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Coupled coincidence point theorems for contractions in generalized fuzzy metric spaces
Fixed Point Theory and Applications volume 2012, Article number: 196 (2012)
Abstract
In this paper, we introduce the concept of a mixed gmonotone mapping and prove coupled coincidence and common coupled fixed point theorems for mappings under ϕcontractive conditions in partially ordered generalized fuzzy metric spaces. We also give an example to illustrate the theorems.
MSC:47H10, 54H25.
1 Introduction
The theory of fuzzy sets has evolved in many directions after investigation of the notion of fuzzy sets by Zadeh [1]. Many authors have introduced the concept of a fuzzy metric space in different ways [2, 3]. George and Veeramani [4, 5] modified the concept of a fuzzy metric space introduced by Kramosil and Michalek [3] and defined a Hausdorff topology on this fuzzy metric space. They showed also that every metric induces a fuzzy metric. Later, many fixed point theorems in fuzzy metric spaces and probabilistic metric spaces have been obtained by [6–10].
Nieto and Lopez [11], Ran and Reurings [12], Petrusel and Rus [13] presented some new results for contractions in partially ordered metric spaces. The main idea in [11, 12] involves combining the ideas of the iterative technique in the contractive mapping principle with those in the monotone technique, discussing the existence of a solution to firstorder ordinary differential equations with periodic boundary conditions and some applications to linear and nonlinear matrix equations.
Bhaskar and Lakshmikantham [14], Lakshmikantham and Ćirić [15] discussed coupled coincidence and coupled fixed point theorems for two mappings F and g, where F has the mixed gmonotone property and F and g commute. The results were used to study the existence of a unique solution to a periodic boundary value problem. In [16], Choudhury and Kundu established a similar result under the condition that F and g are compatible mappings and the function g is monotone increasing. For more details on ordered metric spaces, we refer to [17–19] and references mentioned therein.
Alternatively Mustafa and Sims [20] introduced a new notion of a generalized metric space called Gmetric space. Mujahid Abbas et al. [23] proved a unique fixed point of four Rweakly commuting maps in Gmetric spaces, and Mujahid Abbas et al. [24] obtained some common fixed point results of maps satisfying the generalized (\phi ,\psi )weak commuting condition in partially ordered Gmetric spaces. Rao et al. [22] proved two unique common coupled fixedpoint theorems for three mappings in symmetric Gfuzzy metric spaces. Sun and Yang [21] introduced the concept of Gfuzzy metric spaces and proved two common fixedpoint theorems for four mappings. Some interesting references on Gmetric spaces are [22–25].
In this paper, we introduce the concept of a mixed gmonotone mapping, which is a generalization of the mixed monotone mapping, and prove coupled coincidence point and coupled common fixed point theorems for mappings under ϕcontractive conditions in partially ordered Gfuzzy metric spaces. The work is an extension of the fixed point result in fuzzy metric spaces and the condition is different from [14–16] even in metric spaces. We also give an example to illustrate the theorems.
Recall that if (X,\le ) is a partially ordered set and F:X\to X satisfies that for x,y\in X, x\le y implies F(x)\le F(y), then a mapping F is said to be nondecreasing. Similarly, a nonincreasing mapping is defined.
Before giving our main results, we recall some of the basic concepts and results in Gmetric spaces and Gfuzzy metric spaces.
2 Preliminaries
Definition 2.1 [20]
Let X be a nonempty set, and let G:X\times X\times X\to [0,+\mathrm{\infty}) 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.
The function G is called a generalized metric or a Gmetric on X and the pair (X,G) is called a Gmetric space.
Definition 2.2 [20]
The Gmetric space (X,G) is called symmetric if G(x,x,y)=G(x,y,y) for all x,y\in X.
Definition 2.3 [20]
Let (X,G) be a Gmetric space, and let \{{x}_{n}\} be a sequence in X. A point x\in X is said to be the limit of \{{x}_{n}\} if and only if {lim}_{n,m\to \mathrm{\infty}}G(x,{x}_{n},{x}_{m})=0. In this case, the sequence \{{x}_{n}\} is said to be Gconvergent to x.
Definition 2.4 [20]
Let (X,G) be a Gmetric space, and let \{{x}_{n}\} be a sequence in X. \{{x}_{n}\} is called a GCauchy sequence if and only if {lim}_{n,m,l\to \mathrm{\infty}}G({x}_{n},{x}_{m},{x}_{l})=0. (X,G) is called Gcomplete if every GCauchy sequence in (X,G) is Gconvergent in (X,G).
Proposition 2.1 [20]
In a Gmetric space (X,G), the following are equivalent:

(i)
The sequence \{{x}_{n}\} is GCauchy.

(ii)
For every \epsilon >0, there exists N\in \mathbb{N} such that G({x}_{n},{x}_{m},{x}_{m})<\epsilon for all n,m\ge N.
Proposition 2.2 [20]
Let (X,G) be a Gmetric space; then the function G(x,y,z) is jointly continuous in all three of its variables.
Proposition 2.3 [20]
Let (X,G) be a Gmetric space; then for any x,y,z,a\in X, it follows that

(i)
if G(x,y,z)=0, then x=y=z,

(ii)
G(x,y,z)\le G(x,x,y)+G(x,x,z),

(iii)
G(x,y,y)\le 2G(x,x,y),

(iv)
G(x,y,z)\le G(x,a,z)+G(a,y,z),

(v)
G(x,y,z)\le \frac{2}{3}(G(x,a,a)+G(y,a,a)+G(z,a,a)).
Let (X,d) be a metric space. One can verify that (X,G) is a Gmetric space, where
or
If (X,G) is a Gmetric space, it easy to verify that (X,{d}_{G}) is a metric space, where {d}_{G}(x,y)=\frac{1}{2}(G(x,x,y)+G(x,y,y)).
Definition 2.5 [26]
A binary operation \ast :[0,1]\times [0,1]\to [0,1] is a continuous tnorm if ∗ satisfies the following conditions:

(i)
∗ is commutative and associative;

(ii)
∗ is continuous;

(iii)
a\ast 1=a for all a\in [0,1];

(iv)
a\ast b\le c\ast d whenever a\le c and b\le d for all a,b,c,d\in [0,1].
Definition 2.6 [21]
A 3tuple (X,G,\ast ) is said to be a Gfuzzy metric space (denoted by GF space) if X is an arbitrary nonempty set, ∗ is a continuous tnorm and G is a fuzzy set on {X}^{3}\times (0,+\mathrm{\infty}) satisfying the following conditions for each t,s>0:
(GF1) G(x,x,y,t)>0 for all x,y\in X with x\ne y;
(GF2) G(x,x,y,t)\ge G(x,y,z,t) for all x,y,z\in X with y\ne z;
(GF3) G(x,y,z,t)=1 if and only if x=y=z;
(GF4) G(x,y,z,t)=G(p(x,y,z),t), where p is a permutation function;
(GF5) G(x,a,a,t)\ast G(a,y,z,s)\le G(x,y,z,t+s) (the triangle inequality);
(GF6) G(x,y,z,\cdot ):(0,\mathrm{\infty})\to [0,1] is continuous.
Remark 2.1 Let x=w, y=u, z=u, a=v in (GF5), we have
which implies that
for all u,v,w\in X and s,t>0.
A GF space is said to be symmetric if G(x,x,y,t)=G(x,y,y,t) for all x,y\in X and for each t>0.
Example 2.1 Let X be a nonempty set, and let G be a Gmetric on X. Define the tnorm a\ast b=min\{a,b\} and for all x,y,z\in X and t>0, G(x,y,z,t)=\frac{t}{t+G(x,y,z)}. Then (X,G,\ast ) is a GF space.
Remark 2.2 If (X,M,\ast ) is a fuzzy metric space [4], then (X,G,\ast ) is a GF space, where
In fact, we only need to verify (GF5). Since
we have
which implies that (GF5) holds.
Remark 2.3 If (X,G,\ast ) is a symmetric GF space, let M(x,y,t)=G(x,y,y,t), then (X,M,\ast ) is a fuzzy metric space [4].
Let (X,G,\ast ) be a GF space. For t>0, the open ball {B}_{G}(x,r,t) with center x\in X and radius 0<r<1 is defined by
A subset A\subset X is called an open set if for each x\in A, there exist t>0 and 0<r<1 such that {B}_{G}(x,r,t)\subset A.
Definition 2.7 [21]
Let (X,G,\ast ) be a GF space, then

(1)
a sequence \{{x}_{n}\} in X is said to be convergent to x (denoted by {lim}_{n\to \mathrm{\infty}}{x}_{n}=x) if
\underset{n\to \mathrm{\infty}}{lim}G({x}_{n},{x}_{n},x,t)=1
for all t>0.

(2)
a sequence \{{x}_{n}\} in X is said to be a Cauchy sequence if
\underset{n,m\to \mathrm{\infty}}{lim}G({x}_{n},{x}_{n},{x}_{m},t)=1,
that is, for any \epsilon >0 and for each t>0, there exists {n}_{0}\in \mathbb{N} such that
for n,m\ge {n}_{0}.

(3)
A GF space (X,G,\ast ) is said to be complete if every Cauchy sequence in X is convergent.
Lemma 2.1 [21]
Let (X,G,\ast ) be a GF space. Then G(x,y,z,t) is nondecreasing with respect to t for all x,y,z\in X.
Lemma 2.2 [21]
Let (X,G,\ast ) be a GF space. Then G is a continuous function on {X}^{3}\times (0,+\mathrm{\infty}).
In the rest of the paper, (X,G,\ast ) will denote a GF space with a continuous tnorm ∗ defined as a\ast b=min\{a,b\} for all a,b\in [0,1], and we assume that
Define \mathrm{\Phi}=\{\varphi :{R}^{+}\to {R}^{+}\}, where {R}^{+}=[0,+\mathrm{\infty}) and each \varphi \in \mathrm{\Phi} satisfies the following conditions:
(Φ1) ϕ is strict increasing;
(Φ2) ϕ is upper semicontinuous from the right;
(Φ3) {\sum}_{n=0}^{\mathrm{\infty}}{\varphi}^{n}(t)<+\mathrm{\infty} for all t>0, where {\varphi}^{n+1}(t)=\varphi ({\varphi}^{n}(t)).
Let {\varphi}_{1}(t)=\frac{t}{t+1}, {\varphi}_{2}(t)=kt, where 0<k<1, then {\varphi}_{1},{\varphi}_{2}\in \mathrm{\Phi}.
It is easy to prove that if \varphi \in \mathrm{\Phi}, then \varphi (t)<t for all t>0.
Using (P), one can prove the following lemma.
Lemma 2.3 Let (X,G,\ast ) be a GF space. If there exists \varphi \in \mathrm{\Phi} such that if G(x,y,z,\varphi (t))\ge G(x,y,z,t) for all t>0, then x=y=z.
Lemma 2.4 Let (X,G,\ast ) be a GF space. If we define {E}_{\lambda}:X\times X\times X\to [0,\mathrm{\infty}) by
for all \lambda \in (0,1] and x,y,z\in X, then we have:

(1)
for each \lambda \in (0,1], there exists \mu \in (0,1] such that
{E}_{\lambda}({x}_{1},{x}_{1},{x}_{n})\le \sum _{i=1}^{n1}{E}_{\mu}({x}_{i},{x}_{i},{x}_{i+1}),\phantom{\rule{1em}{0ex}}\mathrm{\forall}{x}_{1},\dots ,{x}_{n}\in X. 
(2)
The sequence {\{{x}_{n}\}}_{n\in \mathbb{N}} in X is convergent if and only if {E}_{\lambda}({x}_{n},{x}_{n},x)\to 0 as n\to \mathrm{\infty} for all \lambda \in (0,1].
Proof (1) For any \lambda \in (0,1], let \mu \in (0,1] and \mu <\lambda, and so, by the triangular inequality (GF5) and Remark 2.1, for any \delta >0, we have
which implies, by Definition 2.1 of {E}_{\mu}, that
Since \delta >0 is arbitrary, we have

(2)
Since G is continuous in its fourth argument, by Definition 2.1 of {E}_{\mu}, we have
G({x}_{n},{x}_{n},x,\eta )>1\lambda \phantom{\rule{1em}{0ex}}\text{for all}\eta 0.
This proved the lemma. □
Lemma 2.5 Let (X,G,\ast ) be a GF space and \{{y}_{n}\} be a sequence in X. If there exists \varphi \in \mathrm{\Phi} such that
for all t>0 and n=1,2,\dots, then \{{y}_{n}\} is a Cauchy sequence in X.
Proof Let {\{{E}_{\lambda}(x,y,z)\}}_{\lambda \in (0,1]} be defined by (2.1). For each \lambda \in (0,1] and n\in \mathbb{N}, putting {a}_{n}={E}_{\lambda}({y}_{n1},{y}_{n1},{y}_{n}), we will prove that
Since ϕ is upper semicontinuous from right, for given \epsilon >0 and each {a}_{n}, there exists {p}_{n}>{a}_{n} such that \varphi ({p}_{n})<\varphi ({a}_{n})+\epsilon. From the definition of {E}_{\lambda} by (2.1), it follows from {p}_{n}>{a}_{n}={E}_{\lambda}({y}_{n1},{y}_{n1},{y}_{n}) that G({y}_{n1},{y}_{n1},{y}_{n},{p}_{n})>1\lambda for all n\in \mathbb{N}.
Thus, by (2.2), (2.3) and Lemma 2.1, we get
Again by Definition 2.1, we get
By the arbitrariness of ε, we have
So, we can infer that {a}_{n+1}\le \varphi ({a}_{n}). If not, then by (2.4), we have {a}_{n+1}\le \varphi ({a}_{n+1})<{a}_{n+1}, which is a contradiction. Hence, (2.4) implies that {a}_{n+1}\le \varphi ({a}_{n}), and (2.3) is proved.
Again and again using (2.3), we get
By Lemma 2.4, for each \lambda \in (0,1], there exists \mu \in (0,\lambda ] such that
Since \varphi \in \mathrm{\Phi}, by condition (Φ3) we have {\sum}_{n=0}^{\mathrm{\infty}}{\varphi}^{n}({E}_{\mu}({y}_{0},{y}_{0},{y}_{1}))<+\mathrm{\infty}. So, for given \epsilon >0, there exists {n}_{0}\in \mathbb{N} such that {\sum}_{i={n}_{0}}^{\mathrm{\infty}}{\varphi}^{i}({E}_{\mu}({y}_{0},{y}_{0},{y}_{1}))<\epsilon. Thus, it follows from (2.5) that
which implies that G({y}_{n},{y}_{n},{y}_{m},\epsilon )>1\lambda for all m,n\in \mathbb{N} with m>n\ge {n}_{0}. Therefore, \{{y}_{n}\} is a Cauchy sequence in X. □
3 Main results
Definition 3.1 [14]
Let (X,\le ) be a partially ordered set. The mapping F is said to have the mixed monotone property if F is monotone nondecreasing in its first argument and is monotone nonincreasing in its second argument; that is, for any x,y\in X,
and
Definition 3.2 [14]
An element (x,y)\in X\times X is called a coupled fixed point of the mapping F:X\times X if
Definition 3.3 [15]
Let (X,\le ) be a partially ordered set and F:X\times X\to X and g:X\to X. We say 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 any x,y\in X,
and
Note that if g is the identity mapping, then Definition 3.3 reduces to Definition 3.1.
Example 3.1 Let X=[1,1] with the natural ordering of real numbers. Let g:X\to X and F:X\times X\to X be defined as
Then F is not mixed monotone but mixed gmonotone.
Definition 3.4 [15]
Let X be a nonempty set, F:X\times X\to X and g:X\to X, then

(1)
An element (x,y)\in X\times X is called a coupled coincidence point of the mappings F and g if
F(x,y)=g(x),\phantom{\rule{2em}{0ex}}F(y,x)=g(y). 
(2)
An element (x,y)\in X\times X is called a common coupled fixed point of the mappings F and g if
F(x,y)=g(x)=x,\phantom{\rule{2em}{0ex}}F(y,x)=g(y)=y.
Definition 3.5 The mappings F:X\times X\to X and g:X\to X are said to be compatible if
and
for all t>0 whenever \{{x}_{n}\} and \{{y}_{n}\} are sequences in X such that
for all x,y\in X are satisfied.
Definition 3.6 [16]
The mappings F:X\times X\to X and g:X\to X are called wcompatible if
whenever g(x)=F(x,y) and g(y)=F(y,x) for some (x,y)\in X\times X.
Remark 3.1 It is easy to prove that if F and g are compatible then they are wcompatible.
Theorem 3.1 Let (X,\le ) be a partially ordered set and (X,G,\ast ) be a complete GF space. Let F:X\times X\to X and g:X\to X be two mappings such that F has the mixed gmonotone property and there exists \varphi \in \mathrm{\Phi} such that
for all x,y,u,v\in X, t>0 for which g(x)\le g(u) and g(y)\ge g(v), or g(x)\ge g(u) and g(y)\le g(v).
Suppose F(X\times X)\subseteq g(X), g is continuous and F and g are compatible. Also suppose

(a)
F is continuous or

(b)
X has the following properties:
(3.6)(3.7)
If there exists {x}_{0},{y}_{0}\in X such that g({x}_{0})\le F({x}_{0},{y}_{0}) and g({y}_{0})\ge F({y}_{0},{x}_{0}), then there exist x,y\in X such that g(x)=F(x,y) and g(y)=F(y,x); that is, F and g have a coupled coincidence point in X.
Proof Let {x}_{0},{y}_{0}\in X be such that g({x}_{0})\le F({x}_{0},{y}_{0}) and g({y}_{0})\ge F({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}). Continuing in this way, we construct two sequences \{{x}_{n}\} and \{{y}_{n}\} in X such that
We shall show that
for all n\ge 0.
We shall use the mathematical induction. Let n=0. Since g({x}_{0})\le F({x}_{0},{y}_{0}) and g({y}_{0})\ge F({y}_{0},{x}_{0}), and as g({x}_{1})=F({x}_{0},{y}_{0}) and g({y}_{1})=F({y}_{0},{x}_{0}), we have g({x}_{0})\le g({x}_{1}) and g({y}_{0})\ge g({y}_{1}). Thus, (3.9) and (3.10) hold for n=0. Suppose now that (3.9) and (3.10) hold for some fixed n\ge 0. Then since g({x}_{n})\le g({x}_{n+1}) and g({y}_{n})\ge g({y}_{n+1}), and as F has the mixed gmonotone property, from (3.8) and (3.3),
and from (3.8) and (3.4),
Now from (3.11) and (3.12), we get g({x}_{n+1})\le g({x}_{n+2}) and g({y}_{n+1})\ge g({y}_{n+2}). Thus, by mathematical induction, we conclude that (3.9) and (3.10) hold for all n\ge 0. Therefore,
and
By putting (x={x}_{n1}, y={y}_{n1}, u={x}_{n}, v={y}_{n}) in (3.5), we get
So, by (3.8), we have
Now, by Lemma 2.5, \{g({x}_{n})\} is a Cauchy sequence.
By putting (x={y}_{n}, y={x}_{n}, u={y}_{n1}, v={x}_{n1}) in (3.5), we get
So, by (3.8), we have
Now, by Lemma 2.5, \{g({y}_{n})\} is also a Cauchy sequence.
Since X is complete, there exist x,y\in X such that
Since F and g are compatible, we have by (3.15)
and
for all t>0. Next, we prove that g(x)=F(x,y) and g(y)=F(y,x).
Let (a) hold. Since F and g are continuous, by Lemma 2.2, taking limits as n\to \mathrm{\infty} in (3.16) and (3.17), we get
for all t>0. We have g(x)=F(x,y), g(y)=F(y,x).
Next, we suppose that (b) holds. By (3.9), (3.10), (3.15), we have for all n\ge 0
Since F and g are compatible and g is continuous, by (3.16) and (3.17), we have
and
Now, we have
for all 0\le k<1. Taking the limit as n\to \mathrm{\infty} in the above inequality, by continuity of G, using (3.8) and (3.19), we have
By (3.5), (3.19) and the above inequality, we have that
Letting k\to 1, which implies that gx=F(x,y) by Lemma 2.3, and similarly, by the virtue of (3.8), (3.15) and (3.20), we get gy=F(y,x). Thus, we have proved that F and g have a coupled coincidence point in X.
This completes the proof of Theorem 3.1. □
Taking g=I (the identity mapping) in Theorem 3.1, we get the following consequence.
Corollary 3.1 Let (X,\le ) be a partially ordered set and (X,G,\ast ) be a complete GF space. Let F:X\times X\to X be a mapping such that F has the mixed monotone property and there exists \varphi \in \mathrm{\Phi} such that
for all x,y,u,v\in X, t>0 for which x\le u and y\ge v. Suppose

(a)
F is continuous or

(b)
X has the following properties:

(i)
if a nondecreasing sequence {x}_{n}\to x, then {x}_{n}\le x for all n,

(ii)
if a nonincreasing sequence {y}_{n}\to y, then {y}_{n}\ge y for all n.
If there exists {x}_{0},{y}_{0}\in X such that {x}_{0}\le F({x}_{0},{y}_{0}) and {y}_{0}\ge F({y}_{0},{x}_{0}), then there exist x,y\in X such that x=F(x,y) and y=F(y,x); that is, F has a coupled fixed point in X.
Now, we shall prove the existence and uniqueness theorem of a coupled common fixed point. Note that if (S,\le ) is a partially ordered set, then we endow the product S\times S with the following partial order:
Theorem 3.2 In addition to the hypotheses of Theorem 3.1, suppose that for every (x,y),({x}^{\star},{y}^{\star})\in X\times X, there exists a (u,v)\in X\times X satisfying g(u)\le g(v) or g(v)\le g(u) such that (F(u,v),F(v,u))\in X\times X is comparable to (F(x,y),F(y,x)), (F({x}^{\star},{y}^{\star}),F({y}^{\star},{x}^{\star})). Then F and g have a unique common coupled fixed point; that is, there exists a unique (x,y)\in X\times X such that
Proof From Theorem 3.1, the set of coupled coincidence points is nonempty. We shall show that if (x,y) and ({x}^{\star},{y}^{\star}) are coupled coincidence points, that is, if g(x)=F(x,y), g(y)=F(y,x) and g({x}^{\star})=F({x}^{\star},{y}^{\star}), g({y}^{\star})=F({y}^{\star},{x}^{\star}), then
By assumption, there is (u,v)\in X\times X such that (F(u,v),F(v,u)) is comparable with (F(x,y),F(y,x)), (F({x}^{\star},{y}^{\star}),F({y}^{\star},{x}^{\star})). Put {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
With the similar proof as in Theorem 3.1, we can prove that the limits of \{g({u}_{n})\} and \{g({v}_{n})\} exist.
Since (F(x,y),F(y,x))=(g({x}_{1}),g({y}_{1}))=(g(x),g(y)) and (F(u,v),F(v,u))=(g({u}_{1}),g({v}_{1})) are comparable, it is easy to show that (g(x),g(y)) and (g({u}_{n}),g({v}_{n})) are comparable for all n\ge 1. Thus, from (3.5),
for each n\ge 1. Letting n\to \mathrm{\infty}, we get
Similarly, one can prove that
By (3.22) and (3.23), we have
which shows that g(x)=g({x}^{\star}).
Similarly, one can prove that g(y)=g({y}^{\star}). Thus, we proved (3.21).
Since g(x)=F(x,y) and g(y)=F(y,x), by the compatibility of F and g, we can get the wcompatibility of F and g, which implies
and
Denote g(x)=z, g(y)=w. Then from (3.24) and (3.25),
Thus, (z,w) is a coupled coincidence point. From (3.21) with {x}^{\star}=z, {y}^{\star}=w, it also follows g(z)=g(x), g(w)=g(y), that is,
From (3.26) and (3.27), we get
Therefore, (z,w) is a common coupled fixed point of F and g. To prove the uniqueness, assume that (p,q) is another coupled common fixed point. Then by (3.21) we have p=g(p)=g(z)=z and q=g(q)=g(w)=w. □
From Remark 2.3, let (X,G,\ast ) be a symmetric GF space. From Theorem 3.1, we get the following
Corollary 3.2 Let (X,\le ) be a partially ordered set and (X,F,\ast ) be a complete fuzzy metric space. Let F:X\times X\to X and g:X\to X be two mappings such that F has the mixed gmonotone property and there exists \varphi \in \mathrm{\Phi} such that
for all x,y,u,v\in X, t>0, for which g(x)\le g(u) and g(y)\ge g(v), or g(x)\ge g(u) and g(y)\le g(v).
Suppose F(X\times X)\subseteq g(X), g is continuous and F and g are compatible. Also suppose

(a)
F is continuous or

(b)
X has the following properties:

(i)
if a nondecreasing sequence {x}_{n}\to x, then {x}_{n}\le x for all n,

(ii)
if a nonincreasing sequence {y}_{n}\to y, then {y}_{n}\ge y for all n.
If there exist {x}_{0},{y}_{0}\in X such that g({x}_{0})\le F({x}_{0},{y}_{0}) and g({y}_{0})\ge F({y}_{0},{x}_{0}), then there exist x,y\in X such that g(x)=F(x,y) and g(y)=F(y,x), that is, F and g have a coupled coincidence point in X.
Remark 3.2 Compared with the results in [15, 16], we can find that Theorem 3.1 is different in the following aspects:

(1)
We assume that F and g are compatible, which is weaker than the conditions in [15, 16], where Theorem 2.1 in [15] assumes commutation for F and g, and Theorem 3.1 in [16] requires g to be a monotone function.
(2) We have a different contractive condition from [15, 16] even in a metric space.

(3)
In our paper, we assume that \varphi \in \mathrm{\Phi}, which is a stronger condition than that in [15, 16]. But we would like to point out that in the case of \varphi (t)=kt, where 0<k<1, the two conditions are equivalent.
Next, we give an example to demonstrate Theorem 3.1.
Example 3.2 Let X=[0,1], a\ast b=min\{a,b\}. Then (X,\le ) is a partially ordered set with the natural ordering of real numbers. Let
for all x,y,z\in [0,1]. Then (X,G,\ast ) is a complete GF space.
Let g:X\to X and F:X\times X\to X be defined as
F obeys the mixed gmonotone property.
Let \varphi (t)=\frac{t}{3} for t\in [0,\mathrm{\infty}). Let \{{x}_{n}\} and \{{y}_{n}\} be two sequences in X such that
then a=0, b=0. Now, for all n\ge 0,
and
Then it follows that
Hence, the mappings F and g are compatible in X. Also, {x}_{0}=0 and {y}_{0}=c are two points in X such that
and
We next verify the inequality of Theorem 3.1. We take x,y,u,v\in X such that g(x)\le g(u) and g(y)\ge g(v), that is, {x}^{2}\le {u}^{2}, {y}^{2}\ge {v}^{2}.
We consider the following cases:
Case 1: x\ge y and u\ge v, then
Case 2: x\ge y, u<v. Since x\le u, then u<v cannot happen.
Case 3: x<y and u\ge v, then
Case 4: x<y and u<v with {x}^{2}\le {u}^{2} and {y}^{2}\ge {v}^{2}, then F(x,y)=0 and F(u,v)=0, that is, G(F(x,y),F(x,y),F(u,v),\varphi (t))=0. Obviously, (3.5) is satisfied.
Thus, it is verified that the functions F, g, ϕ satisfy all the conditions of Theorem 3.1. Here (0,0) is the coupled coincidence point of F and g in X, which is also their common coupled fixed point.
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The authors thank the referees for useful comments and suggestions for the improvement of the paper. This work was supported by the National Natural Science Foundation of China (71171150).
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Hu, XQ., Luo, Q. Coupled coincidence point theorems for contractions in generalized fuzzy metric spaces. Fixed Point Theory Appl 2012, 196 (2012). https://doi.org/10.1186/168718122012196
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DOI: https://doi.org/10.1186/168718122012196