#### 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 finitely -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.

#### 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 -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 finitely -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 to as , and empty set as .

*Definition 1 (see [18]). *Let be a set. A “binary relation” is a subset of . The subsets and of are called the “universal relation” and “empty relation,” respectively.

*Definition 2 (see [7]). *Let be a set with a binary relation . If either or for , then and are called as “-comparative.” is the notion for it.

*Definition 3 (see [18–23]). *Let be a set with a binary relation . Then, the relation is called
(a)“amorphous” if has no precise attribute(b)“reflexive” if (c)“symmetric” if (d)“anti-symmetric” if and (e)“transitive” if and (f)“complete”, “connected” or “dichotomous” if (g)“partial order” if is “reflexive”, “anti-symmetric” and “transitive”

*Definition 4 (see [18]). *Let be a binary relation on a set . Then,
are called inverse relation and symmetric closure of , respectively.

Proposition 5 (see [7]). *Let be a set with a binary relation . Then, for ,
*

*Definition 6 (see [24]). *Let be a set with a binary relation and . Then, the set is defined as the restriction of to .

*Definition 7 (see [7]). *Let be a set with a binary relation . A sequence is called -preserving if

*Definition 8 (see [7, 25]). *Let and be two self-mappings on a set and a binary relation on . Then,
(a) is said to be -closed if(b) is said to be -closed if

*Remark 9. *Under , the identity mapping on , the notion of -closedness coincides with the notion of -closedness of .

*Definition 10 (see [25]). *Let be a set with a metric together with a binary relation . If every -preserving Cauchy sequence in converges, we say is -complete.

*Definition 11 (see [25]). *Let be a set with a metric together with a binary relation and a self-mapping on . If for any -preserving sequence converging to an element , we have , then the mapping is said to be -continuous.

*Definition 12 (see [2]). *Let be a set with a metric together with a binary relation and two self-mappings on . Let be a sequence satisfying . Then, the mappings and are compatible if

*Definition 13 (see [25]). *Let be a set with a metric together with a binary relation and two self-mappings on . Let be a sequence such that and are -preserving sequence satisfying . Then, the mappings and are “-compatible” if

*Remark 14 (see [25]). *Let be a set with a metric together with a binary relation . Then, the following relation holds:

*Definition 15 (see [7, 25]). *Let be a set with a metric together with a binary relation and two self-mappings on . Consider the -preserving sequence such that . Then,
(a) is called “-self-closed” if there exists a subsequence of with (b) is called “-self-closed” if there exists a subsequence of with

*Definition 16 (see [26–29]). *Let be set with a binary relation and a self-mapping on (a)If for any , then is called “-transitive”(b)If for any where is a natural number , we havethen is called -transitive
(c)If for each denumerable subset of , there exists , such that is -transitive, then is called “locally finitely transitive”(d)If for each denumerable subset of , there exists , such that is -transitive, then is called “locally finitely -transitive”

Proposition 17 (see [29]). *Let be a nonempty set, a binary relation on and a self-mapping on . Then,
*(a)* is “-transitive” is “transitive”*(b)* is “locally finitely -transitive” is “locally finitely transitive”*(c)* is “transitive” is “finitely transitive” is “locally finitely transitive” is “locally finitely -transitive”*(d)* is “transitive” is “-transitive” is “locally finitely -transitive”*

*Definition 18 (see [23]). *Let be a nonempty set and a binary relation on . A subset of is called -directed if for each , there exists such that and .

*Definition 19 (see [24]). *Let be a binary relation defined on a nonempty set . Then, for , a finite sequence satisfying the following conditions:
is said to be a path of length in from to .

*Definition 20 (see [7]). *Let be a binary relation on a nonempty set , and a subset of . If there exists a path in from to for each , then is called -connected.

Lemma 21 (see [28]). *Let be a binary relation on a nonempty set , and a sequence satisfying . Now, if for some natural number , is -transitive on the set , then
*

#### 3. Main Results

The first result in this section is on the existence of CP for two mapping and . For a nonempty set and two self-mappings and on , the notations we use herein are as follows:

Theorem 22. *Let be a nonempty set together with a metric , a binary relation on and two self-mappings on . Suppose the following conditions hold:
*(a)*(b)** is -complete*(c)*there exists such that *(d)* is -closed and locally finitely -transitive*(e)* and are -compatible*(f)* is -continuous*(g)* is -continuous or is -self-closed*(h)*for every and , there exists such that**Then, and have a CP.*

*Proof. *Assumption confirms the existence of such that . Now, if then nothing is left to be proved. Otherwise, by assumption , we can pick such that . Again, there will be such that . In this way, we construct a sequence such that
Now, we assert that is -preserving, i.e.,
We will adopt the induction method to prove this fact. In view of assumption , equation (12) holds for , i.e.,
Now, suppose that equation (12) holds for , i.e.,
Then, we have to show that
In view of the fact that is -closed, it is clear that
implying thereby
which guarantees the fact that equation (2) holds for . Therefore, is -preserving sequence. Notice that is also a -preserving sequence due to equation (1), i.e.,
Now, if there exists such that , then, in view of equation (1), turns out to be a CP of and . As an alternative, consider that for all , i.e., .

Denote . Now, in view of assumption , we get
which gives
Therefore, the sequence is decreasing. As is also bounded below by (as a lower bound), we can find satisfying
Now, let us assume that . So, there will always be a such that
Since is decreasing sequence converging to , there exists such that
Thus, in view of assumption , we have
which contradicts the fact that . Hence, we conclude that
Now, we establish that the sequence is Cauchy. Utilizing equation (1), since , we get that the range is a denumerable subset of . Hence, in view of assumption , there exist , such that is -transitive. Let be an arbitrary and fixed real number and let corresponds to verifying the assumption . WLOG, we may consider that In view of (2), there exists satisfying
For all and for all , using triangular inequality, we get
Now, we claim that
This is demonstrated herein using the mathematical induction method. From (27), it is clear that (28) holds for all . Suppose that the conclusion holds for all , where . We have to show that (28) holds for also. As , so . By division algorithm, there exists unique integers and such that
Denoting , the above equation reduces to
so that
Now, using (27), we get
Now, using Lemma 21, we get
As , using inductive hypothesis, we get
Using (33) and (34) and applying contractive condition , we get
Now, using triangular inequality, (25), (32), and (35), we get
Thus, by induction, (28) is verified. From (28), it embraces that the sequence is Cauchy. Now, the -completeness property of and -preserving property of confirm the availability of an element such that
Also, from (11),
Now, by dint of the -continuity of , we acquire
Utilizing (38) and -continuity of ,
Since and are -preserving and
by assumption ,
The next step is to establish that . From assumption , we first consider that is “-continuous.” Using (12), (37), and -continuity of ,
Applying (40) and (42), we get
yielding thereby , which establishes our claim.

Instead of -continuity of , we now suppose that is -self-closed, based on assumption . Then, being -preserving sequence guarantees the existence of a subsequence such that . If for some , then using (11) and by the -preserving property of , we get . Otherwise, suppose , i.e., for all . In this case, in view of assumption , assuming and using assumption , we get
Using triangle inequality, we get
Now, using (40), (42), and (45) in the previous equation, we obtain
which establishes that .

It is clear that Theorem 22 solely considers the existence of a CP of and . 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 Theorem 22 are met. Let the following condition holds additionally:
*(i)* is -connected**then and have a unique point of coincidence.*

*Proof. *From Theorem 22, we get that . Consider that . Then, there exist such that
It is now our goal to prove that . Since , by assumption , there exists a path of some finite length in from to . Now, in view of (48), WLOG we can choose and . Thus, we have
Define the constant sequences and , then in view of equation (48), we have and for all . Put Now, since , we can define sequences , , ..., such that , , ..., for all . Hence, we have
Now, we claim that
This is demonstrated herein using the mathematical induction method. equation (51) holds for as a result of (49). Assume that equation (51) is true for , i.e.,
As is -closed, we obtain
which on using (51) gives us that
Therefore, equation (51) holds. Now, for each and for each , define
We show that
Now, we look at two scenarios in which is fixed. Firstly, suppose that
which gives rise to . Now applying (11), we have . Continuing this process, we get
which establishes that .

Alternatively, assume that . For any , assume . Then,
which gives
As a result, the sequence is decreasing. As is also bounded below by (as a lower bound), there exists such that
Now, we prove that . Assume, on the other hand that . So, there will always be a such that
Since is decreasing sequence converging to , there exists such that
Thus, in view of assumption , we have
which contradicts the fact that . Hence, we conclude that
Thus, equation (56) holds . Now, in light of equation (56) and triangle inequality, we get
Therefore, , which ends the proof.

Theorem 24. *Assume that all of the criteria of Theorem 22 are met. Let the following condition holds additionally:
*(i)* and are “weakly compatible”**then and have a unique CFP.*

*Proof. *Assume such that . Therefore, there exists such that
In light of the Remark 14, the concept -compatibility coincides with the weak compatibility. Hence, . Utilizing in Theorem 23, we obtain yielding thereby
Hence, is a CFP of and .

Now, we assume that is another CFP of and in order to assert the uniqueness. Applying Theorem 23, we get
which finishes the proof.

Theorem 25. *Assume that all of the criteria of Theorem 22 are met. Suppose either of the mappings and is one-to-one. Then, and have a unique CP.*

*Proof. *From Theorem 22, it is evident that . Let, . Then, Theorem 23 permits us to write
Now, since or is one-to-one, we have, which finishes the proof.

Theorem 22 has the following implication when we apply Proposition 17.

Corollary 26. *If either of the below conditions:
*(a)* is “transitive”*(b)* is “-transitive”*(c)* is “finitely transitive”*(d)* is “locally finitely transitive”**is utilized in Theorem 22 instead of the locally finitely -transitivity condition; then, the validity of Theorem 22 remains the same.*

Corollary 27. *If either of the below conditions:**(). is -directed**(). is complete**holds in place of condition of Theorem 23, then the validity of Theorem 23 remains the same.*

*Proof. *If condition is satisfied, then, for each , we get satisfying and . Notice that the sequence works as a path of length in from to , which establishes the fact that is -connected. Now, applying Theorem 23, we obtain the uniqueness of point of coincidence.

Alternately, from assumption , we get which assents that constitutes a path of length in . As a result, is -connected, which wrap up the proof when Theorem 23 is applied.

Under , the identity map, we obtain the following result which is proved by Sk et al. [17].

Corollary 28 (see [17]). *Let be a -complete metric space endowed with a binary relation on and a self-mapping on . Suppose that the following conditions hold:
*(a)*there exists such that ,*(b)* is -closed and locally finitely -transitive*(c)*either is -continuous or is -self-closed*(d)*for every there exists such that**Then, has a fixed point. Further, if we impose an additional hypothesis:
*(e)* is -connected**then has a unique fixed point.*

*Remark 29. *Under the universal relation and , 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 relation , and , 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. *Let together with the usual Euclidean metric . Consider the following relation endowed with