Research Article | Open Access
Fixed-Point Theorem for Multivalued Quasi-Contraction Maps in a V-Fuzzy Metric Space
In this paper, we introduce the concept of a set-valued or multivalued quasi-contraction mapping in V-fuzzy metric spaces. Using this new concept, a fixed-point theorem is established. We also provide an example verifying and illustrating the fixed-point theorem in action.
1. Introduction and Preliminaries
The concept of a metric space has been a fascinating subject that allows distance measurement between two points. Over the years, many mathematicians have been trying to further generalize the notion of a metric space. Two such mathematicians were Mustafa and Sims. In their paper , they provided a way to interpret distance using three points instead of two points as was the case in metric spaces. They called their new approach a G-metric space.
Definition 1. The pair is called a G-metric space if X is a nonempty set and d is a G-metric on X. That is, such that for all , we have(i) if and only if (ii) with (iii) for all with (iv)(v)Sedghi et al. in  modified the G-metric space and introduced their version of interpretation of distance between three points which is called the S-metric space.
Definition 2. The pair is called an S-metric space if X is a nonempty set and d is an S-metric on X. That is, such that for all , we have(i)(ii)(iii)In view of the work of Sedghi et al. in , Abbas et al. in  further generalized the S-metric space giving an interpretation of distance between n points.
Definition 3. The pair is called an A-metric space if X is a nonempty set and d is an A-metric on X. That is, such that for all , we have(i)(ii)(iii)In 2016, Gupta and Kanwar in  used the same approach in the fuzzy setting to extend a fuzzy metric space which uses two points to a V-fuzzy metric space which uses n points. We now outline this sequel that leads up to V-fuzzy metric spaces.
A fuzzy set was first proposed in the 1960s by an electrical engineer named Lotfi Zadeh in . It is an extension of the classical notion of a set.
It should be noted that a fuzzy set is always considered with respect to a nonempty set X called its base set. Each is assigned a membership grade . The formal definition of a fuzzy set is now given.
Definition 4. A fuzzy set is a pair , where X is a nonempty set and .
The value is called the membership grade of or the degree of membership of . The nearer the value of to unity, the higher the degree of membership of . Conversely, the nearer the value of to zero, the lower the degree of membership of . A fuzzy set is sometimes read as, M is a fuzzy set on X.
Definition 5. A t-norm is a function, such that for all , the following are satisfied:(i) (1 acts as the identity element)(ii) (symmetry)(iii) whenever and (non-decreasing)(iv) (associative).Additionally, is said to be a continuous t-norm if is a t-norm and for all sequences and in , where , we have that the limit of these sequences exist andMore specifically, is called left continuous if for each ,Right continuity is analogously defined.
Definition 6. The triple is called a fuzzy metric space if X is a nonempty set, is a continuous t-norm, and M is a fuzzy set on satisfying the following for each and :(i)(ii)(iii)(iv)(v), for all (vi) is continuous
Remark 1 (see ). Informally, we can think of as the degree of nearness between x and y with respect to t.
In 2010, Sun and Yang in  made the first step in generalizing the fuzzy metric space. They called it the Q-fuzzy metric space, and they proved several fixed-point theorems in this space. We define the Q-fuzzy metric space as follows.
Definition 7 (see ). The triplet is called a Q-fuzzy metric space if be a continuous t-norm and Q is a fuzzy set on satisfying the following for each and :(i) with (ii) with (iii)(iv)(v)(vi) is continuousAt this point, Gupta and Kanwar was ready to provide an interpretation of fuzzy metric spaces using n points. They called it the V-fuzzy metric space.
Definition 8 (see ). The triplet is called a V-fuzzy metric space if be a continuous t-norm and V is a fuzzy set on satisfying the following for each and :(i) , (ii), (iii)(iv) where is a permutation of (v)(vi) as (vii) is continuous
Lemma 1 (see ). Let be a V-fuzzy metric space. Then, is nondecreasing. That is, for some , we have
Proof. Since , we have . Now,Hence, for all , we have
Lemma 2 (see ). Let be a V-fuzzy metric space. If for all and with , we haveThen, .
We now examine convergence and Cauchy sequences in V-fuzzy metric spaces.
2. Fixed-Point Theorem in V-Fuzzy Metric Spaces
In this section, we extend the concepts in the existing literature. We introduce the concept of multivalued quasi-contraction maps in V-fuzzy metric spaces, and we also give a fixed-point theorem. We begin with the following concepts in the V-fuzzy setting.
Notation 1. We denote the set of all nonempty closed and bounded subsets of X in a V-fuzzy metric space by .
Definition 12. Let and . Then,
Lemma 3. Let be a V-fuzzy metric space. If for all and with and , we haveThen, .
Definition 13. Let and . The Hausdorff V-fuzzy metric or Hausdorff V-fuzzy metric distance is denoted by and is defined bywhere
Remark 2. Informally, we can think of the Hausdorff V-fuzzy metric as the greatest degree of nearness from a point in one set to the closest point in the other sets with respect to t.
Definition 14. Let be a V-fuzzy metric space. The mapping is said to be a q-multivalued quasi-contraction if there exist such that for all
3. Main Result
We now proceed to give a fixed-point theorem which is our main result.
Theorem 1. Let be a complete V-fuzzy metric space. If the mapping is a q-multivalued quasi-contraction, then T has a fixed point in X. That is, there exists such that .
Proof. T is given to be a q-multivalued quasi-contraction. This implies that there exists such that for all It is clear that for some , with , we have . Using this fact and setting , there exists , Inequality 2 becomesSimilarly setting , there exists , Inequality 2 becomesContinuing in this manner by induction, we obtain a sequence such thatWe now show that is Cauchy. If , we get , where , and therefore will be trivially Cauchy. Thus, without loss of generality assume and . We haveNow, we consider the five cases.
Case 1. IfThen, as and using the fact that , we haveThis implies that as . Therefore, is Cauchy.
Case 2. Ifthen as and using the fact that , we haveThis implies that as . Therefore, is Cauchy.
The other three cases can be done in a similar manner.
Since is complete, there exists such thatThat is, as . Now, let . Then,Therefore, by Lemma 3, we have that . Hence, u is a fixed point of T in X.
Example 1. Let be an A-metric space, and let be a continuous t-norm defined by . If for all and , letNow, and ; therefore, ; hence, condition of Definition 8 holds.
Also, since . Then,Therefore,That is, .
For condition of Definition 8, suppose , then , this implies , Hence, .
Conversely, if , then ; thus, .
Since the A-metric space is symmetric, condition of Definition 8 readily follows.
Now, we show condition of Definition 8This is condition of Definition 8.
Note thatNext, using L’Hopital’s rule,This is of Definition 8.
Finally, to show continuity, let be a sequence in . Then,Hence, the triplet is a V-fuzzy metric space. Let be an A-metric space such thatLet and define byIf , then and this implies that and Inequality (17) holds.
If for some , then .
Also,Therefore, Inequality (17) is satisfied. Here, we choose . The fixed points of the mapping T are and .
Definition 15. Let be a fuzzy metric space. The multivalued map is said to be a q-multivalued quasi-contraction, if there exist such that for all ,
Corollary 1. Let be a complete fuzzy metric space. If the mapping is a q-multivalued quasi-contraction where , then T has a fixed point in X. That is, there exist such that .
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
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Copyright © 2020 Matthew Brijesh Sookoo and Sreedhara Rao Gunakala. 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.