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Coupled fixed point theorems for generalized contractive mappings in partially ordered Gmetric spaces
Fixed Point Theory and Applications volume 2012, Article number: 172 (2012)
Abstract
In this paper, we establish some coupled coincidence and coupled common fixed point theorems for nonlinear contractive mappings having the mixed monotone property in partially ordered Gmetric spaces. The results on fixed point theorems are generalizations of the recent results of Choudhury and Maity (Math. Comput. Model. 54:7379, 2011) and Luong and Thuan (Math. Comput. Model. 55:16011609, 2012).
1 Introduction and preliminaries
One of the simplest and the most useful results in the fixed point theory is the BanachCaccioppoli contraction [1] mapping principle, a power tool in analysis. This principle has been generalized in different directions in different spaces by mathematicians over the years (see [2–10] and references mentioned therein). On the other hand, fixed point theory has received much attention in metric spaces endowed with a partial ordering. The first result in this direction was given by Ran and Reurings [11] and they presented applications of their results to matrix equations. Subsequently, Nieto and RodríguezLópez [12] extended the results in [11] for nondecreasing mappings and obtained a unique solution for a firstorder ordinary differential equation with periodic boundary conditions (see also [13–19]).
In recent times, fixed point theory has developed rapidly in partially ordered metric spaces, that is, metric spaces endowed with a partial ordering. Some of these works are noted in [13, 17, 20, 21]. Bhaskar and Lakshmikantham [21] introduced the concept of a coupled fixed point and the mixed monotone property. Furthermore, they proved some coupled fixed point theorems for mappings which satisfy the mixed monotone property and gave some applications in the existence and uniqueness of a solution for a periodic boundary value problem. After the publication of this work, several coupled fixed point and coincidence point results have appeared in the recent literature. Works noted in [22–25] are some examples of these works.
Mustafa and Sims [26, 27] introduced a new structure of generalized metric spaces, which are called Gmetric spaces, as a generalization of metric spaces to develop and introduce a new fixed point theory for various mappings in this new structure. Later, several fixed point theorems in Gmetric spaces were obtained by [28–34].
To fix the context in which we are placing our results, recall the following notions. Throughout this article, $(X,\u2aaf)$ denotes a partially ordered set with the partial order ⪯. By $x\prec y$, we mean $x\u2aafy$ but $x\ne y$. A mapping $g:X\to X$ is said to be nondecreasing (nonincreasing) if for all $x,y\in X$, $x\u2aafy$ implies $g(x)\u2aafg(y)$ ($g(y)\u2aafg(x)$, respectively).
The concept of a mixed monotone property has been introduced by Bhaskar and Lakshmikantham [21].
Definition 1.1 [21]
Let $(X,\u2aaf)$ be a partial ordered set. A mapping $F:X\times X\to X$ is said to be have the mixed monotone property if $F(x,y)$ is monotone nondecreasing in x and is monotone nonincreasing in y, that is, for any $x,y\in X$,
and
The following concepts were introduced in [35].
Definition 1.2 [35]
Let $(X,\u2aaf)$ be a partial ordered set and $F:X\times X\to X$ and $g:X\to X$ be two mappings. We say that F has the mixed gmonotone property if $F(x,y)$ is gmonotone nondecreasing in x and it is gmonotone nonincreasing in y, that is, for any $x,y\in X$
and
Let $F:X\times X\to X$ and $g:X\to X$ be mappings. An element $(x,y)\in X\times X$ is said to be:

(i)
a coupled fixed point of a mapping F if
$$x=F(x,y)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}y=F(y,x);$$ 
(ii)
a coupled coincidence point of mapping F and g if
$$g(x)=F(x,y)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}g(y)=F(y,x);$$ 
(iii)
a coupled common fixed point of mappings F and g if
$$x=g(x)=F(x,y)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}y=g(y)=F(y,x).$$
Consistent with Mustafa and Sims [26, 27], the following definitions and results will be needed in the sequel.
Definition 1.4 (Gmetric space [27])
Let X be a nonempty set. Let $G:X\times X\times X\to {\mathbb{R}}_{+}$ be a function satisfying the following properties:
(G1) $G(x,y,z)=0$ if $x=y=z$;
(G2) $G(x,x,y)>0$ 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 $z\ne y$;
(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 Gmetric on X and the pair $(X,G)$ is called a Gmetric space.
Definition 1.5 [27]
Let X be a Gmetric space, and let $\{{x}_{n}\}$ be a sequence of points of X, a point $x\in X$ is said to be the limit of a sequence $\{{x}_{n}\}$ if $G(x,{x}_{n},{x}_{m})\to 0$ as $n,m\to \mathrm{\infty}$ and sequence $\{{x}_{n}\}$ is said to be Gconvergent to x.
From this definition, we obtain that if ${x}_{n}\to x$ in a Gmetric space X, then for any $\epsilon >0$, there exists a positive integer N such that $G(x,{x}_{n},{x}_{m})<\epsilon $ for all $n,m\ge N$.
It has been shown in [27] that the Gmetric induces a Hausdorff topology and the convergence described in the above definition is relative to this topology. So, a sequence can converge, at the most, to one point.
Definition 1.6 [27]
Let X be a Gmetric space, a sequence $\{{x}_{n}\}$ is called GCauchy if for every $\epsilon >0$, there is a positive integer N such that $G({x}_{n},{x}_{m},{x}_{l})<\epsilon $ for all $n,m,l\ge N$, that is, if $G({x}_{n},{x}_{m},{x}_{l})\to 0$, as $n,m,l\to \mathrm{\infty}$.
We next state the following lemmas.
Lemma 1.7 [27]
If X is a Gmetric space, then 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)
$G({x}_{m},{x}_{n},x)\to 0$ as $n,m\to \mathrm{\infty}$.
Lemma 1.8 [27]
If X is a Gmetric space, then the following are equivalent:

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

(2)
for every $\epsilon >0$, there exists a positive integer N such that $G({x}_{n},{x}_{m},{x}_{m})<\epsilon $, for all $n,m\ge N$.
Lemma 1.9 [27]
If X is a Gmetric space, then $G(x,y,y)\le 2G(y,x,x)$ for all $x,y\in X$.
Lemma 1.10 If X is a Gmetric space, then $G(x,x,y)\le G(x,x,z)+G(z,z,y)$ for all $x,y,z\in X$.
Definition 1.11 [27]
Let $(X,G)$, $({X}^{\mathrm{\prime}},{G}^{\mathrm{\prime}})$ be two generalized metric spaces. A mapping $f:X\to {X}^{\mathrm{\prime}}$ is Gcontinuous at a point $x\in X$ if and only if it is G sequentially continuous at x, that is, whenever $\{{x}_{n}\}$ is Gconvergent to x, $\{f({x}_{n})\}$ is ${G}^{\mathrm{\prime}}$convergent to $f(x)$.
Definition 1.12 [27]
A Gmetric space X is called a symmetric Gmetric space if
for all $x,y\in X$.
Definition 1.13 [27]
A Gmetric space X is said to be Gcomplete (or a complete Gmetric space) if every GCauchy sequence in X is convergent in X.
Definition 1.14 Let X be a Gmetric space. A mapping $F:X\times X\to X$ is said to be continuous if for any two Gconvergent sequences $\{{x}_{n}\}$ and $\{{y}_{n}\}$ converging to x and y, respectively, $\{F({x}_{n},{y}_{n})\}$ is Gconvergent to $F(x,y)$.
Definition 1.15 Let X be a nonempty set and $F:X\times X\to X$ and $g:X\to X$ two mappings. We say F and g are commutative (or that F and g commute) if
Recently, Choudhury and Maity [36] studied necessary conditions for the existence of a coupled fixed point in partially ordered Gmetric spaces. They obtained the following interesting result.
Theorem 1.16 [36]
Let $(X,\u2aaf)$ be a partially ordered set such that X is a complete Gmetric space and $F:X\times X\to X$ be a mapping having the mixed monotone property on X. Suppose there exists $k\in [0,1)$ such that
for all $x,y,z,u,v,w\in X$ for which $x\u2ab0u\u2ab0w$ and $y\u2aafv\u2aafz$, where either $u\ne w$ or $v\ne z$. If there exists ${x}_{0},{y}_{0}\in X$ such that
and either

(a)
F is continuous or

(b)
X has the following property:

(i)
if a nondecreasing sequence $\{{x}_{n}\}$ is such that ${x}_{n}\to x$, then ${x}_{n}\u2aafx$ for all n,

(ii)
if a nondecreasing sequence $\{{y}_{n}\}$ is such that ${y}_{n}\to y$, then $y\u2aaf{y}_{n}$ for all n,
then F has a coupled fixed point.
Let Θ denote the class of all functions $\phi :[0,\mathrm{\infty})\times [0,\mathrm{\infty})\to [0,\mathrm{\infty})$ satisfying the following condition:
for all $({r}_{1},{r}_{2})\in [0,\mathrm{\infty})\times [0,\mathrm{\infty})$ with ${r}_{1}+{r}_{2}>0$.
Remark 1.17 If the function $\phi :[0,\mathrm{\infty})\times [0,\mathrm{\infty})\to [0,\mathrm{\infty})$ satisfies (1.4), then, for any ${t}_{1},{t}_{2}\in [0,\mathrm{\infty})$ with either ${t}_{1}\ne 0$ or ${t}_{2}\ne 0$, $\phi ({t}_{1},{t}_{2})>0$. Indeed, suppose that ${t}_{1}\ne 0$, we have ${t}_{1}+{t}_{2}>0$. Taking ${t}_{n}^{1}={t}_{1}$ and ${t}_{n}^{2}={t}_{2}$ for all , we have, by (1.4), that
Example 1.18 The following are some examples of φ, for all $({t}_{1},{t}_{2})\in [0,\mathrm{\infty})\times [0,\mathrm{\infty})$,

(1)
$\phi ({t}_{1},{t}_{2})=kmax\{{t}_{1},{t}_{2}\}$ for $k>0$;

(2)
$\phi ({t}_{1},{t}_{2})=a{t}_{1}^{p}+b{t}_{2}^{q}$ for $a,b,p,q>0$;

(3)
$\phi ({t}_{1},{t}_{2})=\frac{1k}{2}({t}_{1}+{t}_{2})$ for some $k\in [0,1)$.
Using basically these concepts, Luong and Thuan [37] proved the following coupled fixed point theorem for nonlinear contractive mappings having the mixed monotone property in partially ordered Gmetric spaces.
Theorem 1.19 [[37], Theorem 2.1]
Let $(X,\le )$ be a partially ordered set and suppose that there exists a Gmetric G on X such that $(X,G)$ is a complete Gmetric space. Let $F:X\times X\to X$ be a mapping having the mixed monotone property on X. Suppose that there exists $\phi \in \mathrm{\Theta}$ such that
for all $x\u2ab0u\u2ab0w$ and $y\u2aafv\u2aafz$. Suppose that either

(a)
F is continuous or

(b)
X has the following property:

(i)
if a nondecreasing sequence $\{{x}_{n}\}$ is such that ${x}_{n}\to x$, then ${x}_{n}\u2aafx$ for all n,

(ii)
if a nondecreasing sequence $\{{y}_{n}\}$ is such that ${y}_{n}\to y$, then $y\u2aaf{y}_{n}$ for all n.
If there exist ${x}_{0},{y}_{0}\in X$ such that ${x}_{0}\u2aafF({x}_{0},{y}_{0})$ and ${y}_{0}\u2ab0F({y}_{0},{x}_{0})$, then F has a coupled fixed point in X.
Starting from the results in Choudhury and Maity [36] and Luong and Thuan [37], our main aim in this paper is to obtain more general coincidence point theorems and coupled common fixed point theorems for mixed monotone operators $F:X\times X\to X$ satisfying a contractive condition which is significantly more general that the corresponding conditions (1.3) and (1.5) in [36] and [37], respectively, thus extending many other related results in literature. We also provide an illustrative example in support of our results.
2 Coupled coincidence points
The first main result in this paper is the following coincidence point theorem which generalizes [[36], Theorem 3.1] and [[37], Theorem 2.1].
Theorem 2.1 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 $g:X\to X$ be a mapping and $F:X\times X\to X$ be a mapping having the mixed gmonotone property on X. Suppose that there exists $\phi \in \mathrm{\Theta}$ such that
for all $x,y,z,u,v,w\in X$ for which $g(x)\u2ab0g(u)\u2ab0g(w)$ and $g(y)\u2aafg(v)\u2aafg(z)$ where
If there exists ${x}_{0},{y}_{0}\in X$ such that
and suppose $F:X\times X\subseteq g(X)$, g is continuous and commutes with F, and also suppose either

(a)
F is continuous or

(b)
X has the following property:

(i)
if a nondecreasing sequence $\{{x}_{n}\}$ is such that ${x}_{n}\to x$, then ${x}_{n}\u2aafx$ for all n,

(ii)
if a nondecreasing sequence $\{{y}_{n}\}$ is such that ${y}_{n}\to y$, then $y\u2aaf{y}_{n}$ for all n,
then F and g have a coupled coincidence point, that is, there exists $(x,y)\in X\times X$ such that $g(x)=F(x,y)$ and $g(y)=F(y,x)$.
Proof Let ${x}_{0},{y}_{0}\in X$ such that $g({x}_{0})\u2aafF({x}_{0},{y}_{0})$ and $F({y}_{0},{x}_{0})\u2aafg({y}_{0})$. Since $F(X\times X)\subseteq g(X)$, we can choose ${x}_{1},{y}_{1}\in X$ such that $g({x}_{1})=F({x}_{0},{y}_{0})$ and $g({y}_{1})=F({y}_{0},{x}_{0})$. Again since $F(X\times X)\subseteq g(X)$, we can choose ${x}_{2},{y}_{2}\in X$ such that $g({x}_{2})=F({x}_{1},{y}_{1})$ and $g({y}_{2})=F({y}_{1},{x}_{1})$. Continuing this process, we can construct sequences $\{{x}_{n}\}$ and $\{{y}_{n}\}$ in X such that
Next, we show that
Since $g({x}_{0})\u2aafF({x}_{0},{y}_{0})=g({x}_{1})$ and $g({y}_{0})\u2ab0F({y}_{0},{x}_{0})=g({y}_{1})$, therefore, (2.3) holds for $n=0$. Next, suppose that (2.3) holds for some fixed $n\ge 0$, that is,
Since F has the mixed gmonotone property, from (2.4) and (1.1), we have
for all $x,y\in X$, and from (2.4) and (1.2), we have
for all $x,y\in X$. If we take $y={y}_{n}$ and $x={x}_{n}$ in (2.5), then we obtain
If we take $y={y}_{n+1}$ and $x={x}_{n+1}$ in (2.6), then
Now, from (2.7) and (2.8), we have
Therefore, by the mathematical induction, we conclude that (2.3) holds for all $n\ge 0$. Since $g({x}_{n})\u2aafg({x}_{n+1})$ and $g({y}_{n})\u2ab0g({y}_{n+1})$ for all $n\ge 0$ so from (2.1), we have
Setting
and
we have, by (2.10), that
As $\phi ({t}_{1},{t}_{2})\ge 0$ for all $({t}_{1},{t}_{2})\in [0,\mathrm{\infty})\times [0,\mathrm{\infty})$, we have
Then the sequence $\{{\omega}_{n}^{x}+{\omega}_{n}^{y}\}$ is decreasing. Therefore, there exists $\omega \ge 0$ such that
Now, we show that $\omega =0$. Suppose, to contrary, that $\omega >0$. From (2.12), the sequences $\{G(g({x}_{n+1}),g({x}_{n+1}),g({x}_{n}))\}$ and $\{G(g({y}_{n+1}),g({y}_{n+1}),g({y}_{n}))\}$ have convergent subsequences $\{G(g({x}_{n(j)+1}),g({x}_{n(j)+1}),g({x}_{n(j)}))\}$ and $\{G(g({y}_{n(j)+1}),g({y}_{n(j)+1}),g({y}_{n(j)}))\}$, respectively. Assume that
and
which gives that ${\omega}_{1}+{\omega}_{2}=\omega $. From (2.11), we have
Then taking the limit as $j\to \mathrm{\infty}$ in the above inequality, we obtain
which is a contradiction. Thus $\omega =0$; that is,
Next, we show that $\{g({x}_{n})\}$ and $\{g({y}_{n})\}$ are GCauchy sequences. On the contrary, assume that at least one of $\{g({x}_{n})\}$ or $\{g({y}_{n})\}$ is not a GCauchy sequence. By Lemma 1.8, there is an $\epsilon >0$ for which we can find subsequences $\{g({x}_{n(k)})\}$, $\{g({x}_{m(k)})\}$ of $\{g({x}_{n})\}$ and $\{g({y}_{n(k)})\}$, $\{g({y}_{m(k)})\}$ of $\{g({y}_{n})\}$ with $n(k)>m(k)\ge k$ such that
Further, corresponding to $m(k)$, we can choose $n(k)$ in such a way that it is the smallest integer with $n(k)>m(k)\ge k$ and satisfies (2.16). Then
By Lemma 1.10, we have
and
In view of (2.16)(2.19), we have
Then letting $k\to \mathrm{\infty}$ in the last inequality and using (2.15), we have
By Lemma 1.9 and Lemma 1.10, we have
and
It follows from (2.21) and (2.22) that
Since $n(k)>m(k)$, we get
and also, from (2.1),
From (2.23) and (2.24), we have
This implies that
From (2.20), the sequences $\{G(g({x}_{n(k)}),g({x}_{n(k)}),g({x}_{m(k)}))\}$ and $\{G(g({y}_{n(k)}),g({y}_{n(k)}),g({y}_{m(k)}))\}$ have subsequences converging to, say, ${\epsilon}_{1}$ and ${\epsilon}_{2}$, respectively, and ${\epsilon}_{1}+{\epsilon}_{2}=\epsilon >0$. By passing to subsequences, we may assume that
Taking $k\to \mathrm{\infty}$ in (2.25) and using (2.15), we have
which is a contradiction. Therefore, $\{g({x}_{n})\}$ and $\{g({y}_{n})\}$ are GCauchy sequences. By Gcompleteness of X, there exists $x,y\in X$ such that
This together with the continuity of g implies that
Now, suppose that assumption (a) holds. From (2.2) and the commutativity of F and g, we obtain
Similarly, we have
Hence, $(x,y)$ is a coupled coincidence point of F and g.
Finally, suppose that assumption (b) holds. Since $\{g({x}_{n})\}$ is nondecreasing satisfying $g({x}_{n})\to x$ and $\{g({y}_{n})\}$ is nonincreasing satisfying $g({y}_{n})\to y$, we have
Using the rectangle inequality and (2.1), we get
Letting $n\to +\mathrm{\infty}$ in the above inequality, we obtain that
which gives that $G(F(x,y),g(x),g(x))=G(F(y,x),g(y),g(y))=0$; that is, $F(x,y)=g(x)$ and $F(y,x)=g(y)$. Therefore, $(x,y)$ is a coupled coincidence point of F and g. The proof is complete. □
Setting $g(x)=x$ in Theorem 2.1, we obtain the following new result:
Theorem 2.2 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 mapping having the mixed monotone property on X. Suppose that there exists $\phi \in \mathrm{\Theta}$ such that
for all $x\u2ab0u\u2ab0w$ and $y\u2aafv\u2aafz$. Suppose that either

(a)
F is continuous or

(b)
X has the following property:

(i)
if a nondecreasing sequence $\{{x}_{n}\}$ is such that ${x}_{n}\to x$, then ${x}_{n}\u2aafx$ for all n,

(ii)
if a nondecreasing sequence $\{{y}_{n}\}$ is such that ${y}_{n}\to y$, then $y\u2aaf{y}_{n}$ for all n.
If there exists ${x}_{0},{y}_{0}\in X$ such that
then F has a coupled fixed point in X.
Remark 2.3 Theorem 2.2 is more general than [[37], Theorem 2.1] since the contractive condition (2.29) is weaker than (1.5), a fact which is clearly illustrated by the following example.
Example 2.4 Let be a set endowed with order $x\u2aafy\iff x\le y$. Let the mapping $G:X\times X\times X\to {\mathbb{R}}_{+}$ be defined by
for all $x,y,z\in X$. Then G is a Gmetric on X. Define the mapping $F:X\times X\to X$ by
Then the following properties hold:

(1)
F is mixed monotone;

(2)
F satisfies condition (2.29) but F does not satisfy condition (1.5).
Indeed, we first show that F does not satisfy condition (1.5). Assume to the contrary, that there exists $\phi \in \mathrm{\Phi}$, such that (1.5) holds. This means
Setting $x=u=w$ and $y\ne v\text{or}v\ne z\text{or}z\ne y$, by Remark 1.17, we get
which gives a contradiction. Hence, F does not satisfy condition (1.5). Now, we prove that (2.29) holds. Indeed, we have
By the above, we get exactly (2.29) with $\phi ({t}_{1},{t}_{2})=\frac{1}{8}({t}_{1}+{t}_{2})$.
By Theorem 2.1, we also obtain the following new result for the coupled coincidence point theorem for mixed gmonotone operators F satisfying a contractive condition.
Theorem 2.5 Let $(X,\u2aaf)$ be a partially ordered set and suppose that there exists a Gmetric G on X such that $(X,G)$ is a complete Gmetric space. Let $F:X\times X\to X$, $g:X\to X$ so that F is a mapping having the mixed gmonotone property on X. Suppose that there exists $\phi \in \mathrm{\Theta}$ such that
for all $g(x)\u2ab0g(u)\u2ab0g(w)$ and $g(y)\u2aafg(v)\u2aafg(z)$. Suppose that either

(a)
F is continuous or

(b)
X has the following property:

(i)
if a nondecreasing sequence $\{{x}_{n}\}$ is such that ${x}_{n}\to x$, then ${x}_{n}\u2aafx$ for all n,

(ii)
if a nondecreasing sequence $\{{y}_{n}\}$ is such that ${y}_{n}\to y$, then $y\u2aaf{y}_{n}$ for all n.
If there exists ${x}_{0},{y}_{0}\in X$ such that
then F and g have a coupled coincidence point.
Let Ψ denote the class of all functions $\psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ satisfying
Corollary 2.6 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 $g:X\to X$ be a mapping and $F:X\times X\to X$ be a mapping having the mixed gmonotone property on X. Suppose that there exists $\psi \in \mathrm{\Psi}$ such that
for all $x,y,z,u,v,w\in X$ for which $g(x)\u2ab0g(u)\u2ab0g(w)$ and $g(y)\u2aafg(v)\u2aafg(z)$ where
If there exists ${x}_{0},{y}_{0}\in X$ such that
and either

(a)
F is continuous or

(b)
X has the following property:

(i)
if a nondecreasing sequence $\{{x}_{n}\}$ is such that ${x}_{n}\to x$, then ${x}_{n}\u2aafx$ for all n,

(ii)
if a nondecreasing sequence $\{{y}_{n}\}$ is such that ${y}_{n}\to y$, then $y\u2aaf{y}_{n}$ for all n,
then F and g have a coupled coincidence point.
Proof In Theorem 2.1, taking $\phi ({t}_{1},{t}_{2})=\psi (max\{{t}_{1},{t}_{2}\})$ for all $({t}_{1},{t}_{2})\in [0,\mathrm{\infty}){\phantom{\rule{0.05em}{0ex}}}^{2}$, we get the desired results. □
Corollary 2.7 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 $g:X\to X$ be a mapping and $F:X\times X\to X$ be a mapping having the mixed gmonotone property on X. Suppose that there exists $\psi \in \mathrm{\Psi}$ such that
for all $x,y,z,u,v,w\in X$ for which $g(x)\u2ab0g(u)\u2ab0g(w)$ and $g(y)\u2aafg(v)\u2aafg(z)$ where
If there exists ${x}_{0},{y}_{0}\in X$ such that
and either

(a)
F is continuous or

(b)
X has the following property:

(i)
if a nondecreasing sequence $\{{x}_{n}\}$ is such that ${x}_{n}\to x$, then ${x}_{n}\u2aafx$ for all n,

(ii)
if a nondecreasing sequence $\{{y}_{n}\}$ is such that ${y}_{n}\to y$, then $y\u2aaf{y}_{n}$ for all n,
then F and g have a coupled coincidence point.
Proof In Theorem 2.1, taking $\phi ({t}_{1},{t}_{2})=\psi ({t}_{1}+{t}_{2})$ for all $({t}_{1},{t}_{2})\in [0,\mathrm{\infty}){\phantom{\rule{0.05em}{0ex}}}^{2}$, we get the desired results. □
3 Coupled common fixed point
Now, we shall prove the existence and uniqueness theorem of a coupled common fixed point. If $(X,\u2aaf)$ is a partially ordered set, we endow the product set $X\times X$ with the partial order relation:
Theorem 3.1 In addition to the hypotheses of Theorem 2.1, suppose that for all $(x,y),({x}^{\ast},{y}^{\ast})\in X\times X$, there exists $(u,v)\in X\times X$ such that $(F(u,v),F(v,u))$ is comparable with $(F(x,y),F(y,x))$ and $(F({x}^{\ast},{y}^{\ast}),F({y}^{\ast},{x}^{\ast}))$. Then F and g have a unique coupled common fixed point.
Proof From Theorem 2.1, the set of coupled coincidences is nonempty. Assume that $(x,y)$ and $({x}^{\ast},{y}^{\ast})$ are coupled coincidence points of F and g. We shall show that
By assumption, there exists $(u,v)\in X\times X$ such that $(F(u,v),F(v,u))$ is comparable with $(F(x,y),F(y,x))$ and $(F({x}^{\ast},{y}^{\ast}),F({y}^{\ast},{x}^{\ast}))$. Putting ${u}_{0}=u$, ${v}_{0}=v$ and choosing ${u}_{1},{v}_{1}\in X$ such that
Then, similarly as in the proof of Theorem 2.1, we can inductively define sequences $\{g({u}_{n})\}$ and $\{g({v}_{n})\}$ in X by
Since $(F({x}^{\ast},{y}^{\ast}),F({y}^{\ast},{x}^{\ast}))=(g({x}^{\ast}),g({y}^{\ast}))$ and $(F(u,v),F(v,u))=(g({u}_{1}),g({v}_{1}))$ are comparable, without restriction to the generality, we can assume that
and
This actually means that
and
Using that F is a mixed gmonotone mapping, we can inductively show that
and
Thus, from (2.1), we get
which implies that
that is, the sequence $\{G(g({u}_{n}),g(x),g(x))+G(g({v}_{n}),g(y),g(y))\}$ is decreasing. Therefore, there exists $\alpha \ge 0$ such that
We shall show that $\alpha =0$. Suppose, to the contrary, that $\alpha >0$. Therefore, $\{G(g({u}_{n}),g(x),g(x))\}$ and $\{G(g({v}_{n}),g(y),g(y))\}$ have subsequences converging to ${\alpha}_{1}$, ${\alpha}_{2}$, respectively, with
Taking the limit up to subsequences as $n\to \mathrm{\infty}$ in (3.2), we have
which is a contradiction. Thus, $\alpha =0$; that is,
which implies that
Similarly, one can show that
Therefore, from (3.3), (3.4) and the uniqueness of the limit, we get $g(x)=g({x}^{\ast})$ and $g(y)=g({y}^{\ast})$. So, (3.1) holds. Since $g(x)=F(x,y)$ and $g(y)=F(y,x)$, by commutativity of F and g, we have
Denote $g(x)=z$ and $g(y)=w$, then by (3.5), we get
Thus, $(z,w)$ is a coincidence point. Then from (3.1) with ${x}^{\ast}=z$ and ${y}^{\ast}=w$, we have $g(x)=g(z)$ and $g(y)=g(w)$, that is,
From (3.6) and (3.7), we get
Then, $(z,w)$ is a coupled common fixed point of F and g. To prove the uniqueness, assume that $(p,q)$ is another coupled common fixed point. Then by (3.1), we have $p=g(p)=g(z)=z$ and $q=g(q)=g(w)=w$. The proof is complete. □
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Acknowledgements
The authors would like to thank the referees for reading this paper carefully, providing valuable suggestions and comments, and pointing out a major error in the original version of this paper. Finally, the first author is supported by the ‘Centre of Excellence in Mathematics’ under the Commission on Higher Education, Ministry of Education, Thailand.
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Wangkeeree, R., Bantaojai, T. Coupled fixed point theorems for generalized contractive mappings in partially ordered Gmetric spaces. Fixed Point Theory Appl 2012, 172 (2012). https://doi.org/10.1186/168718122012172
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Keywords
 Gmetric space
 ordered set
 coupled coincidence point
 coupled fixed point
 mixed gmonotone property