Some <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.5268pt" id="M1" height="16.3568pt" version="1.1" viewBox="-0.0657574 -11.83 37.4278 16.3568" width="37.4278pt"><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,-0.017,13.584,0)"><path id="g117-33" d="M535 230V280H52V230H535Z"/></g><g transform="matrix(.017,0,0,-0.017,27.492,0)"><path id="g113-243" d="M542 242C542 369 469 429 379 468C370 472 372 496 376 513L416 681L400 691L344 648L219 51C167 69 113 128 113 217C113 324 166 387 266 428L252 459C124 418 23 336 23 202C23 95 90 13 205 -8L182 -132C165 -222 191 -254 209 -261L215 -258L270 -7C415 -4 542 98 542 242ZM469 232C469 126 403 33 279 39L355 409C399 394 469 331 469 232Z"/></g></svg>-</span>Fuzzy Cone Contraction Results with Integral Type Application

Rehman, Saif Ur; Jabeen, Shamoona; Khan, Sami Ullah; Jaradat, Mohammed M. M.

doi:https://doi.org/10.1155/2021/1566348

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Volume 2021 | Article ID 1566348 | https://doi.org/10.1155/2021/1566348

Some -Fuzzy Cone Contraction Results with Integral Type Application

Saif Ur Rehman,¹Shamoona Jabeen,²Sami Ullah Khan,¹and Mohammed M. M. Jaradat³

Academic Editor: Naeem Saleem

Received01 Jul 2021

Revised31 Jul 2021

Accepted17 Aug 2021

Published19 Oct 2021

Abstract

In this paper, we define -admissible and --fuzzy cone contraction in fuzzy cone metric space to prove some fixed point theorems. Some related sequences with contraction mappings have been discussed. Ultimately, our theoretical results have been utilized to show the existence of the solution to a nonlinear integral equation. This application is also illustrative of how fuzzy metric spaces can be used in other integral type operators.

1. Introduction

The concept of fuzzy metric space (FM space) was first introduced by Kramosil and Michale [1] while George and Veeramani [2] illustrated some well-known FM space properties. In the sense of Kramosil and Michale [1], George and Sapena [3] and Grabice [4] introduced the idea of fuzzy contraction of complete FM spaces and developed some fixed point (FP) results. Some more related results can be found in [5–8]. Samet et al. [9] proposed the concept of --contraction in complete metric spaces in 2012. Later, Gopal and Vetro [10] presented the concepts of and fuzzy contractive mappings, as well as several novel FP theorems in FM spaces. More FP results in the context of FM spaces can be found in references (see, for example, [11–15]). Mohammadi et al. [16] proved some generalized contraction results in FM spaces with application in integral equations. Recently, the rational type fuzzy contraction concept in complete FM space is given by Rehman et al. [17], and they proved some FP results with an application.

Oner et al. [18] introduced the idea of fuzzy cone metric space (FCM space), proved some basic properties, and developed the first version of “Banach contraction principle for fixed point” in FCM spaces which is stated as follows: “let be a complete FCM space in which fuzzy cone contractive sequences are Cauchy and let be a fuzzy cone contractive mapping being the contractive constant. Then, has a unique fixed point.” Ur Rehman et al. [19] presented some extended “fuzzy cone Banach contraction results” in FCM spaces for some weaker conditions. In topology and analysis, the definition of FCM spaces with different contractive conditions has been commonly used. For further reading, refer to [20–28].

The definition of admissibility has been applied to certain directions by several writers. For a pair of functions, some authors expanded the concept of admissibility. We may advise readers to look for more work in the field of admissibility, as well as references (see [29–33]). Recently, Islam et al. [34] established some FP results in cone -metric space by using generalized -admissible Hardy–Rogers’ contractions over Banachs algebras with application.

In this paper, we define that a mapping is -admissible with respect to and fuzzy cone contraction in FCM space. By using the concept of -admissibility with respect to under mapping , we establish some FP theorems under the fuzzy cone contraction conditions in FCM space with an example. In support of our work, we present an integral type application. By using this concept, one can prove different contractive type FP results for nonlinear mappings with different types of applications in the context of FM spaces. The paper is organized as follows. In Section 2, we introduce the preliminary concepts to support our work. In Section 3, we present some FP results by using different types of contraction conditions in FCM spaces with an illustrative example. In Section 4, we present an integral equation application to validate the concept defined in the paper. Finally, in Section 5, we discuss the conclusion.

2. Preliminaries

The continuous -norm is defined by by Schweizer and Sklar [35].

Definition 1 (see [35]). An operation is called continuous -norm if it satisfies the following conditions:(i) is commutative, associative, and continuous(ii) and , whenever, and , for each The basic -norms, i.e., product, minimum, and Lukasiewicz continuous -norms are defined as follows (see [35]):(i)(ii)(iii)

Definition 2 (see [36]). A subset of a real Banach space is called cone if(i), closed, and (ii) and , then (iii), then A cone and is a partial ordering on via which is defined by iff . stands for and , while stands for . All the cones in this paper have a nonempty interior.

Definition 3 (see [18]). A 3-tuple is called FCM space if is a cone of , is an arbitrary set, is a continuous -norm, and a mapping satisfies the following axioms:(i) and iff (ii)(iii)(iv) is continuous and .

Definition 4 (see [18]). Let be a FCM space, and and be a sequence in . Then,(i)The sequence converges to , if , and so that , . We denote this by or as .(ii)The sequence is Cauchy sequence if , , and so that , .(iii)The sequence is - auchy sequence if , and so that , .(iv) is -complete if every - auchy sequence is convergent in .(v) is said to be fuzzy cone contractive if so that

Definition 5 (see [19]). Let a FCM be triangular in FCM space if and .

Lemma 1 (see [18]). Let in a FCM space and be a sequence in . Then, as , for .

Definition 6 (see [18]). Let be a FCM space and . Then, is called fuzzy cone contractive if , so that

3. Main Result

Definition 7. Let be a FCM space, and let be two functions. We say that is -admissible w.r.t ifNote that in a special case, if we take , then Definition 7 is reduced to the -admissible mapping (i.e., Definition 3.4 in the study by Gopal and Vetro [10]), and also if we take , then we can say that is a -subadmissible mapping.
In the following, denotes the family of all right continuous functions with .

Definition 8. Let be a FCM space, and the mapping is called --fuzzy cone contractive if there exist three functions and , so thatwhere

Theorem 1. Let a FM be triangular in a -complete FCM space and let be an --fuzzy cone contractive if the following axioms hold:(1) is an -admissible w.r.t (2) such that , for (3)The sequence in with and as , then , Then, has a FP such that .

Proof. Let , such thatWe choose a sequenceIf for some , then is a FP of in . Otherwise, we assume that . Since the mapping is an -admissible w.r.t and , for , we have to find thatContinuing this process , we getFrom (5), we havewhereNow, from (11), for , we haveIf is minimum for some , then by (13), we obtainwhich is not possible. Hence, and , we getThis implies thatThus, is an increasing sequence in . Let , we have to show that , for . Let .Using the right side continuity of a function and let the limit , we obtained the contradiction as follows:This implies that , for . Let , where and , for a fixed ,This implies that . It is proved that the sequence is - auchy. Since is -complete, then such that as , i.e.,Now, by the view of Definition 4 (iii),If , i.e., , then . Since is triangular, we have thatThen, from (5) and (21), we havewhereThus, to avoid the contradiction with , for ,Thus, together with (20) and (22), we conclude that .

Corollary 1. Let be a -complete FCM space in which is triangular, and let be an -admissible. Assume that , so thatwhereAssume that the following assertions hold:(i) such that , for (ii)Any sequence in with , and as , then , Then, has a FP in .

Corollary 2. Let a FM be triangular in a -complete FCM space , and let be an -admissible. Assume that , so thatwhereSuppose that the following axioms hold:(i) such that , for (ii)Any sequence in with , and as , then , Then. has a FP in .
Now, to establish the unique FP of an --fuzzy cone contraction map, let the hypothesis (H) is given as follows:(H)For all , , such that

Theorem 2. Adding the hypothesis (H) in Theorem 1, we obtain the uniqueness of a FP of provided the function is nondecreasing.

Proof. Assume that and be the two FPs of in . If , then by (5), we get . Suppose , then from (H), , so thatSince is an -admissible w.r.t , then we getNow, we have to show that , as , for . Since is triangular, then from (5) and (31),where is the FP of and for ,Let such that , and by the hypothesis . Thus,Again by the triangular property of and the view of (5) and (31),where is the FP of and for ,Let such that , , and by the hypothesis . Thus,By continuing the same argument, we obtainThen, by taking the limit , we get thatSimilarly, we can prove thatNow, from (40) and (41), we get the uniqueness, i.e., . Subsequently, we use the following classes of functions in our results without triangularity condition. Suppose thatwhere is nondecreasing and continuous, andwhere is lower semicontinuous, where .

Theorem 3. Let be a -complete FCM space, and let be an -admissible w.r.t . Suppose that and , so that and . Let the following axioms hold:(i) such that , for (ii)Any sequence in with , and , as , then and , Then, has an FP in .

Proof. Let such thatWe define a sequence in such that . If for some , then is an FP of in .
Otherwise, we assume that . However, the mapping is an -admissible w.r.t and . Now, we have to deduce thatContinuing this process , we may obtainClearly,Now, by (44), for ,If , then . Otherwise, if , thenSince is nondecreasing, we may obtain that , and, . Thus, is nondecreasing sequence in . Let , . Now, we have to show that , for . If not, then such that . Therefore, by taking the limit on (50), we getwhich is a contradiction. Thus, we get thatLet such that and , where . We have thatThus, is a - auchy sequence, and since the space is -complete, therefore, . Now, for any sequence in with , and . By the completeness of , as , and , and . Then, easily we may obtainNow, from (44),If , then . If , for , then we havefor . Thus, we get that .

Example 1. Let and -norm is a continuous norm. Let a FM be defined asThen, is a -complete FCM space and a FCM is triangular. Now, we define a mapping byNext, we define and , and we haveHowever, is a -complete FCM space. Now, first, we have to show that is -admissible w.r.t . Since in (59) and (60), and for all and and , which shows that is admissible w.r.t and is an -admissible w.r.t , that is, for all and . On the other hand, for all . If , then, again by using (59) and (60), we have that and for and so that and , for , which is contradiction. Similarly, if , then again we get that and , for , which is contradiction. Hence, and , for . It follows that the mapping is both -admissible and -admissible and , for .
Now, if in a sequence in such that and for all and as , then and hence . This implies that and for all and . Next, we prove that inequality (44) of Theorem 3 is satisfied by using (59)–(61), for all and :That is,Hence, the conditions of Theorem 3 are satisfied, and the mapping has a FP in , i.e., 0.
Note. In special case, by using metric space, the main result of Dutta and Choudhury [37] for the mapping is not applicable; if we put and , then we haveNote. if we take in Theorem 3, then we obtain the following two corollaries.

Corollary 3. Let be a -complete FCM space, and let be an -admissible. Suppose that and , so that and . Let the following axioms hold:(i) such that , for (ii)Any sequence in with , and as , then and , Then, has a FP in .

Corollary 4. Let be a -complete FCM space, and let be an -admissible. Suppose that and , so that and . Let the following conditions hold:(i) such that , for (ii)Any sequence in with , and as , then and , Then, has a FP in .

4. Supportive Application

In this section, we present an integral type application for FP to support our result.

Letwhere . Let be a Banach space of all continuous functions defined on . The induced metric is defined by

Let and consider the fuzzy metric be defined as follows:for and . The space is -complete FM space indeed by the Banach space . Now, here we discuss an integral type application for FCM space and prove the existing solution for the integral equation (67).

Theorem 4. . Let be an integral operator defined as follows:where , , and , that is, satisfies the following: such that , , and , and we have thatwherewhere is bounded on and the . Then, the integral equation (67) has a solution .

Proof. The integral operator is defined in (70). Now, we have to apply Corollary 1, for all , and by the view of (69)–(71), we haveThen, we have the following four cases:(1)If is the minimum term in (72), then . Now, by the view of (69), (71), and (73), for , we have where , for .(2)If is the minimum term in (72), then . Now, by the view of (69), (71), and (73), for , we have where , for .(3)If is the minimum term in (72), then . Now, by the view of (69), (71), and (73), for , we have where , for .(4)If is the minimum term in (72), then . Now, by the view of (69), (71), and (73), for , we have where , for .Hence, in all the cases, it is obvious that (26) holds with and and . As we have mentioned above, is complete and then the FCM space is -complete in which is triangular. The other conditions of Corollary 1 are fulfilled immediately. We deduce that the operator has a FP which is the required solution of the integral equation (67).

5. Conclusion

In this paper, we have presented the mappings, and is -admissible w.r.t and --fuzzy cone contraction in FCM spaces. Using this kind of contractions, we proved FP theorems for -complete FCM space in the sense of George and Veeramani with an illustrative example. Moreover, some extended results in the form of corollaries are discussed. An application is presented to support the concepts defined in the paper. This integral type application is also illustrative of how fuzzy metrics can be used in other integral type operators.

Data Availability

No data set were generated or analyzed during this current study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All the authors have equally contributed to the final manuscript. All the authors have read and approved the manuscript.

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Copyright © 2021 Saif Ur Rehman 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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