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 Open Access
Relationtheoretic coincidence and common fixed point results under \((F,\mathcal{R})_{g}\)contractions with an application
 Waleed M. Alfaqih^{1, 2}Email author,
 Mohammad Imdad^{2},
 Rqeeb Gubran^{3} and
 Idrees A. Khan^{4}
https://doi.org/10.1186/s1366301906627
© The Author(s) 2019
 Received: 23 January 2019
 Accepted: 28 May 2019
 Published: 1 July 2019
Abstract
In this paper, we begin with some observations on Fcontractions. Thereafter, we introduce the notion of \((F,\mathcal{R})_{g}\)contractions and utilize the same to prove some coincidence and common fixed point results in the setting of metric spaces endowed with binary relations. An example is also given to exhibit the utility of our results. We also deduce some consequences in the setting of ordered metric spaces. As an application, we investigate the existence and uniqueness of a solution of integral equation of Volterra type.
Keywords
 Fixed point
 Fcontraction
 Coincidence point
 Common fixed point
 Binary relation
 Integral equation
MSC
 47H10
 54E50
 47H09
 54H25
1 Introduction and preliminaries
The celebrated principle, namely Banach contraction principle, is one of the pivotal results of nonlinear analysis. It establishes that, given a complete metric space \((M,d)\) and a selfmapping T on M, if there exists \(h\in (0,1)\) such that \(d(Tx,Ty)\leq hd(x,y)\) for all \(x,y\in M\), then T has a unique fixed point z in M. Moreover, \(\lim_{n\to \infty }T^{n}x=z\) for all \(x\in M\), where \(T^{n}\) (\(n \geq 1\)) is the ntimes composition of T. This principle is a very popular tool for guaranteeing the existence and uniqueness of solutions for considerable problems arising in several branches of mathematics. Banach contraction has been extended and generalized in many directions (see [16, 18, 20, 23, 25, 28, 31, 36, 37] and the references therein). In this respect, Ran and Reurings [31] and Nieto and Rodriguez–Lopez [25, 26] extended the Banach contraction principle in a very interesting way by showing that if the metric space is endowed with an ordered binary relation, then it is enough to assume that the contraction condition holds only for those comparable elements. Recently, this branch of fixed point theory has been developed through many research works (e.g., Bhaskar and Lakshmikantham [5], BenElMechaiekh [4], Samet and Turinici [33], and Imdad et al. [3, 11, 13–15, 19, 24]). Here it can be pointed out that the first result concerning coincidence point was reported in Machuca [22], which was further improved by Goebel [10], while Jungck [17] proved the first ever common fixed point theorem in 1976.
On the other hand, several authors, such as Boyd and Wong [6], Browder [7], Wardowski [37], Jleli and Samet [16], and several others, generalized the Banach contraction principle by employing various types of control functions. In this regard, Wardowski [37] in 2012 introduced the notion of Fcontractions as follows.
Definition 1.1
([37])
 (\(F_{1}\)):

F is strictly increasing;
 (\(F_{2}\)):

For every sequence \(\{\beta _{n}\}\subset (0,\infty )\),$$ \lim_{n\to \infty }\beta _{n}=0 \quad \textit{if and only if } \lim_{n\to \infty }F(\beta _{n})=\infty ; $$
 (\(F_{3}\)):

There exists \(k\in (0,1)\) such that \(\lim_{\beta \to 0^{+}}\beta ^{k}F(\beta )=0\).
Example 1.1
 (i)
\(F(\beta )=\ln \beta \);
 (ii)
\(F(\beta )=\beta \frac{1}{\beta } \);
 (iii)
\(F(\beta )= \ln (\frac{\beta }{3}+\sin \beta )\).
Definition 1.2
([37])
Theorem 1.1
([37])
Every Fcontraction mapping T defined on a complete metric space \((M,d)\) has a unique fixed point (say z). Moreover, for any \(x\in M\), the sequence \(\{T^{n}x\}\) converges to z.
It was remarked that (see [37]) the monotonicity of F implies that every Fcontraction mapping is contractive and hence continuous. Secelean [35] observed that the continuity of an Fcontraction can be obtained from condition \((F_{2})\). In view of these observations, the proof of Theorem 1.1 can be done without condition \((F_{1})\). Piri and Kumam [27] replaced condition \((F_{3})\) by assuming the continuity of F and proved a theorem, analogous to Wardowski’s, which can also be proved without condition \((F_{1})\). Durmaz et al. [9] proved ordertheoretic fixed point results using Fcontraction. Sawangsup et al. [34] introduced the notion of \(F_{R}\)contraction and utilized the same to prove some relationtheoretic fixed point results. Imdad et al. [15] introduced the notion of \((F, \mathcal{R})\)contraction as follows.
Definition 1.3
([15])
Under some suitable assumptions (see [15]) Imdad et al. proved that every \((F,\mathcal{R})\)contraction mapping possesses a unique fixed point.
On the other hand, Wardowski and Dung [38] used the same class of auxiliary functions given in Definition 1.1 to introduce the notion of Fweak contractions as follows.
Definition 1.4
([38])
Wardowski and Dung [38] proved the following theorem.
Theorem 1.2
 (a)
T has a unique fixed point (say \(z\in M\)),
 (b)
\(\lim_{n\to \infty } T^{n}x=z\) for all \(x\in M\).
We observe that Theorem 1.2 can survive without assumptions \((F_{1})\) and \((F_{3})\) besides removing one way implication of assumption \((F_{2})\).
For the sake of completeness, we collect here some basic definitions and fundamental results needed in our subsequent discussions.
From now on, \(\mathbb{N}\) is the set of natural numbers and \(\mathbb{N}_{0}=\{0\} \cup \mathbb{N}\). We write \(\{x_{n}\} \to x\) whenever \(\{x_{n}\}\) converges to x. In the sequel, M stands for a nonempty set and \(T,g: M \to M\). For the sake of brevity, we write Tx instead of \(T(x)\).

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

coincidence point of \((T,g)\) if \(Tx=gx\) (\(\operatorname{Coin}(T,g)\) stands for the set of all such points);

common fixed point of \((T,g)\) if \(x=Tx=gx\).
Recall that the pair \((T,g)\) is commuting on M if \(Tgx=gTx\) for all \(x \in M\) and weakly compatible if \(Tgx=gTx\) for all \(x \in \operatorname{Coin}(T,g)\). The pair \((T,g)\) is called compatible if \(\lim_{n\to \infty }d(gTx_{n},Tgx_{n})=0\), whenever \(\{x_{n}\}\) is a sequence in M such that \(\lim_{n\to \infty }gx_{n}=\lim_{n\to \infty }Tx_{n}\). The mapping T is called gcontinuous at a point \(x\in M\) if for all \(\{x_{n}\}\subseteq M\), \(\{gx_{n}\}\to gx \) implies \(\{Tx_{n}\}\to Tx\).
In this paper, we introduce the notion of \((F,\mathcal{R})_{g}\)contractions and utilize the same to prove some coincidence and common fixed point results in the setting of related metric spaces. An example is given to exhibit the utility of our results. Some consequences in the setting of ordered metric spaces are also obtained. Our results extend and generalize many results of the existing literature (e.g., [1, 10, 17, 31, 37]). As an application, we investigate the existence and uniqueness of a solution of integral equation of Volterra type. Finally, we provide two examples to exhibit the utility of our application.
2 Relation theoretic notions and auxiliary results
A nonempty subset \(\mathcal{R}\) of \(M\times M\) is said to be a binary relation on M. Trivially, \(M\times M\) is a binary relation on M known as the universal relation. For simplicity, we will write \(x\mathcal{R}y\) whenever \((x,y) \in \mathcal{R}\) and write \(x\mathcal{R}^{\nshortparallel }y\) whenever \(x\mathcal{R}y\) and \(x\neq y\). Observe that \(\mathcal{R}^{\nshortparallel }\) is also a binary relation on M and \(\mathcal{R}^{\nshortparallel } \subseteq \mathcal{R}\). The elements x and y of M are said to be \(\mathcal{R}\)comparable if \(x\mathcal{R}y\) or \(y\mathcal{R}x\), this is denoted by \([x,y]\in \mathcal{R}\).

reflexive if \(x\mathcal{R}x\) for all \(x \in M\);

transitive if, for any \(x, y, z \in M\), \(x\mathcal{R}y\) and \(y\mathcal{R}z\) imply \(x\mathcal{R}z\);

antisymmetric if, for any \(x, y \in M\), \(x\mathcal{R}y\) and \(y\mathcal{R}x\) imply \(x=y\);

preorder if it is reflexive and transitive;

partial order if it is reflexive, transitive, and antisymmetric.
Let M be a nonempty set, \(\mathcal{R}\) be a binary relation on M, and \(E\subseteq M\). Then the restriction of \(\mathcal{R}\) to E is denoted by \(\mathcal{R}\arrowvert _{E}\) and is defined by \(\mathcal{R} \cap E^{2}\). The inverse of \(\mathcal{R}\) is denoted by \(\mathcal{R} ^{1}\) and is defined by \(\mathcal{R}^{1}=\{(u,v)\in M\times M: (v,u) \in \mathcal{R}\}\) and \(\mathcal{R}^{s}=\mathcal{R}\cup \mathcal{R} ^{1}\).
Definition 2.1
([1])
Let M be a nonempty set and \(\mathcal{R}\) be a binary relation on M. A sequence \(\{x_{n}\}\subseteq M\) is said to be an \(\mathcal{R}\)preserving sequence if \(x_{n} \mathcal{R}x_{n+1}\) for all \(n\in \mathbb{N}_{0}\).
Definition 2.2
([1])
Let M be a nonempty set and \(T:M\to M\). A binary relation \(\mathcal{R}\) on M is said to be Tclosed if for all \(x,y\in M\), \(x\mathcal{R}y\) implies \(Tx\mathcal{R}Ty\).
Definition 2.3
([2])
Let M be a nonempty set and \(T,g:M\to M\). A binary relation \(\mathcal{R}\) on M is said to be \((T,g)\)closed if for all \(x,y\in M\), \(gx\mathcal{R}gy\) implies \(Tx\mathcal{R}Ty\).
Definition 2.4
([2])
Let \((M,d)\) be a metric space and \(\mathcal{R}\) be a binary relation on M. We say that M is \(\mathcal{R}\)complete if every \(\mathcal{R}\)preserving Cauchy sequence in M converges to a limit in M.
Remark 2.1
Every complete metric space is \(\mathcal{R}\)complete, whatever the binary relation \(\mathcal{R}\). Particularly, under the universal relation, the notion of \(\mathcal{R}\)completeness coincides with the usual completeness.
Definition 2.5
([2])
Let \((M,d)\) be a metric space, \(\mathcal{R}\) be a binary relation on M, \(T:M\to M\), and \(x\in M\). We say that T is \(\mathcal{R}\)continuous at x if, for any \(\mathcal{R}\)preserving sequence \(\{x_{n}\}\subseteq M\) such that \(\{x_{n}\}\to x\), we have \(\{Tx_{n}\}\to Tx\). Moreover, T is called \(\mathcal{R}\)continuous if it is \(\mathcal{R}\)continuous at each point of M.
Remark 2.2
Every continuous mapping is \(\mathcal{R}\)continuous, whatever the binary relation \(\mathcal{R}\). Particularly, under the universal relation, the notion of \(\mathcal{R}\)continuity coincides with the usual continuity.
Definition 2.6
([2])
Let \((M,d)\) be a metric space, \(\mathcal{R}\) be a binary relation on M, \(T,g:M\to M\), and \(x\in M\). We say that T is \((g,\mathcal{R})\)continuous at x if, for any sequence \(\{x_{n}\}\subseteq M\) such that \(\{gx_{n}\}\) is \(\mathcal{R}\)preserving and \(\{gx_{n}\}\to gx\), we have \(\{Tx_{n}\} \to Tx\). Moreover, T is called \((g,\mathcal{R})\)continuous if it is \((g,\mathcal{R})\)continuous at each point of M.
Observe that on setting \(g=I\), Definition 2.6 reduces to Definition 2.5.
Remark 2.3
Every gcontinuous mapping is \((g,\mathcal{R})\)continuous, whatever the binary relation \(\mathcal{R}\). Particularly, under the universal relation, the notion of \((g,\mathcal{R})\)continuity coincides with the usual gcontinuity.
Definition 2.7
([2])
Let \((M,d)\) be a metric space, \(\mathcal{R}\) be a binary relation on M, and \(T,g:M\to M\). We say that the pair \((T,g)\) is \(\mathcal{R}\)compatible if for any sequence \(\{x_{n}\} \subseteq M\) such that \(\{Tx_{n}\}\) and \(\{gx_{n}\}\) are \(\mathcal{R}\)preserving and \(\lim_{n\to \infty }gx_{n}= \lim_{n\to \infty }Tx_{n}=x\in M\), we have \(\lim_{n\to \infty }d(gTx _{n},Tgx_{n})=0\).
Remark 2.4
Every compatible pair is \(\mathcal{R}\)compatible, whatever the binary relation \(\mathcal{R}\). Particularly, under the universal relation, the notion of \(\mathcal{R}\)compatibility coincides with the usual compatibility.
Definition 2.8
([1])
Let \((M,d)\) be a metric space. A binary relation \(\mathcal{R}\) on M is said to be dselfclosed if for any \(\mathcal{R}\)preserving sequence \(\{x_{n}\}\subseteq M\) such that \(\{x_{n}\}\to x\), there exists a subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) such that \([x_{n_{k}},x]\in \mathcal{R}\) for all \(k\in \mathbb{N}_{0}\).
Definition 2.9
([21])
For \(x,y \in X\), a path of length p (\(p \in \mathbb{N}\)) in \(\mathcal{R}\) from x to y is a finite sequence \(\{u_{0}, u_{1}, \ldots, u_{p}\}\subseteq X\) such that \(u_{0}=x\), \(u_{p}=y\) and \((u_{i}, u_{i+1}) \in \mathcal{R}\) for each \(i \in \{0,1, \ldots, p1\}\).
Definition 2.10
([2])
A subset \(E\subseteq X\) is said to be \(\mathcal{R}\)connected if, for each \(x,y \in E\), there exists a path in \(\mathcal{R}\) from x to y.
The following lemmas are needed in the sequel.
Lemma 2.1
Lemma 2.2
([12])
Let M be a nonempty set and \(g:M\to M\). Then there exists a subset \(E\subseteq M\) such that \(g(E)=g(M)\) and \(g:E\to E\) is one–one.
3 Main results
We begin this section by introducing the notion of \((F,\mathcal{R})_{g}\)contractions as follows.
Definition 3.1
The following proposition is immediate due to the symmetricity of the metric d.
Proposition 3.1
 (a)for all \(x,y\in M\) such that \((gx,gy) \in \mathcal{R}\) and \((Tx,Ty)\in \mathcal{R}\),$$ \tau + F\bigl(d(Tx, Ty)\bigr)\leq F\bigl(d(gx,gy)\bigr); $$
 (b)for all \(x,y\in M\) such that either \((gx,gy),(Tx,Ty) \in \mathcal{R}\) or \((gy,gx),(Ty,Tx)\in \mathcal{R}\),$$ \tau + F\bigl(d(Tx, Ty)\bigr)\leq F\bigl(d(gx,gy)\bigr). $$
Now, we are equipped to state and prove our main result on the existence of coincidence points as follows.
Theorem 3.1
 (a)
there exists \(x_{0}\in M\) such that \(gx_{0}\mathcal{R}Tx _{0}\);
 (b)
\(\mathcal{R}\) is \((T,g)\)closed;
 (c)
T is an \((F,\mathcal{R})_{g}\)contraction;
 (d)
 \((\mathrm{d}_{1})\) :

there exists a subset X of M such that \(T(M)\subseteq X \subseteq g(M)\) and X is \(\mathcal{R}\)complete;
 \((\mathrm{d}_{2})\) :

one of the following conditions is satisfied:
 (i)
T is \((g,\mathcal{R})\)continuous; or
 (ii)
T and g are continuous; or
 (iii)
\(\mathcal{R}\arrowvert _{X}\) is dself closed provided (3.1) holds for all \(x,y \in M\) with \(gx\mathcal{R}gy\) and \(Tx\mathcal{R}^{\nshortparallel }Ty\);
 (i)
 \((\mathrm{d}^{\prime })\) :

 \((\mathrm{d}^{\prime }_{1})\) :

there exists a subset Y of M such that \(T(M)\subseteq g(M)\subseteq Y\) and Y is \(\mathcal{R}\)complete;
 \((\mathrm{d}^{\prime }_{2})\) :

\((T,g)\) is an \(\mathcal{R}\)compatible pair;
 \((\mathrm{d}^{\prime }_{3})\) :

T and g are \(\mathcal{R}\)continuous.
Proof
Next, we present a corresponding uniqueness result of Theorem 3.1.
Theorem 3.2
If, in addition to hypotheses (a)–(d) of Theorem 3.1, we assume that, for all distinct coincidence points \(u,v\in \operatorname{Coin}(T, g)\), gu and gv are \(\mathcal{R}\)comparable and one of T and g is oneone, then \((T,g)\) has a unique coincidence point.
Proof
Now, we present a common fixed point result as follows.
Theorem 3.3
If, in addition to the hypotheses of Theorem 3.2, we assume that \((T,g)\) is a weakly compatible pair, then the pair \((T, g)\) has a unique common fixed point.
Proof
Theorem 3.2 guarantees the existence of a unique coincidence point of the pair \((T, g)\); let u be such a point, and let \(z\in M\) be such that \(z = Tu = gu\). As T and g are weakly compatible, we have \(Tz = Tgu = gTu = gz\). Thus, z is a coincidence point of T and g. As u is unique, we must have \(u = z\). Therefore, u is a common fixed point of \((T, g)\) which is indeed unique (in view of the uniqueness of the coincidence point of \((T, g)\)). This completes the proof. □
The following example shows the utility of our results.
Example 3.1
On setting \(g=I\) in Theorem 3.3, we deduce the following corresponding fixed point result.
Theorem 3.4
 (a)
there exists \(x_{0}\in M\) such that \(x_{0}\mathcal{R}Tx _{0}\);
 (b)
\(\mathcal{R}\) is Tclosed;
 (c)
T is \((F,\mathcal{R})\)contraction;
 (d)
 \((\mathrm{d}_{1})\) :

there exists a subset X of M such that \(T(M)\subseteq X\) and X is \(\mathcal{R}\)complete;
 \((\mathrm{d}_{2})\) :

one of the following holds:
 (i)
T is \(\mathcal{R}\)continuous; or
 (ii)
\(\mathcal{R}\arrowvert _{X}\) is dselfclosed provided (1.1) holds for all \(x,y\in M\) with \(x\mathcal{R}y\) and \(Tx\mathcal{R}^{\nshortparallel }Ty\).
 (i)
 (e)
\(u,v\in \operatorname{Fix}(T)\) implies that \([u,v]\in \mathcal{R}\),
The following result presents a weaker assumption to guarantee the uniqueness of the fixed point.
Theorem 3.5
 \((\mathrm{e}^{\prime })\) :

\(\operatorname{Fix}(T)\) is \(\mathcal{R}^{s}\)connected,
Proof
4 Some consequences in ordered metric spaces
Recall that a triplet \((M,d,\preceq )\) is called an ordered metric space if \((M,d)\) is a metric space and \((M,\preceq )\) is an ordered set. By \(x\prec y\) we mean that \(x\preceq y\) and \(x\neq y\).
Definition 4.1
([8])
Let \((M,\preceq )\) be an ordered set and \(T,g:M\to M\). Then T is said to be gincreasing if, for any \(x,y\in M\), \(gx\preceq gy\) implies that \(Tx\preceq Ty\).
Remark 4.1
Observe that the notion T is gincreasing is equivalent to saying that ⪯ is \((T,g)\)closed.
On setting \(\mathcal{R}=\preceq \) in Theorems 3.1–3.3 and using Remark 4.1, we deduce the following result.
Corollary 4.1
 (a)
there exists \(x_{0}\in M\) such that \(gx_{0}\preceq Tx_{0}\);
 (b)
T is gincreasing;
 (c)there exist \(\tau >0\) and a continuous function F satisfying \((F_{2})\) such that$$ \tau + F\bigl(d(Tx, Ty)\bigr)\leq F\bigl(d(gx,gy)\bigr) \quad \textit{for all } x,y \in M \textit{ with } gx\prec gy \textit{ and } Tx \prec Ty; $$
 (d)
there exists a subset X of M such that \(T(M)\subseteq X \subseteq g(M)\) and X is ⪯complete;
 (e)
either T is \((g,\preceq )\)continuous or T and g are continuous;
 (f)
for all distinct coincidence points \(u,v\in \operatorname{Coin}(T, g)\), Tu and gv are ⪯comparable,
On setting \(\mathcal{R}=\,\preceq \) in Theorem 3.4 and using Remark 4.1, we deduce the following result.
Corollary 4.2
 (a)
there exists \(x_{0}\in M\) such that \(x_{0}\preceq Tx_{0}\);
 (b)
T is ⪯increasing;
 (c)there exist \(\tau >0\) and a continuous function F satisfying \((F_{2})\) such that$$ \tau + F\bigl(d(Tx, Ty)\bigr)\leq F\bigl(d(x,y)\bigr) \quad \textit{for all } x,y \in M \textit{ with } x\prec y \textit{ and } Tx \prec Ty; $$
 (d)
there exists a subset X of M such that \(T(M)\subseteq X\) and X is ⪯complete;
 (e)
T is ⪯continuous.
 (f)
\(u,v\in \operatorname{Fix}(T)\) implies that \([u,v]\in \,\preceq \),
5 Application to integral equations
The following definitions are needed in the sequel.
Definition 5.1
Definition 5.2
Now, we are equipped to state and prove our results in this section which run as follows.
Theorem 5.1
Proof
Next, we provide the following theorem in the presence of an upper solution.
Theorem 5.2
Proof
Finally, to exhibit the utility of Theorems 5.2 and 5.1, we adopt the following examples.
Example 5.1
Proof

the function \(K(s,v,x(v))=\ln (1+x(v))\) is nondecreasing in the third variable.

\(\frac{s}{2}\leq \frac{3}{2}s(1+s)\ln (1+s)+\int _{0}^{s}\ln (1+x(v)) \,dv\), \(s\in [0,1]\) so that \(x(s)=\frac{s}{2}\) is a lower solution for (5.4).

in view of its graph (see Fig. 1), the following inequality holds true for all \(x,y\in [0,1]\):Using the nondecreasing function \(s\mapsto \frac{s}{1+0.01s}\), we have$$ \bigl\vert \ln (1+x)\ln (1+y) \bigr\vert \leq \frac{ \vert xy \vert }{1+0.01 \vert xy \vert }. $$(5.5)$$ \bigl\vert \ln (1+x)\ln (1+y) \bigr\vert \leq \frac{ \vert xy \vert }{1+0.01 \vert xy \vert } \leq \frac{ \max_{s\in [0,1]} \vert xy \vert }{1+0.01\max_{s\in [0,1]} \vert xy \vert }= \frac{ \Vert xy \Vert }{1+0.01 \Vert xy \Vert }. $$
Example 5.2
Proof
Declarations
Acknowledgements
All the authors are very grateful to the anonymous referees for their excellent suggestions, which greatly improved the presentation of the paper.
Availability of data and materials
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
Funding
Not applicable.
Authors’ contributions
All the authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Competing interests
All the authors declare that they have no competing interests.
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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