A New Approach to Fuzzy Differential Equations Using Weakly-Compatible Self-Mappings in Fuzzy Metric Spaces

Shamas, Iqra; Rehman, Saif Ur; Jan, Naeem; Gumaei, Abdu; Al-Rakhami, Mabrook

doi:https://doi.org/10.1155/2021/6123154

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Function Spaces, Hardy-Type Inequalities, and their Applications

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Research Article | Open Access

Volume 2021 | Article ID 6123154 | https://doi.org/10.1155/2021/6123154

A New Approach to Fuzzy Differential Equations Using Weakly-Compatible Self-Mappings in Fuzzy Metric Spaces

Iqra Shamas,¹Saif Ur Rehman,¹Naeem Jan,¹Abdu Gumaei,²and Mabrook Al-Rakhami³

Academic Editor: Douadi Drihem

Received27 Jun 2021

Revised28 Sept 2021

Accepted29 Sept 2021

Published12 Oct 2021

Abstract

The key objective of this research article includes the study of some rational type coincidence point and deriving common fixed point (CFP) results for rational type weakly-compatible three self-mappings in fuzzy metric (FM) space. The “triangular property of FM” is used as a fundamental tool. Moreover, some unique coincidence points and CFP theorems were presented for three self-mappings in an FM space under the conditions of rational type weakly-compatible fuzzy-contraction. In addition, some suitable examples are also given. Furthermore, an application of fuzzy differential equations is provided in the aid of the proposed work. Hence, the innovative direction of rational type weakly-compatible fuzzy-contraction with the application of fuzzy differential equations in FM space will certainly play a vital role in the related fields. It has the potential to be extended in any direction with different types of weakly-compatible fuzzy-contraction conditions for self-mappings with different types of differential equations.

1. Introduction

In 1922, Banach [1] proved a “Banach contraction principle for fixed point (FP)” which is stated as “A self-mapping in a complete metric space satisfy the contraction condition has a unique FP.” After the publication of the “Banach contraction principle,” many researchers contributed in flourishing the FP theory. Thus, different types of contractive results were established for FP and CFP for different types of mappings in the context of metric spaces. Kannan [2], Chatterjea [3, 4], Ali et al. [5], Covitz and Nadler [6], Altun et al. [7], Khan [8], Rehman et al. [9], and Sahin [10] proved some single-valued and multivalued contractive type FP, CFP, and best proximity point results in different types of spaces.

In 1965, the concept of fuzzy sets was given by Zadeh [11], and this concept is investigated, used, and applied in many directions. In [12], Kramosil and Michalek used the concept of fuzzy sets together with metric space and introduced the notion of FM space and some more notions. In 1988, Grabiec [13] used the concept of Kramosil and Michalek [12] and proved two FP theorems of “Banach and Edelstein contraction mapping theorems on complete and compact fuzzy metric spaces, respectively.” Later on, George and Veeramani [14] presented the stronger form of metric fuzziness. In 2002, Gregory and Sapena [15] proved some contractive type FP theorems in complete FM spaces in the sense of [12, 14]. In [16], Sadeghi et al. extended and improved the result of Gregory and Sapena [15] and proved some FP and coincidence point theorems by using an implicit relation for set-valued-mappings on complete partially ordered FM spaces. Bari and Vetro [17] established some FP results by using attractors and weak-fuzzy contractive mappings in FM space. While Hadzic and Pap [18] proved “a FP theorem for multi-valued mappings in probabilistic metric spaces with an application in FM spaces.” Imdad and Ali [19], Shamas et al. [20], Som [21], and Pant and Chauhan [22] proved FP and CFP theorems for different contractive type mappings in FM spaces. Saleh et al. [23, 24] established different contractive type FP results in FM spaces. Recently, Rehman et al. [25] introduced the rational type fuzzy-contraction condition in FM spaces and proved some FP theorems with an application. In 2015, Oner et al. [26] introduced the notion of fuzzy cone metric (FCM) space. They proved some basic properties and a “fuzzy cone Banach contraction theorem” for FP with the assumption that “the fuzzy cone contractive sequences are Cauchy.” Later on, Rehman and Li [27] established some FP theorems in FCM space without the assumption that the “fuzzy cone contractive sequences are Cauchy.” Jabeen et al. [28] proved some common FP results in FCM spaces by using the contractive type weakly-compatible self-mappings with an application. By using the concept of [27], Rehman and Aydi [29] proved some rational type CFP theorems in FCM spaces with an application to Fredholm integral equations.

The chief purpose behind this article is the introduction of a new concept of rational type weakly-compatible fuzzy-contraction maps for three self-mappings in FM spaces. We used the “triangular property of fuzzy metric” as an elementary tool and proved some unique coincidence points and CFP theorems under the rational type weakly compatible fuzzy-contraction conditions for three self-mappings in FM spaces with some suitable examples. Moreover, we applied the fuzzy differential equations for a unique solution in order to support our study. As a result, the current novel direction of rational type weakly-compatible fuzzy-contraction with the application of fuzzy differential equations in FM space will play a vital role in the related fields. It has the potential to be extended in any direction with different types of weakly-contraction conditions for self-mappings with different types of differential equations.

This paper is organized as follows: Section 2 named as preliminaries consists of the basic concepts related to our main work. In Section 3, we established some coincidence points and CFP theorems under the rational type weakly-compatible fuzzy-contraction conditions for three self-mappings in FM spaces with examples to verify the validity of our work. Section 4 deals with the application of the fuzzy differential equations to support our work. Finally, the last section, Section 5 concludes this article.

2. Preliminaries

In this section, we present some preliminaries related to our main work such as continuous -norm, FM space, Cauchy sequence, complete FM space, fuzzy-contraction, fuzzy-contractive sequence, and weakly-compatible self-mappings.

Definition 1 (see [30]). An operation is called a continuous -norm, if (i) continuous(ii) is commutative and associative(iii) and , whenever and ,

The basic continuous -norms of minimum, product, and Lukasiewicz are defined by Schweizer and Sklar [30], respectively, as follows (i)The minimum -norm is (ii)The product -norm is (iii)The Lukasiewicz -norm is

Definition 2 (see [14]). A 3-tuple is said to be a FM space if is an arbitrary set, is a continuous -norm and is a fuzzy set on satisfying the following conditions: (i)(ii)(iii)(iv)(v) is continuous

Lemma 3 (see [14]). is nondecreasing .

Definition 4 (see [14, 15]). Let be a FM space, and is a sequence in . Then, (i) converges to if and , such that , . We may write this or as (ii) is a Cauchy sequence if and such that , (iii) is complete if every Cauchy sequence is convergent in (iv)Fuzzy-contractive if and satisfyingThroughout this paper, represents the set of natural numbers.

Lemma 5 (see [14]). Let be a FM space. A sequence in converges to if and only if as , for .

Definition 6 (see [17]). Let be a FM space. The fuzzy metric is triangular, if

Definition 7 (see [15]). Let be a FM space. A mapping is said to be a fuzzy-contractive if such that

Definition 8 (see [31]). Let and be two self-mappings on a nonempty set (i.e., ). If there exists and for some . Then, is called a coincidence point of and , and is called a point of coincidence of the mappings and . The mappings and are said to be weakly-compatible if they commute at their coincidence point, i.e., for some , then .

Proposition 9 (see [31]). Let and be weakly-compatible self-mappings on a nonempty set . If and have a unique point of coincidence such that , then, is known as the unique common FP of and .

3. Main Result

This section deals with the main results of our paper, here, we establish some coincidence point and CFP theorems under the rational type weakly-compatible fuzzy-contractive for three self-mappings in FM spaces with some suitable examples. Throughout main results, we use the concept of a binary operation is a continuous product -norm which is defined as:

Now we are in the position to present our first main result.

Theorem 10. Let a fuzzy metric is triangular in a complete FM space and let be three self-mappings, satisfies for all , for and with . If , where is a complete subspace of . Then , and have a unique point of coincidence. Moreover, if the pairs and are weakly compatible. Then, , and have a unique CFP in .

Proof. Let be the arbitrary point of . Using the condition choose a sequence in such that Now, by (6), for , Now by using Definition 2 (iv), for , After simplification, we obtain where . Similarly, again by the view of (6), for , Now by using Definition 2 (iv), for , After simplification, we obtain where value is same as in (10). Now from (10), (13), and by induction, Hence, is a fuzzy-contractive sequence in , therefore, Since is triangular, , This shows that is a Cauchy sequence, and is a complete subspace of . Hence, such that as , i.e., Since is triangular, Now from (6), (15), (17), and by using Definition 2 (iv), for , Then, Now, from (17), (18), and (20), we obtain Notice that , where , therefore, for . Next, we have to prove that . Since, is triangular, Now, again from (6), (15), (17), and by using Definition 2 (iv), for , Then, Now, from (17), (22), and (24), we obtain Note that , where , therefore, for . Hence, is a common coincidence point of the mappings such that .
Next, we prove the uniqueness of a coincidence point in for the mappings . Let be the other common coincidence point in such that for some . Then, from (6) and by using Definition 2 (iv), for , note that , where . Thus, we get that , that is, . By using the weak compatibility of the pair and by using Preposition 9, we can get a unique CFP of the mappings , and . Let such that, . Hence, we get that , for .

Corollary 11. Let a fuzzy metric is triangular in a complete FM space and let be three self-mappings, satisfies for all , for and with . If , where is a complete subspace of . Then, , and have a unique point of coincidence. Moreover, if the pairs and are weakly compatible. Then, , and have a unique CFP in .

If we use identity map instead of , i.e., , in Theorem 10, we can get the following corollary:

Corollary 12. Let a fuzzy metric is triangular in a complete FM space and let be two self-mappings, satisfies for all , for , with . Then, the mappings and have a unique CFP in .

Remark 13. If we put the mappings and (identity map) with constant in Theorem 10, we obtained (Theorem 1 of [25]).

Example 14. Let , is a product continuous -norm on which is defined as for all and a fuzzy metric is defined by Then, it is easy to prove that is triangular and is a complete FM space. The mappings be defined as Then, from (29), we have Hence, the self-mappings and are satisfied the weakly-compatible fuzzy-contraction condition in FM spaces. Next, we simplify the second term of (6), then, by using Definition 2 (iv) and from (29), for , we have Lastly, we simplify the third term of (6), then from (29), for , Hence, all the conditions of Theorem 10 are satisfied with and . The mappings and have a unique CFP, that is, .

Theorem 15. Let a fuzzy metric is triangular in a complete FM space and let be three self-mappings, satisfies for all , where for and with . If , where is a complete subspace of . Then , and have a coincidence point in .

Proof. Let be the arbitrary point of . Using the condition choose a sequence in such that Now, by (34), for , where Now from (37), (38), and by using Definition 2 (iv), for , we obtain After simplification, for , where . Similarly, again by view of (34), for , where Now from (41), (42), and by using Definition 2 (iv), Then after simplification, for , Now, from (40), (44), and by induction, Hence, is a fuzzy-contractive sequence in , therefore Since is triangular, , This shows that is a Cauchy sequence and is a complete subspace of . Hence, such that as , i.e., Since is triangular, Now, from (34), (46), (48), and by using Definition 2 (iv), for , we have that where Now from (50) and (51), for , we have By using the value (48) and (52) in (49) with limit , we get that for . Next, we have to prove that . Since, is triangular, Now, from (34), (46), (48), and by using Definition 2 (iv), for , we have that where Now from (54) and (55), for , By using the value (48) and (56) in (53) with limit , we obtain for . Hence, we obtain that is a common coincidence point of the mappings and in such that .

Corollary 16. Let a fuzzy metric is triangular in a complete FM space and let be three self-mappings, satisfies for all , where for and with . If , where is a complete subspace of . Then , and have a coincidence point in . Moreover, if the pairs and are weakly compatible. Then, , and have a unique CFP in .

Proof. From the proof of Theorem 15, is a common coincidence point of the mappings such that for some . Now we prove the uniqueness of the coincidence point, let is another common coincidence point of and in such that for some . Then from (57), for , where Thus, Noticing that where , therefore, for . By using the weak compatibility of the pair and by Preposition 9, we can get a unique CFP of the mappings and . Let such that, . Hence, we get that , for .

Example 17. From Example 14, it is proved that the three self-mappings , and are weakly compatible fuzzy-contractive in FM-spaces, that is, Next, we calculate the value of the second term, present in (57). Then, we have the following cases: (1)If the maximum value of , for . Then, by using (29) (2)If the maximum value of , for . Then, by using (29) (3)If the maximum value of , for . Then, by using (29) (4)If the maximum value of , for . Then, by using (29) (5)If the maximum value of , for . Then, by using (29) Hence, all the cases together with (62) and with contacts and , we have where Thus, all the hypotheses of Corollary 16 are satisfied with , , and the mappings and have a unique CFP, namly, 0.

4. Application to the Fuzzy Differential Equations

In this section, we present an application of the fuzzy differential equations (FDEs) to support our main work. From the book of Lakshmikantham and Mohapatra [32], we have the following FDEs.

Let be the space of all fuzzy subsets of where where is a continuous function. This problem is equivalent to the integral equation where Green’s function is given by

And satisfies . Here, we recall some properties of , that is,

Let , is a continuous -norm, and a fuzzy metric be defined as

, and . Then, it is easy to prove that is triangular and is a complete FM-space.

Now, we prove the existing result for the above boundary value problem by using Corollary 16.

Theorem 18. Assume that and let there exist with such that for all satisfies Let there exists such that where Then the integral equations have a unique common solution in .

Proof. Suppose that with metric The space is a complete metric space. Now, we define the operators as where where , and . Now by the properties of , and from (79), (80) and by using the hypothesis, we have Now, from the above and by view of (75), and (79), we have that Now, from (76), we have that where . Now we are in the position to apply Corollary 16 to get that the mappings and have a unique CFP , i.e., is a solution of the BVP. We have the following cases. (1)If is the maximum term in (77), then . Now from (74) and (84), we have This implies that for all . Thus, the operators , and satisfy all the conditions of Corollary 16 with in (57). Then, the operators , and have a unique CFP , i.e., is a solution of the BVP (4.1). (2)If is the maximum term in (77), then . Now from (74) and (84), we have This implies that Similarly, if is the maximum term in (77), then, . Now, by using (74) and (84), we get that Again, if is the maximum term in (77), then, . Now, by using (74) and (84), we get that Next, if is the maximum term in (77), then, . Now by using (74) and (84), we get that Hence, from (88), (89), (90), and (91), the operators and satisfy all the conditions of Corollary 16 with and in (57). Thus, the operators and have a unique CFP , i.e., is a solution of the BVP (70).

5. Conclusion

In this paper, we established the concept of rational type weakly-compatible fuzzy-contraction for three self-mappings in complete FM spaces and proved some coincidence point and CFP results under the rational type weakly-compatible fuzzy-contraction conditions by using the “triangular property of fuzzy metric” in complete FM spaces with examples. Further, as an application, we presented fuzzy differential equations and proved that the solution of the FDEs has a unique CFP of the integral operators, that is, and . This new direction of rational type weakly-compatible fuzzy-contraction with the application of FDEs in FM space will play a very important role. This idea can be extended and generalized in different directions with different types of weakly-compatible fuzzy-contraction conditions for self-mappings with different types of differential equations in the context of FM spaces.

Data Availability

Data sharing is not applicable to this article as no data set were generated or analysed during the current study.

Conflicts of Interest

The authors declare that there is no conflict 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 Iqra Shamas 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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