Some fixed/coincidence point theorems under $(\psi ,\phi )$contractivity conditions without an underlying metric structure
 AntonioFrancisco RoldánLópezdeHierro^{1} and
 Naseer Shahzad^{2}Email author
https://doi.org/10.1186/168718122014218
© RoldánLópezdeHierro and Shahzad; licensee Springer. 2014
Received: 21 July 2014
Accepted: 26 September 2014
Published: 22 October 2014
Abstract
In this paper we prove a coincidence point result in a space which does not have to satisfy any of the classical axioms that define a metric space. Furthermore, the ambient space need not be ordered and does not have to be complete. Then, this result may be applied in a wide range of different settings (metric spaces, quasimetric spaces, pseudometric spaces, semimetric spaces, pseudoquasimetric spaces, partial metric spaces, Gspaces, etc.). Finally, we illustrate how this result clarifies and improves some wellknown, recent results on this topic.
Keywords
1 Introduction
Fixed point theory plays a crucial role in nonlinear functional analysis since, among other reasons, fixed point results are used to prove the existence (and also uniqueness) of solution when solving various types of equations. The Banach contraction principle is considered to be the pioneering result of the fixed point theory, and it is the most celebrated result in this field. The simplicity of its proof and the possibility of attaining the fixed point by using successive approximations let this theorem become a very useful tool in analysis and in applied mathematics. The great significance of Banach’s principle, and the reason it is possibly one of the most frequently cited fixed point theorems in all of analysis, lies in the fact that its proof contains elements of fundamental importance to the theoretical and practical treatment of mathematical equations. After the appearance of this result in Banach’s thesis in 1922, a great number of extensions (in many occasions, as wellknown as the original result, such as those by Krasnoselskii and Zabreiko, Edelstein, Browder, Schauder, Göhde, Kirk, and Caristi; a comprehensive study can be found in [1]) have been proved in various different frameworks (see [2, 3] in partial metric spaces, [4–7] in Gmetric spaces, [8, 9] in fuzzy metric spaces, [10, 11] in intuitionistic fuzzy metric spaces, [12, 13] in probabilistic metric spaces and [14, 15] in Menger spaces).
In recent times, one of the most attractive research topics in fixed point theory is to prove the existence of a fixed point in metric spaces endowed with partial orders. An initial result in this direction was given by Turinici [16] in 1986. Following this line of research, Ran and Reurings [17] (and later Nieto and RodríguezLópez [18]) used a partial order on the ambient metric space to introduce a slightly different contractivity condition, which must be only verified by comparable points. Thus, they reported two versions of the Banach contraction principle in partially ordered sets and applied them to the study of some applications to matrix equations. Their proofs involved combining the ideas of the iterative technique in the contractive mapping principle with those in the monotone technique. This approach led to a very recent branch of this field, with applications to matrix equations and ordinary differential equations. The literature on this topic has exponentially risen in recent years. To mention some advances on this topic, we highlight the following ones. Firstly, in order to guarantee the existence and uniqueness of a solution of periodic boundary value problems, GnanaBhaskar and Lakshmikantham [19] (and, subsequently, Lakshmikantham and Ćirić [20]) proved, in 2006, the existence and uniqueness of a coupled fixed point (a notion introduced by Guo and Lakshmikantham in [21]) in the setting of partially ordered metric spaces by introducing the notion of mixed monotone property. Later, the notions of tripled fixed point, quadruple fixed point and multidimensional fixed point were introduced by Berinde and Borcut [22], by Karapinar and Luong [23] and by Berzig and Samet [24] (see also [25]), respectively.
But the two main ingredients of all extensions are, basically, the same that we can find in the Banach contraction principle: a complete metric space and a selfmapping verifying a contractive condition. Although modern versions use, in many cases, different kind of mappings, the more intensively studied condition is based on the idea that the distance between the images of any two points (comparable or not) is upper bound by the product of a constant (small enough) and the distance between those points. The main aim of this manuscript is to provide a result powerful enough to guarantee that a nonlinear operator T has, at least, a fixed point, even when we consider that a measure mapping does not have to be an underlying metric structure on the ambient space X and the binary relationship is not necessarily a partial order on X. To do this, we present a result which can be applied in the following adverse conditions: the framework is a set X provided with a preorder and a measure mapping $d:X\times X\to \mathbb{R}$ that does not necessarily verify any of the four classical properties of a metric space (in fact, it need not be one of the following metric structures: a metric, a pseudometric, a quasimetric, a pseudoquasimetric or a semimetric). Furthermore, d has not to be symmetric and the triangular inequality must only be verified by a kind of comparable points. Even if d would verify some of the classical properties of a metric, $(X,d)$ would not be a complete space. In this setting, none of the theorems proved until now can be applied to guarantee that a nonlinear operator (even if it is a contractive mapping) has, at least, a fixed point. We illustrate our results with a particular example. Finally, we show that they extend and improve some wellknown fixed point theorems.
2 Preliminaries
Preliminaries and notation about coincidence points can also be found in [25]. Let n be a positive integer. Henceforth, X will denote a nonempty set and ${X}^{n}$ will denote the product space $X\times X\times \stackrel{n}{\cdots}\times X$. Throughout this manuscript, m and k will denote nonnegative integers and $i,j,s\in \{1,2,\dots ,n\}$. Unless otherwise stated, ‘for all m’ will mean ‘for all $m\ge 0$’ and ‘for all i’ will mean ‘for all $i\in \{1,2,\dots ,n\}$’. In the sequel, let $F:{X}^{N}\to X$ and $T,g:X\to X$ be three mappings. For brevity, $T(x)$ will be denoted by Tx.
Definition 2.1 A binary relation on X is a nonempty subset ℛ of $X\times X$. 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.
All partial orders and equivalence relations are preorders, but preorders are more general. From now on, $(X,\preccurlyeq )$ will always denote a preordered space.
for all $x,y,z\in X$. If d is a metric on X, we say that $(X,d)$ is a metric space.
The function d is a premetric if it satisfies (${M}_{1}$); a pseudometric if it satisfies (${M}_{1}$), (${M}_{3}$) and (${M}_{4}$); a quasimetric (or a nonsymmetric metric) if it satisfies (${M}_{1}$), (${M}_{2}$) and (${M}_{4}$); a quasipseudometric if it satisfies (${M}_{1}$) and (${M}_{4}$); and a semimetric if it satisfies (${M}_{1}$), (${M}_{2}$) and (${M}_{3}$).
Remark 2.1 We point out that there exist different notions of premetric that are not universally accepted. For instance, Kasahara [26] used the term premetric to refer to a quasipseudometric defined on a subset of $X\times X$. However, Kim [27], in the same issue as Kasahara, preferred using the term quasipseudometric (see also Reilly et al. [28]). For our purposes and for the sake of clarity, we prefer using the previous definitions because we consider that it is a more modern nomenclature.
Definition 2.3 A fixed point of a selfmapping $T:X\to X$ is a point $x\in X$ such that $T(x)=x$. A coincidence point between two mappings $T,g:X\to Y$ is a point $x\in X$ such that $T(x)=g(x)$. A common fixed point of $T,g:X\to X$ is a point $x\in X$ such that $T(x)=g(x)=x$.
Remark 2.2 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.
Definition 2.4 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 (w.r.t. ≼).
3 An illustrative example
 1.
The binary relation ≼ is a preorder on $\mathbb{X}$, but it is not a partial order on $\mathbb{X}$.
 2.
The measure mapping d does not hold any of the four classical properties (${M}_{1}$)(${M}_{4}$) that define a metric space. Indeed, it is not a metric on $\mathbb{X}$, neither a premetric nor any of the following: a pseudometric, a quasimetric, a pseudoquasimetric, a semimetric or a partial metric.
 3.
Even if d would verify some of the metric properties of Definition 2.2, $(\mathbb{X},d)$ would not be a complete space.
 4.
T is not a dcontraction (that is, there is no $k\in [0,1)$ such that $d(Tx,Ty)\le kd(x,y)$ for all $x,y\in X$) because $d(T4,T5)=d(1,1)=2$, but $d(4,5)=1$.
Therefore, none of the theorems proved until now can be applied to the quadruple $(\mathbb{X},\preccurlyeq ,d,T)$ in order to guarantee that T has a fixed point.
4 Test functions
One of the most important ingredients of a contractivity condition is the kind of involved functions. Recently, many classes of families have been introduced, like altering distance functions, comparison functions, $(c)$comparison functions, Geraghty functions, etc. In this section, we present the kind of functions we will use and we show how other classes can be seen as particular cases.
Definition 4.1 (Agarwal et al. [29])
We will denote by ℱ the family of all pairs $(\psi ,\phi )$, where $\psi ,\phi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ are functions, verifying the following three conditions.

(${\mathcal{F}}_{1}$) ψ is nondecreasing.

(${\mathcal{F}}_{2}$) If there exists ${t}_{0}\in [0,\mathrm{\infty})$ such that $\phi ({t}_{0})=0$, then ${t}_{0}=0$ and ${\psi}^{1}(0)=\{0\}$.

(${\mathcal{F}}_{3}$) If $\{{a}_{k}\},\{{b}_{k}\}\subset [0,\mathrm{\infty})$ are sequences such that $\{{a}_{k}\}\to L$, $\{{b}_{k}\}\to L$ and verifying $L<{b}_{k}$ and $\psi ({b}_{k})\le (\psi \phi )({a}_{k})$ for all k, then $L=0$.
Example 4.1 If $k\in [0,1)$ and we define ${\psi}_{k}(t)=t$ and ${\phi}_{k}(t)=(1k)t$ for all $t\ge 0$, then $({\psi}_{k},{\phi}_{k})\in \mathcal{F}$. Furthermore, ${\psi}_{k}(t){\phi}_{k}(t)=kt$ for all $t\ge 0$.
Notice that axiom (${\mathcal{F}}_{2}$) does not necessarily imply the well known condition $\psi (t)=0\iff t=0\iff \phi (t)=0$. Furthermore, we do not impose any continuity condition neither on ψ nor on φ. In order to prove that the family ℱ is very general, next we will show a variety of pairs of functions in ℱ that have been previously considered by other authors in the past.
A function $\varphi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ is lower semicontinuous if $\varphi (t)\le {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}\varphi ({t}_{n})$ for all sequence $\{{t}_{n}\}\subset [0,\mathrm{\infty})$ such that $\{{t}_{n}\}\to t$. Similarly, ϕ is upper semicontinuous if, in the same conditions, $\varphi (t)\ge {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}\varphi ({t}_{n})$.
Definition 4.2 (Khan et al. [30])
An altering distance function is a continuous, nondecreasing function $\varphi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ such that $\varphi (t)=0$ if and only if $t=0$.
Proposition 4.1 If ϕ is an altering distance function and $\{{a}_{m}\}\subset [0,\mathrm{\infty})$ verifies $\{\varphi ({a}_{m})\}\to 0$, then $\{{a}_{m}\}\to 0$.
The following lemma shows some examples of pairs in ℱ.
Lemma 4.1 (see [29])
 1.
If φ is lower semicontinuous and ${\phi}^{1}(\{0\})=\{0\}$, then $(\psi ,\phi )\in \mathcal{F}$.
 2.
If φ is continuous and verifies ${\phi}^{1}(\{0\})=\{0\}$, then $(\psi ,\phi )\in \mathcal{F}$.
 3.
If ψ and φ are altering distance functions, then $(\psi ,\phi )\in \mathcal{F}$.
Notice that the condition $\phi \le \psi $ is not necessary.
Proof We prove item (1). Conditions (${\mathcal{F}}_{1}$) and (${\mathcal{F}}_{2}$) are obvious. Next, assume that $\{{a}_{k}\},\{{b}_{k}\}\subset [0,\mathrm{\infty})$ are sequences such that $\{{a}_{k}\}\to L$, $\{{b}_{k}\}\to L$ and verify $L<{b}_{k}$ and $\psi ({b}_{k})\le (\psi \phi )({a}_{k})$ for all k. Therefore, $\psi ({b}_{k})\le (\psi \phi )({a}_{k})=\psi ({a}_{k})\phi ({a}_{k})\le \psi ({a}_{k})$. Hence $0\le \phi ({a}_{k})\le \psi ({a}_{k})\psi ({b}_{k})$ for all k. Letting $k\to \mathrm{\infty}$ and taking into account that ψ is continuous, we deduce that ${lim}_{k\to \mathrm{\infty}}\phi ({a}_{k})=0$. As $\{{a}_{k}\}\to L$ and φ is lower semicontinuous, we deduce that $\phi (L)\le lim{inf}_{t\to L}\phi (t)\le {lim}_{k\to \mathrm{\infty}}\phi ({a}_{k})=0$. Hence $L=0$. The other two items immediately follow from item 1. □
Example 4.2 (see [29])
 1.
If $a,b>0$ and we define $\psi (t)=at$ and $\phi (t)=bt$ for all $t\ge 0$, then $(\psi ,\phi )\in \mathcal{F}$. The case $a\ge b$ is usually included in other papers, but the case $a<b$ is new.
 2.
If $\psi (t)=\phi (t)=t+1$ for all $t\ge 0$, then $(\psi ,\phi )\in \mathcal{F}$. Notice that, in this case, (${\mathcal{F}}_{3}$) holds because it is impossible to find such kind of sequences since $1\le 1+{b}_{k}=\psi ({b}_{k})\le (\psi \phi )({a}_{k})=0$. In this case, the condition $\psi (t)=0\iff t=0$ does not hold.
Corollary 4.1 Let $\psi ,\varphi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ be two functions such that ψ is an altering distance function and ϕ is upper semicontinuous verifying ${\varphi}^{1}(\{0\})=\{0\}$ and $\varphi (t)<\psi (t)$ for all $t>0$. Then $(\psi ,\phi )\in \mathcal{F}$, where $\phi =\psi \varphi $.
Proof It follows from item 1 of Lemma 4.1 because $\phi =\psi \varphi $ is lower semicontinuous and verifies ${\phi}^{1}(\{0\})=\{0\}$. □
A function $\alpha :[0,\mathrm{\infty})\to [0,1)$ is a Geraghty function if the condition $\{\alpha ({t}_{n})\}\to 1$ implies that $\{{t}_{n}\}\to 0$.
Lemma 4.2 (Agarwal et al. [29])
If α is a Geraghty function and we define $\psi (t)=t$ and $\phi (t)=(1\alpha (t))t$ for all $t\ge 0$, then $(\psi ,\phi )\in \mathcal{F}$.
where ψ is an altering distance function and β is a Geraghty function.
Lemma 4.3 If ψ is an altering distance function and β is a Geraghty function, then $(\psi ,(1\beta \circ \psi )\cdot \psi )\in \mathcal{F}$.
Proof Let $\phi =(1\beta \circ \psi )\cdot \psi $, that is, $\phi (t)=(1\beta (\psi (t)))\psi (t)$ for all $t\ge 0$. Notice that the image of $\beta \circ \psi $ is contained in the image of β, which is in $[0,1)$. Therefore, $\beta (\psi (s))\psi (s)\le \psi (s)$ for all $s\ge 0$ (if $\psi (s)=0$, both members are equal, and if $\psi (s)>0$, then $\beta (\psi (s))\cdot \psi (s)<\varphi (s)$ since $\beta (\psi (s))<1$).
(${\mathcal{F}}_{1}$) Since ψ is an altering distance function, then it is nondecreasing.
(${\mathcal{F}}_{2}$) Assume that there exists ${t}_{0}\in [0,\mathrm{\infty})$ such that $\phi ({t}_{0})=0$. Then $(1\beta (\psi ({t}_{0})))\psi ({t}_{0})=0$. Since $1\beta (\psi ({t}_{0}))>0$, then $\psi ({t}_{0})=0$, which means that ${t}_{0}=0$. In such a case, ${\psi}^{1}(0)=\{0\}$ because it is an altering distance function.
Letting $n\to \mathrm{\infty}$, we deduce that $\{\beta (\psi ({a}_{k}))\}\to 1$. Since β is a Geraghty function, $\psi (L)={lim}_{n\to \mathrm{\infty}}\psi ({a}_{k})=0$, which contradicts that $\psi (L)>0$ because $L>0$ and ψ is an altering distance function. □
ψ is continuous and φ verifies that $\{\phi ({t}_{n})\}\to 0$ implies that $\{{t}_{n}\}\to 0$.
Then $(\psi ,\phi )\in \mathcal{F}$.
Proof (${\mathcal{F}}_{1}$) Since ψ is an altering distance function, then it is nondecreasing.
(${\mathcal{F}}_{2}$) Assume that there exists ${t}_{0}\in [0,\mathrm{\infty})$ such that $\phi ({t}_{0})=0$. Letting ${t}_{n}={t}_{0}$ for all $n\ge 1$ and applying (2), we deduce that ${t}_{0}=0$. In such a case, ${\psi}^{1}(0)=\{0\}$ because it is an altering distance function.
(${\mathcal{F}}_{3}$) Let $\{{a}_{k}\},\{{b}_{k}\}\subset [0,\mathrm{\infty})$ be sequences such that $\{{a}_{k}\}\to L$, $\{{b}_{k}\}\to L$ and verify $L<{b}_{k}$ and $\psi ({b}_{k})\le (\psi \phi )({a}_{k})$ for all k. Hence $0\le \phi ({a}_{k})\le \psi ({a}_{k})\psi ({b}_{k})$ for all k. Letting $k\to \mathrm{\infty}$ and taking into account that ψ is continuous, we deduce that ${lim}_{k\to \mathrm{\infty}}\phi ({a}_{k})=0$. By condition (2), $L={lim}_{k\to \mathrm{\infty}}{a}_{k}=0$. □
A comparison function is a nondecreasing function $\varphi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ such that ${lim}_{n\to \mathrm{\infty}}{\varphi}^{n}(t)=0$ for all $t>0$.
Lemma 4.5 If ϕ is a continuous comparison function, and we define $\psi (t)=t$ and $\phi (t)=t\varphi (t)$ for all $t\ge 0$, then $(\psi ,\phi )\in \mathcal{F}$.
Proof It is clear that every comparison function ϕ verifies $\varphi (t)<t$ for all $t>0$. In such a case, if ϕ is continuous, then $\varphi (0)=0$, so $\varphi (t)\le t$ for all $t\ge 0$. Moreover, if $\varphi (t)=t$, then $t=0$.
(${\mathcal{F}}_{1}$) Since ψ is an altering distance function, then it is nondecreasing.
(${\mathcal{F}}_{2}$) Assume that there exists ${t}_{0}\in [0,\mathrm{\infty})$ such that $\phi ({t}_{0})=0$. Then $\varphi ({t}_{0})={t}_{0}$, so ${t}_{0}=0$. In such a case, ${\psi}^{1}(0)=\{0\}$ because it is an altering distance function.
Letting $k\to \mathrm{\infty}$, we deduce that ${lim}_{k\to \mathrm{\infty}}\varphi ({a}_{k})=L$. Therefore, as ϕ is continuous, $\varphi (L)={lim}_{k\to \mathrm{\infty}}\varphi ({a}_{k})=L$, which is only possible when $L=0$. □
 (a1)
ψ is continuous;
 (a2)
ψ is nondecreasing;
 (a3)
(a3) ${lim}_{n\to \mathrm{\infty}}\varphi ({t}_{n})=0\u27f9{lim}_{n\to \mathrm{\infty}}{t}_{n}=0$.
The condition (a3) was introduced by Popescu in [34] and Moradi and Farajzadeh in [35]. Notice that the above conditions do not determine the values $\psi (0)$ and $\varphi (0)$. If $\varphi <\psi $ in $(0,\mathrm{\infty})$, then $(\psi ,\phi =\psi \varphi )\in \mathcal{F}$. This is the case of Lemma 4.4.
After we have shown many different contexts in which some pairs of ℱ appear, we present some of their useful properties.
Lemma 4.6 (Agarwal et al. [29])
 1.
If $t,s\in [0,\mathrm{\infty})$ and $\psi (t)\le (\psi \phi )(s)$, then either $t<s$ or $t=s=0$. In any case, $t\le s$.
 2.
If $t\in [0,\mathrm{\infty})$ and $\psi (t)\le (\psi \phi )(t)$, then $t=0$.
 3.
If $\{{a}_{k}\},\{{b}_{k}\}\subset [0,\mathrm{\infty})$ are such that $\psi ({a}_{k})\le (\psi \phi )({b}_{k})$ for all k and $\{{b}_{k}\}\to 0$, then $\{{a}_{k}\}\to 0$.
 4.
If $\{{a}_{k}\}\subset [0,\mathrm{\infty})$ and $\psi ({a}_{k+1})\le (\psi \phi )({a}_{k})$ for all k, then $\{{a}_{k}\}\to 0$.
 (i)
for $u,v\in [0,\mathrm{\infty})$, if $\psi (u)\le \varphi (v)$, then $u\le v$;
 (ii)
for $\{{u}_{n}\},\{{v}_{n}\}\subset [0,\mathrm{\infty})$ with ${lim}_{n\to \mathrm{\infty}}{u}_{n}={lim}_{n\to \mathrm{\infty}}{v}_{n}=\omega $, if $\psi ({u}_{n})\le \varphi ({v}_{n})$ for all $n\in \mathbb{N}$, then $\omega =0$.
Pairs of functions in ℱ are intimately related with the class ${\mathcal{F}}_{shi}$ of pairs of shifting distance functions, but they are different. On the one hand, pairs in ${\mathcal{F}}_{shi}$ can take values in ℝ, but pairs in ℱ take values in $[0,\mathrm{\infty})$. On the other hand, if a pair $(\psi ,\phi )$ verifies (${\mathcal{F}}_{1}$) and (${\mathcal{F}}_{2}$), then the pair $(\psi ,\varphi =\psi \phi )$ satisfies (i). Furthermore, if $(\psi ,\varphi =\psi \phi )$ satisfies (ii), then $(\psi ,\phi )$ satisfies (${\mathcal{F}}_{3}$).
5 A fixed point theorem without an underlying metric structure
The main aim of this section is to show sufficient conditions in order to ensure that T and g (given in Section 3) have a coincidence point. To set the framework, throughout this section, let $(X,\preccurlyeq )$ be a preordered space, and let $d:X\times X\to \mathbb{R}$ and $T,g:X\to X$ be three mappings. The following definitions are usually considered when X has a metric structure. However, we do not suppose, a priori, any condition on the mapping d. Indeed, we will only be able to prove that d takes nonnegative values as a consequence of a particular version of the triangular inequality. However, in general, we do not consider necessary to assume this sign constraint.
Definition 5.1 We will say that a sequence $\{{x}_{m}\}\subseteq X$:

dconverges to ${x}_{0}\in X$ (and we will write $\{{x}_{m}\}\stackrel{d}{\to}{x}_{0}$ or simply $\{{x}_{m}\}\to {x}_{0}$) if for all $\epsilon >0$ there exists ${m}_{0}\in \mathbb{N}$ such that $d({x}_{m},{x}_{0})\le \epsilon $ for all $m\ge {m}_{0}$;

is dCauchy if for all $\epsilon >0$ there exists ${m}_{0}\in \mathbb{N}$ such that $d({x}_{m},{x}_{{m}^{\prime}})\le \epsilon $ for all ${m}^{\prime}\ge m\ge {m}_{0}$.
We will say that $(X,d)$ is complete if every dCauchy sequence in X is dconvergent in X.
With respect to the previous notions, the following remarks must be done.
Remark 5.1

When the distance measure d is not symmetric (that is, it does not verify axiom (${M}_{3}$)), the definition of convergence or Cauchyness of sequences usually depends on the side, because $d({x}_{n},{x}_{m})$ and $d({x}_{m},{x}_{n})$ can be different. This is the case, for instance, of quasimetric spaces. In such cases, the previous definitions correspond to the idea of rightconvergence and rightCauchyness (see Jleli and Samet [37]) because the more advanced term (which is nearer to the limit) is placed at the right argument of d. Similarly, it can be defined the notions of leftconvergence (using $d({x}_{0},{x}_{m})$) and leftCauchyness (using $d({x}_{{m}^{\prime}},{x}_{m})$) of sequences. Notice that some of the concepts we will present can also be introduced by the right side or by the left side. However, in order not to complicate the notation, we prefer avoiding the term right in all definitions and theorems.

In [[38], Section 3], the authors did a complete study (completion, topology and powerdomains) of spaces verifying axioms (${M}_{1}$) and (${M}_{4}$), which they called generalized metric spaces. They solved the previous discussion using the terms forward convergent sequences (for rightconvergent sequences) and backward convergent sequences (for leftconvergent sequences). However, as we will not assume (${M}_{1}$) nor (${M}_{4}$), we also prefer avoiding these prefixes.

Notice that if d does not verify (${M}_{2}$), then the limit of a sequence, if there exists, might not be unique.

And if d only takes nonpositive values, then all sequences converge to all points.
Similarly can be defined the concepts of $(d,\preccurlyeq )$ nonincreasingclosed set and $(d,\preccurlyeq )$ monotoneclosed set, and, more generally, a dclosed set, when any dlimit of any convergent sequence of points of A is also in A.
Definition 5.3 A mapping $T:X\to X$ is $(d,\preccurlyeq )$ nondecreasingcontinuous at ${x}_{0}\in X$ if we have that $\{T{x}_{m}\}$ dconverges to $T{x}_{0}$ for all ≼nondecreasing sequence $\{{x}_{m}\}$ dconvergent to ${x}_{0}$.
In a similar way, the concepts of $(d,\preccurlyeq )$ nonincreasingcontinuous mapping and $(d,\preccurlyeq )$ monotonecontinuous mapping may be considered and, more generally, a dcontinuous mapping, when $\{T{x}_{m}\}$ dconverges to $T{x}_{0}$ for all sequence $\{{x}_{m}\}$ dconvergent to ${x}_{0}$.
Definition 5.4 We will say that a point $x\in X$ is a dprecoincidence point of T and g if $d(Tx,gx)=d(gx,Tx)=0$.
 (a)
$T(X)\subseteq g(X)$.
 (b)
T is $(g,\preccurlyeq )$nondecreasing.
 (c)
There exists ${x}_{0}\in X$ such that $g{x}_{0}\preccurlyeq T{x}_{0}$.
 (d)There exists $(\psi ,\phi )\in \mathcal{F}$ such that$\psi (d(Tx,Ty))\le (\psi \phi )(d(gx,gy))\phantom{\rule{1em}{0ex}}\text{for all}x,y\in X\text{for which}gx\preccurlyeq gy.$(3)
 (e)
$d(x,z)\le d(x,y)+d(y,z)$ for all $x,y,z\in X$ such that $x\preccurlyeq y\preccurlyeq z$.
 (f)
$d(x,y)\le d(x,z)+d(y,z)$ for all $x,y,z\in X$ such that $x\preccurlyeq y\preccurlyeq z$.
 (g)
Every ≼nondecreasing, dCauchy sequence in X is dconvergent in X.
 (h)
If $\{{x}_{m}\}$ is a nondecreasing sequence and $\{{x}_{m}\}$ dconverges to $x\in X$, then ${x}_{m}\preccurlyeq x$ for all m.
 (i)
$d(x,z)\le d(y,x)+d(y,z)$ for all $x,y,z\in X$ such that $y\preccurlyeq x$ and $y\preccurlyeq z$.
 (j)
Every dprecoincidence point of T and g is a coincidence point of T and g (that is, if $d(Tx,gx)=d(gx,Tx)=0$, then $Tx=gx$).
As we have shown in Section 3, notice that a mapping d verifying (a)(i) does not have to verify any of the conditions that define a metric on X.
Remark 5.2 A priori, d can take negative values in ℝ. However, condition (f) lets us prove some constraints about the sign of d. Indeed, if we take $x=y=z$ in condition (f), we deduce that $d(x,x)\ge 0$ for all $x\in X$. Furthermore, if $y=x$ in (f), it follows that $d(x,z)\ge 0$ for all $x,z\in X$ such that $x\preccurlyeq z$. This does not mean that d is nonnegative because d could take negative values when $z\prec x$ or x and z are not ≼comparable.
Remark 5.3 As we shall show in the proofs, the mapping d could only be considered on the set $\mathrm{\Omega}=\{(x,y)\in {X}^{2}:x\preccurlyeq y\}$, that is, we will only use $d{}_{\mathrm{\Omega}}:\mathrm{\Omega}\to \mathbb{R}$. In this case, the previous remark shows that, as usual, $d(\mathrm{\Omega})\subseteq [0,\mathrm{\infty}[$.
Proof Since $T{x}_{0}\in T(X)\subseteq g(X)$, there exists ${x}_{1}\in X$ such that $T{x}_{0}=g{x}_{1}$. Then $g{x}_{0}\preccurlyeq T{x}_{0}=g{x}_{1}$. Since T is $(g,\preccurlyeq )$nondecreasing, $T{x}_{0}\preccurlyeq T{x}_{1}$. Now $T{x}_{1}\in T(X)\subseteq g(X)$, so there exists ${x}_{2}\in X$ such that $T{x}_{1}=g{x}_{2}$. Then $g{x}_{1}=T{x}_{0}\preccurlyeq T{x}_{1}=g{x}_{2}$. Since T is $(g,\preccurlyeq )$nondecreasing, $T{x}_{1}\preccurlyeq T{x}_{2}$. Repeating this argument, there exists a sequence ${\{{x}_{m}\}}_{m\ge 0}$ such that $g{x}_{m+1}=T{x}_{m}\preccurlyeq T{x}_{m+1}=g{x}_{m+2}$ for all $m\ge 0$. □
Theorem 5.2 Let $(X,\preccurlyeq )$ be a preordered space and let $d:X\times X\to \mathbb{R}$ and $T,g:X\to X$ be three mappings verifying (a)(e). Then any sequence ${\{g{x}_{m}\}}_{m\ge 0}$ such that $T{x}_{m}=g{x}_{m+1}$ for all $m\ge 0$, is dCauchy ($\{{x}_{m}\}$ is given as in Theorem 5.1).
Letting $k\to \mathrm{\infty}$ and using (4) and (6), we deduce that the sequence ${\{{a}_{k}=d(g{x}_{n(k)1},g{x}_{m(k)1})\}}_{k\ge 1}$ also verifies $\{{a}_{k}\}\to {\epsilon}_{0}$, and by (8), we have that $\psi ({b}_{k})\le (\psi \phi )({a}_{k})$ for all k. Since $(\psi ,\phi )\in \mathcal{F}$, axiom (${\mathcal{F}}_{3}$) guarantees that ${\epsilon}_{0}=0$, which is a contradiction with the fact that ${\epsilon}_{0}>0$. This contradiction shows that $\{g{x}_{m}\}$ is a dCauchy sequence. □
After the previous technical results, we give the main results of this manuscript.
Theorem 5.3 Let $(X,\preccurlyeq )$ be a preordered space and let $d:X\times X\to \mathbb{R}$ and $T,g:X\to X$ be three mappings which fulfil conditions (a)(h). Assume that the following condition holds.
(p) $g(X)$ is $(d,\preccurlyeq )$nondecreasingclosed.
Then there exists $z\in X$ such that the sequence $\{g{x}_{m}\}$ (defined in Theorem 5.2) dconverges to gz and to Tz. Furthermore, if (i) holds, then z is a dprecoincidence point of T and g.
Notice that, in the previous result, g and T need not be continuous.
Since $\{d(g{x}_{m},y)\}\to 0$, item 3 of Lemma 4.6 guarantees that $\{d(g{x}_{m+2},Tz)\}\to 0$, that is, $\{g{x}_{m+2}\}$ dconverges to Tz.
Therefore $d(gz,Tz)=d(Tz,gz)=0$. □
 (p′)T and g are $(d,\preccurlyeq )$nondecreasingcontinuous and commuting and, at least, one of the following conditions holds:

(${\mathrm{p}}_{1}^{\prime}$) T is a ≼nondecreasing mapping.

(${\mathrm{p}}_{2}^{\prime}$) g is a ≼nondecreasing mapping.

(${\mathrm{p}}_{3}^{\prime}$) If $z,\omega \in X$ and $\{{z}_{m}\}\subseteq X$ is a ≼nondecreasing sequence such that $\{g{z}_{m}\}$ dconverges to z and to ω at the same time, then $d(z,\omega )=d(\omega ,z)=0$.

Then there exists $y\in X$ such that the sequence $\{g{y}_{m}\}$ (defined in Theorem 5.2) dconverges to gy and to Ty at the same time. Furthermore, if (i) holds, then y is a dprecoincidence point of T and g.
In any case, T and g have, at least, a dprecoincidence point.
Proof Theorem 5.2 guarantees that $\{g{x}_{m+1}\}$ is dCauchy. Since it is ≼nondecreasing, condition (g) implies that there exists $y\in X$ such that $\{g{x}_{m}\}$ dconverges to y. Thus, taking into account that g and T are $(d,\preccurlyeq )$nondecreasingcontinuous, $\{gg{x}_{m}\}$ dconverges to gy and $\{Tg{x}_{m}\}$ dconverges to Ty. Furthermore, since T and g are commuting, $Tg{x}_{m+1}=gT{x}_{m+1}=gg{x}_{m+1}$ for all m, which means that $\{gg{x}_{m}\}$ dconverges, at the same time, to gy and to Ty.
Next, assume that (i) holds, and we claim that y is a dprecoincidence point of T and g. Firstly, if T (or g) is a ≼nondecreasing mapping, then the sequence $\{gg{x}_{m}\}=\{Tg{x}_{m1}\}$ is ≼nondecreasing. Since it dconverges to gy and to Ty, property (h) implies that $gg{x}_{m}\preccurlyeq gy$ and $gg{x}_{m}\preccurlyeq Ty$ for all m. Reasoning as in (9), we conclude that y is a dprecoincidence point of T and g. Secondly, if T and g are not necessarily ≼nondecreasing mappings, we could apply (${\mathrm{p}}_{3}^{\prime}$) to the sequence $\{g{x}_{m}\}$ in order to deduce that $d(gy,Ty)=d(Ty,gy)=0$, that is, y is a dprecoincidence point of T and g. □
We summarize and improve all previous results in the following theorem.
 (a)
$T(X)\subseteq g(X)$.
 (b)
T is $(g,\preccurlyeq )$nondecreasing.
 (c)
There exists ${x}_{0}\in X$ such that $g{x}_{0}\preccurlyeq T{x}_{0}$.
 (d)There exist $(\psi ,\phi )\in \mathcal{F}$ such that$\psi (d(Tx,Ty))\le (\psi \phi )(d(gx,gy))\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}x,y\in X\mathit{\text{for which}}gx\preccurlyeq gy.$
 (e)
$d(x,z)\le d(x,y)+d(y,z)$ for all $x,y,z\in X$ such that $x\preccurlyeq y\preccurlyeq z$.
 (f)
$d(x,y)\le d(x,z)+d(y,z)$ for all $x,y,z\in X$ such that $x\preccurlyeq y\preccurlyeq z$.
 (g)
Every ≼nondecreasing, dCauchy sequence in X is dconvergent in X.
 (h)
If $\{{x}_{m}\}$ is a ≼nondecreasing sequence and $\{{x}_{m}\}$ dconverges to $x\in X$, then ${x}_{m}\preccurlyeq x$ for all m.
 (i)
$d(x,z)\le d(y,x)+d(y,z)$ for all $x,y,z\in X$ such that $y\preccurlyeq x$ and $y\preccurlyeq z$.
 (j)
Every dprecoincidence point of T and g is a coincidence point of T and g.
 (p)
$g(X)$ is $(d,\preccurlyeq )$nondecreasingclosed, or

(${\mathrm{p}}_{1}^{\prime}$) T is a ≼nondecreasing mapping.

(${\mathrm{p}}_{2}^{\prime}$) g is a ≼nondecreasing mapping.

(${\mathrm{p}}_{3}^{\prime}$) If $z,\omega \in X$ and $\{{z}_{m}\}\subseteq X$ is a ≼nondecreasing sequence such that $\{g{z}_{m}\}$ dconverges to z and to ω at the same time, then $d(z,\omega )=d(\omega ,z)=0$.
Then T and g have, at least, a coincidence point.
Notice that condition (j) does not mean that $d(x,y)=0$ implies $x=y$.
Remark 5.4 If g is the identity mapping on the ambient space, then the quadruple $(\mathbb{X},\preccurlyeq ,d,T)$ introduced in Section 3 verifies all conditions (a)(j) and (p)(${\mathrm{p}}^{\prime}$) (see more details in Appendix 2).
Remark 5.5 Obviously, similar results can be stated changing the following hypothesis.
($\tilde{b}$) T is $(g,\preccurlyeq )$nonincreasing.
($\tilde{c}$) There exists ${x}_{0}\in X$ such that $g{x}_{0}\succcurlyeq T{x}_{0}$.
($\tilde{g}$) Every nonincreasing, dCauchy sequence in X is dconvergent in X.
($\tilde{h}$) If $\{{x}_{m}\}$ is a nonincreasing sequence and $\{{x}_{m}\}$ dconverges to $x\in X$, then ${x}_{m}\succcurlyeq x$ for all m.
($\tilde{p}$) $g(X)$ is $(d,\preccurlyeq )$nonincreasingclosed.
(${\tilde{p}}^{\prime}$) T and g are $(d,\preccurlyeq )$nonincreasingcontinuous and commuting and, at least, one of the following conditions holds:
(${\tilde{p}}_{1}^{\prime}$) T is a ≼nonincreasing mapping.
(${\tilde{p}}_{2}^{\prime}$) g is a ≼nonincreasing mapping.
(${\tilde{p}}_{3}^{\prime}$) If $z,\omega \in X$ and $\{{z}_{m}\}\subseteq X$ is a ≼nonincreasing sequence such that $\{g{z}_{m}\}$ dconverges to z and to ω at the same time, then $d(z,\omega )=d(\omega ,z)=0$.
The unicity of the coincidence point cannot be guaranteed unless additional conditions are imposed. A result in this direction is the following.
Theorem 5.6 Under the hypothesis of Theorem 5.5, let $x,y\in X$ be two coincidence points of T and g verifying that there exists $u\in X$ such that $gu\preccurlyeq gx$ and $gu\preccurlyeq gy$. Then $d(Tx,Ty)=d(Ty,Tx)=d(gx,gy)=d(gy,gx)=0$.
Proof Define ${u}_{0}=u$. Since $T{u}_{0}\in T(X)\subseteq g(X)$, there exists ${u}_{1}\in X$ such that $g{u}_{1}=T{u}_{0}$. Repeating this process, there exists a sequence ${\{{u}_{m}\}}_{m\ge 0}$ such that $g{u}_{m+1}=T{u}_{m}$ for all $m\ge 0$. We claim that $\{g{u}_{m}\}\stackrel{d}{\to}gx$ and $\{g{u}_{m}\}\stackrel{d}{\to}gy$. Firstly, we reason using x, but the same argument is valid for y.
Indeed, notice that $g{u}_{0}=gu\preccurlyeq gx$. As T is $(g,\preccurlyeq )$nondecreasing, then $T{u}_{0}\preccurlyeq Tx=gx$, which means that $g{u}_{1}\preccurlyeq gx$. Again, $g{u}_{1}\preccurlyeq gx$ implies $T{u}_{1}\preccurlyeq Tx=gx$, which means that $g{u}_{2}\preccurlyeq gx$. By induction, it is possible to prove that $g{u}_{m}\preccurlyeq gx$ for all $m\ge 0$. Using condition (d), it follows that $\psi (d(g{u}_{m+1},gx))=\psi (d(T{u}_{m},Tx))\le (\psi \phi )(d(g{u}_{m},gx))$ for all $m\ge 0$. Thus, by item 4 of Lemma 4.6, $\{d(g{u}_{m},gx)\}\to 0$. The same argument proves that $g{u}_{m}\preccurlyeq gy$ for all $m\ge 0$ and $\{d(g{u}_{m},gy)\}\to 0$. As a consequence, by (i), $d(Tx,Ty)=d(gx,gy)\le d(g{u}_{m},gx)+d(g{u}_{m},gy)$ for all m, which lets us conclude that $d(Tx,Ty)=d(gx,gy)=0$. □
Corollary 5.1 Under the hypothesis of Theorem 5.5, assume the following conditions.

For all coincidence points $x,y\in X$ of T and g, there exists $u\in X$ such that $gu\preccurlyeq gx$ and $gu\preccurlyeq gy$.

g is injective on the set of all coincidence points of T and g.

If $z,\omega \in T(X)$ verify $d(z,\omega )=d(\omega ,z)=0$, then $z=\omega $.
Then T and g have a unique coincidence point. Furthermore, if T and g are commuting, it is a common fixed point of T and g.
Proof Let $x,y\in X$ be two coincidence points of T and g. Then $gx=Tx\in T(X)$ and $gy=Ty\in T(X)$. By Theorem 5.6, $d(gx,gy)=d(gy,gx)=0$. Therefore $gx=gy$. As g is injective on the set of all coincidence points of T and g, we conclude that $x=y$.
Now let $x\in X$ be a coincidence point of T and g, and let $z=Tx$. By Remark 2.2, z is also a coincidence point of T and g. Then $x=z=Tx=gx$, so x is a common fixed point of T and g. □
Taking $\psi (t)=t$ and $\phi (t)=(1k)t$ for all $t\ge 0$ in the previous results, we obtain the following particular case.
Corollary 5.2 Theorems 5.1, 5.2, 5.3, 5.4, 5.5, 5.6 and Corollary 5.1 also hold if we replace condition (d) by the following one.
(${\mathrm{d}}^{\prime}$) There exists $k\in [0,1)$ such that $d(Tx,Ty)\le k\phantom{\rule{0.3em}{0ex}}d(gx,gy)$ for all $x,y\in X$ for which $gx\preccurlyeq gy$.
6 Consequences
This section is devoted to show how to apply Theorem 5.5 in many different contexts, and how to deduce unidimensional, coupled, tripled, quadruple and multidimensional fixed point theorems (for completeness, they are included in Appendix 1).
6.1 Fixed/coincidence point theorems in partially ordered metric spaces
In this subsection we show that different results in partially ordered metric spaces, including unidimensional, coupled, tripled, quadruple and multidimensional fixed point theorems, can be seen as simple consequences of Theorem 5.5.
Corollary 6.1 Theorems A.1, A.2 and A.3 follow from Theorem 5.5.
However, Theorems A.1 and A.2 cannot be applied to the example of Section 3.
Corollary 6.2 Theorem A.4 follows from Theorem 5.5.
Theorem 5.5 assures us that ${T}_{F}$ and G have a coincidence point, that is, F has a coupled fixed point. □
Tripled, quadruple and multidimensional theorems can be proved similarly using ${X}^{3}$, ${X}^{4}$ and ${X}^{n}$, obtaining the following result.
Corollary 6.3 Theorems A.5, A.6 and A.7 follow from Theorem 5.5.
We remark that the techniques used in this paper might be applied in order to prove other coupled, tripled, quadruple, ntupled fixed point theorems in the framework of various abstract spaces, e.g., partial metric spaces, cone metric spaces, fuzzy metric spaces, bmetric spaces, etc.
6.2 Fixed/coincidence point theorems in quasimetric spaces

(${q}_{1}$) $q(x,y)=0$ 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. Some preliminaries about convergence, Cauchy sequences and completeness in quasimetric spaces can be found in [29, 37].
Then T and g have a unique coincidence point.
which is a contradiction. □
If g is the identity mapping on X, we have the following statement.
Then T has a unique fixed point.
If $\psi (t)=t$ for all $t\ge 0$, we have the following particular case.
Corollary 6.5 (Jleli and Samet [37], Theorem 3.2)
where $\phi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ is continuous with ${\phi}^{1}(0)=\{0\}$. Then T has a unique fixed point.
6.3 Fixed/coincidence point theorems in Gmetric spaces
Following [7, 39], recall that a generalized metric on X (or, more specifically, a Gmetric on X) is a mapping $G:X\times X\times X\to [0,\mathrm{\infty})$ verifying the following properties.
Definition 6.1
(${G}_{1}$) $G(x,y,z)=0$ 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 $x,y,z,a\in X$.
In such a case, the pair $(X,G)$ is called a Gmetric space. Some preliminaries about convergence, Cauchy sequences and completeness in quasimetric spaces can be found in [7, 29, 37, 39].
Lemma 6.1 (see, e.g., [29, 37])
 1.${q}_{G}$ and ${q}_{G}^{\prime}$ are quasimetrics on X. Moreover,${q}_{G}^{\prime}(x,y)\le 2{q}_{G}(x,y)\le 4{q}_{G}^{\prime}(x,y)\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}x,y\in X.$(10)
 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.
Then T and g have a unique coincidence point.
Proof It follows from Theorem 6.1 applied to the quasimetric ${q}_{G}(x,y)=G(x,y,y)$ for all $x,y\in X$ (as in Lemma 6.1). □
To conclude the paper, we include two appendices: in the first one, we recall some celebrated theorems that can be seem as particular cases of our main results; in the second one, we prove the statements announced in Section 3 and why our results can be applied.
Appendix 1: Some recent results we generalize
The following statements are wellknown fixed point theorems in partially ordered metric spaces.
Theorem A.1 (Ran and Reurings [17])
 (a)
$(X,d)$ is complete.
 (b)
T is nondecreasing (w.r.t. ≼).
 (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.
Nieto and RodríguezLópez [18] slightly modified the hypothesis of the previous result obtaining the following theorem.
Theorem A.2 (Nieto and RodríguezLópez [18])
 (a)
$(X,d)$ is complete.
 (b)
T is nondecreasing (w.r.t. ≼).
 (c)
If a nondecreasing sequence $\{{x}_{m}\}$ in X converges to some point $x\in X$, then ${x}_{m}\preccurlyeq x$ for all m.
 (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.
Theorem A.3 (Harjani and Sadarangani [40])
 (i)
T is continuous, or
 (ii)
if a nondecreasing sequence $\{{x}_{n}\}$ in X converges to some point $x\in X$, then ${x}_{n}\preccurlyeq x$ for all n.
If there exists ${x}_{0}\in X$ with ${x}_{0}\preccurlyeq T{x}_{0}$, then T has a fixed point.
Theorem A.4 (Bhaskar and Lakshmikantham [19])
 (i)
$(X,d)$ is complete;
 (ii)
F has the mixed monotone property;
 (iii)F is continuous or X has the following properties:

(${X}_{1}$) if a nondecreasing sequence $\{{x}_{n}\}$ in X converges to some point $x\in X$, then ${x}_{n}\preccurlyeq x$ for all n,

(${X}_{2}$) if a decreasing sequence $\{{y}_{n}\}$ in X converges to some point $y\in X$, then ${y}_{n}\succcurlyeq y$ for all n;

 (iv)
there exist ${x}_{0},{y}_{0}\in X$ such that ${x}_{0}\preccurlyeq F({x}_{0},{y}_{0})$ and ${y}_{0}\succcurlyeq F({y}_{0},{x}_{0})$;
 (v)there exists a constant $k\in (0,1)$ such that for all $(x,y),(u,v)\in X\times X$ with $x\succcurlyeq u$ and $y\preccurlyeq v$,$d(F(x,y),F(u,v))\le \frac{k}{2}[d(x,u)+d(y,v)].$
Then F has a coupled fixed point $({x}^{\ast},{y}^{\ast})\in X\times X$. Moreover, if for all $(x,y),(u,v)\in X\times X$ there exists $({z}_{1},{z}_{2})\in X\times X$ such that $(x,y){\preccurlyeq}_{2}({z}_{1},{z}_{2})$ and $(u,v){\preccurlyeq}_{2}({z}_{1},{z}_{2})$, we have uniqueness of the coupled fixed point and ${x}^{\ast}={y}^{\ast}$.
Theorem A.5 (Berinde and Borcut [22])
 (a)
if a nondecreasing sequence $\{{x}_{m}\}\to x$, then ${x}_{m}\preccurlyeq x$ for all m;
 (b)
if a nondecreasing sequence $\{{y}_{m}\}\to y$, then ${y}_{m}\preccurlyeq y$ for all m.
A quadruple version was obtained by Karapınar and Luong in [23].
Theorem A.6 (Karapınar and Luong [23])
 (a)
if a nondecreasing sequence $\{{x}_{m}\}\to x$, then ${x}_{m}\preccurlyeq x$ for all m;
 (b)
if a nondecreasing sequence $\{{y}_{m}\}\to y$, then ${y}_{m}\preccurlyeq y$ for all m.
Later, Berzig and Samet extended the previous result to the multidimensional case in the following way.
Theorem A.7 (Berzig and Samet [24])
Theorem A.8 (Dutta and Choudhury [41])
for all $x,y\in X$, where $\psi ,\phi :[0,\mathrm{\infty}[\to \mathrm{\infty}$ are both continuous and monotone nondecreasing functions with $\psi (t)=\phi (t)=0$ if and only if $t=0$. Then T has a unique fixed point.
Theorem A.9 (Luong and Thuan [42])
 (a)
F is continuous, or
 (b)X has the following properties:
 (i)
if a nondecreasing sequence $\{{x}_{m}\}\to x$, then ${x}_{m}\preccurlyeq x$ for all m,
 (ii)
if a nonincreasing sequence $\{{y}_{m}\}\to y$, then ${y}_{m}\succcurlyeq y$ for all m.
 (i)
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 in X.
Appendix 2: Proof of statements of Section 3 and Remark 5.4
We firstly prove some properties of this space.
If $x\preccurlyeq y\preccurlyeq z$, then either $x,y,z\in \mathbb{I}$ or $x,y,z\in \{2,3\}$ or $x=y=z\in \{4,5,6,7\}$. The same is true if $y\preccurlyeq x$ and $y\preccurlyeq z$.
(${P}_{2}$) d does not verify (${M}_{1}$) because $d(5,5)=0.5$.
(${P}_{3}$) d does not verify (${M}_{2}$) since $d(2,3)=0$ but $2\ne 3$.
(${P}_{4}$) d does not verify (${M}_{3}$) since $d(6,7)=1$ and $d(7,6)=2$.
(${P}_{5}$) d does not verify (${M}_{4}$): if $x=2$, $y=4$ and $z=3$, then $d(x,y)=2$ and $d(z,x)+d(z,y)=1$.
(${P}_{6}$) $(\mathbb{X},d)$ is not complete. Since $\mathbb{I}=\phantom{\rule{0.2em}{0ex}}]2,1]\subset \mathbb{X}$ and $d{}_{\mathbb{I}\times \mathbb{I}}$ is the Euclidean metric on $\mathbb{I}$, then the sequence ${\{{x}_{m}=2+1/m\}}_{m\in \mathbb{N}}$ is dCauchy but it is not dconvergent in $\mathbb{X}$.
(${P}_{7}$) If $\{{x}_{m}\}$ is a ≼nondecreasing sequence in $\mathbb{X}$, then one, and only one, of the following cases holds.
⊳ ${x}_{m}\in \mathbb{I}$ for all m.
⊳ ${x}_{m}\in \{2,3\}$ for all m.
⊳ There exists ${z}_{0}\in \{4,5,6,7\}$ such that ${x}_{m}={z}_{0}$ for all m (that is, ${x}_{m}$ is a constant sequence).
It follows from that fact that points of $\mathbb{I}$ (respectively, $\{2,3\}$, $\{4,5,6,7\}$) are only ≼related with points of $\mathbb{I}$ (respectively, $\{2,3\}$, themselves), and ${x}_{1}\preccurlyeq {x}_{m}$ for all $m\in \mathbb{N}$.
(${P}_{8}$) If $\{{x}_{m}\}$ is a ≼nondecreasing, dconvergent sequence in $\mathbb{X}$, then one, and only one, of the following cases holds.
⊳ ${x}_{m}\in \mathbb{I}$ for all m. In this case, its dlimit is also in $\mathbb{I}$.
⊳ ${x}_{m}\in \{2,3\}$ for all m. In this case, its dlimit is also in $\{2,3\}$.
⊳ There exists ${z}_{0}\in \{4,6,7\}$ such that ${x}_{m}={z}_{0}$ for all m. In this case, its dlimit is ${z}_{0}$.
It follows from (${P}_{7}$). Notice that ${z}_{0}\ne 5$ since $d(5,5)=0.5>0$.
(${P}_{9}$) T is not a dcontraction (that is, there is no $k\in [0,1)$ such that $d(Tx,Ty)\le kd(x,y)$ for all $x,y\in \mathbb{X}$) because $d(T4,T5)=d(1,1)=2$ but $d(4,5)=1$.
 (a)
$T(\mathbb{X})\subseteq \mathbb{X}$. It is obvious.
 (b)
T is ≼nondecreasing. Let $x,y\in \mathbb{X}$ be such that $x\preccurlyeq y$ and $x\ne y$. By (${P}_{1}$), if $x\in \mathbb{I}$, then $y\in \mathbb{I}$ and $x\le y$. Then $Tx=x/2\le y/2=Ty$, being $Tx,Ty\in \mathbb{I}$, so $Tx\preccurlyeq Ty$. If $x\in \{2,3\}$, then $y\in \{2,3\}$, so $Tx=Ty$. The case $x\in \{4,5,6,7\}$ is impossible since $y\ne x$.
 (c)
There exists ${x}_{0}\in \mathbb{X}$ such that ${x}_{0}\preccurlyeq T{x}_{0}$. If ${x}_{0}=1$, then ${x}_{0}=1\preccurlyeq 1/2=T{x}_{0}$.
 (d)
There exists $k\in [0,1[$ such that $d(Tx,Ty)\le kd(x,y)$ for all $x,y\in X$ for which $x\preccurlyeq y$. Let $k=1/2$ and assume that $x,y\in X$ are such that $x\preccurlyeq y$. By (${P}_{1}$), if $x\in \mathbb{I}$, then $y,Tx,Ty\in \mathbb{I}$ and $d(Tx,Ty)=x/2y/2=xy/2=(1/2)d(x,y)$. If $x\in \{2,3\}$, then $y\in \{2,3\}$ and $d(Tx,Ty)=0=d(x,y)$. If $x\in \{4,5,6,7\}$, then $y=x$ and $Tx=Ty\in \{1,0,1\}\subset \mathbb{I}$. Therefore $d(Tx,Ty)=0$.
 (e)
$d(x,z)\le d(x,y)+d(y,z)$ for all $x,y,z\in \mathbb{X}$ such that $x\preccurlyeq y\preccurlyeq z$. By (${P}_{1}$), there are only three cases. If $x,y,z\in \mathbb{I}$, the triangular inequality holds because it is true in ℝ provided with the Euclidean metric. If $x,y,z\in \{2,3\}$, then all distances are zero. Finally, if $x=y=z\in \{4,5,6,7\}$, all distances are either zero (if $x=y=z\in \{4,6,7\}$) or 0.5 (if $x=y=z=5$).
 (f)
$d(x,y)\le d(x,z)+d(y,z)$ for all $x,y,z\in \mathbb{X}$ such that $x\preccurlyeq y\preccurlyeq z$. It is similar to the previous one using (${P}_{1}$).
 (g)
Every dCauchy, ≼nondecreasing sequence in $\mathbb{X}$ is dconvergent. It follows from (${P}_{7}$).
 (h)
If $\{{x}_{m}\}$ is a ≼nondecreasing sequence and $\{{x}_{m}\}$ dconverges to $x\in \mathbb{X}$, then ${x}_{m}\preccurlyeq x$ for all m. It also follows from (${P}_{7}$).
 (i)
$d(x,z)\le d(y,x)+d(y,z)$ for all $x,y,z\in \mathbb{X}$ such that $y\preccurlyeq x$ and $y\preccurlyeq z$. It is similar to (e) using (${P}_{1}$).
 (j)
Every dprecoincidence point of T and g is a coincidence point of T and g. Assume that $d(Tx,gx)=d(gx,Tx)=0$. Since $Tx\in T(\mathbb{X})\subset \mathbb{I}$, it is impossible $Tx,gx\in \{2,3\}$ or $Tx,gx\in \{4,5\}$ (the only cases, far from $\mathbb{I}$, in which the distance between them can be zero). Then $d(Tx,gx)=0$ implies that $Tx,gx\in \mathbb{I}$ and $0=d(Tx,gx)=Txgx$, so $Tx=gx$.
 (p)
$g(\mathbb{X})=\mathbb{X}$ is ≼nondecreasing dclosed. It is immediate: indeed, $g(\mathbb{X})=\mathbb{X}$ is dclosed.
(p′) T and g are $(d,\preccurlyeq )$ nondecreasingcontinuous and commuting, and g is ≼nondecreasing. It is only necessary to prove that T is $(d,\preccurlyeq )$nondecreasingcontinuous. Actually, it follows from (${P}_{8}$) since there are only three cases: if ${x}_{m}\in \mathbb{I}$ for all m, its dlimit ${x}_{0}$ is also in $\mathbb{I}$, and $\{T{x}_{m}={x}_{m}/2\}$ dconverges to $T{x}_{0}={x}_{0}/2$; if ${x}_{m}\in \{2,3\}$ for all m, then ${x}_{0}\in \{2,3\}$ and $T{x}_{m}=0=T{x}_{0}$ for all m; finally, if there exists ${z}_{0}\in \{4,6,7\}$ such that ${x}_{m}={z}_{0}$ for all m, then ${x}_{0}={z}_{0}={x}_{m}$ for all m.
We point out that T is not dcontinuous: if ${x}_{m}=4$ and ${x}_{0}=5$, then $\{{x}_{m}\}\to {x}_{0}$ but $\{T{x}_{m}=T4=1\}$ does not dconverge to $T5=1$.
Declarations
Acknowledgements
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The second author acknowledges with thanks DSR for financial support. The first author has been partially supported by Junta de Andalucía by project FQM268 of the Andalusian CICYE.
Authors’ Affiliations
References
 Brooks, RM, Schmitt, K: The contraction mapping principle and some applications. Electron. J. Differ. Equ., Monograph 09 (2009). Available in http://ejde.math.txstate.edu or http://ejde.math.unt.edu.Google Scholar
 Aydi H: Some coupled fixed point results on partial metric spaces. Int. J. Math. Math. Sci. 2011., 2011: Article ID 647091Google Scholar
 Shatanawi W, Samet B, Abbas M: Coupled fixed point theorems for mixed monotone mappings in ordered partial metric spaces. Math. Comput. Model. 2012, 55(3–4):680–687. 10.1016/j.mcm.2011.08.042View ArticleMathSciNetGoogle Scholar
 Agarwal RP, Karapınar E: Remarks on some coupled fixed point theorems in G metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 2Google Scholar
 Agarwal RP, Kadelburg Z, Radenović S: On coupled fixed point results in asymmetric G metric spaces. J. Inequal. Appl. 2013., 2013: Article ID 528Google Scholar
 Kadelburg Z, Nashine HK, Radenović S: Common coupled fixed point results in partially ordered G metric spaces. Bull. Math. Anal. Appl. 2012, 4: 51–63.MathSciNetGoogle Scholar
 Mustafa Z, Sims B: Fixed point theorems for contractive mappings in complete G metric spaces. Fixed Point Theory Appl. 2009., 2009: Article ID 917175Google Scholar
 Fang JX: On fixed point theorem in fuzzy metric spaces. Fuzzy Sets Syst. 1992, 46: 107–113. 10.1016/01650114(92)902715View ArticleGoogle Scholar
 Gregori V, Sapena A: On fixed point theorem in fuzzy metric spaces. Fuzzy Sets Syst. 2002, 125: 245–252. 10.1016/S01650114(00)000889View ArticleMathSciNetGoogle Scholar
 Alaca C, Turkoglu D, Yildiz C: Fixed points in intuitionistic fuzzy metric spaces. Chaos Solitons Fractals 2006, 29: 1073–1078. 10.1016/j.chaos.2005.08.066View ArticleMathSciNetGoogle Scholar
 Cho YJ, Roldán A, MartínezMoreno J, Roldán C: Coupled coincidence point theorems in (intuitionistic) fuzzy normed spaces. J. Inequal. Appl. 2013., 2013: Article ID 104Google Scholar
 Hadžić O, Pap E: The Fixed Point Theory in Probabilistic Metric Spaces. Kluwer Academic, Dordrecht; 2001.Google Scholar
 Pap E, Hadžić O, Mesiar R: A fixed point theorem in probabilistic metric spaces and an application. J. Math. Anal. Appl. 1996, 202: 433–449. 10.1006/jmaa.1996.0325View ArticleMathSciNetGoogle Scholar
 Fang JX: Fixed point theorems of local contraction mappings on Menger spaces. Appl. Math. Mech. 1991, 12: 363–372. 10.1007/BF02020399View ArticleGoogle Scholar
 Fang JX: Common fixed point theorems of compatible and weakly compatible maps in Menger spaces. Nonlinear Anal. 2009, 71(5–6):1833–1843. 10.1016/j.na.2009.01.018View ArticleMathSciNetGoogle Scholar
 Turinici M: Abstract comparison principles and multivariable GronwallBellman inequalities. J. Math. Anal. Appl. 1986, 117: 100–127. 10.1016/0022247X(86)902519View ArticleMathSciNetGoogle Scholar
 Ran ACM, Reurings MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 2004, 132: 1435–1443. 10.1090/S0002993903072204View ArticleMathSciNetGoogle Scholar
 Nieto JJ, RodríguezLópez R: Contractive mapping theorem in partially ordered sets and applications to ordinary differential equations. Order 2005, 22: 223–239. 10.1007/s1108300590185View ArticleMathSciNetGoogle Scholar
 Gnana Bhaskar T, Lakshmikantham V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 2006, 65(7):1379–1393. 10.1016/j.na.2005.10.017View ArticleMathSciNetGoogle Scholar
 Lakshmikantham V, Ćirić LJ: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal. 2009, 70(12):4341–4349. 10.1016/j.na.2008.09.020View ArticleMathSciNetGoogle Scholar
 Guo D, Lakshmikantham V: Coupled fixed points of nonlinear operators with applications. Nonlinear Anal. 1987, 11: 623–632. 10.1016/0362546X(87)900770View ArticleMathSciNetGoogle Scholar
 Borcut M, Berinde V: Tripled coincidence theorems for contractive type mappings in partially ordered metric spaces. Appl. Math. Comput. 2012, 218: 5929–5936. 10.1016/j.amc.2011.11.049View ArticleMathSciNetGoogle Scholar
 Karapınar E, Luong NV: Quadruple fixed point theorems for nonlinear contractions. Comput. Math. Appl. 2012, 64: 1839–1848. 10.1016/j.camwa.2012.02.061View ArticleMathSciNetGoogle Scholar
 Berzig M, Samet B: An extension of coupled fixed point’s concept in higher dimension and applications. Comput. Math. Appl. 2012, 63: 1319–1334. 10.1016/j.camwa.2012.01.018View ArticleMathSciNetGoogle Scholar
 Roldán A, MartínezMoreno J, Roldán C: Multidimensional fixed point theorems in partially ordered complete metric spaces. J. Math. Anal. Appl. 2012, 396: 536–545. 10.1016/j.jmaa.2012.06.049View ArticleMathSciNetGoogle Scholar
 Kasahara S: A remark on the contraction principle. Proc. Jpn. Acad. 1968, 44: 21–26. 10.3792/pja/1195521362View ArticleMathSciNetGoogle Scholar
 Kim YW: Pseudo quasi metric spaces. Proc. Jpn. Acad. 1968, 44: 1009–1012. 10.3792/pja/1195520936View ArticleGoogle Scholar
 Reilly IL, Subrahmanjam PV, Vamanamurthy MK: Cauchy sequences in quasipseudometric spaces. Monatshefte Math. 1982, 93: 127–140. 10.1007/BF01301400View ArticleGoogle Scholar
 Agarwal, RP, Karapınar, E, RoldánLópezdeHierro, AF: Fixed point theorems in quasimetric spaces and applications. J. Nonlinear Convex Anal. (accepted)Google Scholar
 Khan MS, Swaleh M, Sessa S: Fixed point theorems by altering distances between the points. Bull. Aust. Math. Soc. 1984, 30(1):1–9. 10.1017/S0004972700001659View ArticleMathSciNetGoogle Scholar
 Kadelburg Z, Radenović S: A note on some recent best proximity point results for nonself mappings. Gulf J. Math. 2013, 1: 36–41.Google Scholar
 Karapınar E, Moradi S:Fixed point theory for cyclic generalized $(\phi \varphi )$contraction mappings. Ann. Univ. Ferrara 2013, 59: 117–125. 10.1007/s1156501201584View ArticleGoogle Scholar
 Berzig M, Karapınar E, Roldán A:Discussion on generalized$(\alpha \psi ,\beta \phi )$contractive mappings via generalized altering distance function and related fixed point theorems. Abstr. Appl. Anal. 2014., 2014: Article ID 259768Google Scholar
 Popescu O:Fixed points for $(\psi ,\phi )$weak contractions. Appl. Math. Lett. 2011, 24(1):1–4. 10.1016/j.aml.2010.06.024View ArticleMathSciNetGoogle Scholar
 Moradi S, Farajzadeh A:On the fixed point of $(\psi ,\phi )$ weak and generalized $(\psi ,\phi )$weak contraction mappings. Appl. Math. Lett. 2012, 25(10):1257–1262. 10.1016/j.aml.2011.11.007View ArticleMathSciNetGoogle Scholar
 Berzig, M: Generalization of the Banach contraction principle. Preprint arXiv:1310.0995 (2013)Google Scholar
 Jleli M, Samet B: Remarks on G metric spaces and fixed point theorems. Fixed Point Theory Appl. 2012., 2012: Article ID 210Google Scholar
 Bonsangue MM, van Breugel F, Rutten JJMM: Generalized metric spaces: completion, topology, and powerdomains via the Yoneda embedding. Theor. Comput. Sci. 1998, 193: 1–51. 10.1016/S03043975(97)00042XView ArticleMathSciNetGoogle Scholar
 Mustafa Z, Sims B: A new approach to generalized metric spaces. J. Nonlinear Convex Anal. 2006, 7: 289–297.MathSciNetGoogle Scholar
 Harjani J, Sadarangani K: Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations. Nonlinear Anal. 2010, 72: 1188–1197. 10.1016/j.na.2009.08.003View ArticleMathSciNetGoogle Scholar
 Dutta PN, Choudhury BS: A generalization of contraction principle in metric spaces. Fixed Point Theory Appl. 2008., 2008: Article ID 406368Google Scholar
 Luong NV, Thuan NX: Coupled fixed points in partially ordered metric spaces and application. Nonlinear Anal. 2011, 74: 983–992. 10.1016/j.na.2010.09.055View ArticleMathSciNetGoogle Scholar
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