Recent Fixed-Point Results for <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.4618998pt" id="M1" height="12.7225pt" version="1.1" viewBox="-0.0657574 -12.2606 18.8199 12.7225" width="18.8199pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-230" d="M475 507C475 612 440 712 326 712C139 712 23 420 23 215C23 96 58 -12 180 -12C369 -12 475 293 475 507ZM391 522C391 486 387 448 379 394H126C155 538 222 677 310 677C386 677 391 571 391 522ZM373 346C344 193 283 22 189 22C126 22 106 114 106 196C106 243 111 293 118 346H373Z"/></g><g transform="matrix(.017,0,0,-0.017,8.547,0)"><path id="g117-33" d="M535 230V280H52V230H535Z"/></g></svg>Contraction Mappings in Rectangular <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 27.2082 11.8613" width="27.2082pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-78" d="M962 650H795L470 145L347 650H176L170 622C268 613 275 606 244 503L190 322C150 188 132 126 118 91C102 50 80 33 18 28L12 0H245L251 28C175 35 170 48 174 93C177 128 191 180 220 284L292 542H294C331 392 383 150 409 4H432L774 555H776L714 137C700 40 694 34 612 28L606 0H868L874 28C793 34 784 37 797 137L849 533C859 612 863 616 956 622L962 650Z"/></g><g transform="matrix(.017,0,0,-0.017,16.905,0)"><use xlink:href="#g117-33"/></g></svg>Metric Spaces with Supportive Application

Mudhesh, Mustafa; Hammad, Hasanen A.; Alsamir, Habes; Arshad, Muhammad; Ameer, Eskandar

doi:https://doi.org/10.1155/2021/5564248

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Abstract Introduction Preliminaries Conclusions Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

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Recent Advances in Fixed Point Theory in Abstract Spaces

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Research Article | Open Access

Volume 2021 | Article ID 5564248 | https://doi.org/10.1155/2021/5564248

Recent Fixed-Point Results for Contraction Mappings in Rectangular Metric Spaces with Supportive Application

Mustafa Mudhesh,¹Hasanen A. Hammad,²Habes Alsamir,³Muhammad Arshad,¹and Eskandar Ameer⁴

Academic Editor: Naeem Saleem

Received08 Jan 2021

Revised29 Jan 2021

Accepted06 Feb 2021

Published23 Feb 2021

Abstract

The goal of this manuscript is to present a new fixed-point theorem on contraction mappings in the setting of rectangular M-metric spaces (RMMSs). Also, a nontrivial example to illustrate our main result has been given. Moreover, some related sequences with contraction mappings have been discussed. Ultimately, our theoretical result has been implicated to study the existence and uniqueness of the solution to a nonlinear integral equation (NIE).

1. Introduction

Fixed-point theory is one of the branches of functional analysis known as “Banach contraction principle (BCP)” [1] which plays a main role in several branches of mathematics and applied sciences. Its core subject is concerned with the stipulations for the existence of one or more fixed points of a mapping from a topological space into itself, whereas we can obtain a fixed point such that and . Later, this principle has been extended by using different forms of contractive conditions in various spaces (for example, see [2–7]).

The idea of a generalized metric space (GMS) (rectangular metric space or Branciari metric space) was introduced by Branciari [8] in 2000, where the triangle inequality is replaced by the inequality for all pairwise distinct points . Under this space, many fixed-point results were proposed and nice applications were discussed (see [9–13]).

Recently, GMSs extended to recent versions such as metric spaces (MMSs), rectangular metric spaces (RMSs), extended rectangular b-metric spaces (ERbMSs), partial rectangular b-metric spaces (PRbMSs), and partial rectangular metric spaces (PRMSs) (see [8, 14–17]).

In 2018, a new version of a metric space was introduced by Özg ür et al. [18], called a rectangular metric space (RMMS) which is also extended (MMS). Shukla [17] defined a PRMS which is the generalized RMS. In such spaces, the authors [17, 18] discussed and presented some fixed-point theorems for self-mappings. They presented some topological concepts about open balls and convergence and defined the notions of a circle and a fixed circle using these concepts with an application to fixed-circle problems. For more details of generalizing fixed-point results, see [8, 14, 17, 19–27].

Continuing in the same direction, in this manuscript, a new fixed-point theorem for contraction mapping in RMMSs is proved and some related sequences are introduced. To support the theoretical result, a nontrivial example is obtained and the existence of a solution to an NIE has been found.

2. Preliminaries

In this part, we consider for all , , , , and .

Now, we give some previous results in the gradation of spaces.

Definition 1. (see [17], PRMS). Let be a nonempty set. A mapping is said to be a PRM on if for any and all distinct points , the following stipulations hold:(i) (ii) (iii) (iv) The pair is called a PRMS.

The above space was extended to an MMS by Asadi et al. [14] as follows.

Definition 2. (MMS). Assume that is a nonempty set. If the function satisfies the following hypotheses, for all , then the pair is called an MMS:(i)(ii)(iii)(iv)

Özgür et al. [18] generalized an MMS to a RMMS as follows.

Definition 3. (RMMS). Let be a nonempty set. If the function verifies the following assumptions, for all , then the pair is called a RMMS:(i) (ii) (iii) (iv) , for all

It is clear that every MMS is a RMMS.

The results given follow from Özgür et al. [18].

Remark 1. Let be a RMMS. Clearly, for every , we have(1)(2) Also, it can be easily verified under special cases that(3)

Example 1. Assume that is the set of all complex numbers, and consider the set for a fixed and define the function by for all . Then, is a RMMS.

Proposition 1. (see [18]). Suppose that is a RMS and a function is a one-to-one and nondecreasing function with so that , for all . Then, the function which is defined by , for all with the set , is a RMMS.

Example 2. Let be a RMS and a function be defined as , with for all , and is a RMMS with the set .

Definition 4. Assume that is a RMMS; a sequence is said to be(1)Converging to a point iff(2)-Cauchy sequence iff exist and is finite.(3)A RMMS called complete if every Cauchy sequence converges to a point so that

Lemma 1. (see [18]). Assume that and as in a RMMS . Then,

Lemma 2. (see [18]). Assume that as in a RMMS . Then,

Lemma 3. (see [18]). Suppose that and as in a RMMS . Then, . Furthermore, if , then .

Lemma 4. (see [18]). Let be a sequence in a RMMS . If there is so that for all , then(i)(ii)(iii)(iv) is an -Cauchy sequence

Lemma 5. (see [18]). Let be a RMMS and be a self-mapping on . If there is so that for all , then as , where is a sequence defined by so that as .

Theorem 1. (see [18]). Suppose that is a self-mapping on a RMMS . If there is so thatthen has a unique fixed point in , where .

Recently, Jleli and Samet [10] established the fixed-point theorem as follows.

Theorem 2. (see [10]). Let be a complete GMS and be a given map. Suppose that there is and such thatthen has a unique fixed point.

Theorem 2 was extended by Jleli et al. [11] as follows.

Theorem 3. (see [11]). Let be a complete GMS and be a given map. Suppose that there is and so thatwherethen has a unique fixed point.

In Theorems 2 and 3, the authors considered is the set of all functions so that the following conditions hold:(i) is nondecreasing(ii) for each sequence , iff (iii) there is and so that (iv) is continuous

3. Main Result

We introduce in this part some new fixed-point results via contraction mapping in RMMSs which generalizes the results of Jleli et al. [11].

Theorem 4. Let be a complete RMMS and be a given mapping. Suppose that there is and so thatwhereand is as defined in the above section. Then, has a unique fixed point.

Proof. Let be an arbitrary value and define a sequence by letting , . If for some , we have , then is a FP of . So we suppose that . Now, from (10), we haveIf , then from (12), one can writeTaking in both sides, we obtainwhich is a contradiction because . Thus, we haveIt follows from (12) thatThis leads toAs in (19), we obtain thatwhich implies from that . Now, from condition , there are and so thatSuppose that . In this case, let . From the definition of the limit, there is such thatThis implies thatThus,where .
Now, consider and as an arbitrary positive number. From the definition of the limit, there is so thatThis leads towhere .
Thus, in both cases, there exist and so thatUsing (14), we obtainTaking limit as in the above inequality, one can obtainThus, there is so thatNow, we will discuss the following two steps.
St₁. When for some integers with , we have . Choose and . Then, and is a periodic point of . By (10) and (14), we obtainAs , we have . On the other hand, we can writewhich leads toor equivalentlyThis yields . Hence, , that is, is a fixed point of .
St₂. When for every with , we shall prove that has a periodic point. Suppose on the contrary, utilizing (10) and (11), we obtainwhereSince is continuous and nondecreasing, using (35) and (36), one can writeLet be the set of such thatIf , there exists such that for every ,In this case, we obtain from (37) thatfor all . Letting in the above inequality and using (20), we obtainIf , we can find a subsequence of , which we denote also by , such thatIn this case, we obtain from (37) thatfor large enough. Letting in the above inequality, we haveThen, in both cases, (44) holds. Using (44) and property , we obtainSimilarly, from condition , there exists such thatLet . If is odd, then we consider , where , Using (30) for all , we can obtainNow, if is even, then writing , where and using (30) and (46), for all , we obtainCombining (47) and (48), we haveSince the series is convergent, we find that is also convergent as , which implies thatThus, the sequence is Cauchy in the complete RMMS . Hence, there is some so that as . Without restriction of the generality, we can suppose that for all .
Assume that . Using (10), we obtainwhereTaking limit as in the above inequality and using and Lemmas 1–3, we havewhich is a contradiction; therefore, , which is also a contradiction with the supposition that does not have a periodic point. Thus, has a periodic point, say , of period . Assume that the set of all fixed points of is empty. Then, we haveUsing (10), we obtain thatwhich is a contradiction. Thus, the set of fixed points of is nonempty, that is, has more than one fixed point.
Now, assume that are two fixed points of so that . Utilizing (10), we obtainwhich is a contradiction. Then, has a unique fixed point. This completes the proof.

Theorem 4 is still valid if we replace condition (10) with one of the following stipulations: for all with ,(i)There is so that(ii)There is so that(iii)There is so that(iv)There are and so that

We note that contains a large class of functions. For instance,

So we conclude the following corollary from Theorem 4.

Corollary 1. Let be a complete RMMS and be a self-mapping. If there exists and such that for all and ,whereThen, has a unique fixed point.

The example below justifies all requirements of Theorem 4.

Example 3. Let for a fixed , where . Define the function by for all . Then, is a RMMS. Take and the RMMS becomes such that and for all . Let be defined by ; it is clear that . Define byNow, for , , we haveThen, satisfies the contraction with .

On the other hand, for , we have

Similarly, for , , we obtainwhere . Hence, the conditions of Theorem 4 are fulfilled and is a unique fixed point of .

4. Supportive Application

In this section, we apply the theoretical results of Theorem 4 to discuss the existence of solution to a NIE:where and are continuous functions.

Define the operator bywhere is the space of all continuous functions defined from to equipped with

Obviously, is a complete RMMS.

We will consider Problem (68) under the following assumptions:(i)The function is continuous and (ii)The function is continuous, where and (iii)For all , and so that the pair is a complete RMMS with (iv)For all , define the function by , where

Now, we can state and prove the important theorem in this part as follows.

Theorem 5. Via hypotheses (i)–(iv), Problem (68) has a unique solution in .

Proof. Define the operator as in (69). If has a unique fixed point, then the NIE (68) has a unique solution. From Hypotheses (i)–(iii), we can writeConsequently, we haveThis implies thatBy Hypothesis (iv), we obtainyieldingwhere with . Thus, all conditions of Theorem 4 are fulfilled. So has a unique fixed point which is a unique solution to NIE (68).

5. Conclusions

In this paper, we established fixed points for contraction self-mapping in RMMS. Our new results are extensions of recent fixed-point theorems of Jalili et al. [11]. Also, we gave an example to clarify the obtained results. Finally, we applied our main result to study the existence and uniqueness of a solution for a NIE. The new concepts lead to further investigations and applications.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interests regarding the publication of this article.

Authors’ Contributions

All authors contributed equally and significantly in writing this article.

Acknowledgments

The authors extend their appreciation to the Deanship of Post Graduate and Scientific Research at Dar Al Uloom University for funding this work.

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Copyright © 2021 Mustafa Mudhesh 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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