Fuzzy Triple Controlled Metric Like Spaces with Applications

: In this article, we introduce the concept of a fuzzy triple controlled metric like space in the sense that the self distance may not be equal to one. We have used three functions in our space that generalize fuzzy controlled rectangular, extended fuzzy rectangular, fuzzy b − rectangular and fuzzy rectangular metric like spaces. Various examples are given to justify our deﬁnitions and results. As for the topological aspect, we prove a fuzzy triple controlled metric like space is not Hausdorff. We also apply our main result to solve the uniqueness of the solution of a fractional differential equation.


Introduction
Fixed point theory provides powerful tools for proving the existence and uniqueness of solutions to various types of problems and Banach fixed point theorem is a fundamental result in fixed point theory [1]. There are several variations of the Banach fixed point theorem that use different types of contractions. For example, the Kannan fixed point theorem [2] is a generalization of the Banach fixed point theorem that uses a more relaxed contraction condition known as Kannan contraction. Similarly, the Nadler fixed point [3] theorem uses a weaker contraction condition called Nadler contraction. The F−contraction [4] is a more general class of contractions than the Banach contraction and it allows for a wider range of functions to be used as contractions. Many variations of the Banach contraction theorem have been developed over the years, and most of them require the self-map to be continuous. The Suzuki-type contraction [5] has been used to prove the existence and uniqueness of fixed points for self-maps of metric spaces that are not necessarily complete or continuous. However, it should be noted that the Suzuki-type contraction has more restrictive conditions than the F−contraction. In particular, the Suzuki-type contraction requires the function φ to satisfy certain growth conditions, while the F−contraction only requires φ to be a nondecreasing function with φ(0) = 0. In 2008, Berinde et al. [6] introduced the concept of an almost contraction, which is a self-map on a metric space that is continuous at its fixed points. This is useful in applications where the self-map may have discontinuities or singularities. The authors in [7] introduced a new type of contraction called the generalized Suzuki-type F−contraction for fuzzy mappings, which is a generalization of the Suzuki-type contraction and the F−contraction introduced by Wardowski. In 2019, Saleem et al. [8] introduced the Suzuki-type generalized multi-valued almost contraction mapping that combines the concepts of Suzuki-type contraction and almost contraction. The authors in [9] introduced Suzuki-type (α, β, γ g )−generalized and modified proximal contractive mapping. This new type of contraction generalizes the concept of proximal contractive mapping that combines the concepts of Suzuki-type contraction, almost contraction and modified proximal contraction. The authors in [10] used F−contraction, F−Suzuki contraction, and F−expanding mappings to prove the existence and uniqueness of fixed points for self-mappings of complete metric spaces. The results generalize some of the well-known fixed point theorems in the literature, such as the Banach fixed point theorem and the Suzuki fixed point theorem. In 2020, Fatemah et al. [11] utilized multi-valued mapping and showed that their results can be applied to linear systems, which have important applications in various areas of engineering and control theory.
In 1965, Zadeh [12] generalized the definition of a crisp set by defining the concept of a fuzzy set that takes the membership value of the elements in the interval [0, 1]. Since a fuzzy set addresses the uncertainty and gives more accurate results as compared to crisp set, so researchers have used fuzzy sets in almost every branch of mathematics ( [13][14][15]). In 1975, Kramosil and Mich'alek [16] utilized fuzzy sets and gave the notion of a fuzzy metric space, which is considered a generalization of Menger's statistical metric spaces [17]. As the topological properties of metric spaces play a vital role so, George and Veeramani [18] generalized the definition of a fuzzy metric space. They discussed the topology of a fuzzy metric space and proved that a fuzzy metric space is Hausdorff. In 1983, Grabiec [19] established the fuzzy version of Banach fixed point theorem.
Hitzler et al. [20] introduced dislocated topologies that can be characterized in terms of certain axioms that generalize the usual axioms of a metric space. They also introduced the concept of dislocated convergence, which is a generalization of the usual notion of convergence in metric spaces. In 2013, the author in [21] introduced the idea of a metric like space. Alghamdi et al. [22] introduced b−metric like space that is a generalization of metric like space. Mlaiki et al. [23] used the controlled function and introduced the concept of controlled metric type spaces. The work of Mlaiki et al. [24] and Asim et al. [25] deals with the mathematical concept of rectangular metric like and extended rectangular b-metric spaces respectively. Abdeljawad et al. [26] used two controlled functions, α and µ and defined the idea of double controlled metric type spaces.
The concept of fuzzy metric like space is introduced by Shukla et al. [27] which is a generalization of [18]. They proved fixed point results by using fuzzy contractive mappings. Javed et al. [28] introduced the notion of a fuzzy b−metric like space in the sense of Kramosil and Michálek. Saleem et al. [29] introduced the notion of fuzzy double controlled metric space and proved Banach fixed point results while Furqan et al. [30] gave the notion of a fuzzy triple controlled metric space. In addition, they show that these spaces are not necessarily Hausdorff, which is an interesting result.
The homotopy method is a powerful numerical technique that can be used to solve a wide range of nonlinear problems ( [31][32][33][34][35][36]). The authors in ( [37,38]) have utilized the homotopy method in estimation of thermal parameters within annular fins. Homotopy method has also been used in inverse analysis of flow problem [39]. In [40], the author has investigated the weighted homotopy analysis method in the solution of one dimensional wave equation and showed the accuracy of the method by means of examples. Liu [41] proposed the multigrid homotopy technique for nonlinear inverse problem. In [42], the author combined the wavelet multiscale method and homotopy method and introduced multiscale-homotopy method for the parameter identification problem in partial differential equation. An other homotopy method, homotopy continuation technique, is applied to crack identification of beam structures [43]. In 2021, Courbot and Colicchio [44] applied the homotopy technique to find the solution of constrained BLASSO and applied their results to 3D tomographic diffractive microscopy images. Słota et al. [45] applied the homotopy technique to the one face fractional inverse Stefan design problem, while in [46], the authors utilized the homotopy method in porosity reconstruction on the basis of Biot elastic model [47].
The concept of like spaces in fuzzy triple controlled metric spaces seems to be a generalization of the notion of like spaces in metric spaces, where the self-distance between points is fixed to be one. By allowing the self-distance to vary in fuzzy triple controlled metric spaces, we have introduced a new level of flexibility in the concept of like spaces, which could have potential applications in various fields. Our proposed fuzzy triple controlled metric like space generalizes many existing results. For example, it generalizes rectangular metric like, b−rectanglar metric like, extended and controlled metric like spaces in fuzzy environment, that can be regarded as the main advantage of our proposed methods. We discuss the topology properties and prove that a fuzzy triple controlled metric like space is not Hausdorff. We prove the Banach contraction principle in the settings of newly defined space and apply our results to fractional differential equation.

Preliminaries
In this section, we will cover the essential concepts related to metric like spaces. In 2012, Samet et al. [49] introduced a new type of contraction, called α − ψ−contraction. They used a function ψ : −→ with the properties: 1.
∑ ∞ n=1 ψ n (t) < ∞ for all t, where ψ n is the n−th iterative of ψ.

Example 3. Consider the function defined by
clearly ψ ∈ Ψ.

Main Results
This section consists the definition of fuzzy triple controlled metric like space and some of its topological aspects. In this section, we will prove fixed point results with the help of α − ψ−contraction and later with the help of standard Banach contraction. Some remarks and examples are also given that illustrates our results. We start with the definition of a fuzzy triple controlled metric like space as follows: 1], together with a continuous t−norm * , is called a fuzzy triple controlled metric like, if for any κ 1 , κ 2 ∈ and all distinct κ 3 , κ 4 ∈ \ {κ 1 , κ 2 }, M satisfies: is called a fuzzy triple controlled metric like space.
Following remarks are immediate consequence of Definition (7).
triple controlled metric like space, then it reduces to an extended fuzzy b−rectangular metric like space [51].
controlled metric like space, then it reduces to fuzzy b−rectangular metric like space [52].
Then ( , M, * ) is a fuzzy triple controlled metric like space with product t−norm.
Working in the similar manner, one can prove remaining results. Hence ( , M, * ) is a fuzzy triple controlled metric like space.
The following example demonstrates that a fuzzy triple controlled metric space may not satisfy the conditions of a fuzzy triple controlled metric-like space Then ( , M, * ) is a fuzzy triple controlled metric space with product t−norm [31]. However it is not a fuzzy triple controlled metric like space because if we take, k 1 = k 2 = 3, then M(3, 3, t) = min{3,3}+t max{3,3}+t = 1, which contradicts (M2), because in like spaces we only have one sided condition, On the other hand, if k 1 = k 2 , then M(k 1 , k 2 , t) can not always be equal to 1 but in this example, for all k 1 , k 2 ∈ , we have M(k 1 , k 2 , t) = 1. Hence a fuzzy triple controlled metric space is not a fuzzy triple controlled metric like space.
Next example shows a fuzzy triple controlled metric like space need not to be a fuzzy triple controlled metric space.
Then ( , M, * ) is a fuzzy triple controlled metric like space. We will prove only (M4). Consider Hence ( , M, * ) is a fuzzy triple controlled metric like space. However it is not a fuzzy triple controlled metric space. For this we will show that the self distance is not equal to one for all k ∈ . Consider Hence ( , M, * ) is not a fuzzy triple controlled metric space.
The following are the definitions of a convergent sequence and a Cauchy sequence in the context of a fuzzy triple controlled metric like space: The following example demonstrates that a fuzzy triple controlled metric like space may not satisfy the Hausdorff property.
Let 3 ∈ F, then M(1, 3, 6) = 6 6 + max(1, 3) Thus a fuzzy triple controlled metric like space is not Hausdorff. (1) and (2), it can be concluded that extended fuzzy rectangular b−metric like spaces, fuzzy rectangular b−metric like spaces, fuzzy rectangular metric like spaces, fuzzy b−metric like spaces, and fuzzy metric like spaces are also not Hausdorff.

Remark 4. According to Remarks
Now we prove Banach contraction principle in the frame work of fuzzy triple controlled metric like space using α − ψ−contraction.
continuing in this way, we have Taking limit n −→ ∞, we have On the same line we can prove continuing in this way, we have Taking limit n −→ ∞, we have Let {κ n } be a sequence in , then we have following cases: Taking limit n −→ ∞ and using (4), we have Hence lim n−→∞ M(κ n , κ n+2m+1 , t) = 1. Case 2. If p = 2m(say) is even, then Taking limit n −→ ∞ and using (4) and (6), we have Thus in both cases, we have lim which shows {κ n } is Cauchy in and converges to some κ in (as is complete), so lim n→∞ M(κ n , κ, t) = M(κ, κ, t).
We will now demonstrate the Banach contraction principle in a fuzzy triple controlled metric like space. This principle will be utilized in the application section to establish the existence and uniqueness of a fractional differential equation.
Then T has a unique fixed point.
Then the boundary value problem (17) has a unique solution.

Conclusions
We have proposed the idea of a fuzzy triple controlled metric like space involving three functions in the rectangular inequality which is a generalization of many metric like spaces in fuzzy set theory. We have demonstrated that a fuzzy triple controlled metric like space does not necessarily satisfy the Hausdorff axiom. Various example are given to validate our definition and results. By utilizing fuzzy α − ψ−contraction, we have demonstrated the validity of Banach's fixed point theorem. We have utilized our results to establish the uniqueness of the solution for a fractional differential equation. The newly obtained results can be applied to explore and analyze various existing results in the literature. For example, one can find fixed point by using fuzzy proximal quasi contraction [59] and (α, β) implicit contractions [60] in our defined space. In future, scholars may also discover the optimal proximity points using the contraction defined in [61] and can use the fuzzy technique in dislocated quasi-metric spaces [62] to find the fixed points. From an application point of view, Our proposed techniques can be employed to establish the singularity of the solution for a nonlinear potential Kadomtsev-Petviashvili and Calogero-Degasperis equations [53], as well as a nonlinear fractional time-space telegraph (FNLTST) equation [55]. The obtained results can also be apply to the domain of words [63] in which the authors have apply their approach to establish the presence of a solution for certain recurrence equations that are linked to the examination of Quicksort algorithms and Divide and Conquer algorithms, respectively. There are other applications to computer sciences, see also ( [64,65]). As for applications in economics, our results can also be apply to dynamic market equilibrium [51], which is a sub-class of our defined fuzzy triple controlled metric like space.