Global Optimal Solutions for Proximal Fuzzy Contractions Involving Control Functions

Recent advancements in fixed point theory are one of the central and active research areas of nonlinear functional analysis, which provides a variety of mathematical methods, principles, and techniques for solving a variety of problems arising from various branches of mathematics as well as various fields in science and engineering. &e Banach fixed point theorem is considered as one of the most fruitful results in this theory. Due to its vast and significant applicability in pure and applied mathematics, this principle has been generalized and developed in various approaches (see, e.g., [1–22]). In particular, Khojasteh et al. [23] presented an impressive technique to the investigation of fixed point theory by developing the notion of simulation functions, which exhibit a significant unifying power. &e idea of simulation functions has been generalized, improved, and extended in different metric spaces (see, e.g., [11, 14, 24, 25]). &e best proximity theory is another expanding and prominent aspect of fixed point theory which plays a fundamental role in the investigation of requirements that guarantee the existence of an optimal approximate fixed point when the functional equationLx � x has no solution. Indeed, a non-self-mapping L: U⟶V does not possess necessarily a fixed point, with U and V are two nonempty subsets of a classical metric space (Λ, d). Best proximity theory is a remarkable generalization of fixed point theorems. In fact, the best proximity point turned out to be a fixed point in a natural way if the mapping in question is a self-mapping. For more recent developments in best proximity theory and related techniques, refer to [9–11, 19, 26–32]. In the present study, following this line of research interest, we present a simulation function approach to best proximity point problems in fuzzy metric spaces. We initiate new concepts of α − ψ − FZ-contraction, α − FZ-contraction, and generalized α − FZ-contraction, and we discuss existence results of best proximity point of such classes of non-self-mappings involving control functions in the structure of complete fuzzy metric spaces. &e furnished results enrich, generalize, and extend various existing findings in the literature.


Introduction
Recent advancements in fixed point theory are one of the central and active research areas of nonlinear functional analysis, which provides a variety of mathematical methods, principles, and techniques for solving a variety of problems arising from various branches of mathematics as well as various fields in science and engineering. e Banach fixed point theorem is considered as one of the most fruitful results in this theory. Due to its vast and significant applicability in pure and applied mathematics, this principle has been generalized and developed in various approaches (see, e.g., ). In particular, Khojasteh et al. [23] presented an impressive technique to the investigation of fixed point theory by developing the notion of simulation functions, which exhibit a significant unifying power. e idea of simulation functions has been generalized, improved, and extended in different metric spaces (see, e.g., [11,14,24,25]). e best proximity theory is another expanding and prominent aspect of fixed point theory which plays a fundamental role in the investigation of requirements that guarantee the existence of an optimal approximate fixed point when the functional equation Lx � x has no solution. Indeed, a non-self-mapping L: U ⟶ V does not possess necessarily a fixed point, with U and V are two nonempty subsets of a classical metric space (Λ, d). Best proximity theory is a remarkable generalization of fixed point theorems. In fact, the best proximity point turned out to be a fixed point in a natural way if the mapping in question is a self-mapping. For more recent developments in best proximity theory and related techniques, refer to [9-11, 19, 26-32].
In the present study, following this line of research interest, we present a simulation function approach to best proximity point problems in fuzzy metric spaces. We initiate new concepts of α − ψ − FZ-contraction, α − FZ-contraction, and generalized α − FZ-contraction, and we discuss existence results of best proximity point of such classes of non-self-mappings involving control functions in the structure of complete fuzzy metric spaces. e furnished results enrich, generalize, and extend various existing findings in the literature. Definition 1 (see [33] ree standard instances are as follows: Definition 2 (see George and Veeramani [34]). Let Λ be an arbitrary set, * is a continuous t-norm, and D is a fuzzy set on Λ × Λ × (0, ∞). e ordered triple (Λ, D, * ) is said to be a fuzzy metric space if for all ϑ, θ, ω ∈ Λ and ς, σ > 0. For ς > 0, the open ball with centre ϑ ∈ Λ and radius ρ, where 0 < ρ < 1, is defined by A subset O of a fuzzy metric space (Λ, D, * ) is said to be open if given any point θ ∈ O, there exists 0 < ρ < 1 and ς > 0 such that B(θ, ρ, ς) ⊆ O. Let τ denote the collection of all open subsets of Λ; hence, τ is a topology on Λ. is topology is Hausdorff and first countable. For further topological results, refer to [2,34].
(1) A sequence ϑ n ⊆ Λ is said to be convergent to ϑ ∈ Λ if an only if lim n⟶∞ D(ϑ n , ϑ, ς) � 1 for all ς > 0 (2) A sequence ϑ n ⊆ Λ is said to be a Cauchy sequence iff for each ε ∈ (0, 1) and ς > 0, there exists n 0 ∈ N such that D(ϑ n , ϑ m , ς) > 1 − ε for all n, m ≥ n 0 (3) A fuzzy metric space is called complete if every Cauchy sequence in Λ has a limit in Λ In [2], Gregori and Sapena initiated the notion of a fuzzy contractive mapping as follows.
Definition 5 (see [4]). Let Ψ be the class of nondecreasing functions ψ: (0, 1] ⟶ (0, 1] fulfilling the following two conditions: Afterwards, Wardowski [5] proposed the idea of a fuzzy H-contractive mapping as follows. Definition 6 (see [5]). Let H be the set of functions η: (0, 1] ⟶ (0, ∞] satisfying the two conditions (W 1 ) and (W 2 ) given by A self-mapping L: Λ ⟶ Λ on a fuzzy metric space (Λ, D, * ) is called a fuzzy H-contractive with respect to the function η ∈ H if there exists a ∈ (0, 1) such that the following inequality holds: e following class of control functions has been introduced in [8], where we used the term class FZ instead of the present FZ-simulation functions.
Definition 8 (see [8]). Let (Λ, D, * ) be a fuzzy metric space, L: Λ ⟶ Λ a mapping, and ξ ∈ FZ. en, L is said to be a FZ-contraction with respect to ξ if the following condition is satisfied: Example 3 (see [8]). e type of fuzzy contractive mappings developed by Gregori and Sapena [2] is a perfect example of FZ-contraction. It can be expressed facilely from the previous definition by taking the FZ-simulation function as where a ∈ (0, 1).
Definition 11 (see [19]). Let U and V be nonempty subsets of a fuzzy metric space (Λ, D, * ). Define U 0 (ς) and V 0 (ς) by the following sets: where Note that, a point ω ∈ U is said to be a fuzzy best proximity point of the mapping L, where L: U ⟶ V, U and V are nonempty subsets of an abstract nonempty set Λ if D(ω, Lω, ς) � D(U, V, ς) for all ς > 0.

Main Results
Firstly, we define the following concepts.

Remark 2.
Note that Definition 14 cannot be reduced to Definition 13 since ψ(t) � t does not belong to Ψ.
Definition 15. Let U and V be nonempty subsets of fuzzy metric space (Λ, D, * ) and α: We say that L: U ⟶ V is a generalized α − FZ-contraction with respect to ξ ∈ FZ if L is an α-proximal admissible such that Journal of Mathematics . (17) Next, we give our first main result.
Making use of (20) and (21), we obtain Regarding that L is an α − ψ − FZ-contraction with respect to ξ ∈ FZ, together with (20), (21), and (ξ 2 ), we obtain Consequently, we have which means that D(ϑ n , ϑ n+1 , ς) is a nondecreasing sequence of positive real numbers in (0, 1]. en, there exists We shall prove that c(ς) � 1. Reasoning by contradiction, suppose that c(ς 0 ) < 1 for some ς 0 > 0. Now, if we take the sequences τ n � D(ϑ n , ϑ n+1 , ς 0 ) and δ n � D(ϑ n− 1 , ϑ n , ς 0 )} and considering (ψ 2 ) and (ξ 3 ) and that ξ is nonincreasing with respect to its second argument, we obtain which is a contradiction and yields Next, we show that the sequence ϑ n is Cauchy. Reasoning by contradiction, suppose that ϑ n is not a Cauchy sequence. us, there exists ε ∈ (0, 1), ς 0 > 0, and two subsequences ϑ n k and ϑ m k of ϑ n with n k > m k ≥ k for all k ∈ N such that Taking into account Lemma 1, we derive By choosing m k as the smallest index satisfying (29), we have On account of (28), (30), and (MS 4 ), we have Taking limit as k ⟶ ∞ and employing (27), we derive On the other hand, we have which imply that Furthermore, given that L is triangular weak-α-admissible and taking into account (20), we deduce that α ϑ n , ϑ m , ς ≥ 1, for all n, m ∈ N with n > m.
So that Regarding the fact that L is an α − ψ − FZ-contraction with respect to ξ ∈ FZ and making use of (35) and (36), we have From (32) and (34), we see that the sequences μ k � D(ϑ m k , ϑ n k , ς 0 ) and ] k � D(ϑ m k − 1 , ϑ n k − 1 , ς 0 ) have the same limit 1 − ε < 1, taking into consideration that ξ is nonincreasing with respect to its second argument; by the property ξ 3 , we conclude that which is a contradiction. So that ϑ n is a Cauchy sequence in U. As U is closed subset of a complete fuzzy metric space (Λ, D, * ), there exists ω ∈ U such that lim n⟶∞ D ϑ n , ω, ς � 1.
In the next theorem, we substitute the continuity of L in eorem 1 with the following condition.
Hence, we deduce that D(ϑ n , ϑ n+1 , ς) is a nondecreasing sequence in (0, 1]. us, there exists s(ς) ≤ 1 such that lim n⟶∞ D(ϑ n , ϑ n+1 , ς) � s(ς) for all ς > 0. We shall prove that s(ς) � 1. Reasoning by contradiction, suppose that s(ς 0 ) < 1 for some ς 0 > 0. Now, if we take the sequences T n � D(ϑ n , ϑ n+1 , ς 0 ) and S n � D(ϑ n− 1 , ϑ n , ς 0 ) and consider (ξ 3 ), we obtain Journal of Mathematics which is a contradiction. erefore, lim n⟶∞ D ϑ n , ϑ n+1 , ς � 1, for all ς > 0. (60) Next, we show that ϑ n is Cauchy sequence. On the contrary, assume that ϑ n is not a Cauchy. Hence, there exist ε ∈ (0, 1), ς 0 > 0, and two subsequences ϑ n k and ϑ m k of ϑ n with n k > m k ≥ k for all k ∈ N such that Taking into account Lemma 1, we derive that By choosing m k as the smallest index satisfying (29), we have Making use of (61) and (63) and the triangular inequality, we get Passing to the limit k ⟶ ∞ and using (60), we derive that On the other hand, which imply that Furthermore, since L is triangular weak-α-admissible, we deduce that α ϑ n , ϑ m , ς ≥ 1, for all n, m ∈ N with n > m. (68) us, for all k ∈ N. By the fact that L is a generalized α − FZ-contraction with respect to ξ ∈ FZ and using (69) and (70), we obtain that where Letting k ⟶ ∞ in equality (72) and using (60), we derive Journal of Mathematics 7 (73) which is a contradiction. en, ϑ n is a Cauchy sequence in U. Given that U is closed subset of a complete fuzzy metric space (Λ, D, * ), there exists ω ∈ U such that lim n⟶∞ D ϑ n , ω, ς � 1.
Proof. Pursuant to the same arguments as those given in the proof of eorem 3, we know that there exists a Cauchy sequence ϑ n in U which converges to ω ∈ U. Further, lim k⟶∞ D ϑ n , ω, ς � 1, for all n ∈ N, ς > 0.
We must point to the fact that, by defining the control function ξ and the admissible mapping α(ϑ, θ) in a proper way, it is possible to particularize and derive a number of varied consequences of our main results. We skip making such a number of corollaries since they seem clear.

Conclusion
is paper has dealt with a FZ-simulation function approach to best proximity point problems in fuzzy metric spaces. We have initiated some classes of non-self-mappings and discussed existence results of the best proximity points of such types of non-self-mappings. Our results can be further extended by replacing the fuzzy metric space by various settings (e.g., partially ordered fuzzy metric spaces and complex valued fuzzy metric spaces), and more generalization can be obtained by the study of optimal coincidence points, optimal best proximity coincidence points, and the setting of cyclic mappings.

Data Availability
e data used to support the findings of this study are included in the references within the article.