# Relational Meir-Keeler Contractions and Common Fixed Point Theorems

Authors: Faizan Ahmad Khan; Faruk Sk; Maryam Gharamah Alshehri; Qamrul Haq Khan; Aftab Alam
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this article, we prove some coincidence and common fixed point theorems under the relation-theoretic Meir-Keeler contractions in a metric space endowed with a locally finitelyT-transitive binary relation. Our newly proved results generalize, extend, and sharpen some existing coincidence point as well as fixed point theorems existing in the literature. Moreover, we give some examples to affirm the efficacy of our results.


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## Body

## 1. Introduction

Banach [1], a Polish mathematician, established the most successful result in fixed point theory, the Banach contraction principle (in short, BCP), in 1922, which says that a contraction mapping on a complete metric space has a unique fixed point. One of the noted generalizations of BCP comprising the concept of coincidence point (in short, CP) and common fixed point (in short, CFP) theorems was established by Jungck [2] in 1976. In succeeding years, many researchers introduced relatively weaker version of commuting mappings and developed exciting CFP results, see [3, 4].On the other hand, generalizations of the underlying space have been trending since some decades. One of such important generalizations was initiated by Turinici [5, 6] in 1986, where he proved fixed point results in a partial ordered set. In this continuation, Alam and Imdad [7] generalized the BCP using a binary relation. Since then, many relation-theoretic fixed point theorems are being studied regularly, see [8, 9] and references therein.Several researchers reported numerous fixed point results employing relatively more generalized contractions. One of such vital contractions was due to Meir and Keeler [10] in 1969, which was further extended by Rao and Rao [11]. In 2013, Patel et al. [12] established some CFP theorems for three and four self-mappings satisfying generalized Meir-Keeler α-contraction in metric spaces. Some generalizations of Meir-Keeler contraction in the framework of different types of spaces have also been reported, see [13–16]. Recently, Sk et al. [17] introduced the Meir-Keeler contraction in relation-theoretic sense and extended relation-theoretic contraction principle to relation-theoretic Meir-Keeler contraction principle.In this paper, we prove some coincidence and common fixed point theorems using the relation-theoretic Meir-Keeler contraction in a metric space endowed with a locally finitelyT-transitive binary relation. We also equip several examples to exhibit the significance of these new findings.


## 2. Preliminaries

We will go over some basic definitions in this section that will help us to prove our primary results. Throughout the paper, we pertain toℕ∪0 as K0, and empty set as ∅.Definition 1 (see [18]).
LetX≠∅ be a set. A “binary relation” is a subset R of X2. The subsets X2 and ∅ of X2 are called the “universal relation” and “empty relation,” respectively.Definition 2 (see [7]).
LetX≠∅ be a set with a binary relation R. If either ϱ,σ∈R or σ,ϱ∈R for ϱ,σ∈X, then ϱ and σ are called as “R-comparative.” ϱ,σ∈R is the notion for it.Definition 3 (see [18–23]).
LetX≠∅ be a set with a binary relation R. Then, the relation R is called
(a)
“amorphous” ifR has no precise attribute(b)
“reflexive” ifϱ,ϱ∈R∀ϱ∈X(c)
“symmetric” ifϱ,σ∈Rσ,ϱ∈R(d)
“anti-symmetric” ifϱ,σ∈R and σ,ϱ∈Rϱ=σ(e)
“transitive” ifϱ,σ∈R and σ,w∈Rϱ,w∈R(f)
“complete”, “connected” or “dichotomous” ifϱ,σ∈R∀ϱ,σ∈X(g)
“partial order” ifR is “reflexive”, “anti-symmetric” and “transitive”Definition 4 (see [18]).
LetR be a binary relation on a set X≠∅. Then,
(1)R−1=ϱ,σ∈X2:σ,ϱ∈RandRs=R∪R−1,
are called inverse relation and symmetric closure ofR, respectively.Proposition 5 (see [7]).
LetX≠∅ be a set with a binary relation R. Then, for ϱ,σ∈X,
(2)ϱ,σ∈Rs⟹ϱ,σ∈R.Definition 6 (see [24]).
LetX≠∅ be a set with a binary relation R and S⊆X. Then, the set RS=R∩S2 is defined as the restriction of R to S.Definition 7 (see [7]).
LetX≠∅ be a set with a binary relation R. A sequence ϱk⊂X is called R-preserving if
(3)ϱk,ϱk+1∈R∀k∈K0.Definition 8 (see [7, 25]).
LetT and H be two self-mappings on a set X≠∅ and R a binary relation on X. Then,
(a)
R is said to be T-closed if
∀ϱ,σ∈X,ρ,σ∈R⟹Tϱ,Tσ∈R(b)
R is said to be T,H-closed if(4)∀ϱ,σ∈X,Hϱ,Hσ∈R⟹Tϱ,Tσ∈RRemark 9.
UnderH=I, the identity mapping on X, the notion of T,H-closedness coincides with the notion of T-closedness of R.Definition 10 (see [25]).
LetX≠∅ be a set with a metric d together with a binary relation R. If every R-preserving Cauchy sequence in X converges, we say X,d is R-complete.Definition 11 (see [25]).
LetX≠∅ be a set with a metric d together with a binary relation R and T a self-mapping on X. If for any R-preserving sequence ϱk⊂X converging to an element ϱ∈X, we have Tϱk⟶dTϱ, then the mapping T is said to be R-continuous.Definition 12 (see [2]).
LetX≠∅ be a set with a metric d together with a binary relation R and T,H two self-mappings on X. Let ϱk⊂X be a sequence satisfying limk⟶∞Hϱk=limk⟶∞Tϱk. Then, the mappings T and H are compatible if limk⟶∞dHTϱk,THϱk=0.Definition 13 (see [25]).
LetX≠∅ be a set with a metric d together with a binary relation R and T,H two self-mappings on X. Let ϱk⊂X be a sequence such that Tϱk and Hϱk are R-preserving sequence satisfying limk⟶∞Hϱk=limk⟶∞Tϱk. Then, the mappings T and H are “R-compatible” if limk⟶∞dHTϱk,THϱk=0.Remark 14 (see [25]).
LetX≠∅ be a set with a metric d together with a binary relation R. Then, the following relation holds:
(5)"commutativity⟹compatibility⟹R−compatibility⟹weakcompatibility".Definition 15 (see [7, 25]).
LetX≠∅ be a set with a metric d together with a binary relation R and T,H two self-mappings on X. Consider the R-preserving sequence ϱk⊂X such that ϱk⟶dϱ. Then,
(a)
R is called “d-self-closed” if there exists a subsequence ϱkp of ϱk with ϱkp,ϱ∈R∀p∈K0(b)
R is called “H−d-self-closed” if there exists a subsequence ϱkp of ϱk with Hϱkp,Hϱ∈R∀p∈K0Definition 16 (see [26–29]).
LetX≠∅ be set with a binary relation R and T a self-mapping on X(a)
If for anyϱ,σ,ς∈X, Tϱ,Tσ∈RandTσ,Tς∈R⟹Tϱ,Tς∈R, then R is called “T-transitive”(b)
If for anyϱ0,ϱ1,⋯,ϱK∈X where K is a natural number ≥2, we have(6)ϱℓ−1,ϱℓ∈Rforeachℓ1≤ℓ≤K⟹ϱ0,ϱK∈R,
thenR is called K-transitive
(c)
If for each denumerable subsetS of X, there exists K=KS≥2, such that RS is K-transitive, then R is called “locally finitely transitive”(d)
If for each denumerable subsetS of TX, there exists K=KS≥2, such that RS is K-transitive, then R is called “locally finitely T-transitive”Proposition 17 (see [29]).
LetX be a nonempty set, R a binary relation on X and T a self-mapping on X. Then,
(a)
R is “T-transitive” ⟺RTX is “transitive”(b)
R is “locally finitely T-transitive” ⟺RTX is “locally finitely transitive”(c)
R is “transitive” ⟹R is “finitely transitive” ⟹R is “locally finitely transitive” ⟹R is “locally finitely T-transitive”(d)
R is “transitive” ⟹R is “T-transitive” ⟹R is “locally finitely T-transitive”Definition 18 (see [23]).
LetX be a nonempty set and R a binary relation on X. A subset S of X is called R-directed if for each ϱ,σ∈S, there exists ς∈X such that ϱ,ς∈R and σ,ς∈R.Definition 19 (see [24]).
LetR be a binary relation defined on a nonempty set X. Then, for ϱ,σ∈X, a finite sequence ϱ0,ϱ1,⋯,ϱp⊂X satisfying the following conditions:
(7)ϱℓ,ϱℓ+1∈Rforeachℓ0≤ℓ≤p−1,ϱ0=ϱandϱp=σ,is said to be a path of length p in R from ϱ to σ.Definition 20 (see [7]).
LetR be a binary relation on a nonempty set X, and Y a subset of X. If there exists a path in R from ρ to σ for each ϱ,σ∈Y, then Y is called R-connected.Lemma 21 (see [28]).
LetR be a binary relation on a nonempty set X, and ϱk⊂X a sequence satisfying ϱk,ϱk+1∈R. Now, if for some natural number K≥2, R is K-transitive on the set L=ϱk:k∈K0, then
(8)ϱk,ϱk+1+rK−1∈Rforallk,r∈K0.


## 3. Main Results

The first result in this section is on the existence of CP for two mappingT and H. For a nonempty set X and two self-mappings T and H on X, the notations we use herein are as follows:
(9)ΘT,H=ρ∈X:Tϱ=Hϱ,Θ¯T,H=ϱ¯∈X:ϱ¯=Tϱ=Hϱ,ϱ∈X.Theorem 22.
LetX be a nonempty set together with a metric d, R a binary relation on X and T,H two self-mappings on X. Suppose the following conditions hold:
(a)
TX⊂HX(b)
X,d is R-complete(c)
there existsϱ0∈X such that Hϱ0,Tϱ0∈R(d)
R is T,H-closed and locally finitely T-transitive(e)
T and H are R-compatible(f)
H is R-continuous(g)
T is R-continuous or R is H−d-self-closed(h)
for everyε>0 and ϱ,σ∈X, there exists δ>0 such that(10)Hρ,Hσ∈Randε≤dHϱ,Hσ<ε+δ⟹dTϱ,Tσ<ε
Then,T and H have a CP.Proof.
Assumptionc confirms the existence of ϱ0∈X such that Hϱ0,Tϱ0∈R. Now, if Hϱ0=Tϱ0 then nothing is left to be proved. Otherwise, by assumption a, we can pick ϱ1∈X such that Tϱ0=Hϱ1. Again, there will be ϱ2∈X such that Hϱ2=Tϱ1. In this way, we construct a sequence ϱk⊂X such that
(11)Hϱk+1=Tϱk∀k∈K0.
Now, we assert thatHϱk is R-preserving, i.e.,
(12)Hϱk,Hϱk+1∈R∀k∈K0.
We will adopt the induction method to prove this fact. In view of assumptionc, equation (12) holds for k=0, i.e.,
(13)Hϱ0,Hϱ1∈R.
Now, suppose that equation (12) holds for k=p>0, i.e.,
(14)Hϱp,Hϱp+1∈R.
Then, we have to show that(15)Hϱp+1,Hϱp+2∈R.
In view of the fact thatR is T,H-closed, it is clear that
(16)Hϱp,Hϱp+1∈RTϱp,Tϱp+1∈R,implying thereby
(17)Hϱp+1,Hϱp+2∈R,which guarantees the fact that equation (2) holds for k=p+1. Therefore, Hϱk is R-preserving sequence. Notice that Tϱk is also a R-preserving sequence due to equation (1), i.e.,
(18)Tϱk,Hϱk+1∈R.
Now, if there existsn0∈K such that Hϱn0=Hϱn0+1, then, in view of equation (1), ϱn0 turns out to be a CP of T and H. As an alternative, consider that Hϱk≠Hϱk+1 for all k∈K0, i.e., dHϱk,Hϱk+1≠0.
Denoteμk≔dHϱk,Hϱk+1. Now, in view of assumption h, we get
(19)μk+1=dHϱk+1,Hϱk+2=dTϱk,Tϱk+1<dHϱk,Hϱk+1=μk,which gives
(20)μk+1<μk.
Therefore, the sequenceμk is decreasing. As μk is also bounded below by 0 (as a lower bound), we can find r≥0 satisfying
(21)limk⟶∞μk=r=infk∈K0μk.
Now, let us assume thatr>0. So, there will always be a δr>0 such that
(22)Hϱ,Hσ∈R,r≤dHϱ,Hσ<r+δr⟹dTϱ,Tσ<r.
Sinceμk is decreasing sequence converging to r, there exists p∈K such that
(23)r≤dHϱp,Hϱp+1<r+δr.
Thus, in view of assumptionh, we have
(24)μp+1=dHϱp+1,Hϱp+2<r,which contradicts the fact that r=infk⟶K0μk. Hence, we conclude that
(25)limk⟶∞dHϱk,Hϱk+1=0.
Now, we establish that the sequenceHϱk is Cauchy. Utilizing equation (1), since Hϱk⊂TX, we get that the range S=Hϱk:k∈K0 is a denumerable subset of TX. Hence, in view of assumption d, there exist K=KS≥2, such that RS is K-transitive. Let ε>0 be an arbitrary and fixed real number and let δ>0 corresponds to ε verifying the assumption h. WLOG, we may consider that δ<ε. In view of (2), there exists n0δ∈ℕ satisfying
(26)dHϱk,Hϱk+1<δ4K∀k≥n0δ.
For allk≥n0δ and for all p1≤p≤K, using triangular inequality, we get
(27)dHϱk,Hϱk+ϱ≤dHϱk,Hϱk+1+dHϱk+1,Hϱk+2⋯+dHϱk+p−1,Hϱk+p≤δ4K+δ4K+⋯+δ4K=pδ4K.
Now, we claim that(28)dHϱk,Hϱk+p<ε+δ2∀k≥n0δand∀p∈K.
This is demonstrated herein using the mathematical induction method. From (27), it is clear that (28) holds for all p∈1,2,3,⋯,K. Suppose that the conclusion holds for all p∈1,2,3,⋯,m, where m≥K. We have to show that (28) holds for k=m+1 also. As m≥K, so m−1≥K−1>0. By division algorithm, there exists unique integers μ and η0≤η≤K−1 such that
(29)m−1=K−1μ+ηm=1+K−1μ+η.
Denotingq≕1+K−1μ, the above equation reduces to
(30)m=q+η,so that
(31)2≤K≤q≤m=q+η.
Now, using (27), we get
(32)dHϱk+q+1,Hϱk+m+1=dHϱk+q+1,Hϱk+q+η+1≤ηδ4K.
Now, using Lemma21, we get
(33)Hϱk,Hϱk+q∈R.
Asq∈K,K+1,.⋯,m, using inductive hypothesis, we get
(34)0<dHϱk,Hϱk+q<ε+δ2<ε+δ.
Using (33) and (34) and applying contractive condition h, we get
(35)dHϱk+1,Hϱk+q+1=dTϱk,Tϱk+q<ε.
Now, using triangular inequality, (25), (32), and (35), we get
(36)dHρk,Hρk+m+1≤dHϱk,Hϱk+1+dHϱk+1,Hϱk+q+1+dHϱk+q+1,Hϱk+m+1<δ4K+ε+ηδ4K<δ4K+ε+δ4KK−1asK≥2andη<K−1=ε+δ4<ε+δ2.
Thus, by induction, (28) is verified. From (28), it embraces that the sequence Hϱk is Cauchy. Now, the R-completeness property of X and R-preserving property of Hϱk confirm the availability of an element ς∈X such that
(37)limk⟶∞Hϱk=ς.
Also, from (11),
(38)limk⟶∞Tϱk=ς.
Now, by dint of theR-continuity of H, we acquire
(39)limk⟶∞HHϱk=Hlimk⟶∞Hϱk=Hς.
Utilizing (38) and R-continuity of H,
(40)limk⟶∞HTϱk=Hlimk⟶∞Tϱk=Hς.
SinceTϱk and Hϱk are R-preserving and
(41)limk⟶∞Tϱk=limk⟶∞Hϱk=ς,by assumption e,
(42)limk⟶∞dHTϱk,THϱk=0.
The next step is to establish thatς∈ΘT,H. From assumption g, we first consider that T is “R-continuous.” Using (12), (37), and R-continuity of T,
(43)limk⟶∞THϱk=Tlimk⟶∞Hϱk=Tς.
Applying (40) and (42), we get
(44)dHς,Tς=dlimk⟶∞HTϱk,limk⟶∞THϱk=limk⟶∞dHTϱk,THϱk=0,
yielding therebyHς=Tς, which establishes our claim.
Instead ofR-continuity of T, we now suppose that R is H,d-self-closed, based on assumption g. Then, Hϱk being R-preserving sequence guarantees the existence of a subsequence Hϱkp such that Hϱkp,ς∈R. If Hϱkk0=ς for some k0∈K, then using (11) and by the R-preserving property of Hϱk, we get Hϱkk0∈ΘT,H. Otherwise, suppose Hϱnp≠ς, i.e., dHϱnp,ς≠0 for all p∈K. In this case, in view of assumption h, assuming ε=dHϱkp,ς and using assumption h, we get
(45)dTHϱnp,Tς<ε.
Using triangle inequality, we get(46)dHς,Tς≤dHς,HTϱkp+dHTϱkp,THϱkp+dTHϱkp,Tς.
Now, using (40), (42), and (45) in the previous equation, we obtain
(47)dHς,Tς=0,which establishes that Tς=Hς.It is clear that Theorem22 solely considers the existence of a CP of T and H. As a result, we must add extra conditions to the hypothesis of Theorem 22 to obtain the uniqueness of point of coincidence, CP and CFPs. This is the purpose of our next theorems.Theorem 23.
Assume that all of the criteria of Theorem22 are met. Let the following condition holds additionally:
(i)
TX is RHXs-connected
thenT and H have a unique point of coincidence.Proof.
From Theorem22, we get that ΘT,H≠∅. Consider that ϱ,σ∈ΘT,H. Then, there exist σ¯,σ¯∈X such that
(48)Tϱ=Hϱ=ϱ¯andTσ=Hσ=σ¯.
It is now our goal to prove thatϱ¯=σ¯. Since Tϱ,Tσ∈TX⊆HX, by assumption i, there exists a path Hς0,Hς1,Hς2,⋯,Hςp of some finite length p in RHXs from Tρ to Tσ. Now, in view of (48), WLOG we can choose ς0=ϱ and ςp=σ. Thus, we have
(49)Hςℓ,Hςℓ+1∈RHXforeachℓ0≤ℓ≤p−1.
Define the constant sequencesςk0=ρ and ςkp, then in view of equation (48), we have Hςk+10=Tςk0=ϱ¯ and Hςk+1p=Tςkp=σ¯ for all k∈K0. Put ς01=ς1,ς02=ς2,ς03=ς3,⋯,ς0p−1=ςp−1. Now, since TX⊂HX, we can define sequences ςk1, ςk2, ..., ςkp−1 such that Hςk+11=Tςk1, Hςk+12=Tςk2, ..., Hςk+1p−1=Tςkp−1 for all k∈K0. Hence, we have
(50)Hςk+1ℓ=Tςkℓ∀k∈K0andforeachℓ0≤ℓ≤p.
Now, we claim that(51)Hςkℓ,Hςkℓ+1∈R∀k∈K0andforeachℓ0≤ℓ≤p−1.
This is demonstrated herein using the mathematical induction method. equation (51) holds for k=0 as a result of (49). Assume that equation (51) is true for k=r, i.e.,
(52)Hςrℓ,Hςrℓ+1∈R.
AsR is T,H-closed, we obtain
(53)Tςrℓ,Tςrℓ+1∈R,which on using (51) gives us that
(54)Hςr+1ℓ,Hςr+2ℓ+1∈Rk∈K0andforeachℓ0≤ℓ≤p−1.
Therefore, equation (51) holds. Now, for each k∈K0 and for each 0≤ℓ≤p−1, define
(55)tkℓ=dHςkℓ,Hςkℓ+1.
We show that(56)limk⟶∞tkℓ=0.
Now, we look at two scenarios in whichℓ is fixed. Firstly, suppose that
(57)tn0ℓ=dHςn0ℓ,Hςn0ℓ+1=0forsomen0∈K0,
which gives rise toHςn0ℓ=Hςn0ℓ+1. Now applying (11), we have tn0+1ℓ=0. Continuing this process, we get
(58)ςkℓ=0∀k≥n0,which establishes that limk⟶∞ςkℓ=0.
Alternatively, assume thatςkℓ>0∀k∈K0. For any ε>0, assume tkℓ=dHςkℓ,Hςkℓ+1=ε. Then,
(59)tk+1ℓ=dHςk+1ℓ,Hςk+1ℓ+1=dTςkℓ,Tςkℓ+1<ε=tkℓ,which gives
(60)tk+1ℓ<tkℓ.
As a result, the sequencetkℓ is decreasing. As tkℓ is also bounded below by 0 (as a lower bound), there exists r≥0 such that
(61)limk⟶∞tkℓ=r=infk∈K0tkℓ.
Now, we prove thatr=0. Assume, on the other hand that r>0. So, there will always be a δr>0 such that
(62)Hϱ,Hσ∈Randr≤dHϱ,Hσ<r+δrdTϱ,Tσ<r.
Sincetkℓ is decreasing sequence converging to r, there exists p∈K such that
(63)r≤dHςpℓ,Hςpℓ+1<r+δr.
Thus, in view of assumptionh, we have
(64)tp+1ℓ=dHςp+1ℓ,Hςp+1ℓ+1<r,which contradicts the fact that r=infk⟶∞tkℓ. Hence, we conclude that
(65)limk⟶∞tkℓ=0.
Thus, equation (56) holds ∀ℓ0≤ℓ≤p−1. Now, in light of equation (56) and triangle inequality, we get
(66)dϱ¯,σ¯≤tk0+tk1+⋯+tkp−1⟶0ask⟶∞.
Therefore,ϱ¯=σ¯, which ends the proof.Theorem 24.
Assume that all of the criteria of Theorem22 are met. Let the following condition holds additionally:
(i)
T and H are “weakly compatible”
thenT and H have a unique CFP.Proof.
Assumeρ∈X such that ϱ∈ΘT,H. Therefore, there exists ρ¯∈X such that
(67)Hϱ=Tϱ=ϱ¯.
In light of the Remark14, the concept R-compatibility coincides with the weak compatibility. Hence, ϱ¯∈ΘT,H. Utilizing ς=ϱ¯ in Theorem 23, we obtain Hϱ=Hϱ¯ yielding thereby
(68)ϱ¯=Hϱ¯=Tϱ¯.
Hence,ϱ¯ is a CFP of T and H.
Now, we assume thatϱ′ is another CFP of T and H in order to assert the uniqueness. Applying Theorem 23, we get
(69)ϱ′=Hϱ′=Hϱ¯=ϱ¯,which finishes the proof.Theorem 25.
Assume that all of the criteria of Theorem22 are met. Suppose either of the mappings T and H is one-to-one. Then, T and H have a unique CP.Proof.
From Theorem22, it is evident that ΘT,H≠∅. Let, ϱ,σ∈ΘT,H. Then, Theorem 23 permits us to write
(70)Tϱ=Hϱ=Tσ=Hσ.
Now, sinceT or H is one-to-one, we have, ϱ=σ which finishes the proof.Theorem22 has the following implication when we apply Proposition 17.Corollary 26.
If either of the below conditions:(a)
R is “transitive”(b)
R is “T-transitive”(c)
R is “finitely transitive”(d)
R is “locally finitely transitive”
is utilized in Theorem22 instead of the locally finitely T-transitivity condition; then, the validity of Theorem 22 remains the same.Corollary 27.
If either of the below conditions:
(i′). TX is Rs-directed
(i′"). RTX is complete
holds in place of conditioni of Theorem 23, then the validity of Theorem 23 remains the same.Proof.
If conditioni′ is satisfied, then, for each ϱ,σ∈TX, we get ς∈X satisfying ρ,ς∈R and σ,ς∈R. Notice that the sequence ϱ,ς,σ works as a path of length 2 in Rs from ρ to σ, which establishes the fact that TX is Rs-connected. Now, applying Theorem 23, we obtain the uniqueness of point of coincidence.
Alternately, from assumptioni′′, we get ϱ,σ∈R∀ϱ,σ∈TX, which assents that ρ,σ constitutes a path of length 1 in Rs. As a result, TX is Rs-connected, which wrap up the proof when Theorem 23 is applied.UnderH=I, the identity map, we obtain the following result which is proved by Sk et al. [17].Corollary 28 (see [17]).
LetX,d be a R-complete metric space endowed with a binary relation R on X and T a self-mapping on X. Suppose that the following conditions hold:
(a)
there existsϱ0∈X such that ϱ0,Tϱ0∈R,(b)
R is T-closed and locally finitely T-transitive(c)
eitherT is R-continuous or R is d-self-closed(d)
for everyε>0 there exists δ>0 such that(71)ϱ,σ∈Randε≤dϱ,σ<ε+δ⟹dTϱ,Tσ<ε
Then,T has a fixed point. Further, if we impose an additional hypothesis:
(e)
TX is Rs-connected
thenT has a unique fixed point.Remark 29.
Under the universal relationR and H=I, the identity map, Theorem 22, and Theorem 23 reduce to the classical fixed point theorem of Meir and Keeler [10].Remark 30.
Under partial order the relationℝ=°, and H=I, the identity map, Theorem 22, and Theorem 23 reduces to fixed point theorem of Harjani et al. [30].


## 4. Examples

Now, we equip two examples to show how important our results are in comparison to other results in the literature.Example 1.
LetX=0,1,1,0,1,1,0,0⊂ℝ2 together with the usual Euclidean metric d. Consider the following relation endowed with X:
(72)R=1,1,0,0.
Then,X,d is a R-complete metric space. Now consider that T,H:X⟶X are two mappings defined by
(73)T1,0=0,1;T0,1=1,0;T1,1=1,1;T0,0=0,0,H0,1=1,0;H0,0=0,1;H1,1=1,1;H1,0=0,0.
Notice that forε=d0,1,1,0=2, we have
(74)dT0,1,T1,0=d1,0,0,1=2<ε,which is absurd. Further, 1,1,0,0∈R and d1,1,0,0=2 but the inequality
(75)dT1,1,T0,0=d1,1,,0,0=2<ε,does not hold. Hence, the existing theorems cannot be applied for this example. Now, assume that ε=dH1,1,H1,0=d1,1,0,0=2. Then, the inequality
(76)dT1,1,T1,0=d1,1,0,1=1<ε,holds. As a result, assumption h of Theorem 22 holds. It can also be seen that all of the conditions of Theorem 22 are met using regular calculation. Therefore, T and H have a CP, namely, 1,1.Although it does not satisfy Theorem23, the CP of T and H in Example 1 is unique, proving that condition i of Theorem 23 is not a necessary condition for the uniqueness of CPs.Example 2.
LetX=0,1,1,0,1,1,0,0⊂ℝ2 together with the usual Euclidean metric d. Consider the following relation endowed with X,
(77)R=ϱ,σ:ϱ,σ∈0,1,1,1.
Then,X,d is a R-complete metric space. Now consider that T,H:X⟶X are two mappings defined by
(78)T1,0=1,0;T0,1=0,1;T1,1=1,0;T0,0=0,1,H1,0=1,0;H0,1=0,1;H1,1=0,1,H0,0=1,1.
Now, forε=dH0,1,H0,0=1, we have
(79)dT0,1,T0,0=d0,1,0,1=0<ε,holds. As a result, assumption h of Theorem 22 holds. It can also be seen that all of the conditions of Theorem 22 are met using regular calculation. Therefore, T and H have CPs, namely, 0,1,1,0. The availability of more than one fixed point certifies the eminence of Theorem 23.
Notice that forε=d0,1,1,0=2, we have
(80)dT0,1,T1,0=d1,0,0,1=2<ε,which is absurd. Further, 0,1,1,1∈R and d0,1,1,1=1 but the inequality
(81)dT0,1,T1,1=d0,1,,1,0=2<ε,does not hold. Hence, the existing theorems cannot be applied for this example.


## 5. Conclusion

In this paper, we have established some coincidence point theorems for two mappings employing the relation-theoretic Meir-Keeler contractions in a metric space endowed with a class of transitive binary relation. Our findings have also led to the deduction of certain related fixed point results. Furthermore, some examples are given to demonstrate the significant progress made in this area.



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*Source: 3550923-2022-06-25.xml*