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Coupled coincidence point results for $(\psi ,\phi )$weakly contractive mappings in partially ordered ${G}_{b}$metric spaces
 Zead Mustafa^{1, 2}Email author,
 Jamal Rezaei Roshan^{3} and
 Vahid Parvaneh^{4}
https://doi.org/10.1186/168718122013206
© Mustafa et al.; licensee Springer 2013
 Received: 27 December 2012
 Accepted: 18 July 2013
 Published: 2 August 2013
Abstract
In this paper we present some coupled coincidence point results for $(\psi ,\phi )$weakly contractive mappings in the setup of partially ordered ${G}_{b}$metric spaces. Our results extend the results of Cho et al. (Fixed Point Theory Appl. 2012:8, 2012) and the results of Choudhury and Maity (Math. Comput. Model. 54:7379, 2011). Moreover, examples of the main results are given.
MSC: 47H10, 54H25.
Keywords
 ${G}_{b}$metric space
 partially ordered set
 coupled coincidence point
 common coupled fixed point
 mixed monotone property
1 Introduction
Existence of coupled fixed points in partially ordered metric spaces was first investigated in 1987 by Guo and Lakshmikantham [1]. Also, Bhaskar and Lakshmikantham [2] established some coupled fixed point theorems for a mixed monotone mapping in partially ordered metric spaces.
Recently, Lakshmikantham and Ćirić [3] introduced the notions of mixed gmonotone mapping and coupled coincidence point and proved some coupled coincidence point and common coupled fixed point theorems in partially ordered complete metric spaces.
Definition 1.1 [3]
Let $(X,\u2aaf)$ be a partially ordered set, and let $F:X\times X\to X$ and $g:X\to X$ be two mappings. F has the mixed gmonotone property if F is monotone gnondecreasing in its first argument and is monotone gnonincreasing in its second argument, that is, for all ${x}_{1},{x}_{2}\in X$, $g{x}_{1}\u2aafg{x}_{2}$ implies $F({x}_{1},y)\u2aafF({x}_{2},y)$ for any $y\in X$ and for all ${y}_{1},{y}_{2}\in X$, $g{y}_{1}\u2aafg{y}_{2}$ implies $F(x,{y}_{1})\u2ab0F(x,{y}_{2})$ for any $x\in X$.
Definition 1.2 [2]
An element $(x,y)\in X\times X$ is called a coupled fixed point of the mapping $F:X\times X\to X$ if $x=F(x,y)$ and $y=F(y,x)$.
Definition 1.3 [3]
 (1)
a coupled coincidence point of mappings $F:X\times X\to X$ and $g:X\to X$ if $g(x)=F(x,y)$ and $g(y)=F(y,x)$;
 (2)
a common coupled fixed point of mappings $F:X\times X\to X$ and $g:X\to X$ if $x=g(x)=F(x,y)$ and $y=g(y)=F(y,x)$.
Recently, Abbas et al. [4] introduced the concept of wcompatible mappings to obtain some coupled coincidence point results in a cone metric space.
Definition 1.4 [4]
Two mappings $F:X\times X\to X$ and $g:X\to X$ are called wcompatible if $g(F(x,y))=F(gx,gy)$, whenever $g(x)=F(x,y)$ and $g(y)=F(y,x)$.
The concept of generalized metric space, or a Gmetric space, was introduced by Mustafa and Sims [5]. Mustafa and others studied fixed point theorems for mappings satisfying different contractive conditions (see [5–22]).
Definition 1.5 (Gmetric space [5])
 (G1)
$G(x,y,z)=0$ iff $x=y=z$;
 (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 $z\ne y$;
 (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 1.6 [5]
Let $(X,G)$ 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 the sequence $\{{x}_{n}\}$ if ${lim}_{n,m\to \mathrm{\infty}}G(x,{x}_{n},{x}_{m})=0$ and one says that the sequence $\{{x}_{n}\}$ is Gconvergent to x. Thus, if ${x}_{n}\to x$ in a Gmetric space $(X,G)$, then for any $\epsilon >0$, there exists a positive integer N such that $G(x,{x}_{n},{x}_{m})<\epsilon $ for all $n,m\ge N$.
Definition 1.7 [5]
Let $(X,G)$ 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 $n,m,l\to \mathrm{\infty}$.
Lemma 1.8 [5]
 (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 $n\to \mathrm{\infty}$.
Lemma 1.9 [23]
If $(X,G)$ is a Gmetric space, then $\{{x}_{n}\}$ is a GCauchy sequence if and only if for every $\epsilon >0$, there exists a positive integer N such that $G({x}_{n},{x}_{m},{x}_{m})<\epsilon $ for all $m>n\ge N$.
Definition 1.10 [5]
A Gmetric space $(X,G)$ is said to be Gcomplete if every GCauchy sequence in $(X,G)$ is convergent in X.
Definition 1.11 [5]
Let $(X,G)$ and $({X}^{\mathrm{\prime}},{G}^{\mathrm{\prime}})$ be two Gmetric spaces. Then a function $f:X\to {X}^{\mathrm{\prime}}$ is Gcontinuous at a point $x\in X$ if and only if it is Gsequentially 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 [23]
Let $(X,G)$ be a Gmetric space. A mapping $F:X\times X\to X$ is said to be continuous at $(x,y)$ 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 1.13 [3]
Let X be a nonempty set. We say that the mappings $F:X\times X\to X$ and $g:X\to X$ are commutative if $g(F(x,y))=F(gx,gy)$ for all $x,y\in X$.
Choudhury and Maity [23] established some coupled fixed point results for mappings with mixed monotone property in partially ordered Gmetric spaces. They obtained the following results.
Theorem 1.14 ([23], Theorem 3.1)
for all $x\u2aafu\u2aafw$ and $y\u2ab0v\u2ab0z$, 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 ${y}_{0}\u2ab0F({y}_{0},{x}_{0})$, then F has a coupled fixed point in X, that is, there exist $x,y\in X$ such that $x=F(x,y)$ and $y=F(y,x)$.
Theorem 1.15 ([23], Theorem 3.2)
 (i)
if a nondecreasing sequence $\{{x}_{n}\}\to x$, then ${x}_{n}\u2aafx$ for all n, and
 (ii)
if a nonincreasing sequence $\{{y}_{n}\}\to y$, then ${y}_{n}\u2ab0y$ for all n,
then F has a coupled fixed point.
The concept of an altering distance function was introduced by Khan et al. [24] as follows.
 1.
ψ is continuous and nondecreasing.
 2.
$\psi (t)=0$ if and only if $t=0$.
In [25], Cho et al. studied coupled coincidence and coupled common fixed point theorems in ordered generalized metric spaces for a nonlinear contractive condition related to a pair of altering distance functions.
Theorem 1.17 ([25], Theorem 3.1)
for all $x,y,u,v,w,z\in X$ with $gw\u2aafgu\u2aafgx$ and $gy\u2aafgv\u2aafgz$. Also, suppose 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.
Definition 1.18 [25]
 (i)
If $\{{x}_{n}\}$ is a nondecreasing sequence with ${x}_{n}\to x$, then ${x}_{n}\u2aafx$ for all $n\in \mathbb{N}$.
 (ii)
If $\{{x}_{n}\}$ is a nonincreasing sequence with ${x}_{n}\to x$, then ${x}_{n}\u2ab0x$ for all $n\in \mathbb{N}$.
Theorem 1.19 ([25], Theorem 3.2)
Let $(X,\u2aaf)$ be a partially ordered set, and let G be a Gmetric on X such that $(X,G,\u2aaf)$ is regular. Assume that there exist altering distance functions ψ, φ and mappings $F:X\times X\to X$ and $g:X\to X$ satisfying (1.2) for all $x,y,u,v,w,z\in X$ with $gw\u2aafgu\u2aafgx$ and $gy\u2aafgv\u2aafgz$. Suppose also that $(g(X),G)$ is Gcomplete, 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.
So far, many authors have discussed fixed point results, periodic point results, coupled and tripled fixed point results and many other related topics in fixed point theory in different extensions of the concept of metric spaces such as bmetric spaces, partial metric spaces, cone metric spaces, Gmetric spaces, etc. (see, e.g., [6, 14, 20, 26–33]). Motivated by the work in [34], Aghajani et al., in a submitted paper [35], extended the notion of Gmetric space to the concept of ${G}_{b}$metric space (see Section 2). In this paper, we obtain some coupled coincidence point theorems for nonlinear $(\psi ,\phi )$weakly contractive mappings in partially ordered ${G}_{b}$metric spaces. These results generalize and modify several comparable results in the literature.
2 Mathematical preliminaries
Aghajani et al. in [35] introduced the concept of generalized bmetric spaces (${G}_{b}$metric spaces) and then they presented some basic properties of ${G}_{b}$metric spaces.
The following is their definition of ${G}_{b}$metric spaces.
Definition 2.1 [35]
Let X be a nonempty set, and let $s\ge 1$ be a given real number. Suppose that a mapping $G:X\times X\times X\to {\mathbb{R}}^{+}$ satisfies:

(${G}_{b}1$) $G(x,y,z)=0$ if $x=y=z$,

(${G}_{b}2$) $0<G(x,x,y)$ for all $x,y\in X$ with $x\ne y$,

(${G}_{b}3$) $G(x,x,y)\le G(x,y,z)$ for all $x,y,z\in X$ with $y\ne z$,

(${G}_{b}4$) $G(x,y,z)=G(p\{x,y,z\})$, where p is a permutation of $x,y,z$ (symmetry),

(${G}_{b}5$) $G(x,y,z)\le s[G(x,a,a)+G(a,y,z)]$ for all $x,y,z,a\in X$ (rectangle inequality).
Then G is called a generalized bmetric and the pair $(X,G)$ is called a generalized bmetric space or a ${G}_{b}$metric space.
It should be noted that the class of ${G}_{b}$metric spaces is effectively larger than that of Gmetric spaces given in [5]. Indeed, each Gmetric space is a ${G}_{b}$metric space with $s=1$ (see also [36]).
The following example shows that a ${G}_{b}$metric on X need not be a Gmetric on X.
Example 2.2 [35]
Let $(X,G)$ be a Gmetric space, and let ${G}_{\ast}(x,y,z)=G{(x,y,z)}^{p}$, where $p>1$ is a real number.
So, ${G}_{\ast}$ is a ${G}_{b}$metric with $s={2}^{p1}$.
for all $x,y,z\in \mathbb{R}$ (see [5]). Then ${G}_{\ast}(x,y,z)=G{(x,y,z)}^{2}=\frac{1}{9}{(xy+yz+xz)}^{2}$ is a ${G}_{b}$metric on ℝ with $s={2}^{21}=2$, but it is not a Gmetric on ℝ. To see this, let $x=3$, $y=5$, $z=7$ and $a=\frac{7}{2}$. Hence, we get ${G}_{\ast}(3,5,7)=\frac{64}{9}$, ${G}_{\ast}(3,\frac{7}{2},\frac{7}{2})=\frac{1}{9}$, ${G}_{\ast}(\frac{7}{2},5,7)=\frac{49}{9}$. Therefore, ${G}_{\ast}(3,5,7)=\frac{64}{9}\nleqq \frac{50}{9}={G}_{\ast}(3,\frac{7}{2},\frac{7}{2})+{G}_{\ast}(\frac{7}{2},5,7)$.
So, $G(0,0,2)>G(0,2,1)$.
However, $G(x,y,z)=max\{d(x,y),d(y,z),d(z,x)\}$ is a ${G}_{b}$metric on ℝ with $s=2$.
Now we present some definitions and propositions in a ${G}_{b}$metric space.
Definition 2.4 [35]
By some straightforward calculations, we can establish the following.
Proposition 2.5 [35]
 (1)
if $G(x,y,z)=0$, then $x=y=z$,
 (2)
$G(x,y,z)\le s(G(x,x,y)+G(x,x,z))$,
 (3)
$G(x,y,y)\le 2sG(y,x,x)$,
 (4)
$G(x,y,z)\le s(G(x,a,z)+G(a,y,z))$.
Definition 2.6 [35]
Let X be a ${G}_{b}$metric space. We define ${d}_{G}(x,y)=G(x,y,y)+G(x,x,y)$ for all $x,y\in X$. It is easy to see that ${d}_{G}$ defines a bmetric d on X, which we call the bmetric associated with G.
Proposition 2.7 [35]
Let X be a ${G}_{b}$metric space. Then, for any ${x}_{0}\in X$ and any $r>0$, if $y\in {B}_{G}({x}_{0},r)$, then there exists $\delta >0$ such that ${B}_{G}(y,\delta )\subseteq {B}_{G}({x}_{0},r)$.
Proof For $s=1$, see Proposition 4 in [5]. Suppose that $s>1$ and let $y\in {B}_{G}({x}_{0},r)$. If $y={x}_{0}$, then we choose $\delta =r$. If $y\ne {x}_{0}$, then $0<G({x}_{0},y,y)<r$. Let $A=\{n\in \mathbb{N}\mid \frac{r}{4{s}^{n+2}}<G({x}_{0},y,y)\}$. Since ${lim}_{n\to \mathrm{\infty}}\frac{1}{4{s}^{n+2}}=0$, hence, for $0<\u03f5=\frac{G({x}_{0},y,y)}{r}<1$, there exists ${n}_{0}\in \mathbb{N}$ such that $\frac{1}{4{s}^{{n}_{0}+2}}<\frac{G({x}_{0},y,y)}{r}$ or $\frac{r}{4{s}^{{n}_{0}+2}}<G({x}_{0},y,y)$. Hence, ${n}_{0}\in A$ and A is a nonempty set, then by the wellordering principle, A has a least element m. Since $m1\notin A$, we have $G({x}_{0},y,y)\le \frac{r}{4{s}^{m+1}}$. Now, if $G({x}_{0},y,y)=\frac{r}{4{s}^{m+1}}$, then we choose $\delta =\frac{r}{4{s}^{m+1}}$ and if $G({x}_{0},y,y)<\frac{r}{4{s}^{m+1}}$, we choose $\delta =\frac{r}{4{s}^{m+1}}G({x}_{0},y,y)$. □
is a base of a topology $\tau (G)$ on X, which we call ${G}_{b}$metric topology.
Now, we generalize Proposition 5 in [5] for a ${G}_{b}$metric space as follows.
Proposition 2.8 [35]
Thus, every ${G}_{b}$metric space is topologically equivalent to a bmetric space. This allows us to readily transport many concepts and results from bmetric spaces into the ${G}_{b}$metric space setting.
Definition 2.9 [35]
 (1)
${G}_{b}$Cauchy if for each $\epsilon >0$ there exists a positive integer ${n}_{0}$ such that for all $m,n,l\ge {n}_{0}$, $G({x}_{n},{x}_{m},{x}_{l})<\epsilon $;
 (2)
${G}_{b}$convergent to a point $x\in X$ if for each $\epsilon >0$ there exists a positive integer ${n}_{0}$ such that for all $m,n\ge {n}_{0}$, $G({x}_{n},{x}_{m},x)<\epsilon $.
Using the above definitions, we can easily prove the following two propositions.
Proposition 2.10 [35]
 (1)
The sequence $\{{x}_{n}\}$ is ${G}_{b}$Cauchy.
 (2)
For any $\epsilon >0$, there exists ${n}_{0}\in \mathbb{N}$ such that $G({x}_{n},{x}_{m},{x}_{m})<\epsilon $ for all $m,n\ge {n}_{0}$.
Proposition 2.11 [35]
 (1)
$\{{x}_{n}\}$ is ${G}_{b}$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 $n\to +\mathrm{\infty}$.
Definition 2.12 [35]
A ${G}_{b}$metric space X is called ${G}_{b}$complete if every ${G}_{b}$Cauchy sequence is ${G}_{b}$convergent in X.
Definition 2.13 Let $(X,G)$ and $({X}^{\mathrm{\prime}},{G}^{\mathrm{\prime}})$ be two ${G}_{b}$metric spaces. Then a function $f:X\to {X}^{\mathrm{\prime}}$ is ${G}_{b}$continuous at a point $x\in X$ if and only if it is ${G}_{b}$sequentially continuous at x, that is, whenever $\{{x}_{n}\}$ is ${G}_{b}$convergent to x, $\{f({x}_{n})\}$ is ${G}_{b}^{\mathrm{\prime}}$convergent to $f(x)$.
Definition 2.14 Let $(X,G)$ be a ${G}_{b}$metric space. A mapping $F:X\times X\to X$ is said to be continuous if for any two ${G}_{b}$convergent sequences $\{{x}_{n}\}$ and $\{{y}_{n}\}$ converging to x and y, respectively, $\{F({x}_{n},{y}_{n})\}$ is ${G}_{b}$convergent to $F(x,y)$.
Mustafa and Sims proved that each Gmetric function $G(x,y,z)$ is jointly continuous in all three of its variables (see Proposition 8 in [5]). But, in general, a ${G}_{b}$metric function $G(x,y,z)$ for $s>1$ is not jointly continuous in all its variables. Now, we present an example of a discontinuous ${G}_{b}$metric.
Thus, $(X,D)$ is a bmetric space with $s=\frac{5}{2}$ (see corrected Example 3 from [37]).
Hence, ${lim}_{n\to \mathrm{\infty}}G({x}_{n},{y}_{n},{z}_{n})\ne G(x,y,z)$.
So, from the above discussion, we need the following simple lemma about the ${G}_{b}$convergent sequences in the proof of our main result.
In particular, if $x=y=z$, then we have ${lim}_{n\to \mathrm{\infty}}G({x}_{n},{y}_{n},{z}_{n})=0$.
Taking the lower limit as $n\to \mathrm{\infty}$ in the first inequality and the upper limit as $n\to \mathrm{\infty}$ in the second inequality, we obtain the desired result. □
3 Main results
Our first result is the following.
for every pair $(x,y),(u,v),(w,t)\in X\times X$ such that $gx\u2aafgu\u2aafgw$ and $gy\u2ab0gv\u2ab0gt$, or $gw\u2aafgu\u2aafgx$ and $gt\u2ab0gv\u2ab0gy$, where $\psi ,\phi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ are altering distance functions.
 1.
$F(X\times X)\subseteq g(X)$.
 2.
F has the mixed gmonotone property.
 3.
F is continuous.
 4.
g is continuous and commutes with F.
If there exist ${x}_{0},{y}_{0}\in X$ such that $g{x}_{0}\u2aafF({x}_{0},{y}_{0})$ and $g{y}_{0}\u2ab0F({y}_{0},{x}_{0})$, then F and g have a coupled coincidence point in X.
Proof Let ${x}_{0},{y}_{0}\in X$ be such that $g{x}_{0}\u2aafF({x}_{0},{y}_{0})$ and $g{y}_{0}\u2ab0F({y}_{0},{x}_{0})$. 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 $g{y}_{1}=F({y}_{0},{x}_{0})$. Then, $g{x}_{0}\u2aafF({x}_{0},{y}_{0})=g{x}_{1}$ and $g{y}_{0}\u2ab0F({y}_{0},{x}_{0})=g{y}_{1}$. Since F has the mixed gmonotone property, we have $F({x}_{0},{y}_{0})\u2aafF({x}_{1},{y}_{0})\u2aafF({x}_{1},{y}_{1})$ and $F({y}_{0},{x}_{0})\u2ab0F({y}_{1},{x}_{0})\u2ab0F({y}_{1},{x}_{1})$, that is, $g{x}_{0}\u2aafg{x}_{1}$ and $g{y}_{0}\u2ab0g{y}_{1}$. In this way, we construct the sequences $\{{z}_{n}\}$ and $\{{t}_{n}\}$ as ${z}_{n}=g{x}_{n}=F({x}_{n1},{y}_{n1})$ and ${t}_{n}=g{y}_{n}=F({y}_{n1},{x}_{n1})$ for all $n\ge 1$, inductively.
One can easily show that for all $n\in \mathbb{N}$, ${z}_{n1}\u2aaf{z}_{n}$ and ${t}_{n1}\u2ab0{t}_{n}$.
We complete the proof in three steps.
we shall prove that ${lim}_{n\to \mathrm{\infty}}{\delta}_{n}=0$.
that is, $\phi ({\delta}_{n1})\le 0$. By our assumptions, we have ${\delta}_{n1}=0$, which contradicts (3.3). Therefore, for all $n\ge 1$, we deduce that ${\delta}_{n+1}\le {\delta}_{n}$, that is, $\{{\delta}_{n}\}$ is a nonincreasing sequence of nonnegative real numbers. Thus, there exists $r\ge 0$ such that ${lim}_{n\to \mathrm{\infty}}{\delta}_{n}=r$.
or, equivalently, ${lim\hspace{0.17em}inf}_{k\to \mathrm{\infty}}max\{G({z}_{m(k)},{z}_{n(k)1},{z}_{n(k)1}),G({t}_{m(k)},{t}_{n(k)1},{t}_{n(k)1})\}=0$, which is a contradiction to (3.8). Consequently, $\{{z}_{n}\}$ and $\{{t}_{n}\}$ are ${G}_{b}$Cauchy.
Step III. We shall show that F and g have a coupled coincidence point.
Now, we prove that $(z,t)$ is a coupled coincidence point of F and g.
From the continuity of F, $\{g(g{x}_{n+1})\}$ is ${G}_{b}$convergent to $F(z,t)$ and $\{g(g{y}_{n+1})\}$ is ${G}_{b}$convergent to $F(t,z)$. By uniqueness of the limit, we have $F(z,t)=gz$ and $F(t,z)=gt$. That is, g and F have a coupled coincidence point. □
In the following theorem, we omit the continuity and commutativity assumptions of g and F.
Theorem 3.2 Let $(X,\u2aaf)$ be a partially ordered set, and let G be a ${G}_{b}$metric on X such that $(X,G,\u2aaf)$ is a regular ${G}_{b}$metric space. Suppose that $F:X\times X\to X$ and $g:X\to X$ are two mappings satisfying (3.1) for every pair $(x,y),(u,v),(w,t)\in X\times X$ such that $gx\u2aafgu\u2aafgw$ and $gy\u2ab0gv\u2ab0gt$, or $gw\u2aafgu\u2aafgx$ and $gt\u2ab0gv\u2ab0gy$, where ψ and φ are the same as in Theorem 3.1.
Let $F(X\times X)\subseteq g(X)$, $g(X)$ is a ${G}_{b}$complete subset of X and F has the mixed gmonotone property.
If there exist ${x}_{0},{y}_{0}\in X$ such that $g{x}_{0}\u2aafF({x}_{0},{y}_{0})$ and $g{y}_{0}\u2ab0F({y}_{0},{x}_{0})$, then F and g have a coupled coincidence point in X.
Moreover, if $g{y}_{0}$ and $g{x}_{0}$ are comparable, then $gu=F(u,v)=F(v,u)=gv$, and if F and g are wcompatible, then F and g have a coupled coincidence point of the form $(t,t)$.
Now, we prove that $F(u,v)=gu$ and $F(v,u)=gv$.
Since $\{g{x}_{n}\}$ is nondecreasing and $\{g{y}_{n}\}$ is nonincreasing, from regularity of X we have $g{x}_{n}\u2aafgu$ and $g{y}_{n}\u2ab0gv$ for all $n\ge 0$.
and hence, $gu=F(u,v)$ and $F(v,u)=gv$.
Now, let $g{y}_{0}\u2aafg{x}_{0}$. Then $gv\u2aafg{y}_{n}\u2aafg{y}_{0}\u2aafg{x}_{0}\u2aafg{x}_{n}\u2aafgu$ for all $n\in \mathbb{N}$. We shall show that $gu=gv$.
Therefore, $max\{G(gu,gu,gv),G(gv,gv,gu)\}=0$. Hence, we get that $G(gv,gv,gu)=G(gu,gu,gv)=0$, and this means that $gu=gv$.
Now, let $t=gu=gv$. Since F and g are wcompatible, then $gt=g(gu)=g(F(u,v))=F(gu,gv)=F(t,t)$. Thus, F and g have a coupled coincidence point of the form $(t,t)$. □
Remark 3.3 In Theorems 3.1 and 3.2, we have extended the results of Cho et al. [25] (Theorems 1.17 and 1.19).
for all $(x,y),(u,v)\in X\times X$ [3].
In the following theorem, we give a sufficient condition for the uniqueness of the common coupled fixed point. Similar conditions were introduced by many authors (see, e.g., [2, 3, 9, 20, 38–45]).
Theorem 3.4 Let all the conditions of Theorem 3.1 be fulfilled, and let the following condition hold:
For arbitrary two points $(x,y)$, $(z,t)$, there exists $(u,v)$ such that $(F(u,v),F(v,u))$ is comparable with $(F(x,y),F(y,x))$ and $(F(z,t),F(t,z))$.
Then F and g have a unique common coupled fixed point.
We shall show that $g(x)=g(z)$ and $g(y)=g(t)$.
Suppose that $(x,y)$ and $(z,t)$ are not comparable. Choose an element $(u,v)\in X\times X$ such that $(F(u,v),F(v,u))$ is comparable with $(F(x,y),F(y,x))$ and $(F(z,t),F(t,z))$.
Let ${u}_{0}=u$, ${v}_{0}=v$ and choose ${u}_{1},{v}_{1}\in X$ so that $g{u}_{1}=F({u}_{0},{v}_{0})$ and $g{v}_{1}=F({v}_{0},{u}_{0})$. Then, similarly as in the proof of Theorem 3.1, we can inductively define sequences $\{g{u}_{n}\}$ and $\{g{v}_{n}\}$ such that $g{u}_{n+1}=F({u}_{n},{v}_{n})$ and $g{v}_{n+1}=F({v}_{n},{u}_{n})$. Since $(gx,gy)=(F(x,y),F(y,x))$ and $(F(u,v),F(v,u))=(g{u}_{1},g{v}_{1})$ are comparable, we may assume that $(gx,gy)\u2aaf(g{u}_{1},g{v}_{1})$. Then $gx\u2aafg{u}_{1}$ and $gy\u2ab0g{v}_{1}$. Using the mathematical induction, it is easy to prove that $gx\u2aafg{u}_{n}$ and $gy\u2ab0g{v}_{n}$ for all $n\in \mathbb{N}$.
Let ${\gamma}_{n}=max\{G(gx,gx,g{u}_{n}),G(gy,gy,g{v}_{n})\}$. We shall show that ${lim}_{n\to \mathrm{\infty}}{\gamma}_{n}=0$. First, assume that ${\gamma}_{n}=0$ for an $n\ge 1$.
So, from the properties of ψ and φ, we deduce that ${\gamma}_{n+1}=0$. Repeating this process, we can show that ${\gamma}_{m}=0$ for all $m\ge n$. So, ${lim}_{n\to \mathrm{\infty}}{\gamma}_{n}=0$.
Now, let ${\gamma}_{n}\ne 0$ for all n, and let ${\gamma}_{n}<{\gamma}_{n+1}$ for some n.
This implies that ${\gamma}_{n}=0$, which is a contradiction.
So, $\{g{u}_{n}\}\to gx$ and $\{g{v}_{n}\}\to gy$.
that is, $\{g{u}_{n}\}\to gz$ and $\{g{v}_{n}\}\to gt$. Finally, since the limit is unique, $gx=gz$ and $gy=gt$.
Since $gx=F(x,y)$ and $gy=F(y,x)$, by the commutativity of F and g, we have $g(gx)=g(F(x,y))=F(gx,gy)$ and $g(gy)=g(F(y,x))=F(gy,gx)$. Let $gx=a$ and $gy=b$. Then $ga=F(a,b)$ and $gb=F(b,a)$. Thus, $(a,b)$ is another coupled coincidence point of F and g. Then $a=gx=ga$ and $b=gy=gb$. Therefore, $(a,b)$ is a coupled common fixed point of F and g.
To prove the uniqueness of a coupled common fixed point, assume that $(p,q)$ is another coupled common fixed point of F and g. Then $p=gp=F(p,q)$ and $q=gq=F(q,p)$. Since $(p,q)$ is a coupled coincidence point of F and g, we have $gp=ga$ and $gq=gb$. Thus, $p=gp=ga=a$ and $q=gq=gb=b$. Hence, the coupled common fixed point is unique. □
The following corollary can be deduced from our previous obtained results.
for every pair $(x,y),(u,v),(w,t)\in X\times X$ such that $x\u2aafu\u2aafw$ and $y\u2ab0v\u2ab0t$, or $w\u2aafu\u2aafx$ and $t\u2ab0v\u2ab0y$, where $\psi ,\phi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ are altering distance functions.
 (a)
F is continuous, or
 (b)
X is regular.
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.
Proof If F satisfies (3.11), then F satisfies (3.1). So, the result follows from Theorems 3.1 and 3.2. □
In Theorems 3.1 and 3.2, if we take $\psi (t)=t$ and $\phi (t)=(1k)t$ for all $t\in [0,\mathrm{\infty})$, where $k\in [0,1)$, we obtain the following result.
for every pair $(x,y),(u,v),(w,t)\in X\times X$ such that $x\u2aafu\u2aafw$ and $y\u2ab0v\u2ab0t$, or $w\u2aafu\u2aafx$ and $t\u2ab0v\u2ab0y$, where $k\in [0,1)$.
 (a)
F is continuous, or
 (b)
X is regular.
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.
The following corollary is an extension of the results by Choudhury and Maity (Theorems 1.14 and 1.15).
for every pair $(x,y),(u,v),(w,t)\in X\times X$ such that $x\u2aafu\u2aafw$ and $y\u2ab0v\u2ab0t$, or $w\u2aafu\u2aafx$ and $t\u2ab0v\u2ab0y$.
 (a)
F is continuous, or
 (b)
X is regular.
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.
Proof If F satisfies (3.12), then F satisfies (3.11). □
Now, we present an example to illustrate Theorem 3.1.
Example 3.8 Let $X=\mathbb{R}$ be endowed with the usual ordering, and let ${G}_{b}$metric on X be given by $G(x,y,z)={(xy+yz+xz)}^{2}$, where $s=2$.
and $g(x)=x$ for all $x,y\in X$.
Define $\psi ,\phi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ by $\psi (t)=bt$, $\phi (t)=(b1)t$, where $1\le b\le \frac{72}{4}=18$.
Obviously, all the conditions of Theorem 3.1 are satisfied. Moreover, $(0,0)$ is a coupled coincidence point of F and g.
4 Applications
In this section, we obtain some coupled coincidence point theorems for mappings satisfying some contractive conditions of integral type in an ordered complete ${G}_{b}$metric space.
 (I)
μ is a positive Lebesgue integrable mapping on each compact subset of $[0,+\mathrm{\infty})$.
 (II)
For all $\epsilon >0$, ${\int}_{0}^{\epsilon}\mu (t)\phantom{\rule{0.2em}{0ex}}dt>0$.
Corollary 4.1 Replace the contractive condition (3.1) of Theorem 3.1 by the following condition:
If the other conditions of Theorem 3.1 hold, then F and g have a coupled coincidence point.
Taking ${\psi}_{1}=\mathrm{\Gamma}o\psi $ and ${\phi}_{1}=\mathrm{\Gamma}o\phi $ and applying Theorem 3.1, we obtain the proof. □
Corollary 4.2 Substitute the contractive condition (3.1) of Theorem 3.1 by the following condition:
Then F and g have a coupled coincidence point if the other conditions of Theorem 3.1 hold.
Now, if we define ${\psi}_{1}=\psi o\mathrm{\Gamma}$ and ${\phi}_{1}=\phi o\mathrm{\Gamma}$ and apply Theorem 3.1, then the proof is obtained. □
We have the following result.
Assume further that the other conditions of Theorem 3.1 are also satisfied. Then F and g have a coupled coincidence point.
Now, applying Theorem 3.1, we obtain the desired result. □
Another consequence of our theorems is the following result.
Corollary 4.4 Replace the contractive condition (3.1) of Theorem 3.1 by the following condition:
Let the other conditions of Theorem 3.1 be satisfied. Then F and g have a coupled coincidence point.
5 Conclusions
We saw that the results of Cho et al. [25] and the results of Choudhury and Maity [23] also hold in the context of ${G}_{b}$metric spaces with some simple changes in the contractive conditions. The most difference between the concepts of Gmetric and ${G}_{b}$metric is that the ${G}_{b}$metric function is not necessarily continuous in all its three variables (see, Example 2.15). On the other hand, by a simple but essential lemma (Lemma 2.16), we can prove many fixed point results in this new structure.
Declarations
Acknowledgements
The authors express their gratitude to the referees for their useful remarks, comments and suggestions.
Authors’ Affiliations
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