[Retracted] Rational Type Fuzzy-Contraction Results in Fuzzy Metric Spaces with an Application

Rehman, Saif Ur; Chinram, Ronnason; Boonpok, Chawalit

doi:https://doi.org/10.1155/2021/6644491

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

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Volume 2021 | Article ID 6644491 | https://doi.org/10.1155/2021/6644491

[Retracted] Rational Type Fuzzy-Contraction Results in Fuzzy Metric Spaces with an Application

Saif Ur Rehman,¹Ronnason Chinram,²and Chawalit Boonpok³

Academic Editor: Ali Jaballah

Received22 Dec 2020

Revised04 Jan 2021

Accepted17 Mar 2021

Published08 Apr 2021

Abstract

This paper aims to introduce the new concept of rational type fuzzy-contraction mappings in fuzzy metric spaces. We prove some fixed point results under the rational type fuzzy-contraction conditions in fuzzy metric spaces with illustrative examples to support our results. This new concept will play a very important role in the theory of fuzzy fixed point results and can be generalized for different contractive type mappings in the context of fuzzy metric spaces. Moreover, we present an application of a nonlinear integral type equation to get the existing result for a unique solution to support our work.

1. Introduction

The theory of fixed point is one of the most interesting areas of research in mathematics. In the last decades, a lot of work was dedicated to the theory of fixed point. A point belonging to a nonempty set is called a fixed point of a mapping if and only if . In 1922, Stefan Banach, a well-known mathematician, proved a Banach contraction principle in [1], which is stated as “A self-mapping in a complete metric space satisfying the contraction condition has a unique fixed point.” After the publication of this principle, many researchers contributed their ideas to the theory of fixed point and proved different contractive type mapping results for single and multivalued mappings in the context of metric spaces for fixed point, coincidence point, and common fixed point. Some of these results can be found in [2–13].

In 1965, the theory of fuzzy set was introduced by Zadeh [14]. Recently, this theory is used, investigated, and applied in many directions. One direction is the evaluation of test results which is the application of fuzzy logic in the processing of students evaluation; moreover, the application is expected to represent the mechanisms of human thought processes capable of resolving the problem of evaluation of students, which can be directly monitored by the teacher (for example, see [15–19]). Many researchers have extensively developed the theory of fuzzy sets and their applications in different fields. Some of their results can be found in [20–29] the references therein.

The other direction is the generalization of metric spaces to fuzzy metric spaces. In [30], Kramosil and Michalek introduced the concept of fuzzy metric spaces (-space) and some more notions. Later on, the stronger form of the metric fuzziness was given by George and Veeramani [31]. In 2002, Gregory and Sapena [32] proved some contractive type fixed point theorems in -spaces. Some more fixed point results in the said space can be found in [33–41].

This research work aims to present the new concept of rational type fuzzy-contraction mappings in -complete -spaces. We use the concept of Gregory and Sapena [32] and the “triangular property of fuzzy metric” presented by Bari and Vetro [33] and prove some unique fixed point theorems under the rational type fuzzy-contraction conditions in -complete -spaces with some illustrative examples. This new theory will play a very important role in the theory of fuzzy fixed point results and can be generalized for different contractive type mappings in the context of fuzzy metric spaces. Moreover, we present an integral type application in the sense of Jabeen et al. [42] to prove a result for a unique solution to support our work. The application section of the paper is more important; one can use this concept and present different types of nonlinear integral type equations for the existence of unique solutions for their results. Some integral type application results in the theory of fixed point can be found in [43–46].

2. Preliminaries

Definition 1. (see [47]). An operation is called a continuous -norm, if(i) is commutative, associative, and continuous.(ii) and , whenever and , .The basic -norms, the minimum, the product, and the Lukasiewicz continuous -norms are defined as follows (see [47]):

Definition 2. (see [31]). A 3-tuple is said to be a -space if is an arbitrary set, is a continuous -norm, and is a fuzzy set on satisfying the following conditions:(i) and (ii)(iii)(iv) is continuous, and .

Lemma 1 (see [31]). is nondecreasing .

Definition 3. (see [31]). Let be a -space, , and a sequence in is(i)Converges to if and , such that , . We may write this or as .(ii)Cauchy sequence if and such that , .(iii) is complete if every Cauchy sequence is convergent in .(iv)[32] fuzzy-contractive if such thatIn the sense of Gregori and Sapena [32], a sequence in a -space is said to be -Cauchy if , for and . A -space is called -complete if every -Cauchy sequence is convergent.
Throughout this paper, represents the set of natural numbers.

Lemma 2 (see [31]). Let be a -space and let a sequence in converge to a point iff , as , for .

Definition 4. (see [33]). Let be a -space. The fuzzy metric is triangular, if

Definition 5 (see [32]). Let be a -space and . Then, is said to be fuzzy-contractive if such thatIn the following, we present some rational type fixed point results under the rational type fuzzy-contraction conditions in -complete -spaces by using the “triangular property of fuzzy metric.” We present illustrative examples to support our results. In the last section of this paper, we present an integral type application for a unique solution to support our work.

3. Main Result

In this section, we define rational type fuzzy-contraction maps and prove some unique fixed point theorems under the rational type fuzzy-contraction mappings in -complete -spaces.

Definition 6. Let be a -space; a mapping is called a rational type fuzzy-contraction if such that, .

Theorem 1. Let be a -complete -space in which is triangular and a mapping is a rational type fuzzy-contraction satisfying (5) with . Then, has a unique fixed point in .

Proof. Fix and . Then, by (5), for ,and after simplification,Similarly,Now, from (7) and (8) and by induction, for , we have thatHence, is a fuzzy-contractive sequence in ; therefore,Now, we show that is a -Cauchy sequence; let , and there is a fixed such thatHence, it is proved that is a -Cauchy sequence. Since is -complete, such that , as , i.e.,Since is triangular, from (5), (10), and (12), for , we haveHence, , for .
Uniqueness. Let such that and ; then, from (5) and by using Definition 2 (iii), for , we haveHence, it is proved that , and this implies that .

Corollary 1 (fuzzy Banach contraction principle). Let be a -complete -space in which is triangular and a mapping is a fuzzy-contraction satisfying (4) with . Then, has a unique fixed point in .

Example 1. Let , be a continuous -norm, and be defined asThen, one can easily verify that is triangular and is a -complete -space. Now we define a mapping asThen, we haveHence, a mapping is a fuzzy contraction. Now, from Definition 2 (iii), for ,Hence, all the conditions of Theorem 1 are satisfied with and . A mapping has a fixed point, i.e., .
Next, we present a generalized rational type fuzzy-contraction theorem.

Theorem 2. Let be a -complete -space in which is triangular and a mapping satisfies with . Then, has a unique fixed point.

Proof. Fix and . Then, by (19), for ,From Definition 2 (iii), , and after simplification, we haveSimilarly, for , we haveNow, from (21) and (22) and by induction, for , we haveHence, is a rational type fuzzy-contractive sequence in such thatNow we have to show that is a -Cauchy sequence; let , and there is a fixed such thatHence, it is proved that is a -Cauchy sequence. Since is -complete, then such that , as , i.e.,Since is triangular,Now from (19), (24), and (26), for , we haveFrom Definition 2 (iii), and we haveThen,Now, from (26), (27), and (30), as , we get thatand where , and hence , i.e., , for .
Uniqueness. Let such that and . Then, from (19) and from Definition 2 (iii), for , we haveHence, , and this implies that , for .

Corollary 2. Let be a -complete -space in which is triangular and a mapping satisfies, , with . Then, has a unique fixed point.

Corollary 3. Let be a -complete -space in which is triangular and a mapping satisfies, , with . Then, has a unique fixed point.

Corollary 4. Let be a -complete -space in which is triangular and a mapping satisfies, , with . Then, has a unique fixed point.

Example 2. From Example 1, we define asThen, one can easily show that is triangular and is -complete -space. Now we define a mapping asThen, we haveA mapping satisfies (4), and hence is a fuzzy contraction. Now, from Definition 2 (iii), for , and after simplification, we get the following:Hence, all the conditions of Theorem 2 are satisfied with , , and , and has a unique fixed point, i.e., .

4. Application

In this section, we present an integral type application to support our work. Let be the space of all -valued continuous functions on the interval , where . The nonlinear integral equation iswhere and . The induced metric can be defined as

The binary operation is defined by , . A standard fuzzy metric can be defined as

Then, one can easily verify that is triangular and is a -complete -space.

Theorem 3. Let the integral equation be defined in (40), and there exists , satisfyingwhereThen, the integral equation in (40) has a unique solution in .

Proof. Define the integral operator byNotice that is well defined and (40) has a unique solution if and only if has a unique fixed point in . Now we have to show that Theorem 1 applies to the integral operator . Then, , we have the following two cases:(a)If in (44), then, from (42) and (43), we have and this implies that such that . Inequality (47) holds if . Thus, the integral operator satisfies all the conditions of Theorem 1 with and in (5). The integral operator has a unique fixed point, ., (40) has a solution in .(b)If in (44), then, from (42) and (43), we haveand this implies thatHere, we simplify the term , and by using Definition 2 (iii) and (42), for , we haveand this implies thatNow from (49) and (51), we have such that . Inequality (52) holds if . Thus, the integral operator satisfies all the conditions of Theorem 1 with and in (5). The integral operator has a unique fixed point, ., (40) has a solution in .

5. Conclusion

In this paper, we have presented the concept of rational type fuzzy-contraction maps in -spaces and proved some rational type fixed point theorems in -complete -spaces under the rational type fuzzy-contraction conditions by using the “triangular property of fuzzy metric.” In the last section, we presented an integral type application for rational type fuzzy-contraction maps and proved a result of a unique solution for an integral operator in -space. In this direction, one can prove more rational type fuzzy-contraction results in -complete -spaces with different types of applications.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to this study.

Acknowledgments

This research was financially supported by Mahasarakham University.

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