On Intuitionistic Fuzzy N b Metric Space and Related Fixed Point Results with Application to Nonlinear Fractional Differential Equations

: This manuscript contains several new notions including intuitionistic fuzzy N b metric space, intuitionistic fuzzy quasi-S b -metric space, intuitionistic fuzzy pseudo-S b -metric space, intuitionistic fuzzy quasi-N -metric space and intuitionistic fuzzy pseudo N b fuzzy metric space. We prove decomposition theorem and ﬁxed-point results in the setting of intuitionistic fuzzy pseudo N b fuzzy metric space. Further, we provide several non-trivial examples to show the validity of introduced notions and results. At the end, we solve an integral equation, system of linear equations and nonlinear fractional differential equations as applications.


Introduction
There are numerous generalizations pertaining to metric space (MS) found in the literature of mathematical analysis.Gahlar [1] introduced the famous notion called 2-MS but while it is not continuous, MS is.Subsequently, Dhage [2] provided the concept of D-MS but Mustafa and Sims [3] proved that several topological properties were not true of D-MS and introduced the notion of G-MS.The majority of the fixed-point results in the setting of G-MS can be quickly determined by employing MS and quasi-MS, according to Jleli and Samet [4].In 2012, Sedghi et al. [5] introduced the concept of S-MS and proved several fixed-point results in the framework of complete S-MS.In 1989, Bakhtin [6] defined b-MS by multiplying a real number τ ≥ 1 on the right side of triangle inequality.If we consider τ = 1 in b-MS then it becomes MS.After that, Sedghi et al. [7] introduced the notion of S b -MS by combining the definitions of b-MS and S-MS; however, problematically, S b -MS is not continuous.
The fuzzy logic paradigm was established by Zadeh [8].Contrary to traditional logic, some numbers not contained within the set are defined as an element within the interval [0, 1] according to the parameters of fuzzy logic.Uncertainty, the mandatory factor of real difficulty, has helped Zadeh learn theories of fuzzy sets (FSs) that bear the difficulty of indefinity.The theory is seen as a fixed point in the fuzzy metric space (FMS) for various processes, one of them utilizing a fuzzy logic.Later on, building upon Zadeh's outcomes, Heilpern [9] established the fuzzy mapping notion and a theorem on an FP for fuzzy contraction mapping in linear MS, expressing a fuzzy general form of Banach's contraction theory.In the definition of FMSs provided by Kaleva and Seikkala [10], the imprecision is introduced when the distance between the elements is not a precise integer.After that first work by Kramosil and Michalek [11], and further work by George and Veeramani [12], the notation of an FMS was introduced.Nadaban [13] defined Fuzzy b-MS and proved several of its properties.Malviya [14] introduced the concepts of N-FMS, Pseudo N-FMS and proved several fixed-point results by using contraction mappings.Recently, Fernandez et al. [15] defined the concept of N b -FMS and proved its topological properties with several fixed-point results for contraction mappings.In 2004, Park [16] introduced intuitionistic fuzzy metric spaces (IFMSs) as a generalization of FMSs using the concepts of intuitionistic FSs, continuous t-norm (CTN) and continuous t-conorm (CTCN).After that, several authors [17][18][19][20][21][22] worked on generalization of an IFMS and proved fixed point results for contraction mappings.Ionescu et al. [23] and Mehmood et al. [24] worked on intuitionistic fuzzy metric spaces and extended b-metric spaces and proved several fixed-point results for contraction mappings under several new conditions.
In this paper, we introduce the definitions of intuitionistic fuzzy N b metric space (IFNBMS), intuitionistic fuzzy quasi-S b -metric space (IFQSbMS), intuitionistic fuzzy pseudo-S b -metric space (IFPSbMS), intuitionistic fuzzy quasi-N-metric space (IFQNMS) and intuitionistic fuzzy pseudo N b metric space (IFPNbMS).We prove decomposition theorem and fixed-point results in new setting.At the end, we provide applications to integral equation, system of linear equations and nonlinear fractional differential equations.

Preliminaries
In this section, we recall several basic definitions from the existing literature.

Definition 4 ([7]
). Suppose Ξ = ∅ is an arbitrary set and µ ≥ 1 a real number.A function S b : Ξ 3 → [0, +∞) is called S b -metric if and only if it verifies the below axioms for all κ, , ω, a ∈ Ξ : Then, a pair (Ξ, S b ) is known an S b -MS.
Remark 1.If we consider µ = 1 in the above definition, then we get the definition of IFNMS.
Example 1.Let Ξ = R be the set of real numbers.Let N and Θ are the functions on Ξ 3 × (0, +∞) defined by    Proof.Here, we prove only (iv) and (viii), others are obvious.Firstly, we investigate (iv), Hence (iv) and (viii) are satisfied. and for all ,  ∈  and  > 0. and for all κ, ∈ Ξ and σ > 0.
Example 3. Let Ξ = R be the set of real numbers.Let N and Θ are the functions on Ξ 3 × (0, +∞) defined by
Remark 2. If we consider µ = 1 in the above definition, then we get the definition of IFQSbMS.

Application in Fixed Point Theory
Now, we present a Banach contraction principle as an application via intuitionistic fuzzy q-contraction in symmetric IFNbMSs.
and f be an intuitionistic fuzzy q−contraction.Then f has a unique fixed point.
Since f is intuitionistic fuzzy q-contraction and f (κ n ) = κ n+1 as n → +∞ On the other hand, we have Since f is intuitionistic fuzzy q-contraction and f (κ n ) = κ n+1 as n → +∞ That is f (κ) = κ, hence, κ is a fixed point of f .Now, we examine the uniqueness, suppose f ( ) = for some ∈ Ξ, then as n → +∞.On the other hand, we have as n → +∞.That is κ = .

Application to Integral Equations
The most significant and engaging activities in mathematics are those involving solving equations of any form.There are numerous methods for dealing with various types of equations.Seeking solutions to the issue at hand and determining whether they are singular or multiple.Since fixed-point theory is an iterative process with a wide range of contexts, it is one of the key techniques that has achieved considerable success in the field of integral equations.
Fixed-point theory is crucial in the investigation of whether there is a solution to a differential or integral problem.In this part, we demonstrate how Theorem 1 applies to a specific nonlinear integral problem.The solution to the question "The solution for a specific nonlinear integral Equation (6) exists or not?" is provided by the following theorem.

Application to Linear Equations
Suppose Ξ = R n and define a complete symmetric IFNbMS on Ξ 3 × (0, +∞) by for all κ, , ω ∈ R n and µ = 2, if There is only one solution to the set of linear equations below.For κ, ∈ R n , we get Using the Equation ( 12), we have Using the Equation ( 12) Hence, f is an intuitionistic fuzzy q-contraction and Theorem 1 hold.That is, the system of linear Equation ( 13) has a unique solution in Ξ.

Application to Nonlinear Fractional Differential Equation
In this section, we apply Theorem 1 is to determine the existence and uniqueness of a solution to nonlinear fractional differential equation given by with boundary conditions κ(0) = 0, κ (0) = Iκ( ) ∈ (0, 1), where D α c means caputo fractional derivative of order α, defined by |κ( )|.The Riemann-Liouville fractional integral of order α is given by Firstly, we provide an acceptable form for a nonlinear fractional differential equation before investigating the existence of a solution.Now, we suppose the following fractional differential equation with the boundary conditions for all ∈ [0, 1] and L is a constant with LЛ < 1, where .
Then the Equation ( 14) has a unique solution.

Conclusions
In this paper, we introduced the notions of IFNbMS, IFQSbMS, IFPSbMS, IFQNMS and IFPNbMS.We proved decomposition theorem and fixed-point results in the setting of IFPNbMS.Further, we provided several non-trivial examples to show the validity of introduced notions and results.At the end, we solved an integral equation, system of linear equations and nonlinear fractional differential equations as applications.This work is extendable by introducing the notions of neutrosophic N b metric spaces, neutrosophic quasi N b metric spaces and neutrosophic pseudo N b metric spaces.

Remark 4 .
If we consider µ = 1 in the above definition, then we get the definition of IFPSbMS.Example 5. Suppose Ξ = R + .Define N τ and Θ τ by

Example 6 .
Let R equipped with a usual metric and Ξ = {{κ n }.{κ n is convergent in R}}.Define CTN by a * b * c = abc and CTCN by a b c = max{a, b, c} for all a, b, c ∈ [0, 1] and