1. Introduction
Zadeh introduced FSs in 1965 [
1], and the concept of FMSs has been extensively explored by numerous researchers through various approaches. Kramosil and Michalek initially introduced FMS in 1975 [
2], laying the foundation for further developments. Later, in 1994, George and Veeramani refined this concept by introducing significant modifications and investigating the Hausdorff topology for these spaces [
3]. Over the years, researchers have expanded on these ideas, introducing statistical FMS, fuzzy pseudo metric spaces, and other related notions [
2,
4,
5,
6]. Various studies have contributed to the advancement of fuzzy metric concepts, drawing from different perspectives [
5,
7,
8,
9,
10,
11].
Several researchers, including [
2,
12,
13,
14,
15,
16,
17,
18,
19], have contributed to the study of commuting mappings and FPTs for compatible and subcompatible maps in FMSs [
20,
21]. These theorems serve as a significant extension and generalization of various FPTs originally formulated for contractive-type mappings in metric spaces and similar structures. Furthermore, the work of [
22] was instrumental in introducing the concept of type
compatibility within FMS. Their research established links between compatible maps and those classified under types
and
, while also formulating a series of FPTs specifically tailored for type
compatible maps in FMS.
Grabiec [
19], in 1988, expanded the classical FPTs initially proposed by Banach and later refined by Edelstein in 1962 [
23], extending their applicability to FMS as outlined by the definitions provided by Kramosil and Michalek.
In 1983, Krassimir Atanassov [
24,
25] introduced the concept of IFSs as an extension of Lotfi Zadeh’s FS theory, broadening the conventional understanding of set structures. Later, in 2004, Park [
26] extended FMSs, originally formulated by George and Veeramani in 1994 [
3], by proposing IFM-spaces. This development provided a natural progression in the field, linking these spaces to existing results in metric spaces (MS) [
24,
25,
26,
27].
Further advancements followed in 2006 when Gregori et al. [
28] analyzed the convergence of topologies induced by fuzzy metrics and IFM. Expanding on this, Saadati et al. [
29] gave the idea of modified IFMSs. After that Saadati, Sedghi, and Shobe [
29] refined the IFMS framework of Park’s work [
26] and the foundational ideas of George and Veeramani [
3], leading to the development of modified IFMSs. Numerous researchers have since explored various aspects of IFMS, employing diverse methodologies [
30,
31]. Several other researchers [
32] have explored additional fixed point results within the context of NMSs.
A major advancement in this field came with Smarandache’s work in 1999, which introduced the concept of measuring dilemmas while also generalizing existing mathematical structures that represent imprecision. Many researchers have contributed to the development of neutrosophic theory, including those who formulated neutrosophic logic [
33] and the Dombi neutrosophic graph [
34]. Ongoing research continues to expand the notion of NSs and their wide-ranging applications.
In neutrosophic set theory, each event is characterized by three key components: truthfulness (T), indeterminacy (I), and falsity (F), as originally introduced by Smarandache in 2003. NMSs further extend these ideas by utilizing continuous triangular norms (CTN) and continuous triangular conorms (CTC) to facilitate their study and application. In the present paper,
Section 2 presents the idea of NS, and NMS [
35] is then defined with the help of CTC and CTN. Additionally, a definition of Cauchy sequences in NMS is provided.
Moving to
Section 3, some definitions and examples of weakly commuting, compatible mappings, as well as occasionally weakly compatible, subsequentially continuous, and subcompatible mappings in NMS, are introduced. In
Section 4, FPT is established by using subsequentially continuous and subcompatible mappings in NMS.
Section 5 focuses on illustrating the practical applications of the results, presenting examples to validate the findings.
Section 6 provides the conclusion of the study. Lastly, the abbreviations and references used in the paper are given.
3. Subcompatibility and Subsequential Continuity in NMS
Definition 7. Let ζ and υ be two maps within NMS . We define ζ and υ as weakly commuting when the following conditions holds: , for all and
Example 3. Let and consider the NMS , where represents the neutrosophic metric functions given as , , and . Define the mappings as and
Now, the following conditions are verified: AS and . Thus,andClearly, holds.andClearly, holds. Since , it can be verified that Thus, ζ and υ satisfy all the conditions.
Definition 8. Assume an NMS . Assume ζ and υ are maps. These maps are compatible, if, for any , the following axioms are fulfilled:andwhenever a sequence in such that both and converge to the same point . Example 4. Assume and the maps and for . Consider a sequence defined by . Then for ,Also, Both mappings and converge to 0. Thus, these maps are compatible in NMS.
Definition 9. Two mappings ζ and υ of an NMS are called reciprocal continuous if and whenever is a sequence such that for some b in .
Example 5. Let and consider the NMS , where
represents the neutrosophic metric functions given as
, , and .
Define the mappings as and
Now, let be a sequence in such that As , both and .
Now, the condition is verified for some : Let . For the given sequence , , and , which satisfies the condition.
As , and .
Since both limits are consistent, ζ and υ are reciprocal continuous.
Remark 1. When both ζ and υ are continuous, it is evident that they exhibit reciprocal continuity. However, it is important to note that this reciprocity does not guarantee continuity in return. Moreover, within the context of the typical FPT applied to a compatible pair of mappings that satisfy contractive conditions, the continuity of one of the mappings either ζ or υ implies their reciprocal continuity, but the converse is not true.
Example 6. Let , and define NMS where , , and . Define mappings as
and .
Both ζ and υ are continuous.
Since both mappings ζ and υ are continuous, they will exhibit reciprocal continuity. That is, for any sequence such that and as , we will have For example, if , then and .
Thus, we have and .
Therefore, the mappings satisfy reciprocal continuity.
However, ζ and υ are continuous, and this does not guarantee that they will be reciprocal continuous in every case without additional conditions.
Thus, the continuity of ζ and υ ensures their reciprocal continuity, but reciprocal continuity does not necessarily imply continuity for these mappings.
Definition 10. Consider an NMS . Let ζ and υ be self-maps on the set . A point a within is defined as a CP of ζ and υ if and only if . In such a case, the value is known as a coincidence point for both ζ and
Definition 11. Consider an NMS denoted as . A pair of self-mappings denoted as is said to be WC if it exhibits commutativity at the point of coincidence. In other words, if ∃ a point in such that , then .
Example 7. Let , and define an NMS where
, , and .
Define two mappings as and . To check if are weakly compatible, we need to verify if ∃ a point such that , and if so, whether . Thus, to find where Clearly, for all , because both mappings are identical. Now commutativity can be verified at . For any ,andThus, holds for all . Since for all , and for all , the pair is weakly compatible.
Thus, the mappings and are weakly compatible because they satisfy the condition of commutativity at the point of coincidence. Specifically, since for all , they exhibit commutativity at every point, satisfying the definition of weak compatibility in an NMS.
Remark 2. Two maps that are compatible also satisfy the condition of being weakly compatible, but the reverse statement is not necessarily true.
Definition 12. Two self-maps denoted as ζ and υ within the framework of an NMS represented as are termed occasionally weakly compatible (OWC) if ∃ a point that is a coincidence point for both ζ and υ, and at this point both ζ and υ exhibit commutativity.
Example 8. Let , and define an NMS , where
, , and .
Define two mappings as and .
Now,Clearly, for all , so every is a coincidence point. Again, for any ,and Since , ζ and υ exhibit commutativity at every coincidence point.
Thus, both maps are OWC.
Definition 13. Consider an NMS , self-maps ζ and υ on the set are defined to be subcompatible if ∃ a sequence in such that and satisfy .
Example 9. Consider the set , and define the neutrosophic metric functions as follows:
, , .
Let the operations ⊎ and ⊙ be defined as the standard addition and multiplication on modulo 1, respectively.
Define the self-maps ζ and υ on A as and .
Now, take the sequence in as for .
Now, convergence of and can be verified: and .
As , and .
Thus, , where .
Now, subcompatibility conditions can be evaluated:
and .
Thus,
, , and .
Thus, the maps ζ and υ satisfy the conditions for subcompatibility in the neutrosophic metric space with the given sequence .
Thus, it is evident that two occasionally weakly compatible maps are also subcompatible, but the reverse statement is not always true. The example provided below illustrates that there can be instances of subcompatible maps that do not qualify as occasionally weakly compatible (OWC).
Example 10. Consider an MS , where d is defined as , ϰ, ς∈ and, for any positive value , we have the operators ⊎ and ⊙ defined as follows: The operator is denoted by ⊎ with its default operation being the minimum, i.e., , and the operator is denoted by ⊙ with its default operation being the maximum, i.e., . Define , and . Clearly, is a complete NMS.
Define ζ and υ as follows: and .
Consider the sequence in set defined as for .
As n approaches infinity, the limits are as follows:
, so 4 is an element of set .
Additionally, approaches 16 and also tends to 16 as n approaches infinity.
Thus, and
.
Thus, the subcompatibility of ζ and υ is proved. However, if and only if and . Therefore, ζ and υ do not satisfy the OWC property.
Our next goal is to introduce the concept of subsequential continuity within the framework of NMS. This concept is more relaxed than the previously established notion of reciprocal continuity, which was firstly given by Pant [
36]. A similar notion was also introduced by H. Boudhadjera [
20] within the framework of MS.
Definition 14. Let be NMS. Self-maps ζ and υ defined on the set are called subsequentially continuous iff ∃ a sequence in such that and satisfy
It is evident that, if both and are continuous or reciprocally continuous, then they are subsequentially continuous. However, the subsequent example demonstrates that ∃ pairs of maps that are subsequentially continuous, but do not meet the criteria of being continuous or reciprocally continuous.
Example 11. Consider an MS , where the metric d is defined as . For any elements ϰ and ς in and for any positive value , we have the operators ⊎ and ⊙ defined as follows:
and . Define , and . Clearly, is a complete NMS.
Define ζ and υ as follows: and . Clearly ζ and υ are discontinuous at . Consider the sequence in set , defined as for .
As n approaches infinity, the limits are as follows:
, so 1 is an element of set .
Additionally, approaches 2 (which is ) and also tends to 1 (which is ) as n approaches infinity.
Therefore, ζ and υ exhibit subsequential continuity.
Now, consider the sequence in set defined as for . In this case, as n approaches infinity for the sequence in set , defined as for :
, so 1 is an element of set . However, approaches 1 (which is ), not 2.
As a result, when n approaches infinity, ζ and υ do not exhibit reciprocal continuity as does not converge to .
Lemma 1. Consider the sequence within the NMS . If ∃ a constant k in the open interval such thatand and . Then is a Cauchy sequence in . Proof. Consider the following conditions: , , and .
These imply that
is a non-decreasing sequence bounded above by 1, and
and are non-increasing sequences bounded below by 0.
By the monotone convergence theorem,
where
,
, and
.
Additionally, for
, using the properties of the neutrosophic metric, we have
Additionally, for
,
N is chosen such that, for all
,
By the triangle inequality,
Thus, is a Cauchy sequence in . □
Lemma 2. Consider the NMS denoted as . For all elements and belonging to as well as for any positive number ϱ, if ∃ a constant k in such thatthen . Proof. If
,
and
for all
and some constant
, then
and
for all
and
. Letting
, we have
,
, and
, so
□
Now, we move forward to formulate the main theorem by leveraging the concepts of subcompatible and subsequentially continuous maps. These definitions, introduced earlier, form the basis for deriving significant fixed point results in NMS.
5. Application
Let us take an example of volterra I.E and apply FPT to find the solution.
Consider the following I.E, which is a Volterra I.E of the first kind:
where •
is the unknown function,
We want to prove that this I.E has a unique solution on the interval .
- Step 1:
Define the Operator:
We define an operator
as follows:
To prove that is continuous, we will show that, if a sequence of functions converges to some function u (pointwise or in some other suitable norm), then the sequence of transformed functions converges to .
Let us consider a sequence of functions
that converges to
u pointwise:
We will show that
For this, we can break down
:
Now, we will use the concept of dominated convergence. For this we will show that
converges pointwise to as for each in the interval .
∃ a dominating function such that ∀n and in and is integrable on .
The dominated convergence theorem (DCT) is used to justify interchanging the limit and the integral sign in .
By the DCT, we can conclude that
This establishes the subsequential continuity of the operator when sequence of functions converges pointwise to u.
- Step 2:
Show is a Contraction Mapping:
To apply the FPT, we have to show that is a contraction mapping on a suitable MS. For this purpose, we can use the norm, which measures the supremum of the absolute values of the function over the interval .
Now, to prove that
is a contraction mapping, we will prove that ∃ a constant
such that
where
is the
norm.
By analyzing , we can derive an expression involving the difference . By using the Mean Value Theorem for integrals, we can find a k such that the contraction condition holds.
- Step 3:
Apply the Fixed-Point Theorem:
Once we have established that is a contraction mapping, we can apply the FPT. This theorem guarantees that ∃ a unique function u in the MS of continuous functions on such that . This function u is the unique solution to the original integral equation.
This method shows that the I.E has a unique solution on the interval by applying the FPT and using the concept of a contraction mapping with subcompatibility or subsequential continuity.
Implementation by taking a mathematical example
- Step 1:
Defining the Operator:
Consider the sequence of functions for on the interval . We aim to show that converges pointwise to on .
(i) Pointwise Convergence:
For any fixed in , as n approaches infinity, converges to 0. Since the numerator grows slower than the denominator, n goes to infinity. Therefore, converges pointwise to .
(ii) Existence of Dominating Function:
We will find a dominating function such that for all n and in .
Since and , we have for all n.
Now, choose . Then for all n and in . Hence, is a dominating function.
(iii) Dominated Convergence Theorem (DCT):
With , which is integrable over , we can apply the DCT to justify the interchange of the limit and the integral sign.
Since is bounded on , we can replace with by the DCT. Therefore, converges to as .
This establishes the subsequential continuity of the operator when functions converges pointwise to u.
- Step 2:
Show is a Contraction Mapping:
To prove that
is a contraction mapping, we need to find a constant
such that for any two functions
and
in the function space, the operator
satisfies the contraction condition:
where
represents the supremum norm over the interval
.
Let
and
be
Thus, the difference
is
Now,
is
Substituting the expression for
,
The absolute value of the integral can be bounded by the integral of the absolute value:
Since
for all
, we have
By the definition of the supremum norm,
Thus,
Since
is independent of
, it can be factored out:
Simplifying the integral
, it follows that
Since
for
, we get
If
, we can set
, where
. Thus,
satisfies the contraction condition:
- Step 3:
Apply the Fixed-Point Theorem:
The Fixed-Point Theorem states:
If Y is a complete metric space and is a contraction mapping with contraction constant , then has a unique FP in Y such that .
Here,
Y is the space of continuous functions equipped with the supremum norm
over the interval
, and
is the operator defined as
We have already shown that is a contraction mapping with contraction constant , which satisfies .
Now, as
is a contraction mapping on a complete MS, ∃ a unique FP
such that
. In other words, ∃ a unique function
satisfying the I.E:
To find the solution, let
by the Picard iteration method. Find
:
Since
, the integral term becomes zero, and we have
Therefore, is the first approximation in the Picard iteration method when the initial guess is .
To find
, the second approximation in the Picard iteration method, we use
obtained from the previous iteration as the input function in the iteration formula:
We will compute this integral to find
:
Therefore, . This is the second approximation in the Picard iteration method.
By using the boundary conditions, we can calculate the , which is the unique solution to the I.E on .
Therefore, by applying the FPT, we have established the existence and uniqueness of the solution to the I.E. This completes the proof that the given Volterra I.E has a unique solution on the interval .
Now we are giving an example in decision theory.
Example 12. Application in Decision Theory:
In decision-making problems involving uncertainty, neutrosophic metric spaces (NMSs) provide a robust framework to evaluate alternatives by considering degrees of membership (T), non-membership (F), and indeterminacy (I). This framework is particularly useful in situations where the available information is incomplete, inconsistent, or vague, such as selecting suppliers for a sustainable supply chain.
For example, a company needs to choose the best supplier from four candidates () based on three criteria:
- 1.
cost efficiency (f),
- 2.
delivery time (g), and
- 3.
environmental sustainability (h).
Each supplier’s performance on these criteria is evaluated in the neutrosophic metric space, where
T represents the degree of membership (e.g., how well a supplier meets the criterion),
F represents the degree of non-membership (e.g., how poorly a supplier performs), and
I represents the degree of indeterminacy (e.g., uncertainty or ambiguity in the evaluation).
The overall performance of a supplier is mapped using k, which combines the three criteria into a single score.
Now, the mappings f (cost efficiency) and g (delivery time) are subcompatible if their evaluations align sufficiently, meaning there is no strong contradiction between their outputs. For example, if a supplier scores high on cost efficiency (f) but moderately on delivery time (g), their neutrosophic evaluations should overlap in a way that allows meaningful aggregation into k (overall performance). Subcompatibility ensures that combining f, g, and h into k results in a coherent overall evaluation. Again, if a sequence of evaluations for a supplier (e.g., periodic assessments over time) converges to a certain neutrosophic value under mappings f and g, then k should reflect this consistency; i.e., gradual changes in criterion evaluations should lead to gradual changes in the overall performance k, avoiding abrupt or discontinuous shifts.
Numerical Example
Consider supplier evaluated as follows:
for cost efficiency,
for delivery time, and
for environmental sustainability.
The overall performance is calculated as Using the component-wise maximum, Here,
: the supplier has a high overall membership degree;
: there is a moderate degree of non-membership;
: some indeterminacy exists in the evaluation.
Now, subcompatibility and sequential continuity are used:
- 1.
Subcompatibility: and are subcompatible if their neutrosophic evaluations do not strongly contradict each other. For example, and align well because their T, F, and I values are close and compatible.
- 2.
Sequential Continuity: If the evaluations of , , and change gradually over time (e.g., ), then should also change gradually, ensuring that the overall performance mapping remains consistent.
Thus, in NMSs, the decision-making process accounts for uncertainty and inconsistency in supplier evaluations. Subcompatibility ensures the coherent aggregation of criteria into overall performance, while sequential continuity maintains consistency in evaluations over time, making NMS an effective tool for solving complex, uncertain problems such as supplier selection.