Kaleva-Seikkala’s Type Fuzzy <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2723999pt" id="M1" height="12.533pt" version="1.1" viewBox="-0.0657574 -12.2606 8.20973 12.533" width="8.20973pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-99" d="M452 333C452 398 425 448 375 448C329 448 235 404 140 292H138L229 683C233 701 234 712 225 712C208 712 149 686 66 677L64 652H97C135 652 140 651 129 598L38 167C23 96 29 57 44 33C60 7 98 -12 132 -12C169 -12 232 3 290 41C383 102 452 223 452 333ZM365 316C365 199 308 87 239 49C221 39 200 35 183 35C137 35 99 70 112 163C116 191 123 215 129 233C190 316 290 392 330 392C348 392 365 372 365 316Z"/></g></svg>-</span>Metric Spaces and Several Contraction Mappings

Li, Shu-Fang; He, Fei; Lu, Shu-Min

doi:https://doi.org/10.1155/2022/2714912

Journal of Function Spaces

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Research Article | Open Access

Volume 2022 | Article ID 2714912 | https://doi.org/10.1155/2022/2714912

Kaleva-Seikkala’s Type Fuzzy -Metric Spaces and Several Contraction Mappings

Shu-Fang Li,¹Fei He,¹and Shu-Min Lu¹

Academic Editor: Shrideh K. Q. Al-Omari

Received12 Mar 2022

Revised03 May 2022

Accepted17 Jun 2022

Published23 Jul 2022

Abstract

In this paper, we introduce the concept of Kaleva-Seikkala’s type fuzzy -metric spaces as a generalization of the notion of -metric spaces and fuzzy metric spaces. In such spaces, we establish Banach type, Reich type, and Chatterjea type fixed-point theorems, which improve the relevant results in fuzzy metric spaces. Two technical lemmas are employed to ensure that a Picard sequence is a Cauchy sequence. Finally, various applications are given to testify the fact that our main theorems extend the cases of -metric spaces.

1. Introduction

In 1965, the theory of fuzzy sets was introduced by Zadeh in [1]. Henceforth, several researchers have discussed and developed this theory and applied the results to various different areas, such as mathematical programming, modeling theory, cybernetics, neural networks, statistics, construction machinery, and image processing (see, e.g., [2–4]). After this pioneering work, some types of fuzzy metric spaces (briefly, ) were presented by numerous authors (refer to [5–7]). In particular, Kramosil and Michalek [7], in 1975, gave the notion of as a modification of the notion of probabilistic metric space initiated by Menger [8]. More detailed information about such spaces and various fixed-point theorems in these can be seen in [9–16]. In 1984, another type of fuzzy metric spaces called Kaleva-Seikkala’s type fuzzy metric space (briefly, -) was initiated by Kaleva and Seikkala [17], which generalized the metric space by defining a nonnegative fuzzy number as the distance between two points and applying a new triangle inequality which is analogous to the common triangle inequality. Drawing inspiration from [17], much work has been done in - (see, e.g., [18–22] and the references therein). Throughout this paper, we denote by , , and , the sets of natural numbers, positive integer numbers, and positive real numbers, respectively. All the concepts about - not given in this paper are the same as in [23, 24].

As a prevalent generalization of the metric spaces, Bakhtin [25] in 1989 introduced the notion of -metric spaces (briefly, -), which was formally defined by Czerwik [26] in 1993. In the last decades, many authors investigated the existence and uniqueness of the fixed point for various contractions in - (see, e.g., [27–33]). Furthermore, Aghajani et al. [34] generalized the concept of the - to the -metric space and established several fixed-point theorems in such spaces. Very recently, Gupta et al. [35] extended various existing results in -metric spaces.

Regarding the concepts of the - and several classical contractions in -, we suggest refer to [25, 26, 32, 33].

In 2012, Sedghi and Shobe [36] initiated the definition of - as a generalization of presented by George and Veeramani in [11]. There are some results in such spaces (see, for example, [36, 37]). Following this trend, Chauhan and Gupta [38] introduced the notion of George and Veeramani’s type fuzzy cone - and established new version of Banach contraction principle. As far as we know, there is no paper devoted to propose Kaleva-Seikkala’s type fuzzy -metric spaces. Due to the existing results mentioned above and application potential, it is significant to focus on this research topic.

In this paper, we introduce the concept of Kaleva-Seikkala’s type fuzzy -metric spaces (briefly, -) which generalizes the notions of - and -. In Section 2, some basic properties and lemmas of - were presented, which will be used later. In Sections 3–5, we establish and prove the fixed-point theorems concerning Banach type contractions, Reich type contractions, and Chatterjea type contractions in such spaces, respectively. It is worth mentioning that the range of all contraction constants in our main results are independent of the space coefficient . These results improve and generalize the corresponding results in -. Moreover, two techinical lemmas for the proof of Cauchy sequence play a pivotal role in the above theorems. In the final section, we give a lemma to show that a - is a special . Applying this lemma, some applications are presented to illustrate the fact that our main results extend the cases of -.

2. Kaleva-Seikkala’s Type Fuzzy -Metric Spaces

In this section, we introduce the concept of - and present some elementary lemmas which will be applied in later sections.

Definition 1. Let be a nonempty set, be a mapping, and be a real number. Suppose that be two nondecreasing and symmetric functions such that For , define

The following conditions are satisfied:

(BM1) ;

(BM2) for each , ;

(BM3) for each :

(BM3) , whenever and ;

(BM3) , whenever and .

Then, is called a fuzzy -metric, and the quintuple is called a fuzzy -metric space with the coefficient . If and satisfies (BM1)-(BM3), then is called a generalized fuzzy -metric space (briefly, ).

Lemma 2. Let be a . For each and , Then, (1) and (2)For each , is a nonincreasing and left continuous function(3) is a nonincreasing and left continuous function for

Lemma 3. Suppose that be a , and if
(-1) ;
(-2) , s.t. for all ;
(-3) .
Then, (-1)(-2)(-3).

Lemma 4. Let be a . Then,
(-1)
(-2) for each , there exists such that for all , (-3) for each , there exists such that for all

Proof. (1)Suppose that, on the contrary, for some and , . We can find such that , , and , which implies thatFrom (BM3) and the condition (-1), we obtain that which is a contradiction. (2)Assume that (-2) holds, i.e., for every , there is such that for all . Since is left continuous and nonincreasing, it is sufficient to prove that for all . If for some and , we have for some . Then, we can find and such that , and . It follows thatBy means of (BM3), we have which is a contradiction. (3)Suppose that (-3) holds, i.e., for each , we have Then, there is such that for all . Assume that, on the contrary, for some and , we have for some . Then, we can find such that , and , which deduces thatOn account of (BM3) and the condition (-3), which is a contradiction.

Definition 5. Let be a and be a suquence in . (1)If , i.e., for each , is said to converge to ( as or )(2)If , equivalently, for any given and , there exists such that , whenever , is said to be a Cauchy sequence(3)If every Cauchy sequence in converges, is called complete

Lemma 6. Let be a with (-2) and be a sequence. If the sequence converges to both and , then .

We end this section by giving an example to illustrate that a is obviously not a or a -.

Example 1. Let and a mapping. If , we define for all . If with , is defined by Define that Then, the assertions hold: (1) is complete, and the coefficient is ;(2) is not a .

Proof. (1)First, we show that (BM1) in Definition 1 is satisfied for all . From the definition of , it is sufficient to verify that implies . Note thatSuppose that there exist such that . Taking , we have , which is a contradiction. It is easy to verify that for each , (BM2) in Definition 1 holds. Next, for each , we prove that the condition (BM3) is satisfied. By a simple calculation, we get for all . (i)We prove (BM3) with . Let satisfyingSince , if or , then Assume that and , we obtain that Thus, we conclude that That completes the proof of (BM3).

We prove (BM3) with . Let satisfying

Now, we consider the following three cases.

Case 1. Assume that we have

Case 2. If or (Here, we discuss the previous assumption), then

Case 3. Suppose that For each , (a)If and , then(b)If or , without loss of generality, let , then(c)If . Note thatMight as well set , which implies that and Thus, we conclude that The part of (BM3) is completed. Therefore, is a .
Finally, we prove is complete. Let be a Cauchy sequence in . For any , there exists such that for all . It implies that . Thus, is a Cauchy sequence in . Due to is complete, we can find such that . For any and , by virtue of as , there is such that Then, Thus, as in . Therefore, is complete.
Let , , , and . Obviously, , and . By the definition of , Since , we have . Thus, Therefore, is not a .

3. Banach Type Contractions in -

In this section, we will state and prove a fixed-point theorem for Banach type contractions in -. This theorem extends Banach’s results in .

Theorem 7. Suppose that be a complete with (-2). Let be a mapping. If there exists such that for all , then admits a unique fixed point in .

Proof. Since , there exists such that . Suppose that , applying (36), we derive for all . Clearly, Continuing this process, we deduce Let , . From the above inequality, for each we obtain that for all .
Now, we shall prove that admits a unique fixed point in . Taking , we construct a sequence by (). For each , , by (40), we deduce Thus, for each , due to satisfies (-2), it follows that we can find such that For with , using (41) and (42), we derive Due to , we can conclude that is a Cauchy sequence. Note that is a complete . Thus, there exists such that , equivalently, for all .
For each and , by virtue of (40), we obtain that Thus, as , that is, as . From Lemma 6, we deduce that . Therefore, is a fixed point of . This completes the proof of the existence of the fixed point.
If has another fixed point , i.e., and , then for some . Using (40), we have a contradiction. Therefore, , and the fixed point of is unique. Hence, . Then, , which deduce that is a fixed point of . By the uniqueness of fixed point of , we get . Furthermore, if there exists such that , then is a fixed point of . Again, by the uniqueness of the fixed point of , we obtain that . Thus, admits a unique fixed point in .

The following corollary is an immediate consequence of Theorem 7.

Corollary 8. Suppose that be a complete with (-2), be a mapping. For each and , where . Then, has a unique fixed point in .

Proof. Obviously, a is a generalized , and thus, taking in Theorem 7, we obtain the result.

To support our results, we give an illustrative example in .

Example 2. Assume that , a mapping. If , we define , . If with , is defined by Define by

Let defined by

Then, is a Banach type contraction with the contraction constant .

Proof. We can obtain that is a complete with the coefficient , which is similar to Example 1. By the definition of , it is easy to prove that has the unique fixed point . Note that for all . Then, Now, we divide it into three cases.

Case 1. If , we can see that

Case 2. Suppose that , we can derive that

Case 3. Assume that and , we get Note that . Thus, we deduce then In conclusion, is a Banach type contraction, and the contraction constant is .

4. Reich Type Contractions in -

In this section, our main contribution is to establish a fixed-point theorem concerning Reich type contraction. Firstly, we introduce a crucial lemma to show that a Picard sequence is a Cauchy sequence.

Lemma 9. Suppose that be a () with (-2) and a sequence. Assume that there exists such that for each , we can find satisfying then is a Cauchy sequence.

Proof. Owing to , we can find such that . For each and with , we will show that Since is with (-2), there exists such that (i)If , then taking account of (57) and (59), we getwhere . (ii)If , take , where , then . Thus, . Then, making the most of (5) and (6), we getTherefore, for each and (), we conclude that The proof is completed.

Remark 10. In fact, we cannot prove the uniqueness of the fixed point concerning the Reich type contractions without additional conditions. To figure this out, we consider the assumption that has the Fatou property.

Let us review the definition of the Fatou property.

Definition 11. Suppose that be a () with (-2) and be a sequence. We say that has the Fatou property if, for each , whenever as and any .

Remark 12. It is obvious that has Fatou property if is a -.

Now, we establish and prove our main contents of this section.

Theorem 13. Suppose that be a complete () with (-2). Assume that be a mapping. If there exist and such that for all . If has the Fatou property, then admits a unique fixed point in .

Proof. Let , we can construct a sequence by . Assume that . By (64), we have It immediately follows that where . Since , we obtain that . Again by (64), we get where . Using Lemma 9, we can conclude that is a Cauchy sequence. Since is complete, we can find such that . Next, we prove that is a fixed point of . Suppose that, on the contrary, , that is, for some . By means of (64), we deduce that Using the fact that as and has the Fatou property, we get Note that . Combining (68) and (69), we obtain that which is a contradiction. Thus, .
If is a fixed point of and , that is, and for some , by virtue of (64), we derive which contradicts the assumption that . Therefore, and has a unique fixed point in .

Theorem 13 can deduce the following corollary in .

Corollary 14. Suppose that be a complete with (-2). Assume that be a mapping. If there exist and such that for all . Then, has a unique fixed point in .

Remark 15. Taking in Theorem 13, we can obtain a result for Banach contractions in . If we take and in Theorem 13, the fixed-point theorem for Kannan type contractions is derived in the same context.

Corollary 16. Let be a complete () with (-2) and . If there exists such that for all . If has the Fatou property, then has a unique fixed point in .

5. Chatterjea Type Contractions in -

The following lemma is crucial in order to prove a Picard sequence is a Cauchy sequence.

Lemma 17. Let be a complete () with (-2). Assume that be a mapping. Define the sequence by . Suppose that for each where . Let such that . Then, the following assertions hold: (i)For each there exists such that , for all , (ii)For each , there exists and such that , for all (iii)The sequence is a Cauchy sequence

Proof. (i)Firstly, we show by induction that, for each ,for all , and for some .
Clearly, (75) holds for all . Assume that (75) holds for all (). Notice that satisfies the condition (-2). By Lemma 4 (2), we can find such that Next, in order to prove (75) for , we divide it into two cases.
Case 1. Assume that . Then, by (74) and (76), we have which implies that Case 2. Suppose that . In this case, taking account of (74), we deduce that From case 1, we derive Therefore, from the above two cases, we prove that (75) holds for all . (ii)For each and , by means of (74) and (76), we obtain thatwhere , for some . (iii)For each and (), from (74) and (ii), we haveThus, we can obtain that . Therefore, we conclude that is a Cauchy sequence.

By Lemma 17, we establish a fixed-point theorem for Chatterjea type contraction in .

Theorem 18. Suppose that be a complete () with (-2). Assume that be a mapping. For each , where . Then, has a unique fixed point , and for any , the sequence of iterates converges to .

Proof. Let , we construct a sequence by . Owing to , there exists such that . Using the Lemma 17, we can obtain that is a Cauchy sequence. Due to is complete, there exists such that . Next, we show that is a fixed point of . For each , from Lemma 4 (40), we can find such that By virtue of (83) and (84), we have It follows that Note that and , . Therefore, we obtain that , that is, converges to , by Lemma 6, we have . This completes the proof of the existence of the fixed point of .
If there exists such that and , that is, for some . Taking (83) into account, we have which contradicts the fact that . Therefore, and has a unique fixed point in .

Corollary 19. Suppose that be a complete with (-2), . If there exists such that for all . Then, has a unique fixed point , and for any , the sequence of iterates converges to .

Proof. Taking in Theorem 18, the desired result is obtained immediately.

6. Applications

The aim of the following lemma is to prove that a - is a special . And then, we can establish the relevant fixed-point theorems in - as corollaries of our main results presented in Sections 3–5.

Lemma 20. Let be a - and a mapping defined by Then, is a , where

Proof. In view of (89), we have It is easy to verify (BM1) and (BM2) in Definition 1. Next, for every , we prove that (BM3) holds.
First, we prove (BM3). Let , satisfying Note that . Suppose that or , we can obtain that Assume that and , we deduce Note that . Thus, we conclude that That completes the proof of (BM3).
Now, we verify (BM3). Let satisfying Assume that or , we get If and , we derive Thus, we conclude that The part of (BM3) is completed.

Remark 21. From Lemma 20, it is obvious that and are homeomorphic, and for each , . Therefore, we give the following three results as corollaries of Theorem 7, Theorem 13, and Theorem 18, respectively.

Corollary 22 (see [32]). Suppose that be a complete - (), . If , where . Then, has a unique fixed point in .

Corollary 23. Let be a complete - (), . Suppose that there exist and such that If has the Fatou property, then has a unique fixed point in .

Corollary 24 (see [33]). Let be a complete - (), . If where , then has a unique fixed point , and for any , the sequence of iterates converges to .

Data Availability

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

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The research was supported by the National Natural Science Foundation of China (12061050) and the Natural Science Foundation of Inner Mongolia (2020MS01004).

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Copyright © 2022 Shu-Fang Li 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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