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(H,F)Closed set and coupled coincidence point theorems for a generalized compatible in partially Gmetric spaces
Journal of Inequalities and Applications volume 2014, Article number: 342 (2014)
Abstract
In this work, we present a notion of an (H,F)closed set and prove the existence of a coupled coincidence point theorem for a pair \{F,H\} of mappings F,H:X\times X\to X with φcontraction mappings in partially ordered metric spaces without Hincreasing property of F and mixed monotone property of H. We give some examples of a nonlinear contraction mapping, which is not applied to the existence of coupled coincidence point by H using the mixed monotone property and Hincreasing property of F. We also show the uniqueness of a coupled coincidence point of the given mappings. Further, we apply our results to the existence and uniqueness of a coupled coincidence point of the given mappings in partially ordered Gmetric spaces with Hincreasing property of F and mixed monotone property of H. These results generalize some recent results in the literature.
1 Introduction
The existence of a fixed point for the contraction type of mappings in partially ordered metric spaces has been first studied by Ran and Reurings [1]. Moreover, they established some new results and presented some applications to matrix equations. In 1987, Guo and Lakshmikantham [2] introduced the concept of a coupled fixed point. Later, Bhaskar and Lakshmikantham [3] introduced the concept of the mixed monotone property for contractive operators. They also showed some applications on the existence and uniqueness of the coupled fixed point theorems for mappings which satisfy the mixed monotone property in partially ordered metric spaces. Lakshimikantham and Ćirić [4] extended the results in [3] by defining the mixed gmonotonicity and studied the existence and uniqueness of coupled coincidence point for such a mappings which satisfy the mixed monotone property in partially ordered metric spaces. As a continuation of this work, many authors conducted research on the coupled fixed point theory and coupled coincidence point theory in partially ordered metric spaces and different spaces. We refer the reader for example to [4–31].
In 2006, Mustafa and Sims [32] introduced the notion of a Gmetric spaces as a generalization of the concept of a metric spaces and proved the analog of the Banach contraction mapping principle in the context of Gmetric spaces. For examples of extensions and applications of these works see [33–46]. In 2011, Choudhury and Maity [47] proved the existence of a coupled fixed point theorem of nonlinear contraction mappings with mixed monotone property in partially ordered Gmetric spaces. Aydi et al. [48] established coupled coincidence and coupled common fixed point results for a mixed gmonotone mapping satisfying nonlinear contractions in partially ordered Gmetric spaces. They generalized the results obtained by Choudhury and Maily [47]. Later, Karapınar et al. [49] extended the results of coupled coincidence and coupled common fixed point theorem for a mixed gmonotone mapping obtained by Aydi et al. [48]. Many authors have studied coupled coincidence point and coupled common fixed point results for a mixed gmonotone mapping satisfying nonlinear contractions in partially ordered Gmetric spaces (see, for example, [47–64]).
One of the interesting ways to developed coupled fixed point theory is to consider the mapping F:X\times X\to X without the mixed monotone property. Recently, Sintunavarat et al. [29, 30] proved some coupled fixed point theorems for nonlinear contractions without mixed monotone property which extended the results of Bhaskar and Lakshmikantham [3] by using the concept of an Finvariant set due to Samet and Vetro [28]. Later, Batra and Vashistha [6] introduced an (F,g)invariant set which is a generalization of an Finvariant set. Recently, Kutbi et al. [22] introduced the concept of an Fclosed set which is weaker than the concept of an Finvariant set and proved some coupled fixed point theorems without the condition of Finvariant set and mixed monotone property. Very recently, Charoensawan and Thangthong [55] generalized and extended the coupled coincidence point theorem of nonlinear contraction mappings in partially ordered Gmetric spaces without the mixed gmonotone property by using the concept of ({F}^{\ast},g)invariant set in partially ordered Gmetric spaces which are generalizations of the results of Aydi et al. [48]. In 2014, Hussain et al. [16] presented the new concept of generalized compatibility of a pair \{F,G\} of mappings F,G:X\times X\to X and proved some coupled coincidence point results of such a mapping without the mixed Gmonotone property of F in partially ordered metric spaces which generalized some recent comparable results in the literature.
In this work, we introduce the concept of (H,F)closed set and the notion of generalized compatibility of a pair \{F,H\} of mapping F,H:X\times X\to X in the setting of Gmetric spaces. We also obtain a coupled coincidence point theorem for a pair \{F,H\} with φcontraction mappings in partially ordered metric spaces without Hincreasing property of F and mixed monotone property of H. Our theorem generalizes and extends the very recent results obtained by Hussain et al. [16] and Karapınar et al. [49].
2 Preliminaries
In this section, we give some definitions, propositions, examples and remarks which are useful for main results in our paper. Throughout this paper, (X,\u2aaf) denotes a partially ordered set with the partial order ⪯. By x\u2aafy, we mean y\u2ab0x. Let (x,\u2aaf) be a partially ordered set, the partial order ⪯_{2} for the product set X\times X defined in the following way, for all (x,y),(u,v)\in X\times X:
where H:X\times X\to X is oneone.
We say that (x,y) is comparable to (u,v) if either (x,y){\u2aaf}_{2}(u,v) or (u,v){\u2aaf}_{2}(x,y).
Definition 2.1 [32]
Let X be a nonempty set and G:X\times X\times X\to {\mathbb{R}}^{+} be a function satisfying the following properties:

(G1) G(x,y,z)=0 if x=y=z.

(G2) 0<G(x,x,y) for all x,y\in X with x\ne y.

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

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

(G5) G(x,y,z)\le G(x,a,a)+G(a,y,z) for all x,y,z,a\in X (rectangle inequality).
Then the function G is called a generalized metric, or more specially, a Gmetric on X, and the pair (X,G) is called a Gmetric space.
Example 2.2 Let (X,d) be a metric space. The function G:X\times X\times X\to [0,+\mathrm{\infty}), defined by G(x,y,z)=d(x,y)+d(y,z)+d(z,x), for all x,y,z\in X, is a Gmetric on X.
Definition 2.3 [32]
Let (X,G) be a Gmetric space, and let ({x}_{n}) be a sequence of points of X. We say that ({x}_{n}) is Gconvergent to x\in X if {lim}_{n,m\to \mathrm{\infty}}G(x,{x}_{n},{x}_{m})=0, that is, for any \epsilon >0, there exists N\in \mathbb{N} such that G(x,{x}_{n},{x}_{m})<\epsilon, for all n,m\ge N. We call x the limit of the sequence ({x}_{n}) and write {x}_{n}\to x or {lim}_{n\to \mathrm{\infty}}{x}_{n}=x.
Proposition 2.4 [32]
Let (X,G) be a Gmetric space, the following are equivalent:

(1)
({x}_{n}) is Gconvergent to x.

(2)
G({x}_{n},{x}_{n},x)\to 0 as n\to +\mathrm{\infty}.

(3)
G({x}_{n},x,x)\to 0 as n\to +\mathrm{\infty}.

(4)
G({x}_{n},{x}_{m},x)\to 0 as n,m\to +\mathrm{\infty}.
Definition 2.5 [32]
Let (X,G) be a Gmetric space. A sequence ({x}_{n}) is called a GCauchy sequence if, for any \epsilon >0, there exists N\in \mathbb{N} such that G({x}_{n},{x}_{m},{x}_{l})<\epsilon, for all n,m,l\ge N. That is, G({x}_{n},{x}_{m},{x}_{l})\to 0 as n,m,l\to +\mathrm{\infty}.
Proposition 2.6 [32]
Let (X,G) be a Gmetric space, the following are equivalent:

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

(2)
for any \epsilon >0, there exists N\in \mathbb{N} such that G({x}_{n},{x}_{m},{x}_{m})<\epsilon, for all n,m\ge N.
Proposition 2.7 [32]
Let (X,G) be a Gmetric space. A mapping f:X\to X is Gcontinuous at 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 Gconvergent to f(x).
Definition 2.8 [32]
A Gmetric space (X,G) is called Gcomplete if every GCauchy sequence is Gconvergent in (X,G).
Definition 2.9 [47]
Let (X,G) 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).
In 2009, Lakshmikantham and Ćirić [4] introduced the concept of a mixed gmonotone mapping and a coupled coincidence point as follows.
Definition 2.10 [4]
Let (X,\u2aaf) 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 for any x,y\in X
and
Definition 2.11 [4]
An element (x,y)\in X\times X is called a coupled coincidence point of mappings F:X\times X\to X, and g:X\to X if F(x,y)=g(x) and F(y,x)=g(y).
Definition 2.12 [4]
Let X be a nonempty set and F:X\times X\to X and g:X\to X. We say F and g are commutative if g(F(x,y))=F(g(x),g(y)) for all x,y\in X.
Let Φ denote the set of functions \phi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) satisfying

1.
\phi (t)<t for all t>0,

2.
{lim}_{r\to {t}^{+}}\phi (r)<t for all t>0.
In 2012, Karapınar et al. [49] proved the following theorems.
Theorem 2.13 [49]
Let (X,\u2aaf) be a partially ordered set and G be a Gmetric on X such that (X,G) is a complete Gmetric space. Suppose that there exist \phi \in \mathrm{\Phi}, F:X\times X\to X, and g:X\to X such that
for all x,y,u,v,z,w\in X for which g(x)\u2ab0g(y)\u2ab0g(z) and g(u)\u2aafg(v)\u2aafg(w). Suppose also that F is continuous and has the mixed gmonotone property, F(X\times X)\subseteq g(X), and g is continuous and commutes with F. If there exists ({x}_{0},{y}_{0})\in X\times X such that
then there exists (x,y)\in X\times X such that g(x)=F(x,y) and g(y)=F(y,x), that is, F and g have a coupled coincidence point.
Hussain et al. [16] introduced the concept of Hincreasing and \{F,H\} generalized compatible as follows.
Definition 2.14 [16]
Suppose that F,H:X\times X\to X are two mappings. F is said to be Hincreasing with respect to ⪯ if for all x,y,u,v\in X, with H(x,y)\u2aafH(u,v), we have F(x,y)\u2aafF(u,v).
Definition 2.15 [16]
An element (x,y)\in X\times X is called a coupled coincidence point of mappings F,H:X\times X\to X if F(x,y)=H(x,y) and F(y,x)=H(y,x).
Definition 2.16 [16]
Let (X,d) be a metric space and F,H:X\times X\to X. We say that the pair \{F,H\} is generalized compatible if
whenever ({x}_{n}) and ({y}_{n}) are sequences in X such that
Definition 2.17 [16]
Let F,H:X\times X\to X be two maps. We say that the pair \{F,H\} is commuting if
It is easy to see that a commuting pair is generalized compatible but the converse is not true in general.
Let ϒ denote the set of all functions \varphi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) such that:

(i)
ϕ is continuous and increasing,

(ii)
\varphi (t)=0 if and only if t=0,

(iii)
\varphi (t+s)\le \varphi (t)+\varphi (s), for all t,s\in [0,\mathrm{\infty}).
Let Ψ be the set of all functions \varphi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) such that {lim}_{t\to r}\psi (t)>0 for all r>0 and {lim}_{t\to {0}^{+}}\psi (t)=0.
Recently, Hussain et al. [16] proved the coupled coincidence point for such mappings involving (\psi ,\varphi )contractive condition as follows.
Theorem 2.18 [16]
Let (X,\u2aaf) be a partially ordered set and M be a nonempty subset of {X}^{4} and let there exists d, a metric on X such that (X,d) is a complete metric space. Assume that F,H:X\times X\to X are two generalized compatible mappings such that F is Hincreasing with respect to ⪯, H is continuous and has the mixed monotone property. Suppose that for any x,y\in X, there exist u,v\in X such that F(x,y)=H(u,v) and F(y,x)=H(v,u). Suppose that there exist \varphi \in \mathrm{\Upsilon} and \psi \in \mathrm{\Psi} such that the following holds:
for all x,y,u,v\in X with H(x,y)\u2aafH(u,v) and H(y,x)\u2ab0H(v,u).
Also suppose that either

(a)
F is continuous or

(b)
X has the following properties: for any two sequences \{{x}_{n}\} and \{{y}_{n}\} with

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

(ii)
if a nonincreasing sequence \{{y}_{n}\}\to y, then y\u2aaf{y}_{n} for all n.

(i)
If there exists ({x}_{0},{y}_{0})\in X\times X with
then there exists (x,y)\in X\times X such that H(x,y)=F(x,y) and H(y,x)=F(y,x), that is, F and H have a coupled coincidence point.
In order to remove the mixed monotone property, Batra and Vashistha [6] introduced the following property.
Definition 2.19 [6]
Let (X,d) be a metric space and F:X\times X\to X, g:X\to X be given mappings. Let M be a nonempty subset of {X}^{4}. We say that M is an (F,g)invariant subset of {X}^{4} if and only if, for all x,y,z,w\in X,

(i)
(x,y,z,w)\in M\iff (w,z,y,x)\in M.

(ii)
(g(x),g(y),g(z),g(w))\in M\Rightarrow (F(x,y),F(y,x),F(z,w),F(w,z))\in M.
Kutbi et al. [22] introduced the notion of Fclosed set which extended the notion of Finvariant set as follows.
Definition 2.20 [22]
Let F:X\times X\to X be a mapping, and let M be a subset of {X}^{4}. We say that M is an Fclosed subset of {X}^{4} if, for all x,y,u,v\in X,
Inspired by above definitions, we give the notion of a (H,F)closed set which is useful for our main results.
Definition 2.21 Let F,H:X\times X\to X be two mappings and let M be a subset of {X}^{6}. We say that M is an (H,F)closed subset of {X}^{6} if, for all x,y,z,u,v,w\in X,
Definition 2.22 Let H:X\times X\to X be a mapping and M be a subset of {X}^{6}. We say that M satisfies the transitive property if and only if for all x,y,z,u,v,w,a,b,c,d\in X,
Definition 2.23 Let F,H:X\times X\to X be two mappings. We say that the pair \{F,H\} is generalized compatible if ({x}_{n}) and ({y}_{n}) are sequences in X such that for some x,y\in X
imply
Remark The set M={X}^{6} is trivially (H,F)closed set, which satisfies the transitive property.
Example 2.24 Let (X,G) be a Gmetric space endowed with a partial order ⪯. Let F,H:X\times X\to X are two generalized compatible mappings such that F is Hincreasing with respect to ⪯, H is continuous and has the mixed monotone property. Define a subset M\subseteq {X}^{6} by
Let (H(x,u),H(u,x),H(y,v),H(v,y),H(z,w),H(w,z))\in M. It is easy to see that, since F is Hincreasing with respect to ⪯, we have F(x,u)\u2ab0F(y,v)\u2ab0F(z,w) and F(u,x)\u2aafF(v,y)\u2aafF(w,z), this implies that
Then M is (H,F)closed subset of {X}^{6}, which satisfies the transitive property.
3 Main results
Now, we state our first result which successively guarantees a coupled coincidence point.
Theorem 3.1 Let (X,\le ) be a partially ordered set and G be a Gmetric on X such that (X,G) is a complete Gmetric space and M be a nonempty subset of {X}^{6}. Assume that F,H:X\times X\to X are two generalized compatible mappings such that H is continuous and for any x,y\in X, there exists u,v\in X such that F(x,y)=H(u,v) and F(y,x)=H(v,u). Suppose that there exists \phi \in \mathrm{\Phi} such that the following holds:
for all x,y,z,u,v,w\in X with (H(x,u),H(u,x),H(y,v),H(v,y),H(z,w),H(w,z))\in M.
Suppose also that either

(a)
F is continuous;

(b)
for any two sequences \{{x}_{n}\} and \{{y}_{n}\} with for all n\u2a7e1
\begin{array}{r}({x}_{n+1},{y}_{n+1},{x}_{n+1},{y}_{n+1},{x}_{n},{y}_{n})\in M\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\\ H({x}_{n},{y}_{n})\to H(x,y),\phantom{\rule{2em}{0ex}}H({y}_{n},{x}_{n})\to H(y,x)\\ \mathit{\text{implies}}\\ (H({x}_{n},{y}_{n}),H({y}_{n},{x}_{n}),H(x,y),H(y,x),H(x,y),H(y,x))\in M,\end{array}
If there exists ({x}_{0},{y}_{0})\in X\times X such that
and M is an (H,F)closed, then there exists (x,y)\in X\times X such that H(x,y)=F(x,y) and H(y,x)=F(y,x), that is, F and H have a coupled coincidence point.
Proof Let {x}_{0},{y}_{0}\in X be such that
From the assumption, there exists ({x}_{1},{y}_{1})\in X\times X such that
Again from the assumption, we can choose {x}_{2},{y}_{2}\in X such that
By repeating this argument, we can construct two sequences {\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}} and {\{{y}_{n}\}}_{n=1}^{\mathrm{\infty}} in X such that
Since
and M is an (H,{F}^{\ast})closed, we get
Again, using the fact that M is a (H,F)closed, we have
Continuing this process, for all n\ge 0 we obtain
Let
We can suppose that {\delta}_{n}>0 for all n\ge 0. If not, ({x}_{n},{y}_{n}) will be a coupled coincidence point and the proof is finished. From (1), (2), and (3), we have
This implies that
Since \varphi (t)<t for all t>0, it follows that \{{\delta}_{n}\} is decreasing sequence. Therefore, there is some \delta \ge 0 such that {lim}_{n\to \mathrm{\infty}}{\delta}_{n}=\delta.
We shall prove that \delta =0. Assume, to the contrary, that \delta >0. Then by letting n\to \mathrm{\infty} in (6) and using the properties of the map φ, we get
A contradiction, thus \delta =0, and hence
Next, we prove that {\{H({x}_{n},{y}_{n})\}}_{n=1}^{\mathrm{\infty}} and {\{H({y}_{n},{x}_{n})\}}_{n=1}^{\mathrm{\infty}} are Cauchy sequences in the Gmetric space (X,G). Suppose, to the contrary, that at least of {\{H({x}_{n},{y}_{n})\}}_{n=1}^{\mathrm{\infty}} and {\{H({y}_{n},{x}_{n})\}}_{n=1}^{\mathrm{\infty}} is not Cauchy sequence in (X,G). Then there exists an \epsilon >0 for which we can find subsequences \{H({x}_{m(k)},{y}_{m(k)})\}, \{H({x}_{n(k)},{y}_{n(k)})\} of {\{H({x}_{n},{y}_{n})\}}_{n=1}^{\mathrm{\infty}} and \{H({y}_{m(k)},{x}_{m(k)})\}, \{H({y}_{n(k)},{x}_{n(k)})\} of {\{H({y}_{n},{x}_{n})\}}_{n=1}^{\mathrm{\infty}}, respectively, with n(k)>m(k)\ge k such that
Further, corresponding to n(k), we can choose m(k) in such a way that it is the smallest integer with m(k)>n(k)\ge K and satisfying (8). Then
Using the rectangle inequality and (9), we have
Letting k\to +\mathrm{\infty} and using (7), we obtain
Again, by the rectangle inequality, we have
Using the fact that G(x,x,y)\le 2G(x,y,y) for any x,y\in X, we obtain
Since m(k)>n(k) and using (3), we have
and
From the fact that M is an (H,F)closed set which satisfies the transitive property, we have
By this process, we can get
Now, using (1), we have
From (12) and (13), it follows that
Letting k\to +\mathrm{\infty} in (14) and using (7) and (11) and {lim}_{r\to {t}^{+}}\varphi (r)<t for all t>0, we have
which is a contradiction. This shows that {\{H({x}_{n},{y}_{n})\}}_{n=1}^{\mathrm{\infty}} and {\{H({y}_{n},{x}_{n})\}}_{n=1}^{\mathrm{\infty}} are Cauchy sequences in the Gmetric space (X,G). Since (X,G) is complete and from (2), {\{H({x}_{n},{y}_{n})\}}_{n=1}^{\mathrm{\infty}} and {\{H({y}_{n},{x}_{n})\}}_{n=1}^{\mathrm{\infty}} are Gconvergent, there exist x,y\in X such that
Since the pair \{F,G\} satisfies the generalized compatibility, from (15), we have
Suppose that assumption (a) holds. For all n\ge 0, from (16), we have
and
We have
Therefore, (x,y) is a coupled coincidence point of F and H.
Suppose now assumption (b) holds. Since {\{H({x}_{n},{y}_{n})\}}_{n=1}^{\mathrm{\infty}} converges to x, {\{H({y}_{n},{x}_{n})\}}_{n=1}^{\mathrm{\infty}} converges to y, the pair \{F,G\} satisfies the generalized compatibility, H is continuous and by (15), we have
and
From (3), (17), (18), and assumption (b), for all n\u2a7e1, we have
Then, by (1), (2), (19), and the triangle inequality, we have
Letting now n\to \mathrm{\infty} in the above inequality and using property of φ such that {lim}_{r\to {0}^{+}}\phi (r)=0, we have
which implies that H(x,y)=F(x,y) and H(y,x)=F(y,x). □
Next, we give an example to validate Theorem 3.1.
Example 3.2 Let X=[0,1], G(x,y,z)=xy+xz+yz, and F,H:X\times X\to X be defined by
and
Clearly, H does not satisfy the mixed monotone property and if x>y, u=v\ne 0, consider
Then F is not Hincreasing.
Now, we prove that for any x,y\in X, there exist u,v\in X such that F(x,y)=H(u,v) and F(y,x)=H(v,u). It is easy to see that we have the following cases.
Case 1: If x=y, then we have F(y,x)=F(x,y)=0=H(0,0).
Case 2: If x>y, then (xy)x>(xy)y and we have
and
Case 3: If y>x, then (yx)y>(yx)x and we have
and
Now, we prove that the pair \{F,G\} satisfies the generalized compatibility hypothesis. Let {\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}} and {\{{y}_{n}\}}_{n=1}^{\mathrm{\infty}} be two sequences in X such that
Then we must have {t}_{1}=0={t}_{2} and it is easy to prove that
Now, for all x,y,z,u,v,w\in X with (H(x,u),H(u,x),H(y,v),H(v,y),H(z,w),H(w,z))\in M={X}^{6} and let \phi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty}) be a function defined by \phi (t)=\frac{t}{8}, we have
Therefore, condition (1) is satisfied. Thus, all the requirements of Theorem 3.1 are satisfied and (0,0) is a coupled coincidence point of F and G.
Next, we show the uniqueness of the coupled coincidence point and coupled fixed point of F and G.
Theorem 3.3 In addition to the hypotheses of Theorem 3.1, suppose that for every (x,y),(z,t)\in X\times X, there exists (u,v)\in X\times X such that
Then F and H have a unique coupled coincidence point. Moreover, if the pair \{F,H\} is commuting, then F and H have a unique coupled fixed point, that is, there exists a unique (a,b)\in {X}^{2} such that
Proof From Theorem 3.1, we know that F and H have a coupled coincidence point. Suppose that (x,y), (z,t) are coupled coincidence points of F and H, that is,
Now, we show that H(x,y)=H(z,t) and H(y,x)=H(t,z). By the hypothesis there exists (u,v)\in X\times X such that
We put {u}_{0}=u and {v}_{0}=v and define two sequences {\{H({u}_{n},{v}_{n})\}}_{n=1}^{\mathrm{\infty}} and {\{H({v}_{n},{u}_{n})\}}_{n=1}^{\mathrm{\infty}} as follows:
Since M is (H,F)closed and
we have
From (H({u}_{1},{v}_{1}),H({v}_{1},{u}_{1}),H(x,y),H(y,x),H(x,y),H(y,x))\in M, if we use again the property of (H,F)closedness, then
By repeating this process, we get
Using (1), (20), and (21), for all n\ge 0, we have
Using property that \phi (t)<t and repeating this process, for all n\ge 0, we get
From \phi (t)<t and {lim}_{r\to {t}^{+}}\phi (r)<t, it follows that {lim}_{n\to \mathrm{\infty}}{\phi}^{n}(t)=0 for each t>0. Therefore, from (23), we have
This implies that
Similarly, we show that
From (25) and (26), we have
Now let the pair \{F,H\} be commuting, we shall prove that F and H have a unique coupled fixed point. Since
and F and H commutes, we have
Denote H(x,y)=a and H(y,x)=b. Then, by (28) and (29), one gets
Therefore, (a,b) is a coupled coincidence point of F and H. Then, by (27) with z=a and t=b, it follows that
Thus, (a,b) is a coupled fixed point of H, by (28), (a,b) is also a coupled fixed point of F. To prove the uniqueness, assume (p,q) is another coupled fixed point of F and H. Then, by (27) and (31), we have
□
Next, we give some applications of our results to coupled coincidence point theorems.
Corollary 3.4 Let (X,\u2aaf) be a partially ordered set and M be a nonempty subset of {X}^{6} and let there exists G be a Gmetric on X such that (X,G) is a complete Gmetric space. Assume that F,H:X\times X\to X are two generalized compatible mappings such that F is Hincreasing with respect to ⪯, H is continuous and has the mixed monotone property. Suppose that for any x,y\in X, there exist u,v\in X such that F(x,y)=H(u,v) and F(y,x)=H(v,u). Suppose that there exists \phi \in \mathrm{\Phi} such that the following holds:
for all x,y,z,u,v,w\in X with F(x,u)\u2ab0F(y,v)\u2ab0F(z,w) and F(u,x)\u2aafF(v,y)\u2aafF(w,z).
Also suppose that either

(a)
F is continuous or

(b)
X has the following properties: for any two sequences \{{x}_{n}\} and \{{y}_{n}\} with

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

(ii)
if a nonincreasing sequence \{{y}_{n}\}\to y, then y\u2aaf{y}_{n} for all n.

(i)
If there exists ({x}_{0},{y}_{0})\in X\times X with
Then there exists (x,y)\in X\times X such that H(x,y)=F(x,y) and H(y,x)=F(y,x), that is, F and H have a coupled coincidence point.
Proof We define the subset M\subseteq {X}^{6} by
From Example 2.24, M is a (H,F)closed set which satisfies the transitive property. For all x,y,z,u,v,w\in X with H(x,u)\u2ab0H(y,v)\u2ab0H(z,w) and H(u,x)\u2aafH(v,y)\u2aafH(w,z), we have (H(x,u),H(u,x),H(y,v),H(v,y),H(z,w),H(w,z))\in M. By (1), we get
Since ({x}_{0},{y}_{0})\in X\times X with
We have
Assumption (a) holds, and F is continuous. By assumption (a) of Theorem 3.1, we have H(x,y)=F(x,y) and H(y,x)=F(y,x).
Next, assumption (b) holds; since F is Hincreasing with respect to ⪯, using (32) and (2), we have
Therefore
From H is continuous and by (15), we have
For any two sequences {\{H({x}_{n},{y}_{n})\}}_{n=1}^{\mathrm{\infty}} and {\{H({y}_{n},{x}_{n})\}}_{n=1}^{\mathrm{\infty}} such that {\{H({x}_{n},{y}_{n})\}}_{n=1}^{\mathrm{\infty}} is a nondecreasing sequence in X with H({x}_{n},{y}_{n})\to x and {\{H({y}_{n},{x}_{n})\}}_{n=1}^{\mathrm{\infty}} is a nonincreasing sequence in X with H({y}_{n},{x}_{n})\to y. Using assumption (b), we have
Since H has the mixed monotone property, we have
Therefore, we have
and so assumption (b) of Theorem 3.1 holds. Now, since all the hypotheses of Theorem 3.1 hold, then F and H have a coupled coincidence point. The proof is completed. □
Corollary 3.5 In addition to the hypotheses of Corollary 3.4, suppose that for every (x,y),(z,t)\in X\times X, there exists (u,v)\in X\times X which is comparable to (x,y) and (z,t). Then F and H have a unique coupled coincidence point.
Proof We define the subset M\subseteq {X}^{6} by
From Example 2.24, M is an (H,F)closed set which satisfies the transitive property. Thus, the proof of the existence of a coupled coincidence point is straightforward by following the same lines as in the proof of Corollary 3.4.
Next, we show the uniqueness of a coupled coincidence point of F and H.
Since for all (x,y),(z,t)\in X\times X, there exists (u,v)\in X\times X such that
and
we can conclude that
Therefore, since all the hypotheses of Theorem 3.3 hold, F and H have a unique coupled coincidence point. The proof is completed. □
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