Coincidence point theorems in quasimetric spaces without assuming the mixed monotone property and consequences in Gmetric spaces
 AntonioFrancisco RoldánLópezdeHierro^{1},
 Erdal Karapınar^{2, 3}Email author and
 Manuel de la Sen^{4}
https://doi.org/10.1186/168718122014184
© RoldánLópezdeHierro et al.; licensee Springer. 2014
Received: 31 May 2014
Accepted: 14 August 2014
Published: 2 September 2014
Abstract
In this paper, we present some coincidence point theorems in the setting of quasimetric spaces that can be applied to operators which not necessarily have the mixed monotone property. As a consequence, we particularize our results to the field of metric spaces, partially ordered metric spaces and Gmetric spaces, obtaining some very recent results. Finally, we show how to use our main theorems to obtain coupled, tripled, quadrupled and multidimensional coincidence point results.
1 Introduction
In recent times, one of the branches of fixed point theory that has attracted much attention is the field devoted to studying this kind of results in the setting of partially ordered metric spaces. After the appearance of the first works in this sense (by Ran and Reurings [1], by Nieto and RodríguezLópez [2], by GnanaBhaskar and Lakshmikantham [3], and by Lakshmikantham and Ćirić [4], to cite some of them), the literature on this topic has expanded significantly. In [3], the authors introduced the notion of mixed monotone property, which has been one of the most usual hypotheses in this kind of results. However, some theorems avoiding these conditions have appeared very recently (see, for instance, [5]). One of the results on this line of study was given by Charoensawan and Thangthong in [6]. To understand their statement, the following notions were considered.
 1.
$(x,u,y,v,z,w)\in M\iff (w,z,v,y,u,x)\in M$;
 2.
$(gx,gu,gy,gv,gz,gw)\in M\Rightarrow (F(x,u),F(u,x),F(y,v),F(v,y),F(z,w),F(w,z))\in M$.
 1.
${\phi}^{1}(\{0\})=\{0\}$,
 2.
$\phi (t)<t$ for all $t>0$,
 3.
${lim}_{s\to {t}^{+}}\phi (s)<t$ for all $t>0$.
Using the previous preliminaries, they proved the following result in the context of Gmetric spaces, which is recalled in Section 2.1.
Theorem 1.1 (Charoensawan and Thangthong [6], Theorem 3.1)
for all $(gx,gu,gy,gv,gz,gw)\in M$.
and M is an $({F}^{\ast},g)$invariant set which satisfies the transitive property, then there exist $x,y\in X$ such that $gx=F(x,y)$ and $gy=F(y,x)$.
First of all, notice that the partial order ≼ in the hypothesis has no sense in the statement of Theorem 1.1. This is only a mistake that proves the special importance of partial orders in this class of results.
In this paper, we show that Theorem 1.1 can be easily deduced from a unidimensional version of the same result. In fact, we prove that the middle variables of $M\subseteq {X}^{6}$ are unnecessary. But the main aim of this work is to obtain some coincidence point theorems in the context of quasimetric spaces that can be applied in several frameworks, including metric spaces and Gmetric spaces. The hypotheses of our main results are very general, and they can be particularized in a variety of different contexts, unidimensional or multidimensional ones, even if the involved mappings do not have the mixed monotone property. Our results also extend and unify some recent theorems that can be found in [7]. As a consequence, we prove that many results in this field of study can be easily derived from our statements.
2 Preliminaries
For the sake of completeness, we collect in this section some basic definitions and wellknown results in this field. Firstly, let ℕ and ℝ denote the sets of all positive integers and all real numbers, respectively. Furthermore, we let ${\mathbb{N}}_{0}=\mathbb{N}\cup \{0\}$. If $A\subseteq \mathbb{R}$ is a nonempty subset of ℝ, the Euclidean metric on A is $d(x,y)=xy$ for all $x,y\in A$. In the sequel, let X be a nonempty set. Given a natural number n, we use ${X}^{n}$ to denote the nth Cartesian power of X, that is, $X\times X\times \cdots \times X$ (n times).
Given a point $x\in X$, the Picard sequence of the operator T (based on x) is the sequence ${\{{T}^{n}x\}}_{n\ge 0}$, which we will denote by $\{{x}_{n}\}$.
The main aim of this manuscript is to show some sufficient conditions to ensure existence and uniqueness of the following kinds of points. A coincidence point of two mappings $T,g:X\to X$ is a point $x\in X$ such that $Tx=gx$. And a coupled coincidence point of two mappings $F:{X}^{2}\to X$ and $g:X\to X$ is a point $(x,y)\in {X}^{2}$ such that $F(x,y)=gx$ and $F(y,x)=gy$. If g is the identity mapping on X, then both kinds of points are called coupled fixed point of T and coupled fixed point of F, respectively.
In such a case, the pair $(X,d)$ is called a metric space.
We say that two mappings $T,g:X\to X$ are commuting if $gTx=Tgx$ for all $x\in X$. We say that $F:{X}^{n}\to X$ and $g:X\to X$ are commuting if $gF({x}_{1},{x}_{2},\dots ,{x}_{n})=F(g{x}_{1},g{x}_{2},\dots ,g{x}_{n})$ for all ${x}_{1},\dots ,{x}_{n}\in X$.
A binary relation on X is a nonempty subset ℛ of ${X}^{2}$. For simplicity, we will write $x\preccurlyeq y$ if $(x,y)\in \mathcal{R}$, and we will say that ≼ is the binary relation. We will write $x\prec y$ when $x\preccurlyeq y$ and $x\ne y$, and we will write $y\succcurlyeq x$ when $x\preccurlyeq y$. We will say that x and y are ≼comparable if $x\preccurlyeq y$ or $y\preccurlyeq x$.
A binary relation ≼ on X is transitive if $x\preccurlyeq z$ for all $x,y,z\in X$ such that $x\preccurlyeq y$ and $y\preccurlyeq z$. 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. 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 $y\preccurlyeq x$ implies $x=y$), then ≼ is called a partial order, and $(X,\preccurlyeq )$ is a partially ordered space.
If $(X,\preccurlyeq )$ is a preordered space and $T,g:X\to X$ are two mappings, we 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 w.r.t. ≼ (or it is ≼nondecreasing).
If $(X,d)$ is a metric space, a mapping $T:X\to X$ is continuous if $\{T{x}_{n}\}\to Tz$ for all sequences $\{{x}_{n}\}\subseteq X$ such that $\{{x}_{n}\}\to z\in X$. If ≼ is a binary relation on X, we say that T is $(g,\preccurlyeq )$ nondecreasingcontinuous if $\{T{x}_{n}\}\to Tz$ for all sequences $\{{x}_{n}\}\subseteq X$ such that $\{{x}_{n}\}\to z\in X$ verifying that $g{x}_{n}\preccurlyeq g{x}_{n+1}$ for all $n\in \mathbb{N}$. If g is the identity mapping on X, we say that T is ≼nondecreasingcontinuous.
2.1 Gmetric spaces
The notion of Gmetric space is defined as follows.
Definition 2.1 (Mustafa and Sims [8])
Let X be a nonempty set, and let $G:X\times X\times X\to {\mathbb{R}}^{+}$ be a function satisfying the following properties:
(G_{1}) $G(x,y,z)=0$ if $x=y=z$;
(G_{2}) $0<G(x,x,y)$ for all $x,y\in X$ with $x\ne y$;
(G_{3}) $G(x,x,y)\le G(x,y,z)$ for all $x,y,z\in X$ with $y\ne z$;
(G_{4}) $G(x,y,z)=G(x,z,y)=G(y,z,x)=\cdots $ (symmetry in all three variables);
(G_{5}) $G(x,y,z)\le G(x,a,a)+G(a,y,z)$ (rectangle inequality) for all $x,y,z,a\in X$.
Then the function G is called a generalized metric, or, more specifically, a Gmetric on X, and the pair $(X,G)$ is called a Gmetric space.
For a better understanding of the subject, we give the following examples of Gmetrics.
for all $x,y,z\in X$, is a Gmetric on X.
Example 2.2 (see, e.g., [8])
for all $x,y,z\in X$, is a Gmetric on X.
In their initial paper, Mustafa and Sims [8] also defined the basic topological concepts in Gmetric spaces as follows.
Definition 2.2 (Mustafa and Sims [8])
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 we write $\{{x}_{n}\}\to x$ or ${lim}_{n\to \mathrm{\infty}}{x}_{n}=x$.
It is clear that the limit of a convergent sequence is unique.
Proposition 2.1 (Mustafa and Sims [8])
 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 $n\to \mathrm{\infty}$.
Definition 2.3 (Mustafa and Sims [8])
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 $m,n,l\ge N$, that is, $G({x}_{n},{x}_{m},{x}_{l})\to 0$ as $n,m,l\to \mathrm{\infty}$.
Proposition 2.2 (Mustafa and Sims [8])
 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 $m,n\ge N$.
Definition 2.4 (Mustafa and Sims [8])
A Gmetric space $(X,G)$ is called Gcomplete if every GCauchy sequence is Gconvergent in $(X,G)$.
Definition 2.5 Let $(X,G)$ be a Gmetric space. A mapping $T:X\to X$ is said to be Gcontinuous if $\{T{x}_{n}\}$ Gconverges to Tx for any Gconvergent sequence $\{{x}_{n}\}$ to $x\in X$. In general, given $m\in \mathbb{N}$, a mapping $F:{X}^{m}\to X$ is said to be Gcontinuous if $\{F({x}_{n}^{1},{x}_{n}^{2},\dots ,{x}_{n}^{m})\}$ Gconverges to $F({x}^{1},{x}^{2},\dots ,{x}^{m})$ for any Gconvergent sequences $\{{x}_{n}^{1}\},\{{x}_{n}^{2}\},\dots ,\{{x}_{n}^{m}\}\subseteq X$ such that $\{{x}_{n}^{i}\}\to {x}^{i}\in X$ for all $i\in \{1,2,\dots ,m\}$.
The following lemma shows a simple way to consider some Gmetrics on ${X}^{2}$ from a Gmetric on X.
Lemma 2.1 (Agarwal et al. [9])
 (a)
G is a ${G}^{\ast}$metric on X.
 (b)
${G}_{s}^{2}$ is a ${G}^{\ast}$metric on ${X}^{2}$.
 (c)
${G}_{m}^{2}$ is a ${G}^{\ast}$metric on ${X}^{2}$.
 1.
Every sequence $\{({x}_{n},{y}_{n})\}\subseteq {X}^{2}$ verifies: $\{({x}_{n},{y}_{n})\}\stackrel{{G}_{s}^{2}}{\u27f6}(x,y)\u27fa\{({x}_{n},{y}_{n})\}\stackrel{{G}_{m}^{2}}{\u27f6}(x,y)\u27fa[\{{x}_{n}\}\stackrel{G}{\u27f6}x\mathit{\text{and}}\{{y}_{n}\}\stackrel{G}{\u27f6}y]$.
 2.
$\{({x}_{n},{y}_{n})\}\subseteq {X}^{2}$ is ${G}_{s}^{2}$Cauchy ⟺ $\{({x}_{n},{y}_{n})\}$ is ${G}_{m}^{2}$Cauchy ⟺ $[\{{x}_{n}\}\mathit{\text{and}}\{{y}_{n}\}\mathit{\text{are}}G\mathit{\text{Cauchy}}]$.
 3.
$(X,G)$ is Gcomplete ⟺ $({X}^{2},{G}_{s}^{2})$ is Gcomplete ⟺ $({X}^{2},{G}_{m}^{2})$ is Gcomplete.
2.2 Quasimetric spaces
Definition 2.6 A mapping $q:X\times X\to [0,\mathrm{\infty})$ is a quasimetric on X if it satisfies (M_{1}), (M_{2}) and (M_{4}), that is, if it verifies, for all $x,y,z\in X$:
(q_{1}) $q(x,y)=0$ if and only if $x=y$,
(q_{2}) $q(x,y)\le q(x,z)+q(z,y)$.
In such a case, the pair $(X,q)$ is called a quasimetric space.
Remark 2.1 Any metric space is a quasimetric space, but the converse is not true in general.
Now, we recollect some basic topological notions and related results about quasimetric spaces (see also, e.g., [10–13]).
Definition 2.7 Let $(X,q)$ be a quasimetric space, $\{{x}_{n}\}$ be a sequence in X, and $x\in X$. We will say that:

$\{{x}_{n}\}$ converges to x (and we will denote it by $\{{x}_{n}\}\stackrel{q}{\u27f6}x$ or by $\{{x}_{n}\}\to x$) if ${lim}_{n\to \mathrm{\infty}}q({x}_{n},x)={lim}_{n\to \mathrm{\infty}}q(x,{x}_{n})=0$;

$\{{x}_{n}\}$ is a Cauchy sequence if for all $\epsilon >0$, there exists ${n}_{0}\in \mathbb{N}$ such that $q({x}_{n},{x}_{m})<\epsilon $ for all $n,m\ge {n}_{0}$.
The quasimetric space $(X,q)$ is said to be complete if every Cauchy sequence is convergent on $(X,q)$.
As q is not necessarily symmetric, some authors distinguished between left/right Cauchy/convergent sequences and completeness.
Definition 2.8 (Jleli and Samet [14])
Let $(X,q)$ be a quasimetric space, $\{{x}_{n}\}$ be a sequence in X, and $x\in X$. We say that:

$\{{x}_{n}\}$ rightconverges to x if ${lim}_{n\to \mathrm{\infty}}q({x}_{n},x)=0$;

$\{{x}_{n}\}$ leftconverges to x if ${lim}_{n\to \mathrm{\infty}}q(x,{x}_{n})=0$;

$\{{x}_{n}\}$ is a rightCauchy sequence if for all $\epsilon >0$ there exists ${n}_{0}\in \mathbb{N}$ such that $q({x}_{n},{x}_{m})<\epsilon $ for all $m>n\ge {n}_{0}$;

$\{{x}_{n}\}$ is a leftCauchy sequence if for all $\epsilon >0$ there exists ${n}_{0}\in \mathbb{N}$ such that $q({x}_{m},{x}_{n})<\epsilon $ for all $m>n\ge {n}_{0}$;

$(X,q)$ is rightcomplete if every rightCauchy sequence is rightconvergent;

$(X,q)$ is leftcomplete if every leftCauchy sequence is leftconvergent;
Remark 2.2 (see, e.g., [14])
A sequence $\{{x}_{n}\}$ in a quasimetric space is Cauchy if and only if it is leftCauchy and rightCauchy.
 1.
The limit of a sequence in a quasimetric space, if it exists, is unique. However, this is false if we consider rightlimits or leftlimits.
 2.If $\{{x}_{n}\}\to x$ and $\{{y}_{n}\}\to y$ in a quasimetric space, then $\{q({x}_{n},{y}_{n})\}\to q(x,y)$, that is, q is continuous in both arguments. It follows from$q(x,y)q(x,{x}_{n})q({y}_{n},y)\le q({x}_{n},{y}_{n})\le q({x}_{n},x)+q(x,y)+q(y,{y}_{n})$
 3.If $\{{x}_{n}\}\to x$, $\{q({x}_{n},{y}_{n})\}\to 0$ and $\{q({y}_{n},{x}_{n})\}\to 0$, then $\{{y}_{n}\}\to x$. It follows from$q({y}_{n},x)\le q({y}_{n},{x}_{n})+q({x}_{n},x)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}q(x,{y}_{n})\le q(x,{x}_{n})+q({x}_{n},{y}_{n}).$
 4.
If a sequence $\{{x}_{n}\}$ has a rightlimit x and a leftlimit y, then $x=y$, $\{{x}_{n}\}$ converges and it has an only limit (from the right and from the left). However, it is possible that a sequence has two different rightlimits when it has no leftlimit.
Then $(X,q)$ is a quasimetric space. Notice that $\{q(1/n,0)\}\to 0$ but $\{q(0,1/n)\}\to 1$. Therefore, $\{1/n\}$ rightconverges to 0 but it does not converge from the left.
The following result shows a simple way to consider quasimetrics from Gmetrics.
Lemma 2.2 (Agarwal et al. [9])
 1.${q}_{G}$ and ${q}_{G}^{\prime}$ are quasimetrics on X. Moreover,${q}_{G}^{\prime}(x,y)\le 2{q}_{G}(x,y)\le 4{q}_{G}^{\prime}(x,y)\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}x,y\in X.$(2)
 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 $\{{x}_{n}\}\stackrel{G}{\u27f6}x\u27fa\{{x}_{n}\}\stackrel{{q}_{G}}{\u27f6}x\u27fa\{{x}_{n}\}\stackrel{{q}_{G}^{\prime}}{\u27f6}x$.
 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.
2.3 Control functions
Functions in Φ (see Definition 1.3) verify the following properties.
 1.
$\phi (t)\le t$ for all $t\ge 0$.
 2.
If $\{{t}_{n}\}\subset [0,\mathrm{\infty})$ is a sequence such that ${t}_{n+1}\le \phi ({t}_{n})$ for all n, then $\{{t}_{n}\}\to 0$.
 3.
If $\{{t}_{n}\},\{{s}_{n}\}\subset [0,\mathrm{\infty})$ are two sequences such that $\{{t}_{n}\}\to 0$ and ${s}_{n}\le \phi ({t}_{n})$ for all n, then $\{{s}_{n}\}\to 0$.
 (3)
It follows from item 2 taking into account that $0\le {s}_{n}\le \phi ({t}_{n})\le {t}_{n}$ for all n. □
Functions in Ψ are more general than those in Φ. The following properties are very useful.
 1.
If ${t}_{m+1}\le \phi ({t}_{m})$ and ${t}_{m}\ne 0$ for all m, then $\{{t}_{m}\}\to 0$.
 2.
Let $\{{t}_{n}\},\{{s}_{n}\}\subset [0,\mathrm{\infty})$ be two sequences such that $\{{t}_{n}\}\to 0$ and ${s}_{n}\le \phi ({t}_{n})$ for all n. Also assume that if ${t}_{n}=0$, then ${s}_{n}=0$. Hence $\{{s}_{n}\}\to 0$.
Proof (1) It is the same proof of item 2 of Lemma 2.3.
(2) It follows from the fact that ${s}_{n}\le \phi ({t}_{n})<{t}_{n}$ if ${t}_{n}>0$, and ${s}_{n}=0$ if ${t}_{n}=0$. In any case, ${s}_{n}\le {t}_{n}$ for all n. □
Remark 2.4 The difference between items 2 and 3 of Lemma 2.3 and items 1 and 2 of Lemma 2.4 is important. If we assume that $\phi \in \mathrm{\Psi}$ and ${t}_{m+1}\le \phi ({t}_{m})$ for all m, then it is impossible to deduce that $\{{t}_{m}\}\to 0$ or $\{\phi ({t}_{m})\}\to 0$ in item 1 of the previous result. For instance, define $\phi (t)=t/2$ if $t>0$, and $\phi (0)=1/2$. Then $\phi \in \mathrm{\Psi}$ and the sequence $\{{t}_{m}\}=\{0,1/2,0,1/2,0,1/2,\dots \}$ verifies ${t}_{m+1}\le \phi ({t}_{m})$ for all m but it does not converge.
3 Coincidence point theorems on quasimetric spaces without the mixed monotone property
In this section, we present some coincidence point theorems in the framework of quasimetric spaces under very general conditions which can be extended to the coupled case and can be applied to mappings that have not necessarily the mixed monotone property.
3.1 Basic notions depending on a subset ℳ
Definition 3.1 (See Kutbi et al. [5])
We say that a nonempty subset ℳ of ${X}^{2}$ is:

reflexive if $(x,x)\in \mathcal{M}$ for all $x\in X$;

antisymmetric if $x=y$ for all $x,y\in X$ such that $(x,y),(y,x)\in \mathcal{M}$;

transitive if $(x,z)\in \mathcal{M}$ for all $x,y,z\in X$ such that $(x,y),(y,z)\in \mathcal{M}$.
Given two mappings $T,g:X\to X$, we say that ℳ is:

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

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

$(T,g)$ 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 $gx=gy$ and $(gx,gy)\in \mathcal{M}$.
Clearly, every transitive subset is also gtransitive. Moreover, ℳ is gclosed if and only if it is $(g,{I}_{X})$closed, where ${I}_{X}$ denotes the identity mapping on X. The following lemma shows a simple way to consider gtransitive, $(T,g)$closed sets.
 1.
If ≼ is a preorder on X, then ${\mathcal{M}}_{\preccurlyeq}$ is reflexive, transitive and gtransitive.
 2.
If ≼ is a partial order on X, then ${\mathcal{M}}_{\preccurlyeq}$ is reflexive, transitive, antisymmetric and gtransitive.
 3.
${\mathcal{M}}_{\preccurlyeq}$ is gclosed if and only if g is ≼nondecreasing.
 4.
${\mathcal{M}}_{\preccurlyeq}$ is $(T,g)$closed if and only if T is $(g,\preccurlyeq )$nondecreasing.
 5.
If ≼ is a partial order on X and ${\mathcal{M}}_{\preccurlyeq}$ is $(T,g)$closed, then ${\mathcal{M}}_{\preccurlyeq}$ is $(T,g)$compatible.
□
It is convenient to highlight that the notion of gtransitive, $(T,g)$closed, nonempty subset $\mathcal{M}\subseteq {X}^{2}$ is more general than the idea of nondecreasing mapping on a preordered space (following the previous lemma), as we show in the following example.
Then ℳ does not come from a preorder (or a partial order) on X because it is not reflexive ($(0,0)\notin \mathcal{M}$), nor transitive ($(0,1),(1,2)\in \mathcal{M}$ but $(0,2)\notin \mathcal{M}$) nor antisymmetric ($(0,1),(0,1)\in \mathcal{M}$ but $0\ne 1$). However, ℳ is gtransitive and $(T,g)$closed.
In the following definitions, we will use sequences $\{{x}_{n}\}\subseteq X$ such that $({x}_{n},{x}_{m})\in \mathcal{M}$ for all $n,m\in \mathbb{N}$ with $n<m$. In this sense, the following notions must be called ‘rightnotions’ because the same concepts could also be introduced involving sequences $\{{x}_{n}\}\subseteq X$ such that $({x}_{n},{x}_{m})\in \mathcal{M}$ for all $n,m\in \mathbb{N}$ with $n>m$ (in this case, they would be ‘leftnotions’). Then we could talk about $(T,g,\mathcal{M})$ rightPicard sequences, ℳrightcontinuity, $(O,\mathcal{M})$ rightcompatibility and rightregularity. However, we advice the reader that, in order not to complicate the notation, we will omit the term ‘right’.
Definition 3.2 Let $(X,q)$ be a quasimetric space, let ℳ be a nonempty subset of ${X}^{2}$, and let $T:X\to X$ be a mapping. We say that T is ℳcontinuous if $\{T{x}_{n}\}\stackrel{q}{\u27f6}Tu$ for all sequences $\{{x}_{n}\}\subseteq X$ such that $\{{x}_{n}\}\stackrel{q}{\u27f6}u\in X$ and $({x}_{n},{x}_{m})\in \mathcal{M}$ for all $n,m\in \mathbb{N}$ with $n<m$.
Remark 3.1 Every continuous mapping from a quasimetric space into itself is also ℳcontinuous, whatever the subset ℳ.
Definition 3.3 Let $T,g:X\to X$ be two mappings, let ${\{{x}_{n}\}}_{n\ge 0}\subseteq X$ be a sequence, and let ℳ be a nonempty subset of ${X}^{2}$. We say that $\{{x}_{n}\}$ is a:

$(T,g)$ Picard sequence if$g{x}_{n+1}=T{x}_{n}\phantom{\rule{1em}{0ex}}\text{for all}n\ge 0;$(3)

$(T,g,\mathcal{M})$ Picard sequence if it is a $(T,g)$Picard sequence and$(g{x}_{n},g{x}_{m})\in \mathcal{M}\phantom{\rule{1em}{0ex}}\text{for all}n,m\in {\mathbb{N}}_{0}\text{such that}nm.$(4)
 1.
If $T(X)\subseteq g(X)$, then there exists a $(T,g)$Picard sequence based on each ${x}_{0}\in X$.
 2.
If ℳ is a gtransitive, $(T,g)$closed, nonempty subset of ${X}^{2}$, then every $(T,g)$Picard sequence ${\{{x}_{n}\}}_{n\ge 0}$ such that $(g{x}_{0},T{x}_{0})\in \mathcal{M}$ is a $(T,g,\mathcal{M})$Picard sequence.
Proof (1) Let ${x}_{0}\in X$ be arbitrary. Since $T{x}_{0}\in T(X)\subseteq g(X)$, then there exists ${x}_{1}\in X$ such that $g{x}_{1}=T{x}_{0}$. Similarly, since $T{x}_{1}\in T(X)\subseteq g(X)$, then there exists ${x}_{2}\in X$ such that $g{x}_{2}=T{x}_{1}$. Repeating this argument by induction, we may consider a $(T,g)$Picard sequence $\{{x}_{n}\}$ based on ${x}_{0}$.
for all $n,m\in \mathbb{N}$ such that $n<m$. □
The following definition extends some ideas that can be found in [17–19].
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 3.5 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 that $({x}_{n},u)\in \mathcal{M}$ 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 $n<m$.
3.2 Coincidence point theorems using $(g,\mathcal{M},\mathrm{\Phi})$contractions of the first kind
Next, we present the kind of contractions we will use.
for all $x,y\in X$ such that $(gx,gy)\in \mathcal{M}$. If $\phi ,{\phi}^{\prime}\in \mathrm{\Psi}$, we say that T is a $(g,\mathcal{M},\mathrm{\Psi})$ contraction of the first kind.
Remark 3.2 It is not necessary that functions in Φ and in Ψ verify all their properties in $[0,\mathrm{\infty})$. In fact, as we shall only use inequalities (5)(6), the properties of functions in Φ and in Ψ must only be verified on the image of the quasimetric q, that is, on $q(X\times X)\subseteq [0,\mathrm{\infty})$, which does not necessarily coincide with $[0,\mathrm{\infty})$ (for instance, if X is qbounded).
Remark 3.3 One of the best advantages of using a subset $\mathcal{M}\subseteq {X}^{2}$ is that a unique condition covers two particularly interesting cases:

$\mathcal{M}={X}^{2}$, in which contractivity conditions (5)(6) hold for all $x,y\in X$; and

$\mathcal{M}={\mathcal{M}}_{\preccurlyeq}$, where ≼ is a preorder or a partial order on X, in which (5)(6) must be assumed for all $x,y\in X$ such that $gx\preccurlyeq gy$.
Both possibilities were independently studied in the past, but this new vision unifies them in an only assumption.
The following one is a first property of this kind of mappings.
 1.
T is a $(g,\mathcal{M},\mathrm{\Phi})$contraction of the first kind.
 2.
T is a $(g,\mathcal{M},\mathrm{\Psi})$contraction of the first kind and ℳ is $(T,g)$compatible.
Then T is ℳcontinuous at every point in which g is ℳcontinuous.
If $\phi ,{\phi}^{\prime}\in \mathrm{\Phi}$, then item 3 of Lemma 2.3 guarantees that $\{q(T{x}_{n},Tz)\}\to 0$ and $\{q(Tz,T{x}_{n})\}\to 0$, so $\{T{x}_{n}\}$ qconverges to Tz. If $\phi ,{\phi}^{\prime}\in \mathrm{\Psi}$ and we additionally assume that ℳ is $(T,g)$compatible, we can use item 2 of Lemma 2.4 applied to the sequences $\{{t}_{n}=q(T{x}_{n},Tz)\}$ and $\{{s}_{n}=q(g{x}_{n},gz)\}$ in order to deduce that $\{q(T{x}_{n},Tz)\}\to 0$ (notice that if ${s}_{n}=0$, then ${t}_{n}=0$) and similarly $\{q(Tz,T{x}_{n})\}\to 0$. □
The first main result of this work is the following one.
 (A)
There exists a $(T,g,\mathcal{M})$Picard sequence on X.
 (B)
T is a $(g,\mathcal{M},\mathrm{\Phi})$contraction of the first 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 X (or $g(X)$) is $(q,\mathcal{M})$regular;
 (d)
$(X,q)$ is complete, $g(X)$ is closed and X (or $g(X)$) is $(q,\mathcal{M})$regular;
 (e)
$(X,q)$ is complete, 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.
Notice that, by Lemma 3.2, the previous result also holds if we replace condition (A) by one of the following stronger hypotheses:
(${\mathrm{A}}^{\prime}$) $T(X)\subseteq g(X)$ and ℳ is gtransitive and $(T,g)$closed.
(${\mathrm{A}}^{\u2033}$) ℳ is gtransitive and $(T,g)$closed, and there exists a $(T,g)$Picard sequence ${\{{x}_{n}\}}_{n\ge 0}$ such that $(g{x}_{0},T{x}_{0})\in \mathcal{M}$.
And by Remark 3.1, the ℳcontinuity of the mappings can be replaced by continuity.
which is a contradiction. This contradiction ensures us that $\{g{x}_{n}\}$ is rightCauchy in $(X,q)$, and Step 2 holds.
Similarly, using the contractivity condition (6), it can be proved that $\{g{x}_{n}\}$ is leftCauchy in $(X,q)$, so we conclude that $\{g{x}_{n}\}$ is a Cauchy sequence in $(X,q)$. Now, we prove that T and g have a coincidence point distinguishing between cases (a)(e).
(the other case is similar). Hence, u is a coincidence point of T and g.
Case (b): X (or $g(X)$ or $T(X)$) is qcomplete and T and g are ℳcontinuous and commuting. It is obvious because (b) implies (a).
Case (c): $(g(X),q)$ is complete and X (or $g(X)$) is $(q,\mathcal{M})$ regular. As $\{g{x}_{m}\}$ is a Cauchy sequence in the complete space $(g(X),q)$, there is $u\in g(X)$ such that $\{g{x}_{m}\}\to u$. Let $v\in X$ be any point such that $u=gv$. In this case, $\{g{x}_{m}\}\to gv$. We are also going to show that $\{g{x}_{m}\}\to Tv$, so we will conclude that $gv=Tv$ (and v is a coincidence point of T and g).
By item 3 of Lemma 2.3, $\{g{x}_{n}\}$ qconverges to Tv.
Case (d): $(X,q)$ is complete, $g(X)$ is closed and X (or $g(X)$) is $(q,\mathcal{M})$ regular. It follows from the fact that a closed subset of a complete quasimetric space is also complete. Then $(g(X),q)$ is complete and case (c) is applicable.
By item 3 of Remark 2.3, as $\{gg{x}_{m}\}\to gu$, the previous properties imply that $\{Tg{x}_{m}\}\to gu$. We are going to show that $\{Tg{x}_{m}\}\to Tu$ and this finishes the proof.
As $\{gg{x}_{n}\}\to gu$, then $\{Tg{x}_{n}\}\to Tu$. □
However, q is not a metric because $q(1,2)\ne q(2,1)$.
 1.
The sequence $\{{x}_{n}\}$, given by ${x}_{n}={\lambda}^{n}$ for all $n\in {\mathbb{N}}_{0}$, is a $(T,g,\mathcal{M})$Picard sequence.
 2.
The function $g:\mathbb{R}\to \mathbb{R}$ is bijective and nondecreasing.
 3.
The range of g, which is $g(X)=\mathbb{R}$, is closed and complete in $(\mathbb{R},q)$.
 4.
We claim that T is a $(g,\mathcal{M},\mathrm{\Phi})$contraction of the first kind. To prove it, let $x,y\in X$ be such that $(gx,gy)\in \mathcal{M}$. If $Tx=Ty$, then (5)(6) are obvious. Next, assume that $Tx\ne Ty$. In particular, $x\ne y$. Hence, $gx\ne gy$ because g is bijective. Therefore, the condition $(gx,gy)\in \mathcal{M}$ leads to two cases.

If $0\le gy<gx\le 1$, then $0\le y<x\le 1$. Therefore$\begin{array}{r}q(Tx,Ty)=q(\lambda x,\lambda y)=2\lambda (yx)={\phi}_{\lambda}(2(yx))={\phi}_{\lambda}(q(x,y));\\ q(Ty,Tx)=q(\lambda x,\lambda y)=\lambda (yx)={\phi}_{\lambda}(yx)={\phi}_{\lambda}(q(y,x)).\end{array}$

If $\{gx,gy\}=\{9,16\}$, then $\{x,y\}=\{3,4\}$. In such a case,$\begin{array}{r}q(T3,T4)=q(3,4)=1<7\lambda =\lambda q(9,16)={\phi}_{\lambda}(q(g3,g4));\\ q(T4,T3)=q(4,3)=2<14\lambda =\lambda q(16,9)={\phi}_{\lambda}(q(g4,g3)).\end{array}$
 5.
Let $\{{x}_{n}\}\subset \mathbb{R}$ be a sequence such that $({x}_{n},{x}_{n+1})\in \mathcal{M}$ for all $n\in {\mathbb{N}}_{0}$. Then one, and only one, of the following cases holds.
(5.a) There exists ${n}_{0}\in \mathbb{N}$ such that ${x}_{{n}_{0}}\in [0,1]$. In this case, ${x}_{n}\in [0,1]$ and ${x}_{n+1}\le {x}_{n}$ for all $n\in {\mathbb{N}}_{0}$.
To prove it, notice that $({x}_{{n}_{0}},{x}_{{n}_{0}+1})\in \mathcal{M}$ is only possible when ${x}_{{n}_{0}}={x}_{{n}_{0}+1}$ or $0\le {x}_{{n}_{0}+1}<{x}_{{n}_{0}}\le 1$. In any case, ${x}_{{n}_{0}+1}\in [0,1]$. Repeating this argument, ${x}_{n}\in [0,1]$ for all $n\ge {n}_{0}$. But if ${n}_{0}1\in \mathbb{N}$, the condition $({x}_{{n}_{0}1},{x}_{{n}_{0}})\in \mathcal{M}$ also leads to ${x}_{{n}_{0}1}\in [0,1]$. And we can again repeat the argument.
(5.b) There exists ${n}_{0}\in \mathbb{N}$ such that ${x}_{{n}_{0}}\in \{9,16\}$. In this case, ${x}_{n}\in \{9,16\}$ for all $n\in {\mathbb{N}}_{0}$.
(5.c) There exists $z\in \mathbb{R}\mathrm{\u2572}([0,1]\cup \{9,16\})$ such that ${x}_{n}=z$ for all $\in {\mathbb{N}}_{0}$. In this case, $\{{x}_{n}\}$ is a constant sequence.
 6.
The range $g(X)=\mathbb{R}$ is $(q,\mathcal{M})$regular. To prove it, let $u\in \mathbb{R}$ and let $\{{x}_{n}\}\subset \mathbb{R}$ be a sequence such that$\{{x}_{n}\}\stackrel{q}{\to}u$ and $({x}_{n},{x}_{n+1})\in \mathcal{M}$ for all $n\in {\mathbb{N}}_{0}$. In particular, $\{{x}_{n}\}\to u$ using the Euclidean metric. We can distinguish the previous three cases.
(6.a) Suppose that ${x}_{n}\in [0,1]$ and ${x}_{n+1}\le {x}_{n}$ for all $n\in {\mathbb{N}}_{0}$. Therefore, $u\in [0,1]$ and $u\le {x}_{n+1}\le {x}_{n}$ for all $n\in {\mathbb{N}}_{0}$, so $({x}_{n},u)\in \mathcal{M}$ for all $n\in {\mathbb{N}}_{0}$.
(6.b) Suppose that ${x}_{n}\in \{9,16\}$ for all $n\in {\mathbb{N}}_{0}$. Then $u\in \{9,16\}$ and, therefore, $({x}_{n},u)\in \mathcal{M}$ for all $n\in {\mathbb{N}}_{0}$.
(6.c) Suppose that ${x}_{n}=z\in \mathbb{R}\mathrm{\u2572}([0,1]\cup \{9,16\})$ for all $n\in {\mathbb{N}}_{0}$. Therefore $u=z$ and $({x}_{n},u)\in \mathcal{M}$ for all $n\in {\mathbb{N}}_{0}$.
The previous properties show that case (c) of Theorem 3.1 is applicable, so T and g have, at least, a coincidence point, which is $x=0$.
We extend the previous theorem to the case in which $\phi \in \mathrm{\Psi}$.
Theorem 3.2 If we additionally assume that ℳ is $(T,g)$compatible, then Theorem 3.1 also holds even if T is a $(g,\mathcal{M},\mathrm{\Psi})$contraction of the first kind.
By item 2 of Lemma 2.4 we conclude that $\{q(g{x}_{n+1},Tv)\}\to 0$. In the same way, $\{q(Tv,g{x}_{n+1})\}\to 0$, so $\{g{x}_{n}\}$ qconverges to Tv.
The same argument is valid when applied to inequalities (15)(16). □
3.3 Coincidence point theorems using $(g,\mathcal{M},\mathrm{\Phi})$contractions of the second kind
Many results on fixed point theory in the setting of Gmetrics can be similarly proved using the quasimetrics ${q}_{G}$ and ${q}_{G}^{\prime}$ associated to G as in Lemma 2.2 (see, for instance, Agarwal et al. [9]). These families of quasimetrics verify additional properties that are not true for an arbitrary quasimetric. Using these properties, it is possible to relax some conditions on the kind of considered contractions, obtaining similar results. This is the case of the following kind of mappings.
for all $x,y\in X$ such that $(gx,gy)\in \mathcal{M}$. If $\phi \in \mathrm{\Psi}$, we say that T is a $(g,\mathcal{M},\mathrm{\Psi})$ contraction of the second kind.
Notice that condition (17) 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 3.8 We say that a quasimetric space $(X,q)$ is:

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 $(X,q)$.
It is convenient not to confuse the previous notions with the concept of left/right complete quasimetric space given in Definition 2.8. Lemma 2.2 guarantees that there exists a wide family of quasimetrics that verify all the previous properties.
Corollary 3.1 Every quasimetric ${q}_{G}$ and ${q}_{G}^{\prime}$ associated to a Gmetric G on X is right and leftCauchy and right and leftconvergent.
Next we prove a similar result to Theorem 3.1. In this case, the contractivity condition is weaker but we suppose additional conditions on the ambient space.
 (A)
There exists a $(T,g,\mathcal{M})$Picard sequence on X.
 (B)
T is a $(g,\mathcal{M},\mathrm{\Phi})$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.
Notice that, by Lemma 3.2, the previous result also holds if we replace condition (A) by one of the following stronger hypotheses:
(${\mathrm{A}}^{\prime}$) $T(X)\subseteq g(X)$ and ℳ is gtransitive and $(T,g)$closed.
(${\mathrm{A}}^{\u2033}$) ℳ is gtransitive and $(T,g)$closed, and there exists a $(T,g)$Picard sequence ${\{{x}_{n}\}}_{n\ge 0}$ such that $(g{x}_{0},T{x}_{0})\in \mathcal{M}$.
And by Remark 3.1, the ℳcontinuity of the mappings can be replaced by continuity.
Proof We can follow, step by step, the lines of the proof of Theorem 3.1 to deduce, in the case $g{x}_{n}\ne g{x}_{n+1}$ for all $n\ge 0$, that $\{g{x}_{n}\}$ is rightCauchy in $(X,q)$. Using that $(X,q)$ is rightCauchy, then it is a Cauchy sequence in $(X,q)$. Now, we prove that T and g have a coincidence point distinguishing between cases (a)(e). Cases (a) and (b) have the same proof as in Theorem 3.1.
Case (c): $(g(X),q)$ is complete and rightconvergent, and X (or $g(X)$) is $(q,\mathcal{M})$ regular. As $\{g{x}_{m}\}$ is a Cauchy sequence in the complete space $(g(X),q)$, there is $u\in g(X)$ such that $\{g{x}_{m}\}\to u$. Let $v\in X$ be any point such that $u=gv$. In this case, $\{g{x}_{m}\}\to gv$. We are also going to show that $\{g{x}_{m}\}\to Tv$, so we will conclude that $gv=Tv$ (and v is a coincidence point of T and g).
By item 3 of Lemma 2.3, we have that $\{q(g{x}_{n+1},Tv)\}\to 0$, which means that $\{g{x}_{n}\}$ rightconverges to Tv. Since $(X,q)$ is rightconvergent, then $\{g{x}_{n}\}$ is a convergent sequence in $(X,q)$, and by item 4 of Remark 2.3, it converges to Tv.
Case (d): $(X,q)$ is complete and rightconvergent, $g(X)$ is closed and X (or $g(X)$) is $(q,\mathcal{M})$ regular. It follows from the fact that a closed subset of a complete quasimetric space is also complete. Then $(g(X),q)$ is complete and case (c) is applicable.
In this case, by item 3 of Lemma 2.3, we have that $\{q(Tg{x}_{n},Tu)\}\to 0$, which means that $\{Tg{x}_{n}\}$ rightconverges to Tu. Since $(X,q)$ is rightconvergent, then $\{Tg{x}_{n}\}$ is a convergent sequence in $(X,q)$, and by item 4 of Remark 2.3, it converges to Tu. □
Example 3.3 Theorem 3.3 can also be applied to mappings given in Example 3.2 because $(\mathbb{R},q)$ is rightconvergent.
Repeating the arguments of Theorem 3.2, we extend the previous theorem to the case in which $\phi \in \mathrm{\Psi}$.
Theorem 3.4 If we additionally assume that ℳ is $(T,g)$compatible, then Theorem 3.3 also holds even if T is a $(g,\mathcal{M},\mathrm{\Psi})$contraction of the second kind.
3.4 Consequences
The previous theorems admit a lot of different particular cases employing continuity, the condition $T(X)\subseteq g(X)$ and the case in which g is the identity mapping on X. We highlight the following one in which a partial order is involved. Preliminaries of the following result can be found in [20].
Corollary 3.2 (AlMezel et al. [20], Theorem 34)
 (i)
$T(X)\subseteq g(X)$;
 (ii)
T is monotone $(g,\preccurlyeq )$nondecreasing;
 (iii)
there exists ${x}_{0}\in X$ such that $g{x}_{0}\preccurlyeq T{x}_{0}$;
 (iv)there exists $\phi \in \mathrm{\Psi}$ verifying$d(Tx,Ty)\le \phi (d(gx,gy))\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}x,y\in X\mathit{\text{such that}}gx\preccurlyeq gy.$
 (a)
$(X,d)$ is complete, T and g are continuous and the pair $(T,g)$ is Ocompatible;
 (b)
$(X,d)$ is complete and T and g are continuous and commuting;
 (c)
$(g(X),d)$ is complete and $(X,d,\preccurlyeq )$ is nondecreasingregular;
 (d)
$(X,d)$ is complete, $g(X)$ is closed and $(X,d,\preccurlyeq )$ is nondecreasingregular;
 (e)
$(X,d)$ is complete, g is continuous and monotone ≼nondecreasing, the pair $(T,g)$ is Ocompatible and $(X,d,\preccurlyeq )$ is nondecreasingregular.
Then T and g have, at least, a coincidence point.
Proof It is only necessary to apply Theorem 3.2 to the subset ${\mathcal{M}}_{\preccurlyeq}=\{(x,y)\in {X}^{2}:x\preccurlyeq y\}$, taking into account the properties given in Lemma 3.1. Notice that in case (e), we use Lemma 3.3 to avoid assuming that T is continuous. □
The following result improves the last one because we do not assume that T is ℳcontinuous in hypothesis (b).
 (a)
T is ℳcontinuous, or
 (b)
ℳ is gclosed and $(X,q,\mathcal{M})$ is regular.
Then T and g have, at least, a coincidence point.
Proof We show that case (b) in Theorem 3.1 is applicable. By item 1 of Lemma 3.2, X contains a $(T,g)$Picard sequence $\{{x}_{n}\}$ based on ${x}_{0}\in X$, and by item 2 of the same lemma, $\{{x}_{n}\}$ is a $(T,g,\mathcal{M})$Picard sequence.
If T is ℳcontinuous, item (b) of Theorem 3.1 (and also Theorem 3.2 in the case of a $(g,\mathcal{M},\mathrm{\Psi})$contraction) can be used to ensure that T and g have, at least, a coincidence point. In other case, if ℳ is gclosed and $(X,q,\mathcal{M})$ is regular, then Lemma 3.3 guarantees that T is ℳcontinuous. □
Another interesting particularization is the following one.
Corollary 3.4 (Karapınar et al. [7], Theorem 33)
 (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.
As a consequence, in the following result, a partial order is not necessary.
Corollary 3.5 (Karapınar et al. [7], Corollary 35)
 (a)
T and g are continuous and commuting, or
 (b)
$(X,d,\preccurlyeq )$ is regular and gX is closed.
If there exists a point ${x}_{0}\in X$ such that $g{x}_{0}\preccurlyeq T{x}_{0}$, then T and g have, at least, a coincidence point.
4 Applications to Gmetric spaces
One of the most interesting, recent lines of research in the field of fixed point theory is devoted to Gmetric spaces. Taking into account Lemma 2.2, we can take advantage of our main results to present some new theorems in this area. The following result is an easy application to Gmetric spaces.
for all $x,y\in X$ such that $(gx,gy)\in \mathcal{M}$. If there exists ${x}_{0}\in X$ such that $(g{x}_{0},T{x}_{0})\in \mathcal{M}$, then T and g have, at least, a coincidence point.
Notice that this result is also valid if $\phi \in \mathrm{\Psi}$ and ℳ is $(T,g)$compatible.
Proof It follows from Theorem 3.3 and Corollary 3.1 using the quasimetric ${q}_{G}^{\prime}$ associated to G (as in Lemma 2.2). Notice that there exists a $(T,g,\mathcal{M})$Picard sequence on X by items 1 and 2 of Lemma 3.2. □
In order not to lose the power and usability of Theorems 3.3 and 3.4, we present the following properties comparing ${q}_{G}$ and ${q}_{G}^{\prime}$.
Definition 4.1 Let $(X,G)$ be a Gmetric space, and let $A\subseteq X$ and $\mathcal{M}\subseteq {X}^{2}$ be two nonempty subsets. We say that $(A,G,\mathcal{M})$ is regular (or A is $(G,\mathcal{M})$ regular) if we have that $({x}_{n},u)\in \mathcal{M}$ for all n provided that $\{{x}_{n}\}$ is a Gconvergent sequence on A, $u\in A$ is its Glimit and $({x}_{n},{x}_{m})\in \mathcal{M}$ for all $n<m$.
 1.
the subset A is $(G,\mathcal{M})$regular;
 2.
the subset A is $({q}_{G},\mathcal{M})$regular;
 3.
the subset A is $({q}_{G}^{\prime},\mathcal{M})$regular.
Proof It follows from the fact that $(X,G)$, $(X,{q}_{G})$ and $(X,{q}_{G}^{\prime})$ have the same convergent sequences, and they converge to the same limits. □
Similarly, the following result can be proved.
Lemma 4.2 Given a Gmetric space $(X,G)$, a nonempty subset $\mathcal{M}\subseteq {X}^{2}$ and two mappings $T,g:X\to X$, we have that the pair $(T,g)$ is $(O,\mathcal{M})$compatible (respectively, $({O}^{\prime},\mathcal{M})$compatible) in $(X,{q}_{G})$ if and only if it is $(O,\mathcal{M})$compatible (respectively, $({O}^{\prime},\mathcal{M})$compatible) in $(X,{q}_{G}^{\prime})$.
Proof It follows from the fact that $(X,{q}_{G})$ and $(X,{q}_{G}^{\prime})$ have the same convergent sequences, and they converge to the same limits. Furthermore, taking into account that ${q}_{G}\le 2{q}_{G}^{\prime}\le 4{q}_{G}$, then $\{{q}_{G}({x}_{n},{y}_{n})\}\to 0$ if and only if $\{{q}_{G}^{\prime}({x}_{n},{y}_{n})\}\to 0$. □
Definition 4.2 Let $(X,G)$ be a Gmetric space, and let ℳ be a nonempty subset of ${X}^{2}$. Two mappings $T,g:X\to X$ are said to be $(O,\mathcal{M})$ compatible if the pair $(T,g)$ is $(O,\mathcal{M})$compatible in $(X,{q}_{G})$ (or, equivalently, in $(X,{q}_{G}^{\prime})$).
Similarly, the notion of $({O}^{\prime},\mathcal{M})$compatibility in a G metric space $(X,G)$ can be defined. We present the following result, which is a complete version of our main results in the context of Gmetric spaces.
 (A)
There exists a $(T,g,\mathcal{M})$Picard sequence on X.
(${\mathrm{A}}^{\prime}$) $T(X)\subseteq g(X)$ and ℳ is gtransitive and $(T,g)$closed.
(${\mathrm{A}}^{\u2033}$) ℳ is gtransitive and $(T,g)$closed, and there exists a $(T,g)$Picard sequence ${\{{x}_{n}\}}_{n\ge 0}$ such that $(g{x}_{0},T{x}_{0})\in \mathcal{M}$.
Also assume that, at least, one of the following two conditions holds.
for all $x,y\in X$ for which $(gx,gy)\in \mathcal{M}$.
for all $x,y\in X$ for which $(gx,gy)\in \mathcal{M}$.
 (a)
X (or $g(X)$ or $T(X)$) is Gcomplete, T and g are ℳcontinuous and the pair $(T,g)$ is $({O}^{\prime},\mathcal{M})$compatible;
(a′) X (or $g(X)$ or $T(X)$) is Gcomplete, T and g are continuous and the pair $(T,g)$ is $({O}^{\prime},\mathcal{M})$compatible;
 (b)
X (or $g(X)$ or $T(X)$) is Gcomplete and T and g are ℳcontinuous and commuting;
 (c)
$(g(X),G)$ is complete and X (or $g(X)$) is $(G,\mathcal{M})$regular;
 (d)
$(X,G)$ is complete, $g(X)$ is closed and X (or $g(X)$) is $(G,\mathcal{M})$regular;
 (e)
$(X,G)$ is complete, g is ℳcontinuous, ℳ is gclosed, the pair $(T,g)$ is $(O,\mathcal{M})$compatible and X is $(G,\mathcal{M})$regular.
(e′) $(X,G)$ is complete, g is continuous, ℳ is gclosed, the pair $(T,g)$ is $(O,\mathcal{M})$compatible and X is $(G,\mathcal{M})$regular.
Then T and g have, at least, a coincidence point.
Proof It follows from Theorems 3.3 and 3.4 taking into account Corollary 3.1, Lemmas 2.2, 4.2 and Definition 4.2. Notice that (A′) ⇒ (${\mathrm{A}}^{\u2033}$) ⇒ (A), (a′) ⇒ (a), (b′) ⇒ (b) and (e′) ⇒ (e). □
We particularize the previous result to the case in which $\mathcal{M}={\mathcal{M}}_{\preccurlyeq}$, associated to a preorder or a partial order ≼ on X. In such a case, Lemma 3.1 is applicable. We leave to the reader to interpret ≼nondecreasingcontinuity as ${\mathcal{M}}_{\preccurlyeq}$continuity, Gregularity as $(G,{\mathcal{M}}_{\preccurlyeq})$compatibility, Ocompatibility as $(O,{\mathcal{M}}_{\preccurlyeq})$compatibility, and ${O}^{\prime}$compatibility as $({O}^{\prime},{\mathcal{M}}_{\preccurlyeq})$compatibility.
Corollary 4.3 Let $(X,G)$ be a Gmetric space provided with a preorder ≼, and let $T,g:X\to X$ be two mappings such that $T(X)\subseteq g(X)$ and T is $(g,\preccurlyeq )$nondecreasing. Assume that, at least, one of the following two conditions holds.
for all $x,y\in X$ for which $gx\preccurlyeq gy$.
for all $x,y\in X$ for which $gx\preccurlyeq gy$.
 (a)
X (or $g(X)$ or $T(X)$) is Gcomplete, T and g are ≼nondecreasingcontinuous and the pair $(T,g)$ is ${O}^{\prime}$compatible;
(a′) X (or $g(X)$ or $T(X)$) is Gcomplete, T and g are continuous and the pair $(T,g)$ is ${O}^{\prime}$compatible;
 (b)
X (or $g(X)$ or $T(X)$) is Gcomplete and T and g are ≼–nondecreasingcontinuous and commuting;
 (c)
$(g(X),G)$ is complete and X (or $g(X)$) is Gregular;
 (d)
$(X,G)$ is complete, $g(X)$ is closed and X (or $g(X)$) is Gregular;
 (e)
$(X,G)$ is complete, g is ≼nondecreasing and ≼nondecreasingcontinuous, the pair $(T,g)$ is Ocompatible and X is Gregular.
(e′) $(X,G)$ is complete, g is ≼nondecreasing and continuous, the pair $(T,g)$ is Ocompatible and X is Gregular.
If there exists ${x}_{0}\in X$ verifying $g{x}_{0}\preccurlyeq T{x}_{0}$, then T and g have, at least, a coincidence point.
We also leave to the reader the task of particularizing the previous results to the case in which g is the identity mapping on X, obtaining fixed points of T.
5 Coupled coincidence point theorems
In this section, we deduce that Theorem 1.1 follows from Theorem 3.3. However, the main aim of this subsection is to describe how Theorems 3.1, 3.2, 3.3 and 3.4 can be employed in order to obtain some coupled coincidence point theorems, because these techniques can be extrapolated to many contexts.
for all $(x,y)\in {X}^{2}$. □
5.1 Charoensawan and Thangthong’s coupled coincidence point result in Gmetric spaces
Notice that ${M}^{\prime}$ is a subset of ${X}^{4}={X}^{2}\times {X}^{2}$.
 1.If there exist ${x}_{0},{y}_{0}\in X$ such that$(F({x}_{0},{y}_{0}),F({y}_{0},{x}_{0}),F({x}_{0},{y}_{0}),F({y}_{0},{x}_{0}),g{x}_{0},g{y}_{0})\in M,$
 2.
 3.
If M verifies the second property of Definition 1.1, then ${M}^{\prime}$ is a $({T}_{F},\mathcal{G})$closed set.
 4.
If M is an $({F}^{\ast},g)$invariant set, then ${M}^{\prime}$ is a $({T}_{F},\mathcal{G})$closed set.
We point out that we will only use the second property of the notion of $({F}^{\ast},g)$invariant set (Definition 1.1). This shows that $(T,g)$closed sets are more general than an $({F}^{\ast},g)$invariant set because the first property will not be employed (this was also established in Kutbi et al. [5]).
Proof (1) By definition, $(F({x}_{0},{y}_{0}),F({y}_{0},{x}_{0}),F({x}_{0},{y}_{0}),F({y}_{0},{x}_{0}),g{x}_{0},g{y}_{0})\in M$ implies that $(g{x}_{0},g{y}_{0},F({x}_{0},{y}_{0}),F({y}_{0},{x}_{0}))\in {M}^{\prime}$, which means that $(\mathcal{G}({x}_{0},{y}_{0}),{T}_{F}({x}_{0},{y}_{0}))\in {M}^{\prime}$.
 (3)Assume that M is an $({F}^{\ast},g)$invariant set, and let $x,u,y,v\in X$ be such that $(\mathcal{G}(x,u),\mathcal{G}(y,v))\in {M}^{\prime}$. By definition, since $(gx,gu,gy,gv)\in {M}^{\prime}$, then $(gy,gv,gy,gv,gx,gu)\in M$. As M is $({F}^{\ast},g)$invariant, then$(F(y,v),F(v,y),F(y,v),F(v,y),F(x,u),F(u,x))\in M.$
In particular, $(F(x,u),F(u,x),F(y,v),F(v,y))\in {M}^{\prime}$, which means that $({T}_{F}(x,u),{T}_{F}(y,v))\in {M}^{\prime}$. Hence, ${M}^{\prime}$ is a $({T}_{F},\mathcal{G})$closed set. □
Using this notation, the following result is obvious.
for all $(x,u),(y,v)\in {X}^{2}$ such that $(\mathcal{G}(x,u),\mathcal{G}(y,v))\in {M}^{\prime}$.
Notice that condition (21) is weaker than condition (1). The previous properties prove the following consequence.
 1.
If F is Gcontinuous, then ${T}_{F}$ is ${q}_{{G}_{2}}$continuous.
 2.
Proof It is a straightforward exercise. □
Corollary 5.1 Theorem 1.1 follows from Theorem 3.3.
Proof Under the hypothesis of Theorem 1.1, let us consider the quasimetric space $({X}^{2},{q}_{{G}_{2}})$, the mappings ${T}_{F}$ and and the subset ${M}^{\prime}$ defined by (20). By item 3 of Lemma 2.1, $({X}^{2},{G}_{2})$ is a complete Gmetric space, and by item 7 of Lemma 2.2, $({X}^{2},{q}_{{G}_{2}})$ is a complete quasimetric space. Furthermore, Corollary 3.1 guarantees that $({X}^{2},{q}_{{G}_{2}})$ is left/rightCauchy and left/rightconvergent. Lemma 5.4 ensures that ${T}_{F}$ and are ${q}_{{G}_{2}}$continuous. Lemma 5.2 proves that ${T}_{F}({X}^{2})\subseteq \mathcal{G}({X}^{2})$ and ${M}^{\prime}$ is a transitive, $({T}_{F},\mathcal{G})$closed, nonempty subset of ${({X}^{2})}^{2}$. Finally, Lemma 5.3 ensures that ${T}_{F}$ is a $(\mathcal{G},{M}^{\prime},\mathrm{\Phi})$contraction of the second kind. As a consequence, case (b) of Theorem 3.3 (replacing condition (A) by (A′), and ℳcontinuity by continuity) guarantees that ${T}_{F}$ and have, at least, a coincidence point, which is a coincidence point of F and g. □
In fact, the previous proof shows that two conditions are not necessary in Theorem 3.3: neither the first property of $({F}^{\ast},g)$invariant sets nor the middle variables of M in ${X}^{6}$.
5.2 Kutbi et al.’s coupled fixed point theorems without the mixed monotone property
In [5], the authors introduced the following notion and proved the following result.
Definition 5.1 (Kutbi et al. [5])
Corollary 5.2 (Kutbi et al. [5], Theorem 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.
5.3 Sintunaravat et al.’s coupled fixed point theorems without the mixed monotone property
Similarly, the following result is a consequence of our main results.
Corollary 5.3 (Sintunaravat et al. [21])
 (a)
F is continuous, or
 (b)for any two sequences $\{{x}_{m}\}$, $\{{y}_{m}\}$ with $({x}_{m+1},{y}_{m+1},{x}_{m},{y}_{m})\in M$,$\{{x}_{m}\}\to x,\phantom{\rule{2em}{0ex}}\{{y}_{m}\}\to y,$