A Note on Orthogonal Fuzzy Metric Space, Its Properties, and Fixed Point Theorems

Abstract

The article generalizes the notion of orthogonal fuzzy metric space into a broader term, named as orthogonal picture fuzzy metric space. The obtained results improve and extend the idea of the orthogonal fuzzy metric space and its related results. However, this article outstretches the above-mentioned notion further into a newly defined concept, named as orthogonal picture fuzzy metric space. A detailed insight is given into the topic by presenting some fixed point results in the frame of the newly defined structure. To elaborate the results more precisely, some concrete examples are given.

1. Introduction

In 2013, Cuong [2] proposed a new concept named picture fuzzy sets (PFS), which is an extension of fuzzy sets and intuitionistic fuzzy sets. In a picture fuzzy set, each element is specified by the degree of membership, the degree of non-membership, and degree of neutrality together with the condition that the sum of these grades should be less or equal to .

In this regard, Phong et al. [7] studied some compositions of picture fuzzy relations. Cuong and Hai [8] investigated main fuzzy logic operators: negations, conjunctions, disjunctions, and implications on picture fuzzy sets, and constructed the main operations for fuzzy inference processes in picture fuzzy systems. Singh [9] studied the correlation coefficients of picture fuzzy sets. Cuong et al. [10] then investigated the classification of representable picture t-norms and picture t-conorms operators for picture fuzzy sets.

Eshaghi et al. [4] presented a new generalization of the Banach fixed point theorem (BFPT) by defining the notion of orthogonal sets. The orthogonal set is a non-empty set equipped with a binary relation (called orthogonal relation) having a special structure (see [4]). The metric defined on the orthogonal set is called orthogonal metric space. The orthogonal metric space contains partially ordered metric space and graphical metric space. Hezarjaribi [5] further extended the results of [4] to orthogonal fuzzy metric space. Also, Ishtiaq et al. [6] extended the results of [4] to orthogonal neutrosophic metric space. Some more details about generalized orthogonal metric spaces have been provided by Javed et al. [11], Uddin et al. [12, 13], and Senapati et al. [14].

In this paper, we introduce orthogonal picture fuzzy metric space which generalize picture fuzzy metric space and orthogonal fuzzy metric spaces. We show that every picture fuzzy metric space is an orthogonal picture fuzzy metric space but not conversely. We investigate different conditions on the picture fuzzy to show the existence of fixed points in various types of contractions. We also present some examples in support of the obtained results. The authors intend to further widen the interesting idea of orthogonality to the intuitionistic fuzzy metric space and spherical fuzzy metric spaces. Some interesting results on the same two topics can be read in the articles [15, 16] and [17], respectively.

2. Preliminaries

Definition 1 (see [1]). A fuzzy set is a pair , where is a non-empty set, is a membership function and for each is called the grade of membership of in

Definition 2 (see [2]). A picture fuzzy set on the universe set is an object of the form where is called the degree of positive membership of in A, is called the “degree of neutral membership of in A,” and is called the degree of negative membership of in A, and satisfy for all . Then, is called the degree of refusal membership of in A.

Definition 3 (see [3]). Suppose is an arbitrary set, assume a five tuple where is a CTN, is a CTCN, andare FSs on. If meet the following circumstances for all
(B1)
(B2)
(B3)
(B4)
(B5)
(B6) is non decreasing (ND) function of and
(B7)
(B8)
(B9)
(B10)
(B11) is non increasing (NI) function of and
Then, is an IFMS.

Definition 4. Suppose, assume five tuples where is a CTN, is a CTCN, andare picture fuzzy set on. If meet the following circumstances for all
(P1)
(P2)
(P3)
(P4)
(P5)
(P6) is non decreasing (ND) function of and
(P7)
(P8)
(P9)
(P10)
(P11) is non increasing (NI) function of and
(P12)
(P13)
(P14)
(P15)
(P16) is non increasing (NI) function of and
(P17) If then and
Then, is a PFMS.

Definition 5 (see [4]). Assume and is a binary relation. Assume there exists such that or for all. Thus, is said to be an OS. Furthermore, we denote OS by.

Definition 6 (see [4]). Suppose that is an OS. A sequence for is called an (OS) if for all or for all .

2.1. Orthogonal Picture Fuzzy Metric Space

Definition 7. Let be called an OPFMS if is a non-empty OS, is a CTN, is a CTCN, and are pfs on if the following condition are satisfied for all with either ( or ), either ( or ), and either ( ):
₁)
2)
3) if and only if
4)
5)
6) is continuous,
7)
8) if and only if
9)
10)
11) is continuous,
12) 0
13) if and only if
14)
15)
16) is continuous,
17) If then and
Then, is called OPFMS.

Remark 8. Every PFMS is an OPFMS but the converse is not true.

Example 1. Let and define a CTN as, CTCN as and define a binary relation by iff . Take for all , then it is OPFS, but not an PFMS.

It is easy to see that for (5), (10), and (15) fails.

Remark 9. The above example is also OPFMS if we take

Definition 10. An OS in an OPFMS is said to be orthogonal convergent (O-C) to if

Definition 11. An OS in an OPFMS is said to be Orthogonal Cauchy (O-CS) if there exists such that for all .

Definition 12. is OC at in an OPFMS whenever for each OS for all in if , , and for all, then , , and for all .

Definition 13. An OPFMS is said to be orthogonally complete (O-complete) if every O-CS is convergent.

Example 2. Assume OPFMS as given in Example 1 and define a sequence in by such that or . Define a CTN as, CTCN as , and define a binary relation by iff . Take

Example 3. From proof of Example 2, is a O-CS in an OPFMS. for all .

Lemma 14. If for some and then.

Definition 15. Let be an OPFMS. A map is an orthogonal contraction if there exists such that for every and with , we have

Theorem 16. Let be an O-complete PFMS such that Let be an OC, O-CON and OPR. Thus, has a unique FP, say . Furthermore,

Proof. Since is an O-complete PFMS, there exists such that That is, Take Since is OPR, is an OS. Now, since is an O-CON, we get for all and Note that is nondecreasing on Therefore, by applying the above expression, we can deduce for all and . Thus, from (15) and (5), we have We know that for all and So, from (19) we get, for all and Therefore, by applying the above expression, we can deduce for all and . Thus, from (21) and (10), we have We know that for all and So, from (22) we get, for all and Therefore, by applying the above expression, we can deduce for all and . Thus, from (24) and (15), we have We know that for all and So, from (25) we get, So, is a O-CS. The O-completeness of the PFMS ensures that there exist such that and as for all Now, since is an , , , and as . Now, we have Taking limit as , we get , and and hence
Now, we show the uniqueness of the FP of the mapping. Assume that and are two FPs of such that. We can get Since is OPR, one writes for all So from (10), we can derive Therefore, So from (11), we can derive Therefore, Similarly, from (12), we can derive Therefore, So, ; hence, is the unique FP.

Corollary 17. Assume be an O-complete PFMS. Assume be O-CON and OPR and if is an OS with then for all Then, has a unique FP, say

Proof. We can similarly derive as in the proof of Theorem 16 that is a O-CS and so it converges to Hence, for all from (10), we can get Then, we can write Taking limit as , we get and from (11), we can get Then, we can write Taking limit as , we get and from (12), we can get Then, we can write Taking limit as , we get so Next proof is similar as in Theorem 16.

Example 4. Let. We define a binary relation by
Define an OPFMS as in Example 1 by for all , with the CTN and CTCN Then, is an O-complete PFMS. Define by Then, the following cases are satisfied: (1)If and, then and(2)If , then and(3)If , then and(4)If and, then andThis clearly implies that. Hence, is OPR. We can easily see that if then then and then for all and Hence, is .
The above four cases for satisfies the below contractive conditions: All conditions of Theorem 16 are satisfied. Also, 0 is FP of .

Theorem 18. Let be an O-complete PFMS such that Let be OC,O-CON and OPR. Assume that there exist and such that for all . Then, has a unique FP, so. Furthermore, , , and for all and

Proof. Since is an O-complete PFMS, there exists such that Thus, Consider Since is OPR, is an OS. We can get Two cases arise:
Case1: If then Then, Then, Then by Lemma 14, for all .
Case2: If then Then, and Then, for all and . Then by Theorem 16, we have a OCS. By completeness of , there exists such that , , and for all
We know that is an OC, then Now, we prove that is a FP of Let and . Then, Taking we get Taking we get Taking we get Here, from Lemma 14, we have. Suppose and the FPs of . We have Because is an OPR, so we can write We can write Hence, we write that Hence, we write that Hence, we write that for all . Hence,

Corollary 19. Let be an O-complete PFMS and be an OC and OPR. Then , we get , Then, has a unique FP.

Proof. It follows from Theorem 16 and Theorem 18.

Example 5. Let and define a binary relation ⊥ by Define by for all and with the CTN and CTCN, respectively;, then is an O-complete PFMS. Note that, , and ∀ Define by We have the following cases: (1)If , then and(2)If , then and(3)If , then and(4)If and, then and(5)If and , then and(6)If and , then and(7)If and , then and(8)If and , then and(9)If and , then andBecause it is clearly implying that. Hence, is OPR. Let be an arbitrary OS in that converges to. We have Note that if , , and , then , , and for all and Hence, is OC. The case is clear. Let. We have Indeed, it is satisfied for all above 9 cases. But, is not a contraction. Assume then for we have It is a contradiction. Hence, all the conditions of Theorem 18 are satisfied and 0 is the unique FP of.

Definition 20. Let be an OPFMS. A mapping is named to be an PF -contractive if ∃ so that for all and Here, is called the PFS -contractive constant of.

Theorem 21. Let be an O-complete PFMS such that Let be an OC, -contraction and OPR. Thus, has a FP, say, and for all

Proof. Let be an O-complete PFMS. For an arbitrary, That is, Consider Since is OPR, is an OS. If for some , then is a FP of. We assume that for all . For all and , we get from (12), We have Implying that Continuing in this way, we get We have Now, for and , we have Again, continuing in this way, we get Continuing in this way, we get Continuing in this way, we get By using (87) in the above inequality, we have using (88), and using (89) we deduce from the above expression that , and for all
Therefore, is a O-CS in By the completeness of , we know that is an OC and there exists such that Now, we prove that is a FP of . For this, we obtain from (78) that That is, Using the above inequality, we obtain Taking limit as and using (97), (98), and (99) in the above expression, we get that , , and that is, Therefore, is a FP of , , , and for all

Corollary 22. Let be a O-complete PFMS and satisfy for all , where . Then, has a FP.

Proof. is the unique FP of by using Theorem 21, and is also a FP of as . From Theorem 21, , is a FP since the FP of is also a FP of

Example 6. Let and define a binary relation by Define by With CTN and CTCN then is an O-complete PFMS. Also observe that, , and, ∀
Define by Therefore, it will satisfy the following cases: (1)If , then and (2)If , then (3)If and , then and (4)If and , then and Because it is clearly implying that. Hence, is OPR. Let be an arbitrary OS in that converges to. as converges to . We can easily see that if , , and , then clearly , , and for all and Hence, is OC. Also above all cases satisfied PFS -contractive mapping All conditions of Theorem 21 are satisfied and 1 is a FP of .

3. Conclusions

A picture fuzzy set is more proficient and more capable than an intuitionistic fuzzy set and fuzzy to cope with uncertain and unpredictable information in realistic issues. Herein, we have introduced the notion of orthogonal picture fuzzy metric space and investigated some new type of fixed point theorems in this new setting. Moreover, we have provided non-trivial examples to demonstrate the viability of the proposed results. Since our structure is more general than the class of picture fuzzy metric spaces, our results and notions expand and generalize several previous results. This work can be easily extended in the structure of orthogonal picture fuzzy cone metric spaces, and orthogonal picture fuzzy bipolar metric spaces.

Data Availability

No data was used during this research

Conflicts of Interest

The authors declare that they have no competing interests.