Pata-Type Fixed-Point Theorems in Kaleva–Seikkala’s Type Fuzzy Metric Space

Abstract

The purpose of this paper is to generalize the fixed-point theorems for Banach–Pata-type contraction and Kannan–Pata-type contraction from metric spaces to Kaleva–Seikkala’s type fuzzy metric spaces. Moreover, two examples are given for the support of our results.

1. Introduction and Preliminaries

In 1984, Kaleva and Seikkala [1] introduced the concept of fuzzy metric spaces for the first time, and they established some fixed-point theorems in a complete fuzzy metric space. This result was investigated by many authors from different points of view, see [2–10] and the references therein.

Definition 1 (see [11]). A mapping is called a fuzzy real number, whose level set is denoted by if it satisfies two axioms:(1)There exists such that (2)For each is a closed interval of , where The set of all such fuzzy real numbers is denoted by . If and whenever , then η is called a nonnegative fuzzy real number, and by , we mean the set of all nonnegative fuzzy real numbers. The notation stands for the fuzzy number satisfying and if . Clearly, . can be embedded in if satisfies .

Definition 2 (see [1]). Let X be a nonempty set and the mappings be symmetric, nondecreasing in both arguments, and satisfy . Let d be the mapping , and write for all and all . The quadruple is called a fuzzy metric space if the following axioms are satisfied: (D1) . (D2) for all , . (D3) for all . (D3L) whenever , , and , . (D3R) whenever , , and , .

Lemma 1 (see [6]). Let be a fuzzy metric space and for all , where are any two fixed elements. Then,(1)(2) is a left continuous and nonincreasing function for (3) is a left continuous and nonincreasing function for

Lemma 2 (see [12]). Let be a fuzzy metric space, and suppose that (R-1)  (R-2) for each , there exists such that for all  (R-3) Then, (R-1)  (R-2)  (R-3).

Lemma 3 (see [6]). Let be a fuzzy metric space. Then,(1)(R-1) implies that, for each ,for all .(2)(R-2) implies that, for each , there exists such thatfor all .(3)(R-3) implies that, for each , there exists such thatfor all .

Definition 3 (see [6]). Let be a fuzzy metric space and . Then,(1) is said to be convergent to x if , i.e., for all (2) is called a Cauchy sequence in X if ; equivalently, for any given and , there exists such that whenever (3) is said to be complete if each Cauchy sequence in X is convergent to some point in X

Lemma 4 (see [6]). Let be a fuzzy metric space with (R-2). Then, for each , is continuous at .

In 2011, Pata [13] extended the Banach contraction principle with weaker hypotheses than those of the Banach contraction principle in the complete metric space. Since then, several other fixed point results in the spirit of Pata have appeared, see [14–18]. In particular, Chakraborty and Samanta [14] proved a generalization of Kannan’s fixed-point theorem based on the result of Pata.

Throughout the following, will be a complete metric space, and will be a complete fuzzy metric space. Fix an arbitrary point , and we denote and for all and . Also, is an increasing function, continuous at zero, with . Given a function .

Theorem 1 (see [13]). Let be a complete metric space. Let ,, and be fixed constants. If the inequalityis satisfied for every and all , then f has a unique fixed point .

Theorem 2 (see [14]). Let be a complete metric space. Let , , and be fixed constants. If the inequalityis satisfied for every and all , then f has a unique fixed point .

In this paper, we prove two further extensions of Pata-type fixed theorems in complete Kaleva–Seikkala’s type fuzzy metric space using contractive condition of Banach type and Kannan type. Afterwards, the fixed-point theorems for the corresponding linear contraction are given as corollaries. Our theorems extend the main results of [13, 14]. Moreover, two nontrivial examples are given to illustrate our two theorems, and our examples show that these two theorems are independent to each other.

2. Main Results

Our results of this paper are stated as follows.

Theorem 3. Let be a complete fuzzy metric space with (R-2). Let and be fixed constants. Let be a mapping such thatfor every , and all . Then, f has a unique fixed point .

Proof. Starting from , construct a sequence such that . For all , we denote . If for some , then is a fixed point of f. Thus, we always assume that for all .
In order to prove this theorem, we divide into the following five steps:

Step 1. We show that the sequence is decreasing. Clearly, suppose that and in (6), we obtain

Step 2. We prove that the sequence is bounded. For any , by Lemma 3, there exists such thatfor all . Then, from Step 1, we haveSince , we haveHence,Suppose that is unbounded. Then, there exist , , and a subsequence of such that and . LetThen, we getwhich implies thatLetting in (14), we have andwhich contradict (14). Thus, is bounded, that is, for any , there exists a constant such that .

Step 3. We shall show thatNote that is a decreasing and bounded sequence. So, assume that for some and . By (6), we obtain thatLetting , we haveNote thatThen, we can see , which contradicts our assumption.

Step 4. We show that is a Cauchy sequence. Suppose not, choose and , and then there exist subsequences and of with such thatBy Lemma 3, there exists such thatPutting , we get . Similarly, we can see thatPassing to the limit as in (22) and (23), we have . By (6), we getTaking in (2), we obtain thatNote that , a contradiction. Thus, is a Cauchy sequence. Since X is complete, there exists such that .

Step 5. We prove that z is the unique fixed point for mapping f.
First, we show that . By (6), for all , there exists , and we haveLetting , we get . Hence, .
Next, we prove the uniqueness of z. Assume that is another fixed point for f. For each , there exists such thatThus, we have as , which implies . Therefore, f has a unique fixed point .

Remark 1. From the proof of Theorem 3, we can see that, to keep the sequence converge to the fixed point, the range of ε in (6) can be limited from to for some given constant .
By Remark 1, letting , we can deduce the following corollary, which is the Banach contraction principle in fuzzy metric spaces.

Corollary 1 (see Corollary 5.3, [6]). Let be a complete fuzzy metric space with (R-2) and be a mapping. If there exists such thatfor all and , then f has a unique fixed point .

Theorem 4 generalizes the result in [14] to fuzzy metric spaces.

Theorem 4. Let be a complete fuzzy metric space with (R-2). Let and be fixed constants. Let be a mapping such thatfor every , , and all . Then, f has a unique fixed point .

Proof. Starting from , construct a sequence such that . For any , we denote . If for some , then is a fixed point of f. Thus, we always assume that for all . Then, the proof is divided into the following five steps:

Step 1. We show that the sequence is decreasing. Suppose that and in (29), we obtainThus, we get

Step 2. The sequence is bounded. For each , there exists , andLetting in (32), there exists a constant such that .

Step 3. We shall prove that for each . For any , we obtainSince is a decreasing and bounded sequence, assume that for some . Letting in (2), we haveFromwe see , a contradiction. For each , we get .

Step 4. We show that for each and all . For any , there exists , and we havePutting and in (36), we get .
Therefore, is a Cauchy sequence. Since X is complete, there exists such that .

Step 5. We prove that z is the unique fixed point for mapping f.
First, we show that . Using (29), for any , there exists , and we haveBy taking limits as and in the inequality above, we get . Hence, which implies .
Finally, we prove the uniqueness of z. Assume that is another fixed point of f. For any , there exists , and we getPassing to the limit as in (38), we have . Thus, . Therefore, f has a unique fixed point .
Following the arguments in Remark 1, from Theorem 4, we can obtain the Corollary 2.

Corollary 2 (see Corollary 5.3, [6]). Let be a complete fuzzy metric space with (R-2) and be a mapping. If there exists such thatfor all and , then f has a unique fixed point .

Now, we construct two examples to illustrate Theorems 3 and 4, respectively.

Example 1. Let , , and . Define the fuzzy metric byLet be a map defined by for . Then, the following hold:(a) is a complete fuzzy metric with (R-2).(b)(6) is satisfied for , , , and .(c)T has the unique fixed point .(d)(29) is not satisfied. So, Theorem 4 cannot be verified by this example.

Proof. First, it is easy to see that is a complete fuzzy metric space, and satisfies (R-2). Note that implies that ; then, conclusion (c) is true. Next, we prove conclusions (b) and (d), respectively.(b)Letting , , , , and , we show that (6) holds for all , and .For any , we can see that . Then, we consider the following two cases:

Case 1. If , then . For any , we have

Case 2. If , then . Note thatThen, for any , we haveFrom Cases 1 and 2, we see that (6) holds for all . In fact, is the unique fixed point for map T. So, Theorem 3 is verified.(d)We show that (29) is not satisfied.Letting , by (29), we deduce thatHowever, let and . For any , we haveSo, this example cannot verify Theorem 4, and we prove conclusion (d).

Example 2. Let , , and . Let be a fuzzy metric such thatLet be a map such thatfor . Then, the following hold:(a) is a complete fuzzy metric with (R-2).(b)(29) is satisfied for , , , and .(c)T has the unique fixed point .(d)(6) is not satisfied. So, this example cannot verify Theorem 3.

Proof. Clearly, we can see that is the unique fixed point for map T, and conclusion (c) is true. Now, we prove conclusions (a), (b), and (d), respectively.(a)We show that is a complete fuzzy metric space with (R-2).Clearly, for any , is a fuzzy number, and (D1) and (D2) in Definition 2 are satisfied. Note that satisfies (R-2). So, it is sufficient to prove that (D3) holds.
For any , it is obvious that . To see (D3L), let us consider the following two cases:(i)If or , then or , leading to thatSo, (D3L) is satisfied.(ii)If and , then . So, , leading to thatThus, (D3L) holds. To see (D3R), we consider the following two cases: Case a1. Suppose that there are at least two points in which are equal. For any and ,(i)If or , without loss of generality, let , then . Since is decreasing as , we have(ii)If , then we have Case a2. Suppose that , , and . For any and , denote and . Without loss of generality, let . Then, we haveSince is decreasing as and is decreasing, we deduce thatFrom Cases a1 and a2, we conclude that (D3R) holds for all .
Note that X is a finite set. Then, is complete, and the proof of conclusion (a) is completed.(b)We prove (29) for , , , , and .For any , we can see that . Then, we consider the following two cases: Case b1. If , it is easy to see that, for any , we have and . So, we can obtain that andfor all . If , then we haveFrom the above two equalities, by calculation, we obtain that, for any , Case b2. If , then we have . Note thatThen, for any , we haveFrom Cases b1 and b2, we see that (29) holds for all . In fact, is the unique fixed point for map T. So, Theorem 4 is verified.(d)We show that (6) is not satisfied for map T in .Finally, we show that (6) is not satisfied for map T in . If for some and some , then by (6), we deduce that, for any ,Letting , we have , which implies that . However, for and , we have and andfor all , which is a contradiction.

Remark 2. From Examples 1 and 2, we can see that Theorems 3 and 4 are independent to each other.
Recently, Jamshaid et al. [19] investigated the fuzzy fixed points of fuzzy mappings via F-contractions. On the basis of their work, a natural question for multivalued mappings can be raised as follows:

Question 1. Is the multivalued case of Theorem 3 true?

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 and significantly in writing this article. All authors read and approved the final manuscript.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (11561049).