A Fixed Point Approach to Lattice Fuzzy Set via F-Contraction

: In this work, we design and demonstrate the occurrence of L -fuzzy common ﬁxed points of L -fuzzy mappings ( L -FM) meeting contractive criteria in the framework of complete b-metric spaces (b-MS) employing F-contractions and a certain class of continuous functions. We also conduct a case study to determine the implementation of our derived principles. A few other concepts which are the direct consequences of our ﬁndings are explained in this paper.


Introduction
In 1965, Zadeh [1] presented the concept of fuzzy set (FS) that expands on the idea of a crisp set by giving all of its elements membership values between [0, 1]. The levels of possession of a particular property are described using hazy ideas. Because it is effective at solving control problems, FS theory has the potential to address situations that crisp set theory finds troublesome. Systems that are vague, complicated, and nonlinear in nature are governed by fuzzy sets. FS theory has made it simpler to resolve real-world issues since it defines and simplifies the concept of fuzziness and flaws. It is currently a generally recognized theory. Due to the flexibility in solving real-world problems with this theory, a number of researchers modified fuzzy concepts in many other fields of science, such as [2][3][4] and references therein.
By replacing the interval [0, 1] with a complete distributive lattice L later in 1967, Goguen [2] advanced this concept to L-FS theory. An FS is a special type of L-FS where L = [0, 1]. For α L -cut sets of L-FM, Rashid et al. [5] studied FP theorems for L-FMs to the discovery of the Hausdorff distance. Common FP results on L-FPs can be seen in [5][6][7] and references therein. Heilpern [8] introduced the theory of FM and established a theorem on FP for FM in metric linear space, which serves as a fuzzy generalization of Banach's contraction principle [9].
The most active and vigorous area of research in pure mathematics is fixed point (FP) theory. For nonlinear analysis, FP theory is an effective technique. Numerous disciplines, including biosciences, chemistry, commerce, finances, astronomy, and game scheme, use FP methods. The most significant result of Banach's work in metric FP theory is Banach's principle [9], which was published in 1922. This rule provides a reliable way of finding FPs of a function that satisfies certain conditions on complete metric spaces (MS) as well as guaranteeing their existence and uniqueness.
Nadler [10] improved Banach's principle for multivalued mappings in complete MS in 1969.
Backhtin [11] proposed the concept of a b-MS for the first time in 1989. The b-MS results were first conceptualized by Czerwik [12] in 1993. By adopting this theory, several researchers 1.
Here, (Ψ, d) is called a b-MS. For w = 1, a b-MS becomes an ordinary MS.
Example [13]. The set ς t with 0 < t < 1, together with the function d : Definition 2 [13]. Consider that (ψ, d) is a b-MS and s j is a sequence in ψ. Then, (1) s j is called a convergent sequence to some s ∈ Ψ, iff for all > 0, ∃ n 0 ( ) ∈ N such that for all j ≥ n 0 ( ), we have d s j , s < . Then, we write lim j→∞ s j = s.
(2) s j is said to be a Cauchy sequence iff for all > 0, ∃ n 0 ( ) ∈ N such that for each k, j ≥ n 0 ( ), we have d s j , s k < .
(3) A b-MS is called complete if every Cauchy sequence is convergent in it.
Note: Throughout this paper, we denote CB(Ψ) as the set of non-empty closed and bounded subsets of Ψ and CL(Ψ) as the set of all non-empty closed subsets of Ψ. Definition 3 [15]. Consider that (ψ, d) is a b-MS; for z ∈ ψ and M, N ∈ CL(ψ), we define The Hausdorff b-metric induced by d can be defined on CB(Ψ) as: for all M, N, ∈ CB(Ψ).
Lemma 1 [18]. Suppose that (ψ, d) is a b-MS. For any M, N, O ∈ CB(ψ) and any l, m ∈ ψ, we have the following: Lemma 2 [18]. Let (ψ, d) be a b-MS. For M ∈ CB(ψ) and a ∈ ψ, we have where M is the closure of the set M in Ψ.
Lemma 5 [19]. If M, N ∈ CB(ψ) with H(M, N) < , then for each a ∈ M, there exists b ∈ N, such that d(a, b) < . Definition 4 [1]. Let ψ be a universal set. A function G : ψ → [0, 1] is known as an FS in ψ. The value G(u) of G at u ∈ ψ stands for the degree of membership of u in G. The set of all FSs in ψ will be denoted by F(ψ). G(u) = 1 means full membership, G(u) = 0 means no membership, and intermediate values between 0 and 1 mean partial membership.
Example. Let A denote the old and B denote the young and ψ = [0, 100]. Then, A and B both are fuzzy sets that are defined by The α-level set of G is denoted by [G] α and defined as Definition 5 [7]. A partially ordered set (poset) is a set X with a binary relation such that for all m, n, q ∈ X , (1) m m (reflexive); (2) m n and n m implies m = n (anti-symmetric); and (3) m n and n q implies m q (transitivity).
Definition 6 [7]. A poset (L, L ) is said to be a (2) Complete lattice if it is lattice and ∨Q ∈ L, ∧Q ∈ L for all Q ⊆ L.
(3) Distributive lattice if it is lattice and for all ξ, η, t ∈ L. (4) Complete distributive lattice if it is lattice and (5) Bounded lattice if it is a lattice along with a maximal element 1 L and a minimal element 0 L , which satisfies 0 L L x L 1 L for every x ∈ L.
Note: ∨ means least upper bound and ∧ means greatest lower bound.

Example 1.
Lattice of Klein four group (L1) and lattice of dihedral group of order 6 (L2) are shown in Figure 1, which are complete and bounded but not distributive lattices.

Definition 8 [2]. The -level set of an L-FS is denoted by [ ] and is defined as below:
where ̅ is the closure of the set (crisp set). Let Definition 7 [2]. An L-FS G on a non-empty set ψ is a function G : ψ → L, where L is a bounded complete distributive lattice, along with 1 L and 0 L .
, then the L-FS becomes an FS in the sense of Zadeh. Hence, the class of L-FSs is larger than the class of FSs.
Definition 8 [2]. The α L -level set of an L-FS G is denoted by [G] α L and is defined as below: [7]. Let ψ 1 be any set and ψ 2 be a metric space. A function g : Definition 10 [19]. Suppose that (ψ, d) is an MS and T : Definition 11 [5]. Consider a b-MS (ψ, d) and T 1 , Banach Contraction Theorem [9]. Consider a complete metric space (ψ, d) and a self-map T on ψ. If T satisfies the following contraction condition, then it has a unique fixed point in ψ : f or all x, y ∈ ψ and for some α ∈ [0, 1). [20]. F s denotes the collection of functions F : (0, +∞) → (−∞, +∞), satisfying:

Remark 2.
Throughout the paper, we assume that functions F ∈ F s are continuous from the right.

Results
This section deals with our investigations regarding the existence of L-fuzzy common fixed points via F-contractions in the environment of complete b-MSs. Moreover, the results are supported with an example. A few corollaries are assembled to generalize our results.
By the second property of F-contractions, lim n→∞ F(s n d(x n , x n+1 )) = 0.

Discussion
Many problems arising in engineering, economics, and other fields of science are solved by converting them into differential or integral equations. The fixed point technique provides an effective environment in which these functional inclusions can be solved by fixed point methods. In the context of complete b-metric spaces applying F-contractions and a certain class of continuous functions, we develop and illustrate the presence of L-fuzzy common fixed points of L-fuzzy mappings (L-FM) satisfying a contractive criterion. A prime example is also provided to show how our derived concepts are put into practice, as well as the explanation of a few additional ideas that directly follow our findings. bmetric space is a generalized form of a metric space and L-fuzzy mapping is a more general form of fuzzy mapping and multivalued mappings. Thus, our results are helpful for future researchers.

Conclusions
In the setting of b-metric spaces, we studied the presence of L-fuzzy common fixed points using F-contractions. Furthermore, the results are backed up by examples. To generalize our result, we assembled a few corollaries.

Conflicts of Interest:
The authors declare no conflict of interest.