Fuzzy fixed point results of generalized almost F$\mathcal{\mathbf{F}}$-contractions in controlled metric spaces

In this paper, we derive some common α-fuzzy fixed point results for fuzzy mappings under generalized almost $\mathcal{\mathbf{F}}$
 F
 -contractions in the context of a controlled metric space, which generalize many preexisting results in the literature. As an application, we establish some multivalued fixed point results. For justification of our results, we provide a nontrivial example.


Introduction
The Banach fixed point theorem (BFPT) [1] is an important tool in fixed point theory.
It guarantees the existence and uniqueness of a fixed point of certain self-mappings on metric spaces. It has various applications in several branches of mathematics. There are many extensions and generalizations of the BFPT in the literature; see [2][3][4][5][6][7]. Berinde [8,9] studied various contractive-type mappings and introduced the concept of almost contractions.
Wardowski [10] introduced a new type of contractions, called F-contractions, and established a related fixed point theorem in the context of complete metric spaces.
The following works deal with F-contractions: [11][12][13][14][15][16]. Afterward, Altun et al. [17] modified Definition 1.3 by adding the following condition: We denote by F the family of all functions F satisfying (C1)-(C4). Nadler [18] derived the multivalued version of Banach fixed point theorem by using the Hausdorff metric over the family of nonempty closed bounded subsets of a complete metric space. We denote by CLB(W) the family of nonempty closed bounded subsets and by CLD(W) the family of nonempty closed subsets of W. Recently, Kamran et al. [19] introduced the concept of an extended b-metric space, which generalized the notion of a b-metric space [20,21] by replacing the constant with a function depending on two variables.

Definition 1.4 ([19]
) Let W be a nonempty set, and let σ : it satisfies the following axioms: Later on, several researchers worked on fixed point results in the context of extended b-metric spaces; see [22][23][24][25]. In the same direction, Mlaiki et al. [26] gave the idea of a controlled-type metric space (for further extensions, see [27]), which generalizes the notion of a b-metric space. Definition 1.5 ([26]) Let W be a nonempty set, and let σ : is called a controlled metric if for all ω 1 , ω 2 , ω 3 ∈ W, it satisfies the following axioms: is called a controlled metric space. Remark 1.1 Every controlled metric space is a generalization of a b-metric space and is different from an extended b-metric space.
For other definitions and information on the topology induced by d σ , see [26]. In [28], Alamgir et al. established a Pompieu-Hausdorff metric over the family of nonempty closed subsets of a controlled metric space W as follows. On the other hand, in 1981, Heilpern [29] used fuzzy sets [30] to introduce a class of fuzzy mappings, which is a generalization of multivalued mappings and proved a fixed point theorem for fuzzy contraction mappings in metric spaces. The result introduced by Heilpern is a fuzzy generalization of the Banach fixed point theorem. Consequently, several authors studied and generalized fuzzy fixed point theorems in many directions; see [31][32][33][34][35][36][37][38]. In this paper, we prove some common α-fuzzy fixed point results for fuzzy mappings under generalized almost F-contractions in the context of controlled metric spaces, which generalize many preexisting results in the literature. At the end, we give an example for the justification of our main result.

Main results
In this section, we define fuzzy sets, fuzzy mappings, and α-fuzzy fixed points and prove some common α fuzzy fixed point results in the context of controlled metric spaces.
which assigns to every member of W a membership grade in A σ .
We denote by F σ (W) the collection of all fuzzy sets in W. Let A σ ∈ F σ (W) and α ∈ [0, 1]. Then the α-level set of A σ is denoted by [A σ ] α and is defined as is a generalized Hausdorff controlled fuzzy metric on CLB(W). Proof Let us suppose on the contrary that for each a ∈ A,

Definition 2.2 Let S, T be fuzzy mappings from W into (W). Then
From Definition 1.6 we have that for each a ∈ A, Hence from equations (4) and (5) we get a contradiction.
be a complete controlled metric space, and let S, T be fuzzy Then there exists a common α-fuzzy fixed point of S and T.
By equation (12) there exists n 1 ∈ N such that n(F(d σ (ω n , ω n+1 ))) l ≤ 1 for all n ≥ n 1 . Thus, for all n ≥ n 1 , we have From the triangle inequality and equation (13) for m > n ≥ n 1 , we have converges by the ratio test for each m ∈ N. Therefore, by taking the limit as n → ∞ in the above inequality we get d σ (ω n , ω m ) → 0. Since W is complete, there exists ρ ∈ W such that lim n→∞ ω n = ρ. Next, we prove that ρ is a fixed point of T. Suppose on the contrary that ρ is not a fixed point of T. Then there exist N 0 ∈ N and a subsequence {ω n r } of {ω n } such that d σ (ω 2n r , [Tρ] α T (ρ) ) > 0 for all n r ≥ N 0 . As d σ (ω 2n r , [Tρ] α T (ρ) ) > 0 for all n r ≥ N 0 , from Lemma 2.1, condition (1) of Definition 1.3, and (6) we have
Proof By taking Ł = 0 in Theorem 2.1 we get the proof. Proof By taking S = T in Theorem 2.1 we get the proof. (W, d σ ) be a complete controlled metric space, and let T be a fuzzy mapping from W into (W). Suppose that for each ω 1 ∈ W, there exist α T (ω 1 ), α T (ω 2 ) ∈ (0, 1] such that [Tω 1 ] α T (ω 1 ) , [Tω 2 ] α T (ω 2 ) are nonempty closed subsets of W. Assume there exist F ∈ F and > 0 such that
Proof By taking S = T and Ł = 0 in Theorem 2.1 we get the proof.

Then there is a common fixed point of A and B.
Proof for all ω 1 , ω 2 ∈ W with H σ (Aω 1 , Aω 2 ) > 0, where

Then there exists a fixed point of A.
Proof Take  We further suppose thatT is a multivalued mapping induced by the fuzzy mapping T : W → (W), that is, T(ω 1 )(T) = μ ∈ W : T(ω 1 )(μ) = max t∈W T(ω 1 )(t) . (W, d σ ) be a complete controlled metric space, μ ∈ W, and let T be a fuzzy mapping from W into (W) such thatT(ω 1 ) is a nonempty compact set for all ω 1 ∈ W. Then μ ∈T(μ) if and only if
Then by the same steps we can show that μ ∈T(μ). (W, d σ ) be a complete controlled metric space, and letŜ,T : W → (W) be fuzzy mappings such that for each ω 1 ∈ W,Ŝ(ω 1 ) andT(ω 1 ) are nonempty closed subsets of W. Assume there exist F ∈ F , > 0, and Ł ≥ 0 such that

Conclusion
In this work, we introduced the concept of fuzzy mappings in a more general space, called a controlled metric space. Further, we derived the existence of common α-fuzzy fixed points for two fuzzy mappings under generalized almost F-contractions in the setting of controlled metric spaces. Our results generalize many well-known results in the literature. For justification of the obtained results, we gave an illustrative example.