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On coupled fixed point theorems on partially ordered Gmetric spaces
Journal of Inequalities and Applications volume 2012, Article number: 200 (2012)
Abstract
In this manuscript, we extend, generalize and enrich some recent coupled fixed point theorems in the framework of partially ordered Gmetric spaces in a way that is essentially more natural.
MSC:46N40, 47H10, 54H25, 46T99.
1 Introduction and preliminaries
In [1] Aydi et al. established coupled coincidence and coupled common fixed point results for a mixed gmonotone mapping satisfying nonlinear contractions in partially ordered Gmetric spaces. These results generalize those of Choudhury and Maity [2].
Here we generalize, improve, enrich and extend the above mentioned coupled fixed point results of Aydi et al.
Throughout this paper, let \mathbb{N} denote the set of nonnegative integers, and {\mathbb{N}}^{\ast} be the set of positive integers.
Definition 1.1 (See [3])
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.
Every Gmetric on X defines a metric {d}_{G} on X by
Example 1.2 Let (X,d) be a metric space. The function G:X\times X\times X\to [0,+\mathrm{\infty}), defined by
or
for all x,y,z\in X, is a Gmetric on X.
Definition 1.3 (See [3])
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 and write {x}_{n}\to x or {lim}_{n\to +\mathrm{\infty}}{x}_{n}=x.
Proposition 1.4 (See [3])
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 1.5 (See [3])
Let (X,G) be a Gmetric space. A sequence \{{x}_{n}\} is called a GCauchy sequence if, for any \epsilon >0, there is N\in \mathbb{N} such that G({x}_{n},{x}_{m},{x}_{l})<\epsilon for all m,n,l\ge N, that is, G({x}_{n},{x}_{m},{x}_{l})\to 0 as n,m,l\to +\mathrm{\infty}.
Proposition 1.6 (See [3])
Let (X,G) be a Gmetric space. Then 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 m,n\ge N.
Proposition 1.7 (See [3])
Let (X,G) be a Gmetric space. A mapping f:X\to X is Gcontinuous at {x}_{0} if and only if it is Gsequentially continuous at {x}_{0}, that is, whenever ({x}_{n}) is Gconvergent to {x}_{0}, the sequence (f({x}_{n})) is Gconvergent to f({x}_{0}).
Definition 1.8 (See [3])
A Gmetric space (X,G) is called Gcomplete if every GCauchy sequence is Gconvergent in (X,G).
Definition 1.9 (See [2])
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, y respectively, \{F({x}_{n},{y}_{n})\} is Gconvergent to F(x,y).
Let (X,\le ) be a partially ordered set and (X,G) be a Gmetric space, g:X\to X be a mapping. A partially ordered Gmetric space, (X,G,\u2aaf), is called gordered complete if for each convergent sequence {\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}\subset X, the following conditions hold:
(O{C}_{1}) if \{{x}_{n}\} is a nonincreasing sequence in X such that {x}_{n}\to {x}^{\ast} implies g{x}^{\ast}\u2aafg{x}_{n}, \mathrm{\forall}n\in \mathbb{N},
(O{C}_{2}) if \{{y}_{n}\} is a nondecreasing sequence in X such that {y}_{n}\to {y}^{\ast} implies g{y}^{\ast}\u2ab0g{y}_{n}, \mathrm{\forall}n\in \mathbb{N}.
Moreover, a partially ordered Gmetric space, (X,G,\u2aaf), is called ordered complete when g is equal to identity mapping in the above conditions (O{C}_{1}) and (O{C}_{2}).
Definition 1.10 (See [4])
An element (x,y)\in X\times X is said to be a coupled fixed point of the mapping F:X\times X\to X if
Definition 1.11 (See [5])
An element (x,y)\in X\times X is called a coupled coincidence point of a mapping F:X\times X\to X and g:X\to X if
Moreover, (x,y)\in X\times X is called a common coupled coincidence point of F and g if
Definition 1.12 Let F:X\times X\to X and g:X\to X be mappings. The mappings F and g are said to commute if
Definition 1.13 (See [4])
Let (X,\le ) be a partially ordered set and F:X\times X\to X be a mapping. Then F is said to have mixed monotone property if F(x,y) is monotone nondecreasing in x and is monotone nonincreasing in y, that is, for any x,y\in X,
and
Definition 1.14 (See [5])
Let (X,\le ) be a partially ordered set and F:X\times X\to X and g:X\to X be two mappings. Then F is said to have mixed gmonotone property if F(x,y) is monotone gnondecreasing in x and is monotone gnonincreasing in y, that is, for any x,y\in X,
and
Let Φ denote the set of functions \varphi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) satisfying

(a)
{\varphi}^{1}(\{0\})=\{0\},

(b)
\varphi (t)<t for all t>0,

(c)
{lim}_{r\to {t}^{+}}\varphi (r)<t for all t>0.
Lemma 1.15 (See [5])
Let \varphi \in \mathrm{\Phi}. For all t>0, we have {lim}_{n\to \mathrm{\infty}}{\varphi}^{n}(t)=0.
Aydi et al. [1] proved the following theorems.
Theorem 1.16 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 \varphi \in \mathrm{\Phi}, F:X\times X\to X and g:X\to X such that
for all x,y,u,v,w,z\in X with gw\u2aafgu\u2aafgx and gy\u2aafgv\u2aafgz. 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 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, that is, there exists (x,y)\in X\times X such that gx=F(x,y) and gy=F(y,x).
Theorem 1.17 Let (X,\u2aaf) be a partially ordered set and G be a Gmetric on X such that (X,G,\u2aaf) is regular. Suppose that there exist \varphi \in \mathrm{\Phi} and mappings F:X\times X\to X and g:X\to X such that
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 complete, 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.
In this manuscript, we generalize, improve, enrich and extend the above coupled fixed point results. We also state some examples to illustrate our results. This paper can be considered as a continuation of the remarkable works of Berinde [6, 7].
2 Main results
We begin with an example to illustrate the weakness of Theorem 1.16 and Theorem 1.17 above.
Example 2.1 Let X=\mathbb{R}. Define G:X\times X\times X\to [0,\mathrm{\infty}) by
for all x,y,z\in X. Then (X,G) is a Gmetric space. Define a map F:X\times X\to X by F(x,y)=\frac{1}{12}x+\frac{7}{12}y and g:X\to X by g(x)=\frac{x}{2} for all x,y\in X. Suppose x=u=z
and
It is clear that there is no \varphi \in \mathrm{\Phi} that provides the statement (1.4) of Theorem 1.16.
Notice that (0,0) is the unique common coincidence point of F and g. In fact, F(0,0)=g(0)=0.
For some coupled fixed point and coupled coincidence point theorems, we refer the reader to [8–34].
We now state our first result which successively guarantees a coupled fixed point.
Theorem 2.2 Let (X,\u2aaf) be a partially ordered set and G be a Gmetric on X such that (X,G) is a complete Gmetric space. Suppose that there exist \varphi \in \mathrm{\Phi}, F:X\times X\to X and g:X\to X such that
for all x,y,u,v,w,z\in X with gw\u2aafgu\u2aafgx and gy\u2aafgv\u2aafgz. 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 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, that is, there exists (x,y)\in X\times X such that gx=F(x,y) and gy=F(y,x).
Proof Given {x}_{0},{y}_{0}\in X satisfying g{x}_{0}\u2aafF({x}_{0},{y}_{0}) and F({y}_{0},{x}_{0})\u2aafg{y}_{0}, we shall construct iterative sequences ({x}_{n}) and ({y}_{n}) in the following way: 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}). Analogously, we 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}) due to the same reasoning. Since F has the mixed gmonotone property, we conclude that g{x}_{0}\u2aafg{x}_{1}\u2aafg{x}_{2} and g{y}_{2}\u2aafg{y}_{1}\u2aafg{y}_{0}. By repeating this process, we derive the iterative sequence
and
If for some {n}_{0} we have (g{x}_{{n}_{0}+1},g{y}_{{n}_{0}+1})=(g{x}_{{n}_{0}},g{y}_{{n}_{0}}), then F({x}_{{n}_{0}},{y}_{{n}_{0}})=g{x}_{{n}_{0}} and F=({y}_{{n}_{0}},{x}_{{n}_{0}})=g{y}_{{n}_{0}}, that is, F and g have a coincidence point. So, we assume that (g{x}_{n+1},g{y}_{n+1})\ne (g{x}_{n},g{y}_{n}) for all n\in \mathbb{N}. Thus, we have either g{x}_{n+1}=F({x}_{n},{y}_{n})\ne g{x}_{n} or g{y}_{n+1}=F({y}_{n},{x}_{n})\ne g{y}_{n}. We set
for all n\in \mathbb{N}. Due to the property (G2), we have {t}_{n}>0 for all n\in \mathbb{N}. By using inequality (2.3), we obtain
Taking (2.4) into account, (2.5) becomes
Since \varphi (t)<t for all t>0, it follows that {t}_{n} is monotone decreasing. Therefore, there is some L\ge 0 such that {lim}_{n\to +\mathrm{\infty}}{t}_{n}=L.
Now, we assert that L=0. Suppose, on the contrary, that L>0. Letting n\to +\mathrm{\infty} in (2.6) and using the properties of the map ϕ, we get
which is contradiction. Thus L=0. Hence
Next, we prove that (g{x}_{n}) and (g{y}_{n}) are Cauchy sequences in the Gmetric space (X,G). Suppose, on the contrary, that at least one of (g{x}_{n}) and (g{y}_{n}) is not a Cauchy sequence in (X,G). Then there exist \epsilon >0 and sequences of natural numbers (m(k)) and (l(k)) such that for every natural number k, m(k)>l(k)\ge k and
Now, corresponding to l(k), we choose m(k) to be the smallest for which (2.8) holds. Hence
Using the rectangle inequality (property (G5)), we get
Letting k\to +\mathrm{\infty} in the above inequality and using (2.7) yields
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 from properties (G2)(G4)
Next, using inequality (2.3), we have
Now, using (2.7),(2.10), the properties of the function ϕ, and letting k\to +\mathrm{\infty} in (2.11), we get
which is a contradiction. Thus, we have proven that (g{x}_{n}) and (g{y}_{n}) are Cauchy sequences in the Gmetric space (X,G). Now, since (X,G) is complete, there are x,y\in X such that (g{x}_{n}) and (g{y}_{n}) are respectively Gconvergent to x and y. That is from Proposition 1.4, we have
Using the continuity of g, we get from Proposition 1.7
Since g{x}_{n+1}=F({x}_{n},{y}_{n}) and g{y}_{n+1}=F({y}_{n},{x}_{n}), employing the commutativity of F and g yields
Now, we shall show that F(x,y)=gx and F(y,x)=gy.
The mapping F is continuous, and since the sequences (g{x}_{n}) and (g{y}_{n}) are respectively Gconvergent to x and y, using Definition 1.9, the sequence (F(g{x}_{n},g{y}_{n})) is Gconvergent to F(x,y). Therefore, from (2.13), (g(g{x}_{n+1})) is Gconvergent to F(x,y). By uniqueness of the limit and using (2.12), we have F(x,y)=gx. Similarly, we can show that F(y,x)=gy. Hence, (x,y) is a coupled coincidence point of F and g. This completes the proof. □
The following example illustrates that Theorem 2.2 is an extension of Theorem 1.16.
Example 2.3 Let us reconsider Example 2.1. Define a map F:X\times X\to X by
and g:X\to X by g(x)=\frac{x}{2} for all x,y\in X. Then F(X\times X)=[0,\mathrm{\infty})=g(X)=X. We observe that
and
Then, the statement (2.3) of Theorem 2.2 is satisfied for \varphi (t)=\frac{2}{3}t and (0,0) is the desired coupled coincidence point.
In the next theorem, we omit the continuity hypothesis of F.
Theorem 2.4 Let (X,\u2aaf) be a partially ordered set and G be a Gmetric on X such that (X,G,\u2aaf) is gordered complete. Suppose that there exist \varphi \in \mathrm{\Phi} and mappings F:X\times X\to X and g:X\to X such that
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 complete, 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.
Proof Proceeding exactly as in Theorem 2.2, we have that (g{x}_{n}) and (g{y}_{n}) are Cauchy sequences in the complete Gmetric space (g(X),G). Then, there exist x,y\in X such that g{x}_{n}\to gx and g{y}_{n}\to gy. Since (g{x}_{n}) is nondecreasing and (g{y}_{n}) is nonincreasing, using the regularity of (X,G,\u2aaf), we have g{x}_{n}\u2aafgx and gy\u2aafg{y}_{n} for all n\ge 0. If g{x}_{n}=gx and g{y}_{n}=gy for some n\ge 0, then gx=g{x}_{n}\u2aafg{x}_{n+1}\u2aafgx=g{x}_{n} and gy\u2aafg{y}_{n+1}\u2aafg{y}_{n}=gy, which implies that g{x}_{n}=g{x}_{n+1}=F({x}_{n},{y}_{n}) and g{y}_{n}=g{y}_{n+1}=F({y}_{n},{x}_{n}), that is, ({x}_{n},{y}_{n}) is a coupled coincidence point of F and g. Then, we suppose that (g{x}_{n},g{y}_{n})\ne (gx,gy) for all n\ge 0. Using the rectangle inequality, (2.15) and property \varphi (t)<t for all t>0, we get
Letting n\to +\mathrm{\infty} in the above inequality, we obtain
which implies that gx=F(x,y) and gy=F(y,x). Thus we proved that (x,y) is a coupled coincidence point of F and g. □
Corollary 2.5 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 k\in [0,1), F:X\times X\to X and g:X\to X such that
for all x,y,u,v,w,z\in X with gw\u2aafgu\u2aafgx and gy\u2aafgv\u2aafgz. Suppose also that F is continuous, has the mixed gmonotone property, F(X\times X)\subseteq g(X) and 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 F({y}_{0},{x}_{0})\u2aafg{y}_{0}, then F and g have a coupled coincidence point.
Proof Taking \varphi (t)=kt with k\in [0,1) in Theorem 2.4, we obtain Corollary 2.5. □
Corollary 2.6 Let (X,\u2aaf) be a partially ordered set and G be a Gmetric on X such that (X,G,\u2aaf) is gordered complete. Suppose that there exist k\in [0,1), F:X\times X\to X and g:X\to X such that
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 complete, F has the mixed gmonotone property, 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.
Proof Taking \varphi (t)=kt with k\in [0,1) in Theorem 2.4, we obtain Corollary 2.6 □
Remark 2.7 Taking g={l}_{x} (the identity mapping) in Corollary 2.5, we obtain [[2], Theorem 3.1]. Taking g={I}_{x} in Corollary 2.6, we obtain [[2], Theorem 3.2].
Now we shall prove the existence and uniqueness theorem of a coupled common fixed point. If (X,\u2aaf) is a partially ordered set, we endow the product set X\times X with the partial order ∇ defined by
Theorem 2.8 In addition to the hypothesis of Theorem 2.2, 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(x,y),F(u,v)) is comparable with (F(x,y),F(y,x)) and (F({x}^{\ast},{y}^{\ast}),F({y}^{\ast},{x}^{\ast})). Suppose also that ϕ is a nondecreasing function. Then F and g have a unique coupled common fixed point, that is, there exists a unique (x,y)\in X\times X such that
Proof From Theorem 2.2, the set of coupled coincidences is nonempty. We shall show that if (x,y) and ({x}^{\ast},{y}^{\ast}) are coupled coincidence points, that is, if gx=F(x,y), g(y)=F(y,x), g{x}^{\ast}=F({x}^{\ast},{y}^{\ast}) and g{y}^{\ast}=F({y}^{\ast},{x}^{\ast}), then
By assumption, there exists (u,v)\in X\times X such that (F(u,v),F(v,u)) is comparable with (F(x,y),F(y,x)) and (F({x}^{\ast},{y}^{\ast}),F({y}^{\ast},{x}^{\ast})). Without loss of generality, we can assume that
and
Put {u}_{0}=u, {v}_{0}=v and choose {u}_{1},{v}_{1}\in X such 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 2.2, we can inductively define sequences (g{u}_{n}) and (g{v}_{n}) in X by g{u}_{n+1}=F({u}_{n},{v}_{n}) and g{v}_{n+1}=F({v}_{n},{u}_{n}).
Further, set {x}_{0}=x, {y}_{0}=y, {{x}_{0}}^{\ast}={x}^{\ast}, {{y}_{0}}^{\ast}={y}^{\ast} and, in the same way, define the sequences (g{x}_{n}), (g{y}_{n}), (g{{x}_{n}}^{\ast}) and (g{{y}_{n}}^{\ast}). Since
then gx\u2aafg{u}_{1} and g{v}_{1}\u2aafgy. Using that F is a mixed gmonotone mapping, one can show easily that gx\u2aafg{u}_{n} and g{v}_{n}\u2aafgy for all n\ge 1. Thus from (2.15), we get
Without loss of generality, we can suppose that (g{u}_{n},g{v}_{n})\ne (gx,gy) for all n\ge 1. Since ϕ is nondecreasing, from the previous inequality, we get
for each n\ge 1. Letting n\to +\mathrm{\infty} in the above inequality and using Lemma 1.15, we obtain
Analogously, we derive that
Hence, from (2.17), (2.18) and the uniqueness of the limit, we get gx=g{x}^{\ast} and gy=g{y}^{\ast}. Hence the equalities in (2.16) are satisfied. Since gx=F(x,y) and gy=F(y,x), by commutativity of F and g, we have
Denote gx=z and gy=w, then by (2.19), we get
Thus, (z,w) is a coincidence point. Then, from (2.16) with {x}^{\ast}=z and {y}^{\ast}=w, we have gx=gz and gy=gw, that is,
From (2.20), (2.21), we get
Then, (z,w) is a coupled common fixed point of F and g.
To prove the uniqueness, assume that (p,q) is another coupled common fixed point. Then by (2.16), we have p=gp=gz=z and q=gq=gw=w. □
Theorem 2.9 Let (X,\u2aaf) be a partially ordered set and G be a Gmetric on X such that (X,G) is a complete Gmetric space and (X,G,\u2aaf) is regular. Suppose that there exist \varphi \in \mathrm{\Phi} and F:X\times X\to X having the mixed monotone property such that
for all x,y,u,v,w,z\in X with w\u2aafu\u2aafx and y\u2aafv\u2aafz. If there exist {x}_{0},{y}_{0}\in X such that {x}_{0}\u2aafF({x}_{0},{y}_{0}) and F({y}_{0},{x}_{0})\u2aaf{y}_{0}, then F has a coupled fixed point. Furthermore, if {y}_{0}\u2aaf{x}_{0}, then x=y, that is, x=F(x,x).
Proof Following the proof of Theorem 2.4 with g={I}_{x}, we have only to show that x=F(x,x). Since {y}_{0}\u2aaf{x}_{0}, we get y\u2aaf{y}_{n}\u2aaf\cdots \u2aaf{y}_{1}\u2aaf{y}_{0}\u2aaf{x}_{0}\u2aaf{x}_{1}\u2aaf\cdots \u2aaf{x}_{n}\u2aafx.
Thus, we have y\u2aafx. Suppose that G(x,x,y)>0. Using inequality (2.15), we have
a contradiction. Thus, G(x,x,y)=0 and x=y=F(x,x). □
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Karapınar, E., Kaymakçalan, B. & Taş, K. On coupled fixed point theorems on partially ordered Gmetric spaces. J Inequal Appl 2012, 200 (2012). https://doi.org/10.1186/1029242X2012200
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DOI: https://doi.org/10.1186/1029242X2012200
Keywords
 coupled fixed point
 coincidence point
 mixed gmonotone property
 ordered set
 Gmetric space