phi (G(gx,gy,gy)).$$(52)
Then T and g have a coincidence point, that is, there exists $w\in X$ such that $gw=Tw$.
Corollary 3.6 Let $(X,\u2aaf)$ be an ordered set endowed with a Gmetric and $T:X\to X$ and $g:X\to X$ be given mappings. Suppose that the following conditions hold:

(i)
$(X,G)$ is Gcomplete;

(ii)
X is gordered complete;

(iii)
T is gnondecreasing;

(iv)
there exists ${x}_{0}\in X$ such that $g{x}_{0}\u2aafT{x}_{0}$;

(v)
$T(X)\subset g(X)$ and g is Gcontinuous and commutes with T;

(vi)
there exists a function $\phi \in \mathrm{\Psi}$ such that for all $x,y\in X$ with $gx\u2ab0gy$,
$$G(Tx,Ty,Ty)\le \phi (G(gx,gy,gy)).$$(53)
Then T and g have a coincidence point, that is, there exists $w\in X$ such that $gw=Tw$.
Finally, we let $g=i{d}_{X}$ in the Theorem 3.7 and Theorem 3.8 and conclude the following theorems.
Theorem 3.9 Let $(X,\u2aaf)$ be an ordered set endowed with a Gmetric and $T:X\to X$ be a given mapping. Suppose that the following conditions hold:

(i)
$(X,G)$ is Gcomplete;

(ii)
T is Gcontinuous;

(iii)
T is nondecreasing (with respect to ⪯);

(iv)
there exists ${x}_{0}\in X$ such that ${x}_{0}\u2aafT{x}_{0}$;

(v)
there exists a function $\phi \in \mathrm{\Psi}$ such that for all $x,y,z\in X$ with $x\u2ab0y\u2ab0z$,
$$G(Tx,Ty,Tz)\le \phi (G(x,y,z)).$$(54)
Then T has a fixed point.
Theorem 3.10 Let $(X,\u2aaf)$ be an ordered set endowed with a Gmetric and $T:X\to X$ be a given mapping. Suppose that the following conditions hold:

(i)
$(X,G)$ is Gcomplete;

(ii)
X is ordered complete;

(iii)
T is nondecreasing;

(iv)
there exists ${x}_{0}\in X$ such that ${x}_{0}\u2aafT{x}_{0}$;

(v)
there exists a function $\phi \in \mathrm{\Psi}$ such that for all $x,y,z\in X$ with $x\u2ab0y\u2ab0z$,
$$G(Tx,Ty,Tz)\le \phi (G(x,y,z)).$$(55)
Then T has a fixed point.
We next consider some equivalence conditions and their implementation on Gmetric spaces. Let $\mathcal{S}$ denote the set of functions $\beta :[0,\mathrm{\infty})\to [0,1)$ satisfying the condition
In 2007, Jachymski and Jóźwik [51] proved that the classes $\mathcal{S}$ and Ψ are equivalent. Regarding this result, we state the following fixed point theorems on Gmetric spaces.
Theorem 3.11 Let $(X,\u2aaf)$ be an ordered set endowed with a Gmetric and $T:X\to X$ be a given mapping. Suppose that the following conditions hold:

(i)
$(X,G)$ is Gcomplete;

(ii)
T is Gcontinuous;

(iii)
T is nondecreasing (with respect to ⪯);

(iv)
there exists ${x}_{0}\in X$ such that ${x}_{0}\u2aafT{x}_{0}$;

(v)
there exists a function $\beta \in \mathcal{S}$ such that for all $x,y,z\in X$ with $x\u2ab0y\u2ab0z$,
$$G(Tx,Ty,Tz)\le \beta (G(x,y,z))G(x,y,z).$$(56)
Then T has a fixed point.
Theorem 3.12 Let $(X,\u2aaf)$ be an ordered set endowed with a Gmetric and $T:X\to X$ be a given mapping. Suppose that the following conditions hold:

(i)
$(X,G)$ is Gcomplete;

(ii)
X is ordered complete;

(iii)
T is nondecreasing (with respect to ⪯);

(iv)
there exists ${x}_{0}\in X$ such that ${x}_{0}\u2aafT{x}_{0}$;

(v)
there exists a function $\beta \in \mathcal{S}$ such that for all $x,y,z\in X$ with $x\u2ab0y\u2ab0z$,
$$G(Tx,Ty,Tz)\le \beta (G(x,y,z))G(x,y,z).$$(57)
Then T has a fixed point.
The two corollaries below are immediate consequences of Theorem 3.11 and Theorem 3.12.
Corollary 3.7 Let $(X,\u2aaf)$ be an ordered set endowed with a Gmetric and $T:X\to X$ be a given mapping. Suppose that the following conditions hold:

(i)
$(X,G)$ is Gcomplete;

(ii)
T is Gcontinuous;

(iii)
T is nondecreasing (with respect to ⪯);

(iv)
there exists ${x}_{0}\in X$ such that ${x}_{0}\u2aafT{x}_{0}$;

(v)
there exists a function $\beta \in \mathcal{S}$ such that for all $x,y\in X$ with $x\u2ab0y$,
$$G(Tx,Ty,Ty)\le \beta (G(x,y,y))G(x,y,y).$$(58)
Then T has a fixed point.
Corollary 3.8 Let $(X,\u2aaf)$ be an ordered set endowed with a Gmetric and $T:X\to X$ be a given mapping. Suppose that the following conditions hold:

(i)
$(X,G)$ is Gcomplete;

(ii)
X is ordered complete;

(iii)
T is nondecreasing (with respect to ⪯);

(iv)
there exists ${x}_{0}\in X$ such that ${x}_{0}\u2aafT{x}_{0}$;

(v)
there exists a function $\beta \in \mathcal{S}$ such that for all $x,y\in X$ with $x\u2ab0y$,
$$G(Tx,Ty,Ty)\le \beta (G(x,y,y))G(x,y,y).$$(59)
Then T has a fixed point.
Denote by Φ the set of functions $\phi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ satisfying the conditions (${\mathrm{\Psi}}_{1}$) and (${\mathrm{\Psi}}_{2}$). Jachymski [52] proved the equivalence of the socalled distance functions (see Lemma 1 in [52]). Inspired by this result, we can state the following theorem.
Theorem 3.13 Let $(X,\u2aaf)$ be an ordered set endowed with a Gmetric and T be a selfmap on a Gcomplete partially ordered Gmetric space $(X,G)$. The following statements are equivalent:

(i)
there exist functions $\psi ,\eta \in \mathrm{\Phi}$ such that
$$\psi (G(Tx,Ty,Tz))\le \psi (G(x,y,z))\eta (G(x,y,z)),$$(60) 
(ii)
there exist $\alpha \in [0,1)$ and a function $\psi \in \mathrm{\Phi}$ such that
$$\psi (G(Tx,Ty,Tz))\le \alpha \psi (G(x,y,z)),$$(61) 
(iii)
there exists a continuous and nondecreasing function $\alpha :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ such that $\alpha (t)<t$ for all $t>0$ such that
$$G(Tx,Ty,Tz)\le \alpha (G(x,y,z)),$$(62) 
(iv)
there exist a function $\psi \in \mathrm{\Phi}$ and a nondecreasing function $\eta :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ with ${\eta}^{1}(0)=0$ such that
$$\psi (G(Tx,Ty,Tz))\le \psi (G(x,y,z))\eta (G(x,y,z)),$$(63) 
(iv)
there exist a function $\psi \in \mathrm{\Phi}$ and a lower semicontinuous function $\eta :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ with ${\eta}^{1}(0)=0$ and ${lim\hspace{0.17em}inf}_{t\to \mathrm{\infty}}\eta (t)>0$ such that
$$\psi (G(Tx,Ty,Tz))\le \psi (G(x,y,z))\eta (G(x,y,z)),$$(64)
for any $x,y,z\in X$ with $x\u2ab0y\u2ab0z$.
As a consequence of Theorem 3.13, we state the next corollary.
Corollary 3.9 Let $(X,\u2aaf)$ be an ordered set endowed with a Gmetric and T be a selfmap on a Gcomplete partially ordered Gmetric space $(X,G)$. The following statements are equivalent:

(i)
there exist functions $\psi ,\eta \in \mathrm{\Phi}$ such that
$$\psi (G(Tx,Ty,Ty))\le \psi (G(x,y,y))\eta (G(x,y,y)),$$(65) 
(ii)
there exist $\alpha \in [0,1)$ and a function $\psi \in \mathrm{\Phi}$ such that
$$\psi (G(Tx,Ty,Ty))\le \alpha \psi (G(x,y,y)),$$(66) 
(iii)
there exists a continuous and nondecreasing function $\alpha :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ such that $\alpha (t)<t$ for all $t>0$ such that
$$G(Tx,Ty,Ty)\le \alpha (G(x,y,y)),$$(67) 
(iv)
there exist a function $\psi \in \mathrm{\Phi}$ and a nondecreasing function $\eta :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ with ${\eta}^{1}(0)=0$ such that
$$\psi (G(Tx,Ty,Ty))\le \psi (G(x,y,y))\eta (G(x,y,y)),$$(68) 
(iv)
there exist a function $\psi \in \mathrm{\Phi}$ and a lower semicontinuous function $\eta :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ with ${\eta}^{1}(0)=0$ and ${lim\hspace{0.17em}inf}_{t\to \mathrm{\infty}}\eta (t)>0$ such that
$$\psi (G(Tx,Ty,Ty))\le \psi (G(x,y,y))\eta (G(x,y,y)),$$(69)
for any $x,y\in X$ with $x\u2aafy$.
4 Remarks on coupled fixed point theorems in Gmetric spaces
In this section, we prove that most of the coupled fixed point theorems on a Gmetric space X can be derived from the wellknown fixed point theorems on Gmetric spaces in the literature provided that $(X,G)$ is a symmetric Gmetric space. In the rest this paper, we shall assume that $(X,G)$ represents a symmetric Gmetric space.
Let $(X,\u2aaf)$ be a partially ordered set endowed with a metric G and $F:X\times X\to X$ and $g:X\to X$ be given mappings. We define a partial order ⪯_{2} on the product set $X\times X$ as follows:
Definition 4.1 F is said to have the mixed gmonotone property if $F(x,y)$ is monotone nondecreasing in x and is monotone nonincreasing in y; that is, for any $x,y\in X$,
If g is an identity mapping, then F is said to have the mixed monotone property, that is,
Let $Y=X\times X$. It is easy to show that the mappings $\mathrm{\Lambda},\mathrm{\Delta}:Y\times Y\times Y\to [0,\mathrm{\infty})$ defined by
for all $(x,y),(u,v),(z,w)\in Y$, are Gmetrics on Y.
Now, define the mapping $T:Y\to Y$ by
The following lemma is obvious.
Lemma 4.1 The following properties hold:

(a)
If $(X,G)$ is Gcomplete, then $(Y,\mathrm{\Lambda})$ and $(Y,\mathrm{\Delta})$ are Λcomplete and Δcomplete, respectively;

(b)
F has the mixed (gmixed) monotone property if and only if T is monotone nondecreasing (gnondecreasing) with respect to ⪯_{2};

(c)
$(x,y)\in X\times X$ is a coupled fixed point of F if and only if $(x,y)$ is a fixed point of T;

(d)
$(x,y)\in X\times X$ is a coupled coincidence point of F and g if and only if $(x,y)$ is a coupled coincidence point of T and g.
4.1 Shatanawi’s coupled fixed point results in a Gmetric space
In [31], Shatanawi proved the following theorems.
Theorem 4.1 (cf. [31])
Let $(X,G)$ be a Gcomplete Gmetric space. Let $F:X\times X\to X$ be a mapping such that
for all $x,y,u,v,z,w\in X$. If $k\in [0,1)$, then there exists a unique $x\in X$ such that $F(x,x)=x$.
In what follows, we prove the following theorem.
Theorem 4.2 Theorem 4.1 follows from Theorem 2.1.
Proof From (73), for all $(x,y),(u,v),(z,w)\in Y$ with $x\u2ab0u\u2ab0z$ and $y\u2aafv\u2aafw$, we have
and
which implies that
that is,
for all $(x,y),(u,v),(z,w)\in Y$, where Λ is defined in (70). From Lemma 4.1, since $(X,G)$ is Gcomplete, $(Y,\mathrm{\Lambda})$ is Λcomplete. In this case, regarding Theorem 2.1, we conclude that T has a fixed point, which due to Lemma 4.1 implies that F has a coupled fixed point.
Analogously, Theorem 4.3 is obtained from Theorem 2.2. □
The following theorem can be derived easily from Theorem 4.1.
Theorem 4.3 Let $(X,G)$ be a Gcomplete Gmetric space. Let $F:X\times X\to X$ be a mapping such that
If $k\in [0,1)$, then there is a unique $x\in X$ such that $F(x,x)=x$.
We note that Theorem 4.3 above is not stated in [31].
Theorem 4.4 Theorem 4.3 follows from Theorem 2.2.
Example 4.1 Let $X=\mathbb{R}$. Define $G:X\times X\times X\to [0,+\mathrm{\infty})$ by
for all $x,y,z\in X$. Then $(X,G)$ is a Gmetric space. Define a map $F:X\times X\to X$ by $F(x,y)=\frac{3}{5}x\frac{1}{5}y$ for all $x,y\in X$. Then, for all $x,y,u,v,w,z\in X$ with $y=v=z$, we have
and
Then it is easy to see that there is no $k\in [0,1)$ such that
for all $x,y,u,v,z,w\in X$. Indeed, the inequality above holds for $k\ge \frac{6}{5}$. Thus, Theorem 4.1 does not apply to this example. However, it is easy to see that 0 is the unique point such that $F(0,0)=0$.
On the other hand, Theorem 2.1 yields the existence of the fixed point. Indeed,
where $k\in [\frac{4}{5},1)$. Thus, all conditions of Theorem 3.1 are satisfied, which guarantees the existence of the fixed point $F(0,0)=0$.
4.2 Choudhury and Maity’s coupled fixed point results in a Gmetric space
Choudhury and Maity [53] proved the following coupled fixed point theorems on ordered Gmetric spaces.
Theorem 4.5 Let $(X,\u2aaf)$ be a partially ordered set and G be a Gmetric on X such that $(X,G)$ is a complete Gmetric space. Let $F:X\times X\to X$ be a Gcontinuous mapping having the mixed monotone property on X. Suppose that there exists a $k\in [0,1)$ such that
for all $x,y,u,v,w,z\in X$ with $x\u2ab0u\u2ab0w$ and $y\u2aafv\u2aafz$, where either $u\ne w$ or $v\ne z$. If there exist ${x}_{0},{y}_{0}\in X$ such that ${x}_{0}\u2aafF({x}_{0},{y}_{0})$ and $F({y}_{0},{x}_{0})\u2aaf{y}_{0}$, then F has a coupled fixed point, that is, there exists $(x,y)\in X\times X$ such that $x=F(x,y)$ and $y=F(y,x)$.
Theorem 4.6 If in the above theorem, instead of Gcontinuity of F, we assume that X is ordered complete, then F has a coupled fixed point.
We will prove the following result.
Theorem 4.7 Theorem 4.5 and Theorem 4.6 follow from Theorems 3.4 and 3.6, respectively.
Proof From (75), for all $(x,y),(u,v),(w,z)\in X\times X$ with $x\u2ab0u\u2ab0w$ and $y\u2aafv\u2aafz$, we have
and
This implies that for all $(x,y),(u,v),(w,z)\in X\times X$ with $x\u2ab0u\u2ab0w$ and $y\u2aafv\u2aafz$,
that is,
for all $(x,y),(u,v),(w,z)\in Y$ with $(x,y){\u2ab0}_{2}(u,v){\u2ab0}_{2}(w,z)$, where Λ is defined in (70).
It follows from Lemma 4.1 that since $(X,G)$ is Gcomplete, then $(Y,\mathrm{\Lambda})$ is Λcomplete. Since F has the mixed monotone property, T is a nondecreasing mapping with respect to ⪯_{2}. The assumption that there exist ${x}_{0},{y}_{0}\in X$ such that ${x}_{0}\u2aafF({x}_{0},{y}_{0})$ and $F({y}_{0},{x}_{0})\u2aaf{y}_{0}$ becomes $({x}_{0},{y}_{0}){\u2aaf}_{2}T({x}_{0},{y}_{0})$ in terms of the order ⪯_{2}. Now, if F is Gcontinuous, then T is Λcontinuous. In this case, applying Theorem 3.4, we get that T has a fixed point, which due to Lemma 4.1 implies that F has a coupled fixed point. If X is ordered complete, then Y satisfies the following property: if a nondecreasing (with respect to ⪯_{2}) sequence $\{{u}_{n}\}$ in Y converges to some point $u\in Y$, then ${u}_{n}{\u2aaf}_{2}u$ for all n. Applying Theorem 3.6, we get that T has a fixed point, that is, F has a coupled fixed point. □
Remark 4.1 (Uniqueness)
If, in addition, we suppose that for all $(x,y),(u,v)\in X\times X$, there exists $({z}_{1},{z}_{2})\in X\times X$ such that $(x,y){\u2aaf}_{2}({z}_{1},{z}_{2})$ and $(u,v){\u2aaf}_{2}({z}_{1},{z}_{2})$, from the last part of Theorems 3.4 and 3.6, we obtain the uniqueness of the fixed point of T, which implies the uniqueness of the coupled fixed point of F. Now, let $({x}^{\ast},{y}^{\ast})\in X\times X$ be the unique coupled fixed point of F. Since $({y}^{\ast},{x}^{\ast})$ is also a coupled fixed point of F, we get ${x}^{\ast}={y}^{\ast}$.
4.3 Coupled fixed point results of Aydi et al. in a Gmetric space
We consider the following fixed point theorems established by Aydi et al. [23]. The following lemma is trivial.
Lemma 4.2 (See [23]) Let $\varphi \in \mathrm{\Psi}$. For all $t>0$, we have ${lim}_{n\to \mathrm{\infty}}{\varphi}^{n}(t)=0$.
Aydi et al. [17] proved the following fixed point theorems.
Theorem 4.8 Let $(X,\u2aaf)$ be a partially ordered set and G be a Gmetric on X such that $(X,G)$ is a Gcomplete Gmetric space. Suppose that there exist $\varphi \in \mathrm{\Psi}$ and $F:X\times X\to X$ such that
for all $x,y,u,v,w,z\in X$ with $x\u2ab0u\u2ab0w$ and $y\u2aafv\u2aafz$. Suppose also that F is Gcontinuous and has the mixed monotone property. If there exist ${x}_{0},{y}_{0}\in X$ such that ${x}_{0}\u2aafF({x}_{0},{y}_{0})$ and $F({y}_{0},{x}_{0})\u2aaf{y}_{0}$, then F has a coupled fixed point, that is, there exists $(x,y)\in X\times X$ such that $x=F(x,y)$ and $y=F(y,x)$.
Replacing the Gcontinuity of F by ordered completeness of X yields the next theorem.
Theorem 4.9 Let $(X,\u2aaf)$ be a partially ordered set and G be a Gmetric on X such that $(X,G,\u2aaf)$ is Gcomplete. Suppose that there exist $\varphi \in \mathrm{\Psi}$ and $F:X\times X\to X$ such that
for all $x,y,u,v,w,z\in X$ with $x\u2ab0u\u2ab0w$ and $y\u2aafv\u2aafz$. Suppose also that F has the mixed monotone property and X is ordered complete. If there exist ${x}_{0},{y}_{0}\in X$ such that ${x}_{0}\u2aafF({x}_{0},{y}_{0})$ and $F({y}_{0},{x}_{0})\u2aaf{y}_{0}$, then F has a coupled fixed point, that is, there exists $(x,y)\in X\times X$ such that $x=F(x,y)$ and $y=F(y,x)$.
We will prove the following result.
Theorem 4.10 Theorem 4.8 and Theorem 4.9 follow from Theorems 3.9 and 3.10, respectively.
Proof From (76), for all $(x,y),(u,v),(w,z)\in X\times X$ with $x\u2ab0u\u2ab0w$ and $y\u2aafv\u2aafz$, we have
and
This implies that for all $(x,y),(u,v),(w,z)\in X\times X$ with $x\u2ab0u\u2ab0w$ and $y\u2aafv\u2aafz$,
holds. Rewrite the above inequality as
for all $(x,y),(u,v),(w,z)\in Y$ with $(x,y){\u2ab0}_{2}(u,v){\u2ab0}_{2}(w,z)$. Here, ${\mathrm{\Lambda}}^{\prime}:Y\times Y\times Y\to [0,\mathrm{\infty})$ is the Gmetric on Y defined by
where Λ is given in (70). Thus, the mapping T satisfies the conditions of Theorem 3.9 (resp. Theorem 3.10). Therefore, T has a fixed point, which implies that F has a coupled fixed point. □
4.4 On coupled fixed point results of Abbas et al. in a Gmetric space
Let Θ be the set of functions $\theta :{[0,\mathrm{\infty})}^{2}\to [0,1)$ which satisfy the condition
The following theorems have been given by Abbas et al. [37].
Theorem 4.11 Let $(X,\u2aaf)$ be a partially ordered set and G be a Gmetric on X. Assume that there is an altering distance function $\theta \in \mathrm{\Theta}$ and a map $F:X\times X\to X$ such that
for all $x,y,u,v,w,z\in X$ with $x\u2ab0u\u2ab0w$ and $y\u2aafv\u2aafz$. Suppose that F is Gcontinuous and has the mixed monotone property. If there exist ${x}_{0},{y}_{0}\in X$ such that ${x}_{0}\u2aafF({x}_{0},{y}_{0})$ and $F({y}_{0},{x}_{0})\u2aaf{y}_{0}$, then F has a coupled fixed point, that is, there exists $(x,y)\in X\times X$ such that $x=F(x,y)$ and $y=F(y,x)$.
Theorem 4.12 Let $(X,\u2aaf)$ be a partially ordered set and G be a Gmetric on X. Assume that there is an altering distance function $\theta \in \mathrm{\Theta}$ and a mapping $F:X\times X\to X$ such that
for all $x,y,u,v,w,z\in X$ with $x\u2ab0u\u2ab0w$ and $y\u2aafv\u2aafz$. Suppose that F has the mixed monotone property and X is ordered complete. If there exist ${x}_{0},{y}_{0}\in X$ such that ${x}_{0}\u2aafF({x}_{0},{y}_{0})$ and $F({y}_{0},{x}_{0})\u2aaf{y}_{0}$, then F has a coupled fixed point, that is, there exists $(x,y)\in X\times X$ such that $x=F(x,y)$ and $y=F(y,x)$.
We will prove the following result.
Theorem 4.13 Theorem 4.11 and Theorem 4.12 follow from Theorem 3.11 and Theorem 3.12.
Proof From (78), for all $(x,y),(u,v),(w,z)\in X\times X$ with $x\u2ab0u\u2ab0w$ and $y\u2aafv\u2aafz$, we have
This is equivalent to
for all $(x,y),(u,v),(z,w)\in Y$ with $(x,y){\u2ab0}_{2}(u,v){\u2ab0}_{2}(w,z)$, where $\theta (t,s)=\beta (t+s)$, which clearly implies that $\beta \in \mathcal{S}$.
From Lemma 4.1, since $(X,G)$ is Gcomplete, $(Y,\mathrm{\Lambda})$ is Λcomplete. Since F has the mixed monotone property, T is a nondecreasing mapping with respect to ⪯_{2}. According to the assumption of Theorem 4.11, we have $({x}_{0},{y}_{0}){\u2aaf}_{2}T({x}_{0},{y}_{0})$. Now, if F is Gcontinuous, then T is Λcontinuous. In this case, applying Theorem 3.11, we get that T has a fixed point, which implies from Lemma 4.1 that F has a coupled fixed point. If X is ordered complete, then Y satisfies the following property: if a nondecreasing (with respect to ⪯_{2}) then the sequence $\{{u}_{n}\}$ in Y converges to some point $u\in Y$, then ${u}_{n}{\u2aaf}_{2}u$ for all n. Applying Theorem 3.12, we get that T has a fixed point, which implies that F has a coupled fixed point. □
5 Remarks on common coupled fixed point theorems in Gmetric spaces
In this last section, we investigate the similarity between most of the common coupled fixed point theorems and ordinary fixed point theorems in the context of Gmetric spaces and we show that the former are immediate consequences of the latter.
5.1 Shatanawi’s common coupled fixed point results in a Gmetric space
We start with two theorems by Shatanawi [31].
Theorem 5.1 (cf. [31])
Let $(X,G)$ be a Gmetric space. Let $F:X\times X\to X$ and $g:X\to X$ be two mappings such that
for all $x,y,u,v,z,w\in X$. Assume that F and g satisfy the following conditions:

(1)
$F(X\times X)\subset g(X)$,

(2)
$g(X)$ is Gcomplete,

(3)
g is Gcontinuous and commutes with F.
If $k\in [0,1)$, then there is a unique $x\in X$ such that $gx=F(x,x)=x$.
Theorem 5.2 (cf. [31])
Let $(X,G)$ be a Gmetric space. Let $F:X\times X\to X$ and $g:X\to X$ be two mappings such that
for all $x,y,u,v\in X$. Assume that F and g satisfy the following conditions:

(1)
$F(X\times X)\subset g(X)$,

(2)
$g(X)$ is Gcomplete,

(3)
g is Gcontinuous and commutes with F.
If $k\in [0,1)$, then there is a unique $x\in X$ such that $gx=F(x,x)=x$.
Theorem 5.3 Theorem 5.1 and Theorem 5.2 follow from Theorem 3.1 and Theorem 3.2, respectively.
Proof From (81), for all $(x,y),(u,v),(w,z)\in X\times X$, we have
and
Therefore, for all $(x,y),(u,v),(w,z)\in X\times X$, we obtain
that is,
for all $(x,y),(u,v),(w,z)\in Y$, where ${T}_{F},{T}_{g}:{X}^{2}\to {X}^{2}$ are mappings such that ${T}_{F}(a,b)=(F(a,b),F(b,a))$ and ${T}_{g}(a,b)=(ga,gb)$ and Λ is defined in (70). From Lemma 4.1, since $(X,G)$ is Gcomplete, $(Y,\mathrm{\Lambda})$ is Λcomplete. In this case, applying Theorem 3.1, we get that ${T}_{F}$ and ${T}_{g}$ have a common fixed point, which implies from Lemma 4.1 that F and g have a common coupled fixed point.
Analogously, Theorem 5.2 is obtained from Theorem 3.2. □
Example 5.1 Let $X=\mathbb{R}$. Define $G:X\times X\times X\to [0,+\mathrm{\infty})$ by
for all $x,y,z\in X$. Then $(X,G)$ is a Gmetric space. Define a map $F:X\times X\to X$ by $F(x,y)=\frac{4}{8}x\frac{1}{8}y$ and $g:X\to X$ by $g(x)=\frac{7x}{8}$ for all $x,y\in X$. Then, for all $x,y,u,v,z,w\in X$ with $y=v=z$, we have
and
Then it is easy to see that there is no $k\in [0,1)$ such that
for all $x,y,u,v,z,w\in X$. Thus, Theorem 5.1 does not provide the existence of the common fixed point for the maps on this example. However, it is easy to see that 0 is the unique point $x\in X$ such that $x=gx=F(x,x)$.
On the other hand, notice that Theorem 3.1 yields the fixed point. Indeed,
and also
Then the condition (6) of Theorem 3.1 holds for $k\in [\frac{5}{7},1)$. Thus, all conditions of Theorem 3.1 are satisfied, which provides the common coupled fixed point of F and g.
5.2 Nashine’s common coupled fixed point results in a Gmetric space
Nashine [54] studied common coupled fixed points on ordered Gmetric spaces and proved the following theorems.
Theorem 5.4 Let $(X,G,\u2aaf)$ be a partially ordered Gmetric space. Let $F:X\times X\to X$ and $g:X\to X$ be mappings such that F has the mixed gmonotone property. Suppose that there exist ${x}_{0},{y}_{0}\in X$ such that $g{x}_{0}\u2aafF({x}_{0},{y}_{0})$ and $F({y}_{0},{x}_{0})\u2aafg{y}_{0}$. Suppose also that there exists $k\in [0,\frac{1}{2})$ such that
holds for all $x,y,u,v,w,z\in X$ satisfying $gx\u2ab0gu\u2ab0gw$ and $gy\u2aafgv\u2aafgz$, where either $gu\ne gz$ or $gv\ne gw$. We assume the following hypotheses:

(i)
$F(X\times X)\subseteq g(X)$,

(ii)
F is Gcontinuous,

(iii)
$g(X)$ is Gcomplete,

(iv)
g is Gcontinuous and commutes with F.
Then F and g have a coupled coincidence point, that is, there exists $(x,y)\in X\times X$ such that $gx=F(x,y)$ and $gy=F(y,x)$. If $gu=gz$ and $gv=gw$, then F and g have a common fixed point, that is, there exists $x\in X$ such that $gx=F(x,x)=x$.
Theorem 5.5 If in the above theorem we replace the Gcontinuity of F by the assumption that X is gordered complete, then F and g have a coupled coincidence point.
Theorem 5.6 Theorem 5.4 and Theorem 5.5 follow from Corollary 3.3 and Corollary 3.4, respectively.
Proof From (83), for all $(x,y),(u,v),(w,z)\in X\times X$ with $gx\u2ab0gu\u2ab0gw$ and $gy\u2aafgv\u2aafgz$, we have
and
This implies that for all $(x,y),(u,v),(z,w)\in X\times X$ with $gx\u2ab0gu\u2ab0gw$ and $gy\u2aafgv\u2aafgz$,
that is,
for all $(x,y),(u,v),(z,w)\in Y$ with $(gx,gy){\u2ab0}_{2}(gu,gv){\u2ab0}_{2}(gw,gz)$.
From Lemma 4.1, since $(X,G)$ is Gcomplete, $(Y,\mathrm{\Lambda})$ is Λcomplete. Also, since F has the mixed gmonotone property, T is a gnondecreasing mapping with respect to ⪯_{2}. From the assumption of Theorem 5.4, we have $(g{x}_{0},g{y}_{0}){\u2aaf}_{2}T({x}_{0},{y}_{0})$. Now, if F is Gcontinuous, then T is Λcontinuous. In this case, due to Corollary 3.3, we deduce that T and g have a coincidence point, which from Lemma 4.1 implies that F and g have a coupled coincidence point. If, on the other hand, X is gordered complete, then Y satisfies the following property: if a nondecreasing (with respect to ⪯_{2}) sequence $\{{u}_{n}\}$ in Y converges to a point $u\in Y$, then ${u}_{n}{\u2aaf}_{2}u$ for all n. According to Corollary 3.4, T and g have a coincidence point, which from Lemma 4.1 implies that F and g have a coupled coincidence point. □
5.3 Common coupled fixed point results of Aydi et al. in a Gmetric space
Recently, Aydi et al. [17] proved the following theorems on Gmetric spaces.
Theorem 5.7 Let $(X,\u2aaf)$ be a partially ordered set and G be a Gmetric on X such that $(X,G)$ is a Gcomplete Gmetric space. Suppose that there exist maps $\varphi \in \mathrm{\Psi}$ and $F:X\times X\to X$ and $g:X\to X$ such that
for all $x,y,u,v,w,z\in X$ with $gx\u2ab0gu\u2ab0gw$ and $gy\u2aafgv\u2aafgz$. Suppose also that F is Gcontinuous and has the mixed gmonotone property, $F(X\times X)\subset g(X)$, and g is Gcontinuous and commutes with F. If there exist ${x}_{0},{y}_{0}\in X$ such that $g{x}_{0}\u2aafF({x}_{0},{y}_{0})$ and $F({y}_{0},{x}_{0})\u2aafg{y}_{0}$, then F and g have a coupled coincidence point, that is, there exists $(x,y)\in X\times X$ such that $gx=F(x,y)$ and $gy=F(y,x)$.
Theorem 5.8 Let $(X,\u2aaf)$ be a partially ordered set and G be a Gmetric on X such that $(X,G,\u2aaf)$ is Gcomplete. Suppose that there exist $\varphi \in \mathrm{\Psi}$ and $F:X\times X\to X$ and $g:X\to X$ such that
for all $x,y,u,v,w,z\in X$ with $gx\u2ab0gu\u2ab0gw$ and $gy\u2aafgv\u2aafgz$. Suppose also that X is gordered complete and F has the mixed gmonotone property, $F(X\times X)\subset g(X)$, and g is Gcontinuous and commutes with F. If there exist ${x}_{0},{y}_{0}\in X$ such that $g{x}_{0}\u2aafF({x}_{0},{y}_{0})$ and $F({y}_{0},{x}_{0})\u2aafg{y}_{0}$, then F and g have a coupled coincidence point, that is, there exists $(x,y)\in X\times X$ such that $gx=F(x,y)$ and $gy=F(y,x)$.
The following result can be also proved easily.
Theorem 5.9 Theorem 4.8 and Theorem 4.9 follow from Theorems 3.7 and 3.8.
Proof We observe from (84) that for all $(x,y),(u,v),(w,z)\in X\times X$ with $gx\u2ab0gu\u2ab0gw$ and $gy\u2aafgv\u2aafgz$, we have
and also
Therefore, for all $(x,y),(u,v),(w,z)\in X\times X$ with $gx\u2ab0gu\u2ab0gw$ and $gy\u2aafgv\u2aafgz$, the following inequality holds:
that is,
for all $(x,y),(u,v),(w,z)\in Y$ with $(x,y){\u2ab0}_{2}(u,v){\u2ab0}_{2}(w,z)$, where ${T}_{F},{T}_{g}:{X}^{2}\to {X}^{2}$ are mappings such that ${T}_{F}(a,b)=(F(a,b),F(b,a))$ and ${T}_{g}(a,b)=(ga,gb)$. Note that here, ${\mathrm{\Lambda}}^{\prime}:Y\times Y\times Y\to [0,\mathrm{\infty})$ is a Gmetric on Y defined by
Thus, we proved that the mappings ${T}_{F}$ and ${T}_{g}$ satisfy the conditions of Theorem 3.7 (resp. Theorem 3.8). Hence, ${T}_{F}$ and ${T}_{g}$ have a coincidence point, which implies that F and g have a coupled coincidence point. □
5.4 Common coupled fixed point results of Cho et al. in a Gmetric space
Finally, we consider the results of Cho et al. [55]. We state their fixed point theorems below.
Theorem 5.10 Let $(X,\u2aaf)$ be a partially ordered set and G be a Gmetric on X such that $(X,G)$ is a Gcomplete Gmetric space. Let $F:X\times X\to X$ and $g:X\to X$ be Gcontinuous mappings such that F has the mixed gmonotone property and g commutes with F. Assume that there exist altering distance functions ϕ and ψ such that
for all $x,y,u,v,w,z\in X$ with $gx\u2ab0gu\u2ab0gw$ and $gy\u2aafgv\u2aafgz$. Suppose also that $F(X\times X)\subseteq g(X)$. If there exist ${x}_{0},{y}_{0}\in X$ such that $g{x}_{0}\u2aafF({x}_{0},{y}_{0})$ and $F({y}_{0},{x}_{0})\u2aafg{y}_{0}$, then F and g have a coupled coincidence point, that is, there exists $(x,y)\in X\times X$ such that $gx=F(x,y)$ and $gy=F(y,x)$.
Theorem 5.11 Let $(X,\u2aaf)$ be a partially ordered set and G be a Gmetric on X and let $F:X\times X\to X$ and $g:X\to X$ be mappings. Assume that there exist altering distance functions ϕ and ψ such that
for all $x,y,u,v,w,z\in X$ with $gx\u2ab0gu\u2ab0gw$ and $gy\u2aafgv\u2aafgz$. Suppose that $g(X)$ is Gcomplete, the mapping F has the mixed gmonotone property and $F(X\times X)\subseteq g(X)$. If there exist ${x}_{0},{y}_{0}\in X$ such that $g{x}_{0}\u2aafF({x}_{0},{y}_{0})$ and $F({y}_{0},{x}_{0})\u2aafg{y}_{0}$, then F and g have a coupled coincidence point, that is, there exists $(x,y)\in X\times X$ such that $gx=F(x,y)$ and $gy=F(y,x)$.
Theorem 5.12 Theorem 5.10 and Theorem 5.11 follow from Theorems 3.7 and 3.8.
Proof From the assumption, for all $(x,y),(u,v),(z,w)\in X\times X$ with $gx\u2ab0gu\u2ab0gw$ and $gy\u2aafgv\u2ab0gz$, we have
and
This implies (since ψ is nondecreasing) that for all $(x,y),(u,v),(z,w)\in X\times X$ with $gx\u2ab0gu\u2ab0gw$ and $gy\u2aafgv\u2aafgw$, we have
that is,
for all $(x,y),(u,v),(z,w)\in Y$ with $(gx,gy){\u2ab0}_{2}(gu,gv){\u2ab0}_{2}(gz,gw)$, where ${T}_{F},{T}_{g}:{X}^{2}\to {X}^{2}$ such that ${T}_{F}(a,b)=(F(a,b),F(b,a))$ and ${T}_{g}(a,b)=(ga,gb)$ and Δ is a Gmetric defined in (71). Regarding Theorem 3.13, the conditions (90) and (32) of Theorem 3.7 are equivalent. Therefore, the mappings ${T}_{F}$ and ${T}_{g}$ satisfy the conditions of Theorem 3.7 (resp. Theorem 3.8) and have a coincidence point. Hence, the maps F and g have a coupled coincidence point. □
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Keywords
 coupled fixed point
 fixed point
 ordered set
 metric space