Observations on relationtheoretic coincidence theorems under Boyd–Wong type nonlinear contractions
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Abstract
In this article, we carry out some observations on existing metrical coincidence theorems of Karapinar et al. (Fixed Point Theory Appl. 2014:92, 2014) and Erhan et al. (J. Inequal. Appl. 2015:52, 2015) proved for Lakshmikantham–Ćirićtype nonlinear contractions involving \((f,g)\)closed transitive sets after proving some coincidence theorems satisfying Boyd–Wongtype nonlinear contractivity conditions employing the idea of \((f,g)\)closed locally ftransitive binary relation.
Keywords
47H10 54H25MSC
\(\mathcal{R}\)complete metric spaces Locally ftransitive binary relations \((g,d)\)selfclosedness \(\mathcal{R}\)connectedness1 Introduction
In last ten decades, the classical Banach contraction principle [3] has been generalized by numerous authors in the different directions by improving the underlying contraction conditions (e.g., [4, 5, 6]), enhancing the number of involved mappings [4, 7], weakening the involved metrical notions [7, 8], and enlarging the class of ambient spaces [9, 10, 11]. In 2004, Ran and Reurings [12] obtained a variant of the classical Banach contraction principle to a complete metric space endowed with partial order relation, which was slightly modified by Nieto and RodríguezLópez [13] in 2005. Later, the trend of utilization of partial order relation in the context of fixed/coincidence point theorems was adopted by various authors [6, 8, 14, 15, 16, 17, 18, 19]. Such generalizations are carried out by improving either the contraction conditions or the underlying spaces keeping the partial order relation fixed; however, several authors adopted another way of improving the Banach contraction principle by using various binary relations, such as preorder (Turinici [20]), transitive relation (BenElMechaiekh [21]), tolerance (Turinici [22, 23]), strict order (Ghods et al. [24]), and symmetric closure (Samet and Turinici [25]).
In 2015, Alam and Imdad [26] obtained yet another generalization of the classical Banach contraction principle employing an amorphous (arbitrary) binary relation and observed that several wellknown metrical fixed point theorems can further be improved up to arbitrary binary relations (instead of partial order, preorder, transitive relation, tolerance, strict order, and symmetric closure). With this in mind, Alam and Imdad [7] introduced relationtheoretic analogues of certain involved metrical notions such as completeness, continuity, gcontinuity, compatibility, etc. and utilized the same to prove coincidence theorems for relationpreserving contractions. It is worth noticing that under the universal relation, such newly defined notions reduce to their corresponding usual notions, and henceforth relationtheoretic metrical fixed/coincidence point theorems reduce to their corresponding classical fixed/coincidence point theorems. Note that relationpreserving contractions remain relatively weaker than usual contractions as they are required to hold merely for the elements that are related in the underlying relation.
Several metrical fixed point theorems under arbitrary binary relations are proved by various authors such as Khan et al. [10], Ayari et al. [27], RoldánLópezdeHierro [28], RoldánLópezdeHierro and Shahzad [29], and Shahzad et al. [30], which are generalizations of the relationtheoretic contraction principle due to Alam and Imdad [26]. Here we can point out that arbitrary binary relation is general enough and often does not work for certain contractions, so that various fixed/coincidence point theorems are proved in metric spaces equipped with different types of binary relations, for example, preorder (RoldánLópezdeHierro and Shahzad [11]), transitive relations (Shahzad et al. [31]), finitely transitive relations (Berzig and Karapinar [32], Berzig et al. [33]), locally finitely transitive relations (Turinici [34, 35]), locally finitely Ttransitive relations (Alam et al. [36]), and locally Ttransitive relations (see Alam and Imdad [37]).
In 2010, Samet and Vetro [38] proved coupled fixed point theorems without using partial ordering employing the idea of an Finvariant set. In 2013, Kutbi et al. [39] weakened the idea of an Finvariant set by introducing the notion of Fclosed sets. Thereafter Karapinar et al. [1] proved some unidimensional versions of earlier coupled fixed point results involving Fclosed sets and then obtained such coupled fixed point results by using the corresponding unidimensional fixed point results. Furthermore, Karapinar et al. [1] observed that the notion of a transitive Fclosed (or Finvariant) set is equivalent to the concept of a preordered set and thereafter also showed that some recent multidimensional results using Finvariant sets can be reduced to wellknown results on ordered metric spaces.
Theorem 1
([1])
 (i)
\(f(X)\subseteq g(X)\),
 (ii)
Mis\((f,g)\)compatible, \((f,g)\)closed, and transitive,
 (iii)
there exists\(x_{0}\in X\)such that\((gx_{0},fx _{0})\in M\),
 (iv)there exists \(\varphi \in \varPhi \) such that$$ d(fx,fy)\leq \varphi \bigl(d(gx,gy) \bigr) \quad \forall x,y \in X\textit{ with }(gx,gy)\in M. $$
 \((a)\)

fandgareMcontinuous and\((O,M)\)compatible,
 \((b)\)

fandgare continuous and commuting,
 \((c)\)

\((X,d,M)\)is regular, and\(g(X)\)is closed.
Theorem 2
([1])
 (i)
\(f(X)\subseteq g(X)\),
 (ii)
fis\((g,\preccurlyeq )\)increasing,
 (iii)
there exists\(x_{0}\in X\)such that\(g(x_{0}) \preccurlyeq f(x_{0})\),
 (iv)there exists \(\varphi \in \varPhi \) such that$$ d(fx,fy)\leq \varphi \bigl(d(gx,gy) \bigr)\quad \forall x,y \in X\textit{ with }g(x)\preccurlyeq g(y), $$
 (v)
\(\varphi (0)=0\), or ≼ is antisymmetric.
 \((a)\)

fandgare continuous and commuting, or
 \((b)\)

\((X,d,\preccurlyeq )\)is regular, and\(g(X)\)is closed.
Erhan et al. [2] slightly modified Theorem 1 by proving the following sharpened version.
Theorem 3
([2])
 (i)
\(f(X)\subseteq g(X)\),
 (ii)
Mis\((f,g)\)closed andgtransitive,
 (iii)
there exists\(x_{0}\in X\)such that\((gx_{0},fx _{0})\in M\),
 (iv)there exists \(\varphi \in \varPhi \) such that$$ d(fx,fy)\leq \varphi \bigl(d(gx,gy) \bigr)\quad \forall x,y \in X\textit{ with }(gx,gy)\in M. $$
 \((a)\)

fandgareMcontinuous and\((O,M)\)compatible,
 \((b)\)

fandgare continuous and\((O,M)\)compatible,
 \((c)\)

fandgare continuous and commuting,
 \((d)\)

\((X,d,M)\)is regular, \(g(X)\)is closed, andMis\((f,g)\)compatible.
Alam and Imdad [7] observed that relationtheoretic metrical fixed/coincidence point results combine the idea contained in Karapinar et al. [1] as the set M (utilized by Karapinar et al. [1]) being subset of \(X^{2}\) is in fact a binary relation on X. The aim of this paper is to prove relatively more sharpened and improved versions of foregoing results using a relationtheoretic approach.
2 Relationtheoretic notions and auxiliary results
In this paper, \(\mathbb{N}\) and \(\mathbb{N}_{0}\) denote the sets of natural numbers and whole numbers, respectively (i.e., \(\mathbb{N} _{0}=\mathbb{N} \cup \{0\}\)). In this section, to make our exposition selfcontained, we give some definitions and basic results related to our main results.
Definition 1
([41])
Let X be a nonempty set. A subset \(\mathcal{R}\) of \(X^{2}\) is called a binary relation on X.
Trivially, \(X^{2}\) and ∅ are binary relations on X, which are respectively called the universal relation (or full relation) and empty relation. Another important relation of this kind is the relation \(\triangle _{X}=\{(x,x):x\in X\}\), called the identity relation (or the diagonal relation) on X.
In this paper, \(\mathcal{R}\) stands for a nonempty binary relation, but for simplicity, we write only “binary relation” instead of “nonempty binary relation.”
Definition 2
([26])
Let \(\mathcal{R}\) be a binary relation defined on a nonempty set X, and let \(x,y\in X\). We say that x and y are \(\mathcal{R}\)comparative and write \([x,y]\in \mathcal{R}\) if either \((x,y)\in \mathcal{R}\) or \({(y,x)\in \mathcal{R}}\).
Definition 3

reflexive if \((x,x)\in \mathcal{R}\)\(\forall x\in X\),

irreflexive if \((x,x)\notin \mathcal{R}\)\(\forall x\in X\),

symmetric if \((x,y)\in \mathcal{R}\) implies \((y,x)\in \mathcal{R}\),

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

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

complete or connected or dichotomous if \([x,y]\in \mathcal{R}\)\(\forall x,y\in X\),

weakly complete or weakly connected or trichotomous if \([x,y]\in \mathcal{R}\) or \(x=y\)\(\forall x,y\in X\).
Definition 4

amorphous if it has no specific properties at all,

a strict order or sharp order if \(\mathcal{R}\) is irreflexive and transitive,

a nearorder if \(\mathcal{R}\) is antisymmetric and transitive,

a pseudoorder if \(\mathcal{R}\) is reflexive and antisymmetric,

a quasiorder or preorder if \(\mathcal{R}\) is reflexive and transitive,

a partial order if \(\mathcal{R}\) is reflexive, antisymmetric, and transitive,

a simple order if \(\mathcal{R}\) is a weakly complete strict order,

a weak order if \(\mathcal{R}\) is a complete preorder,

a total order or linear order or chain if \(\mathcal{R}\) is a complete partial order,

a tolerance if \(\mathcal{R}\) is reflexive and symmetric,

an equivalence if \(\mathcal{R}\) is reflexive, symmetric, and transitive.
Remark 1
Clearly, the universal relation \(X^{2}\) on a nonempty set X remains a complete equivalence relation.
Definition 5
([41])
 (1)
The inverse or transpose or dual relation of \(\mathcal{R}\), denoted by \(\mathcal{R}^{1}\), is defined by \(\mathcal{R}^{1}=\{(x,y)\in X^{2}:(y,x)\in \mathcal{R}\}\).
 (2)
The reflexive closure of \(\mathcal{R}\), denoted by \(\mathcal{R}^{\#}\), is defined as the set \(\mathcal{R}\cup \triangle _{X}\) (i.e., \(\mathcal{R}^{\#}:=\mathcal{R}\cup \triangle _{X}\)). In fact, \(\mathcal{R}^{\#}\) is the smallest reflexive relation on X containing \(\mathcal{R}\).
 (3)
The symmetric closure of \(\mathcal{R}\), denoted by \(\mathcal{R}^{s}\), is defined as the set \(\mathcal{R}\cup \mathcal{R} ^{1}\) (i.e., \(\mathcal{R}^{s}:=\mathcal{R}\cup \mathcal{R}^{1}\)). In fact, \(\mathcal{R}^{s}\) is the smallest symmetric relation on X containing \(\mathcal{R}\).
Proposition 1
([26])
Definition 6
([46])
Let X be a nonempty set, let \(E\subseteq X\), and let \(\mathcal{R}\) be a binary relation on X. Then the restriction of \(\mathcal{R}\) to E, denoted by \(\mathcal{R}\vert _{E}\), is defined as the set \(\mathcal{R}\cap E^{2}\) (i.e., \(\mathcal{R}\vert _{E}:= \mathcal{R}\cap E^{2}\)). In fact, \(\mathcal{R}\vert _{E}\) is a relation on E induced by \(\mathcal{R}\).
Definition 7
([26])
Definition 8
([7])
Note that under the restriction \(g=I\), the identity mapping on X, Definition 8 reduces to the notion of fclosedness of \(\mathcal{R}\) defined in [26].
Proposition 2
([7])
LetXbe a nonempty set, let\(\mathcal{R}\)be a binary relation onX, and letfandgbe selfmappings on X. If\(\mathcal{R}\)is\((f,g)\)closed, then so is\(\mathcal{R}^{s}\).
Definition 9
([7])
Let \((X,d)\) be a metric space, and let \(\mathcal{R}\) be a binary relation on X. We say that \((X,d)\) is \(\mathcal{R}\)complete if every \(\mathcal{R}\)preserving Cauchy sequence in X converges.
Remark 2
Every complete metric space is \(\mathcal{R}\)complete for any binary relation \(\mathcal{R}\). Particularly, under the universal relation, the notion of \(\mathcal{R}\)completeness coincides with usual completeness.
Definition 10
([7])
Let \((X,d)\) be a metric space, and let \(\mathcal{R}\) be binary relation on X. A subset E of X is called \(\mathcal{R}\)closed if every \(\mathcal{R}\)preserving convergent sequence in E converges to a point of E.
Remark 3
Every closed subset of a metric space is \(\mathcal{R}\)closed for any binary relation \(\mathcal{R}\). Particularly, under the universal relation, the notion of \(\mathcal{R}\)closedness coincides with usual closedness.
Proposition 4
([7])
An\(\mathcal{R}\)closed subspace of an\(\mathcal{R}\)complete metric space is\(\mathcal{R}\)complete.
Definition 11
([7])
Let \((X,d)\) be a metric space, let \(\mathcal{R}\) be a binary relation on X, let g be a selfmapping on X, and let \(x\in X\). A mapping \(f:X\rightarrow X\) is called \((g,\mathcal{R})\)continuous at x if for any sequence \(\{x_{n}\}\) such that \(\{gx_{n}\}\) is \(\mathcal{R}\)preserving and \(g(x_{n})\stackrel{d}{ \longrightarrow } g(x)\), we have \(f(x_{n})\stackrel{d}{\longrightarrow } f(x)\). Moreover, f is called \((g,\mathcal{R})\)continuous if it is \((g,\mathcal{R})\)continuous at each point of X.
Note that under the restriction \(g=I\), the identity mapping on X, Definition 11 reduces to the notion of \(\mathcal{R}\)continuity of f defined in [7].
Remark 4
Every continuous (respectively, gcontinuous) mapping is \(\mathcal{R}\)continuous (respectively, \((g,\mathcal{R})\)continuous) for any binary relation \(\mathcal{R}\). Particularly, under the universal relation, the notion of \(\mathcal{R}\)continuity (respectively, \((g,\mathcal{R})\)continuity) coincides with usual continuity (respectively, gcontinuity).
Definition 12
([7])
Remark 5
Definition 13
([7])
Let \((X,d)\) be a metric space, and let g be a selfmapping on X. A binary relation \(\mathcal{R}\) defined on X is called \((g,d)\)selfclosed if for any \(\mathcal{R}\)preserving sequence \(\{x_{n}\}\) such that \(x_{n}\stackrel{d}{ \longrightarrow } x\), there exists a subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) with \([gx_{n_{k}},gx]\in \mathcal{R} \) for all \(k \in \mathbb{N}_{0}\).
Note that under the restriction \(g=I\), the identity mapping on X, Definition 13 reduces to the notion of dselfclosedness of \(\mathcal{R}\) defined in [26].
Definition 14
([47])
Inspired by Turinici [34, 35], Alam and Imdad [37] introduced the following notion by localizing the notion of ftransitivity.
Definition 15
([37])
Let X be a nonempty set, and let f be a selfmapping on X. A binary relation \(\mathcal{R}\) on X is called locally ftransitive if for each (effectively) \(\mathcal{R}\)preserving sequence \(\{x_{n}\}\subset f(X)\) (with range \(E:=\{x_{n}:n \in \mathbb{N}_{0}\}\)), the binary relation \(\mathcal{R}\vert _{E}\) is transitive.
Definition 16
([25])
Let X be a nonempty set, and let \(\mathcal{R}\) be a binary relation on X. A subset E of X is called \(\mathcal{R}\)directed if for each pair \(x,y\in E\), there exists \(z\in X\) such that \((x,z)\in \mathcal{R}\) and \((y,z)\in \mathcal{R}\).
Definition 17
([46])
 (i)
\(z_{0}=x\) and \(z_{k}=y\),
 (ii)
\((z_{i},z_{i+1})\in \mathcal{R}\) for each i (\(0\leq i\leq k1\)).
Definition 18
([7])
Let X be a nonempty set, and let \(\mathcal{R}\) be a binary relation on X. A subset E of X is called \(\mathcal{R}\)connected if for each pair \(x,y\in E\), there exists a path in \(\mathcal{R}\) from x to y.
Inspired by the notion of an \((f,g)\)compatible subset of \(X^{2}\) utilized by Karapinar et al. [1] and RoldánLópezdeHierro et al. [47], we introduce the following notion.
Definition 19
Next, we propose the following fact.
Proposition 5
LetXbe a nonempty set, let\(\mathcal{R}\)be a binary relation onX, and letfandgbe selfmappings on X. If\(\mathcal{R}\)is antisymmetric and\((f,g)\)closed, then\(\mathcal{R}\)is\((f,g)\)compatible.
Proof
Recently, Alam and Imdad [7] proved the following relationtheoretic coincidence theorem under linear contraction.
Theorem 4
([7])
 \((a)\)

\(f(X)\subseteq g(X)\cap Y\),
 \((b)\)

\(\mathcal{R}\)is\((f,g)\)closed,
 \((c)\)

\(X(f,g,\mathcal{R})\)is nonempty,
 \((d)\)
 there exists \(\alpha \in [0,1)\) such that$$ d(fx,fy)\leq \alpha d(gx,gy)\quad \forall x,y\in X \textit{ with }(gx,gy)\in \mathcal{R}, $$
 \((e)\)

 \((e1)\)

fandgare\(\mathcal{R}\)compatible,
 \((e2)\)

gis\(\mathcal{R}\)continuous,
 \((e3)\)

eitherfis\(\mathcal{R}\)continuous, or\(\mathcal{R}\)is\((g,d)\)selfclosed,
 \((e^{\prime })\)

 \((e^{\prime }1)\)

\(Y \subseteq g(X)\),
 \((e^{\prime }2)\)

eitherfis\((g,\mathcal{R})\)continuous, orfandgare continuous, or\(\mathcal{R}\vert _{Y}\)isdselfclosed.
Proposition 6
Finally, we state the following known results, which are needed in the proofs of our main results.
Lemma 1
([18])
Let\(\varphi \in \varOmega \). If\(\{a_{n}\}\subset (0,\infty )\)is a sequence such that\(a_{n+1}\leq \varphi (a_{n})\)for all\(n\in \mathbb{N}_{0}\), then\(\lim_{n\to \infty }a_{n}=0\).
Lemma 2
 (i)
\(k\leq m_{k}< n_{k} \)\(\forall k\in \mathbb{N}\),
 (ii)
\(d(x_{m_{k}},x_{n_{k}})> \epsilon \)\(\forall k \in \mathbb{N}\),
 (iii)
\(d(x_{m_{k}},x_{n_{k1}})\leq \epsilon \)\(\forall k\in \mathbb{N}\).
 (iv)
\(\lim_{k\to \infty }d(x_{m_{k}},x_{n_{k}})= \epsilon \),
 (v)
\(\lim_{k\to \infty } d(x_{m_{k}+1},x_{n_{k}+1})= \epsilon \).
Lemma 3
([50])
LetXbe a nonempty set, and letgbe a selfmapping on X. Then there exists a subset\(E\subseteq X\)such that\(g(E)=g(X)\)and\(g: E \rightarrow X\)is onetoone.
3 Main results
Now, we are equipped to prove the following result on the existence of a coincidence point under the φcontractivity condition.
Theorem 5
 \((a)\)

\(f(X)\subseteq g(X)\cap Y\),
 \((b)\)

\(\mathcal{R}\)is\((f,g)\)closed and locallyftransitive,
 \((c)\)

\(X(f,g,\mathcal{R})\)is nonempty,
 \((d)\)
 there exists \(\varphi \in \varOmega \) such that$$ d(fx,fy)\leq \varphi \bigl(d(gx,gy) \bigr)\quad \forall x,y \in X\textit{ with }(gx,gy)\in \mathcal{R}, $$
 \((e)\)

 \((e1)\)

fandgare\(\mathcal{R}\)compatible,
 \((e2)\)

gis\(\mathcal{R}\)continuous,
 \((e3)\)

eitherfis\(\mathcal{R}\)continuous, or\(\mathcal{R}\)is\((f,g)\)compatible and\((g,d)\)selfclosed,
 \((e^{\prime })\)

 \((e^{\prime }1)\)

\(Y \subseteq g(X)\),
 \((e^{\prime }2)\)

eitherfis\((g,\mathcal{R})\)continuous, orfandgare continuous, or\(\mathcal{R}\)and\(\mathcal{R}\vert _{Y}\)are\((f,g)\)compatible anddselfclosed, respectively.
Proof
 (i)
\(d(ggx_{n_{k}},gz)=0\)\(\forall k\in \mathbb{N} ^{0}\), and
 (ii)
\(d(ggx_{n_{k}},gz)>0\)\(\forall k\in \mathbb{N}^{+}\).
 (i)
\(d(gx_{n_{k}},gu)=0\)\(\forall k\in \mathbb{N} ^{0}\), and
 (ii)
\(d(gx_{n_{k}},gu)>0\)\(\forall k\in \mathbb{N} ^{+}\).
Corollary 1
 \((a)\)

\(f(X)\subseteq g(X)\),
 \((b)\)

\(\mathcal{R}\)is\((f,g)\)closed and locallyftransitive,
 \((c)\)

\(X(f,g,\mathcal{R})\)is nonempty,
 \((d)\)
 there exists \(\varphi \in \varOmega \) such that$$ d(fx,fy)\leq \varphi \bigl(d(gx,gy) \bigr)\quad \forall x,y \in X\textit{ with }(gx,gy)\in \mathcal{R}, $$
 \((e)\)

 \((e1)\)

fandgare\(\mathcal{R}\)compatible,
 \((e2)\)

gis\(\mathcal{R}\)continuous,
 \((e3)\)

eitherfis\(\mathcal{R}\)continuous, or\(\mathcal{R}\)is\((g,d)\)selfclosed and\((f,g)\)compatible,
 \((e^{\prime })\)

 \((e^{\prime }1)\)

there exists an\(\mathcal{R}\)closed subspaceYofXsuch that\(f(X)\subseteq Y \subseteq g(X)\),
 \((e^{\prime }2)\)

eitherfis\((g,\mathcal{R})\)continuous, orfandgare continuous, or\(\mathcal{R}\)and\(\mathcal{R}\vert _{Y}\)are\((f,g)\)compatible anddselfclosed, respectively.
Proof
The result corresponding to part \((e)\) and alternating part \((e^{\prime })\) follows by taking \(Y=X\) in Theorem 5 and using Proposition 4, respectively. □
Remark 6
If g is onto in Corollary 1, then we can remove assumption \((a)\) as in this case it trivially holds. Also, we can omit assumption \((e^{\prime }1)\) as it trivially hods for \(Y=g(X)=X\) using Proposition 3. Whenever f is onto, in view of assumption \((a)\), g must be onto, and hence again the same conclusion is immediate.
Remark 7

The notion of “locally ftransitive binary relation” is weaker than that of “transitive/gtransitive binary relation.”

The notion of “regularity of \((X,d, M)\)” (in the context of hypothesis \((d)\) of Theorem 3) can be replaced by a relatively weaker notion, namely “\((g,d)\)selfclosedness of M.” Further, the notion of “\((g,d)\)selfclosedness of M” is not necessary as it can also alternatively be replaced by either “\((g,M)\)continuity of f” or “continuity of f and g.”

The notion of “\(({O}, M)\)compatibility of f and g” is replaced by a relatively weaker notion, namely “Mcompatibility of f and g.”

There is no need to impose the closedness requirement on \(g(X)\) as it suffices to take an Mclosed subspace Y of X such that \(f(X)\subseteq Y\subseteq g(X)\).
Corollary 2
 (i)
\(\varphi (0)=0\),
 (ii)
gis onetoone,
 (iii)
\(\mathcal{R}\)is antisymmetric.
Proof
Suppose that (i) holds. Take \(x,y\in X\) such that \((gx,gy) \in \mathcal{R}\) and \(g(x)=g(y)\). Utilizing the contractivity condition \((d)\), we get \(d(fx,fy)\leq \varphi (0)=0\), which implies that \(f(x)=f(y)\). It follows that \(\mathcal{R}\) is \((f,g)\)compatible.
Suppose that (ii) holds. Take \(x,y\in X\) such that \((gx,gy)\in \mathcal{R}\) and \(g(x)=g(y)\). As g is onetoone, we get \(x=y\), which implies that \(f(x)=f(y)\). It follows that \(\mathcal{R}\) is \((f,g)\)compatible.
Finally, in case (iii), our result follows from Proposition 5. □
Remark 8

“\((f,g)\)closedness of ≼” is equivalent to “\((g, \preccurlyeq )\)increasingness of f”;

“locally ftransitivity of ≼” is weaker than “transitivity of ≼”;

hypothesis (v) (of Theorem 2) can also be replaced by the injectivity of g;

“commutativity of f and g”(in the context of hypothesis \((a)\) of Theorem 2) is replaced by “≼compatibility of f and g”;

“continuity of f and g” is replaced by the relatively weaker notion “≼continuity of f and g”;

“continuity of f” (in the context of hypothesis \((a)\) of Theorem 2) is replaced by the relatively weaker notion“≼continuity of f”. Further, this notion is also not necessary as it can alternatively be replaced by “\((g,d)\)selfclosedness of ≼”;

“regularity of \((X,d,M)\)” (in the context of hypothesis \((b)\) of Theorem 2) can be replaced by relatively weaker notion namely: “\((g,d)\)selfclosedness of ≼”. Further, this notion is also not necessary as it can alternatively be replaced by “\((g, \preccurlyeq )\)continuity of f” or “continuity of f and \(g''\);

“closedness of whole subspace \(g(X)\)” (in the context of hypothesis \((b)\) of Theorem 2) is also replaced by “≼closedness of any subset Y with \(f(X)\subseteq Y\subseteq g(X)\)”.
The following consequence of Theorem 5 and Corollary 1 is immediate.
Corollary 3
 (i)
\(\mathcal{R}\)is transitive,
 (ii)
\(\mathcal{R}\)isftransitive,
 (iii)
\(\mathcal{R}\)isgtransitive,
 (iv)
\(\mathcal{R}\)is locally transitive.
In view of Remarks 2–5, we conclude that Theorem 5 (also Corollaries 1, 2, and 3) remains true if the usual metrical terms of completeness, closedness, compatibility, continuity, and gcontinuity are used instead of their respective \(\mathcal{R}\)analogues.
Setting \(g = I\), the identity mapping on X, in Theorem 5, we obtain the corresponding fixed point result contained in [7].
 (i)
\(\mathcal{R}\) is locally ftransitive (in assumption \((b)\)),
 (ii)
\(\mathcal{R}\) is \((f,g)\)compatible (in assumptions \((e3)\) and \((e^{\prime }2)\)).
In this case, \(\varphi (0)=0\), and therefore by Corollary 2, \(\mathcal{R}\) is \((f,g)\)compatible. Hence condition (ii) is vacuously met out. Finally, we can accomplish the proof of Theorem 4 by proceeding on the lines of the proof of Theorem 5 (see the proof of Theorem 4.1 in [7]).
4 Uniqueness results
In this section, we present the results regarding the uniqueness of a point of coincidence, coincidence point, and a common fixed point corresponding to some earlier results. Recall that two selfmappings f and g defined on a nonempty set X are called weakly compatible if \(f(x)=g(x)\) implies \(f(gx)=g(fx)\) for all \(x\in X\).
Theorem 6
 \((u_{1})\):

\(f(X)\)is\(\mathcal{R}\vert ^{s}_{g(X)}\)connected, and
 \((u_{2})\):

\(\mathcal{R}\)is\((f,g)\)compatible.
Proof
Corollary 4
 \((u_{1}^{\prime })\)

\(\mathcal{R}\vert _{f(X)}\)is complete, and
 \((u_{1}^{\prime \prime })\)

\(f(X)\)is\(\mathcal{R}\vert ^{s}_{g(X)}\)directed.
Theorem 7
 \((u_{3})\):

one offandgis onetoone.
Theorem 8
 \((u_{4})\):

fandgare weakly compatible.
The proofs of Corollary 4 and of Theorems 7 and 8 are similar to those of Corollary 4.6 and of Theorems 4.7 and 4.8, respectively, which are contained in [7].
5 An illustrative example
In this section, we construct an example to highlight the worth and realized improvements in our newly proved results.
Example 1
Take any \(\mathcal{R}\vert _{Y}\)preserving sequence \(\{x_{n}\}\) such that \(x_{n}\stackrel{d}{\longrightarrow } x\). As \((x_{n},x_{n+1})\in \mathcal{R}\vert _{Y}\) for all \(n\in \mathbb{N}\), there exists \(N \in \mathbb{N}\) such that \(x_{n}=x \in \{0,1\} \) for all \(n \geq N\). Therefore we can choose a subsequence \(\{x_{n_{k}}\}\) of the sequence \(\{x_{n}\}\) such that \(x_{n_{k}}=x\) for all \(k\in \mathbb{N}\), which amounts to saying that \([x_{n_{k}},x]\in \mathcal{R}\vert _{Y}\) for all \(k\in \mathbb{N}\). Hence \(\mathcal{R}\vert _{Y}\) is dselfclosed. We can easily see that contraction condition \((d)\) and the remaining hypotheses of Theorem 5 are also satisfied. Consequently, in view of Theorem 5, f and g have a coincidence point (namely, \(x=0\)).
Furthermore, hypotheses \((u_{1})\), \((u_{2})\), and \((u_{4})\) of Theorem 8 also hold. Thus all the hypotheses of Theorem 8 are satisfied, and hence f and g have a unique common fixed point (namely, \(x=0\)).
Note that the present example cannot be covered by Karapinar et al. [1] and Erhan et al. [2] (i.e., Theorems 1 and 3, respectively), which substantiate the utility of Theorem 5 over Theorems 1 and 3.
6 Conclusions
In view of our newly proved results, we conclude that under relationtheoretic linear contraction, merely an arbitrary binary relation is required. If we extend such results to φcontractions (under the family Ω), then a weaker version of nearorder is required, namely, a “locally ftransitive antisymmetric” (or, more appropriately, “locally ftransitive \((f.g)\)compatible)” binary relation. Particularly, in case \(\varphi (0)=0\) or g is onetoone, a “locally ftransitive binary relation” is sufficient.

\(n =2\): \((x, y)\sqsubseteq _{M^{2}} (u, v) \Longleftrightarrow [(x,y) =(u,v)\text{ or }(u,v,x,y)\in {M^{2}}]\);

\(n=3\): \((x, y, z)\sqsubseteq _{M^{3}} (u,v,w) \Longleftrightarrow [(x,y,z)=(u,v,w)\text{ or }(u,v,w,x,y,z) \in {M^{3}}]\).
 (1)
\(\sqsubseteq _{M}\) is reflexive for whatever M,
 (2)
M satisfies the transitive property if and only if \(\sqsubseteq _{M}\) is a preorder on X.
Therefore, if we redefine \(x\sqsubseteq u\Longleftrightarrow (x,u) \in M \), then it is equivalent to say that \(\sqsubseteq :=M\).
Notes
Acknowledgements
The last author is thankful to University Grant Commission, New Delhi, Government of India, for the financial support in the form of MANF (Moulana Azad National Fellowship). All the authors are thankful to three anonymous learned referees for their encouraging comments on the earlier version of the manuscript.
Availability of data and materials
Not applicable.
Authors’ contributions
All the authors read and approved the final manuscript.
Funding
Not applicable.
Competing interests
The authors declare that they have no competing interests.
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