Common fixed point theorems under \((R,\mathcal {S})\)contractivity conditions
 AntonioFrancisco RoldánLópezdeHierro^{1}Email author and
 Naseer Shahzad^{2}
https://doi.org/10.1186/s1366301605325
© RoldánLópezdeHierro and Shahzad 2016
Received: 19 August 2015
Accepted: 23 March 2016
Published: 26 April 2016
Abstract
Very recently, RoldánLópezdeHierro and Shahzad introduced the notion of Rcontractions as an extension of several notions given by different researchers (for instance, Rcontractions generalize MeirKeeler contractions, \(\mathcal{Z}\)contractions  involving simulation functions  by Khojasteh et al., manageable contractions by Du and Khojasteh, Geraghty’s contractions, Banach contractions, etc.). In this manuscript, we use Rfunctions to present existence and uniqueness coincidence (and common fixed) point results under a contractivity condition that extend some celebrated contractive mappings. In our main theorems, we employ a binary relation on the metric space, which does not have to be a partial order. Finally, we illustrate our technique with an example in which other previous statements cannot be applied: in fact, we show how to apply our main results to a new kind of contractivity conditions which cannot be expressed in separate variables.
Keywords
MSC
1 Introduction
Taking into account its applications to several fields of study, fixed point theory has demonstrated to be a powerful branch of nonlinear analysis. All results in this area are inspired on the Banach contractive mapping principle, introduced in 1922. The way in which the most recent results generalize the initial theorem are diverse. Some manuscripts presented very general contractivity conditions (see [1–4]), especially using auxiliary functions (see [5–8]), other papers were developed in abstract metric spaces (see [9–11]), some contributions involved additional structures like partial orders (see [12, 13]) and even multidimensional fixed/coincidence points were introduced (see [14–17]).
 (1)
On the one hand, most of authors have introduced contractivity conditions each time weaker. Thus, there are fewer requirements for checking that a mapping is contractive.
 (2)
On the other hand, several assumptions as regards the analytic and geometric elements that are considered in the statements have been appearing. For instance, there are many results in which the metric space is not necessarily complete (this condition has been replaced by the completeness of an appropriate subset) and, even more, we do not need to consider a metric space (many results have been demonstrated by using quasimetric spaces and pseudoquasimetric spaces).
Following the first line of research, in recent times, Khojasteh et al. [18] introduced the notion of \(\mathcal{Z}\)contraction by using a new class of auxiliary functions called simulation functions. This kind of functions have attracted much attention because they are useful to express a great family of contractivity conditions that were well known in the field of fixed point theory. Immediately afterward, RoldánLópezdeHierro et al. [19] extended the family of simulation functions by avoiding a symmetric condition that was implicitly considered in the original definition.
Very recently, inspired by \(\mathcal{Z}\)contractions, RoldánLópezdeHierro and Shahzad [20] introduced the notion of Rcontractions as an extension of several notions given by different researchers. Rcontractions do not only extend the class of \(\mathcal {Z}\)contractions but they also generalize manageable contractions by Du and Khojasteh, Geraghty’s contractions, Banach contractions, etc. Furthermore, these authors succeeded in proving that MeirKeerler contractions are also Rcontractions. Like \(\mathcal{Z}\)contractions are based in manageable functions, the key piece of an Rcontraction is its associated underlying Rfunction, which satisfies two independent conditions involving sequences of nonnegative real numbers. Rfunctions have only two arguments, in such way that they are appropriate in order to study contractivity conditions that only involve two elements: the distance between two points and the distance between their images by a selfmapping. Many contractivity conditions were introduced in the past by using these two terms but, in general, they always were conditions in separate variables (that is, these terms were the arguments of different auxiliary functions).
In 1986, Turinici [21] gave an initial result for guaranteeing existence of fixed points by involving a new algebraic structure: a partial order on the metric space. However, the most celebrated results in this line of research, with applications to matrix equations, were given by Ran and Reurings [12], and, later, by Nieto and RodríguezLópez [13]. Following these results, the contractivity condition does not need to hold for all pairs of points: it must only be satisfied by points that are related through the partial order. Thus, the continuity of the involved contractive mapping cannot be derived from the contractivity condition (as in the Banach theorem): in fact, Nieto and RodríguezLópez replaced such condition by the regularity of the partially ordered ambient metric space (which is a condition about the behavior of nondecreasing convergent sequences).
In this manuscript, we use Rfunctions to present existence and uniqueness coincidence (and common fixed) point results under a contractivity condition that extend some celebrated contractive mappings. In our main theorems, we employ a binary relation on the metric space that does not have to be either a partial order nor a transitive relation. Thus, we restrict very much the set of pairs of point for which the contractivity condition must hold. Furthermore, we replace the assumption as regards the completeness of the metric space by precompleteness of an appropriate subspace. Moreover, in our main results, we do not assume the existence of a point that serves as initial condition: we suppose a weaker condition about the existence of a PicardJungck sequence. Finally, we illustrate our technique with an example in which other previous statements (like the Dutta and Choudhury theorem, among others) cannot be applied: in fact, we show how to apply our main results to a new kind of contractivity conditions which cannot be expressed in separate variables.
2 Preliminaries
Let us introduce here basic notions and fundamental results. From now on, \(\mathbb{N}=\{0,1,2, 3,\ldots\}\) stands for the set of all nonnegative integers, \(\mathbb{N}^{\ast}=\mathbb{N}\diagdown\{0\}\) and \(\mathbb{R}\) denotes the set of all real numbers. Henceforth, X stands for a nonempty set and \(T,g:X\rightarrow X\) will denote two selfmappings. For simplicity, we write Tx instead of \(T(x)\).

fixed point of T if \(Tx=x\) (\(\operatorname{Fix}(T)\) will denote the set of all fixed points of T);

coincidence point of T and g if \(Tx=gx\) (\(\operatorname{Coin}(T,g)\) will denote the set of all coincidence points of T and g);

point of coincidence of T and g if there exists \(z\in\operatorname{Coin}(T,g)\) such that \(x=Tz\);

common fixed point of T and g if \(Tx=gx=x\).
Inspired by [22], given a point \(x_{0}\in X\), a PicardJungck sequence of the pair \(( T,g ) \) based on \(x_{0}\) is a sequence \(\{x_{n}\}_{n\in\mathbb{N}}\subseteq X\) such that \(gx_{n+1}=Tx_{n}\) for all \(n\in\mathbb{N}\).
The mappings T and g are commuting if \(Tgx=gTx\) for all \(x\in X\). A pair \(\{T,g\}\) is weakly compatible if \(Tgx=gTx\) for all \(x\in X\) such that \(Tx=gx\).
A binary relation on X is a nonempty subset \(\mathcal{S}\) of \(X\times X\). For simplicity, we denote \(x\, \mathcal{S}\, y\) if \((x,y)\in \mathcal{S}\) (in some cases, we also use the symbol ≺ to denote a binary relation on X because the notation \(x\prec y\) can be more usual for the reader). We say that x and y are \(\mathcal{S}\)comparable if \(x\, \mathcal{S}\, y\) or \(y\, \mathcal{S}\, x\). A binary relation \(\mathcal{S}\) on X is reflexive if \(x\, \mathcal{S}\, x\) for all \(x\in X\); it is transitive if \(x\, \mathcal{S}\, z\) for all \(x,y,z\in X\) such that \(x\, \mathcal{S}\, y\) and \(y\, \mathcal{S}\, z\); and it is antisymmetric if \(x\, \mathcal{S}\, y\) and \(y\, \mathcal{S}\, x\) imply \(x=y\). A preorder (or a quasiorder) is a reflexive and transitive binary relation. And a partial order is an antisymmetric preorder.
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 \(\{x_{n}\}\rightarrow z\). T is continuous if it is continuous at every point of X.
Definition 1
A metric space \(( X,d ) \) endowed with a binary relation \(\mathcal{S}\) is \(\mathcal{S}\)nondecreasingregular if for all \(\mathcal{S}\)nondecreasing sequence \(\{x_{n}\}\subseteq X\) such that \(\{x_{n}\}\rightarrow u\in X\), it follows that \(x_{n}\, \mathcal{S}\, u\) for all \(n\in\mathbb{N}\).
Lemma 2
(Roldán et al. [24], Lemma 16, Berzig et al. [7], Lemma 13)
The following notion was introduced in [20].
Definition 3
(RoldánLópezdeHierro and Shahzad [20])
 (\(\varrho_{1}\)):

If \(\{a_{n}\}\subset ( 0,\infty ) \cap D\) is a sequence such that \(\varrho(a_{n+1},a_{n})>0\) for all \(n\in\mathbb{N}\), then \(\{a_{n}\}\rightarrow0\).
 (\(\varrho_{2}\)):

If \(\{a_{n}\},\{b_{n}\}\subset ( 0,\infty ) \cap D\) are two sequences converging to the same limit \(L\geq0\) and verifying that \(L< a_{n}\) and \(\varrho(a_{n},b_{n})>0\) for all \(n\in\mathbb{N}\), then \(L=0\).
We denote by \(R_{D}\) the family of all Rfunctions whose domain is \(D\times D\).
 (\(\varrho_{3}\)):

If \(\{a_{n}\},\{b_{n}\}\subset ( 0,\infty ) \cap D\) are two sequences such that \(\{b_{n}\}\rightarrow0\) and \(\varrho(a_{n},b_{n})>0\) for all \(n\in \mathbb{N}\), then \(\{a_{n}\}\rightarrow0\).
 (\(\varrho_{4}\)):

If \(\{a_{n}\},\{b_{n}\}\subset [ 0,\infty ) \cap D\) are two sequences such that \(\{b_{n}\}\rightarrow0\) and \(\varrho(a_{n},b_{n})>0\) for all \(n\in \mathbb{N}\), then \(\{a_{n}\}\rightarrow0\).
Notice that (\(\varrho_{4}\)) ⇒ (\(\varrho _{3}\)).
Proposition 4
 1.
If a function \(\varrho:D\times D\rightarrow\mathbb{R}\) verifies \(\varrho ( t,s ) \leq st\) for all \(t,s\in D\cap ( 0,\infty ) \), then (\(\varrho_{3}\)) holds.
 2.
If a function \(\varrho:D\times D\rightarrow\mathbb{R}\) verifies \(\varrho ( t,s ) \leq st\) for all \(t,s\in D\cap [ 0,\infty ) \), then (\(\varrho_{3}\)) and (\(\varrho_{4}\)) holds.
Proposition 5
If \(\varrho\in R_{D}\), then \(\varrho(t,t)\leq0\) for all \(t\in ( 0,\infty ) \cap D\).
Definition 6
(RoldánLópezdeHierro and Shahzad [20])
In [20], the authors give a wide range of Rfunctions and Rcontractions. In fact, MeirKeeler contractions [25, 26], \(\mathcal{Z}\)contractions [18, 19], manageable contractions [27], and Geraghty contractions [5] are particular cases of Rcontractions. Another example is the following one.
Theorem 7
(RoldánLópezdeHierro and Shahzad [20])
As a consequence of the previous result and their main theorems, the authors obtained the following consequence.
Corollary 8
(RoldánLópezdeHierro and Shahzad [20])
The previous statement generalizes the wellknown Dutta and Choudhury theorem.
Theorem 9
(Dutta and Choudhury [20], Theorem 2.1)
Then T has a unique fixed point.
3 Binary relations on a set
Throughout this section, T and g will always denote selfmappings on X and \(\mathcal{S}\) will be a binary relation on X. Recall that we will write \(x\, \mathcal{S}^{\ast}\, y\) when \(x\, \mathcal{S}\, y\) and \(x\neq y\). We introduce some properties that a binary relation can verify.
Definition 10

transitive on A if \(x\,\mathcal{S}\,z\) for all \(x,y,z\in A\) such that \(x\,\mathcal{S}\,y\) and \(y\,\mathcal{S}\,z\);

transitive if it is transitive on X;

gtransitive if it is transitive on \(g ( X ) \) (that is, \(gx\,\mathcal{S}\,gy\) and \(gy\,\mathcal{S}\,gz\) imply \(gx\,\mathcal{S}\,gz\));

\(( T,g ) \)transitive if \(gx\,\mathcal{S}\,Ty\) for all \(x,y\in X\) such that \(gx\,\mathcal{S}\,gy\) and \(gy\,\mathcal{S}\,Ty\);

\((T,g)\)compatible if \(Tx=Ty\) for all \(x,y\in X\) such that \(gx=gy\) and \(gx\,\mathcal{S}\,gy\);

gclosed if \(gx\,\mathcal{S}\,gy\) for all \(x,y\in X\) such that \(x\,\mathcal{S}\,y\).
Proposition 11
Every transitive binary relation is gtransitive and \(( T,g ) \)transitive, whatever T and g.
The following examples show that the notions of gtransitivity and \((T,g ) \)transitivity properly extend the notion of transitivity throughout independent notions.
Example 12
Example 13
Despite the above examples, there are some relationships between gtransitivity and \(( T,g ) \)transitivity.
Proposition 14
If \(\mathcal{S}\) is gtransitive and \(T(X)\subseteq g(X)\), then \(\mathcal{S}\) is \(( T,g ) \)transitive.
Proof
Let \(x,y\in X\) be such that \(gx\,\mathcal{S}\,gy\) and \(gy\,\mathcal{S}\,Ty\). Since \(Ty\in T(X)\subseteq g(X)\), there is \(z\in X\) such that \(Ty=gz\). Hence \(gx\,\mathcal{S}\,gy\) and \(gy\,\mathcal{S}\,gz\). As \(\mathcal{S}\) is gtransitive, then \(gx\,\mathcal{S}\,gz\), so \(gx\,\mathcal{S}\,Ty\). This proves that \(\mathcal{S}\) is \(( T,g ) \)transitive. □
The following example shows that gtransitivity and \(( T,g ) \)transitivity do not imply any of the properties that a partial order satisfies.
Example 15
Definition 16
We say that T is \(( g,\mathcal{S} ) \)nondecreasing if \(Tx\,\mathcal{S}\,Ty\) for all \(x,y\in X\) such that \(gx\,\mathcal{S}\,gy\).
Proposition 17
If \(\mathcal{S}\) is reflexive and antisymmetric, and T is \(( g,\mathcal{S} ) \)nondecreasing, then \(\mathcal{S}\) is \(( T,g ) \)compatible.
Proof
Let \(x,y\in X\) be such that \(gx=gy\) and \(gx\,\mathcal{S}\,gy\). As \(\mathcal {S}\) is reflexive, \(gx\,\mathcal{S}\,gy\) and \(gy\,\mathcal{S}\,gx\). Since T is \(( g,\mathcal{S} ) \)nondecreasing, \(Tx\,\mathcal{S}\,Ty\) and \(Ty\,\mathcal{S}\,Tx\). As \(\mathcal{S}\) is antisymmetric, \(Tx=Ty\). □
Proposition 18
If g is injective, then any binary relation \(\mathcal{S}\) is \(( T,g ) \)compatible, whatever T.
Proof
Let \(x,y\in X\) be such that \(gx=gy\) and \(gx\,\mathcal{S}\,gy\). As g is injective, \(x=y\), so \(Tx=Ty\). □
Definition 19
We will write \(T(X)\subseteq_{\mathcal{S}}g(X)\) if for all \(x\in X\) there exists \(y\in X\) such that \(Tx=gy\) and \(gx\,\mathcal{S}\,gy\).
Clearly, if \(T(X)\subseteq_{\mathcal{S}}g(X)\), then \(T(X)\subseteq g(X)\).
Definition 20
A subset A of a metric space \(( X,d ) \) is precomplete if each Cauchy sequence \(\{a_{n}\}\subseteq A\) is convergent to a point of X.
Remark 21
 1.
The empty subset is precomplete.
 2.
Every complete subset of X is precomplete.
 3.
Every subset of a complete metric space is also precomplete.
Example 22
Although \(X= ( 0,3 ) \), endowed with the Euclidean metric, is not complete, and \(A= ( 1,2 ) \) is not complete, the set A is precomplete.
Proposition 23
If \(A\subseteq B\subseteq X\) and B is precomplete, then A is also precomplete.
Remark 24
If \(T(X)\subseteq g(X)\) and one of X, or \(g(X)\) or \(T(X)\) is complete, then \(T(X)\) is precomplete.
When a binary relation \(\mathcal{S}\) is not symmetric, we can consider ‘rightnotions’ and ‘leftnotions’ depending on the character of the involved sequences. For instance, ‘rightnotions’ corresponds to definitions in which a sequence \(\{x_{n}\}\subseteq X\) satisfies \(x_{n}\,\mathcal{S}\,x_{m}\) for all \(n,m\in\mathbb{N}\) with \(n< m\), and ‘leftnotions’ are associated to condition \(x_{n}\,\mathcal{S}\,x_{m}\) with \(n>m\). In this paper, we consider the first ones, and we introduce rightregularity, \(( O,\mathcal {S} )\)rightcompatibility, \((T,g,\mathcal{S})\)rightPicardJungck sequences and \(\mathcal{S}\)rightcontinuity. However, we will omit the term ‘right’.
Definition 25

\((T,g)\)PicardJungck sequence if(in such a case, we say that \(\{x_{n}\}\) is based on the initial point \(x_{0}\));$$ gx_{n+1}=Tx_{n} \quad \text{for all }n\in \mathbb{N} $$(2)

\((T,g,\mathcal{S})\)PicardJungck sequence if it is a \((T,g ) \)PicardJungck sequence and$$ gx_{n}\,\mathcal{S}\,gx_{m} \quad \text{for all }n,m\in \mathbb{N}\text{ such that }n< m. $$(3)
The following result shows some sufficient conditions in order to guarantee the existence of a \(( T,g,\mathcal{S} ) \)PicardJungck sequence.
Proposition 26
 1.
If \(TX\subseteq g(X)\), T is \(( g,\mathcal{S} ) \)nondecreasing and there exists a point \(x_{0}\in X\) such that \(gx_{0}\,\mathcal{S}\,Tx_{0}\), then there exists a \(( T,g,\mathcal{S} ) \)PicardJungck sequence on X based on \(x_{0}\).
 2.
If \(T(X)\subseteq_{\mathcal{S}}g(X)\), then there exists a \(( T,g,\mathcal{S} ) \)PicardJungck sequence on X based on each arbitrary point \(x_{0}\in X\).
Proof
Step 1. We claim that there exists a sequence \(\{x_{n}\}\subseteq X\) such that \(gx_{n+1}=Tx_{n}\) and \(gx_{n}\,\mathcal{S}\,gx_{n+1}\) for all \(n\in\mathbb{N}\).
(1) Since \(Tx_{0}\in T(X)\subseteq g(X)\), we can find \(x_{1}\in X\) such that \(Tx_{0}=gx_{1}\). As \(Tx_{1}\in T(X)\subseteq g(X)\), there is \(x_{2}\in X\) such that \(Tx_{1}=gx_{2}\). Moreover, as \(gx_{0}\,\mathcal{S}\,Tx_{0}=gx_{1}\) and T is \(( g,\mathcal{S} ) \)nondecreasing, \(Tx_{0}\,\mathcal{S}\,Tx_{1}\), which means that \(gx_{1}\,\mathcal{S}\,gx_{2}\). By induction, we can consider a sequence \(\{x_{n}\}\subseteq X\) such that \(gx_{n+1}=Tx_{n}\) and \(gx_{n}\,\mathcal{S}\,gx_{n+1}\) for all \(n\in\mathbb{N}\).
(2) Let \(x_{0}\in X\) be a point. Since \(Tx_{0}\in T(X)\subseteq _{\mathcal{S}}g(X)\), there is \(x_{1}\in X\) verifying \(Tx_{0}=gx_{1}\) and \(gx_{0}\,\mathcal{S}\,gx_{1}\). Repeating this argument, from \(Tx_{1}\in T(X)\subseteq_{\mathcal{S}}g(X)\) it follows that there is \(x_{2}\in X\) satisfying \(Tx_{1}=gx_{2}\) and \(gx_{1}\,\mathcal{S}\,gx_{2}\). By induction, Step 1 holds.
The same proof is valid when the binary relation is omitted.
Proposition 27
If \(TX\subseteq g(X)\), then there exists a \(( T,g ) \)PicardJungck sequence on X based on each \(x_{0}\in X\).
From now on, let \(( X,d ) \) be a metric space.
Definition 28
The map \(T:X\rightarrow X\) is \(\mathcal{S}\)continuous if \(\{Tx_{n}\}\rightarrow Tu\) for all sequence \(\{x_{n}\}\subseteq X\) such that \(\{x_{n}\}\rightarrow u\in X\) and \(x_{n}\,\mathcal{S}\,x_{m}\) for all \(n,m\in\mathbb{N}\) with \(n< m\).
Remark 29
If \(T:X\rightarrow X\) is continuous and X is endowed with a binary relation \(\mathcal{S}\), then T is \(\mathcal{S}\)continuous.
Definition 30
If A is a subset of a metric space \(( X,d ) \) endowed with a binary relation \(\mathcal{S}\), we say that A is \(( d,\mathcal {S} ) \)regular if for all sequence \(\{x_{n}\}\subseteq X\) such that \(\{x_{n}\}\rightarrow u\in X\) and \(x_{n}\,\mathcal{S}\,x_{m}\) for all \(n,m\in\mathbb{N}\) with \(n< m\), we have \(x_{n}\,\mathcal{S}\,u\) for all \(n\in\mathbb{N}\).
The following definition extends some ideas that can be found on [28–30].
Definition 31
Clearly, commutativity implies \(( O,\mathcal{S} ) \)compatibility.
4 Some coincidence point theorems under \(( R,\mathcal {S} )\)contractivity conditions
 (C_{1}):

\(\varrho ( d ( Tx,Ty ) ,d ( gx,gy ) ) >0 \) for all \(x,y\in X\) such that \(Tx\,\mathcal{S}^{\ast}\,Ty\) and \(gx\, \mathcal{S}^{\ast}\, gy\).
 (C_{2}):

\(\varrho ( d ( Tx,Ty ) ,d ( gx,gy ) ) >0 \) for each \(x,y\in X\) verifying \(gx\, \mathcal{S}^{\ast}\, gy\).
 (C_{3}):

\(\varrho ( d ( Tx,Ty ) ,d ( gx,gy ) ) >0 \) for each \(x,y\in X\) satisfying \(x\neq y\) and \(gx\,\mathcal{S}\,gy\).
Obviously, (C_{3}) ⇒ (C_{2}) ⇒ (C_{1}) because the only difference between them is the kind of points for which the inequality holds. The contractivity condition can also be useful to prove some properties of the binary relation.
Proposition 32
Suppose that \(\varrho:D\times D\rightarrow\mathbb{R}\) is a function for which (C_{3}) holds, and assume that \(\varrho ( t,0 ) \leq0\) for all \(t>0\). Then \(\mathcal{S}\) is \(( T,g ) \)compatible.
Proof
Let \(x,y\in X\) be such that \(gx=gy\) and \(gx\,\mathcal{S}\,gy\). By contradiction, assume that \(Tx\neq Ty\). Then, necessarily, \(x\neq y\). Let \(t_{0}=d ( Tx,Ty ) \in ( 0,\infty ) \cap\operatorname{ran}(d)\subseteq ( 0,\infty ) \cap D\). By (C_{3}), \(\varrho ( d ( Tx,Ty ) ,d ( gx,gy ) ) =\varrho ( t_{0},0 ) >0\), which contradicts \(\varrho ( t_{0},0 ) \leq0\). Thus \(Tx=Ty\). □
4.1 Coincidence point theorems under \(\mathcal{S}\)continuity
The first main result of the present manuscript is the following one, in which we use the weaker contractivity condition.
Theorem 33
 (A):

There is on X a \((T,g,\mathcal{S})\)PicardJungck sequence.
 (B):

There is an Rfunction \(\varrho\in R_{D}\) such that \(\operatorname{ran}(d)\subseteq D\) and$$ \varrho \bigl( d ( Tx,Ty ) ,d ( gx,gy ) \bigr) >0 \quad \textit{for all }x,y\in X\textit{ such that }Tx\, \mathcal{S}^{\ast}\, Ty\textit{ and }gx\, \mathcal{S}^{\ast}\, gy. $$(4)
 (a)
The pair \((T,g)\) is \(( O,\mathcal{S} ) \)compatible; or
 (b)
T and g are commuting.
Then T and g have a coincidence point. In fact, if \(\{x_{n}\}\) is any \((T,g,\mathcal{S})\)PicardJungck sequence, either \(\{gx_{n}\}\) contains a coincidence point of T and g, or \(\{gx_{n}\}\) converges to a coincidence point of T and g.

If T and g are continuous, then T and g are \(\mathcal {S}\)continuous.

If there is a subset A such that \(T(X)\subseteq A\subseteq X\) and A is precomplete or complete, then \(T(X)\) is precomplete. In particular, if one of \(T(X)\), or \(g(X)\), or X is complete, then \(T(X)\) is precomplete.

By Proposition 26, hypothesis (A) is guaranteed under the following conditions.
 (A′):

\(\mathcal{S}\) is \(( T,g ) \)transitive (or gtransitive), \(TX\subseteq g(X)\), T is \(( g,\mathcal{S} ) \)nondecreasing and there exists a point \(x_{0}\in X\) such that \(gx_{0}\,\mathcal{S}\,Tx_{0}\).
 (A^{′′}):

\(\mathcal{S}\) is \(( T,g ) \)transitive (or gtransitive) and \(T(X)\subseteq_{\mathcal {S}}g(X)\).
Proof
Case (b). Assume that T and g are commuting. Then Case (a) is applicable because commutativity implies \((O,\mathcal{S} ) \)compatibility. □
The following consequence is obtained by using the binary relation \(x\,\mathcal{S}\,y\) for all \(x,y\in X\).
Corollary 34
 (A):

There exists on X a \((T,g)\)PicardJungck sequence.
 (B):

There exists an Rfunction \(\varrho\in R_{D}\) such that \(\operatorname{ran}(d)\subseteq D\) and$$\varrho \bigl( d ( Tx,Ty ) ,d ( gx,gy ) \bigr) >0 \quad \textit{for all }x,y\in X\textit{ such that }Tx\neq Ty\textit{ and }gx\neq gy. $$
Additionally, assume that the pair \((T,g)\) is Ocompatible or T and g are commuting.
Then T and g have, at least, a coincidence point. In fact, if \(\{ x_{n}\}\) is any \((T,g)\)PicardJungck sequence, either \(\{gx_{n}\}\) contains a coincidence point of T and g, or \(\{gx_{n}\}\) converges to a coincidence point of T and g.
In the following result, we denote by ≺ a transitive binary relation (for instance, a preorder or a partial order), which is not necessarily reflexive. In this case, we replace hypothesis (A) by (A′) by virtue of Proposition 26 and the new condition (C).
Corollary 35
 (A′):

\(TX\subseteq g(X)\), T is \(( g,\prec ) \)nondecreasing and there exists a point \(x_{0}\in X\) such that \(gx_{0}\prec Tx_{0}\).
 (B):

There exists an Rfunction \(\varrho\in R_{D}\) such that \(\operatorname{ran}(d)\subseteq D\) and$$\varrho \bigl( d ( Tx,Ty ) ,d ( gx,gy ) \bigr) >0\quad \textit{for all }x,y\in X \textit{ such that }Tx\prec Ty\textit{ and }gx\prec gy. $$
 (C):

The binary relation ≺ is transitive (or gtransitive, or \(( T,g ) \)transitive).
Additionally, assume that the pair \((T,g)\) is \(( O,\mathcal {S} ) \)compatible or T and g are commuting.
Then T and g have, at least, a coincidence point. In fact, if \(\{ x_{n}\}\) is any \((T,g,\prec)\)PicardJungck sequence, either \(\{gx_{n}\}\) contains a coincidence point of T and g, or \(\{gx_{n}\}\) converges to a coincidence point of T and g.
If \(g=I_{X}\) is the identity mapping on X, then we derive the following result.
Corollary 36
 (A):

There exists on X a Picard sequence \(\{x_{n+1}=Tx_{n}\}\) such that \(x_{n}\,\mathcal{S}\,x_{m}\) for all \(n,m\in \mathbb{N}\) such that \(n< m\).
 (B):

There exists an Rfunction \(\varrho\in R_{D}\) such that \(\operatorname{ran}(d)\subseteq D\) and$$\varrho \bigl( d ( Tx,Ty ) ,d ( x,y ) \bigr) >0 \quad \textit{for all }x,y\in X \textit{ such that }x\, \mathcal{S}^{\ast}\, y\textit{ and }Tx\, \mathcal{S}^{\ast}\, Ty. $$
Then T has, at least, a fixed point.
4.2 Coincidence point theorems under \(\mathcal{S}\)regularity and condition (C_{2})
In the following result, the contractivity condition is stronger because we do not assume that \(Tx\, \mathcal{S}^{\ast}\, Ty\) and \(gx\, \mathcal{S}^{\ast}\, gy\). However, the following result is applicable even if T and g are not \(\mathcal{S}\)continuous.
Theorem 37
 (A):

There exists on X a \((T,g,\mathcal{S})\)PicardJungck sequence.
 (B):

There exists an Rfunction \(\varrho\in R_{D}\) such that (\(\varrho_{3}\)) holds, \(\operatorname{ran}(d)\subseteq D\) and$$ \varrho \bigl( d ( Tx,Ty ) ,d ( gx,gy ) \bigr) >0 \quad \textit{for all }x,y\in X\textit{ such that }gx\, \mathcal{S}^{\ast }\, gy. $$(5)
 (c)
\((g(X),d)\) is complete;
 (d)
\((X,d)\) is complete and \(g(X)\) is closed;
 (e)
\((X,d)\) is complete, the pair \((T,g)\) is \(( O,\mathcal{S} ) \)compatible, \(\mathcal{S}\) is gclosed and g is injective on \(g ( X ) \) and \(\mathcal{S}\)continuous.
Then T and g have, at least, a coincidence point.
Proof
Since (5) ⇒ (4), the proof of Theorem 33 can be followed, point by point, to prove that the sequence \(\{gx_{n}\}\), which satisfies \(gx_{n}\neq gx_{n+1}\) for all \(n\in\mathbb{N}\), is a Cauchy sequence. Next, we distinguish some cases.
Let \(M=\{ k\in\mathbb{N}:Tx_{n(k)}=Tz \}\).
Subcase (c.1). Assume that M is finite. In this case, there exists \(k_{0}\in\mathbb{N}\) such that \(Tx_{n(k)}\neq Tz\) for all \(k\geq k_{0} \). Let \(\{a_{k}=d ( Tx_{n(k)},Tz ) \}_{k\geq k_{0}}\) and \(\{b_{k}=d ( gx_{n(k)},gz ) \}_{k\geq k_{0}}\). Then \(a_{k}>0\) and \(b_{k}>0\) for all \(k\geq k_{0}\). Moreover, \(\{b_{k}\}\rightarrow0\). As (6) means that \(\varrho ( a_{k},b_{k} ) >0\) for all \(k\geq k_{0}\), condition (\(\varrho_{3}\)) implies that \(\{a_{k}\}\rightarrow0\). It follows that \(\{gx_{n(k)+1}=Tx_{n(k)}\} \rightarrow Tz\). As this is a subsequence of \(\{gx_{n}\}\) and \(\{gx_{n}\} \rightarrow gz\), we conclude that \(Tz=gz\), that is, z is a coincidence point of T and g.
Subcase (c.2). Assume that M is not finite. In this case, there exists a subsequence \(\{Tx_{n^{\prime}(k)}\}\) of \(\{Tx_{n(k)}\}\) such that \(Tx_{n^{\prime}(k)}=Tz\) for all \(k\in\mathbb{N}\). Since \(gx_{n^{\prime }(k)+1}=Tx_{n^{\prime}(k)}=Tz\) for all \(k\in\mathbb{N}\), and \(\{gx_{n}\}\rightarrow gz\), we also conclude that \(Tz=gz\), that is, z is a coincidence point of T and g.
Case (d). Assume that \(g(X)\) is closed and \((X,d)\) is complete. In this case, we can apply item (c) because any closed subsets of complete spaces are also complete.
Subcase (e.2). Assume that \(M^{\prime}\) is not finite. In this case, there exists a subsequence \(\{gx_{n(k)}\}\) of \(\{gx_{n}\}\) such that \(gx_{n(k)}=u\) for all \(k\in\mathbb{N}\). Hence \(\{Tgx_{n(k)}\} \rightarrow Tu\). □
If \(x\,\mathcal{S}\,y\) for all \(x,y\in X\), we derive the following consequence.
Corollary 38
 (A):

There is on X a \((T,g)\)PicardJungck sequence.
 (B):

There exists an Rfunction \(\varrho\in R_{D}\) such that (\(\varrho_{3}\)) holds, \(\operatorname{ran}(d)\subseteq D\) and$$\varrho \bigl( d ( Tx,Ty ) ,d ( gx,gy ) \bigr) >0\quad \textit{for all }x,y\in X \textit{ such that }gx\neq gy. $$
 (c)
\((g(X),d)\) is complete;
 (d)
\((X,d)\) is complete and \(g(X)\) is closed;
 (e)
\((X,d)\) is complete, g is injective on \(g ( X ) \) and continuous, and the pair \((T,g)\) is Ocompatible.
Then T and g have, at least, a coincidence point.
In the following result, we denote by ≺ a transitive binary relation (for instance, a preorder or a partial order), which is not necessarily reflexive.
Corollary 39
 (A):

\(TX\subseteq g(X)\), T is \(( g,\prec ) \)nondecreasing and there exists a point \(x_{0}\in X\) such that \(gx_{0}\prec Tx_{0}\).
 (B):

There exists an Rfunction \(\varrho\in R_{D}\) such that (\(\varrho_{3}\)) holds, \(\operatorname{ran}(d)\subseteq D\), and$$\varrho \bigl( d ( Tx,Ty ) ,d ( gx,gy ) \bigr) >0 \quad \textit{for all }x,y\in X\textit{ such that }gx\, \mathcal{S}^{\ast}\, gy. $$
 (C):

The binary relation ≺ is transitive (or gtransitive, or \(( T,g ) \)transitive).
 (c)
\((g(X),d)\) is complete;
 (d)
\((X,d)\) is complete and \(g(X)\) is closed;
 (e)
\((X,d)\) is complete, the pair \((T,g)\) is \(( O,\prec ) \)compatible, g is ≺nondecreasing, and g is injective on \(g ( X ) \) and \(\mathcal{S}\)continuous.
Then T and g have, at least, a coincidence point.
Furthermore, if \(g=I_{X}\) is the identity mapping on X, then we derive the following fixed point result.
Corollary 40
 (A):

There is on X a Picard sequence \(\{x_{n+1}=Tx_{n}\}\) such that \(x_{n}\,\mathcal{S}\,x_{m}\) for all \(n,m\in\mathbb{N}\) such that \(n< m\).
 (B):

There exists an Rfunction \(\varrho\in R_{D}\) such that (\(\varrho_{3}\)) holds, \(\operatorname{ran}(d)\subseteq D\), and$$\varrho \bigl( d ( Tx,Ty ) ,d ( x,y ) \bigr) >0 \quad \textit{for all }x,y\in X \textit{ such that }x\, \mathcal{S}^{\ast}\, y. $$
Then T has, at least, a fixed point.
4.3 Coincidence point theorems under \(\mathcal{S}\)regularity and condition (C_{3})
If we assume that the contractivity condition is more restrictive, we can avoid the injectivity of g in condition (e) in Theorem 37.
Theorem 41
 (A):

There is on X a \((T,g,\mathcal{S})\)PicardJungck sequence.
 (B):

There exists an Rfunction \(\varrho\in R_{D}\) such that (\(\varrho_{4}\)) holds, \(\operatorname{ran}(d)\subseteq D\), and$$ \varrho \bigl( d ( Tx,Ty ) ,d ( gx,gy ) \bigr) >0 \quad \textit{for each }x,y\in X\textit{ such that }x\neq y\textit{ and }gx\,\mathcal{S}\,gy. $$(7)
 (e′):

\((X,d)\) is complete, \(\mathcal{S}\) is gclosed, \((T,g)\) is \(( O,\mathcal{S} ) \)compatible, g is \(\mathcal {S}\)continuous and X is \((d,\mathcal{S})\)regular.
Then T and g have, at least, a coincidence point.
Proof
The reader may particularize the previous results to the cases: (1) \(x\,\mathcal{S}\,y\) for all \(x,y\in X\); (2) ≺ is a transitive binary relation on X; (3) g is the identity mapping on X.
5 Common fixed point theorems under \(( R,\mathcal{S} ) \)contractivity conditions
 (\(\mathcal{A}_{xy}\)):

there is a \(( T,g ) \)PicardJungck sequence \(\{z_{n}\}\subseteq X\) such that, for all \(n\in\mathbb{N}\), \(gz_{n}\) is \(\mathcal{S}\)comparable, at the same time, to gx and to gy.
The following result shows that this condition can be guaranteed under some usual properties.
Lemma 42
If \(x,y\in\operatorname{Coin}(T,g)\), \(TX\subseteq g(X)\), T is \(( g,\mathcal{S} ) \)nondecreasing and there exists a point \(z_{0}\in X\) such that \(gz_{0}\,\mathcal{S}\,Tz_{0}\) and \(gz_{0}\) are \(\mathcal{S}\)comparable, at the same time, to gx and to gy, then property (\(\mathcal{A}_{xy}\)) holds.
Proof
Suppose, for instance, that \(gx\,\mathcal{S}\,gz_{0}\) and \(gy\,\mathcal{S}\,gz_{0}\) (the order of the arguments is not important). Let \(\{z_{n}\}\) be a \((T,g ) \)PicardJungck sequence on X based on \(z_{0}\) (it exists by Proposition 27). As T is \(( g,\mathcal {S} ) \)nondecreasing, then \(Tx\,\mathcal{S}\,Tz_{0}\) and \(Ty\,\mathcal{S}\,Tz_{0}\), which means that \(gx\,\mathcal{S}\,gz_{1}\) and \(gy\,\mathcal{S}\,gz_{1}\). Repeating this argument by induction, property (\(\mathcal{A}_{xy}\)) holds. □
In the following result, we take advantage of property (\(\mathcal{A}_{xy}\)) in order to give a first step about the uniqueness of the coincidence point.
Lemma 43
 (p)
gx and gy are \(\mathcal{S}\)comparable.
 (q)
Property (\(\mathcal{A}_{xy}\)) holds and \(\mathcal{S}\) is \(( T,g ) \)compatible.
Then \(gx=gy\).
Proof
Taking into account that (5) ⇒ (4), we can use the contractivity condition (4).
Theorem 44
 (a)
gx and gy are \(\mathcal{S}\)comparable.
 (b)
Property (\(\mathcal{A}_{xy}\)) holds and \(\mathcal{S}\) is \(( T,g ) \)compatible.
Hence T and g have a unique point of coincidence.
If we additionally assume that g (or T) is injective on \(\operatorname{Coin}(T,g)\), then T and g have a unique coincidence point.
Proof
By Theorem 33 (or Theorem 37), the set of all coincidence points of T and g is nonempty, so T and g have, at least, a point of coincidence. Let ω and \(\omega^{\prime}\) be two points of coincidence of T and g. By definition, there are two coincidence points \(x,y\in\operatorname{Coin}(T,g)\) such that \(\omega=Tx=gx\) and \(\omega^{\prime}=Ty=gy\). Thus, it follows from Lemma 43 that \(\omega=gx=gy=\omega^{\prime}\), so T and g have a unique point of coincidence.
Additionally, assume that g (or T) is injective on \(\operatorname{Coin}(T,g)\), and let \(x,y\in\operatorname{Coin}(T,g)\) be two arbitrary coincidence points of T and g. In order to prove that \(x=y\), assume that \(x\neq y\). By Lemma 43, \(Tx=gx=gy=Ty\). And as g (or T) is injective on \(\operatorname{Coin}(T,g)\), then \(x=y\), which contradicts the fact that \(x\neq y\). Thus, \(x=y\) and T and g have a unique coincidence point. □
Theorem 45
Under the hypotheses of Theorem 44, assume that T and g are weakly compatible (or commuting). Then T and g have a unique common fixed point.
Proof
Let \(x_{0}\in X\) be a coincidence point of T and g, and let define \(\omega=gx_{0}\). Since \(Tx_{0}=gx_{0}\) and T and g are weakly compatible, \(Tgx_{0}=gTx_{0}\), so \(T\omega=Tgx_{0}=gTx_{0}=g\omega\). Then ω is another coincidence point of T and g. By Theorem 43, \(gx_{0}=g\omega\), so \(\omega=gx_{0}=g\omega\). In particular, \(\omega =g\omega=T\omega\), so ω is a common fixed point of T and g.
The uniqueness of the common fixed point follows from the fact that any common fixed point is a point of coincidence, and Theorem 44 guarantees that there exists a unique point of coincidence. □
6 A new kind of coincidence point theorems involving Rfunctions
Definition 46
 (\(\Omega_{1}\)):

If \(\{a_{n}\}\subset ( 0,\infty ) \) is a sequence such that \(\phi(a_{n+1})<\psi ( a_{n} ) +\varphi ( a_{n} a_{n+1} )\) for all \(n\in \mathbb{N} \), then \(\{a_{n}\}\rightarrow0\).
 (\(\Omega_{2}\)):

If \(\{a_{n}\},\{b_{n}\}\subset ( 0,\infty ) \) are two sequences converging to the same limit \(L\geq 0\) and verifying that \(L< a_{n}\) and \(\phi(a_{n})<\psi ( b_{n} ) +\varphi ( a_{n} b_{n} )\) for all \(n\in\mathbb{N}\), then \(L=0\).
In condition (\(\Omega_{1}\)), the term \(\varphi ( a_{n} a_{n+1} ) \) can be replaced in different ways, depending on the researcher’s interest.
Example 47
If ψ and ϕ are altering distance functions such that \(\psi <\phi\), and \(\varphi(t)=0\) for all \(t\in [ 0,\infty ) \), then \(( \psi,\phi,\varphi ) \in\Omega\).
Example 48
If ψ is an altering distance function, ϕ is a lower semicontinuous function such that \(\psi<\phi\) and \(\phi^{1} ( \{0\} ) =\{0\}\), and \(\varphi(t)=0\) for all \(t\in [ 0,\infty ) \), then \(( \psi,\phi,\varphi ) \in\Omega\).
The following properties are given in order to show Example 51.
Proposition 49
Proof
Proposition 50
Proof
If \(x=0\), the unique solution of equation (9) is \(s_{0}=0\). Henceforth, assume that \(x>0\) is fixed.
By Proposition 49, any solution \(s_{x}\) of equation (9) satisfies \(0< s_{x}< x\). As we have just seen, to find a solution \(s_{x}\) of equation (9) is equivalent to find a solution \(p_{x}=s_{x} x\) of equation \(f ( p ) =x^{4}/ ( 1+x^{2} ) \).
Next we show a nontrivial example of functions ψ, ϕ, and φ such that \(( \psi,\phi,\varphi ) \in\Omega\).
Example 51
Functions in Ω permit us to introduce a new kind of Rfunctions.
Lemma 52
Obviously, \(\varrho_{\psi,\phi,\varphi}\) is an Rfunction that, in general, cannot be decomposed as \(\psi ( s ) \phi ( t ) \). Obviously, the same property can be studied in an appropriate subset \(A\subseteq [ 0,\infty ) \).
Proof
Example 53
The following corollary is a particular case of Theorem 37 using the described functions ψ, ϕ, and φ. Notice that the new contractivity condition has not been studied in the past. This is only an example of how using Rfunctions in order to establish new contractivity conditions.
Corollary 54
 (A):

There is on X a \((T,g,\mathcal{S})\)PicardJungck sequence.
 (B):

For all \(x,y\in X\) such that \(gx\, \mathcal {S}^{\ast }\, gy\), we have$$ d ( Tx,Ty ) \leq\frac{d ( gx,gy ) ^{3}}{1+d ( gx,gy ) ^{2}}+\frac{\sqrt{ d ( Tx,Ty ) d ( gx,gy ) }}{2 ( 1+d ( Tx,Ty ) d ( gx,gy ) ) }. $$(12)
 (c)
\((g(X),d)\) is complete;
 (d)
\((X,d)\) is complete and \(g(X)\) is closed;
 (e)
\((X,d)\) is complete, g is injective on \(g ( X ) \) and \(\mathcal{S}\)continuous, \(\mathcal{S}\) is gclosed, and the pair \((T,g)\) is \(( O,\mathcal{S} ) \)compatible.
Then T and g have, at least, a coincidence point.
Proof
It follows from Theorem 37 using the Rfunction \(\varrho_{\psi,\phi,\varphi}\) described in Lemma 52 and the triple \(( \psi,\phi,\varphi ) \) of Example 51. □
Example 55
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 the 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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