On Generalized Rational <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.4619093pt" id="M1" height="9.70824pt" version="1.1" viewBox="-0.0657574 -9.24633 20.0256 9.70824" width="20.0256pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-223" d="M545 106L524 126C493 85 467 65 455 65C438 65 427 113 405 238C448 295 498 362 543 439L533 448L478 435C453 386 423 331 398 295H395C370 404 347 448 282 448C169 448 23 309 23 153C23 54 65 -12 128 -12C203 -12 283 70 339 155H341C360 29 380 -12 411 -12C444 -12 491 11 545 106ZM333 204C265 95 210 54 169 54C137 54 113 96 113 171C113 302 191 405 252 405C301 405 318 306 333 204Z"/></g><g transform="matrix(.017,0,0,-0.017,9.748,0)"><path id="g117-33" d="M535 230V280H52V230H535Z"/></g></svg>Geraghty Contraction Mappings in <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.4619007pt" id="M2" height="11.8613pt" version="1.1" viewBox="-0.0657574 -11.3994 22.0236 11.8613" width="22.0236pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-72" d="M673 297H417L411 270C517 261 522 258 506 183L487 92C476 38 428 19 360 19C207 19 123 132 123 287C123 472 243 631 441 631C557 631 617 583 617 469L647 472C650 545 655 604 658 632C625 643 554 666 467 666C201 666 23 502 23 278C23 90 156 -16 339 -16C410 -16 501 6 569 24C566 43 567 70 573 101L589 183C604 259 607 262 667 271L673 297Z"/></g><g transform="matrix(.017,0,0,-0.017,11.737,0)"><use xlink:href="#g117-33"/></g></svg>Metric Spaces

Priyobarta, N.; Khomdram, Bulbul; Rohen, Yumnam; Saleem, Naeem

doi:https://doi.org/10.1155/2021/6661045

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Volume 2021 | Article ID 6661045 | https://doi.org/10.1155/2021/6661045

On Generalized Rational Geraghty Contraction Mappings in Metric Spaces

N. Priyobarta,¹Bulbul Khomdram,¹Yumnam Rohen,¹and Naeem Saleem²

Academic Editor: Efthymios G. Tsionas

Received13 Dec 2020

Revised07 Feb 2021

Accepted12 Feb 2021

Published09 Mar 2021

Abstract

In this paper, we discuss about various generalizations of admissible mappings. Furthermore, we extend the concept of admissible to generalize rational Geraghty contraction in metric space. With this new contraction mapping, we establish some fixed-point theorems in metric space. The obtained result is verified with an example.

1. Introduction and Preliminaries

Samet et al. [1] make a remarkable contribution by introducing admissible and contractive mappings. They also showed that the most celebrated result, the Banach contraction principle and various other results, are consequences of their results. Geraghty [2] also made an improvement of the Banach contraction principle. Mustafa and Sims [3] introduced the concept of metric space and established the Banach contraction principle. In this paper, considering both the concepts of Samet et al. [1] and Geraghty [2], we introduce generalized rational Geraghty contraction in the framework of metric space and establish some theorems on fixed points.

Mustafa and Sims [3] give the following definition.

Definition 1. (see [3]). Let be a nonempty set, and satisfies the following:(i) if and only if .(ii) for all with .(iii) for all with .(iv) (symmetry in all three variables).(v), for all .Here, is known as generalized metric or metric. The pair is known as metric space.
The following example shows the relation between metric space and metric space given by Mustafa and Sims [3].

Example 1. (see [3]).(i)Let be an ordinary metric space, then is a metric on .(ii)Let be an ordinary metric space, then is a metric on .(iii)Let be a metric on , then is a metric on .

Definition 2. (see [3]). Consider a metric space and a sequence of points of . is said to be convergent to provided ; that is, there exists satisfying and where . Here, is known as the limit of the sequence and is denoted as or .

Proposition 1. (see [3]). In a metric space , we have the following equivalent statements:(i) is convergent to .(ii) when .(iii) when .(iv) when .

Definition 3. (see [3]). In a metric space , the sequence is said to be Cauchy if for any , there exists satisfying and ; that is, when .

Proposition 2. (see [3]). In a metric space , we have the following equivalent statements:(i)The sequence is Cauchy.(ii)For any , there exists such that for all .

Definition 4. (see [3]). A metric space is said to be complete if every Cauchy sequence is convergent in .

Lemma 1. (see [3]). In a metric space , for , we have the following:(i)If , then .(ii).(iii).(iv).(v).(vi).

Definition 5. (see [3]). In a metric space , a mapping is known as continuous if is convergent to , where is any convergent sequence converging to .
Here firstly, we recall the definition of -admissible mappings and its generalizations in metric space and -metric space.

Definition 6. (see [1]). Let be a self-mapping on a metric space , and let be a function. is said to be an -admissible if , and makes .

Example 2. (see [1]). Consider and define and by for all andThen, is -admissible.

Definition 7. (see [4]). Let and . It is said that the pair is -admissible if such that , then we have and .

Definition 8. (see [5]). Let and . It is said that is a triangular -admissible mapping if the following holds: (T1) makes . (T2), , makes .

Definition 9. (see [4]). Let and . It is said that a pair is a triangular -admissible mapping if the following holds: (T1) makes and . (T2), , makes .

Definition 10. (see [6]). Let be a self-mapping on a metric space and let be two functions. It is said that is -admissible mapping with respect to if , and implies .
It can be noted that if , then the above definition becomes Definition 6. If we take , then is said to be an -subadmissible mapping.

Lemma 2. (see [5]). Let be a triangular -admissible mapping. Let us take such that . Form a sequence as . Then, , where , .

Lemma 3. (see [7]). Let be triangular -admissible mapping. Let us take such that . Form sequences and , where . Then, , where , .

Alghamdi and Karapinar [8] generalized the concept of -admissible mappings in the context of -metric space and called it -admissible. The definition of -admissible given by Alghamdi and Karapinar is defined as follows.

Definition 11. (see [8]). Let and , then is said to be -admissible if for all thenAlghamdi and Karapinar [8] introduced contractive mappings of type-I and type-II. They also introduced contractive mappings of type-. They also gave the relation between these different types of contractions and equivalent Banach contractions.
Alghamdi and Karapinar [9] further generalized the results of Alghamdi and Karapinar [8] by introducing generalized contractive mappings of type-I and type-II.
Kutbi et al. [10] defined rectangular admissible mapping. They also defined weak contractive mappings to establish some coincidence point theorems for coupled and tripled in -metric space.

Definition 12. (see [10]). Let be a -metric space and let and . is said to be a rectangular admissible mapping with respect to if the following holds:(i) implies , .(ii) and imply , .Hussain et al. [11] generalized the concept of rectangular admissible mappings used to obtain coupled and tripled fixed-point theorems.
Hussain et al. [12] established a generalized form of admissible mappings in order to prove coincidence points and common fixed points in the framework of -metric spaces. Furthermore, several authors obtained different kinds of generalization of Banach contraction principle in different spaces (see for details [13–20]).

Definition 13. (see [12]). Let be an arbitrary set, , and . The mapping is called an -dominating map on if or for each in .

Definition 14. (see [12]). In an arbitrary set , let be given mappings and be a function. The pair is said to be partially weakly admissible if and only if for all .

Definition 15. (see [12]). In an arbitrary set , let be given mappings and be a function. The pair is said to be partially weakly admissible with respect to if and only if for all , , where .
In the above definition, if , is said to be partially weakly admissible (or admissible of rank 3) with respect to .
If (the identity mapping on ), then the above definition becomes the definition of partially weakly admissible pair.
Ansari et al. [21] also studied -admissible mappings in -metric space by introducing --subadmissible mapping and -dominating map. They also introduced -subdominating map, -regular in the framework of -metric space, and partially weakly --admissible and partially weakly --subadmissible mappings.

Definition 16. (see [21]). Let be a -metric space, and let be a self-mapping on and be a function. is said to be a subadmissible (or -subadmissible of rank 3) mapping if

Definition 17. (see [21]). Let be an arbitrary set, , and . A mapping is called an -subdominating map on if or for each in .

Definition 18. (see [21]). In a -metric space , let be given mappings and be a function. The pair is said to be partially weakly subadmissible (or subadmissible of rank 3) if and only if for all .

Definition 19. (see [21]). In a -metric space , let be given mappings and be a function. The pair is said to be partially weakly subadmissible (or subadmissible of rank 3) with respect to if and only if for all , , where .
Hussain et al. [22] defined Meir–Keeler contractive mapping and used it in proving fixed-point theorems in the framework of -metric spaces.

Definition 20. (see [22]). Let be a -metric space and . Let be an -admissible mapping satisfying the following: for each , there exists such that implies for all . Then, is known as a Meir–Keeler contractive mapping.

In the above definition, is the collection of nondecreasing functions continuous in such that if and only if and .

The concept of -admissible mappings is extended to -metric space by Zhou et al. [23] and called it -admissible. They are defined as follows.

Definition 21. (see [23]). Let and , then is said to be -admissible if for all :They also extended -admissibility for two mappings. Furthermore, they also introduced concepts of various contractive mappings viz. type A, type B, type C, type D, and type E.
Bulbul et al. [24] also derived the concept of generalized contractive-type mappings on the line of generalized -- contractive-type mappings. Nabil et al. [25] also defined the concept of -admissible mappings in -metric space.
From these, what we observe is that -admissible was for the first time used by Samet et al. [1] to represent -admissible while dealing with coupled fixed point-related problems. Phiangsungnoen et al. [26] also used the name -admissible mapping in order to represent -admissible for fuzzy mappings. On the contrary, -admissible of Alghamdi and Karapinar [9] and -admissible of Zhou et al. [23] are all extended versions of -admissible mappings in -metric space and -metric space, respectively. Thus, we can remark that -admissible and its various forms can be extended to -metric as well as -metric spaces and further to -metric and -metric spaces. With this idea, we introduce various forms of -admissible mappings in the context of -metric space and present following definitions. For notation, we use for -admissible mappings in -metric space.

Definition 22. Let and , then is said to be -admissible if , implies .

Definition 23. Let and . We say that the pair is admissible if such that , then we have and .

Definition 24. Let and . We say that is triangular admissible mapping if the following holds:(i) implies .(ii) and implies .

Definition 25. Let and let be functions. We say that is admissible mapping with respect to if ,Note that if we take , then this definition becomes Definition 22. Also, if we take , then it is said that is an subadmissible mapping.

Definition 26. Let and . We say that the pair is admissible mapping with respect to if such that , then we have and .

Lemma 4. Let are triangular -admissible mappings. Suppose that there exists such that . Define sequencesThen, we have , , .

2. Main Results

Let us take as the collection of functions such that gives , where is a bounded sequence of positive real numbers.

We start our results with the following definitions.

Definition 27. In a metric space , let be a function. We say that mappings is a pair of generalized rational Geraghty contraction mappings of type-I if for all and ,where

Definition 28. In a -metric space , let be a function. We say that the mappings are a pair of generalized rational Geraghty contraction mappings of type-II if for all and ,whereIf , then we have the following.

Definition 29. In a metric space , let be a function. We say that mapping is a generalized rational Geraghty contraction mappings of type-I if there exists such that for all ,where

Definition 30. In a -metric space , let be a function. We say that the mapping is a generalized rational Geraghty contraction mappings of type-II if there exists such that for all ,where

Theorem 1. In a complete -metric space , let be a function. Let be two mappings satisfying the following:(i) is pair of generalized rational -Geraghty contraction mappings of type-I.(ii) is a pair of triangular -admissible mappings.(iii)There exists such that .(iv) and are continuous.Then, a common fixed point exists for the pair .

Proof. Let be such that and . Inductively, we construct a sequence in as follows:where .
By assumption and the pair is -admissible, by Lemma 4, we haveThen,for all .
Now,If , thenwhich is a contradiction. Hence,This implies thatfor all .
So the sequence is nonnegative and nonincreasing. Now we prove that . It is clear that is a decreasing sequence. So, for some , we have .
From (23),Now, by taking limit as , we havethat is,By the property of , we haveWe have to show that is a Cauchy sequence. If possible, let is not a Cauchy sequence. Then, there exist and sequences and such that, for all positive integers , we get :Therefore,Taking ,Therefore,Also, from the triangular inequality, we haveTaking upper limit as above, we obtainThus,By triangle inequality, we haveTaking limit as , we haveFollowing the above process, we haveCombining, we haveBy Lemma 4, , we haveFinally, we conclude thatApplying , we obtainSo,a contradiction. Thus, is a Cauchy sequence. By completeness of , there exists such that implies that and . As and are continuous, we get and . Thus, . Similarly, , and we have . Then, have common fixed point.
In the next theorem, we dropped the continuity condition.

Theorem 2. In a complete -metric space , let be a function. Let be two mappings satisfying the following:(i) is a pair of generalized rational -Geraghty contraction mappings of type-I.(ii) is a pair of triangular -admissible mappings.(iii)There exists such that .(iv)If is a sequence in such that for all and as , then a subsequence of exists satisfying for all .Then, have a common fixed point.

Proof. It follows the similar lines of Theorem 1. Define a sequence and , where converges to . By the hypothesis of , a subsequence of exists satisfying for all . Now, we haveso thatOn the contrary, we obtainLetting , then we haveSuppose that . From (47), for a large , we have , which implies thatThen, we haveLetting in (49), we claim thatwhich is a contradiction. Thus, we find that implies .
Also showing that in is a common fixed point of and .

3. Consequences

Ifand in Theorems 1 and 2, we have the following corollaries.

Corollary 1. In a complete -metric space , let be -admissible mapping satisfying the following:(i) is generalized rational -Geraghty contraction mappings of type-I.(ii) is triangular -admissible.(iii)There exists such that .(iv) is continuous.

Then, has a fixed point , and is a Picard operator; that is, converges to .

Corollary 2. In a complete -metric space , let be -admissible mapping satisfying the following:(i) is a generalized rational -Geraghty contraction mappings of type-I.(ii) is triangular -admissible.(iii)There exists such that .(iv)If is a sequence in such that for all and as , then a subsequence of exists satisfying for all .

Then, has a fixed point and is a Picard operator; that is, converges to .

Ifin Theorems 1 and 2, we can have another result.

Let be a -metric space and let be functions. Mappings are called a pair of generalized rational -Geraghty contraction-type mappings with respect to if there exists such that for all :where

Theorem 3. In a complete -metric space , let be -admissible mapping with respect to satisfying the following:(i) is a pair of a generalized rational -Geraghty contraction type mapping.(ii) is a pair of triangular -admissible mappings.(iii)There exists such that .(iv) and are continuous.

Then, have common fixed point.

Proof. Let be such that and . Inductively, we form a sequence in as follows:where .
By assumption and the pair is -admissible with respect to , we have from which we deduce that which also implies that . Continuing in this way, we obtain for all :Therefore,for all .
Now,From the definition of , the case is impossible.which is a contradiction. Otherwise, in other case,This implies thatfor all .
Following the similar lines of the Theorem 1, we can prove that and have a common fixed point.

Theorem 4. In a complete -metric space , let be a pair of -admissible mappings with respect to satisfying the following:(i)The pair is a generalized rational -Geraghty contraction type mappings.(ii)The pair is triangular -admissible.(iii)There exists such that .(iv)If is a sequence in such that for all and as , then a subsequence of exists satisfying for all .

Then, and have common fixed point.

Proof. It follows the similar line of Theorem 2.
Ifand in Theorems 3 and 4, we get the following corollaries.

Corollary 3. In a complete -metric space , let be -admissible mappings with respect to satisfying the following:(i) is a generalized rational -Geraghty contraction type mapping.(ii) is triangular -admissible.(iii)There exists such that .(iv) is continuous.

Then, has a fixed point and be a Picard operator; that is, converges to .

Corollary 4. In a complete -metric space , let be -admissible mapping with respect to satisfying the following:(i) is a generalized rational -Geraghty contraction type mapping.(ii) is triangular -admissible.(iii)There exists such that .(iv)There exists such that for all and as , then a subsequence of exists satisfying for all .

Then, has a fixed point , and is a Picard operator; that is, converges to .

Example 3. Let with -metric, then , , , , andDefine the mappings as follows for each , , , and , thenLet , , , then condition (i) is not satisfied by the mapping as , whereThus, is not true.
We prove that Theorem 1 can be applied to and . Let ; clearly, is -admissible such that . Let so that and . Hence, is -admissible. We know that condition (i) of Theorem 1 is satisfied.
If , then , and we havewhereand .Hence, the conditions of Theorem 1 are satisfied. So, and have a common fixed point.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2021 N. Priyobarta et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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