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