Some Rational Coupled Fuzzy Cone Contraction Theorems in Fuzzy Cone Metric Spaces with an Application

Rehman, Saif Ur; Talha Waheed, Muhammad; Jan, Naeem; Gumaei, Abdu; Al-Rakhami, Mabrook

doi:https://doi.org/10.1155/2021/4764441

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Recent Advances in Fixed Point Theory in Abstract Spaces

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Volume 2021 | Article ID 4764441 | https://doi.org/10.1155/2021/4764441

Some Rational Coupled Fuzzy Cone Contraction Theorems in Fuzzy Cone Metric Spaces with an Application

Saif Ur Rehman,¹Muhammad Talha Waheed,¹Naeem Jan,¹Abdu Gumaei,²and Mabrook Al-Rakhami³

Academic Editor: Huseyin Isik

Received13 Jun 2021

Accepted03 Aug 2021

Published18 Aug 2021

Abstract

In this paper, we establish the new concept of rational coupled fuzzy cone contraction mapping in fuzzy cone metric spaces and prove some unique rational-type coupled fixed-point theorems in the framework of fuzzy cone metric spaces by using “the triangular property of fuzzy cone metric.” To ensure the existence of our results, we present some illustrative unique coupled fixed-point examples. Furthermore, we present an application of a Lebesgue integral-type contraction mapping in fuzzy cone metric spaces and to prove a unique coupled fixed-point theorem.

1. Introduction

In 1965, the theory of fuzzy sets was introduced by Zadeh [1]. Kramosil and Michalek [2] introduced the notion of FMS by using continuous -norm with fuzzy sets. Afterward, Grabiec [3] established the completeness property of the FMS and proved a “Fuzzy Banach Contraction Principle for a unique fixed point (FP) in complete FMS.” Since then, many contributed to this theory concerning FP results (e.g., see [4–6]). Later on, in 1994, George and Veeramani [7] modified the concept of FMS introduced by Kramosil and Michalek [2], and they presented the topological properties and proved Baire’s theorem on complete FMS. In 2002, some contractive-type FP theorems were proved by Gregory and Sapena [8] on complete FMS by using the concept of [2, 7]. Some related FP concepts in FMSs can be found in [9–12]. Recently, the rational-type fuzzy contraction concept in FMS is given by Rehman et al. [13], and they proved some FP results with an application.

Jaggi [14] proved the rational-type FP result for a contractive condition. However, Harjani et al. [15] modified the concept of Jaggi [14] and proved a generalized result in “partially ordered metric space.” In 2011, Luong and Thuan [16] proved generalized rational weak contraction results in “partially ordered metric space,” which is a generalization of the result of [14]. In [17], Guo and Lakshmikantham presented the concept of coupled FP with applications by using the nonlinear operator. Later on, Bhaskar [18] and Lakshmikantham [19] proved coupled FP results in “partially ordered metric space.” In [20], Sedghi et al. used commuting mappings and established some common coupled FP theorems in FMSs.

In 2007, the notion of cone metric space (CMS) was introduced by Huang and Zhang [21]. They proved some basic convergence properties and FP theorems on CMS. In 2008, Abbas et al. [22] proved some common FP theorems without continuity for noncommuting mappings on CMS. After that, many others contributed their ideas to the problem of FP results in CMS. Some of their FP contributions can be found in [23–25].

Oner et al. [26] introduced the concept of fuzzy cone metric space (FCMS) and proved a “fuzzy cone Banach contraction theorem” for FP in complete FCMSs in which they assumed that the “fuzzy cone contractive sequences are Cauchy.” In [27], Rehman and Li proved some FP theorems in FCMSs without the assumption of “ sequences are Cauchy” by using the “triangular property of FCM.” Some more FP findings in the said space can be found in [28–31]. Recently, Chen et al. [32] and Rehman and Aydi [33] established some coupled FP and common FP results, respectively, in FCMs with integral types of applications. Waheed et al. [34] proved some coupled FP theorems in FCMSs depending on another function with an application to Volterra integral equations.

In this paper, we prove some rational-type unique coupled FP theorems in FCMSs under the rational type conditions with supportive examples. In addition, to verify the validity of our work, we present an application of a Lebesgue integral-type contraction condition theorem to support our work. The layout of this paper is as follows: Section 2 consists of some basic preliminary concepts. In Section 3, we define the rational coupled mapping in FCMS and prove some unique rational coupled FP results in complete FCMSs with suitable examples. Section 4 deals with the application of Lebesgue integral-type contraction mapping to get the existence result of unique coupled FP theorems in complete FCMSs.

2. Preliminaries

In this section, we recall some basic definitions and lemmas related to our main results. Throughout the complete paper, represents a set of natural numbers and -norm represents a continuous -norm as defined in [35].

Definition 1. An operation would be a -norm if fulfils the following conditions:(1) is associative, commutative, and continuous(2), (3) whenever and

Definition 2. Let be a real Banach space, . Then, a subset is called a cone:(1)If , closed, and (2)If and , then (3)If , then .A partial ordering is defined on a given cone by . stands for and , while stands for . In this paper, all cones have a nonempty interior.

Definition 3. A 3-tuple is said to be a FMS if is any set, is a -norm, and is a fuzzy set on which satisfies the following:(1)(2)(3)(4)(5) is continuous; and

Definition 4. A 3-tuple is said to be a FCMS if is a cone of , is an arbitrary set, is a -norm, and is a fuzzy set on which satisfies the following:(1)(2)(3)(4)(5) is continuous; , and

Definition 5. Let a 3-tuple be a FCMS and and a sequence in (1)Converges to if and and there is , for . We may write this or as .(2)Is a Cauchy sequence if and and there is , for .(3) is complete if every Cauchy sequence is convergent in .(4)Is if , satisfying

Lemma 1. Let be a FCMS, and let a sequence in converge to a point if as , for .

Definition 6. Let be a FCMS. The FCM is triangular, if .

Definition 7 (see [26]). Let be a FCMS and . Then, is known as a if such that

Definition 8. An element is known as a coupled FP of a function ifFurthermore, we shall study some unique coupled FP results in FCMSs under the rational coupled conditions with examples. Also, we present an application of the Lebesgue integral-type rational coupled mapping to get a unique rational coupled FP result in FCMSs.

3. Main Results

In this section, we shall present our main results with illustrative examples.

Definition 9. Let be a FCMS. A mapping is called a rational coupled if and such tha

Theorem 1. Assume that be a complete FCMS in which is triangular and a mapping is a rational coupled satisfying (3). Then, has a unique coupled FP in .

Proof. Let any ; we define sequences and in such thatNow, from (3) and (4), for ,This implies thatSimilarly,Now, from (6) and (7) and by induction, for , we have thatThis shows is a sequence; therefore,Now, for and for ,Hence, is a Cauchy sequence. Since, by the completeness of , , so thatNow, for sequence and from (3) and (4), for ,This implies thatSimilarly,Now, from (13) and (14) and by induction, for , we have thatThis shows is a sequence; therefore,Now, for and for , we haveHence, is a Cauchy sequence. Since, by the completeness of , so thatNow, we shall prove that . Since is triangular and by the view of (3), (9), and (11), for ,Hence, for . Next, we have to prove that ; therefore, by the triangular property of and by the view of (3), (16), and (18), for ,Hence, , for .
Uniqueness: suppose and are other coupled fixed-point pairs in such that and . Now, from (3) and by using Definition 4 (4), for ,Hence, we get that for . Similarly, again from (3) and by using Definition 4 (4), for ,Hence, we get that for .

Corollary 1. Let be a mapping on a complete FCMS in which is triangular and satisfies the inequality , and . Then, has a unique coupled FP in .

Example 1. Let ; is a -norm, and is defined asThen, one can easily prove the triangular property of FCM from the above example and is a complete FCMS. We define a mapping byThen, we have. Hence, we proved that is a rational coupled . Now, by using Definition 4 (4), for ,. Hence, from the above, we conclude that all the conditions of Theorem 1 are satisfied with , , and .

Theorem 2. Let be a mapping on a complete FCMS in which is triangular and satisfies the inequality, , , and with . Then, has a unique coupled FP in .

Proof. Let any ; we define sequences and in such thatNow, from (28) and (29), for ,Now, by Definition 4 (4), for ,After simplification, we get thatwhere . Similarly, again, by using (28) and Definition 4 (4), we get thatwhere . Now, from (32) and (33) and by induction, for , we have thatThis shows is a sequence; therefore,Now, for and for , we haveHence, is a Cauchy sequence. Since, by the completeness of , so thatNow, for sequence , from (28) and (29), for ,Now, again by Definition 4 (4), for ,After simplification, we get thatwhere the value of is same as in (32). Similarly, again by using (28) and Definition 4 (4), for , we get thatNow, from (40) and (41) and by induction, for , we have thatThis shows is a sequence; therefore,Now, for and for ,Hence, is a Cauchy sequence. Since, by the completeness of , so thatNow, we shall prove that . Since is triangular,Now, by the view of (28), (35), and (37), and by using Definition 4 (4), for ,Hence,Now, from (35), (46), and (48), for ,which is a contradiction. As , we get that for . Next, we prove that . Now, again from the triangularity of ,Now, by the view of (28), (43), and (45), and by using Definition 4 (4), for ,Hence,Now, from (43), (50), and (52), for ,which is a contradiction. As , we get that for .
Uniqueness: suppose and are other coupled FP pairs in such that and . Now, from (28) and by using Definition 4 (4), for ,Hence, for . Next, we shall show that , again from (28), and by using Definition 4 (4), for , we haveHence, for .

Corollary 2. Let be a mapping on a complete FCMS in which is triangular and satisfies the inequality , , and with . Then, has a unique coupled FP in .

Corollary 3. Let be a mapping on a complete FCMS in which is triangular and satisfies the inequality , , and with . Then, has a unique coupled FP in .

Corollary 4. Let be a mapping on a complete FCMS in which is triangular and satisfies the inequality , , and with . Then, has a unique coupled FP in .

Example 2. From Example 1, we define a FM byThen, one can easily prove the triangular property of FCM from the above example and is a complete FCMS. We define a mapping byThen,. Hence, we proved that a mapping is a coupled . Now, By using Definition 4 (4) to simplify the rational term of (28), for ,. Again, by using Definition 4 (4) and (61) to simplify the rational term of (28), for , we have. After simple routine calculation, we can get the term result of (28) as follows:Hence, from the above, we conclude that all the conditions of the Theorem 2 are satisfied with , , and .

4. Application

In this section, we present an application on Lebesgue integral-type contraction mapping to support our main work. In 2002, Branciari proved the following result on complete metric space for a unique FP (see [36]).

Theorem 3. Let be a complete metric space, , and be a mapping such that for each ,where is a Lebesgue integrable mapping which is summable (i.e., with finite integral on each compact subset of ) and for each ,Then, has a unique FP such that for any , .”
Now, we are in the position to use the above concept and to prove a unique coupled FP theorem in complete FCMSs.

Theorem 4. Let be a mapping on a complete FCMS in which is triangular and satisfiesfor all , , , and , and is a Lebesgue integrable mapping which is summable (i.e., with finite integral on each compact subset of ) and for each ,Then, has a unique coupled FP in .

Proof. Let any ; we define sequences and in such thatNow, from (67) and from the proof of Theorem 1, for ,Similarly, again by using the arguments, we haveNow, from (70) and (71) and by induction, for , we haveThis shows that is a sequence, and therefore,Hence, we get thatNow, for and for ,We get thatHence, is a Cauchy sequence. Since, by the completeness of , so thatNow, for sequence from (67) and from the proof of Theorem 1, for , we haveSimilarly, again by using the same arguments, we haveNow, from (78) and (79) and by induction, for , we haveThis shows that is a sequence, and therefore,Hence, we get thatNow, for and for , we haveWe get thatHence, is a Cauchy sequence. Since, by the completeness of , so thatNow, we prove . Then, from (67), (74), and (77), for ,Hence, for . Next, we shall prove that ; again from (67), (82), and (85), for ,Hence, which implies for .
Uniqueness: suppose and are other coupled fixed-point pairs in such that and . Now, from (67) and from the proof of Theorem 1, for ,Hence, we get that for . Next, we have to prove , and now, by using (67) and from the proof of Theorem 1, for , we have thatHence, we get that for .

5. Conclusion

We established the new concept of rational coupled fc-contraction mapping in FCMSs and proved some unique rational coupled FP theorems in FCMSs under the rational coupled fc-contraction conditions by using the “triangular property of fuzzy cone metric” with the help of some suitable examples to unify our work. In the last section, we presented an application of the Lebesgue integral-type coupled contraction theorem for unique rational coupled FP in complete FCMSs. By using this concept, one can prove more rational coupled-type fc-contraction results in complete FCMSs with different integral types of application to prove unique coupled FP results.

Data Availability

Data sharing does not apply to this article as no data sets were generated or analyzed during the current study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The authors are grateful to the Deanship of Scientific Research, King Saud University for funding through Vice Deanship of Scientific Research Chairs.

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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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