Abstract
In this paper, we introduce fuzzy multiplicative metric space and prove some best proximity point theorems for single-valued and multivalued proximal contractions on the newly introduced space. As corollaries of our results, we prove some fixed-point theorems. Also, we present best proximity point theorems for Feng-Liu-type multivalued proximal contraction in fuzzy metric space. Moreover, we illustrate our results with some interesting examples.
1. Introduction and Preliminaries
Best proximity point is the generalization of fixed point and is useful when contraction map is not a self-map that is where . A point is known as best proximity point if . Fan [1] presented best approximation theorem which is stated as follows: “If is a nonempty compact convex subset of a Hausdorff locally convex topological vector space and is a continuous non-self-mapping, then there exists an element in such a way that .” A best proximity point theorem is more applicable than best approximation theorem, as it provides optimal approximate solution. Therefore, best proximity point theory seeks attention of authors such as [2–7]. Many research works done on multivalued non-self-maps use Nadler’s approach [8]. Nadler’s theorem is stated as follows: “Let be a complete metric space and be a mapping from into , where is the collection of all closed and bounded subsets of , such that for all , where . Then, has a fixed point.” Another way of defining multivalued contraction is approached by Feng and Liu [9]. They proved a fixed-point theorem for newly defined multivalued contraction which is stated as follows: “Let be a complete metric space, , where is the collection of all closed subsets of , be a multivalued mapping. If there exists a constant such that for any , there is (where for some ) satisfying . Then, has a fixed point in provided that and is lower semicontinuous”. With the help of example, in the same article, they also have shown that Feng-Liu-type multivalued contraction is more general than Nadler’s multivalued contraction. Recently, Sahin et al. [10] proved best proximity point theorem for Feng-Liu-type multivalued map.
On the other hand, fuzzy metric space was firstly defined by Kramosil and Michalek [11] and then modified by George and Veeramani [12]. The modified definition is given as follows.
Definition 1 (see [12]). A 3-tuple is called fuzzy metric space if is an arbitrary set, is continuous -norm, and is a fuzzy set on satisfying the following conditions for all and :
FM1:
FM2: if and only if
FM3:
FM4:
FM5: is continuous
The norm is defined as follows.
Definition 2 (see [12]). A continuous -norm is a binary operation if the pair is a topological monoid, that is, (1) satisfies associative and commutative laws(2) is continuous(3), (4)for every , whenever and Some known examples of a continuous -norm are .
Many researches have been produced on fixed-point theory in fuzzy metric spaces [4, 13–19]. Vetro and Salimi [20] proved best proximity point theorem in fuzzy metric spaces. Due to the development of new calculus by Grossman and Katz [21], known as multiplicative calculus, a metric was introduced by Bashirov et al. [22] called multiplicative metric defined as follows.
Definition 3 (see [22]). Assume a nonempty set . Regard multiplicative metric as a mapping obeying the following assertions:
M1: for all and if and only if
M2:
M3: for all
Getting inspiration from all the work mentioned above, we firstly introduce fuzzy multiplicative metric space and prove some of its topological properties. Moreover, we obtain some best proximity point theorems for Feng-Liu-type multivalued non-self-maps on fuzzy multiplicative metric space.
2. Fuzzy Multiplicative Metric Spaces
This section introduces a new type of metric space which is fuzzy analogy of multiplicative metric space. We give an example to show the existence of such space.
Definition 4. A triplet is termed as fuzzy multiplicative metric space if is continuous -norm, is arbitrary set, and is fuzzy set on fulfilling the accompanying conditions for all and .
FMM1:
FMM2: if and only if
FMM3:
FMM4:
FMM5: is continuous
Here, we have an example of fuzzy multiplicative metric which cannot be fuzzy metric.
Example 5. Let and , consider a continuous -norm as . Then, is fuzzy multiplicative metric space.
Remark 6. (1)Let be a fuzzy multiplicative metric space. Whenever for and , , we can find , such that (2)Let . For any , we can find such that , and for any , we can find such that
We now discuss some topological properties of fuzzy multiplicative metric space.
Definition 7. Let be a fuzzy multiplicative metric space and , ; then, an open ball having center and radius is defined as
Proposition 8. Every open ball is an open set in fuzzy multiplicative metric space.
Proof. Consider an open ball and let . This implies that . Since , using Remark 6, we can find , , such that . Let . Since , therefore by using Remark 6, we can find , , such that . Now, for a given and such that , we can find such that . Now, consider the ball . We claim that .
Now, implies that . Therefore,
Therefore, , and hence,
☐
Proposition 9. Let be a fuzzy multiplicative metric space. Define .
Then, is a topology on .
Theorem 10. Every fuzzy multiplicative metric space is Hausdorff.
Proof. Assume that is a given fuzzy multiplicative metric space. Let be two distinct points of , and then, . Let , . For each , , using Remark 6, we can find such that . Now, consider the open balls and . Clearly, For if there exists Then, which is a contradiction. Therefore, is Hausdorff.☐
Definition 11. In a fuzzy multiplicative metric space , a sequence is a convergent sequence which converges to if and only if there exist with , for all and for each .
Theorem 12. Let be a fuzzy multiplicative metric space, and be a sequence in . Then, converges to if and only if as for each .
Proof. Suppose that . Then, for each and , there exists a natural number such that for all . We have . Hence, as .
Conversely, suppose that as . Then, for each and , there exist a natural number such that for all . In that case, . Hence, as .☐
Definition 13. Consider a sequence in a fuzzy multiplicative metric space . If for each , there exist such that for all , and then, is termed as Cauchy sequence in .
Theorem 14. Let be a fuzzy multiplicative metric space, and be a sequence in . Then, is Cauchy if and only if as for each .
Proof. Suppose that is a Cauchy sequence in . Then, for each and , there exists a natural number such that for all . We have . Hence, as .
Conversely, suppose that as . Then, for each and , there exists a natural number such that for all . In that case, . Hence, is a Cauchy sequence.☐
Proposition 15. In a fuzzy multiplicative metric space , if a sequence converges in , then is Cauchy.
Proof. Let and be real numbers with . Since , there is some such that . Also, suppose that converges in , say it converges to . Then, there exists such that for each , Thus, for , we have ☐
That is is a Cauchy sequence.
Definition 16. A fuzzy multiplicative metric space is termed as complete if and only if every sequence in which is Cauchy must converge in .
Definition 17. Let be a fuzzy multiplicative metric space. A subset of is closed if for each sequence in which is convergent with , we have .
Remark 18. Let be a complete fuzzy multiplicative metric space. A subset of is closed if and only if is complete.
The following lemma is the analogue of Kiany’s lemma [16] in the setting of newly defined space.
Lemma 19. Let be a fuzzy multiplicative metric space such that for and Suppose is a sequence in such that for all where . Then, is a Cauchy sequence.
Proof. For each and , we have Thus, for each , we get ☐
Choosing constants and such that and . Therefore, for ,
Using (12) in above inequality, we have
That is
The above expression can be simplified as
Then, from the above, we have
for each . Hence, for each ,
which shows that is a Cauchy sequence.
Definition 20. Consider a fuzzy multiplicative metric space and ; then, for all , where for all .
Definition 21. Let be a fuzzy multiplicative metric space and . If every sequence of , fulfilling the condition that for some in and for all , has a convergent subsequence, then is termed as approximatively compact with respect to .
3. Best Proximity Point Theorems in Fuzzy Multiplicative Metric Spaces
In the present section, we prove some best proximity point theorems for single-valued and multivalued proximal contractions. First, we define the analogous of proximal contractions in the setting of fuzzy multiplicative metric space and then proceed to the main results.
Definition 22. Let be a fuzzy multiplicative metric space and . A mapping is named as multiplicative contraction of first kind if there exists , such that for all
Theorem 23. Let be a complete fuzzy multiplicative metric space and such that is approximatively compact with respect to . Assume that , be multiplicative contraction of first kind and . Then, possesses best proximity point.
Proof. Let then for , there exist such that Further, since , there exist such that Similarly, for , there exist such that By continuing the similar steps, we get By successive application of fuzzy multiplicative contraction, we have for all For any , Using (27) in above inequality, we obtain By assumption, , we get that Hence, is a Cauchy sequence. The completeness of fuzzy multiplicative metric space implies that converges to , that is, for all . Notice that ☐
Therefore, as . Since is approximatively compact with respect to , so has a convergent subsequence converging to some . Further, for each , we have
Letting , we get , which implies that and implies that , there exist , such that . From this and equation (26) implies that
Applying limit to above inequality gives which implies that . Hence, , which shows that possesses best proximity point .
Example 24. Let and where for and . Then, is complete fuzzy multiplicative metric space with defined as . Let and then and are closed subsets of and , . Define as
Let and then and such that . It can be easily checked that is proximal contraction in fuzzy multiplicative metric space with . Also, the condition holds.
Since all statements of Theorem 23 hold, therefore possesses best proximity point. We can see that is best proximity point of .
If in Theorem 23, then we obtain the following corollary which is the fixed-point theorem for fuzzy multiplicative contraction in fuzzy multiplicative metric space.
Corollary 25. Let be a complete fuzzy multiplicative metric space. A mapping satisfying has fixed point provided that .
Now, we prove a best proximity theorem for Feng-Liu-type multivalued contraction in fuzzy multiplicative metric space.
Theorem 26. Let be complete fuzzy multiplicative metric space. be two nonempty closed subsets of having -property and . Let be a mapping such that and for all and , there exist satisfying for some and . Assume that satisfy for every and . Then, has best proximity point in provided that is upper semicontinuous.
Proof. Let be arbitrary point. Choose . Then, by assumption, there exist such that Presently, let , and then, we can discover such that Again by assumption, there exist such that Also, we can find such that ☐
Proceeding in similar manner, we develop two sequences and in and , respectively, with , and
for all and . Then again, since and have -property, so from inequality (43), we get
Therefore, from inequality (44), we have
From inequality (44), we have
Combining inequalities (46) and (47), we get
for all and .
Let and then . The inequality (48) gives
for and . By our assumption (37) and Lemma 19, is Cauchy sequence.
Now, from inequalities (44) and (46), we have
Also, from inequalities (44) and (56), we have
for and . Hence, is Cauchy sequence.
As subsets and are closed in , therefore converges to points of and , respectively. Thus, there exist and such that and as .
Letting in inequality (43), we have
for . The inequality (56) shows that the sequence is increasing and it converges to 1. Since is upper semicontinuous, so implies to the fact that , that is, , and hence, . Therefore, (FM1)
that is, . This shows that possesses best proximity point .
If in Theorem 26, then we obtain the following corollary which is the fixed-point theorem for Feng-Liu-type contraction in fuzzy multiplicative metric space.
Corollary 27. Let be complete fuzzy multiplicative metric space. Let be a mapping, for all and (where for some ) satisfying for some and . Then, possesses fixed point provided that and is upper semicontinuous.
4. Best Proximity Point Theorems of Feng-Liu-Type Mappings in Fuzzy Metric Space
Getting motivation from the work of Sahin et al. [10], we prove the following result.
Theorem 28. Let be complete fuzzy metric space. be closed and nonempty having -property and . Let be a mapping such that and for all and there exist satisfying for some and . Assume that satisfy for every and . Then, possesses best proximity point in provided that is upper semicontinuous.
Proof. Let be arbitrary point. Choose . Then, by assumption, there exist such that and .
Presently, let , then we can discover such that
Again by assumption, there exist such that and .
Also, we can find such that
☐
Proceeding in a similar manner, we develop two sequences and in and , respectively, with , and
for all and . Then again, since and have -property, so from inequality (61), we get