Coincidence point theorems in quasi-metric spaces without assuming the mixed monotone property and consequences in G-metric spaces

In this paper, we present some coincidence point theorems in the setting of quasi-metric 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 G-metric 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.


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 [], by Nieto and Rodríguez-López [], by Gnana-Bhaskar and Lakshmikantham [], and by Lakshmikantham and Ćirić [], to cite some of them), the literature on this topic has expanded significantly. In [], 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, []). One of the results on this line of study was given by Charoensawan  Using the previous preliminaries, they proved the following result in the context of Gmetric spaces, which is recalled in Section ..
Theorem . (Charoensawan and Thangthong [], Theorem .) Let (X, ) be a partially ordered set and G be a G-metric on X such that (X, G) is a complete G-metric space, and let M be a nonempty subset of X  . Assume that there exists ϕ ∈ and also suppose that F : X × X → X and g : X → X such that G F(x, u), F(y, v), F(z, w) + G F(u, x), F(v, y), F(w, z) ≤ φ G(gx, gy, gz) + G(gu, gv, gw) () for all (gx, gu, gy, gv, gz, gw) ∈ M. Suppose also that F is continuous, F(X × X) ⊆ g(X) and g is continuous and commutes with F. If there exist x  , y  ∈ X such that F(x  , y  ), F(y  , x  ), F(x  , y  ), F(y  , x  ), gx  , gy  ∈ M and M is an (F * , g)-invariant set which satisfies the transitive property, then there exist x, y ∈ 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 .. 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 . can be easily deduced from a unidimensional version of the same result. In fact, we prove that the middle variables of M ⊆ X  are unnecessary. But the main aim of this work is to obtain some coincidence point theorems in the context of quasi-metric spaces that can be applied in several frameworks, including metric spaces and G-metric 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 []. As a consequence, we prove that many results in this field of study can be easily derived from our statements.

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 N  = N ∪ {}. If A ⊆ R is a nonempty subset of R, the Euclidean metric on A is d(x, y) = |x -y| for all x, y ∈ 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 × X × · · · × X (n times). http://www.fixedpointtheoryandapplications.com/content/2014/1/184 Definition . (Mustafa and Sims []) Let X be a nonempty set, and let G : X × X × X → R + be a function satisfying the following properties: (x, x, y) for all x, y ∈ X with x = y; (G  ) G(x, x, y) ≤ G(x, y, z) for all x, y, z ∈ X with y = z; (G  ) G(x, y, z) = G(x, z, y) = G(y, z, x) = · · · (symmetry in all three variables); (G  ) G(x, y, z) ≤ G(x, a, a) + G(a, y, z) (rectangle inequality) for all x, y, z, a ∈ X.
Then the function G is called a generalized metric, or, more specifically, a G-metric on X, and the pair (X, G) is called a G-metric space.
Note that every G-metric on X induces a metric d G on X defined by d G (x, y) = G(x, y, y) + G(y, x, x) for all x, y ∈ X.
For a better understanding of the subject, we give the following examples of G-metrics.
Example . Let (X, d) be a metric space. The function G : X × X × X → [, +∞), defined by G(x, y, z) = max d(x, y), d(y, z), d (z, x) for all x, y, z ∈ X, is a G-metric on X.
In their initial paper, Mustafa and Sims [] also defined the basic topological concepts in G-metric spaces as follows.
Definition . (Mustafa and Sims []) Let (X, G) be a G-metric space, and let {x n } be a sequence of points of X. We say that that is, for any ε > , there exists N ∈ N such that G(x, x n , x m ) < ε for all n, m ≥ N . We call x the limit of the sequence, and we write {x n } → x or lim n→∞ x n = x.
It is clear that the limit of a convergent sequence is unique.

Proposition . (Mustafa and Sims [])
In a G-metric space (X, G), the following conditions are equivalent.

Definition . (Mustafa and Sims
Proposition . (Mustafa and Sims []) In a G-metric space (X, G), the following conditions are equivalent.
The following lemma shows a simple way to consider some G-metrics on X  from a G-metric on X.
Then the following conditions are equivalent.
In such a case, the following properties hold. . Every sequence {(x n , y n )} ⊆ X  verifies:

Quasi-metric spaces
In such a case, the pair (X, q) is called a quasi-metric space.
Remark . Any metric space is a quasi-metric space, but the converse is not true in general.
Now, we recollect some basic topological notions and related results about quasi-metric spaces (see also, e.g., [-]).
Definition . Let (X, q) be a quasi-metric space, {x n } be a sequence in X, and x ∈ X. We will say that: • {x n } converges to x (and we will denote it by all n, m ≥ n  . The quasi-metric 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 . (Jleli and Samet []) Let (X, q) be a quasi-metric space, {x n } be a sequence in X, and x ∈ X. We say that: Remark . . The limit of a sequence in a quasi-metric space, if it exists, is unique. However, this is false if we consider right-limits or left-limits. . If {x n } → x and {y n } → y in a quasi-metric space, then {q(x n , y n )} → 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) ≤ q(x n , y n ) ≤ q(x n , x) + q(x, y) + q(y, y n ) for all n. In particular, {q(x n , z)} → q(x, z) and {q(z, x n )} → q(z, x) for all z ∈ X. http://www.fixedpointtheoryandapplications.com/content/2014/1/184 . If {x n } → x, {q(x n , y n )} →  and {q(y n , x n )} → , then {y n } → x. It follows from q(y n , x) ≤ q(y n , x n ) + q(x n , x) and q(x, y n ) ≤ q(x, x n ) + q(x n , y n ).
. If a sequence {x n } has a right-limit x and a left-limit 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 right-limits when it has no left-limit.
Example . Let X be a subset of R containing [, ] and define, for all x, y ∈ X, Then (X, q) is a quasi-metric space. Notice that {q(/n, )} →  but {q(, /n)} → . Therefore, {/n} right-converges to  but it does not converge from the left.
The following result shows a simple way to consider quasi-metrics from G-metrics.

Lemma . (Agarwal et al. [])
Let (X, G) be a G-metric space, and let us define q G , q G : Then the following properties hold. . q G and q G are quasi-metrics on X. Moreover, . In (X, q G ) and in (X, q G ), a sequence is right-convergent (respectively, left-convergent) if and only if it is convergent. In such a case, its right-limit, its left-limit and its limit coincide. . In (X, q G ) and in (X, q G ), a sequence is right-Cauchy (respectively, left-Cauchy) if and only if it is Cauchy. . In (X, q G ) and in (X, q G ), every right-convergent (respectively, left-convergent) sequence has a unique right-limit (respectively, left-limit).

Control functions
Functions in (see Definition .) verify the following properties. Proof () By item , t n+ ≤ ϕ(t n ) ≤ t n for all n, so {t n } is a nonincreasing sequence of nonnegative real numbers. Then it is convergent. Let L = lim n→∞ t n ≥ . We claim that L = . If L > , then {t n } is a sequence of numbers greater than L that converges to L. Hence, which is a contradiction.
() It follows from item  taking into account that  ≤ s n ≤ ϕ(t n ) ≤ t n for all n. Proof () It is the same proof of item  of Lemma .. () It follows from the fact that s n ≤ ϕ(t n ) < t n if t n > , and s n =  if t n = . In any case, s n ≤ t n for all n.
Remark . The difference between items  and  of Lemma . and items  and  of Lemma . is important. If we assume that ϕ ∈ and t m+ ≤ ϕ(t m ) for all m, then it is impossible to deduce that {t m } →  or {ϕ(t m )} →  in item  of the previous result. For instance, define ϕ(t) = t/ if t > , and ϕ() = /. Then ϕ ∈ and the sequence {t m } = {, /, , /, , /, . . .} verifies t m+ ≤ ϕ(t m ) for all m but it does not converge.

Coincidence point theorems on quasi-metric 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.

Basic notions depending on a subset M Definition . (See Kutbi et al. [])
We say that a nonempty subset M of X  is: Given two mappings T, g : X → X, we say that M is: for all x, y ∈ X such that (gx, gy) ∈ M; http://www.fixedpointtheoryandapplications.com/content/2014/1/184 • (T, g)-compatible if Tx = Ty for all x, y ∈ X such that gx = gy and (gx, gy) ∈ M.
Clearly, every transitive subset is also g-transitive. Moreover, M is g-closed 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 g-transitive, (T, g)-closed sets.
Lemma . Given a binary relation on X, let us consider M = {(x, y) ∈ X  : x y}, and let T, g : X → X be two mappings.
. If is a preorder on X, then M is reflexive, transitive and g-transitive.
. If is a partial order on X, then M is reflexive, transitive, antisymmetric and g-transitive.
Proof First four properties are obvious. We prove the last one. Since T is (g, )-nondecreasing, It is convenient to highlight that the notion of g-transitive, (T, g)-closed, nonempty subset M ⊆ X  is more general than the idea of nondecreasing mapping on a preordered space (following the previous lemma), as we show in the following example.
In the following definitions, we will use sequences {x n } ⊆ X such that (x n , x m ) ∈ M for all n, m ∈ N with n < m. In this sense, the following notions must be called 'right-notions' because the same concepts could also be introduced involving sequences {x n } ⊆ X such that (x n , x m ) ∈ M for all n, m ∈ N with n > m (in this case, they would be 'left-notions').
Then we could talk about (T, g, M)-right-Picard sequences, M-right-continuity, (O, M)right-compatibility and right-regularity. However, we advice the reader that, in order not to complicate the notation, we will omit the term 'right'.
Definition . Let (X, q) be a quasi-metric space, let M be a nonempty subset of X  , and let T : X → X be a mapping. We say that Remark . Every continuous mapping from a quasi-metric space into itself is also Mcontinuous, whatever the subset M. http://www.fixedpointtheoryandapplications.com/content/2014/1/184 Definition . Let T, g : X → X be two mappings, let {x n } n≥ ⊆ X be a sequence, and let M be a nonempty subset of X  . We say that {x n } is a: for all n, m ∈ N such that n < m.
The following definition extends some ideas that can be found in [-]. Definition . Let (X, q) be a quasi-metric space, and let A ⊆ X and M ⊆ X  be two nonempty subsets. We say that

Coincidence point theorems using (g, M, )-contractions of the first kind
Next, we present the kind of contractions we will use. Definition . Let (X, q) be a quasi-metric space, let T, g : X → X be two mappings, and let M ⊆ X  be a nonempty subset of X  . We say that T is a (g, M, )-contraction of the first kind if there exist ϕ, ϕ ∈ such that q(Tx, Ty) ≤ ϕ q(gx, gy) and () for all x, y ∈ X such that (gx, gy) ∈ M. If ϕ, ϕ ∈ , we say that T is a (g, M, )-contraction of the first kind.
Remark . It is not necessary that functions in and in verify all their properties in [, ∞). In fact, as we shall only use inequalities ()-(), the properties of functions in and in must only be verified on the image of the quasi-metric q, that is, on Remark . One of the best advantages of using a subset M ⊆ X  is that a unique condition covers two particularly interesting cases: • M = X  , in which contractivity conditions ()-() hold for all x, y ∈ X; and • M = M , where is a preorder or a partial order on X, in which ()-() must be assumed for all x, y ∈ X such that gx 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.
Lemma . Let (X, q) be a quasi-metric space, let T, g : X → X be two mappings, and let M ⊆ X  be a g-closed, nonempty subset of X  such that (X, q, M) is regular. Suppose that, at least, one of the following conditions holds. regular, then (gx n , gz) ∈ M for all n ∈ N. Applying the contractivity conditions ()-(), we have that, for all n, q(Tx n , Tz) ≤ ϕ q(gx n , gz) and q(Tz, Tx n ) ≤ ϕ q(gz, gx n ) .
If ϕ, ϕ ∈ , then item  of Lemma . guarantees that {q(Tx n , Tz)} →  and {q(Tz, Tx n )} → , so {Tx n } q-converges to Tz. If ϕ, ϕ ∈ and we additionally assume that M is (T, g)compatible, we can use item  of Lemma . applied to the sequences {t n = q(Tx n , Tz)} and {s n = q(gx n , gz)} in order to deduce that {q(Tx n , Tz)} →  (notice that if s n = , then t n = ) and similarly {q(Tz, Tx n )} → .
The first main result of this work is the following one.
Theorem . Let (X, q) be a quasi-metric space, let T, g : X → X be two mappings, and let M be a nonempty subset of X  . Suppose that the following conditions are fulfilled. And by Remark ., the M-continuity of the mappings can be replaced by continuity.
Proof Let {x n } be an arbitrary (T, g, M)-Picard sequence on X, and let ϕ, ϕ ∈ be such that ()-() hold. If there exists some n  ∈ N such that gx n  = gx n  + , then gx n  = gx n  + = Tx n  , so x n  is a coincidence point of T and g, and the proof is finished. On the contrary, assume that gx n = gx n+ for all n ≥ . Therefore, q(gx n , gx n+ ) >  and q(gx n+ , gx n ) >  for all n ≥ . () Step . We claim that lim n→∞ q(gx n , gx n+ ) = lim n→∞ q(gx n+ , gx n ) = . Taking into account (), if we apply the contractivity condition () to x = gx n+ and y = gx n+ , we obtain that q(gx n+ , gx n+ ) = q(Tx n , Tx n+ ) ≤ ϕ q(gx n , gx n+ ) for all n ≥ . http://www.fixedpointtheoryandapplications.com/content/2014/1/184 By item  of Lemma ., we have that {q(gx n , gx n+ )} → . Similarly, using x = gx n+ and y = gx n+ and the contractivity condition (), we could deduce that {q(gx n+ , gx n )} → . Therefore, we have proved that lim n→∞ q(gx n , gx n+ ) = lim n→∞ q(gx n+ , gx n ) = . () Step . We claim that {gx n } is right-Cauchy in (X, q), that is, for all ε > , there is n  ∈ N such that q(gx n , gx m ) ≤ ε for all m > n ≥ n  . We reason by contradiction. If {gx n } is not right-Cauchy, there exist ε  >  and two subsequences {gx n(k) } k∈N  and {gx m(k) } k∈N  verifying that Taking m(k) as the smallest integer, greater than n(k), verifying this property, we can suppose that gx m(k) ), and taking limit as k → ∞, it follows from () that lim k→∞ q(gx n(k) , gx m(k) ) = ε  .
Joining both inequalities we deduce that, for all k, Letting k → ∞, it follows from () that Since q(gx n(k) , gx m(k) ) > ε  for all n, {q(gx n(k) , gx m(k) )} → ε  and ϕ ∈ , then lim k→∞ ϕ q(gx n(k) , gx m(k) ) = lim Letting k → ∞ in () and taking into account () and (), it follows that which is a contradiction. This contradiction ensures us that {gx n } is right-Cauchy in (X, q), and Step  holds. Similarly, using the contractivity condition (), it can be proved that {gx n } is left-Cauchy in (X, q), so we conclude that {gx n } is a Cauchy sequence in (X, q). Now, we prove that T and g have a coincidence point distinguishing between cases (a)-(e). Case

(X) or T(X)) is q-complete and T and g are M-continuous and commuting. It is obvious because (b) implies (a).
Case (c): (g(X), q) is complete and X (or g(X)) is (q, M)-regular. As {gx m } is a Cauchy sequence in the complete space (g(X), q), there is u ∈ g(X) such that {gx m } → u. Let v ∈ X be any point such that u = gv. In this case, {gx m } → gv. We are also going to show that {gx m } → Tv, so we will conclude that gv = Tv (and v is a coincidence point of T and g).
Indeed, as {gx n } is a convergent sequence in g(X) such that (gx n , gx m ) ∈ M for all n < m, and X (or g(X)) is (q, M)-regular, then (gx n , gv) ∈ M for all n, where gv = u ∈ g(X) is the limit of {gx n }. Applying the contractivity conditions ()-(), q(gx n+ , Tv) = q(Tx n , Tv) ≤ ϕ q(gx n , gv) and () q(Tv, gx n+ ) = q(Tv, Tx n ) ≤ ϕ q(gv, gx n ) for all n ≥ .
By item  of Lemma ., {gx n } q-converges to Tv. Case (d): (X, q) is complete, g(X) is closed and X (or g(X)) is (q, M)-regular. It follows from the fact that a closed subset of a complete quasi-metric space is also complete. Then (g(X), q) is complete and case (c) is applicable. Case Clearly, M does not come from any partial order on X as in Lemma . because it is not antisymmetric: (, ), (, ) ∈ M but  = . Let us consider on X the function q : X × X → [, ∞) given, for all x, y ∈ X, by Then q is a complete quasi-metric on R. In fact, it has the same convergent sequences to the same limits as the Euclidean metric d(x, y) = |x -y| for all x, y ∈ R because However, q is not a metric because q(, ) = q(, ). Now, given a real number λ ∈ (., ), let us consider the mappings T, g : R → R defined, for all x ∈ R, by Also consider the function ϕ λ : [, ∞) → [, ∞) defined by ϕ λ (t) = λt for all t ∈ [, ∞). Clearly, ϕ λ ∈ ∩ . We are going to show that Theorem . is applicable to the previous setting, because the previous properties hold. http://www.fixedpointtheoryandapplications.com/content/2014/1/184 . The sequence {x n }, given by x n = λ n for all n ∈ N  , is a (T, g, M)-Picard sequence. . The function g : R → R is bijective and nondecreasing. . The range of g, which is g(X) = R, is closed and complete in (R, q). . We claim that T is a (g, M, )-contraction of the first kind. To prove it, let x, y ∈ X be such that (gx, gy) ∈ M. If Tx = Ty, then ()-() are obvious. Next, assume that Tx = Ty. In particular, x = y. Hence, gx = gy because g is bijective. Therefore, the condition (gx, gy) ∈ M leads to two cases. . Let {x n } ⊂ R be a sequence such that (x n , x n+ ) ∈ M for all n ∈ N  . Then one, and only one, of the following cases holds. (.a) There exists n  ∈ N such that x n  ∈ [, ]. In this case, x n ∈ [, ] and x n+ ≤ x n for all n ∈ N  . To prove it, notice that (x n  , x n  + ) ∈ M is only possible when x n  = x n  + or  ≤ x n  + < x n  ≤ . In any case, x n  + ∈ [, ]. Repeating this argument, x n ∈ [, ] for all n ≥ n  . But if n  - ∈ N, the condition (x n  - , x n  ) ∈ M also leads to x n  - ∈ [, ]. And we can again repeat the argument. (.b) There exists n  ∈ N such that x n  ∈ {, }. In this case, x n ∈ {, } for all n ∈ N  . (.c) There exists z ∈ R ([, ] ∪ {, }) such that x n = z for all ∈ N  . In this case, {x n } is a constant sequence.
. The range g(X) = R is (q, M)-regular. To prove it, let u ∈ R and let {x n } ⊂ R be a sequence such that{x n } q → u and (x n , x n+ ) ∈ M for all n ∈ N  . In particular, {x n } → u using the Euclidean metric. We can distinguish the previous three cases. We extend the previous theorem to the case in which ϕ ∈ .
Theorem . If we additionally assume that M is (T, g)-compatible, then Theorem . also holds even if T is a (g, M, )-contraction of the first kind.
Proof We can follow, point by point, the proof of the previous result and obtain inequalities ()-(). In this case, we cannot use Lemma ., but we may use the fact that M is (T, g)-compatible. Therefore, we know that, as (gx n , gv) ∈ M for all n, then q(gx n , gv) =  ⇒ gx n = gv ⇒ Tx n = Tv ⇒ q(gx n+ , Tv) = q(Tx n , Tv) = .
By item  of Lemma . we conclude that {q(gx n+ , Tv)} → . In the same way, {q(Tv, gx n+ )} → , so {gx n } q-converges to Tv. The same argument is valid when applied to inequalities ()-().

Coincidence point theorems using (g, M, )-contractions of the second kind
Many results on fixed point theory in the setting of G-metrics can be similarly proved using the quasi-metrics q G and q G associated to G as in Lemma . (see, for instance, Agarwal et al. []). These families of quasi-metrics verify additional properties that are not true for an arbitrary quasi-metric. 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.
Definition . Let (X, q) be a quasi-metric space, let T, g : X → X be two mappings, and let M ⊆ X  be a nonempty subset of X  . We say that T is a (g, M, )-contraction of the second kind if there exists ϕ ∈ such that q(Tx, Ty) ≤ ϕ q(gx, gy) () for all x, y ∈ X such that (gx, gy) ∈ M. If ϕ ∈ , we say that T is a (g, M, )-contraction of the second kind.
Notice that condition () is not symmetric on x and y because (gx, gy) ∈ M does not imply (gy, gx) ∈ M. In order to compensate this absence of symmetry, we will suppose an additional condition on the ambient space.
Definition . We say that a quasi-metric space (X, q) is: • right-Cauchy if every right-Cauchy sequence in (X, q) is, in fact, a Cauchy sequence in (X, q); • left-Cauchy if every left-Cauchy sequence in (X, q) is, in fact, a Cauchy sequence in (X, q); • right-convergent if every right-convergent sequence in (X, q) is, in fact, a convergent sequence in (X, q); http://www.fixedpointtheoryandapplications.com/content/2014/1/184 • left-convergent if every left-convergent 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 quasi-metric space given in Definition .. Lemma . guarantees that there exists a wide family of quasi-metrics that verify all the previous properties.

Corollary . Every quasi-metric q G and q G associated to a G-metric G on X is right and left-Cauchy and right and left-convergent.
Next we prove a similar result to Theorem .. In this case, the contractivity condition is weaker but we suppose additional conditions on the ambient space.
Theorem . Let (X, q) be a right-Cauchy quasi-metric space, let T, g : X → X be two mappings, and let M be a nonempty subset of X  . Suppose that the following conditions are fulfilled. And by Remark ., the M-continuity of the mappings can be replaced by continuity.
Proof We can follow, step by step, the lines of the proof of Theorem . to deduce, in the case gx n = gx n+ for all n ≥ , that {gx n } is right-Cauchy in (X, q). Using that (X, q) is right-Cauchy, 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 .. Case (c): (g(X), q) is complete and right-convergent, and X (or g(X)) is (q, M)-regular. As {gx m } is a Cauchy sequence in the complete space (g(X), q), there is u ∈ g(X) such that {gx m } → u. Let v ∈ X be any point such that u = gv. In this case, {gx m } → gv. We are also going to show that {gx m } → Tv, so we will conclude that gv = Tv (and v is a coincidence point of T and g). http://www.fixedpointtheoryandapplications.com/content/2014/1/184 Indeed, as {gx n } is a convergent sequence in g(X) such that (gx n , gx m ) ∈ M for all n < m, and X (or g(X)) is (q, M)-regular, then (gx n , gv) ∈ M for all n, where gv = u ∈ g(X) is the limit of {gx n }. Applying the contractivity condition (), q(gx n+ , Tv) = q(Tx n , Tv) ≤ ϕ q(gx n , gv) for all n.
By item  of Lemma ., we have that {q(gx n+ , Tv)} → , which means that {gx n } rightconverges to Tv. Since (X, q) is right-convergent, then {gx n } is a convergent sequence in (X, q), and by item  of Remark ., it converges to Tv.
Case (d): (X, q) is complete and right-convergent, g(X) is closed and X (or g(X)) is (q, M)regular. It follows from the fact that a closed subset of a complete quasi-metric space is also complete. Then (g(X), q) is complete and case (c) is applicable.
Case In this case, by item  of Lemma ., we have that {q(Tgx n , Tu)} → , which means that {Tgx n } right-converges to Tu. Since (X, q) is right-convergent, then {Tgx n } is a convergent sequence in (X, q), and by item  of Remark ., it converges to Tu.
Example . Theorem . can also be applied to mappings given in Example . because (R, q) is right-convergent.
Repeating the arguments of Theorem ., we extend the previous theorem to the case in which ϕ ∈ .
Theorem . If we additionally assume that M is (T, g)-compatible, then Theorem . also holds even if T is a (g, M, )-contraction of the second kind.

Consequences
The previous theorems admit a lot of different particular cases employing continuity, the condition T(X) ⊆ 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 [].  X, d, ) be an ordered metric space, and let T, g : X → X be two mappings such that the following properties are fulfilled. Proof It is only necessary to apply Theorem . to the subset M = {(x, y) ∈ X  : x y}, taking into account the properties given in Lemma .. Notice that in case (e), we use Lemma . 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).
Corollary . Let (X, q) be a complete quasi-metric space, let T, g : X → X be two mappings such that T(X) ⊆ g(X), and let M be a g-transitive, (T, g)-closed, nonempty subset of X  . Suppose that T is a (g, M, )-contraction (respectively, T is a (g, M, )-contraction and M is (T, g)-compatible), g is M-continuous, T and g are commuting and there exists x  ∈ X such that (gx  , Tx  ) ∈ M. Also assume that, at least, one of the following conditions holds. Another interesting particularization is the following one. As a consequence, in the following result, a partial order is not necessary.
Corollary . (Karapınar et al. [], Corollary ) Let (X, d) be a complete metric space, and let be a transitive relation on X. Let T, g : X → X be two mappings such that TX ⊆ gX and T is (g, )-nondecreasing. Suppose that there exists ϕ ∈ such that d(Tx, Ty) ≤ ϕ d(gx, gy) for all x, y ∈ X such that gx gy.
(   ) Also suppose that Assume that either (a) T and g are continuous and commuting, or (b) (X, d, ) is regular and gX is closed.
If there exists a point x  ∈ X such that gx  Tx  , then T and g have, at least, a coincidence point.

Applications to G-metric spaces
One of the most interesting, recent lines of research in the field of fixed point theory is devoted to G-metric spaces. Taking into account Lemma ., we can take advantage of our main results to present some new theorems in this area. The following result is an easy application to G-metric spaces.
Corollary . Let (X, G) be a complete G-metric space, let T, g : X → X be two mappings such that T(X) ⊆ g(X), and let M ⊆ X  be a g-transitive, (T, g)-closed, nonempty subset of X  . Assume that T and g are continuous and commuting, and there exists ϕ ∈ such that G(Tx, Tx, Ty) ≤ ϕ G (gx, gx, gy) for all x, y ∈ X such that (gx, gy) ∈ M. If there exists x  ∈ X such that (gx  , Tx  ) ∈ M, then T and g have, at least, a coincidence point.
Notice that this result is also valid if ϕ ∈ and M is (T, g)-compatible.
Proof It follows from Theorem . and Corollary . using the quasi-metric q G associated to G (as in Lemma .). Notice that there exists a (T, g, M)-Picard sequence on X by items  and  of Lemma ..
In order not to lose the power and usability of Theorems . and ., we present the following properties comparing q G and q G .
Definition . Let (X, G) be a G-metric space, and let A ⊆ X and M ⊆ X  be two nonempty subsets. We say that (A, G, M) is regular (or A is (G, M)-regular) if we have that (x n , u) ∈ M for all n provided that {x n } is a G-convergent sequence on A, u ∈ A is its G-limit and (x n , x m ) ∈ M for all n < m. http://www.fixedpointtheoryandapplications.com/content/2014/1/184 (b) X (or g(X) or T(X)) is G-complete and T and g are -nondecreasing-continuous and commuting; (b ) X (or g(X) or T(X)) is G-complete and T and g are continuous and commuting; (c) (g(X), G) is complete and X (or g(X)) is G-regular; closed and X (or g(X)) is G-regular; (e) (X, G) is complete, g is -nondecreasing and -nondecreasing-continuous, the pair , g is -nondecreasing and continuous, the pair (T, g) is O-compatible and X is G-regular.
If there exists x  ∈ X verifying gx  Tx  , 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.

Coupled coincidence point theorems
In this section, we deduce that Theorem . follows from Theorem .. However, the main aim of this subsection is to describe how Theorems ., ., . and . can be employed in order to obtain some coupled coincidence point theorems, because these techniques can be extrapolated to many contexts. We introduce the following notation. Given two mappings F : X  → X and g : X → X, we define T F , G : X  → X  , for all (x, y) ∈ X  , by T F (x, y) = F(x, y), F(y, x) and G(x, y) = (gx, gy).
Lemma . Let F : X  → X and g : X → X be two mappings.
. If F(X  ) ⊆ g(X), then T F (X  ) ⊆ G(X  ). . If F and g are commuting, then T F and G are also commuting.
for all (x, y) ∈ X  .

Charoensawan and Thangthong's coupled coincidence point result in G-metric spaces
One of the key objectives of this subsection is to prove that, in Theorem ., the middle variables of M are not necessary. Indeed, given a nonempty subset M ⊆ X  , let us define Notice that M is a subset of X  = X  × X  . http://www.fixedpointtheoryandapplications.com/content/2014/1/184 Lemma . Let F : X  → X and g : X → X be two mappings, and let M ⊆ X  . . If there exist x  , y  ∈ X such that We point out that we will only use the second property of the notion of (F * , g)-invariant set (Definition .). This shows that (T, g)-closed sets are more general than an (F * , g)invariant set because the first property will not be employed (this was also established in Kutbi et al. []).
() Assume that M is transitive, and let x, u, y, v, z, w ∈ X be such that (x, u, y, v), As M is transitive, then (z, w, z, w, x, u) ∈ M, so (x, u, z, w) ∈ M . Therefore, M is transitive, and it is also G-transitive because every transitive subset is also G-transitive, whatever G.
In the following result, we use the quasi-metric q G  on X  associated, by Lemma ., to the G-metric G  : , y, y) + G(u, v, v).
Using this notation, the following result is obvious.
Notice that condition () is weaker than condition (). The previous properties prove the following consequence.
Lemma . Let (X, G) be a G-metric space, and let F : X  → X and g : X → X be two mappings.
. If F is G-continuous, then T F is q G  -continuous. . If g is G-continuous, then G is q G  -continuous.
Proof It is a straightforward exercise.
Proof Under the hypothesis of Theorem ., let us consider the quasi-metric space (X  , q G  ), the mappings T F and G and the subset M defined by (). By item  of Lemma ., (X  , G  ) is a complete G-metric space, and by item  of Lemma ., (X  , q G  ) is a complete quasi-metric space. Furthermore, Corollary . guarantees that (X  , q G  ) is left/right-Cauchy and left/right-convergent. Lemma . ensures that T F and G are q G continuous. Lemma . proves that T F (X  ) ⊆ G(X  ) and M is a transitive, (T F , G)-closed, nonempty subset of (X  )  . Finally, Lemma . ensures that T F is a (G, M , )-contraction of the second kind. As a consequence, case (b) of Theorem . (replacing condition (A) by (A ), and M-continuity by continuity) guarantees that T F and G 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 .: neither the first property of (F * , g)-invariant sets nor the middle variables of M in X  .

Kutbi et al.'s coupled fixed point theorems without the mixed monotone property
In [], the authors introduced the following notion and proved the following result.  (y, v) .
Then F has a coupled fixed point.

Sintunaravat et al.'s coupled fixed point theorems without the mixed monotone property
Similarly, the following result is a consequence of our main results. for all m ≥ , then (x, y, x m , y m ) ∈ M for all m ≥ . If there exists (x  , y  ) ∈ X × X such that (F(x  , y  ), F(y  , x  ), x  , y  ) ∈ M and M is an Finvariant set which satisfies the transitive property, then there exist x, y ∈ X such that x = F(x, y) and y = F(y, x), that is, F has a coupled fixed point.

Choudhury and Kundu's coupled coincidence point theorems under the mixed g-monotone property
Although our main results in Section  do not need the mixed monotone property, we show in this subsection how to interpret that property using a subset M ⊆ X  , so that our main results are also applicable to this context. We start recalling this notion.
Definition . Let be a binary relation on X, and let F : X  → X and g : X → X be two mappings. We say that F has the mixed g-monotone property (with respect to ) if F(x, y) is monotone g-nondecreasing in x and monotone g-nonincreasing in y, that is, for any x, y ∈ X , x  , x  ∈ X, gx  gx  ⇒ F(x  , y) F(x  , y) and y  , y  ∈ X, gy  gy  ⇒ F(x, y  ) F(x, y  ).