[Retracted] On Multivalued Fuzzy Contractions in Extended -Metric Spaces

Abstract

In this paper, we establish a Hausdorff metric over the family of nonempty closed subsets of an extended -metric space. Thereafter, we introduce the concept of multivalued fuzzy contraction mappings and prove related -fuzzy fixed point theorems in the context of extended -metric spaces that generalize Nadler’s fixed point theorem as well as many preexisting results in the literature. Further, we establish -fuzzy fixed point theorems for Ćirić type fuzzy contraction mappings as a generalization of previous results. Moreover, we give some examples to support the obtained results.

1. Introduction

In 1928, Von Neumann [1] introduced the concept of fixed points for multivalued mappings because of its applications in several branches of mathematics. The development of the geometric fixed point theory for multivalued mappings was initiated by the work of Nadler [2]. He used the Pompeiu–Hausdorff metric to prove the multivalued contraction principle over the collection of nonempty closed and bounded subsets of a metric space. After that, several researchers have studied and generalized Nadler’s contraction principle in many directions.

In 1965, Zadeh [3] initiated the concept of fuzzy set theory. After that, several authors extended the Banach contraction principle for single and multivalued mappings in the context of fuzzy sets. In 1981, Heilpern [4] proved a fixed point theorem for fuzzy contraction mappings as a generalization of Nadler’s contraction principle. Consequently, several authors studied and generalized fuzzy fixed point theorems in many directions (see [5–11]). In 2015, Phiangsungnoen and Kumam [12] established the concept of multivalued fuzzy contraction mappings in -metric spaces and proved a related -fuzzy fixed point theorem. In [13], Anita extended -fuzzy fixed point theorems involving Ćirić type fuzzy contraction mappings.

Meanwhile, Kamran in [14] introduced the concept of an extended -metric, as a generalization of a -metric, and proved fixed point results on such space. Thereafter, many researchers have studied and generalized fixed point results for single and multivalued mappings (see [15–24]).

In this paper, we establish a Hausdorff metric over the family of nonempty closed subsets of an extended -metric space. After that, we introduce the concept of multivalued fuzzy contraction mappings and prove -fuzzy fixed point theorems for such mappings in the context of extended -metric spaces that generalize many preexisting results in the literature. To justify our results, we give some examples. In the last section, we further establish the fact that, by utilizing these concepts, we can also derive results for multivalued mappings. Throughout this paper, we will denote by the collection of nonempty closed and bounded subsets of and by the collection of nonempty closed subsets of .

Definition 1. (see [14]). Let be a nonempty set with . Then, a mapping is called an extended -metric, if for all , it satisfies the following:(1)iff(2)(3)

Clearly, every -metric space is an extended -metric space with .

Example 1. (see [14]). Let . Define byfor all . Thus, is an extended -metric on , where is defined as .

Samreen et al. in [25] established the concept of an extended -comparison function as an extension of a -comparison function and generalized the concept of --contraction mappings in the framework of extended -metric spaces.

Definition 2. (see [25]). Let be an extended -metric space. Then, a function is called an extended -comparison function if it is increasing, and there exists a mapping such that, for some , , converges for all and for every . Here, for and is an orbit at a point . We say that is an extended -comparison function for at . It is known that, for each extended -comparison function , we have for all and for .

We denote by the collection of all extended -comparison functions. If we put in Definition 2, we get , which is a -comparison function. We denote by the collection of all -comparison functions.

In [17], Subashi and Gjini introduced the concept of a Pompeiu–Hausdorff metric on the collection of all compact subsets of extended -metric spaces. On the other hand, Subashi in [18] initiated Pompeiu–Hausdorff metric on the collection of all nonempty closed and bounded subsets of extended -metric spaces.

Next, recall definitions of fuzzy sets, fuzzy mappings, -fuzzy fixed point, Ćirić type contraction, and related fixed point theorem in -metric spaces from [12, 13, 26].

Let be a -metric space. Then a fuzzy set in is characterized by a membership functionwhich assigns every member of a membership grade in . Denote by the collection of all fuzzy sets in . Let us take and . The -level set of is denoted by and is defined as follows:where denotes the closure of . Clearly, and are subsets of .

For , a fuzzy set is said to be more accurate than a fuzzy set and is denoted by , if and only if for each . Now, for , , , and , define

Then Hausdorff fuzzy -metric is denoted by and is defined as follows:

Remark 1. (see [12]). The function is a generalized Hausdorff fuzzy -metric induced by and is defined as follows:where .

Definition 3. (see [12]). Let be a nonempty set and be a -metric space. Then(1)A mapping is called a fuzzy mapping(2)For a fuzzy mapping , an element is called a fuzzy fixed point of , if

Theorem 1 (see [12]). Let be a complete -metric space with . Let and such that, for each , is a nonempty closed subset of and such thatfor all . Then, has an -fuzzy fixed point.

Definition 4. (see [26]). A self-mapping on a metric space is called a Ćirić type contraction if and only if for all , there exists such that

Theorem 2 (see [13]). Let be a complete -metric space with . Let and such that, for each , is a nonempty closed subset of and such thatwherefor all . Then, has an -fuzzy fixed point.

Remark 2. If in Theorem 2, we get Theorem 1. Hence, Theorem 2 is an extension of Theorem 1.

2. Main Results

For , and . The following lemma is essential in the sequel.

Lemma 1. Let be an extended -metric space. Thenfor all and , where .

Proof. Since is an extended -metric space, therefore by using the triangle inequality one writesThis implies thatfor all . By taking infimum over in equation (13), we obtainSince and , thus from equation (14), we have

Lemma 2. Let . Then

Proof. From triangle inequality, we haveThis implies thatAgain, by the triangle inequality,By continuing in this fashion, we have

Theorem 3. Let be a sequence in an extended -metric space with the property that, for all , and , where is a real constant. Then, is a Cauchy sequence.

Proof. Let and choose a positive integer . Then, from the triangle inequality for all , we haveSince , the series converges by the ratio test for each . LetThus, for , the above inequality impliesLet us take . Hence, we conclude that is a Cauchy sequence.

Now we will introduce the Pompeiu–Hausdorff metric.

Definition 5. Consider nonempty subsets of extended -metric space , and we define

By following the same procedure as Lemma 2 of [27], we state the following lemma.

Lemma 3. For all , we have

Same as Theorem 2.1 of [27], we have the following theorem.

Theorem 4. Let is an extended -metric space, then the function is a generalized extended -metric space in .

Definition 6. , where is the closure of a set , if and only if there exists a sequence in such that .

Denote for and

Lemma 4. If , then , where

Proof. Let , then there exists a sequence in , where such that . Hence, by Lemma 1,This implies thatwhich proves the lemma.

Definition 7. The upper topological limit of a sequence in the extended -metric space is denoted by which is determined by

Following Theorem 2.2 of [27], we have the following.

Theorem 5. A point if and only if there exists a subsequence such that and , for

As Theorem 2.3 of [27], we have the following theorem.

Theorem 6. is closed.

Similar to Corollary 2.1 of [27], we state the following.

Corollary 1.

Again, as Corollary 2.2 of [27], we have the following.

Corollary 2.

By applying the same procedure as Theorem 2.4 of [27], we state the following.

Theorem 7. Let us consider a complete extended -metric space with for all , where . Then, is complete.

Definition 8. Let be an extended -metric space with . Then, a fuzzy set in is characterized by a membership functionwhich assigns every member of a membership grade in .

We denote by the collection of all fuzzy sets in . Let us take and . The -level set of is denoted by and is defined as follows:where denotes the closure of . Clearly, and are subsets of the extended -metric space . For , a fuzzy set is said to be more accurate than a fuzzy set and is denoted by , if and only if for each . Now, for , , , and , define

Remark 3. From Theorem 4, the function defined byis a Hausdorff extended fuzzy -metric on .

Theorem 8. Let be a complete extended -metric space with . Let be a fuzzy mapping and such that, for each , is a nonempty closed subset of and such thatfor all . Then, has an -fuzzy fixed point.

Proof. Let us take an arbitrary point . Suppose there exists . As is a nonempty closed subset of , thus clearly if and , we get as an -fuzzy fixed point of and the proof is complete. Hence, throughout our proof, we will assume and . Thus, . From the definition of Hausdorff metric, equation (37), and , we havewhere is a real number. Suppose there exists with such thatAs is a nonempty closed subset of , therefore we assume , so . From the definition of Hausdorff metric, equation (37), and , we haveSuppose there exists with such thatAs is a nonempty closed subset of , so we assume , so . From the definition of Hausdorff metric, equation (37), and , we haveBy induction, we can construct a sequence in such that , , andfor all . From the triangle inequality, for all , we haveSince the series converges, therefore is a Cauchy sequence. As is complete, so there exists such that . Next, we will show that is an -fuzzy fixed point. From the triangle inequality, we haveBy letting and , we get . Since is closed, we get . Hence, is an -fuzzy fixed point of .

By putting , where in Theorem 8, we get the following corollary.

Corollary 3. Let be a complete extended -metric space with . Let be a fuzzy mapping and such that, for each , is a nonempty closed subset of and such thatfor all , where . Assume , where is a real constant. Then, has an -fuzzy fixed point.

Remark 4. Theorem 8 generalizes Theorem 3.1 of [12]. Also, Corollary 3 generalizes Corollary 3.2 and 3.4 of [12].

Example 2. Let . Define by and by , for all . Then, is a complete extended -metric space, which is not a -metric space. Define byDefine by , for all . Clearly, we can see that, for all ,Hence, is a nonempty closed subset of . Also, , where . Therefore, all the conditions of Theorem 8 are satisfied, and hence, is an -fuzzy fixed point of .

Next, we will prove the existence of -fuzzy fixed point for multivalued fuzzy contraction mapping under Ćirić type contractive condition in the setting of complete extended -metric spaces.

Theorem 9. Let be a complete extended -metric space with . Let be a fuzzy mapping and such that, for each , is a nonempty closed subset of and such thatwherefor all , whereThen, has an -fuzzy fixed point.

Proof. Let us take an arbitrary point . Suppose there exists . Recall that is a nonempty closed subset of . Thus, clearly if and , we get as an -fuzzy fixed point of and the proof is complete. Hence, throughout our proof, we will assume and . Thus, . From the definition of Hausdorff metric, equation (49), and , we haveThis implies thatNow, we will take the following two cases: Case 1: if , then we have which is a contradiction. Case 2: if , then we have where is a real number.This ensures that there exists with such thatAs is a nonempty closed subset of , therefore, we assume , so . From the definition of Hausdorff metric, equation (49), and , we haveIf , then we havewhich is a contradiction. Thus, . As is increasing, we havewhere is a real number. By induction, we can construct a sequence in such that , , andfor all . By applying the same procedure as Theorem 8, we prove that is an -fuzzy fixed point of .

Remark 5. If we put in Theorem 9, we get Theorem 8. Hence, Theorem 9 is an extension of Theorem 8.

Example 3. Let . Define byHence, is a complete extended -metric space, where is defined by . Define byDefine by , for all . Clearly, we can see that, for all ,Now, for all , we get and .
Also since for all For any , we haveThis implies that . Similarly, . Define by for all . Hence, for all , we haveHence, all the conditions of Theorem 9 hold, and therefore, there exists such that is an -fuzzy fixed point of .

Next, we will show that, by utilizing Corollary 3, we can prove fixed point results for multivalued mappings.

Corollary 4. Let be a complete extended -metric space with . Let be a multivalued mapping such thatfor all , where . Assume , where is a real constant. Then, has a fixed point.

Proof. Let be any arbitrary mapping and define byHence, we obtainThus, condition (37) becomes (49). Therefore, Corollary 3 can be applied to get an -fuzzy fixed point , where . Hence, the multivalued mapping has a fixed point.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare no conflicts of interest.

Authors’ Contributions

All authors contributed equally and significantly in writing this article.

Acknowledgments

The fourth author would like to acknowledge that his contribution to this work was carried out with the aid of a grant from the Carnegie Corporation provided through the African Institute for Mathematical Sciences.