A note on ‘$(G,F)$Closed set and tripled point of coincidence theorems for generalized compatibility in partially metric spaces’
 Erdal Karapınar^{1, 2}Email author and
 AntonioFrancisco RoldánLópezdeHierro^{3}
https://doi.org/10.1186/1029242X2014522
© Karapınar and RoldánLópezdeHierro; licensee Springer. 2014
Received: 11 July 2014
Accepted: 21 November 2014
Published: 17 December 2014
Abstract
Recently, some (common) multidimensional fixed theorems in partially ordered complete metric spaces have appeared as a generalization of existing (usual) fixed point results. Unexpectedly, we realized that most of such (common) coupled fixed theorems are either weaker or equivalent to existing fixed point results in the literature. In particular, we prove that the results included in the very recent paper (Charoensawan and Thangthong in Fixed Point Theory Appl. 2014:245, 2014) can be considered as a consequence of existing fixed point theorems on the topic in the literature.
MSC:47H10, 54H25.
Keywords
1 Introduction and preliminaries
Multidimensional fixed point theory was initiated in 2006 by Gnana Bhaskar and Lakshmikantham [1]. In fact, the authors [1] investigated the existence and uniqueness of a coupled fixed point of certain operators in the context of a partially ordered set to solve a periodic boundary value problem. Since then, multidimensional fixed point theorems have been investigated heavily by several authors; see, e.g., [1–29] and related references therein.
In this short note, we underline the fact that most of the multidimensional fixed point theorems can be derived from the existing (unidimensional) fixed point results in the literature. In particular, we shall show that the result in the recent report [6] can be considered in this frame.
For the sake of completeness, we recollect some basic definitions, notations and results on the topic in the literature. Throughout the paper, let X be a nonempty set. Given a positive integer n, let ${X}^{n}$ be the product space $X\times X\times \stackrel{n}{\cdots}\times X$. Let $\mathbb{N}=\{0,1,2,\dots \}$ be the set of all nonnegative integers. In the sequel, n, m and k will be used to denote nonnegative integers. Unless otherwise stated, ‘for all n’ will mean ‘for all $n\ge 0$’.
Definition 1.1 (Roldán and Karapınar [22])
A preorder (or a quasiorder) ≼ on X is a binary relation on X that is reflexive (i.e., $x\preccurlyeq x$ for all $x\in X$) and transitive (if $x,y,z\in X$ verify $x\preccurlyeq y$ and $y\preccurlyeq z$, then $x\preccurlyeq z$). In such a case, we say that $(X,\preccurlyeq )$ is a preordered space (or a preordered set). If a preorder ≼ is also antisymmetric ($x\preccurlyeq y$ and $y\preccurlyeq x$ imply $x=y$), then ≼ is called a partial order.
Throughout this manuscript, let $(X,d)$ be a metric space, and let ≼ be a preorder (or a partial order) on X. In the sequel, $T,g:X\to X$ and $F:{X}^{n}\to X$ will denote mappings.
Definition 1.2 A point $({x}_{1},{x}_{2},\dots ,{x}_{n})\in {X}^{n}$ is:

a coupled coincidence point of F and g if $n=2$,$F({x}_{1},{x}_{2})=g{x}_{1}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}F({x}_{2},{x}_{1})=g{x}_{2};$

a tripled coincidence point of F and g if $n=3$,$F({x}_{1},{x}_{2},{x}_{3})=g{x}_{1},\phantom{\rule{2em}{0ex}}F({x}_{2},{x}_{1},{x}_{2})=g{x}_{2}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}F({x}_{3},{x}_{2},{x}_{1})=g{x}_{3};$

a quadrupled coincidence point of F and g if $n=4$,$\begin{array}{r}F({x}_{1},{x}_{2},{x}_{3},{x}_{4})=g{x}_{1},\phantom{\rule{2em}{0ex}}F({x}_{2},{x}_{3},{x}_{4},{x}_{1})=g{x}_{2},\\ F({x}_{3},{x}_{4},{x}_{1},{x}_{2})=g{x}_{3}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}F({x}_{4},{x}_{1},{x}_{2},{x}_{3})=g{x}_{4}.\end{array}$
Notice that when we take g as the identity mapping on X, then a point verifying the related conditions above is a coupled (respectively, tripled, quadrupled) fixed point of F due to Gnana Bhaskar and Lakshmikantham [1] (respectively, Berinde and Borcut [9], Karapınar [13]).
Definition 1.3 (AlMezel et al. [21])
If $(X,\preccurlyeq )$ is a preordered space and $T,g:X\to X$ are two mappings, we will say that T is a $(g,\preccurlyeq )$nondecreasing mapping if $Tx\preccurlyeq Ty$ for all $x,y\in X$ such that $gx\preccurlyeq gy$. If g is the identity mapping on X, T is ≼nondecreasing.
In [28], $(g,\preccurlyeq )$nondecreasing mappings were called gisotone mappings (in particular, when X is a product space ${X}^{n}$).
Definition 1.4 A fixed point of a selfmapping $T:X\to X$ is a point $x\in X$ such that $Tx=x$. A coincidence point between two mappings $T,g:X\to X$ is a point $x\in X$ such that $Tx=gx$. A common fixed point of $T,g:X\to X$ is a point $x\in X$ such that $Tx=gx=x$.
Definition 1.5 We will say that T and g are commuting if $gTx=Tgx$ for all $x\in X$, and we will say that F and g are commuting if $gF({x}_{1},{x}_{2},\dots ,{x}_{n})=F(g{x}_{1},g{x}_{2},\dots ,g{x}_{n})$ for all ${x}_{1},\dots ,{x}_{n}\in X$.
Remark 1.1 If $T,g:X\to X$ are commuting and ${x}_{0}\in X$ is a coincidence point of T and g, then $T{x}_{0}$ is also a coincidence point of T and g.
In 2003, Ran and Reurings characterized the Banach contraction mapping principle in the context of partially ordered metric space.
Theorem 1.1 (Ran and Reurings [20])
 (a)
$(X,d)$ is complete.
 (b)
T is ≼nondecreasing.
 (c)
T is continuous.
 (d)
There exists ${x}_{0}\in X$ such that ${x}_{0}\preccurlyeq T{x}_{0}$.
 (e)
There exists a constant $k\in (0,1)$ such that $d(Tx,Ty)\le kd(x,y)$ for all $x,y\in X$ with $x\succcurlyeq y$.
Then T has a fixed point. Moreover, if for all $(x,y)\in {X}^{2}$ there exists $z\in X$ such that $x\preccurlyeq z$ and $y\preccurlyeq z$, we obtain uniqueness of the fixed point.
After Ran and Reurings’ result, fixed point theorems have been investigated heavily. One of the interesting results in this direction was reported by Nieto and RodríguezLópez in [19], who slightly modified the hypothesis of the previous result swapping condition (c) with the fact that $(X,d,\preccurlyeq )$ is nondecreasingregular as follows.
Definition 1.6 Let $(X,\preccurlyeq )$ be an ordered set endowed with a metric d. We will say that $(X,d,\preccurlyeq )$ is nondecreasingregular (respectively, nonincreasingregular) if any ≼nondecreasing (respectively, ≼nonincreasing) sequence $\{{x}_{m}\}$ is dconvergent to $x\in X$, we have that ${x}_{m}\preccurlyeq x$ (respectively, ${x}_{m}\succcurlyeq x$) for all m. And $(X,d,\preccurlyeq )$ is regular if it is both nondecreasingregular and nonincreasingregular.
and proved a version of the following result in which the space is not necessarily endowed with a partial order (but the contractivity condition holds over all pairs of points of the space).
 (a)
$(X,d)$ is complete.
 (b)
T is ≼nondecreasing.
 (c)
Either T is continuous or $(X,d,\preccurlyeq )$ is nondecreasingregular.
 (d)
There exists ${x}_{0}\in X$ such that ${x}_{0}\preccurlyeq T{x}_{0}$.
 (e)
There exists $\phi \in \mathrm{\Phi}$ such that $d(Tx,Ty)\le \phi (d(x,y))$ for all $x,y\in X$ with $x\succcurlyeq y$.
Then T has a fixed point. Moreover, if for all $(x,y)\in {X}^{2}$ there exists $z\in X$ such that $x\preccurlyeq z$ and $y\preccurlyeq z$, we obtain uniqueness of the fixed point.
An interesting generalization of the previous result was given by Wang in [28] using this extended partial order on ${X}^{n}$.
Theorem 1.3 (Wang [28], Theorem 3.4)
 (a)
T is continuous, G is continuous and commutes with T, or
 (b)
$(X,d,\preccurlyeq )$ is regular and $G({X}^{n})$ is closed.
If there exists ${Y}_{0}\in {X}^{n}$ such that $G({Y}_{0})$ and $T({Y}_{0})$ are ⊑comparable, then T and G have a coincidence point.
For further generalizations of the previous result, we refer readers to papers of Romaguera [25] and in AlMezel et al. [21].
Gnana Bhaskar and Lakshmikantham introduced the following condition in order to guarantee the existence of coupled fixed points
Definition 1.7 (Gnana Bhaskar and Lakshmikantham [1])
On the other hand, Samet and Vetro [26] succeeded in proving some results in which the mapping F did not necessarily have the mixed monotone property.
Definition 1.8 (Samet and Vetro [26])
 (i)
$(x,y,z,w)\in M\u27fa(w,z,y,x)\in M$;
 (ii)
$(x,y,z,w)\in M\u27f9(F(x,y),F(y,x),F(z,w),F(w,z))\in M$.
The following theorem is the main result in [26].
Theorem 1.4 (Samet and Vetro [26])
 (i)
M is Finvariant;
 (ii)
there exists $({x}_{0},{y}_{0})\in {X}^{2}$ such that $(F({x}_{0},{y}_{0}),F({y}_{0},{x}_{0}),{x}_{0},{y}_{0})\in M$;
 (iii)for all $(x,y,u,v)\in M$, we have$\begin{array}{r}d(F(x,y),F(u,v))\\ \phantom{\rule{1em}{0ex}}\le \frac{\alpha}{2}[d(x,F(x,y))+d(y,F(y,x))]\\ \phantom{\rule{2em}{0ex}}+\frac{\beta}{2}[d(u,F(u,v))+d(v,F(v,u))]+\frac{\theta}{2}[d(x,F(u,v))+d(y,F(v,u))]\\ \phantom{\rule{2em}{0ex}}+\frac{\gamma}{2}[d(u,F(x,y))+d(v,F(y,x))]+\frac{\delta}{2}[d(x,u)+d(y,v)],\end{array}$
where α, β, θ, γ, δ are nonnegative constants such that $\alpha +\beta +\theta +\gamma +\delta <1$.
Then F has a coupled fixed point, i.e., there exists $(x,y)\in X\times X$ such that $F(x,y)=x$ and $F(y,x)=y$.
Furthermore, Sintunavarat et al. [27] introduced the notion of transitive property to reconsider the Lakshmikantham and Ćirić’s theorem (see [17]) without the mixed monotone property.
Definition 1.9 (Sintunavarat et al. [27])
Then they proved the following result.
Theorem 1.5 (Sintunavarat et al. [27])
 (a)
F is continuous or
 (b)if for any two sequences $\{{x}_{m}\}$, $\{{y}_{m}\}$ with $({x}_{m+1},{y}_{m+1},{x}_{m},{y}_{m})\in M$,$\{{x}_{m}\}\to x,\phantom{\rule{2em}{0ex}}\{{y}_{m}\}\to y,$
for all $m\ge 1$, then $(x,y,{x}_{m},{y}_{m})\in M$ for all $m\ge 1$.
If there exists $({x}_{0},{y}_{0})\in X\times X$ such that $(F({x}_{0},{y}_{0}),F({y}_{0},{x}_{0}),{x}_{0},{y}_{0})\in M$ and M is an Finvariant set which satisfies the transitive property, 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.
Recently, Charoensawan [11], based on Batra and Vashistha’s results, introduced the tripled case as follows.
Definition 1.10 (Charoensawan [11])
The following concept is an extension of Definition 1.9.
Definition 1.11 (Charoensawan [11])
Definition 1.12 (Charoensawan [11])
In the previous definitions, it is not necessary to consider either a metric or a partial order on X.
Theorem 1.6 (Charoensawan [11], Theorem 3.7)
for all $x,y,z,u,v,w\in X$ with $(gx,gy,gz,gu,gv,gw)\in M$ or $(gu,gv,gw,gx,gy,gz)\in M$. Suppose that $F({X}^{3})\subseteq gX$, g commutes with F.
Meanwhile, Kutbi et al. [16] used a bidimensional extension of an Finvariant subset as follows.
Definition 1.13 (Kutbi et al. [16])
The following one is the main result of Kutbi et al. [16].
Theorem 1.7 (Kutbi et al. [16])
 (i)
M is Fclosed;
 (ii)
there exists $({x}_{0},{y}_{0})\in {X}^{2}$ such that $(F({x}_{0},{y}_{0}),F({y}_{0},{x}_{0}),{x}_{0},{y}_{0})\in M$;
 (iii)there exists $k\in [0,1)$ such that for all $(x,y,u,v)\in M$, we have$d(F(x,y),F(u,v))+d(F(y,x),F(v,u))\le k(d(x,u)+d(y,v)).$
Then F has a coupled fixed point.
2 Main results
Notice that this family of control functions was employed by Sintunavarat et al. in Theorem 1.5 and by Charoensawan in Theorem 1.6. Here, we should mention that it is not as general as Wang’s family Φ since the value $\phi (0)$ is not necessarily determined if $\phi \in \mathrm{\Phi}$. Thus, we have ${\mathrm{\Phi}}^{\prime}\subset \mathrm{\Phi}$ in this sense.
Secondly, we pay attention to the following fact: Charoensawan’s notion of Finvariant set is similar to Kutbi et al.’s notion of Fclosed set, but it is different from Samet and Vetro’s original concept because property (i) in Definition 1.8 is not imposed. Then, coherently with Definition 1.13, we prefer calling these subsets employing the term Fclosed.
Definition 2.1 Let $T,g:X\to X$ be two mappings and let $M\subseteq {X}^{2}$ be a subset. We will say that M is:

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

$(T,g)$compatible if $Tx=Ty$ for all $x,y\in X$ such that $gx=gy$.
Definition 2.2 ([29])
We will say that a subset $M\subseteq {X}^{2}$ is transitive if $(x,y),(y,z)\in M$ implies that $(x,z)\in M$.
Definition 2.3 ([29])
Let $(X,d)$ be a metric space and let $M\subseteq {X}^{2}$ be a subset. We will say that $(X,d,M)$ is regular if for all sequence $\{{x}_{m}\}\subseteq X$ such that $\{{x}_{m}\}\to x$ and $({x}_{m},{x}_{m+1})\in M$ for all m, we have that $({x}_{m},x)\in M$ for all m.
Remark 2.1 If T and g are commuting, then they are also $(O,M)$compatible, whatever M.
The main result in [29], using the previous notions, is the following one.
Theorem 2.1 (Karapınar et al. [29])
 (a)
T and g are Mcontinuous and $(O,M)$compatible;
 (b)
T and g are continuous and commuting;
 (c)
$(X,d,M)$ is regular and gX is closed.
If there exists a point ${x}_{0}\in X$ such that $(g{x}_{0},T{x}_{0})\in M$, then T and g have, at least, a coincidence point.
The following one is the main result of [6].
Theorem 2.2 (Charoensawan and Thangthong [6], Theorem 3.1)
 (a)
F is continuous or
 (b)for any three sequences $\{{x}_{n}\}$, $\{{y}_{n}\}$ and $\{{z}_{n}\}$ with$\begin{array}{r}(G({x}_{n},{y}_{n},{z}_{n}),G({y}_{n},{z}_{n},{x}_{n}),G({z}_{n},{x}_{n},{y}_{n}),\\ \phantom{\rule{1em}{0ex}}G({x}_{n+1},{y}_{n+1},{z}_{n+1}),G({y}_{n+1},{z}_{n+1},{x}_{n+1}),G({z}_{n+1},{x}_{n+1},{y}_{n+1}))\in M\end{array}$
and M is an $(G,F)$closed, then there exist $(x,y,z)\in X\times X\times X$ such that $G(x,y,z)=F(x,y,z)$, $G(y,z,x)=F(y,z,x)$, and $G(z,x,y)=F(z,x,y)$, that is, F and G have a tripled point of coincidence.
The following remarks must be done in order to clarify some facts stated in [6] to the reader.

In the previous theorem, the authors assumed that $(X,\preccurlyeq )$ is a partially ordered set. Clearly, it is a superfluous hypothesis.

We understand that ‘${x}_{0},{y}_{0},{z}_{0}\in X\times X$’ is an erratum and that it must be replaced by ‘${x}_{0},{y}_{0},{z}_{0}\in X$’.

In [6], Example 3.2 is invalid since $G(x,y,z)=x+y+z$ does not necessarily belong to $X=[0,1]$ when $x,y,z\in X$ are arbitrary.
for all $(x,y,z),(u,v,w)\in Y$, are metrics on Y.
It is simple to show the following properties.
Lemma 2.1 (see, e.g., [7])
 (1)
$(X,d)$ is complete if and only if $(Y,\eta )$ (and $(Y,\delta )$) is complete;
 (2)
F has the mixed monotone property if and only if ${T}_{F}$ is monotone nondecreasing with respect to ⪯;
 (3)
$(x,y,z)\in X\times X\times X$ is a tripled fixed point of F if and only if $(x,y,z)$ is a fixed point of ${T}_{F}$.
 (4)
$(x,y,z)\in X\times X\times X$ is a tripled coincidence point of F and G if and only if $(x,y,z)$ is a coincidence point of ${T}_{F}$ and ${T}_{G}$.
 (5)
$(x,y,z)\in X\times X\times X$ is a tripled common fixed point of F and G if and only if $(x,y,z)$ is a common fixed point of ${T}_{F}$ and ${T}_{G}$.
As a consequence of the previous facts, next we show that Theorem 2.2 is not a true extension: indeed, it can be seen as a simple corollary of Theorem 2.1.
Theorem 2.3 Theorem 2.2 follows from Theorem 2.1.
for all $(x,y,z),(u,v,w)\in Y$ such that $({T}_{G}(x,y,z),{T}_{G}(u,v,w))\in M$ (notice that $M\subseteq {X}^{6}={Y}^{2}$). By Lemma 2.1, all conditions of Theorem 2.1 are satisfied. □
3 Final remarks
In this section, we underline that the common/coincidence point theorem in [6] can be concluded as a fixed point theorem. For this purpose, we first recall the following crucial lemma.
Lemma 3.1 ([30])
Let X be a nonempty set and $T:X\to X$ be a function. Then there exists a subset $E\subseteq X$ such that $T(E)=T(X)$ and $T:E\to X$ is onetoone.
 (a)
T is continuous, or
 (b)
$(X,d,M)$ is regular.
If there exists a point ${x}_{0}\in X$ such that $({x}_{0},T{x}_{0})\in M$, then T and g have, at least, a fixed point.
We skip the proof of this theorem since it can be considered as a special case of Theorem 2.1. Indeed, if we take g as the identity map on X, we conclude the result. On the other hand, by the following lemma, we shall show that Theorem 2.1 can be derived from Theorem 3.1.
Theorem 3.2 Theorem 2.1 is a consequence of Theorem 3.1.
for all $gx,gy\in g(E)$. Since $g(E)=g(X)$ is complete, by using Theorem 3.1, there exists ${x}_{0}\in X$ such that $h(g{x}_{0})=g{x}_{0}$. Hence, T and g have a point of coincidence. It is clear that T and g have a unique common fixed point whenever T and g are weakly compatible. □
From Theorem 2.3 and Theorem 3.2 we conclude the following result.
Theorem 3.3 Theorem 2.2 is a consequence of Theorem 3.1.
Declarations
Acknowledgements
The second author has been partially supported by Junta de Andalucía by project FQM268 of the Andalusian CICYE.
Authors’ Affiliations
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