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

Some Modified Fixed Point Results in -Fuzzy Metric Spaces

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; moc.liamg@nmg.lahsiv

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 [515] 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 thatDefine , 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 thatSince 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 thatNow, 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 .