Fixed point results via extended FZ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{FZ}$\end{document}-simulation functions in fuzzy metric spaces

In this paper, we introduce a new class of control functions, namely extended FZ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{FZ}$\end{document}-simulation functions, and employ it to define a new contractive condition. We also prove some new fixed and best proximity point results in the context of an M-complete fuzzy metric space. The presented theorems unify, generalize, and improve several existing results in the literature.


Introduction
Fixed point theory is one of the central parts of research in functional analysis that provides several mathematical concepts and fruitful tools for the resolution of many problems arising from different fields of engineering and sciences. Due to its potential applicability, the Banach contraction principle is one of the most crucial results, and it asserts that every self-contraction G defined on a complete metric space X admits a unique fixed point. This influential result has been generalized and extended in different approaches and several abstract spaces (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]). In particular, Khojasteh et al. [2] proposed a new approach to the study of fixed point theory based on the notion of simulation functions which exhibit a significant unifying power over several known results. Roldán et al. [20] slightly revised the previous notion of simulation function by reformulating the definition given in [2]. In sequential study, Demma et al. [13] extended and generalized the concept of simulation functions on a b-metric framework by providing a new concept of b-simulation functions, and then in connection with existing fixed point results of [2], the authors addressed several new ones. Roldán and Samet [11] developed the family of extended simulation functions with respect to a lower semi-continuous mapping. The usefulness and applicability of these control functions have inspirited many authors to diversify it further in different metric spaces (see e.g. [4,8,9,11,13,15,[20][21][22][23]).
On the other hand, the non-self-mapping G : U → V with U ∩ V = ∅ does not have a fixed point. In this case, it is of interest to find an element x in U such that d(x, Gx) is minimum. Since d(U, V ) ≤ d(x, Gx), for all x ∈ U, the point x in U which satisfies the condition d(x, Gx) = d(U, V ) is called best proximity point. A best proximity theorem enunciates sufficient conditions for the existence of a best proximity point of the mapping G. In fact, best proximity theorems are natural generalizations of fixed point theorems.
The concept of fuzzy metric space was introduced by Kramosil and Michalek [24] and further modified by George and Veeramani [25] with the purpose of obtaining a Hausdorff topology. Later on, Gregori and Sapena [26] introduced the concept of fuzzy contractive mappings and proved a fixed point result in the setting of fuzzy metric space. In [27], Mihet proposed the class of ψ-contractive mappings, which is larger than the class of fuzzy contractive mappings given in [26]. Following this direction, Wardowski [28] presented and studied the concept of H-contractive mappings. Very recently, inspired by the approach in [2], Melliani and Moussaoui [3] (see also [4]) initiated the study of FZ-contractions involving a new class of simulation functions which provides a unique and common point of view for several previously known concepts in the context of fuzzy metric spaces such as fuzzy contractive, fuzzy ψ-contractive, and H-contractive mappings.
In the present paper, we introduce a new class of control functions, namely extended FZ-simulation functions, we prove some fixed points results in the context of an Mcomplete fuzzy metric space by defining a new contractive condition via the same class. The presented theorems unify, generalize, and improve several existing results in the literature.
An ordered triple (X, M, * ) is said to be a strong fuzzy metric space if the triangular inequality (FM 4 ) of Definition 2 is replaced by the following one: (FM 4 ) : M(x, y, t) * M(y, z, t) ≤ M(x, z, t) for all x, y, z ∈ X and t > 0. For further details and topological results, the reader is refereed to [24][25][26]30].
for all x, y ∈ X, t > 0. ε ∈ (0, 1) and t > 0, there exists n 0 ∈ N such that M(x n , x m , t) > 1ε for all n, m ≥ n 0 . 3. A sequence {x n } ⊆ X is said to be a G-Cauchy sequence if M(x n , x n+p , t) = 1 for all p ∈ N and t > 0. 4. A fuzzy metric space in which every M-Cauchy (G-Cauchy) sequence is convergent is called an M-complete (G-complete) fuzzy metric space.

Definition 4 ([26]
) Let (X, M, * ) be a fuzzy metric space. A mapping G : X → X is said to be fuzzy contractive mapping if there exists λ ∈ (0, 1) such that for each x, y ∈ X and t > 0.

Definition 5 ([27]
) Let (X, M, * ) be a fuzzy metric space. A mapping G : X → X is said to be a fuzzy ψ-contractive mapping if M(Gx, Gy, t) ≥ ψ M(x, y, t) for all x, y ∈ X, t > 0.
The authors in [3] proved the following result.
Theorem 1 Let (X, M, * ) be an M-complete strong fuzzy metric space and G : X − → X be an FZ-contraction with respect to ζ ∈ FZ. Then G has a unique fixed point.
Let (X, M, * ) be a fuzzy metric space, ϕ : X → (0, 1] be a given function, and G : X → X be a mapping. The set of all fixed points of T will be denoted by The set of all ones of the function ϕ will be denoted by Sezen et al. [33] presented the notion of fuzzy ϕ-fixed point as follows. Definition 9 ([33]) Let X be a nonempty set, ϕ : X → (0, 1] be a given function, and G : X → X. An element z ∈ X is said to be a fuzzy ϕ-fixed point of the mapping G if and only if z ∈ Fix(G) ∩ O ϕ .
Let F : (0, 1] 3 → (0, 1] be a given function, and consider the following axioms: We consider the following classes of functions:  ( The main result of [33] is the following.
Theorem 2 Let (X, M, * ) be a G-complete fuzzy metric space, G : X → X, and ϕ : X → (0, 1] be a lower semi-continuous function. Suppose that there exist two functions ψ ∈ and F ∈ F M such that, for all x, y ∈ X, t > 0, Then G has a unique ϕ-fixed point.

A new class of control functions
In this section, we enlarge the class of FZ-simulation functions by introducing the class of extended FZ-simulation functions.
We denote the collection of all extended FZ-simulation functions by FZ e .
The converse inclusion is not true, we confirm this by the following example.

Main results
First we introduce the following concept of (FZ ϕ e , F)-contraction.

Definition 11
Let (X, M, * ) be a fuzzy metric space, ϕ : X → (0, 1] be a given function, and F ∈ F . A mapping G : X → X is said to be an (FZ ϕ e , F)-contraction, if there exists e ∈ FZ e such that for all x, y ∈ X and all t > 0, where Our first main result is the following theorem.
Next, let x 0 ∈ X be an arbitrary point and {x n } be the Picard sequence defined by x n = G n x 0 , n ∈ N. If there exists some m ∈ N such that x m = x m+1 , then x m is a fixed point of G and hence a fuzzy ϕ-fixed point of G (as Fix(G) ⊆ O ϕ ), which completes the proof. For this reason, assume that x n = x n+1 for all n ∈ N, which means that M(x n , x n+1 , t) < 1 for all t > 0.
If there exists some which is a contradiction. As consequence, Regarding (E1), inequality (4) yields that which means that Therefore ϑ n (t) < ϑ n+1 (t). Then, it follows that {ϑ n (t)} is an increasing sequence of positive real numbers in (0, 1]. Consequently, there exists l(t) ≤ 1 such that lim n→∞ ϑ n (t) = l(t) ≤ 1 for all t > 0. We shall prove that l(t) = 1. On the contrary, we assume that l(t) < 1 for some t > 0. Denote τ n (t) = ϑ n+1 (t) and δ n (t) = min{ϑ n (t), ϑ n+1 (t)} = ϑ n (t), we have Since {δ n (t)} is strictly increasing, we have δ n (t) < l(t). Regarding (E2), we get lim sup n→∞ e τ n (t), δ n (t) < 0, which is in contradiction with e(τ n (t), δ n (t)) ≥ 0 for all n ∈ N. Accordingly, we deduce that Due to (F 1 ), we have Taking n → ∞ and keeping (5) in mind, we obtain Next, we show that {x n } is an M-Cauchy sequence in X. Arguing by contradiction, we assume that {x n } is not an M-Cauchy sequence. Then there exist ∈ (0, 1), t 0 > 0 and two Taking into account Lemma 1, we have By choosing m k as the smallest index satisfying (9), we get On account of (8) and (10), the triangular inequality yields Taking the limit of both sides as k → ∞, using (6) and (T 3 ), we derive that Since G is an (FZ ϕ e , F)-contraction, we have that, for all k ∈ N, which implies that where Passing to the limit as k → ∞ in the above equality, using (6), (7), (F 2 ) and taking into account the continuity of F, we obtain Therefore, (12) gives rise to By the triangular inequality, we have Letting k → ∞ in the last inequality and using (6) and (10), we get From (13) and (14), we derive that On the other hand, by (6), (10) and regarding (F 2 ) , we have In particular, it follows from (12), (F 1 ), and (8) that ) for all k ∈ N. From the above observations, (11) and (15), we conclude that lim k→∞ α k = lim k→∞ β k = 1 -and β k < 1 -. Thus, we can apply axiom (E2) to these sequences; as a consequence 0 ≤ lim k→∞ sup e(α k , β k ) < 0, which is a contradiction. Thus, we deduce that {x n } is an M-Cauchy sequence. Since (X, M, * ) is an M-complete fuzzy metric space, there exists u ∈ X such that lim n→∞ M(x n , u, t) = 1, ∀t > 0. (16) Due to the lower semi-continuity of ϕ, (7) and (16), we derive that Therefore, u ∈ O ϕ . Next, we shall show that u is a fixed point of G arguing by contradiction. Suppose that M(u, Gu, t) < 1 for some t > 0. Let us define β n (t) = N ϕ F (x n , u, t) for all n ∈ N.
Using (F 1 ), we obtain Taking the limit as n → ∞ and using the continuity of F lim n→∞ά n (t) = lim On the other hand, As F is continuous, we have and {ά n (t)} n≥n 0 ⊂ (0, 1] is a sequence converging to μ(t) < 1 such that, for all n ≥ n 0 , e ά n (t), μ(t) = e ά n (t),β n (t) Regarding (E3), the last inequality yields that μ(t) = 1, which contradicts (18). As a consequence, M(u, Gu, t) = 1, which together with (17) means that u is a fuzzy ϕ-fixed point of G.
As a final step, we shall show the uniqueness of a fuzzy ϕ-fixed point of G. We argue by contradiction. Suppose that there are two distinct ϕ-fixed points u, v ∈ X of the mapping G. Then M(u, v, t) < 1 for all t > 0. Since we have Fix(G) ⊆ O ϕ , it follows that ϕ(u) = ϕ(v) = 1. Now, using (2), we have where Regarding (E1), inequality (19) yields that a contradiction, thus u = v. Therefore, the fuzzy ϕ-fixed point of G is unique. This completes the proof.
To support our result, we provide an illustrative example. Precisely, we show that our result (Theorem 4) can be used to cover this example, while Theorem 2 is not applicable. In order to show that G is an (FZ ϕ e , F)-contraction mapping, we distinguish the following cases: Case I: Hence, G is an (FZ ϕ e , F)-contraction mapping. Therefore, all the hypotheses of Theorem 4 are satisfied, and hence G has a ϕ-fixed point (namely x = 0).
Finally, we show that Theorem 2 is not applicable in this example. In fact, suppose that there is ψ ∈ such that the contraction condition (1) of Theorem 2 holds, that is, for all x, y ∈ X, we have F M (Gx, Gy, t), ϕ(Gx), ϕ(Gy) ≥ ψ F M(x, y, t), ϕ(x), ϕ(y) . Choose x = 0 and y = 1 2 and take into the account that ψ(t) > t for all t ∈ (0, 1), we have which is a contradiction. This shows that it is impossible to find a function ψ ∈ such that the contraction condition (1) holds. Therefore, Theorem 2 is not applicable.

Corollary 1 ([34])
Let (X, M, * ) be an M-complete fuzzy metric space and G : X → X be a given mapping such that, for all x, y ∈ X, t > 0 and for some λ ∈ (0, 1), Then G has a unique fixed point.  F(a, b, c) = a · b · c for all a, b, c ∈ (0, 1] and ϕ(x) = 1 for all x ∈ X in Theorem 4.

Corollary 3
Let (X, M, * ) be an M-complete fuzzy metric space, and let G : X → X be a given mapping. Suppose that there exists some ψ ∈ such that, for all x, y ∈ X, t > 0, Gy, t) .
Then G has a unique fixed point.

Corollary 4
Let (X, M, * ) be an M-complete fuzzy metric space, and let G : X → X be a given mapping and η ∈ H such that, for all x, y ∈ X, t > 0, where k ∈ (0, 1). Then G has a unique ϕ-fixed point.

Corollary 5
Let (X, M, * ) be an M-complete fuzzy metric space and G : X → X be a given mapping. Assume that for all x, y ∈ X and t > 0, where φ : [0, ∞) − → [0, ∞) is a right-continuous function with φ(t) < t for all t > 0. Then G has a unique fuzzy ϕ-fixed point.

Best proximity point results
In this section, we obtain a sufficient condition to ensure the existence of a ϕ-best proximity point in the setting of fuzzy metric spaces. Our results can be viewed as an extension of some related results in the existing literature.
Let U and V be two nonempty subsets of a fuzzy metric space (X, M, * ) and G : U → V be a non-self-mapping. We will use the following notations: The set of all best proximity points of the non-self-mapping G : U → V will be denoted by Definition 12 ([33]) Let X be a nonempty set, ϕ : X → (0, 1] be a given function, and G : U → V be a non-self-mapping. An element u * ∈ U is said to be a fuzzy ϕ-best proximity point of G if and only if u * is a best proximity point of G and ϕ(u * ) = 1.

Definition 13
Let U and V be two nonempty closed subsets of a fuzzy metric space (X, M, * ). We say that the operator G : U → V is an (FZ ϕ e , F)-fuzzy proximal contraction with respect to e ∈ FZ e if there exist a function ϕ : X → (0, 1] and F ∈ F such that for all u, v, x, y ∈ U and t > 0, where Theorem 5 Let U and V be two nonempty subsets of an M-complete fuzzy metric space (X, M, * ) such that U 0 (t) is nonempty and ϕ : X → (0, 1], F ∈ F . Suppose that G : U → V is an (FZ ϕ e , F)-fuzzy proximal contraction with respect to e ∈ FZ e . Suppose also (i) U 0 (t) is closed with respect to the topology induced by M; Then G has a unique fuzzy ϕ-best proximity point, that is, there exists x * ∈ U such that We shall indicate that F(1, ϕ(σ ), ϕ(σ )) = 1. Reasoning by contradiction, suppose that F(1, ϕ(σ ), ϕ(σ )) < 1, and using (E1) we derive which is a contradiction. Therefore, F(1, ϕ(σ ), ϕ(σ )) = 1.
Using property (E1), we deduce that which yields γ n (t) < γ n+1 (t). Therefore, we deduce that {γ n (t)} is an increasing sequence of real numbers in (0, 1]. Thus, there exists h(t) ≤ 1 such that lim n→∞ γ n (t) = h(t) ≤ 1 for all n ∈ N. In particular, as {γ n (t)} is strictly increasing, then h(t) > γ n (t). We shall prove that h(t) = 1 for all t > 0. Suppose, on the contrary, that h(t) < 1 for some t > 0. If we choose the sequences n (t) = γ n+1 (t) and θ n (t) = min{γ n (t), γ n+1 (t)}, we have lim n→∞ n (t) = lim n→∞ θ n (t) = h(t) and θ n (t) < h(t). By condition (E2), we derive that lim sup n→∞ e n (t), θ n (t) < 0, which contradicts equation (22). Accordingly, we deduce that Moreover, using (F 1 ) we get and which implies for all k ∈ N. So, by (20), we have where By following a similar reasoning to that in the proof of Theorem 4, one can show that Particularly, it follows from (27), (F 1 ), and (24) that On account of the above observations, we deduce that lim k→∞ r k = lim k→∞ s k = 1 -and s k < 1 -. Regarding axiom (E2), we obtain 0 ≤ lim sup k→∞ e(r k , s k ) < 0, which is a contradiction. This contradiction proves that {x n } is an M-Cauchy sequence.
Since U 0 (t) is a closed subset of the M-complete fuzzy metric space (X, M, * ), there exists x * ∈ U 0 (t) such that lim n→∞ M x n , x * , t = 1 ∀t > 0.
By the lower semi-continuity of ϕ, (23) and (28), we have As G(U 0 (t)) ⊆ V 0 (t) and x * ∈ U 0 (t), there exists ω ∈ U 0 (t) such that Now, we shall prove that x * = w, reasoning by contradiction. Suppose that M(x * , w, t) < 1 for some t > 0. Define s n (t) = N ϕ F x n , x * , t for all n ∈ N.
Using (F 1 ), we obtain Taking the limit as n → ∞ and using the continuity of F, we have On the other hand, Due to the continuity of F, we have As consequence, there exists n 0 ∈ N such that s n (t) = F M x * , w, t , 1, ϕ(w) = a(t), n ≥ n 0 .
By (29), we conclude that x * is a fuzzy ϕ-best proximity point of G. Finally, we shall show the uniqueness of the fuzzy ϕ-best proximity point of G, that is, B est (G) ∩ O ϕ is singleton. We argue by contradiction. Suppose that x * , w * ∈ X are two distinct fuzzy ϕ-best proximity fixed points of the mapping G. Then M(x * , w * , t) < 1 for all t > 0. Hence M x * , Gx * , t = M(U, V , t) and M w * , Gw * , t = M(U, V , t).

Corollary 6
Let U and V be two nonempty subsets of an M-complete fuzzy metric space (X, M, * ) such that U 0 (t) is nonempty. Assume that the mappings G : X → X, ϕ : X → (0, 1], ψ ∈ , and F ∈ F satisfy the following conditions: (i) U 0 (t) is closed with respect to the topology induced by M; (ii) G(U 0 (t)) ⊆ V 0 (t); (iii) ϕ is continuous. Then G has a unique fuzzy ϕ-best proximity point, that is, there exists x * ∈ U such that B est (G) ∩ O ϕ = {x * }.

Conclusion
In this study, we established the concept of extended FZ-simulation functions with a view to consider a new class of fuzzy contractions, namely (FZ ϕ e , F)-contractions. Such a family generalized, extended, and unified several results and enriched various classical types of fuzzy contractions in the literature. We must underline that by properly specifying the control function e, we can particularize and derive different consequences of our main results. Nevertheless, further research is needed in this regard, because it is plausible to explore the existence and uniqueness of a common fixed point or a coincidence point of two self-mappings in a more general setting, for example, partially ordered fuzzy metric spaces.