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# Coupled fixed point theorems for generalized contractive mappings in partially ordered *G*-metric spaces

- Rabian Wangkeeree
^{1, 2}Email author and - Thanatporn Bantaojai
^{1}

**2012**:172

https://doi.org/10.1186/1687-1812-2012-172

© Wangkeeree and Bantaojai; licensee Springer 2012

**Received:**27 July 2012**Accepted:**17 September 2012**Published:**3 October 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 *G*-metric spaces. The results on fixed point theorems are generalizations of the recent results of Choudhury and Maity (Math. Comput. Model. 54:73-79, 2011) and Luong and Thuan (Math. Comput. Model. 55:1601-1609, 2012).

## Keywords

*G*-metric space- ordered set
- coupled coincidence point
- coupled fixed point
- mixed
*g*-monotone property

## 1 Introduction and preliminaries

One of the simplest and the most useful results in the fixed point theory is the Banach-Caccioppoli 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 Rodríguez-López [12] extended the results in [11] for non-decreasing mappings and obtained a unique solution for a first-order 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 *G*-metric 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 *G*-metric 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 non-decreasing (non-increasing) 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]

*mixed monotone property*if $F(x,y)$ is monotone non-decreasing in

*x*and is monotone non-increasing in

*y*, that is, for any $x,y\in X$,

The following concepts were introduced in [35].

**Definition 1.2** [35]

*F*has the

*mixed*

*g-monotone property*if $F(x,y)$ is

*g*-monotone non-decreasing in

*x*and it is

*g*-monotone non-increasing in

*y*, that is, for any $x,y\in X$

- (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** (*G*-metric space [27])

Let *X* be a non-empty 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 *G*-metric on *X* and the pair $(X,G)$ is called a *G*-metric space.

**Definition 1.5** [27]

Let *X* be a *G*-metric 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 *G*-convergent to *x*.

From this definition, we obtain that if ${x}_{n}\to x$ in a *G*-metric 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 *G*-metric 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 *G*-metric space, a sequence $\{{x}_{n}\}$ is called *G*-Cauchy 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*

*G*-

*metric space*,

*then 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)
$G({x}_{m},{x}_{n},x)\to 0$

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

**Lemma 1.8** [27]

*If*

*X*

*is a*

*G*-

*metric space*,

*then the following are equivalent*:

- (1)
*the sequence*$\{{x}_{n}\}$*is**G*-*Cauchy*; - (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* *G*-*metric space*, *then* $G(x,y,y)\le 2G(y,x,x)$ *for all* $x,y\in X$.

**Lemma 1.10** *If* *X* *is a* *G*-*metric 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 *G*-continuous at a point $x\in X$ if and only if it is *G* sequentially continuous at *x*, that is, whenever $\{{x}_{n}\}$ is *G*-convergent to *x*, $\{f({x}_{n})\}$ is ${G}^{\mathrm{\prime}}$-convergent to $f(x)$.

**Definition 1.12** [27]

*G*-metric space

*X*is called a symmetric

*G*-metric space if

for all $x,y\in X$.

**Definition 1.13** [27]

A *G*-metric space *X* is said to be *G*-complete (or a complete *G*-metric space) if every *G*-Cauchy sequence in *X* is convergent in *X*.

**Definition 1.14** Let *X* be a *G*-metric space. A mapping $F:X\times X\to X$ is said to be continuous if for any two *G*-convergent sequences $\{{x}_{n}\}$ and $\{{y}_{n}\}$ converging to *x* and *y*, respectively, $\{F({x}_{n},{y}_{n})\}$ is *G*-convergent to $F(x,y)$.

**Definition 1.15**Let

*X*be a non-empty 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 *G*-metric 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*

*G*-

*metric 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 non*-*decreasing sequence*$\{{x}_{n}\}$*is such that*${x}_{n}\to x$,*then*${x}_{n}\u2aafx$*for all**n*, - (ii)
*if a non*-*decreasing 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*.

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)
$\phi ({t}_{1},{t}_{2})=\frac{1-k}{2}({t}_{1}+{t}_{2})$ 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 *G*-metric spaces.

**Theorem 1.19** [[37], Theorem 2.1]

*Let*$(X,\le )$

*be a partially ordered set and suppose that there exists a G*-

*metric*

*G*

*on*

*X*

*such that*$(X,G)$

*is a complete G*-

*metric 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 non*-*decreasing sequence*$\{{x}_{n}\}$*is such that*${x}_{n}\to x$,*then*${x}_{n}\u2aafx$*for all**n*, - (ii)
*if a non*-*decreasing 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*

*G*-

*metric on*

*X*

*such that*$(X,G)$

*is a complete*

*G*-

*metric space*.

*Let*$g:X\to X$

*be a mapping and*$F:X\times X\to X$

*be a mapping having the mixed*

*g*-

*monotone 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 non*-*decreasing sequence*$\{{x}_{n}\}$*is such that*${x}_{n}\to x$,*then*${x}_{n}\u2aafx$*for all**n*, - (ii)
*if a non*-*decreasing 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

*F*has the mixed

*g*-monotone property, from (2.4) and (1.1), we have

*G*-Cauchy sequences. On the contrary, assume that at least one of $\{g({x}_{n})\}$ or $\{g({y}_{n})\}$ is not a

*G*-Cauchy 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

*G*-Cauchy sequences. By

*G*-completeness of

*X*, there exists $x,y\in X$ such that

*g*implies that

*F*and

*g*, we obtain

Hence, $(x,y)$ is a coupled coincidence point of *F* and *g*.

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*

*G*-

*metric on*

*X*

*such that*$(X,G)$

*is a complete*

*G*-

*metric 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 non*-*decreasing sequence*$\{{x}_{n}\}$*is such that*${x}_{n}\to x$,*then*${x}_{n}\u2aafx$*for all**n*, - (ii)
*if a non*-*decreasing 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

*G*is a

*G*-metric on

*X*. Define the mapping $F:X\times X\to X$ by

- (1)
*F*is mixed monotone; - (2)
*F*satisfies condition (2.29) but*F*does not satisfy condition (1.5).

*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

*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 *g*-monotone operators *F* satisfying a contractive condition.

**Theorem 2.5**

*Let*$(X,\u2aaf)$

*be a partially ordered set and suppose that there exists a*

*G*-

*metric*

*G*

*on*

*X*

*such that*$(X,G)$

*is a complete*

*G*-

*metric space*.

*Let*$F:X\times X\to X$, $g:X\to X$

*so that*

*F*

*is a mapping having the mixed g*-

*monotone 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 non*-*decreasing sequence*$\{{x}_{n}\}$*is such that*${x}_{n}\to x$,*then*${x}_{n}\u2aafx$*for all**n*, - (ii)
*if a non*-*decreasing 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*.

**Corollary 2.6**

*Let*$(X,\u2aaf)$

*be a partially ordered set and*

*G*

*be a*

*G*-

*metric on*

*X*

*such that*$(X,G)$

*is a complete*

*G*-

*metric space*.

*Let*$g:X\to X$

*be a mapping and*$F:X\times X\to X$

*be a mapping having the mixed*

*g*-

*monotone 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 non*-*decreasing sequence*$\{{x}_{n}\}$*is such that*${x}_{n}\to x$,*then*${x}_{n}\u2aafx$*for all**n*, - (ii)
*if a non*-*decreasing 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*

*G*-

*metric on*

*X*

*such that*$(X,G)$

*is a complete*

*G*-

*metric space*.

*Let*$g:X\to X$

*be a mapping and*$F:X\times X\to X$

*be a mapping having the mixed*

*g*-

*monotone 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 non*-*decreasing sequence*$\{{x}_{n}\}$*is such that*${x}_{n}\to x$,*then*${x}_{n}\u2aafx$*for all**n*, - (ii)
*if a non*-*decreasing 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

**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 non-empty. Assume that $(x,y)$ and $({x}^{\ast},{y}^{\ast})$ are coupled coincidence points of

*F*and

*g*. We shall show that

*X*by

*F*is a mixed

*g*-monotone mapping, we can inductively show that

*F*and

*g*, we have

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. □

## Declarations

### 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.

## Authors’ Affiliations

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