Unified relationtheoretic metrical fixed point theorems under an implicit contractive condition with an application
 Md Ahmadullah^{1},
 Javid Ali^{1}Email author and
 Mohammad Imdad^{1}
https://doi.org/10.1186/s1366301605316
© Ahmadullah et al. 2016
Received: 4 November 2015
Accepted: 9 March 2016
Published: 25 March 2016
Abstract
The main purpose of this article is to establish relationtheoretic metrical fixed point theorems via an implicit contractive condition which is general enough to yield a multitude of corollaries corresponding to several well known contraction conditions (e.g. Banach (Fundam. Math. 3:133181, 1922), Kannan (Am. Math. Mon. 76:405408, 1969), Reich (Can. Math. Bull. 14:121124, 1971), Bianchini (Boll. Unione Mat. Ital. 5:103108, 1972), Chatterjea (C. R. Acad. Bulg. Sci. 25:727730, 1972), Hardy and Rogers (Can. Math. Bull. 16:201206, 1973), Ćirić (Proc. Am. Math. Soc. 45:267273, 1974) and several others) wherein even such corollaries are new results on their own. As an example we utilize our main results, to prove a theorem on the existence and uniqueness of the solution of an integral equation besides providing an illustrative example.
Keywords
MSC
1 Introduction
In 1920, Banach formulated the classical contraction mapping principle in his Ph.D. thesis which was later published in Banach [1]. It is one of the most fruitful and applicable theorems ever proved in classical functional analysis. In the course of the last century, this theorem has been generalized and improved by numerous researchers chiefly by replacing contraction mappings with a relatively more general contractive mappings and this practice is still going on. Rhoades [8] carried out a comparative study of various classes of utilized mappings which include Kannan [2], Reich [3], Bianchini [4], Chatterjea [5], Sehgal [9], Hardy and Rogers [6], Ćirić [7] besides several other ones. The survey article due to Rhoades [8] is generally consulted by every researcher of this domain and also continues to serve as a standard reference.
In 1997, Popa [10] initiated the idea of an implicit function which is designed to cover several well known contraction conditions of the existing literature in one go besides admitting several new ones. Indeed, the strength of an implicit function lies in their unifying power besides being general enough to yield new contraction conditions. Here, it is fascinating to point out that some of the presented examples (in Section 2) are of nonexpansive type and Lipschitzian type. For further details about implicit functions, one can consult [10–17].
In recent years, a multitude of ordertheoretic metrical fixed point theorems have been proved for orderpreserving contractions. This trend appears to be initiated (in 1986) by Turinici [18]. In 2004, unknowingly, Ran and Reurings [19] rediscovered a slightly more natural ordertheoretic version of the Banach contraction principle and utilized his result well to establish the existence and uniqueness of the solution of a system of linear equations under a suitable set of conditions. In the recent past, this result of Ran and Reurings has been generalized and improved by several researchers and by now there exists a considerable literature around this theorem. Out of all such extensions and generalizations, the results due to Nieto and RodríguezLópez [20, 21] and Jachymski [22] deserve special mention. Thereafter, several authors utilized various variants of binary relations namely: preorder, transitive, tolerance, strict order, symmetric closure etc. to prove their respective fixed point theorems. Most recently, Alam and Imdad [23, 24] established a new relationtheoretic version of the Banach contraction principle employing general binary relation which in turn generalizes several well known relevant ordertheoretic fixed point theorems.
The aim of this paper is to prove some unified metrical fixed point theorems employing an arbitrary binary relation under an implicit contractive condition which is general enough to cover a multitude of well known contraction conditions in one go besides yielding several new ones. We also provide an example to demonstrate the generality of our results over several well known corresponding results. Finally, we utilize our results to prove the existence and uniqueness of the solution of an integral equation.
2 Implicit relation
 (F_{1}):

F is nonincreasing in the fifth variable; and \(F(s,t,t,s,s+t,0)\leq0\) for \(s, t \geq0\) implies that there exists \(h \in[0, 1)\) such that \(s\leq ht\);
 (F_{2}):

\(F(s, 0, s, 0, 0,s) > 0\), for all \(s > 0\).
 (F_{3}):

F is nonincreasing in the sixth variable; and \(F(s, s, 0, 0, s,s) > 0\), for all \(s > 0\).
Example 1
The functions \(F :\mathbb{R}^{6}_{+} \to\mathbb{R}\) defined below satisfy the foregoing requirements (see [11, 13, 16, 17]):
(1) \(F (s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6})= s_{1}  ks_{2}\), where \(k\in[0, 1)\);
(2) \(F (s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6})= s_{1}  k(s_{3}+ s_{4})\), where \(k\in[0, 1/2)\);
(3) \(F (s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6})= s_{1}  k(s_{5}+ s_{6})\), where \(k\in[0, 1/2)\);
(4) \(F (s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6})= s_{1}  a_{1}s_{2}  a_{2}(s_{3}+s_{4})  a_{3}(s_{5}+s_{6})\), where \(a_{1},a_{2},a_{3}\in[0, 1)\) and \(a_{1}+2a_{2}+2a_{3}<1\);
(5) \(F (s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6})= s_{1}  ks_{2}  L \min\{s_{3}, s_{4}, s_{5}, s_{6}\}\), where \(k\in[0, 1)\) and \(L\geq0\);
(6) \(F (s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6})= s_{1} (a_{1}s_{2}+a_{2}s_{3}+a_{3}s_{4}+a_{4}(s_{5}+s_{6}) )\), where \(a_{1}, a_{2}, a_{3}, a_{4} \geq0\) and \(a_{1}+ a_{2}+ a_{3}+2 a_{4}< 1\);
(7) \(F (s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6})= s_{1}  k \max \{s_{2}, s_{3}, s_{4}, \frac{s_{5} +s_{6}}{2} \} L \min\{s_{3}, s_{4}, s_{5}, s_{6}\}\), where \(k\in[0, 1)\) and \(L\geq0\);
(8) \(F (s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6})= s_{1}  k \max\{s_{2}, s_{3}, s_{4}, s_{5}, s_{6}\}\), where \(k\in[0, 1/2)\);
(9) \(F (s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6})= s_{1}(a_{1}s_{2}+a_{2}s_{3}+a_{3}s_{4}+a_{4}s_{5}+a_{5}s_{6})\), where \(a_{i}\)’s >0 (for \(i=1,2,3,4,5\)) and sum of them is strictly less than 1;
(10) \(F (s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6})= s_{1}k \max \{ s_{2},s_{3},s_{4},\frac{s_{5}}{2},\frac{s_{6}}{2} \} \), where \(k\in[0,1)\);
(11) \(F (s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6})= s_{1}k \max \{s_{2}, s_{3}, s_{4}\}(1k)(as_{5}+ bs_{6})\), where \(k\in[0,1)\) and \(0\leq a,b <{1/2}\);
(12) \(F (s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6})= s_{1}^{2}s_{1} (a_{1}s_{2}+a_{2}s_{3}+a_{3}s_{4} )a_{4}s_{5}s_{6}\), where \(a_{1}>0\); \(a_{2},a_{3},a_{4}\geq 0\); \(a_{1}+a_{2}+a_{3}<1\) and \(a_{1}+a_{4}<1\);
(14) \(F (s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6})= s_{1}^{2}a_{1} \max\{s_{2}^{2},s_{3}^{2},s_{4}^{2}\}a_{2} \max\{s_{3}s_{5},s_{4}s_{6}\}a_{3}s_{5}s_{6}\), where \(a_{i}\)’s ≥0 (for \(i=1,2,3\)); \(a_{1}+2a_{2}<1\) and \(a_{1}+a_{3}<1\);
(15) \(F (s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6})= s^{3}_{1} k (s^{3}_{2}+ s^{3}_{3}+ s^{3}_{4}+ s^{3}_{5}+ s^{3}_{6} )\), where \(k\in[0,1/11)\);
3 Relevant relationtheoretic notions
In this section, we present some basic definitions, propositions and relevant relationtheoretic variants of some metrical notions namely: completeness and continuity.
Definition 1
[25]
A binary relation on a nonempty set X is defined as a subset of \(X\times X\), which will be denoted by \(\mathcal{R}\). We say that ‘x relates to y under \(\mathcal{R}\)’ iff \((x,y)\in \mathcal{R}\).
In the following, \(\mathcal{R}\) stands for a nonempty binary relation while \(\mathbb{N}_{0}\) denotes the set of whole numbers, i.e., \(\mathbb{N}_{0}= \mathbb{N}\cup\{0\}\). In this presentation, we always employ a nonempty binary relation (i.e., \(\mathcal{R} \ne \emptyset\)).
Definition 2
[23]
Let \(\mathcal{R}\) be a binary relation defined on a nonempty set X. Then any pair of points x, y in X is said to be \(\mathcal{R}\)comparative if either \((x,y)\in \mathcal{R}\) or \((y,x)\in\mathcal{R}\), which is together written as \([x,y]\in\mathcal{R}\).
Definition 3
[26]
A binary relation \(\mathcal{R}\) is called complete if every elements are comparable under that relation (i.e., \([x,y]\in \mathcal{R}\) \(\forall x,y\in X\)).
Definition 4
[25]
 (i)the inverse (or transpose or dual) relation of \(\mathcal{R}\), is defined as$$\mathcal{R}^{1}=\bigl\{ (x,y)\in X^{2}:(y,x)\in\mathcal{R} \bigr\} \mbox{ which is denoted by } \mathcal{R}^{1}; $$
 (ii)
the symmetric closure of \(\mathcal{R}\) is defined as the smallest symmetric relation containing \(\mathcal{R}\) (i.e., \(\mathcal{R}^{s}:=\mathcal{R}\cup\mathcal{R}^{1}\)). Often, it is denoted by \(\mathcal{R}^{s}\).
Proposition 1
[23]
Definition 5
[23]
Definition 6
[23]
Alam and Imdad [24] introduced relationtheoretic variants of some metrical notions namely: completeness and continuity.
Definition 7
Let \((X,d,\mathcal{R})\) be a metric space equipped with a binary relation \(\mathcal{R}\) defined on X. We say that \((X,d)\) is \(\mathcal{R}\)complete if every \(\mathcal {R}\)preserving Cauchy sequence in X converges to a point in X.
Remark 1
Under any binary relation \(\mathcal{R}\), every complete metric space is \(\mathcal{R}\)complete. Particularly, under the universal relation the notion of \(\mathcal{R}\)completeness coincides with usual completeness.
Definition 8
Let \((X,d,\mathcal{R})\) be a metric space equipped with a binary relation \(\mathcal{R}\) defined on X. Then a mapping \(T:X\rightarrow X\) is called \(\mathcal{R}\)continuous at x if for any \(\mathcal{R}\)preserving sequence \(\{x_{n}\}\) with \(x_{n}\stackrel{d}{\longrightarrow} x\), we have \(T(x_{n})\stackrel{d}{\longrightarrow} T(x)\). As usual, T is called \(\mathcal{R}\)continuous if it is \(\mathcal{R}\)continuous on the whole of X.
Remark 2
Under any binary relation \(\mathcal{R}\), every continuous mapping is \(\mathcal{R}\)continuous. Particularly, under the universal relation the notion of \(\mathcal{R}\)continuity coincides with usual continuity.
Definition 9
[23]
Let \((X,d,\mathcal {R})\) be a metric space equipped with a binary relation \(\mathcal{R}\) defined on X. Then \(\mathcal{R}\) is called dselfclosed if for any \(\mathcal{R}\)preserving sequence \(\{x_{n}\}\) with \(x_{n}\stackrel{d}{\longrightarrow} x\), there is a subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) such that \([x_{n_{k}},x]\in\mathcal{R}\), \(\forall k\in\mathbb{N}_{0}\).
Definition 10
[27]
Let \((X,d,\mathcal {R})\) be a metric space equipped with a binary relation \(\mathcal{R}\) defined on X. Then a subset D of X is called \(\mathcal{R}\)directed if for every pair of points x, y in D, there is z in X such that \((x,z)\in\mathcal{R}\) and \((y,z)\in\mathcal{R}\).
Definition 11
[28]
Let \(\mathcal{R}\) be a binary relation defined on a nonempty set X and a pair of points x, y in X. If there is a finite sequence \(\{z_{0},z_{1},z_{2},\ldots,z_{l}\}\subset X\) such that \(z_{0}=x\), \(z_{l}=y\) and \((z_{i},z_{i+1})\in\mathcal{R}\) for each \(i\in\{0,1,2,\ldots,l1\}\), then this finite sequence is called a path of length l (where \(l\in\mathbb{N}\)) joining x to y in \(\mathcal{R}\).
Observe that a path of length l involves \((l+1)\) elements of X that need not be distinct in general.

\(F(T)\): the collection of all fixed points of T;

\(X(T,\mathcal{R})\): the set of all points x in X such that \((x,Tx)\in\mathcal{R}\);

\(\Delta(D,\mathcal{R}) :=\bigcup_{x,y\in D} \{z\in X: (x,z)\in\mathcal{R} \mbox{ and } (y,z)\in\mathcal{R} \}\);

\(\Upsilon(x,y,\mathcal{R})\): the collection of all paths joining x to y in \(\mathcal{R}\) where \(x,y \in X\);

\(\Upsilon_{T}(x,y,\mathcal{R})\): the collection of all paths \(\{ z_{0},z_{1},z_{2},\ldots,z_{l}\}\) joining x to y in \(\mathcal{R}\) such that \([z_{i}, Tz_{i}]\in\mathcal{R}\) for each \(i\in\{1,2,3,\ldots,l1\}\).
4 Fixed point theorems
Now, we are equipped to prove the main result of this paper.
Theorem 1
 (a)
\((X,d)\) is \(\mathcal{R}\)complete,
 (b)
\(X(T,\mathcal{R})\) is nonempty,
 (c)
\(\mathcal{R}\) is Tclosed,
 (d)
either T is \(\mathcal{R}\)continuous or \(\mathcal{R}\) is dselfclosed,
 (e)there exists an implicit function \(F\in \mathcal{F}\) withfor all \(x,y\in X\) such that \((x,y)\in \mathcal{R}\).$$F\bigl(d(Tx,Ty), d(x,y), d(x,Tx), d(y,Ty), d(x,Ty), d(y,Tx)\bigr)\leq 0, $$
Proof
Hence, owing to (F_{1}), we obtain \(d(x, Tx)=0\), so that \(Tx=x\), i.e., x is the fixed point of T.
Similarly, if \((x, x_{n_{k}})\in\mathcal{R}\), \(\forall k\in\mathbb {N} _{0}\), then owing to (F_{2}), we obtain \(d(Tx, x)=0\), so that \(Tx=x\), i.e., x is the fixed point of T.
Thus, in all the cases T has a fixed point. □
Theorem 2
Proof
Define \(d^{i}_{n} := d(z^{i}_{n}, z^{i+1}_{n})\), for all \(n\in \mathbb{N}_{0}\) and for each \(i\in\{0,1,2,\ldots,l1\}\).
If \(\mathcal{R}\) is complete or X is \(\mathcal{R}^{s}\)directed, then the following corollary is worth recording.
Corollary 1
 (\(\mathrm{f}^{\prime}\)):

\(\mathcal{R}\) is complete;
 (\(\mathrm{f}^{\prime\prime}\)):

X is \(\mathcal{R}^{s}\)directed and \(\Delta(X,\mathcal{R}^{s})\subset X(T,\mathcal{R}^{s})\).
Proof
Suppose that the condition (\(\mathrm{f}^{\prime}\)) holds. Then for any pair of points x, y in X, we have \([x,y]\in\mathcal{R}\), which implies that \(\{x,y\}\) is a path of length 1 from x to y in \(\mathcal{R}^{s}\), so that \(\Upsilon_{T}(x,y,\mathcal{R}^{s})\) is nonempty. Finally, proceeding along the lines of the proof of Theorem 2, we complete the proof.
Alternatively, if (\(\mathrm{f}^{\prime\prime}\)) holds, then for any pair of points x, y in X, there is z in X such that \([x,z]\in\mathcal{R}\) and \([y,z]\in\mathcal{R}\) so that \(\{x,z,y\}\) is a path of length 2 joining x to y in \(\mathcal{R}^{s}\). As \(z\in\Delta(X,\mathcal{R}^{s})\subset X(T,\mathcal{R}^{s})\), therefore \([z,Tz]\in\mathcal{R}\). Thus, for each x, y in X, \(\Upsilon_{T}(x,y,\mathcal{R}^{s})\) is nonempty and hence in view of Theorem 2 the result follows. □
From Theorems 1 and 2, we can deduce a host of corollaries which are embodied in the following.
Corollary 2
Proof
The proof of Corollary 2 follows from Theorems 1, 2, and the examples of the implicit function, (1)(16). □
Remark 3

Corollary 2 corresponding to condition (8) can be viewed a relationtheoretic version of the Banach contraction principle which was established by Alam and Imdad [23].

Corollary 2 corresponding to condition (9) is a relationtheoretic version of famous Kannan fixed point theorem proved in [2], which remains a new result.

Corollary 2 corresponding to condition (10) is a relationtheoretic version of a fixed point theorem of Chatterjea [5], which is not reported in the literature till date.

Corollary 2 corresponding to condition (12) is a relationtheoretic version of a fixed point theorem due to Bianchini [4], which is new to the existing literature.

Corollary 2 corresponding to condition (16) with \(a_{4}=0\) is a relationtheoretic version of a fixed point theorem of Reich [3], which is indeed new.

Corollary 2 corresponding to condition (18) is merely a partial (due to the fact \(k\in[0,1/2)\)) relationtheoretic version of Ćirić [7], which has remained unreported in the literature.

Corollary 2 corresponding to condition (19) is a relationtheoretic version of Hardy and Rogers [6], which is yet another addition to the existing literature.
As specified in Corollary 2, results corresponding to (11), (13)(15), (17), (20), (21) are relationtheoretic versions of several known fixed point theorems of the existing literature, whereas, the results corresponding to (22)(26) are new.
We utilize the following example to demonstrate the genuineness of our extension.
Example 2
5 An application
Now, we give the following definitions.
Definition 12
Definition 13
Theorem 3
Proof
(vi) Finally, let x and y be arbitrary elements of \(C(I,\mathbb {R})\) and \(z:=\max\{x,y\}\). Then \(x(t)\leq z(t)\) and \(y(t)\leq z(t)\) for all \(t\in I\). This implies that \((x,z)\in\mathcal{R}\) and \((y,z)\in\mathcal{R}\). Therefore, the finite sequence \(\{x,z,y\}\) describes a path which joins x to y in \(\mathcal{R}\).
Now, on using Corollary 2 corresponding to (8) (see Remark 3), the mapping T admits a unique fixed point, which also remains a unique solution of the problem described by (27). □
Declarations
Acknowledgements
All the authors thank the referees for their valuable suggestions and comments bringing about several improvements. The second author is thankful to UGCIndia for a Startup Grant (No. F.3062/2014(BSR)).
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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