Advances in Fuzzy Systems
Volume 2019 (2019), Article ID 6923937, 10 pages
https://doi.org/10.1155/2019/6923937

Research Article

Some Modified Fixed Point Results in -Fuzzy Metric Spaces

Vishal Gupta,¹ Manu Verma,¹ and Mohammad Saeed Khan²

¹Department of Mathematics, Maharishi Markandeshwar (Deemed to be University), Mullana, India
²Department of Mathematics and Statistics, Sultan Qaboos University, P.O. Box 36, Al-Khoud 123, Muscat, Oman

Correspondence should be addressed to Vishal Gupta; vishal.gmn@gmail.com

Received 31 May 2018; Accepted 21 February 2019; Published 27 March 2019

Academic Editor: Antonin Dvorák

Copyright © 2019 Vishal Gupta et al. 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.

Abstract.

The present research paper focuses on the existence of fixed point in -fuzzy metric space. The presentation of -fuzzy metric space in -tuple encourages us to define different mapping in the symmetric -fuzzy metric space. Here, the properties of fuzzy metric space are extended to -fuzzy metric space. The introduction of notion for pair of mappings on -fuzzy metric space called -weakly commuting of type and weakly commuting of type is given. This proved fixed point theorem in -fuzzy metric space employing the effectiveness of E.A. property and CLRg property. For the justification of the results, some examples are illustrated.

1. Introduction

Metric space is one of the important basic areas of research for the mathematicians. Many researchers accelerated the concept of metric space either by introducing different contractions in different fields or by extending number of variables in the metric space. Different types of mappings are introduced to facilitate the fixed point in metric spaces such as weakly commuting pair of mappings [1], compatible mappings [2], and weakly compatible mappings [3]. Subsequently, Aamri and Moutawakil [4] introduced the notion of E.A. property. In 2011, Sintunavarat and Kumam[5] stamped the idea of common limit in the range of (called CLRg property) which relaxes the requirement of completeness (or closedness) of the underlying subspace. Fixed point results are proved through the same concept in fuzzy metric spaces. Many authors [5–15] have given results about the common fixed point results in several spaces. On the basis of number of variables, there are many different generalizations, such as generalized metric space by Mustafa and Sims [16], generalized fuzzy metric spaces by Sun and Yang [17], new generalized metric space called -metric space by Sedghi [18], and -metric spaces by Abbas et al. [19], which is generalization of -metric spaces. Also, -fuzzy metric spaces were introduced by Gupta and Kanwar [20], which are based on fuzzy metric for -tuples.

The above mentioned generalizations of metric spaces are described below.

Definition 1 ([16]). Let be a nonempty set and let be a function satisfying the following conditions for all :

(G-1) if ,(G-2) with ,(G-3) with ,(G-4),(G-5).

The function is called a generalized metric on and the pair is called a -metric space.

In 2012, Sedghi et al. [18] introduced a new generalized metric space called -metric space.

Definition 2 ([18]). Let be a nonempty set. Suppose a function satisfies the following conditions:

(S-1),(S-2) if and only if ,(S-3) for any .

Then the ordered pair is called -metric space.

Abbas et al. [19] established the notion of -metric spaces, which is considered as generalizations of -metric space.

Definition 3 ([19]). Let be a nonempty set. A function is called an -metric on , if for any , , the following conditions hold:

(A-1),(A-2) if and only if ,(A-3).

The pair is called -metric space.

Fuzzy sets introduced by Zadeh [21] are the engender for all the research in different fields. Kramosil and Michalek [22] introduced the concept of fuzzy metric spaces.

Definition 4 ([23]). A binary operation is called continuous -norms; it satisfies following conditions:

(T-1) is commutative and associative,(T-2) is continuous,(T-3), ,(T-4) whenever and for all .

Definition 5 ([22]). The -tuple is called a fuzzy metric space if is an arbitrary set, is continuous -norm, and is a fuzzy set in satisfying the following conditions:

for all and ,

(FM-1),(FM-2), if and only if ,(FM-3),(FM-4),(FM-5) is left continuous.

Note that can be thought of as the degree of nearness between and with respect to .

Example 6. Let be a metric space. Define -norm or . For all , .

Then is a fuzzy metric space.

Lemma 7 ([24]). Let be a fuzzy metric space. If there exists for all , such that

for all , , then .

In the process of generalization of fuzzy metric space, Sun and Yang [17] presented the notion of -fuzzy metric space as follows.

Definition 8 ([17]). A -tuple is said to be -fuzzy metric space (denoted by GF space) if is an arbitrary nonempty set, is continuous -norm, and is a fuzzy set on satisfying the following conditions:
for each and ,

(GF-1) with ,(GF-2) with ,(GF-3) if and only if ,(GF-4),(GF-5).(GF-6) is left continuous.

Lemma 9 ([17]). Let be a GF space. Then is nondecreasing with respect to for all .

2. -Fuzzy Metric Space

These all generalizations advocate -fuzzy metric spaces. In 2016, Gupta and Kanwar [20] stamped the move of these generalization to -tuples as discussed below.

Definition 10 ([20]). Let be nonempty set. A -tuple is said to be a -fuzzy metric space (denoted by -space), where is a continuous -norm and is a fuzzy set on satisfying the following conditions for each :

(VF-1) for all with ;(VF-2) for all with ;(VF-3) if ;(VF-4), where is a permutation function;(VF-5);(VF-6);(VF-7) is continuous.

Example 11 ([20]). Let be a -metric space. Define -norm or . For all , ,

then is a -fuzzy metric space.

Lemma 12 ([20]). Let be a -fuzzy metric space; then is nondecreasing with respect to .

Lemma 13 ([20]). Let be a -fuzzy metric space such that

with ; then .

Definition 14 ([20]). Let be a -fuzzy metric space. A sequence is said to converge to a point if as for all ; that is, for each , there exists such that for all we have and we write .

Definition 15 ([20]). Let be a -fuzzy metric space. A sequence is said to be a Cauchy sequence if as for all ; that is, for each there exists such that for all , we have .

Definition 16 ([20]). A -fuzzy metric space is said to be complete if every Cauchy sequence in is convergent.

In the present research paper, topology is induced by -fuzzy metric spaces. The introduction of concepts of -weakly commuting of type and - weakly commuting of type in -fuzzy metric spaces is given which helps in determining the fixed point theorem for symmetric -fuzzy metric spaces.

3. Topology Induced by -Fuzzy Metric Space

Definition 17. Let be a -fuzzy metric space. For , the open ball with center and radius is defined as

Result 1. Every open ball is an open set.
Consider an open ball . Now

Since , we can find , , such that .
Let .
Since , we can find , , such that .
Further for a given and such that we can find , such that .
Consider the Ball

We will show that

Now implies .
Therefore

Therefore and hence

Result 2. Let be a -fuzzy metric space. Define

Then is a topology on .

Definition 18. Let be a -fuzzy metric space. The following condition is satisfied:

whenever , , , , and

then is called continuous function on .

Lemma 19. Let be a -fuzzy metric space. Then is a continuous function .

Proof. Since

and

then there exists such that

As is nondecreasing with respect to , we have

and

Considering continuity of the function with respect to and letting , we have

Therefore, is continuous function on .

Remark 20. In the present paper, will denote an -space with a continuous -norm defined as for all and we assume that

Define , where and each satisfying the following conditions:

(ϕ-1) is strict increasing,(ϕ-2) is upper semicontinuous from the right,(ϕ-3) for all .

Lemma 21. Let be an -space. If there exists such that

then .

Proof. Since

and also , by using Lemma 12, we have

From (22) and (23) and definition of -fuzzy metric space, one can get .

Remark 22. Let , , in (VF-5); we have

which implies that

for all and .

An -space is said to be symmetric if for all and for each .

Lemma 23. Let be an -metric space; if we define by

for all and , then we have

(i)for each , there exists such that

(ii)the sequence in is convergent if and only if as for all .

Proof. (i) For any , let and . Therefore, by the triangular inequality (VF-5) and Remark 22

which gives, using (26),

Since is arbitrary, we have

(ii) Since is continuous in its th argument (by (26)), we have

This proves the lemma.

Lemma 24. Let be a -Fuzzy metric space and be a sequence in . If there exists such that

for all and , then is a Cauchy sequence in .

Proof. Let be defined by (26).
For each and , put

We will prove that

Since is upper semicontinuous from right, for given and each , there exists such that . From (26), it follows from that for all .
Thus, by (32), (34), and Lemma 12, we get

Again, by (26), we get

By the arbitrariness of , we have

So, we can interpret that .
If not, then by (37), we have ; this is a contradiction. Hence (37) implies that , and (34) is proved.
By repeated application of (34), we get

By Lemma 23, for each , there exists such that

Since , by condition (-3), we have

So, for given , there exists such that . Thus, it follows from (39) that

which implies that for all with . Therefore is a Cauchy sequence in .

Definition 25. A pair of self-mappings of V-fuzzy metric space is said to be -weakly commuting of type if

for all and .

Definition 26. A pair of self-mappings of a V-fuzzy metric space is said to be V-R weakly commuting of type if there exists some positive real number such that

for all and .

Remark 27. If we interchange and in above definitions, then the pair of self-mappings of V-fuzzy metric space is said to be -weakly commuting of type and -R weakly commuting of type , respectively.

For proving our main results, we use the following relation.

The following example shows that a pair of mapping that is V-weakly commuting of type does not need to be V-weakly commuting of type .

Example 28. Let be the V-fuzzy metric space with

for all .
Define , .
Then we find

and

Then, one can get

which implies

and

Hence the pair is not V-weakly commuting of type , but it is V-weakly commuting of type .

Lemma 29. If and are V-weakly commuting of type or V-R-weakly commuting of type , then and are weakly compatible.

Proof. Let be a coincidence point of and ; i.e., ; then if pair is V-weakly commuting of type , we have

It follows that . Hence and commute at their coincidence point.
Similarly, if pair is V-R weakly commuting of type , we have

and thus ; then the pair is weakly compatible.
The converse of the lemma need not be true.

Example 30. and .
Define by and , ; we see that is the only coincidence point and and , so and are weakly compatible.
But by easy calculation, for , one can have

and

i.e.,

Therefore, and are not V-weakly commuting of type .

Definition 31 ([4]). A pair of self-mappings on X is said to satisfy the property E.A. if there exists a sequence such that

Definition 32 ([25]). A pair of self-mappings on X is said to satisfy the CLRg property if there exists a sequence such that

Now, we are ready to prove our results for symmetric -fuzzy metric spaces.

Theorem 33. Let be a symmetric -space and mappings satisfying the following conditions:

(i) and are V-weakly commuting of type ;(ii);(iii) is a V-complete subspace of ;(iv)these exists such that for all and ,

Then and have common fixed point.

Proof. Let such that and , where , and then by induction we can define a sequence as follows

We will prove that is a Cauchy sequence in .

which gives

By Lemma 24, the sequence is a V-Cauchy sequence. Since , is a V-Cauchy sequence in .
By hypothesis (iii), we know that is V-complete; then there exists such that

Now , so there exists such that . Therefore

We will prove that :

taking limit as ,

which implies,

Since V-fuzzy metric space is symmetric, we have

which implies (by Lemma 21).
Since pair is V-weakly commuting of type , then

which implies

Hence .
Eventually, we show that is common fixed point of and . Suppose ; then

,
which is the contradiction. Hence, .
To prove the uniqueness, suppose and are such that , and ; then again using condition (iv), we have

Hence, , which gives a contradiction. Hence . Therefore is a unique common fixed point of and .

Example 34. Let be a standard V-fuzzy metric space.
Let and define by , , .
We see that is the only coincidence point and and are weakly compatible.
Let be a sequence such that

where is a coincidence point.
Then the pair is V-weakly commuting of type . Further and have a unique common fixed point of and .

Corollary 35. Theorem 33 remains true if we replace -weakly commuting and -weakly commuting of type by -weakly commuting and -weakly commuting of type (considering the other conditions are the same).

Theorem 36. Let be a symmetric V-fuzzy metric space and suppose mappings are V-weakly commuting of type satisfying the following conditions:

(i) and satisfy the E.A property;(ii) is a closed subspace of ;(iii)there exists a such that for all , and ,

 Then and have a unique common fixed point.

Proof. Since, the mappings and satisfy the E.A. property, then there exists a sequence in satisfying

Since is closed subspace of and , then there exists such that .
Also, .
We will prove, :

and, taking limit as , we have

Since V-fuzzy metric space is symmetric, we have

which implies (by Lemma 21).
Since pair is V-weakly commuting of type , then

which implies .
Hence .
Finally, we show that is a common fixed point of and . Suppose , then

which is a contradiction. Hence .
To prove the uniqueness, suppose that and are such that , , and ; then again using condition (iii), we have

Hence , which gives a contradiction. Hence . Therefore ‘’ is a unique common fixed point of and .

Theorem 37. Let be a symmetric V-fuzzy metric space and suppose mappings are V-weakly commuting of type satisfying the following conditions:

(i) and satisfy the CLRg property;(ii) is a closed subspace of ;(iii)there exists a such that for all , and ,

 Then and have a unique common fixed point.

Proof. Proof follows on the same lines of Theorem 33 and by definition of CLRg property.

Data Availability

The related results, applications, definitions, and all other data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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