On some fixed point theorems under \((\alpha,\psi,\phi)\)contractivity conditions in metric spaces endowed with transitive binary relations
 Naseer Shahzad^{1}Email author,
 Erdal Karapınar^{2, 3} and
 AntonioFrancisco RoldánLópezdeHierro^{4}
https://doi.org/10.1186/s1366301503595
© Shahzad et al. 2015
Received: 20 February 2015
Accepted: 22 June 2015
Published: 22 July 2015
Abstract
After the appearance of Nieto and RodríguezLópez’s theorem, the branch of fixed point theory devoted to the setting of partially ordered metric spaces have attracted much attention in the last years, especially when coupled, tripled, quadrupled and, in general, multidimensional fixed points are studied. Almost all papers in this direction have been forced to present two results assuming two different hypotheses: the involved mapping should be continuous or the metric framework should be regular. Both conditions seem to be different in nature because one of them refers to the mapping and the other one is assumed on the ambient space. In this paper, we unify such different conditions in a unique one. By introducing the notion of continuity of a mapping from a metric space into itself depending on a function α, which is the case that covers the partially ordered setting, we extend some very recent theorems involving control functions that only must be lower/upper semicontinuous from the right. Finally, we use metric spaces endowed with transitive binary relations rather than partial orders.
1 Introduction
In recent times, some extensions of the Banach contractive mapping principle have been introduced using a contractivity condition that involves two different functions. For instance, in 2008, Dutta and Choudhury presented the following generalization.
Theorem 1.1
(Dutta and Choudhury [1])
Functions like ψ verifying the previous properties are known in the literature as altering distance functions (see [2]).
Remark 1.1
Notice that Theorem 1.1 remains true if φ only satisfies the following assumptions: φ is lower semicontinuous and \(\varphi^{1}(\{0\})=\{0\}\) (see, for instance, Abbas and Đorić [3] and Đorić [4]).
The above remark yields the following statement.
Theorem 1.2
We have also an ordered version of Theorem 1.2 (see [3–5]).
Theorem 1.3
The previous results were extended to the case of a contractivity condition involving three different functions. For instance, Eslamian and Abkar [6] established the following result.
Theorem 1.4
(Eslamian and Abkar [6])
Some of the previous results became equivalent.
Choudhury and Kundu [8] also extended Theorem 1.4 to the ordered case.
Theorem 1.6
(Choudhury and Kundu [8])
Following similar arguments as in the proof of Theorem 1.5, the following result was obtained.
In a very recent paper, Shaddad et al. proved the following result, which is a generalization of the previous ones.
Theorem 1.8
(Shaddad et al. [9], Theorem 2.5)
 1.There exist an altering distance function ψ, an upper semicontinuous function \(\theta: [ 0,\infty ) \rightarrow [ 0,\infty ) \), and a lower semicontinuous function \(\varphi: [ 0,\infty ) \rightarrow [ 0,\infty ) \) such thatwhere \(\theta(0)=\varphi(0)=0\) and \(\psi(t)\theta(t)+\varphi(t)>0\) for all \(t>0\).$$\psi \bigl( d(fx,fy) \bigr) \leq\theta \bigl( d(x,y) \bigr) \varphi \bigl( d(x,y) \bigr)\quad \textit{for all }x\succcurlyeq y, $$
 2.
There exists \(x_{0}\in X\) such that \(x_{0}\asymp f(x_{0})\);
 3.
f is nondecreasing;
 4.At least, one of the following conditions holds:
 (a)
f is continuous or
 (b)
if \(\{x_{n}\}\rightarrow x\) when \(n\rightarrow\infty\) in X, then \(x_{n}\asymp x\) for all n.
 (a)
Then f has a fixed point. Moreover, if for each \(x,y\in X\) there exists \(z\in X\) which is comparable to x and y, then the fixed point is unique.
The condition ‘\(\psi(t)\theta(t)+\varphi(t)>0\) for all \(t>0\)’ is not new because, as we have just commented, under some weak continuity conditions, it was firstly considered in Choudhury and Kundu [8], and it can also be found in Razani and Parvaneh [10]. As a consequence of the previous theorem, the authors obtained the following result.
Corollary 1.1
(Shaddad et al. [9], Corollary 2.6)
 1.There exist an altering distance function ψ and a lower semicontinuous function \(\varphi: [ 0,\infty ) \rightarrow [ 0,\infty ) \) such thatwhere \(\varphi(0)=0\);$$ \psi \bigl( d(fx,fy) \bigr) \leq\psi \bigl( d(x,y) \bigr) \varphi \bigl( d(x,y) \bigr)\quad \textit{for all }x\succcurlyeq y, $$(1)
 2.
There exists \(x_{0}\in X\) such that \(x_{0}\asymp f(x_{0})\);
 3.
f is nondecreasing;
 4.At least, one of the following conditions holds:
 (a)
f is continuous or
 (b)
if \(\{x_{n}\}\rightarrow x\) when \(n\rightarrow\infty\) in X, then \(x_{n}\asymp x\) for all n.
 (a)
Then f has a fixed point. Moreover, if for each \(x,y\in X\) there exists \(z\in X\) which is comparable to x and y, then the fixed point is unique.

if \(\{x_{m}\}\subseteq X\) is such that \(\{x_{m}\}\rightarrow x\in X\) and \(x_{m}\preccurlyeq x_{m+1}\) for all \(m\in\mathbb{N}\), then we have that \(x_{m}\preccurlyeq x\) for all \(m\in\mathbb{N}\);

if \(\{x_{m}\}\subseteq X\) is such that \(\{x_{m}\}\rightarrow x\in X\) and \(x_{m}\succcurlyeq x_{m+1}\) for all \(m\in\mathbb{N}\), then we have that \(x_{m}\succcurlyeq x\) for all \(m\in\mathbb{N}\).
On the other hand, in hypothesis 1 of Corollary 1.1, the condition ‘\(\varphi(t)>0\) for all \(t>0\)’ is necessary. For instance, consider the following example.
Example 1.1
This paper has three main aims. On the one hand, we generalize Theorem 1.1 introducing a contractivity condition involving control function that does not have to be continuous nor monotone. In fact, this new kind of control functions only have to verify sequential properties. We will show that two functions are powerful enough to handle contractivity conditions as in hypothesis 1 of Theorem 1.8, in which three functions appear. The new class of control functions includes pairs that are not necessary altering distance functions, and the semicontinuity is imposed only from the right side and on the interval \(( 0,\infty ) \). On the other hand, the second objective is to describe a unified condition to handle two independent hypotheses (the continuity of a mapping and the regularity of the partially ordered metric space), which were initially introduced from Ran and Reurings’ theorem and Nieto and RodríguezLópez’s theorem. Finally, we show that many results obtained in the setting of a partially ordered metric space do not need a partial order, but only a transitive binary relation on a subset of the metric space.
2 Preliminaries
In the sequel, \(\mathbb{N}=\{0,1,2,3,\ldots\}\) denotes the set of all nonnegative integers and \(\mathbb{R}\) denotes the set of all real numbers. Henceforth, X and Y will denote nonempty sets. Elements of X are usually called points.
Let \(T:X\rightarrow Y\) be a mapping. The domain of T is X and it is denoted by DomT. Its range, that is, the set of values of T in Y, is denoted by \(T(X)\). A mapping T is completely characterized by its domain, its range, and the manner in which each origin \(x\in\operatorname{Dom}T\) is applied to its image \(T(x)\in T(X)\). For simplicity, we denote, as usual, \(T(x)\) by Tx. For any set X, we denote the identity mapping on X by \(I_{X}:X\rightarrow X\), which is defined by \(I_{X}x=x\) for all \(x\in X\).
Given two selfmappings \(T,g:X\rightarrow X\), we will say that a point \(x\in X\) is a coincidence point of T and g if \(Tx=gx\). We will denote by \(\operatorname{Coin}(T,g)\) the set of all coincidence points of T and g. If x is a coincidence point of T and g, then the point \(\omega=Tx=gx\) is called a point of coincidence of T and g. A common fixed point of T and g is a point \(x\in X\) such that \(Tx=gx=x\). Given a selfmapping \(T:X\rightarrow X\), we will say that a point \(x\in X\) is a fixed point of T if \(Tx=x\). We will denote by \(\operatorname{Fix}(T)\) the set of all fixed points of T.
The notion of metric space and the concepts of convergent sequence and Cauchy sequence in a metric space can be found, for instance, in [11]. We will write \(\{x_{n}\}\rightarrow x\) when a sequence \(\{x_{n}\}_{n\in\mathbb{N}}\) of points of X converges to \(x\in X\) in the metric space \(( X,d ) \). A metric space \(( X,d ) \) is complete if every Cauchy sequence in X converges to some point of X. The limit of a convergent sequence in a metric space is unique.
In a metric space \(( X,d ) \), a mapping \(T:X\rightarrow X\) is continuous at a point \(z\in X\) if \(\{Tx_{n}\}\rightarrow Tz\) for all sequence \(\{x_{n}\}\) in X such that \(\{x_{n}\}\rightarrow z\). And T is continuous if it is continuous at every point of X.
A binary relation on X is a nonempty subset \(\mathcal{R}\) of \(X\times X\). For simplicity, we denote if \((x,y)\in \mathcal{R}\), and we will say that is the binary relation on X. This notation lets us write \(x\prec y\) when and \(x\neq y\). We write when . We will say that x and y are comparable, and we will write \(x\asymp y\) if or . A binary relation on X is reflexive if for all \(x\in X\); it is transitive if for all \(x,y,z\in X\) such that and ; and it is antisymmetric if and imply \(x=y\).
A reflexive and transitive relation ≼ on X is a preorder (or a quasiorder) on X. In such a case, \((X,\preccurlyeq)\) is a preordered space. If a preorder ≼ is also antisymmetric, then ≼ is called a partial order, and \((X,\preccurlyeq)\) is a partially ordered space (or a partially ordered set). We will use the symbol for a general binary relation on X, and the symbol ≼ for a reflexive binary relation on X (for instance, a preorder or a partial order).
An ordered metric space is a triple \(( X,d,\preccurlyeq ) \) where \(( X,d ) \) is a metric space and ≼ is a partial order on X. And if ≼ is a preorder on X, then \(( X,d,\preccurlyeq ) \) is a preordered metric space.
Definition 2.1

nondecreasingregular if for all sequence \(\{x_{m}\} \subseteq A \) such that \(\{x_{m}\}\rightarrow a\in A\) and for all \(m\in\mathbb{N}\), we have that for all \(m\in\mathbb{N}\);

nonincreasingregular if for all sequence \(\{x_{m}\} \subseteq A \) such that \(\{x_{m}\}\rightarrow a\in A\) and for all \(m\in\mathbb{N}\), we have that for all \(m\in\mathbb{N}\);

regular if it is both nondecreasingregular and nonincreasingregular.
Some authors called ordered complete to a regular ordered metric space (see, for instance, [12]). Furthermore, Roldán et al. called sequential monotone property to nondecreasingregularity (see [13]).
Definition 2.2
Let be a binary relation on X and let \(T,g:X\rightarrow X\) be two mappings. We say that T is nondecreasing if for all \(x,y\in X\) such that . And T is nondecreasing if for all \(x,y\in X\) such that .
Functions in \(\mathcal{F}_{\mathrm{alt}}\) are called altering distance functions (see [2, 14–16]).
Remark 2.1
If \(\phi: [ 0,\infty ) \rightarrow [ 0,\infty ) \) is nondecreasing (for instance, if \(\phi\in{\mathcal {F}_{\mathrm{alt}}}\)) and \(t,s\in [ 0,\infty ) \) verify \(\phi(t)<\phi(s) \), then \(t< s\).
To prove it, assume that \(t\geq s\). As ϕ is nondecreasing, then \(\phi(s)\leq\phi(t)<\phi(s)\), which is impossible.
Definition 2.3

lower semicontinuous from the right at \(s\in [ 0,\infty ) \) if$$\phi(s)\leq\liminf_{t\rightarrow s^{+}}\phi(t); $$

lower semicontinuous from the right on A if it is lower semicontinuous from the right at every \(s\in A\);

lower semicontinuous from the right if it is lower semicontinuous from the right on \([ 0,\infty ) \).
Example 2.1
Lemma 2.1
3 Main results
In this section, we present some results that extend and unify all theorems given in Introduction. To do that, some notions are introduced. In the sequel, X will denote a nonempty set, d will be a metric on X, \(T,g:X\rightarrow X\) will be arbitrary selfmappings and \(\alpha:X\times X\rightarrow{}[0,\infty)\) will denote a function.
Definition 3.1
Given a metric space \((X,d)\), a point \(z_{0}\in X\), a function \(\alpha:X\times X\rightarrow{}[0,\infty)\) and two mappings \(T,g:X\rightarrow X\), we say that T is \((d,g,\alpha )\)rightcontinuous at \(z_{0}\) if we have that \(\{Tx_{n}\}\) converges to \(Tz_{0}\) for all sequence \(\{x_{n}\}\subseteq X\) such that \(\{ gx_{n}\}\) is convergent to \(gz_{0}\) and verifying that \(\alpha ( gx_{n},gx_{n+1} ) \geq1\) for all \(n\in\mathbb{N}\). And T is \((d,g,\alpha )\)rightcontinuous if it is \((d,g,\alpha)\)rightcontinuous at every point of X. If g is the identity mapping on X, we say that T is \((d,\alpha)\)rightcontinuous.
Similarly, a mapping \(T:X\rightarrow X\) is \((d,g,\alpha )\)leftcontinuous at \(z_{0}\) if we have that \(\{Tx_{n}\}\) converges to \(Tz_{0}\) for all sequence \(\{x_{n}\}\) such that \(\{gx_{n}\}\) is convergent to \(gz_{0}\) and verifying that \(\alpha ( gx_{n+1},gx_{n} ) \geq1\) for all \(n\in\mathbb{N}\). And T is \((d,g,\alpha)\)leftcontinuous if it is \((d,g,\alpha)\)leftcontinuous at every point of X. If g is the identity mapping on X, we say that T is \((d,\alpha)\)leftcontinuous.
A mapping \(T:X\rightarrow X\) is \((d,g,\alpha)\)continuous if it is both \((d,g,\alpha)\)rightcontinuous and \((d,g,\alpha )\)leftcontinuous.
If \(\alpha(x,y)=1\) for all \(x,y\in X\), we say that T is \((d,g)\)continuous (notice that both sides lead to the same condition) if \(\{gx_{n}\}\rightarrow gz_{0}\) implies that \(\{Tx_{n}\}\rightarrow Tz_{0}\), whatever the point \(z_{0}\in X\) and the sequence \(\{x_{n}\}\subseteq X\).
It is obvious that every continuous mapping from X into itself is also \(( d,I_{X},\alpha ) \)rightcontinuous and \(( d,I_{X},\alpha ) \)leftcontinuous whatever α, but the converse is false.
Definition 3.2
Given a function \(\alpha:X\times X\rightarrow{}[0,\infty)\), we say that two points \(x,y\in X\) are \((g,\alpha)\)comparable if \(\alpha(gx,gy)\geq1\) or \(\alpha(gy,gx)\geq1\).
Definition 3.3
Obviously, every transitive mapping α is also gtransitive, whatever the mapping g.
Definition 3.4
Example 3.1
If \(\alpha(x,y)\geq1\) for all \(x,y\in X\), then α is transitive and gtransitive, and every selfmapping \(T:X\rightarrow X\) is \((g,\alpha )\)admissible.
Remark 3.1
In the next definition, we present the kind of control functions we will involve in the contractivity condition.
Definition 3.5
 (\(\mathcal{F}_{\mathcal{A}}^{1} \)):

If \(\{a_{n}\}\subset ( 0,\infty ) \) is a sequence such that \(\psi ( a_{n+1} ) \leq\phi(a_{n})\) for all \(n\in\mathbb{N}\), then \(\{a_{n}\}\rightarrow0\);
 (\(\mathcal{F}_{\mathcal{A}}^{2}\)):

If \(\{a_{n}\},\{b_{n}\}\subset [ 0,\infty ) \) are two sequences converging to the same limit L and such that \(L< a_{n}\) and \(\psi ( b_{n} ) \leq\phi(a_{n})\) for all \(n\in\mathbb{N}\), then \(L=0\).
Notice that the previous conditions do not impose any constraint about the continuity nor the monotony of the functions ψ and ϕ, as in the following example.
Example 3.2

Suppose that \(L=1\). As \(1=L< a_{n}\) for all \(n\in\mathbb{N}\), then \(\psi(a_{n})=a_{n}/4\) and \(\{\psi(a_{n})\}\rightarrow L/4=1/4\). On the other hand, as \(1/4\leq\psi(t)\) for all \(t\in(0.9,1.1)\) and \(\{b_{n}\} \rightarrow1\), then there exists \(n_{0}\in\mathbb{N}\) such thatwhich is a contradiction because \(\{a_{n}/8\}\rightarrow1/8\). Hence, the case \(L=1\) is impossible.$$\frac{1}{4}\leq\psi(b_{n})\leq\phi(a_{n})= \frac{\psi(a_{n})}{2}=\frac {a_{n}}{8} \quad \text{for all }n\geq n_{0}, $$

Suppose that \(L\neq1\). As ψ and ϕ are continuous at L, then the inequalityimplies that \(\psi(L)\leq\psi(L)/2\). Hence, \(\psi(L)=0\) and \(L=0\).$$\psi ( b_{n} ) \leq\phi(a_{n})=\frac{\psi(a_{n})}{2} \quad \text{for all }n\in\mathbb{N} $$
Let us show that the class \(\mathcal{F}_{\mathcal{A}}\) includes several types of functions.
Lemma 3.1
If \(\psi,\phi: [ 0,\infty ) \rightarrow [ 0,\infty ) \) are two functions such that ψ is continuous on \(( 0,\infty ) \), ϕ is upper semicontinuous from the right on \(( 0,\infty ) \), and \(\phi<\psi\) on \(( 0,\infty ) \), then \(( \psi,\phi ) \) verifies (\(\mathcal{F}_{\mathcal {A}}^{2}\)).
Furthermore, if, additionally, ψ is nondecreasing on \(( 0,\infty ) \), then \(( \psi,\phi ) \in\mathcal{F}_{\mathcal{A}}\).
Proof
Corollary 3.1
If \(\psi\in{\mathcal{F}_{\mathrm{alt}}}\) and \(\phi: [ 0,\infty ) \rightarrow [ 0,\infty ) \) is an upper semicontinuous from the right function such that \(\phi(0)=0\) and \(\phi(t)<\psi(t)\) for all \(t>0\), then \(( \psi,\phi ) \in\mathcal{F}_{\mathcal{A}}\).
Corollary 3.2
If ψ, θ and φ are three functions as in hypothesis 1 of Theorem 1.8, then \(( \psi,\phi=\theta\varphi ) \in\mathcal{F}_{\mathcal{A}}\).
The main result of the present paper is the following one.
Theorem 3.1
 1.
There exists a subset \(A\subseteq X\) such that \(T(X)\subseteq A\subseteq g(X)\) and \(( A,d ) \) is complete;
 2.
α is gtransitive and T is \((g,\alpha)\)admissible;
 3.There exists \(( \psi,\phi ) \in\mathcal{F}_{\mathcal{A}}\) such that$$ \alpha(gx,gy) \psi \bigl( d(Tx,Ty) \bigr) \leq\phi \bigl( d(gx,gy) \bigr)\quad \textit{for all }x,y\in X; $$(5)
 4.At least, one of the following conditions holds:
 (a)
there exists \(x_{0}\in X\) such that \(\alpha (gx_{0},Tx_{0})\geq1\) and T is \((d,g,\alpha)\)rightcontinuous;
 (b)
there exists \(x_{0}\in X\) such that \(\alpha (Tx_{0},gx_{0})\geq1\) and T is \((d,g,\alpha)\)leftcontinuous.
 (a)
Then T and g have, at least, a coincidence point.
Proof
Theorem 3.2
Under the hypothesis of Theorem 3.1, assume that \(\phi(0)=0\) and \(\psi^{1}(\{0\})=\{0\}\). If x and y are two coincidence points of T and g for which there exists \(z\in X\) such that z is, at the same time, \((g,\alpha)\)comparable to x and to y, then \(Tx=gx=gy=Ty\).
Proof
Let \(x,y\in\operatorname{Coin}(T,g)\) be two coincidence points of T and g for which there exists \(z_{0}\in X\) such that \(z_{0}\) is, at the same time, \((g,\alpha)\)comparable to x and to y. Let \(\{z_{n}\}\) be the PicardJungck sequence of \(( T,g ) \) based on \(z_{0}\), that is, \(gz_{n+1}=Tz_{n}\) for all \(n\in\mathbb{N}\). We are going to show that \(\{gz_{n}\}\rightarrow gx\) and \(\{gz_{n}\}\rightarrow gy\) so, by the uniqueness of the limit, we will conclude that \(gx=gy\).

If there exists some \(n_{0}\in\mathbb{N}\) such that \(d(gz_{n_{0}},gx)=0\), then \(\psi ( d(gz_{n_{0}+1},gx) ) \leq\phi ( 0 ) =0\), so \(gz_{n_{0}+1}=gx\). In this case, by induction, we deduce that \(gz_{n}=gx\) for all \(n\geq n_{0}\), which implies that \(\{gz_{n}\}\rightarrow gx\).

On the contrary case, assume that \(d(gz_{n},gx)>0\) for all \(n\in\mathbb{N}\). In such a case, property (\(\mathcal{F}_{\mathcal {A}}^{1} \)) applied to \(\{a_{n}=d(gz_{n},gx)\}_{n\in\mathbb{N}}\subset ( 0,\infty ) \) guarantees that \(\{d(gz_{n},gx)\} \rightarrow 0\), that is, \(\{gz_{n}\}\rightarrow gx\).
If we had supposed that \(\alpha(gx,gz_{0})\geq1\), we would have obtained the same conclusion. Then, in any case, \(\{gz_{n}\}\rightarrow gx\). Changing the roles of x and y, we also have that \(\{gz_{n}\}\rightarrow gy\). Therefore \(gx=gy\). □
Theorem 3.3
 (U)
For all coincidence points x and y of T and g, there exists \(z\in X\) such that z is, at the same time, \((g,\alpha )\)comparable to x and to y.
Hence T and g have a unique common fixed point \(\omega\in X\). Furthermore, \(\omega=gx\) for all \(x\in\operatorname{Coin}(T,g)\).
Proof
Let \(x\in\operatorname{Coin}(T,g)\) be an arbitrary coincidence point of T and g and let \(\omega=gx\). As T and g commute, then \(T\omega=Tgx=gTx=g\omega\), so ω is another coincidence point of T and g. By hypothesis (U), there exists \(z\in X\) such that z is, at the same time, \((g,\alpha)\)comparable to x and to ω. Hence, Theorem 3.2 guarantees that \(gx=g\omega\), which means that \(\omega=gx=g\omega\). As a result, \(\omega=g\omega=T\omega\), that is, ω is a common fixed point of T and g.
To prove the uniqueness, let \(u,v\in X\) be two common fixed points of T and g. As u and v are coincidence points of T and g, hypothesis (U) implies that there exists \(z\in X\) such that z is, at the same time, \((g,\alpha)\)comparable to u and to v. Thus, Theorem 3.2 guarantees that \(gu=gv\), which means that \(u=gu=gv=v\). As a consequence, T and g have a unique common fixed point, which is ω. □
4 Consequences of the main results
The best advantage of the previous theorems is that they can be particularized in a wide variety of different results. This section is dedicated to deducing some direct consequences of them in the context of metric spaces. For instance, in the following statement we use Lemma 3.1.
Corollary 4.1
 1.
There exists a subset \(A\subseteq X\) such that \(T(X)\subseteq A\subseteq g(X)\) and \(( A,d ) \) is complete;
 2.
α is gtransitive and T is \((g,\alpha)\)admissible;
 3.There exist two functions \(\psi,\phi: [ 0,\infty ) \rightarrow [ 0,\infty ) \) such that ψ is continuous and nondecreasing on \(( 0,\infty ) \), ϕ is upper semicontinuous from the right on \(( 0,\infty ) \), \(\phi<\psi\) on \(( 0,\infty ) \), and the following inequality holds:$$\alpha(gx,gy) \psi \bigl( d(Tx,Ty) \bigr) \leq\phi \bigl( d(gx,gy) \bigr)\quad \textit{for all }x,y\in X; $$
 4.At least, one of the following conditions holds:
 (a)
there exists \(x_{0}\in X\) such that \(\alpha (gx_{0},Tx_{0})\geq1\) and T is \((d,g,\alpha)\)rightcontinuous;
 (b)
there exists \(x_{0}\in X\) such that \(\alpha (Tx_{0},gx_{0})\geq1\) and T is \((d,g,\alpha)\)leftcontinuous.
 (a)
Then T and g have, at least, a coincidence point.
 (U)
For all coincidence points x and y of T and g, there exists \(z\in X\) such that z is, at the same time, \((g,\alpha )\)comparable to x and to y.
Then T and g have a unique common fixed point \(\omega\in X\). Furthermore, \(\omega=gx\) for all \(x\in\operatorname{Coin}(T,g)\).
Proof
It follows from Theorems 3.1 and 3.3 taking into account that, by Lemma 3.1, \(( \psi,\phi ) \in\mathcal{F}_{\mathcal{A}}\). □
In the following result, we use a different contractivity condition involving three control functions by decomposing \(\phi=\theta\varphi\).
Corollary 4.2
 1.
There exists a subset \(A\subseteq X\) such that \(T(X)\subseteq A\subseteq g(X)\) and \(( A,d ) \) is complete;
 2.
α is gtransitive and T is \((g,\alpha)\)admissible;
 3.There exist three functions \(\psi,\theta,\varphi: [ 0,\infty ) \rightarrow [ 0,\infty ) \) such that ψ is continuous and nondecreasing on \(( 0,\infty ) \), θ is upper semicontinuous from the right on \(( 0,\infty ) \), φ is lower semicontinuous from the right on \(( 0,\infty ) \), \(\theta\varphi<\psi\) on \(( 0,\infty ) \), and the following inequality holds:$$\alpha(gx,gy) \psi \bigl( d(Tx,Ty) \bigr) \leq\theta \bigl( d(gx,gy) \bigr)  \varphi \bigl( d(gx,gy) \bigr)\quad \textit{for all }x,y\in X; $$
 4.At least, one of the following conditions holds:
 (a)
there exists \(x_{0}\in X\) such that \(\alpha (gx_{0},Tx_{0})\geq1\) and T is \((d,g,\alpha)\)rightcontinuous;
 (b)
there exists \(x_{0}\in X\) such that \(\alpha (Tx_{0},gx_{0})\geq1\) and T is \((d,g,\alpha)\)leftcontinuous.
 (a)
Then T and g have, at least, a coincidence point.
 (U)
For all coincidence points x and y of T and g, there exists \(z\in X\) such that z is, at the same time, \((g,\alpha )\)comparable to x and to y.
Then T and g have a unique common fixed point \(\omega\in X\). Furthermore, \(\omega=gx\) for all \(x\in\operatorname{Coin}(T,g)\).
Proof
Corollary 4.3
 (3′):

There exist three functions \(\psi\in{\mathcal{F}_{\mathrm{alt}}}\) and \(\theta,\varphi: [ 0,\infty ) \rightarrow [ 0,\infty ) \) such that θ is upper semicontinuous from the right on \(( 0,\infty ) \), φ is lower semicontinuous from the right on \(( 0,\infty ) \), \(\theta\varphi<\psi\) on \(( 0,\infty ) \), and the following inequality holds:$$\alpha(gx,gy) \psi \bigl( d(Tx,Ty) \bigr) \leq\theta \bigl( d(gx,gy) \bigr)  \varphi \bigl( d(gx,gy) \bigr) \quad \textit{for all }x,y\in X. $$
It is also interesting to highlight the case in which \(\theta=\psi\).
Corollary 4.4
 1.
There exists a subset \(A\subseteq X\) such that \(T(X)\subseteq A\subseteq g(X)\) and \(( A,d ) \) is complete;
 2.
α is gtransitive and T is \((g,\alpha)\)admissible;
 3.There exist two functions \(\psi,\varphi: [ 0,\infty ) \rightarrow [ 0,\infty ) \) such that ψ is continuous and nondecreasing on \(( 0,\infty ) \), φ is lower semicontinuous from the right on \(( 0,\infty ) \), \(\varphi >0\) on \(( 0,\infty ) \), and the following inequality holds:$$\alpha(gx,gy) \psi \bigl( d(Tx,Ty) \bigr) \leq\psi \bigl( d(gx,gy) \bigr)  \varphi \bigl( d(gx,gy) \bigr) \quad \textit{for all }x,y\in X; $$
 4.At least, one of the following conditions holds:
 (a)
there exists \(x_{0}\in X\) such that \(\alpha (gx_{0},Tx_{0})\geq1\) and T is \((d,g,\alpha)\)rightcontinuous;
 (b)
there exists \(x_{0}\in X\) such that \(\alpha (Tx_{0},gx_{0})\geq1\) and T is \((d,g,\alpha)\)leftcontinuous.
 (a)
Then T and g have, at least, a coincidence point.
 (U)
For all coincidence points x and y of T and g, there exists \(z\in X\) such that z is, at the same time, \((g,\alpha )\)comparable to x and to y.
Then T and g have a unique common fixed point \(\omega\in X\). Furthermore, \(\omega=gx\) for all \(x\in\operatorname{Coin}(T,g)\).
If we use ψ as the identity mapping on \([0,\infty)\), we deduce the following statement.
Corollary 4.5
 (3′):

There exists a lower semicontinuous from the right on \(( 0,\infty ) \) function \(\varphi: [ 0,\infty ) \rightarrow [ 0,\infty ) \) such that \(\varphi>0\) on \(( 0,\infty ) \), and the following inequality holds:$$\alpha(gx,gy) d(Tx,Ty)\leq d(gx,gy)\varphi \bigl( d(gx,gy) \bigr)\quad \textit{for all }x,y\in X. $$
Furthermore, if \(\lambda\in [ 0,1 ) \) and we use \(\varphi (t)=(1\lambda)t\) for all \(t\in [ 0,\infty ) \), then we derive the following result.
Corollary 4.6
 (3^{′′}):

There exists \(\lambda\in [ 0,1 ) \) such that$$\alpha(gx,gy) d(Tx,Ty)\leq\lambda d(gx,gy) \quad \textit{for all }x,y\in X. $$
In the following result, we use \(\alpha(x,y)=1\) for all \(x,y\in X\).
Corollary 4.7
 1.
There exists a subset \(A\subseteq X\) such that \(T(X)\subseteq A\subseteq g(X)\) and \(( A,d ) \) is complete;
 2.There exists \(( \psi,\phi ) \in\mathcal{F}_{\mathcal{A}}\) such that$$\psi \bigl( d(Tx,Ty) \bigr) \leq\phi \bigl( d(gx,gy) \bigr)\quad \textit{for all }x,y\in X; $$
 3.
T is \((d,g)\)continuous.
Then T and g have, at least, a coincidence point.
Additionally, assume that T and g commute, \(\phi(0)=0\) and \(\psi ^{1}(\{0\})=\{0\}\). Then T and g have a unique common fixed point \(\omega\in X\). Furthermore, \(\omega=gx\) for all \(x\in\operatorname{Coin}(T,g)\).
5 Some coincidence point theorems in metric spaces endowed with a binary relation
As we pointed out in Introduction, one of the branches that have attracted much attention in fixed point theory is dedicated to partially ordered metric spaces. However, some properties of a partial order are not necessary to prove some fixed/coincidence point theorem. In this section we show some consequences of our main results in the setting of metric spaces endowed with a binary relation which has only to be transitive on a subset of the metric space.
Definition 5.1
Let be a binary relation on a set X and let A be a nonempty subset of X. We say that is transitive on A if for all \(a,b,c\in A\) such that and .
Definition 5.2
Let \(( X,d ) \) be a metric space and let be a binary relation on X. Given two mappings \(T,g:X\rightarrow X\) and a point \(z_{0}\in X\), we say that T is nondecreasingcontinuous at \(z_{0}\) is \(\{Tx_{n}\}\rightarrow Tz_{0}\) for all sequence \(\{x_{n}\}\subseteq X\) such that \(\{gx_{n}\}\rightarrow gz_{0}\) and for all \(n\in\mathbb{N}\). And T is nondecreasingcontinuous in \(A\subseteq X\) if T is nondecreasingcontinuous at every point of A.
Similarly, we say that T is nonincreasingcontinuous at \(z_{0}\) is \(\{Tx_{n}\}\rightarrow Tz_{0}\) for all sequence \(\{x_{n}\}\subseteq X\) such that \(\{gx_{n}\}\rightarrow gz_{0}\) and for all \(n\in\mathbb{N}\). And T is nonincreasingcontinuous in \(A\subseteq X\) if T is nonincreasingcontinuous at every point of A.
The following result directly follows from the respective definitions.
Lemma 5.1
The main result of this section is the following one.
Theorem 5.1
 1.
There exists a subset \(A\subseteq X\) such that \(T(X)\subseteq A\subseteq g(X)\) and \(( A,d ) \) is complete;
 2.
 3.
 4.
Then T and g have, at least, a coincidence point.
Then T and g have a unique common fixed point \(\omega\in X\). Furthermore, \(\omega=gx\) for all \(x\in\operatorname{Coin}(T,g)\).
Notice that, in the previous result, the binary relation must only be transitive on \(g(X)\).
Proof
It follows from Theorems 3.1 and 3.3 using the function defined in (2) and taking into account the equivalences given in Lemma 5.1. □
We can repeat Corollaries 4.14.7 in this new framework. However, among them, we only highlight the following one.
Corollary 5.1
 1.
There exists a subset \(A\subseteq X\) such that \(T(X)\subseteq A\subseteq g(X)\) and \(( A,d ) \) is complete;
 2.
 3.There exist three functions \(\psi,\theta,\varphi: [ 0,\infty ) \rightarrow [ 0,\infty ) \) such that ψ is continuous and nondecreasing on \(( 0,\infty ) \), θ is upper semicontinuous from the right on \(( 0,\infty ) \), φ is lower semicontinuous from the right on \(( 0,\infty ) \), \(\theta\varphi<\psi\) on \(( 0,\infty ) \), and the following inequality holds:
 4.
Then T and g have, at least, a coincidence point.
Then T and g have a unique common fixed point \(\omega\in X\). Furthermore, \(\omega=gx\) for all \(x\in\operatorname{Coin}(T,g)\).
6 Fixed point theorems
If we use g as the identity mapping on X, we obtain the following fixed point theorems in metric spaces, endowed with a binary relation or not.
Theorem 6.1
 1.
There exists a subset \(A\subseteq X\) such that \(T(X)\subseteq A\) and \(( A,d ) \) is complete;
 2.
α is transitive and T is αadmissible;
 3.There exists \(( \psi,\phi ) \in\mathcal{F}_{\mathcal{A}}\) such that$$\alpha(x,y) \psi \bigl( d(Tx,Ty) \bigr) \leq\phi \bigl( d(x,y) \bigr)\quad \textit{for all }x,y\in X; $$
 4.At least, one of the following conditions holds:
 (a)
there exists \(x_{0}\in X\) such that \(\alpha(x_{0},Tx_{0})\geq 1\) and T is \((d,\alpha)\)rightcontinuous;
 (b)
there exists \(x_{0}\in X\) such that \(\alpha(Tx_{0},x_{0})\geq 1\) and T is \((d,\alpha)\)leftcontinuous.
 (a)
Then T has, at least, a fixed point.
 (U)
For all fixed points x and y of T, there exists \(z\in X\) such that z is, at the same time, αcomparable to x and to y.
Then T has a unique fixed point.
Corollary 6.1
 1.
There exists a subset \(A\subseteq X\) such that \(T(X)\subseteq A\) and \(( A,d ) \) is complete;
 2.
α is transitive and T is αadmissible;
 3.There exist two functions \(\psi,\phi: [ 0,\infty ) \rightarrow [ 0,\infty ) \) such that ψ is continuous and nondecreasing on \(( 0,\infty ) \), ϕ is upper semicontinuous from the right on \(( 0,\infty ) \), \(\phi<\psi\) on \(( 0,\infty ) \), and the following inequality holds:$$\alpha(x,y) \psi \bigl( d(Tx,Ty) \bigr) \leq\phi \bigl( d(x,y) \bigr)\quad \textit{for all }x,y\in X; $$
 4.At least, one of the following conditions holds:
 (a)
there exists \(x_{0}\in X\) such that \(\alpha(x_{0},Tx_{0})\geq 1\) and T is \((d,\alpha)\)rightcontinuous;
 (b)
there exists \(x_{0}\in X\) such that \(\alpha(Tx_{0},x_{0})\geq 1\) and T is \((d,\alpha)\)leftcontinuous.
 (a)
Then T has, at least, a fixed point.
 (U)
For all fixed points x and y of T, there exists \(z\in X\) such that z is, at the same time, αcomparable to x and to y.
Then T has a unique fixed point.
Proof
It follows from Corollary 4.1 using g as the identity mapping on X. □
In the context of metric spaces that are not endowed with binary relations, we can also highlight the following statement, in which we assume that \(\alpha(x,y)=1\) for all \(x,y\in X\).
Corollary 6.2
In the case of metric spaces endowed with binary relations, we have the following results.
Theorem 6.2
 1.
There exists a subset \(A\subseteq X\) such that \(T(X)\subseteq A\) and \(( A,d ) \) is complete;
 2.
 3.
 4.
Then T has, at least, a fixed point.
Then T has a unique fixed point.
Proof
It follows from Theorem 5.1 using g as the identity mapping on X. □
Corollary 6.3
 1.
There exists a subset \(A\subseteq X\) such that \(T(X)\subseteq A\) and \(( A,d ) \) is complete;
 2.
 3.There exist three functions \(\psi,\theta,\varphi: [ 0,\infty ) \rightarrow [ 0,\infty ) \) such that ψ is continuous and nondecreasing on \(( 0,\infty ) \), θ is upper semicontinuous from the right on \(( 0,\infty ) \), φ is lower semicontinuous from the right on \(( 0,\infty ) \), \(\theta\varphi<\psi\) on \(( 0,\infty ) \), and the following inequality holds:
 4.
Then T has, at least, a fixed point.
Then T has a unique fixed point.
Proof
It follows from Corollary 5.1 using g as the identity mapping on X. □
7 A unified version of Ran and Reurings’ theorem and Nieto and RodríguezLópez’s theorem
After the appearance of Ran and Reurings’ theorem [17] and Nieto and RodríguezLópez’s theorem [18], many fixed point results were introduced in the ambient of metric spaces endowed with partial orders (see, for instance, [19] in which the authors introduced a close contractivity condition in Lspaces). Since them, many coincidence/fixed point theorems have been proved distinguishing between either the involved mappings are continuous or the ambient space is regular. In this section, we show a unified version of both theorems using a unique condition. The following one is the particularization of Definition 5.2 to the case in which g is the identity mapping on X and is an arbitrary binary relation on X.
Definition 7.1
Given a metric space \((X,d)\) endowed with a binary relation , a mapping \(T:X\rightarrow X\) is nondecreasingcontinuous at \(z_{0}\in X\) if we have that \(\{ Tx_{n}\}\) converges to \(Tz_{0}\) for all nondecreasing sequence \(\{ x_{n}\}\) convergent to \(z_{0}\). And T is nondecreasingcontinuous if it is nondecreasingcontinuous at every point of X.
Similarly, a mapping \(T:X\rightarrow X\) is nonincreasingcontinuous at \(z_{0}\in X\) if we have that \(\{Tx_{n}\}\) converges to \(Tz_{0}\) for all nonincreasing sequence \(\{x_{n}\}\) convergent to \(z_{0}\). And T is nonincreasingcontinuous if it is nonincreasingcontinuous at every point of X.
It is obvious that every continuous mapping is also nondecreasingcontinuous, but the converse is false.
Example 7.1
The following one is a particularization of Corollary 6.3 using altering distance functions.
Theorem 7.1
 1.
\(( X,d ) \) (or \(( T(X),d ) \)) is complete;
 2.
 3.
 4.
Then T has, at least, a fixed point.
 (U)
Then T has a unique fixed point.
The previous result improves Theorem 1.8 in three senses: (1) the binary relation does not have to be a partial order, but a transitive binary relation; (2) ϕ has only to be upper semicontinuous from the right; (3) the mapping T must only be nondecreasingcontinuous, which is a condition that unifies and extends hypotheses 4(a) and 4(b) of Theorem 1.8.
Theorem 7.2
 1.There exist an altering distance function ψ, an upper semicontinuous from the right function \(\theta: [ 0,\infty ) \rightarrow [ 0,\infty ) \), and a lower semicontinuous from the right function \(\varphi: [ 0,\infty ) \rightarrow [ 0,\infty ) \) such that where \(\theta(0)=\varphi(0)=0\) and \(\psi(t)\theta(t)+\varphi(t)>0\) for all \(t>0\).
Corollary 7.1
 (3′):

There exists \(x_{0}\in X\) such that \(x_{0}\asymp Tx_{0}\) and T is continuous.
Proof
If follows from the fact that if T is continuous, then it is both nondecreasingcontinuous and nonincreasingcontinuous. □
Corollary 7.2
Proof
The following results are well known in the field of fixed point theory.
Corollary 7.3
(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 \(x_{0}\preccurlyeq Tx_{0}\);
 (e)
There exists a constant \(\lambda\in(0,1)\) such that \(d(Tx,Ty)\leq \lambda d(x,y)\) for all \(x,y\in X\) with \(x\succcurlyeq y\).
Then T has a fixed point. Moreover, if for all \((x,y)\in X^{2}\) there exists \(z\in X\) such that \(x\preccurlyeq z\) and \(y\preccurlyeq z\), we obtain uniqueness of the fixed point.
Proof
It follows from Corollary 7.1. □
Corollary 7.4
(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 \(x_{0}\preccurlyeq Tx_{0}\);
 (e)
There exists a constant \(\lambda\in(0,1)\) such that \(d(Tx,Ty)\leq \lambda d(x,y)\) for all \(x,y\in X\) with \(x\succcurlyeq y\).
Then T has a fixed point. Moreover, if for all \((x,y)\in X^{2}\) there exists \(z\in X\) such that \(x\preccurlyeq z\) and \(y\preccurlyeq z\), we obtain uniqueness of the fixed point.
Proof
It follows from Corollary 7.2. □
Finally, we also prove that the Shaddad et al. theorem is an easy consequence of our main results.
Proof
It is only necessary to apply Corollaries 7.1 and 7.2 (which are immediate consequences of Theorem 7.1), which cover cases 4(a) and 4(b) in Theorem 1.8. □
Finally, we point out that the present techniques can be easily generalized to guarantee the existence and uniqueness of multidimensional coincidence/fixed points following the techniques described in [20–26].
Declarations
Acknowledgements
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. N Shahzad acknowledges with thanks DSR for financial support. AF RoldánLópezdeHierro is grateful to the Department of Quantitative Methods for Economics and Business of the University of Granada. The same author has been partially supported by Junta de Andalucía by project FQM268 of the Andalusian CICYE.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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