# Employing Locally FinitelyT-Transitive Binary Relations to Prove Coincidence Theorems for Nonlinear Contractions

Authors: Mohammad Arif; Idrees A. Khan; Mohammad Imdad; Aftab Alam
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this article, we prove some relation-theoretic results on coincidence and common fixed point for a nonlinear contraction employing a locally finitelyT-transitive binary relation, where T stands for a self-mapping on the underlying metric space. Our newly proved results deduce sharpened versions of certain relevant results of the existing literature. Finally, we adopt some examples to substantiate the genuineness of our proved results herein.


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## Body

## 1. Introduction

The study of order-theoretic metrical fixed point theory was initiated in 1986 by Turinici [1]. After two decades, Ran Reurings [2] extended the classical Banach contraction principle employing partial ordering, which is indeed a natural variant of Turinici’s results (cf. [1]). Thereafter, Neito and Rodrìguez-Lop̀ez [3] sharpened the fixed point theorem of Ran Reurings [2]. In this continuation, several researchers generalized and extended the fixed point theorem of Neito and Rodrìguez-Lop̀ez [3] (see references [4–14]). In 2015, Alam and Imdad [15] proved relatively more natural results of Neito and Rodrìguez-Lop̀ez [3] using an amorphous (arbitrary) binary relation. Most recently, Alam et al. [16] obtained a variant of fixed point theorem of Alam and Imdad [15] under nonlinear contractions. As noticed in [16], results proved under nonlinear contractions cannot be extended to the amorphous binary relation. Often, proving the fixed point result under nonlinear contractions, the underlying binary relation requires transitivity. In order to weaken the transitivity condition, Alam et al. [16] utilized the locally finitely T-transitive binary relation.The aim of this paper is to extend the results of Alam et al. [16] to a pair of self-mappings defined on a relational metric space. The locally finitely T-transitive binary relation is employed, which covers previously known results for nonlinear contractions. Finally, we furnish some illustrative examples to demonstrate the worth of our newly proved results.


## 2. Preliminaries

In this sequel, firstly we recall some known relevant definitions.Definition 1 (see [17, 18]).
LetT1 and T2 be self-mappings on a nonempty set ℳ, then(i)
IfT2u=T1u for some u in ℳ, then u is called a coincidence point of T1 and T2(ii)
Ifu∈ℳ is a coincidence point of T1 and T2 and u¯∈ℳ such that u¯=T2u=T1u, then u¯ is called a point of coincidence of T1 and T2(iii)
Ifu∈ℳ is a coincidence point of T1 and T2 such that u=T2u=T1u, then u is called a common fixed point of T1 and T2(iv)
IfT2T1u=T1T2u for all u in ℳ, then T1 and T2 are called commuting(v)
IfT1 and T2 commute at their coincidence points, i.e., for any u∈ℳ, T2u=T1u⟹T2T1u=T1T2u, then T1 and T2 are called weakly compatible.Definition 2 (see [19–21]).
LetT1 and T2 be self-mappings on a nonempty set ℳ, then(i)
IfdT2T1u,T1T2u≤dT2u,T1u for all u∈ℳ, then T1 and T2 are called weakly commuting.(ii)
Iflimn⟶∞dT2T1un,T1T2un=0, with un being a sequence in ℳ such that limn⟶∞T2un=limn⟶∞T1un, then T1 and T2 are called compatible.(iii)
If for all sequencesun⊂ℳ, with T2un⟶dT2u⟹T1un⟶dT1u, then T1 is said to be T2-continuous at u in ℳ. Moreover, T1 is called T2-continuous if it is T2-continuous at each point of ℳ.


## 3. Relation-Theoretic Notions and Some Auxiliary Results

For a nonempty setℳ, a subset ℛ of ℳ2 is called a binary relation on ℳ. ℛ is defined as uℛv or u,v∈ℛ. Usually, the universal relation and empty relation (void relation), respectively, on ℳ are denoted by ℳ2 and ∅.Definition 3 (see [22, 23]).
A binary relationℛ on a nonempty set ℳ is called(i)
Reflexive ifu,u∈ℛ,∀u∈ℳ(ii)
Symmetric if wheneveru,v∈ℛ and then v,u∈ℛ(iii)
Antisymmetric if wheneveru,v∈ℛ and v,u∈ℛ and then u=v(iv)
Transitive if wheneveru,v∈ℛ and v,z∈ℛ and then u,z∈ℛ(v)
Complete or connected or dichotomous ifu,v∈ℛ,∀u,v∈ℳ(vi)
Weakly complete or weakly connected or trichotomous ifu,v∈ℛ or u=v, ∀u,v∈ℳ.Definition 4 (see [22–27]).
Letℛ be a binary relation defined on a nonempty set ℳ. Then, ℛ is said to be(i)
Amorphous ifℛ has no specific properties at all(ii)
Near order ifℛ is antisymmetric and transitive(iii)
Pseudo-order ifℛ is reflexive and antisymmetric(iv)
Quasiorder or preorder ifℛ is reflexive and transitive(v)
Strict order orsharp order if ℛ is irreflexive and transitive(vi)
Partial order ifℛ is reflexive, antisymmetric, and transitive.(vii)
Tolerance ifℛ is reflexive and symmetric(viii)
Equivalence ifℛ is reflexive, symmetric, and transitive
Throughout this paper,ℛ stands for a nonempty binary relation, but for the sake of simplicity, we often write “binary relation” instead of “nonempty binary relation.” Also, ℕ stands for the set of natural numbers, while ℕ0 for the set of whole numbers (i.e.,ℕ0=ℕ∪0).Remark 1.
Notice that the “universal relationℳ2” on a nonempty set ℳ remains a “complete equivalence relation.”Definition 5 (see [30]).
For a binary relationℛ defined on a nonempty set ℳ, U⊆ℳ such that the restriction of ℛ to U is denoted by ℛ|U and is defined to be the set ℛ∩U2i.e.,ℛ|U:=ℛ∩U2. In fact, ℛ|U is a relation on U induced by ℛ.Definition 6 (see [15]).
A binary relationℛ defined on a nonempty set ℳ is called “ℛ-comparative,” if either u,v∈ℛ or v,u∈ℛ for u,v∈ℳ. We denote it by “u,v∈ℛ.”Definition 7 (see [28]).
GivenN∈ℕ0,N≥2, a binary relation ℛ defined on a nonempty set ℳ is called N-transitive if for any u0,u1,…,uN∈ℳ,(1)ui−1,ui∈ℛfor eachi1≤i≤N⟹u0,uN∈ℛ.
Notice that “2-transitivity” coincides with “transitivity.” Following Turinici [29], ℛ is called “finitely transitive” if it is “N-transitive” for some N≥2.Definition 8 (see [29]).
GivenN∈ℕ0,N≥2, a binary relation ℛ defined on a nonempty set ℳ is called “locally finitely transitive” if for each denumerable (effectively) subset E of ℳ, there exists N=NE≥2 such that ℛ|E is “N-transitive.”Definition 9 (see [31]).
Letℳ be a nonempty set equipped with a binary relation ℛ and T a self-mapping on ℳ. We say that ℛ is “T-transitive” if for any u,v,z∈ℳ,(2)Tu,Tv,Tv,Tz∈ℛ⟹Tu,Tz∈ℛ.
Henceforth, the notions of “T-transitivity” and “locally finitely transitivity” are not only weaker as compared to “transitivity” but also independent of one another.Definition 10 (see [31]).
Letℳ be a nonempty set equipped with a binary relation ℛ and T a self-mapping on ℳ. Then ℛ is called “locally T-transitive” if for each denumerable (effectively) subset E of TM, such that ℛ|E is “transitive.”Definition 11 (see [16]).
Letℳ be a nonempty set equipped with a binary relation ℛ and T a self-mapping on ℳ. Then for a given N∈ℕ0,N≥2, ℛ is called “locally finitely T-transitive” if for each denumerable (effectively) subset E of Tℳ, there exists N=NE such that ℛ|E is “N-transitive.”
In view of Proposition1 [32], the following are predictable.Proposition 1.
Letℳ be a nonempty set equipped with a binary relation ℛ and T a self-mapping on ℳ. Then,(i)
ℛ is T-transitive ⟺ ℛ|Tℳ is transitive(ii)
ℛ is locally finitely T-transitive ⟺ ℛ|Tℳ is locally finitely transitive(iii)
ℛ is transitive ⟹ ℛ is finitely transitive ⟹ ℛ is locally finitely transitive ⟹ ℛ is locally finitely T-transitive(iv)
ℛ is transitive ⟹ ℛ is T-transitive ⟹ ℛ is locally finitely T-transitive.Proposition 2.
Letℳ be a nonempty set, ℛ a binary relation defined on ℳ, and T,T2 a pair of self-mappings on ℳ with T1ℳ⊆T2ℳ. Then,(i)
ℛ is T2-transitive ⟹ ℛ is T1-transitive(ii)
ℛ is locally finitely T2-transitive ⟹ ℛ is locally finitely T1-transitive(iii)
ℛ is transitive ⟹ ℛ is finitely transitive ⟹ ℛ is locally finitely transitive ⟹ ℛ is locally finitely T2-transitive(iv)
ℛ is transitive ⟹ ℛ is finitely T2-transitive ⟹ ℛ is locally finitely T2-transitive(v)
ℛ is transitive ⟹ ℛ is T2-transitive ⟹ ℛ is locally T2-transitive ⟹ ℛ is locally finitely T2-transitive(vi)
ℛ is transitive ⟹ ℛ is T2-transitive ⟹ ℛ is locally finitely T2-transitive.Definition 12 (see [22]).
Letℛ be a binary relation defined on a nonempty set ℳ. Then,(1)
The dual relation ofℛ, denoted by ℛ−1, is defined by ℛ−1:=u,v∈ℳ2:v,u∈ℛ.(2)
“The symmetric closure ofℛ,” denoted by ℛs, is defined to be the set ℛ∪ℛ−1i.e.,ℛs:=ℛ∪ℛ−1. In fact, ℛs is the smallest symmetric relation on ℳ containing ℛ.Proposition 3 (see [15]).
For a binary relationℛ defined on a nonempty set ℳ,(3)u,v∈ℛs⟺u,v∈ℛ.Definition 13 (see [15]).
Letℳ be a nonempty set equipped with a binary relation ℛ. A sequence un⊂ℳ is called “ℛ-preserving” if(4)un,un+1∈ℛ∀n∈ℕ0.
Notice that the term “ℛ-nondecreasing” is utilized by Shahzad et al. [33] and Roldán-López-de-Hierro and Shahzad [34] instead of “ℛ-preserving.”Definition 14 (see [15]).
Letℳ be a nonempty set equipped with a binary relation ℛ and T1 a self-mapping on ℳ. Then, ℛ is called T1-closed if (for u,v∈ℳ)(5)u,v∈ℛ⟹T1u,T1v∈ℛ.Definition 15 (see [35]).
Letℳ be a nonempty set equipped with a binary relation ℛ and T1,T2 a pair of self-mappings on ℳ. Then, ℛ is said to be T1,T2-closed if for all u,v∈ℳ,(6)T2u,T2v∈ℛ⟹T1u,T1v∈ℛ.
Definition15 reduces to Definition 14, if we take T2=I (the identity mapping on ℳ).Proposition 4 (see [35]).
Letℳ be a nonempty set equipped with a binary relation ℛ and T1,T2 a pair of self-mappings on ℳ. If ℛ is “T1,T2-closed,” then so is ℛs.
Now, we recall the metrical notions via a binary relation, namely, “completeness,” “closedness,” “continuity,” “T2-continuity,” and “compatibility.”Definition 16 (see [35]).
Letℳ,d be a metric space equipped with a binary relation ℛ. We say that ℳ,d is “ℛ-complete” if every ℛ-preserving Cauchy sequence in ℳ converges in ℳ.Remark 2.
Complete metric space impliesℛ-complete, for any binary relation ℛ. Converse implication is true with respect to the universal relation.Definition 17 (see [35]).
Letℳ,d be a metric space equipped with a binary relation ℛ. A subset U of ℳ is called “ℛ-closed” if every ℛ-preserving sequence in U converges to a point of U.Remark 3.
Every closed subset of a metric space isℛ-closed for any binary relation ℛ. Indeed, via the universal relation, the notion of ℛ-closedness coincides with usual closedness.Proposition 5 (see [35]).
Everyℛ-complete subspace of a metric space is ℛ-closed.Proposition 6 (see [35]).
Everyℛ-closed subspace of an ℛ-complete metric space is ℛ-complete.Definition 18 (see [35]).
Letℳ,d be a metric space equipped with a binary relation ℛ and u∈ℳ. A mapping T1:ℳ⟶ℳ is called “ℛ-continuous” at u if for any ℛ-preserving sequence un such that un⟶du, we have T1un⟶dT1u. Furthermore, T1 is called ℛ-continuous if it is ℛ-continuous at each point of ℳ.Remark 4.
Notice that “continuity” implies “ℛ-continuity” for any binary relation ℛ. In fact, under the universal relation, the notion of “ℛ-continuity” leads to “continuity.”Definition 19 (see [35]).
Letℳ,d be a metric space equipped with a binary relation ℛ and T2 a self-mapping on ℳ and u∈ℳ. A mapping T1:ℳ⟶ℳ is called “T2,ℛ-continuous” at u if for any sequence un such that T2un is ℛ-preserving and T2un⟶dT2u, we have T1un⟶dT1u. Moreover, T1 is called “T2,ℛ-continuous” if it is “T2,ℛ-continuous” at each point of ℳ.
On settingT2=I (the identity mapping on ℳ), Definition 19 leads to Definition 18.Remark 5.
Notice thatT2-continuity implies T2,ℛ-continuity, for any binary relation ℛ. In fact, via the universal relation, the notion of T2,ℛ-continuity leads to T2-continuity.Definition 20 (see [35]).
Letℳ,d be a metric space equipped with a binary relation ℛ and T1,T2 be a pair of self-mappings defined on ℳ. Then, T1 and T2 are said to be “ℛ-compatible” if for any sequence un⊂ℳ such that T1un and T2un are ℛ-preserving and limn⟶∞T2un=limn⟶∞T1un, we have limn⟶∞dT2T1un,T1T2un=0.Remark 6.
Given a metric spaceℳ,d and a binary relation ℛ on ℳ, “commutativity” ⟹ “weak commutativity”⟹ “compatibility” ⟹ “ℛ-compatibility” ⟹ “weak compatibility.” Via the universal relation, “ℛ-compatibility” reduces to “compatibility.”Definition 21 (see [15]).
Letℳ,d be a metric space. A binary relation ℛ on ℳ is called “d-self-closed” if for any ℛ-preserving sequence un such that un⟶du, there exists a subsequence unkofunwithunk,u∈ℛ,∀k∈ℕ0.Definition 22 (see [35]).
Letℳ,d be a metric space and T2 a self-mapping on ℳ. A binary relation ℛ on ℳ is called “T2,d-self-closed” if for any ℛ-preserving sequence un such that un⟶du, there exists a subsequence unk of un with T2unk,T2u∈ℛ,∀k∈ℕ0.
On takingT2=I (the identity mapping on ℳ), Definition 22 reduces to Definition 21.Definition 23 (see [24]).
Letℳ be a nonempty set equipped with a binary relation ℛ. A subset U of ℳ is called “ℛ-directed” if for each pair u,v∈U, there exists z∈ℳ such that u,z∈ℛ and v,z∈ℛ.Definition 24 (see [30]).
Letℳ be a nonempty set equipped with a binary relation ℛ. Given u,v∈ℳ, a path of length k (where k is a natural number) in ℛ from u to v is a finite sequence z0,z1,z2,…,zk⊂ℳ satisfying the following conditions:(i)
z0=u,andzk=v(ii)
zi,zi+1∈ℛ for each i0≤i≤k−1.
Notice that a path of lengthk involves k+1 elements of ℳ, although they are not necessarily distinct.Definition 25 (see [35]).
Letℳ be a nonempty set equipped with a binary relation ℛ. A subset U of ℳ is called “ℛ-connected” if for each pair u,v∈U, there exists a path in ℛ from u to v.Definition 26 (see [32]).
Letℳ be a nonempty set equipped with a binary relation ℛ and T1,T2 a pair of self-mappings on ℳ. Then, ℛ is called “T1,T2-compatible” if for all u,v∈ℳ,(7)T2u,T2v∈ℛ,andT2u=T2v⟹T1u=T1v.
Given a nonempty setℳ, a binary relation ℛ on ℳ, and a pair of self-mappings T1,T2 on ℳ, we employ the following notations:(i)
“CT1,T2:=u∈ℳ:T2u=T1u,”i.e., the set of all coincidence points of T1 and T2(ii)
“C¯T1,T2:=u¯∈ℳ:u¯=T2u=T1u,u∈ℳ,”i.e., the set of all points of coincidence of T1 and T2(iii)
“ℳT1,ℛ:=u∈ℳ:u,T1u∈ℛ”(iv)
“ℳT1,T2,ℛ:=u∈ℳ:T2u,T1u∈ℛ.”
In order to prove our main results, we utilize the following class of control functions:(8)Φ=ϕ:0,∞⟶0,∞:ϕt<tfor eacht>0andlim supr⟶tϕr<tfor eacht>0.
In view of the symmetry ofd, we can have the following.Proposition 7.
Ifℳ,d is a metric space, ℛ is a binary relation on ℳ, T1,T2 is a pair of self-mappings on ℳ, and φ∈Φ, then the following contractive conditions are equivalent:(i)
dT1u,T1v≤φdT2u,T2vu,v∈ℳ with T2u,T2v∈ℛ(ii)
dT1u,T1v≤φdT2u,T2vu,v∈ℳ with T2u,T2v∈ℛ.
Combining Theorems4 and 5 of Alam et al. [16], we record the following for the future references.Theorem 1 (see [16]).
Letℳ,d be a metric space equipped with a binary relation ℛ and T a self-mapping on ℳ. Assume that the following conditions hold:(a)
ℳ,d is ℛ-complete.(b)
ℛ is locally finitely T-transitive and T-closed.(c)
EitherT is ℛ-continuous or ℛ is d-self-closed.(d)
ℳT,ℛ is nonempty.(e)
There existsϕ∈Φ such that(9)dTu,Tv≤ϕdu,v∀u,v∈ℳwithu,v∈ℛ.
Then,T has a fixed point.(f)
Furthermore, ifuTℳ is ℛs-connected, then T has a unique fixed point.
In this sequel, we recall some results, which are needed in our foregoing discussion.Lemma 1 (see [37]).
Letϕ∈Φ. If tn⊂0,∞ is a sequence such that tn+1≤ϕtn∀n∈ℕ0, then limn⟶∞tn=0.Lemma 2 (see [38]).
Letℳ be a nonempty set and T2 a self-mapping on ℳ. Then, there exists a subset U⊆ℳ such that T2U=T2ℳ and T2:U⟶ℳ is one-to-one.Lemma 3 (see [37]).
Letℳ be a nonempty set and T1,T2 a pair of self-mappings on ℳ. If T1 and T2 are weakly compatible, then every point of coincidence of T1 and T2 is also a coincidence point of T1 and T2.Lemma 4 (see [39]).
Letℳ,d be a metric space and un a sequence in ℳ. If un is not a Cauchy sequence, then there exist ϵ>0 and two subsequences unk and umk of un such that(i)
k≤mk<nk,∀k∈ℕ(ii)
dumk,unk≥ϵ(iii)
dumk,upk<ϵ,∀pk∈mk+1,mk+2,…,nk−2,nk−1.
In addition to this, ifun also verifies limn⟶∞dun,un+1=0, then(10)limk⟶∞dumk,unk+p=ϵ,∀p∈ℕ0.Lemma 5 (see [29]).
Letℳ be a nonempty set, ℛ a binary relation on ℳ, and zn an ℛ-preserving sequence in ℳ. If ℛ is N-transitive on Z:=zn:n∈ℕ0 for some natural number N≥2, then(11)zn,zn+1+rN−1∈ℛ,∀n,r∈ℕ0.


## 4. Main Results

Firstly, we establish a result on the existence of the coincidence point, which runs as follows.Theorem 2.
Letℳ,d be a metric space equipped with a binary relation ℛ and an ℛ-complete subspace N of ℳ. Let T1,T2 be a pair of self-mappings on ℳ. Suppose that the following conditions hold:(a)
T1ℳ⊆T2ℳ∩N(b)
ℛ is locally finitely T1-transitive and T1,T2-closed(c)
ℳT1,T2,ℛ is nonempty(d)
There existsϕ∈Φ such that dT1u,T1v≤ϕdT2u,T2v,∀u,v∈ℳ with T2u,T2v∈ℛ(e)
e1T1 and T2 are ℛ-compatiblee2T2 is ℛ-continuouse3 Either T1 is ℛ-continuous or ℛ is T1,T2-compatible and T2,d-self-closedOr alternately,e′e1′N⊆T2ℳe2′ Either T1 is T2,ℛ-continuous or T1 and T2 are continuous or ℛ and ℛ|N are T1,T2-compatible and d-self-closed, respectively.
Then,T1 and T2 have a coincidence point.Proof.
In lieu of hypothesisa, it is equivalent to say that T1ℳ⊆T2ℳ and T1ℳ⊆N. With a view of hypothesis c, let u0∈ℳ such that T2u0,T1u0∈ℛ. If T2u0=T1u0, then u0 is a coincidence point of T1 and T2, and hence, we are done. Otherwise, as T1ℳ⊆T2ℳ, there exists some u1∈ℳ such that T2u1=T1u0. Again, since T1u1∈ℳ, there exists some u2∈ℳT1ℳ⊆T2ℳ such that T2u2=T1u1. In the similar fashion, we can obtain a sequence un⊆ℳ of joint iterations of T1 and T2 with the initial point u0,i.e.,(12)T2un+1=T1un,for alln∈ℕ0.
Now, we assert thatT2un and T1un are ℛ-preserving sequences, i.e.,(13)T1un,T1un+1∈ℛ,for alln∈ℕ0,(14)T2un,T2un+1∈ℛ,for alln∈ℕ0.
We prove these assertions by mathematical induction, but for the sake of simplicity, we prove only (14). Equation (13) follows with the help of (12) and (14). On utilizing the fact that u0∈ℳT1,T2,ℛ and (12) (for n=0), we have(15)T2u0,T2u1∈ℛ,which implies that (14) is true for n=0. Assume that (14) is true for n=r>0, i.e.,(16)T2ur,T2ur+1∈ℛ.
In view ofT1,T2-closedness of ℛ, we have(17)T1ur,T1ur+1∈ℛ,which on employing (12) gives rise to(18)T2ur+1,T2ur+2∈ℛ.
Hence, (14) is true for n=r+1. Thus, by the induction process, (14) is true for n∈ℕ0. In view of (12) and (14), we obtain (13).
Therefore,T2un and T1un are ℛ-preserving. Applying the contractive condition d to (14) with T2un, not equal to T2un+1for  alln∈ℕ0, we obtain(19)dT2un+1,T2un+2=dT1un,T1un+1≤ϕdT2un,T2un+1,for  alln∈ℕ0.
Owing to Lemma1, we obtain(20)limn⟶∞dT2un,T2un+1=0.
Now, we show thatT2un is a Cauchy sequence. Suppose on the contrary that T2un is not Cauchy. Therefore, owing to Lemma 4, there exist ϵ>0 and two subsequences T2unk and T2umk of the sequence T2un such that k≤mk<nk with dT2unk,T2umk≥ϵ and dT2unk,T2upk<ϵ, where pk∈mk+1,mk+2,…,nk−2,nk−1. Furthermore, in view of (20), Lemma 4 assures us that(21)limn⟶∞dT2umk,T2unk+p=ϵ,∀p∈ℕ0.
In view of (12), T2un⊂T1ℳ⊆N and hence the range T1:=T2un:n∈ℕ0 (of the sequence T2un) is a denumerable subset of T1ℳ. Hence by locally finite T1-transitivity of ℛ, there exists a natural number N=NE≥2 such that ℛ|T1 is N-transitive.
Sincemk<nk and N−1>0, by the division rule, we have(22)nk−mk=N−1μk−1+N−ηkμk−1≥0,0≤N−ηk<N−1⟺nk+ηk=mk+1+N−1μkμk≥1,1<ηk≤N,where μk and ηk are suitable natural numbers such that ηk lies in the interval 1,N. Hence, without loss of generality, we can choose subsequences T2unk and T2umk of T2un (satisfying (21)) such that ηk remains constant, say η, independent of k. We write (23)mk′=nk+η=mk+1+N−1μk,where η1<η≤N is constant. Owing to (21) and (23), we obtain(24)limk⟶∞dT2umk,T2umk′=limk⟶∞dT2umk,T2unk+η=ϵ.
Using triangular inequality, we have(25)dT2umk+1,T2umk′+1≤dT2umk+1,T2umk+dT2umk,T2umk′+dT2umk′,T2umk′+1,which, on letting k⟶∞ and using (20) and (24), yields that(26)limk⟶∞dT2umk+1,T2umk′+1=ϵ.
In view of (23) and Lemma 5, we have T2umk,T2umk′∈ℛ. We denote δk=dT2umk,T2umk′. Now, owing to (12) and hypothesis d, we have(27)dT2umk+1,T2umk′+1=dT1umk,T1umk′≤ϕdT2umk,T2umk′=ϕδk,so that(28)dT2umk+1,T2umk′+1≤ϕδk.
Utilizing the fact thatδk⟶ϵ as k⟶∞ (in view of (26)) and by the definition of Φ, we have(29)lim supk⟶∞ϕδk=lim supt⟶ϵϕt<ϵ.
On making limit superior ask⟶∞ in (28), using (26) and (29), we obtain(30)ϵ=limsupk⟶∞dT2umk+1,T2umk′+1≤limsupk⟶∞ϕδk<ϵ,which shows that our supposition was wrong so that T2un is a Cauchy sequence. Owing to (12), T2un⊂T1ℳ⊆N so that T2un is an ℛ-preserving Cauchy sequence in N. Owing to the ℛ-completeness of N, there exists v∈N such that(31)limn⟶∞T2un=v.
In lieu of (12) and (31), we obtain(32)limn⟶∞T1un=v.
Now, to establish the proof with respect to hypothesese and e′, firstly suppose the hypothesis e holds. T2un is an ℛ-preserving convergent sequence. Therefore, applying e2 (i.e.,ℛ-continuity of T2), (14), and (31), we have(33)limn⟶∞T2T2un=T2limn⟶∞T2un=T2v.
Again, in view of hypothesise2 (i.e.,ℛ-continuity of T2), (13), and (32), we have(34)limn⟶∞T2T1un=T2limn⟶∞T1un=T2v.
Owing to (13) and (14) (i.e.,T1un and T2un are ℛ-preserving, respectively) and limn⟶∞T1un=limn⟶∞T2un=v (due to (31) and (32)), using assumption e1 (i.e., ℛ-compatibility of T1 and T2), we obtain(35)limn⟶∞dT2T1un,T1T2un=0.
Now, we claim thatv is a coincidence point of T1 and T2. To do this, we utilize hypothesis e3. Assume that T1 is ℛ-continuous. Due to (14), (31), and ℛ-continuity of T1, we get(36)limn⟶∞T1T2un=T1limn⟶∞T2un=T1v.
Utilizing (34)–(36) and continuity of d, we get(37)dT2v,T1v=dlimn⟶∞T2T1un,limn⟶∞T1T2un=limn⟶∞dT2T1un,T1T2un=0,so that(38)T2v=T1v.
Therefore,v is a coincidence point of T1 and T2. Alternately, suppose that ℛ is T2,d-self-closed. Owing to (14) and (31) (i.e.,T2un is ℛ-preserving and T2un⟶dv, respectively) and T2,d-self-closedness of ℛ, there exists a subsequence T2unk of T2un such that(39)T2T2unk,T2v∈ℛ,∀k∈ℕ0.
SinceT2unk⟶dv, equations (31)–(36) also survive for unk instead of un. Employing (39), assumption d, definition of Φ, and Proposition 7, we obtain(40)dT1T2unk,T1v≤ϕdT2T2unk,T2v,∀k∈ℕ0,and we claim that(41)dT1T2unk,T1v≤dT2T2unk,T2v,∀k∈ℕ.
In order to establish (41), we take a partition ℕ0,ℕ+ of ℕ,i.e.,ℕ0∪ℕ+=ℕandℕ0∩ℕ+=∅, verifying that(i)
dT2T2unk,T2v=0∀k∈ℕ0(ii)
dT2T2unk,T2v>0∀k∈ℕ+
For case (i), using (39) and the T1,T2-compatibility of ℛ, we get dT1T2unk,T1v=0∀k∈ℕ0, and hence, (41) holds for all k∈ℕ0. For case (ii), owing to definition of Φ, we have dT1T2unk,T1v≤ϕdT2T2unk,T2v<dT2T2unk,T2v,∀k∈ℕ+, and hence, (41) holds for all k∈ℕ+. Thus in all cases, (41) holds for all k∈ℕ.
Applying (33)–(35), (41), continuity of d, and triangular inequality, we get(42)dT1v,T2v≤dT1v,T1T2unk+dT1T2unk,T2T1unk+dT2T1unk,T2vordT1v,T2v<dT2v,T2T2unk+dT1T2unk,T2T1unk+dT2T1unk,T2v⟶0ask⟶∞,so that(43)T2v=T1v.Thus, v is a coincidence point of T1 and T2. Hence, we are through.
Secondly, suppose that hypothesise′ holds. Due to hypothesis e1′ (i.e., N⊆T2ℳ), one can obtain some u∈ℳ such that v=T2u. Hence, (31) and (32), respectively, reduce to(44)limn⟶∞T2un=T2u,(45)limn⟶∞T1un=T2u.
Now, we show thatu is a coincidence point of T1 and T2. To do this, owing to hypothesis e2′, assuming that T1 is T2,ℛ-continuous, and then employing (14) and (44), we get(46)limn⟶∞T1un=T1u.
Due to (45) and (46), we get(47)T2u=T1u.
Hence, conclusion follows.
Next, suppose thatT1 and T2 are continuous. In view of Lemma 2, there exists a subset U⊆ℳ with T2U=T2ℳ and T2:U⟶ℳ is one-to-one. We define H:T2U⟶T2ℳ by(48)HT2a=T1a∀T2a∈T2Uwherea∈U.
SinceT1ℳ⊆T2ℳ and T2:U⟶ℳ is one-to-one, H is well defined. Since T1 and T2 are continuous, so is H. Utilizing T2ℳ=T2U and assumptions a and e1′, respectively, yields that T1ℳ⊆T2U∩N and N⊆T2U, which confirms that, without loss of generality, we can find a sequence such that unn=1∞⊂U verifying (12), enabling us to choose u∈U. Employing (44), (45) and (48) and continuity of H, we get(49)T1u=HT2u=Hlimn⟶∞T2un=limn⟶∞HT2un=limn⟶∞T1un=T2u.
Hence,u is a coincidence point of T1 and T2, and hence again, we are through.
Lastly, suppose thatℛ|N is d-self-closed. Since T2un is ℛ|N-preserving (owing to (14)) and T2un⟶dT2u∈N (in view of (44)), owing to d-self-closedness of ℛ|N, there exists a subsequence T2unk of T2un such that(50)T2unk,T2u∈ℛN,∀k∈ℕ0.
Applying (12), (50), assumption d, and Proposition 7, we obtain(51)dT2unk+1,T1u=dT1unk,T1u≤ϕdT2unk,T2u,∀k∈ℕ0.
We claim that(52)dT2unk+1,T1u≤dT2unk,T2u,∀k∈ℕ.
In order to establish (52), we take a partition ℕ0,ℕ+ of ℕ,i.e.,ℕ0∪ℕ+=ℕandℕ0∩ℕ+=∅, verifying that(i)
dT2unk,T2u=0,∀k∈ℕ0(ii)
dT2unk,T2u>0,∀k∈ℕ+
For case (i), utilizing (50) and T1,T2-compatibility of ℛ, we get dT1unk,T1u=0∀k∈ℕ0, and hence, (52) holds for all k∈ℕ0. For case (ii), owing to the definition of Φ, we have dT2unk+1,T1u=dT1unk,T1u≤ϕdT2unk,T2u<dT2unk,T2u,∀k∈ℕ+, and hence, (52) holds for all k∈ℕ+. Thus, (52) holds for all k∈ℕ.
Applying (44), (52), and continuity of d, we get(53)dT2u,T1u=dlimk⟶∞T2unk+1,T1u=limk⟶∞dT2unk+1,T1u≤limk⟶∞dT2unk,T2u=0,so that(54)T2u=T1u.
Hence,u is a coincidence point T1 and T2. Thus in all cases, T1 and T2 have a coincidence point on T1 and T2, which completes the proof.
In lieu of Propositions1 and 2, we obtain the following consequence of Theorem 2.Corollary 1.
Theorem2 remains true if “locally finitely T1-transitive” (employed in hypothesis b) is replaced by one of the following conditions (besides retaining the rest of the hypotheses):(i)
ℛ is transitive(ii)
ℛ is finitely transitive(iii)
ℛ is T1-transitive(iv)
ℛ is T2-transitive(v)
ℛ is finitely T1-transitive(vi)
ℛ is finitely T2-transitive(vii)
ℛ is locally T1-transitive(viii)
ℛ is locally T2-transitive.Corollary 2.
Letℳ be a nonempty set, ℛ a binary relation, and d a metric such that ℳ,d is an ℛ-complete metric space. Let T1,T2 be a pair of self-mappings on ℳ. Assume that the following conditions hold:(a)
T1ℳ⊆T2ℳ(b)
ℛ is T1,T2-closed and locally finitely T1-transitive(c)
ℳT1,T2,ℛ is nonempty(d)
There existsϕ∈Φ such that dT1u,T1v≤ϕdT2u,T2v∀u,v∈ℳ with T2u,T2v∈ℛ(e)
e1T1 and T2 are ℛ-compatiblee2T2 is ℛ-continuouse3 Either T1 is ℛ-continuous or ℛ is T1,T2-compatible and T2,d-self-closedOr alternately,e′e1′ There exists an ℛ-closed subspace N of ℳ such that T1ℳ⊆N⊆T2ℳe2′ Either T1 is T2,ℛ-continuous or T1 and T2 are continuous or ℛ and ℛ|N are T1,T2-compatible and d-self-closed, respectively.
Then,T1 and T2 have a coincidence point.Proof.
The proof of the parte and the alternative part e′ remains as follows by setting N=ℳ in Theorem 2 and utilizing Proposition 6, respectively.Corollary 3.
Conclusions of Theorem2 (even Corollary 1) remain true, if we replace the assumption T1,T2-compatibility of ℛ (utilized in assumptions (e3) and (e2′)) by one of the following conditions while retaining the rest of the hypotheses:(i)
ϕ0=0(ii)
T2 is one-to-one.Proof.
Assume thati holds. Let u,v,∈ℳ such that T2u,T2v∈ℛ with T2u=T2v. Utilizing the contractive condition d, we get dT1u,T1v≤ϕ0=0, which implies that T1u=T1v. It follows that ℛ is T1,T2-compatible. Next, assume that ii holds. Take u,v∈ℳ such that T2u,T2v∈ℛ and T2u=T2v. Since T2 is one-to-one, we get u=v, which implies that T1u=T1v. Hence, ℛ is T1,T2-compatible.
In lieu of Remarks2–6, we obtain the more natural form of Theorem 2 in the following consequence.Corollary 4.
If the natural metrical notions such as completeness, closedness, compatibility (or commutativity or weak commutativity), continuity, andT2-continuity are used instead of their respective ℛ-analogues, then Theorem 2 (and Corollary 2) remains holding.
Now, we present a uniqueness result corresponding to Theorem2.Theorem 3.
Assume in addition to hypotheses of Theorem2, the following conditions hold:f1: T1ℳ is ℛ|T2ℳs-connectedf2: ℛ is T1,T2-compatible.Then, T1 and T2 have a unique point of coincidence.Proof.
In lieu of Theorem2, we have C¯T1,T2≠∅. Let u¯,v¯∈C¯T1,T2, then there exist u,v∈ℳ such that(55)u¯=T2u=T1u,v¯=T2v=T1v.
Now, we assert thatu¯=v¯. Since T1u,T1v∈T1ℳ⊆T2ℳ, by assumption f1, there exists a path (say T2w0,T2w1,T2w2,…,T2wk) of some finite length k in ℛ|T2ℳs from T1u to T1v (where w0,w1,w2,…,wk∈ℳ). Owing to (55), without loss of generality, we may set w0=u and wk=v, and we have(56)T2wi,T2wi+1∈ℛT2ℳfor eachi0≤i≤k−1.
Now, we construct the constant sequenceswn0=u and wnk=v. Utilizing (55), we have T2wn+10=T1wn0=u¯andT2wn+1k=T1wnk=v¯∀n∈ℕ0. Put w01=w1,w02=w2,…,w0k−1=wk−1. Since T1ℳ⊆T2ℳ, on the patterns similar to that of Theorem 2, we can obtain sequences wn0,wn0,…,wn0 in ℳ way such that T2wn+11=T1wn1,T2wn+12=T1wn2,…,T2wn+1k−1=T1wnk−1∀n∈ℕ0. Hence, we obtain(57)T2wn+1i=T1wni∀n∈ℕ0and for eachi0≤i≤k.
Now, we assert that(58)T2wni,T2wni+1∈ℛ∀n∈ℕ0and for eachi0≤i≤k−1.
We prove the assertion by using the mathematical induction. It follows from (56) that (58) holds for n=0. Assume that (58) holds for n=r>0, i.e.,(59)T2wri,T2wri+1∈ℛfor eachi0≤i≤k−1.
Sinceℛ is T1,T2-closed, using Proposition 4, we obtain(60)T1wri,T1wri+1∈ℛfor eachi0≤i≤k−1.
Employing (57) gives rise to(61)T2wr+1i,T2wr+1i+1∈ℛfor eachi0≤i≤k−1.
Hence, (58) holds for n=r+1. Thus, by induction, (58) holds for all n∈ℕ0. Now for all n∈ℕ0 and for each i0≤i≤k−1, define δni=:dT2wni,T2wni+1. Then, we claim that(62)limn⟶∞δni=0for eachi0≤i≤k−1.
We fixi and distinguish two cases. Firstly, assume that δn0i:=dT2wn0i,T2wn0i+1=0 for some n0∈ℕ0,i.e.,(63)T2wn0i=T2wn0i+1.
Now, owing to (57) and (63) and utilizing the hypothesis f2, we obtain dT1wn0i,T1wn0i+1=0. Consequently owing to (57), δn0+1i=dT2wn0+1i,T2wn0+1i+1=dT1wn0i,T1wn0i+1=0. Thus by induction, we get δni=0∀n≥n0, yielding thereby limn⟶∞δni=0, so that δn0+1imust be “0,” and hence by mathematical induction on n, we get δni=0∀n≥n0 and limn⟶∞δni=0. Secondly, assume that δni>0∀n∈ℕ0. Due to (57), (58), and hypothesis d, we have(64)δn+1i=dT2wn+1i,T2wn+1i+1=dT1wni,T1wni+1≤ϕdT2wni,T2wni+1,so that(65)δn+1i≤ϕδni.
In view of Lemma1 and (65), we have(66)limn⟶∞δni=0.
Thus in all cases, (62) is proved for each i0≤i≤k−1. Utilizing the triangular inequality and (62), we obtain(67)du¯,v¯≤δn0+δn1+⋯+δnk−1⟶0asn⟶∞.
Therefore,u¯=v¯, which exhausts the proof.Corollary 5.
Replace the conditionf1 in Theorem 3 by one of the following conditions:f1′ℛ|T1ℳ is completef1″T1ℳ is ℛ|T2ℳs-directed.Then, the same conclusion of Theorem 3 follows.Theorem 4.
Suppose all the hypotheses of Theorem3 and the following condition hold:f3 Either T1 or T2 is one-to-one.Then, T1 and T2 have a unique point of coincidence.Theorem 5.
Suppose all the hypotheses of Theorem3 and the following condition embodied in condition e′ of Theorem 3 hold:e3′: T1 and T2 are weakly compatible.Then, T1 and T2 have a unique common fixed point.
The proofs of Corollary5 and of Theorems 4 and 5 can be obtained in the similar lines of Corollary 4.6 and Theorems 4.7 and 4.8, respectively, contained in [35].
TakingT2=Iℳ, identity mapping on ℳ, and Theorems 2 and 3, we get the following fixed point result which is a sharpened version corresponding to Theorem 1.Corollary 6.
Letℳ be a nonempty set equipped with a binary relation ℛ and a metric d. Let T1 be a self-mapping on ℳ and N be an ℛ-complete subspace of ℳ such that T1ℳ⊆N. Assume that the following conditions hold:(i)
ℛ is T1-closed and locally finitely T1-transitive.(ii)
EitherT1 is ℛ-continuous or ℛ|N is d-self-closed.(iii)
ℳT1,ℛ is nonempty.(iv)
There existsϕ∈Φ such that(68)dT1u,T1v≤ϕdu,v,∀u,v∈ℳwithu,v∈ℛ.Then, T1 has a fixed point.(v)
Furthermore, ifT1ℳ is ℛs-connected, then T1 has a unique fixed point.


## 5. Examples

In this section, we present some examples to support the worth of our newly proved results.Example 1.
Letℳ=0,4 equipped with a usual metric d and binary relation ℛ=0,0,1,1,0,1,1,0,0,3,2,3. We consider a pair of self-mappings (T1,T2) on ℳ defined by(69)T1u=u,u∈0,1;1,u∈1,4,T2u=u,u∈0,1/1/2;1,u=12;3,otherwise.
LetN=0,1 be ℛ-complete such that T1ℳ=0,1⊂T2ℳ=0,1\1/2∪3 and N⊂T2ℳ=0,1\1/2∪3. Define a mapping ϕ:0,∞⟶0,∞ by ϕt=1/3t. Clearly, 0∈ℳT1,T2,ℛ,ϕ∈Φ, both T1 and T2 are not continuous, and ℛ is T1,T2-closed. It is easy to see that ℛ is locally finitely T1-transitive relation but neither locally T2-transitive nor transitive. For any ℛ-preserving sequence un in N, (70)un,un+1∈ℛN,for alln∈ℕwithun⟶du.
Notice that ifun,un+1∈ℛ|N, for all n∈ℕ, then there exists K∈ℕ such that un=u∈0,1,for alln≥K. So, we are able to choose a subsequence unk of the sequence un such that unk=u (for all k∈ℕ), which amounts to referring that unk,u∈ℛ|N (for all k∈ℕ). Hence, ℛ|N is d-self-closed. It is easy to see that ℛ is T1,T2-compatible. Now, to verify the contractive condition d of Theorem 2. To do this, we need to show that only for T2u,T2v∈0,3 and for the remaining elements can be easily verified. If T2u,T2v=0,3, then u=0 and v∈1,4, which implies that T10=0 and T1v=1:(71)1=dT10,T1v≤ϕdT20,T23=ϕd0,3=ϕ3=1.
Clearly,ℛ is T1,T2-compatible. Thus, all the hypotheses (a, b, c, d, and e′) of Theorem 2 are satisfied; hence, T1 and T2 have a coincidence point (namely, CT1,T2=0). Notice that T1 and T2 are weakly compatible, i.e., commute at their coincidence point. Therefore, all the hypotheses of Theorem 4 are satisfied. Notice that “u=0” is the unique common fixed point of T1 and T2.Example 2.
Considerℳ=0,5 equipped with a usual metric d and binary relation ℛ=0,0,1,2,2,3,3,0,1,0,2,0. On ℳ, we define two self-mappings T1 and T2 by(72)T1u=u2,0≤u≤1;1,u=2,3,u=3,2,otherwise,T2u=u,u∈0,1\1/2,1/3;2,u=12;3,u=13;4,otherwise,where . stands for the greatest integer function. It is easy to see that ℛ is neither locally T1-transitive nor transitive but locally finitely T1-transitive relation (i.e.,ℛ|T1ℳ is 3-transitive). Let N=0,1,2,3 be ℛ-complete such that T1ℳ=0,1,2,3⊂T2ℳ=0,1\1/2,1/3∪2,3,4∩N and N⊂T2ℳ=0,1\1/2,1/3∪2,3,4. Define a mapping ϕ:0,∞⟶0,∞ by ϕt=3/4t. Then, ϕ∈Φ. Notice that as 0∈ℳT1,T2,ℛ, ϕ∈Φ, both T1 and T2 are not continuous, and ℛ is T1,T2-closed. Choose any ℛ-preserving sequence un in N, i.e.,(73)un,un+1∈ℛN,for alln∈ℕwithun⟶du.
As ifun,un+1∈ℛ|N, for all n∈ℕ, then there exists K∈ℕ such that un=u∈0,for alln≥K. Thus, we choose a subsequence unk of the sequence un such that unk=u (for all k∈ℕ), which amounts to saying that unk,u∈ℛ|N (for all k∈ℕ). Hence, ℛ|N is d-self-closed. In view of T1 and T2, it is easy to see that ℛ is T1,T2-compatible. By straightforward calculations, one can verify the underlying contractive condition d of Theorem 2. Clearly, all the hypotheses (a, b, c, d, and e′) of Theorem 2 are satisfied; hence, T1 and T2 have a coincidence point (namely, CT1,T2=0). Notice that T1 and T2 commute at their coincidence point. Therefore, condition e3′ of Theorem 5 is also satisfied. Hence, all the hypotheses of Theorem 5 are satisfied. Observe that x=0 is a unique common fixed point of T1 and T2 (also ϕ lies in nonlinear class of [32]), whereas main results of Alam et al. [32] are not applicable because ℛ is not locally T1-transitive.


## 6. Conclusions

In the context of nonlinear contractions, the use of arbitrary binary relation cannot ensure the existence of coincidence results (e.g., [16]). Alam et al. [32] employed the locally T1-transitive binary relation to prove their results. With a view to have further improvement, we utilized the relatively weaker notion than the locally T1-transitive binary relation, namely, locally finitely T1-transitive relation, but at the same time, the class of nonlinear contractions remains relatively smaller. However, in order to prove their results, Alam et al. [32] utilized Boyd–Wong nonlinear contraction defined by(74)Ω=ϕ:0,∞⟶0,∞:ϕt<tfor eacht>0andlim supr⟶t+ϕr<tfor eacht>0,which enlarges the class Φ (described earlier), i.e.,Φ⊂Ω.Henceforth, our newly proved results and the results due to Alam et al. [32] are independent of each other. None of them is a proper generalization of another.



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