Abstract

Classical sets, fuzzy sets, intuitionistic fuzzy sets, and other sets are all generalized into the neutrosophic sets. A neutrosophic set is a mathematical approach that helps with challenges involving data that is inconsistent, indeterminate, or imprecise. The goal of this manuscript is to present the notion of neutrosophic 2-metric spaces. In this situation, we prove various fixed point theorems. The findings support previous methodologies in the literature and are backed up by various examples and an application.

1. Introduction and Preliminaries

There is a lot of imprecision and vagueness or fuzziness in our daily lives when it comes to sharing knowledge and information. Simple examples include comments like “Umar is tall” and “Khaleel is smart.” Terms like “tall” and “smart” in the examples above are vague in the sense that they cannot be defined precisely. There was a need to compare terms such as “tall” and “smart.” In 1965, Zadeh [1] coined the “fuzzy notion” to depict imprecise terms (such as those in the preceding examples) in stark contrast. In this continuation, Kramosil and Michlek [2] established the approach of fuzzy metric spaces (FMSs). George and Veeramani [3] initiated the approach of FMSs by utilizing continuous t-norms (CTNs). Grabiec [4] gave the fuzzy interpretation of the Banach contraction principle in FMSs. Sharma [5] coined the term fuzzy 2-metric spaces (F2MSs). Cho [6] established a common fixed point theorem for four mappings in FMSs, while Han [7] extended the results to F2MSs. Priyanka and Malviya [8] also derived several common fixed point theorems in F2MSs for occasionally weekly compatible mappings.

The concept of intuitionistic fuzzy 2-metric spaces (IF2MSs) was presented by Mursaleen and Danishlohani [9]. CTNs and continuous triangular conorms (CTCNs) were used to define neutrosophic metric spaces (NMSs). NMSs have been studied for their topological and structural features. Kirişci and Simsek [10] expanded the notion of the intuitionistic fuzzy metric space approach and proposed the notion of NMSs. In the context of NMSs, Simsek and Kirişci [11] and Sowndrarajan et al. [12] demonstrated various fixed point results. Neutrosophic soft linear spaces were first established by Bera and Mahapatra [13]. Bera and Mahapatra [14] established neutrosophic soft normed linear spaces. Ishtiaq et al. [15] introduced the concept of orthogonal neutrosophic metric spaces and proved several interesting fixed point results in the context of orthogonal neutrosophic metric spaces. Jeyaraman and Sowndrarajan [16] used contraction mappings to prove various common fixed point results in the context of neutrosophic metric spaces. Şahin et al. [17] introduced the notion of neutrosophic triple partial metric spaces and proved some fixed point results. Zararsız and Riaz [18] introduced the notion of bipolar fuzzy metric spaces and proved several fixed point results and provided an interesting application towards multiattribute decision-making.

Fathollahi et al. [19] proved several fixed point results for modified weak and rational α-ψ-contractions in ordered 2-metric spaces. Ali et al. [20] solved nonlinear differential equations in the context of neutrosophic metric spaces. Al-Omeri et al. [21] worked on (Φ, Ψ)-weak contractions in the context of neutrosophic cone metric spaces. Naeem et al. [22] worked on strong convergence theorems for a finite family of enriched strictly pseudocontractive mappings and ΦT-enriched Lipschitzian mappings using a new modified mixed-type Ishikawa iteration scheme with error. Al-Omeri et al. [23, 24] worked on numerous interesting contraction mappings in the context of neutrosophic cone metric spaces. Hussain et al. [25] proved several fixed point results for contraction mappings. Salama and Alblowi [26] worked on neutrosophic topological spaces.

In this manuscript, we replace the triangular inequalities of neutrosophic metric spaces by tetrahedron inequalities and introduce a notion of neutrosophic 2-metric spaces. The main objectives of this manuscript are as follows: (i)To introduce the notion of neutrosophic 2-metric spaces (N2MSs)(ii)To prove fixed point results in the context of N2MSs(iii)To enhance the literature of neutrosophic fixed point theory(iv)To prove the uniqueness of the solution of integral equations

Now, we provide some definitions that are helpful for readers to understand the main section.

Definition 1 (see [1]). If a binary relation (◊) on the interval [0,1] fulfills the below criteria, then (◊) is known as CTNs (CTCNs):
(a1) (◊) is commutative and associative
(b1) (◊) is continuous
(c1) For all
for all whenever and

Definition 2 (see [5]). Let A function is said to be 2-metric if fulfills the below criteria: (a)To each pair of there is a point such that (b), when at least two of are equal(c) for all (d) for all Then, the pair is called 2-metric space, and is a 2-metric on .

Example 1 (see [5]). Let A mapping defined by for all and is a 2-metric on .

Definition 3 (see [9]). Let be a set, be a CTN, ◊ be a CTCN, and be mappings. A five tuple is called an intuitionistic fuzzy 2-metric space (IF2MS), if the following conditions are satisfied: (1)(2)(3) for all when at least two of are equal(4)(5)(6) is left continuous(7)(8) for all when at least two of are equal(9)(10)(11) is left continuousfor all and .

Definition 4 (see [10]). Suppose , and assume a six tuple , where is a CTN, is a CTCN, and , , and are neutrosophic sets (NSs) on . If meets the below conditions for all and
(NS1)
(NS2)
(NS3)
(NS4)
(NS5)
(NS6) is continuous
(NS7)
(NS8)
(NS9)
(NS10)
(NS11)
(NS12) is continuous
(NS13)
(NS14)
(NS15)
(NS16)
(NS17)
(NS18) is continuous
(NS19)
(NS20) If , then
Then, is a neutrosophic metric on and be a NMS.

2. Neutrosophic 2-Metric Spaces

Definition 5. Let be a set, be a CTN, ◊ be a CTCN, and be mappings. A six tuple is called a N2MS, if the following conditions are satisfied:
(NM1)
(NM2)
(NM3) for all when at least two of are equal
(NM4)
(NM5)
(NM6) is left continuous
(NM7)
(NM8) for all when at least two of are equal
(NM9)
(NM10)
(NM11) is left continuous
(NM12)
(NM13) for all when at least two of are equal
(NM14)
(NM15)
(NM16) is left continuous
for all and .

Remark 6. The inequalities (NM5), (NM10), and (NM15) correspond to tetrahedron inequality in 2-metric space. The function values of , and be interpreted as the probability that the area of triangle is less than

Remark 7. (i)Every N2MS is nonnegative(ii)We may assume that every N2MS contains at least three distinct points

Example 2. Let be a 2-metric space, and Let be three mappings defined by for all . Then, is known as N2MS.

Definition 8. Assume be a sequence in N2MS. Then, (1) is known as convergent to if for all and . It is denoted by or (2) is known as the Cauchy sequence if for all and (3)If each Cauchy sequence in N2MS is convergent, then N2MS is said to be complete

Note 9. From now, we will assume that is a N2MS with the condition for all

3. Main Results

We establish the basic properties of N2MSs and demonstrate some fixed point findings in this section.

Lemma 10. If is a sequence for all in a given N2MS , then the below inequalities hold for all and :

Lemma 11. Let be a Cauchy sequence in N2MS such that whenever with Then, the sequence converges to at most one limit point.

Lemma 12. Let be a N2MS. If for some and for any Then,

Proof. It is easy to show on the lines of Lemma 1 in [15].

Definition 13. Let be a N2MS and be a self-mapping on If there is such that For all , then is known as contractive mapping.

Theorem 14. Let be a complete N2MS with and suppose that for all and . Let be a contractive mapping in the above definition. Then, has a unique fixed point.

Proof. It is easy to show on the lines of Theorem 1 in [15].

Example 3. Let , , and Let be three mappings defined by for all . Then, is a complete N2MS.
Now, define a self-mapping by Then, all the conditions of Theorem 14 are fulfilled, and 0 is a unique fixed point.

Example 4. Let , , and Let be three mappings defined by for all . Then, is a complete N2MS.
Now, define a self-mapping by Then, all the conditions of Theorem 14 are fulfilled for , and 0 is a unique fixed point.

Definition 15. Let be a N2MS and be a self-mapping on Then, (I) is known as continuous at , if for all implies (II)let and is known as uniform locally contractive if for all . Clearly, each uniform locally contractive mapping is continuous.

Example 5. Suppose be a N2MS and be a self-mapping on given by for all ( is constant). Let and . Assume that for all Therefore, we have that , and for all Hence, is a uniform locally contractive mapping.

Remark 16. In a N2MS , a contractive mapping can be considered as a uniform locally contractive mapping.

Definition 17. A N2MS is said to be metrically convex if for each , there is a for which , and , where for all

Theorem 18. Let be a metrically convex N2MS. If a self-mapping on is uniform locally contractive, then is a contractive mapping with the fuzzy contractive constant .

Proof. Let . Since is metrically convex, there are points and such that where and for Also, As is uniform locally contractive, we have for Hence, we have So is a contractive mapping.

Definition 19. Let be a N2MS and . A finite sequence is known as -chain from to if and A N2MS is known as -chainable if for each , there is a -chain from to .

Theorem 20. Let be a complete and -chainable N2MS. If a self-mapping on is a -uniform locally contractive, then has a unique fixed point in .

Proof. Without loss of generality, let and . Since is -chainable, there is a -chain from to . From here, we have and Utilizing induction, we get for all Now, we deduce for all Utilizing the Lemma 10, we examine that {} is a Cauchy sequence in . As is complete, there is a point such that . Since is continuous, we have Hence, and is a fixed point of . To show uniqueness, assume for some Since is -chainable, there is a -chain from to Now, for any , we have for all We have for all So, we obtain that

Definition 21. Let be a N2MS and be two self-mappings on A pair is said to be weak compatible if for some implies

Theorem 22. Let be a complete -chainable N2MS and the self-mapping on fulfilling the following criteria: (1) and(2)There exist such that , , and for all and (3)The pairs and are weakly compatible(4) and are continuousThen, , and have a unique common fixed point in .

Corollary 23. Let be a complete -chainable N2MS and the self-mapping on fulfilling the following criteria:
(C1) and
(C2) There exist such that and for all and
(C3) The pairs and are weakly compatible
(C4) and are continuous
Then, , and have a unique common fixed point in .
If assume in the preceding corollary, we deduce the below result.

Corollary 24. Let be a complete -chainable N2MS and the self-mapping on fulfilling the following criteria.
There exist such that for all and and and are continuous. Then, and have a unique common fixed point in .
If we suppose in the preceding corollary, then we get the below result.

4. Application

Let be the set of all continuous functions with domain of real values and defined on .

Now, we let the neutrosophic integral equation: where be a neutrosophic function of and . Define , and by with CTN and CTCN define by and Then, be a complete N2MS.

Assume that for , and for all . Also, consider Then, neutrosophic integral Equation (25) has a unique solution.

Proof. Define by Scrutinize that survival of a fixed of the operator has come to the survival of solution of a neutrosophic integral equation.
Now for all , we get That is, the neutrosophic integral equation satisfied the criteria of Theorem 14. Hence, the neutrosophic integral equation has a unique solution.

5. Conclusions

In this manuscript, we established the notion of neutrosophic 2-metric space by replacing the triangular inequalities of neutrosophic metric spaces by tetrahedron inequalities and introduce a notion of neutrosophic 2-metric spaces and proved some interesting results in the context of neutrosophic 2-metric spaces. These results boost the approaches of existing ones in the literature. Several examples and an application to examine the uniqueness of the solution of the integral equation are also imparted. This work can easily be extended in various structures like neutrosophic-controlled 2-metric spaces, neutrosophic triple partial 2-metric spaces, and neutrosophic 3-metric spaces. In the future, we will work on fixed point results for more than two self-mappings and solve differential and integral equations by utilizing neutrosophic 2-metric spaces.

Data Availability

On request, the data used to support the findings of this study can be obtained from the corresponding author.

Conflicts of Interest

There are no competing interests declared by the authors.

Authors’ Contributions

This article was written in collaboration by all of the contributors. The final manuscript was read and approved by all writers.