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Some remarks about the existence of coupled gcoincidence points
 İnci M Erhan^{1},
 AntonioFrancisco RoldánLópezdeHierro^{2} and
 Naseer Shahzad^{3}Email author
https://doi.org/10.1186/s136600150558y
© Erhan et al.; licensee Springer. 2015
 Received: 24 October 2014
 Accepted: 13 January 2015
 Published: 12 February 2015
Abstract
Very recently, in a series of subsequent papers, Nan and Charoensawan introduced the notion of gcoincidence point of two mappings in different settings (metric spaces and Gmetric spaces) and proved some theorems in order to guarantee the existence and uniqueness of such kind of points. Although their notion seems to be attractive, in this paper, we show how this concept can be reduced to the unidimensional notion of coincidence point, and how their main theorems can be seen as particular cases of existing results. Moreover, we prove that the proofs of their main statements have some gaps.
1 Introduction
After the appearance of the works by Turinici [1] and, subsequently, by Ran and Reurings [2] and Nieto and RodríguezLópez [3] in partially ordered metric spaces, the branch of fixed point theory devoted to the study of existence and uniqueness of coupled, tripled, quadrupled, and, in general, multidimensional fixed points has attracted much attention. Unfortunately, many of the presented highdimensional results become simple consequences of their corresponding unidimensional versions (see [4–12] and references therein).
Very recently, Nan and Charoensawan [13] introduced the notion of gcoincidence point of two mappings \(F,H:X\times X\rightarrow X\), and proved existence and uniqueness theorems of such kind of points. In this paper, we show that their results can be seen as simple consequences of existing unidimensional results. The same commentaries about their results can also be done for the statements introduced by the same authors in [14].
In order not to enlarge this shortnote unnecessarily, we only include some basic preliminaries. The rest of definitions and basic facts can be found in the mentioned papers.
2 Preliminaries
In the sequel, we denote by \(\mathbb{N}= \{ 0,1,2,\ldots \} \) the family of all nonnegative integers. Let X denote a nonempty set and, given \(n\in\mathbb{N}\), \(n\geq2\), let \(X^{n}\) be the product space \(X\times X\times\overset{(n)}{\cdots}\times X\). Henceforth, T, g, F, and H stand for mappings as follows: \(T,g:X\rightarrow X\) and \(F,H:X^{2}\rightarrow X\). Consider the following kind of control functions.
Definition 2.1
 (\(\mathcal{P}_{1}\)):

\(\varphi(t)< t\) for all \(t>0\);
 (\(\mathcal{P}_{2}\)):

\(\lim_{s\rightarrow t^{+}}\varphi(s)< t\) for all \(t>0\).
Remark 2.1
(2) By (\(\mathcal{P}_{1}\)), it is clear that \(\lim_{t\rightarrow0^{+}}\varphi(t)=0\) for all \(\varphi\in\Phi\). However, this property does not imply the following one: if \(\{a_{n}\},\{b_{n}\}\subset [ 0,\infty ) \) verify that \(a_{n}\leq\varphi(b_{n})\) for all \(n\in\mathbb{N}\) and \(\{b_{n}\}\rightarrow0\), then \(\{a_{n}\} \rightarrow0 \). For instance, given \(a>0\), if we define \(a_{n}=a\) and \(b_{n}=0\) for all \(n\in\mathbb{N}\), then \(a_{n}=a=\varphi_{a}(0)=\varphi_{a}(b_{n})\) for all \(n\in\mathbb{N}\) and \(\{b_{n}\}\rightarrow0\), but \(\{a_{n}\}\rightarrow a\neq0\).
The following definitions and theorem can be found in [15].
Definition 2.2

\((T,g)\) closed if \((Tx,Ty)\in\mathcal{M}\) for all \(x,y\in X\) such that \((gx,gy)\in\mathcal{M}\);

\((T,g)\) compatible if \(Tx=Ty\) for all \(x,y\in X\) such that \(gx=gy\).
Definition 2.3
We will say that a subset \(\mathcal{M}\subseteq X^{2}\) is transitive if \((x,y),(y,z)\in\mathcal{M}\) implies that \((x,z)\in\mathcal{M}\).
Definition 2.4
Let \((X,d)\) be a metric space and let \(\mathcal{M}\subseteq X^{2}\) be a subset. We will say that \((X,d,\mathcal{M})\) is regular if for all sequence \(\{x_{n}\}\subseteq X\) such that \(\{x_{n}\}\rightarrow x\) and \((x_{n},x_{n+1})\in\mathcal{M}\) for all \(n\in \mathbb{N}\), we have \((x_{n},x)\in\mathcal{M}\) for all \(n\in\mathbb{N}\).
Definition 2.5
Let \((X,d)\) be a metric space, let \(\mathcal{M}\subseteq X^{2}\) be a subset and let \(x\in X\). A mapping \(T:X\rightarrow X\) is said to be ℳcontinuous at x if for all sequence \(\{x_{n}\}\subseteq X\) such that \(\{x_{n}\}\rightarrow x\) and \((x_{n},x_{n+1})\in\mathcal{M}\) for all \(n\in\mathbb{N}\), we have \(\{Tx_{n}\}\rightarrow Tx\). T is ℳcontinuous if it is ℳcontinuous at each \(x\in X\).
The reader can compare the previous notion with the concept of ‘nondecreasingcontinuity’ introduced in [16].
Remark 2.2
Every continuous mapping is also ℳcontinuous, whatever \(\mathcal{M}\subseteq X^{2}\).
Definition 2.6
Remark 2.3
If T and g are commuting (that is, \(Tgx=gTx\) for all \(x\in X\)), then they are also \((O,\mathcal{M})\)compatible, whatever ℳ.
The main result in [15] was the following one.
Theorem 2.1
(Karapınar et al. [15, Theorem 33])
 (a)
T and g are ℳcontinuous and \((O,\mathcal {M})\)compatible;
 (b)
T and g are continuous and commuting;
 (c)
\((X,d,\mathcal{M})\) is regular and \(g(X)\) is closed.
If there exists a point \(x_{0}\in X\) such that \((gx_{0},Tx_{0})\in \mathcal{M} \), then T and g have, at least, a coincidence point.
Next, we show that the proof of Theorem 2.1 given by the authors in [15] demonstrates, point by point, a slightly stronger result. To do that, we notice that some of the definitions involved in the last theorem were subtly modified in [17] in the following sense.
Definition 2.7
(See Kutbi et al. [17])

gtransitive if \((gx,gz)\in\mathcal{M}\) for all \(x,y,z\in X \) such that \(( gx,gy ) , ( gy,gz ) \in\mathcal{M}\);

\((T,g)\) compatible if \(Tx=Ty\) for all \(x,y\in X\) such that \(gx=gy\) and \(( gx,gy ) \in\mathcal{M}\).
With respect to the previous definitions, we point out the following remarks.
(1) In the proof of Theorem 2.1 in [15], the hypothesis ‘ℳ is \((T,g)\)compatible’ was only used in one subcase of case (c) (the subcase in which there exists some \(m_{0}\in\mathbb{N}\) such that \(d(gx_{m_{0}},x)=0\)). Then we can remove it from the general hypotheses of the theorem if we add it to assumption (c).
(2) The notions of \((T,g)\) compatibility in Definitions 2.2 and 2.7 are different. In fact, the notion given in Definition 2.7 is weaker than the concept given in Definition 2.2. However, as the reader can easily check, in the proof of Theorem 2.1 given by the authors in [15], they only used \(( T,g ) \)compatibility in the sense of Definition 2.7 because they assumed that, in the mentioned subcase, we also have \(( gx_{m_{0}},gz ) \in\mathcal{M}\).
(4) By Remark 2.2, we can suppose that T and g are continuous in case (a) because this condition implies that they are also ℳcontinuous.
As a consequence of the previous commentaries, we deduce that the subtle refinement given in Definition 2.7 shows that the proof of Theorem 2.1 given by the authors in [15] demonstrates, point by point, the following stronger result (in which we use \(( T,g ) \)compatibility in the sense of Definition 2.7).
Theorem 2.2
 (a):

T and g are ℳcontinuous and \((O,\mathcal {M})\)compatible;
 (a′):

T and g are continuous and \((O,\mathcal{M})\)compatible;
 (b):

T and g are continuous and commuting;
 (c):

\((X,d,\mathcal{M})\) is regular, \(g(X)\) is closed and ℳ is \((T,g)\)compatible.
If there exists a point \(x_{0}\in X\) such that \((gx_{0},Tx_{0})\in \mathcal{M} \), then T and g have, at least, a coincidence point.
Remark 2.4
The condition ‘\(g(X)\) is closed’ of assumption (c) was only used to guarantee that \(( g(X),d ) \) is a complete metric space. As a consequence, the same thesis can be deduced replacing, in case (c), that ‘\(( X,d ) \) is complete and \(g(X)\) is closed’ by the weaker condition ‘\(g(X)\) is dcomplete’, and the proof is obtained by verbatim. This argument was already employed, for instance, in the proof of Theorem 34 in [11].
3 About some coupled gcoincidence point theorems in metric spaces
From now on, \(g:X\rightarrow X\) and \(F,H:X^{2}\rightarrow X\) will denote arbitrary mappings. In [13], the authors introduced the following notions.
Definition 3.1
An element \((x,y)\in X^{2}\) is called a coupled gcoincidence point of the mappings F and H if \(F(x,y)=H(gx,gy)\) and \(F(y,x)=H(gy,gx)\).
Definition 3.2
Definition 3.3
Definition 3.4
The main theorem in [13] was the following one.
Theorem 3.1
(Nan and Charoensawan [13, Theorem 3.1])
Notice that the authors did not established in the statement of the previous theorem that M is transitive in the sense of Definition 3.3, but they used it throughout their proof. The authors also forgot to say that the pair \(\{F,H\}\) is ggeneralized compatible (they only mentioned generalized compatible pairs, which correspond to a different notion that they had previously commented in [13]). Next, we show that this is not a new result.
Theorem 3.2
Theorem 3.1 (including the facts that ‘M is transitive’ and replacing the hypothesis ‘ \(F,H:X\times X\rightarrow X\) are two generalized compatible mappings’ by ‘ \(F,H:X\times X\rightarrow X\) are two ggeneralized compatible mappings’) immediately follows from Theorem 2.2.
Proof
• As \(( X,d ) \) is complete, then \(( X^{2},D_{2} ) \) is also a complete metric space.
• As H and g are continuous mappings with respect to d, then \(T_{H,g}\) is also continuous with respect to \(D_{2}\).
• As F is a continuous mapping with respect to d, then \(T_{F}\) is also continuous with respect to \(D_{2}\).
• A point \(( x,y ) \in X^{2}\) is a gcoincidence point of F and H (in the sense of Definition 3.1) if, and only if, \(( x,y ) \) is a coincidence point of \(T_{F}\) and \(T_{H,g}\).
As a consequence of the previous facts, by using item (a′) of Theorem 2.2 applied to \(T_{F}\) and \(T_{H,g}\) in \((X^{2},D_{2})\) and \(M\subseteq X^{4}=(X^{2})^{2}\), we conclude that \(T_{F}\) and \(T_{H,g}\) have, at least, a coincidence point, which is a gcoincidence point of F and H. □
In the following result, the continuity of F is not assumed.
Theorem 3.3
(Nan and Charoensawan [13, Theorem 3.2])
The previous statement has the same mistakes as we have pointed out about Theorem 3.1. In fact, the authors assumed the \(( X,d ) \) and \(( g(X),d ) \) are, at the same time, complete, which is unnecessary. In this case, we can follow, point by point, the proof of Theorem 3.2, but the continuity of \(T_{F}\) is not guaranteed because F is not necessarily continuous. Nevertheless, additional mistakes can be found in its proof. We can easily discover them comparing this result with Theorem 2.2.
Before that, let us show a mistake that can be found in some papers, closely related to item (2) of Remark 2.1. When F is not necessarily continuous, it is usual to assume that the metric space is, in some sense, regular (in the previous result, this assumption is condition (3)). In such a case, the existence of a coincidence point can be deduced applying the contractivity condition to the terms of the sequence and the desired limit. In some cases, this is not possible, as in the following example.
Example 3.1
Next, let us show that some hypotheses of Theorem 3.3 are not appropriate. In fact, we claim that, under appropriate conditions, Theorem 3.3 is a consequence of Theorem 2.2. To prove it, we could try to apply item (c) of the last one. But, comparing both results, we observe three important differences.
To overcome the previous drawbacks, it would be convenient to consider the following hypotheses.
(1) \(H(g(X)\times g(X))\) is complete in \((X,d)\), which means that \(T_{H,g}(X^{2})\) is complete in \((X^{2},D_{2})\).
(2) The regularity condition (3) must be replaced by the following one: ‘if \(\{x_{n}\},\{y_{n}\}\subseteq X\) are sequences such that \(\{x_{n}\}\rightarrow x\in X\), \(\{y_{n}\}\rightarrow y\in X\) and \(( x_{n},y_{n},x_{n+1},y_{n+1} ) \in M\) for all \(n\in \mathbb{N}\), then \(( x_{n},y_{n},x,y ) \in M\) for all \(n\in \mathbb{N}\)’. In such a case, \(( X^{2},D_{2},M ) \) is regular in the sense of Definition 2.4.
(3) A kind of \(( T_{F},T_{H,g} ) \)compatibility is necessary to ensure that the limit as \(n\rightarrow\infty\) in (7) is zero. For instance, we propose assuming that if \(( x,y,u,v ) \in M\) is such that \(H(gx,gy)=H(gu,gv)\) and \(H(gy,gx)=H(gv,gu)\), then \(F(x,y)=F(u,v)\) and \(F(y,x)=F(v,u)\). In this case, M is \(( T_{F},T_{H,g} ) \)compatible. This condition can be omitted if we additionally assume that \(\varphi(0)=0\) (see, for instance, Corollary 35 in [15]).
Under these new conditions, Theorem 3.3 becomes a consequence of Theorem 2.2.
4 About some coupled coincidence point theorems in Gmetric spaces
All necessary preliminaries (about quasimetrics, Gmetrics, contractions depending on a subset \(\mathcal{M}\subseteq X^{2}\), etc.) of this part can be found in [18]. Let Ψ be the family of functions \(\varphi \in\Phi\) such that \(\varphi(t)=0\) if, and only if, \(t=0\) (notice that Ψ and Φ were employed in [18] using the contrary notation). The following result shows a simple way to consider quasimetrics from Gmetrics.
Lemma 4.1
(Agarwal et al. [19])
 (1)\(q_{G}\) and \(q_{G}^{\prime}\) are quasimetrics on X. Moreover,$$ q_{G}^{\prime}(x,y)\leq2q_{G}(x,y) \leq4q_{G}^{\prime}(x,y) \quad\textit{for all }x,y\in X. $$
 (2)
In \((X,q_{G})\) and in \((X,q_{G}^{\prime})\), a sequence is rightconvergent (respectively, leftconvergent) if and only if it is convergent. In such a case, its rightlimit, its leftlimit and its limit coincide.
 (3)
In \((X,q_{G})\) and in \((X,q_{G}^{\prime})\), a sequence is rightCauchy (respectively, leftCauchy) if and only if it is Cauchy.
 (4)
In \((X,q_{G})\) and in \((X,q_{G}^{\prime})\), every rightconvergent (respectively, leftconvergent) sequence has a unique rightlimit (respectively, leftlimit).
 (5)
If \(\{x_{n}\}\subseteq X\) and \(x\in X\), then \(\{x_{n}\}\overset {G}{\longrightarrow}x \Longleftrightarrow\{x_{n}\}\overset{q_{G}}{\longrightarrow}x \Longleftrightarrow\{x_{n}\}\overset{q_{G}^{\prime}}{ \longrightarrow}x\).
 (6)
If \(\{x_{n}\}\subseteq X\), then \(\{x_{n}\}\) is GCauchy ⟺ \(\{x_{n}\}\) is \(q_{G}\)Cauchy ⟺ \(\{x_{n}\}\) is \(q_{G}^{\prime}\)Cauchy.
 (7)
\((X,G)\) is complete ⟺ \((X,q_{G})\) is complete ⟺ \((X,q_{G}^{\prime})\) is complete.
Definition 4.1
Clearly, if T and g are commuting, then they are both \(( O,\mathcal{M} ) \)compatible or \(( O^{\prime},\mathcal {M} ) \)compatible. The following notion also extends the regularity of an ordered metric space.
Definition 4.2
Let \((X,q)\) be a quasimetric space and let \(A\subseteq X\) and \(\mathcal {M}\subseteq X^{2}\) be two nonempty subsets. We say that \(( A,q,\mathcal{M} ) \) is regular (or A is \((q,\mathcal{M})\) regular) if we have \(( x_{n},u ) \in\mathcal{M}\) for all n provided that \(\{x_{n}\}\) is a qconvergent sequence on A, \(u\in A\) is its qlimit and \(( x_{n},x_{m} ) \in\mathcal{M}\) for all \(n< m\).
Definition 4.3
Notice that condition (8) is not symmetric on x and y because \(( gx,gy ) \in\mathcal{M}\) does not imply \(( gy,gx ) \in\mathcal{M}\). In order to compensate this absence of symmetry, we will suppose an additional condition on the ambient space.
Definition 4.4

rightCauchy if every rightCauchy sequence in \((X,q)\) is, in fact, a Cauchy sequence in \(( X,q ) \);

leftCauchy if every leftCauchy sequence in \((X,q)\) is, in fact, a Cauchy sequence in \(( X,q ) \);

rightconvergent if every rightconvergent sequence in \((X,q)\) is, in fact, a convergent sequence in \(( X,q ) \);

leftconvergent if every leftconvergent sequence in \((X,q)\) is, in fact, a convergent sequence in \(( X,q ) \).
Theorem 4.1
(RoldánLópezdeHierro et al. [18, Theorem 3.3])
 (A)
There exists a \((T,g,\mathcal{M})\)Picard sequence on X.
 (B)
T is a \(( g,\mathcal{M},\Psi ) \)contraction of the second kind.
 (a)
X (or \(g(X)\) or \(T(X)\)) is qcomplete, T and g are ℳcontinuous and the pair \((T,g)\) is \(( O^{\prime },\mathcal{M} ) \)compatible;
 (b)
X (or \(g(X)\) or \(T(X)\)) is qcomplete and T and g are ℳcontinuous and commuting;
 (c)
\((g(X),q)\) is complete and rightconvergent, and X (or \(g(X)\)) is \((q,\mathcal{M})\)regular;
 (d)
\((X,q)\) is complete and rightconvergent, \(g(X)\) is closed and X (or \(g(X)\)) is \((q,\mathcal{M})\)regular;
 (e)
\((X,q)\) is complete and rightconvergent, g is ℳcontinuous, ℳ is gclosed, the pair \((T,g)\) is \(( O, \mathcal{M} ) \)compatible and X is \((q,\mathcal{M})\)regular.
Then T and g have, at least, a coincidence point.
 (A′):

\(T(X)\subseteq g(X)\), ℳ is gtransitive and \(( T,g ) \)closed, and there exists \(x_{0}\in X\) such that \(( gx_{0},Tx_{0} ) \in\mathcal{M}\).
 (\(\mathrm{A}^{\prime\prime}\)):

ℳ is gtransitive and \(( T,g ) \)closed, and there exists a \(( T,g ) \)Picard sequence \(\{x_{n}\}_{n\geq0}\) such that \(( gx_{0},Tx_{0} ) \in\mathcal{M}\).
The ℳcontinuity of the mappings can be replaced by continuity. The previous result was extended to the more general case in which \(\varphi\in\Phi\) as follows.
Remark 4.1
Notice that the hypothesis ‘there exists \(x_{0}\in X\) such that \(( gx_{0},Tx_{0} ) \in\mathcal{M}\)’ was omitted in condition (A′) by mistake in [18].
Theorem 4.2
If we additionally assume that ℳ is \(( T,g ) \)compatible, then Theorem 4.1 also holds even if T is a \(( g,\mathcal{M},\Phi ) \)contraction of the second kind.
Remark 4.2
As we pointed out in the previous section, throughout the proof of Theorem 4.1 in [18], the \(( T,g ) \)compatibility of ℳ was only used under assumptions (c), (d), and (e), when X (or \(g(X)\)) is \((q,\mathcal{M})\)regular. However, when T and g are continuous (or ℳcontinuous), it is not necessary.
In [14], the authors introduced the following notions in which \(F,H:X\times X\rightarrow X\) are two arbitrary mappings, and they proved the following theorem.
Definition 4.5
An element \((x,y)\in X^{2}\) is called a coupled coincidence point of the mappings F and H if \(F(x,y)=H(x,y)\) and \(F(y,x)=H(y,x)\).
Definition 4.6
Definition 4.7
Definition 4.8
Notice that, although the authors did not remark it in [14], the previous definition needs a metric structure, maybe in a metric space, in a quasimetric space or in a Gmetric space.
The first main result in [14] is the following one.
Theorem 4.3
(Nan and Charoensawan [14, Theorem 3.1])
 (a)
F is continuous;
 (b)for any two sequences \(\{x_{n}\}\) and \(\{y_{n}\}\) with for all \(n\geq1\)$$\begin{aligned} & (x_{n+1},y_{n+1},x_{n+1},y_{n+1},x_{n},y_{n}) \in M\quad \textit{and} \\ & \bigl\{ H(x_{n},y_{n})\bigr\} \rightarrow H(x,y),\qquad \bigl\{ H(y_{n},x_{n})\bigr\} \rightarrow H(y,x) \\ & \quad\textit{implies}\quad \\ & \bigl( H(x_{n},y_{n}),H(y_{n},x_{n}),H(x,y),H(y,x),H(x,y),H(y,x) \bigr) \in M. \end{aligned}$$

The partial order ⪯ is a superfluous hypothesis.

The authors did not assume the transitive property in the statement although they used it throughout the proof.
Following the techniques we have shown in the first part of the manuscript, we may deduce the following statement.
Proof
• As \(( X,G ) \) is a complete Gmetric space, then \(( X^{2},q_{G} ) \) is a complete quasimetric space.
• As \(q_{G}\) comes from a Gmetric, then \(( X^{2},q_{G} ) \) is a rightCauchy quasimetric space.
• As F (respectively, H) is a continuous mapping with respect to G, then \(T_{F}\) (respectively, \(T_{H}\)) is also continuous with respect to \(q_{G}\).
• A point \(( x,y ) \in X^{2}\) is a coupled coincidence point of F and H (in the sense of Definition 4.5) if, and only if, \(( x,y ) \) is a coincidence point of \(T_{F}\) and \(T_{H}\).
As a consequence of Theorem 4.2, \(T_{F}\) and \(T_{H}\) have, at least, a coincidence point, which is a coupled coincidence point of F and H. □
With respect to the case in which F is not necessarily continuous and we assume the regularity condition (hypothesis (b) of Theorem 4.3), we can now repeat the same commentaries that we gave in the previous section.
Declarations
Acknowledgements
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. N Shahzad acknowledges with thanks DSR for financial support. AF RoldánLópezdeHierro has been partially supported by Junta de Andalucía by project FQM268 of the Andalusian CICYE.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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