# Global Optimal Solutions for Proximal Fuzzy Contractions Involving Control Functions

Authors: Abdelhamid Moussaoui; Nawab Hussain; Said Melliani
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this study, we introduce new concepts ofα−FZ-contraction and α−ψ−FZ-contraction and we discuss existence results of the best proximity points of such types of non-self-mappings involving control functions in the structure of complete fuzzy metric spaces. Our results extend, generalize, enrich, and improve diverse existing results in the current literature.


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## Body

## 1. 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. The 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. The idea of simulation functions has been generalized, improved, and extended in different metric spaces (see, e.g., [11, 14, 24, 25]).The 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 ofα−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. The furnished results enrich, generalize, and extend various existing findings in the literature.


## 2. Preliminaries

Throughout this study,ℕ and ℝ will represent natural and real numbers, respectively. First, we start with some notions and main properties of fuzzy metric spaces.Definition 1.
(see [33]). A binary operation ∗:0,1×0,1⟶0,1 is a continuous t-norm if it fulfills the following conditions:(CT1)∗ is continuous(CT2)∗ is commutative and associative(CT3)℘∗1=℘ for all ℘∈0,1(CT4) ℘∗ℓ≤J∗ı whenever ℘≤J and ℓ≤1, for all ℘,ℓ,J,1∈0,1Example 1.
Three standard instances are as follows:(a)
℘∗ℓ=℘.ℓ(b)
℘∗ℓ=min℘,ℓ(c)
℘∗ℓ=max0,℘+ℓ−1Definition 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,∞. The ordered triple Λ,D,∗ is said to be a fuzzy metric space ifℳS1Dϑ,θ,ς>0ℳS2Dϑ,θ,ς=1 if and only if ϑ=θℳS3Dϑ,θ,ς=Dθ,ϑ,ςℳS4Dϑ,θ,ς∗Dθ,ω,σ≤Dϑ,ω,ς+σℳS5Dϑ,θ,.:0,∞⟶0,1 is continuousfor all ϑ,θ,ω∈Λ and ς,σ>0.
Forς>0, the open ball with centre ϑ∈Λ and radius ρ, where 0<ρ<1, is defined by(1)ℬϑ,ρ,ς=θ∈Λ:Dϑ,θ,ς>1−ρ.
A subsetO 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 ℬθ,ρ,ς⊆O. Let τ denote the collection of all open subsets of Λ; hence, τ is a topology on Λ. This topology is Hausdorff and first countable. For further topological results, refer to [2, 34].Example 2.
(see [34]). Let Λ,d be a metric space and ∗ be the product t-norm, and define the function D:Λ2×0,∞⟶0,1 by(2)Dϑ,θ,ς=e−dϑ,θ/ς,for all ϑ,θ∈Λ,ς>0. Then, Λ,D,∗ is a fuzzy metric space on Λ.Lemma 1.
(see [1]). Dϑ,θ,. is nondecreasing for all ϑ,θ in Λ.Definition 3.
(see [34]). Let Λ,D,∗ be a fuzzy metric space.(1)
A sequenceϑn⊆Λ is said to be convergent to ϑ∈Λ if an only if limn⟶∞Dϑn,ϑ,ς=1 for all ς>0(2)
A sequenceϑn⊆Λ is said to be a Cauchy sequence iff for each ε∈0,1 and ς>0, there exists n0∈ℕ such that Dϑn,ϑm,ς>1−ε for all n,m≥n0(3)
A fuzzy metric space is called complete if every Cauchy sequence inΛ has a limit in Λ
In [2], Gregori and Sapena initiated the notion of a fuzzy contractive mapping as follows.Definition 4.
(see [2]). Let Λ,D,∗ be a fuzzy metric space. A mapping ℒ:Λ⟶Λ is called a fuzzy contractive mapping if there exists a∈0,1 such that(3)1Dℒϑ,ℒθ,ς−1≤a1Dϑ,θ,ς−1,for each ϑ,θ∈Λ and ς>0.Definition 5.
(see [4]). Let Ψ be the class of nondecreasing functions ψ:0,1⟶0,1 fulfilling the following two conditions:ψ1ψ is continuousψ2ψc>c for all c∈0,1
A self-mappingℒ:Λ⟶Λ on a fuzzy metric space Λ,D,∗ is called a fuzzy ψ-contractive mapping if(4)Dℒϑ,ℒθ,ς≥ψDϑ,θ,ς,for allϑ,θ∈Λ,ς>0.
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 W1 and W2 given byW1η transforms 0,1 onto 0,∞W2η is strictly decreasing
A self-mappingℒ: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:(5)ηDℒϑ,ℒθ,ς≤aηDϑ,θ,ς,for allϑ,θ∈Λ,ς>0.
The 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 7.
(see [8]). The function ξ:0,1×0,1⟶ℝ is said to be a FZ-simulation function if the following properties hold:ξ1ξ1,1=0ξ2ξμ,ν<1/ν−1/μ for each μ,ν∈0,1ξ3 if μn,νn are sequences in 0,1 such that limn⟶∞μn=limn⟶∞νn<1, then limn⟶∞supξμn,νn<0
ByFZ, we denote the collection of all FZ-simulation functions.Definition 8.
(see [8]). Let Λ,D,∗ be a fuzzy metric space, ℒ:Λ⟶Λ a mapping, and ξ∈FZ. Then, ℒ is said to be a FZ-contraction with respect to ξ if the following condition is satisfied:(6)ξDℒϑ,ℒθ,ς,Dϑ,θ,ς≥0,for allϑ,θ∈Λ,ς>0.Example 3.
(see [8]). The 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(7)ξμ,ν=a1ν−1−1μ+1,for allμ,ν∈0,1,where a∈0,1.Example 4.
(see [8]). The corresponding FZ-simulation function for the fuzzy ψ-contractive mapping is defined by(8)ξμ,ν=1ψν−1μ,for allμ,ν∈0,1withψ∈Ψ.Definition 9.
(see [6]). Let Λ,D,∗ be a fuzzy metric space. We say that a mapping ℒ:Λ⟶Λ is α-admissible if there exists a function α:Λ×Λ×0,+∞⟶0,+∞ such that for all ϑ,θ∈Λ, ς>0,(9)αϑ,θ,ς≥1⟹αℒϑ,ℒθ,ς≥1.
In line with [15] (see also [16]), we use the notion of triangular weak-α-admissible function in the form that is as follows.Definition 10.
Letℒ:Λ⟶Λ be a mapping and α:Λ×Λ×0,∞⟶0,∞ be a function. We say that ℒ is a triangular weak-α-admissible if(10)αϑ,θ,ς≥1,αθ,ω,ς≥1⟹αϑ,ω,ς≥1,for all ϑ,θ,ω∈Λ, ς>0.Definition 11.
(see [19]). Let U and V be nonempty subsets of a fuzzy metric space Λ,D,∗. Define U0ς and V0ς by the following sets:(11)U0ς=u∈U:Du,v,ς=DU,V,ς,for somev∈V,V0ς=v∈V:Du,v,ς=DU,V,ς,for someu∈U,where(12)DU,V,ς=supDu,v,ς:u∈U,v∈V.
Note that, a pointω∈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 Dω,ℒω,ς=DU,V,ς for all ς>0.


## 3. Main Results

Firstly, we define the following concepts.Definition 12.
LetU and V be two nonempty subsets of fuzzy metric space Λ,D,∗ and α:Λ×Λ×0,∞⟶0,∞. We say that ℒ:U⟶V is an α-proximal admissible if(13)αϑ,θ,ς≥1,Du,ℒϑ,ς=Dv,ℒθ,ς=DU,V,ς⟹αu,v,ς≥1,for all u,v,ϑ,θ∈Λ and ς>0.Remark 1.
Note that ifDU,V,ς=1, then Definition 12 reduces to Definition 9 of α-admissibility.Definition 13.
LetU and V be nonempty subsets of fuzzy metric space Λ,D,∗ and α:Λ×Λ×0,∞⟶0,∞. We say that ℒ:U⟶V is an α−FZ-contraction with respect to ξ∈FZ if ℒ is an α-proximal admissible such that(14)αϑ,θ,ς≥1,Du,ℒϑ,ς=DU,V,ς,Dv,ℒθ,ς=DU,V,ς⟹ξDu,v,ς,Dϑ,θ,ς≥0,for all u,v,ϑ,θ∈U and ς>0.Definition 14.
LetU and V be nonempty subsets of fuzzy metric space Λ,D,∗, α:Λ×Λ×0,∞⟶0,∞, and ψ∈Ψ. We say that ℒ:U⟶V is an α−ψ−FZ-contraction with respect to ξ∈FZ if ℒ is an α-proximal admissible such that(15)αϑ,θ,ς≥1,Du,ℒϑ,ς=DU,V,ς,Dv,ℒθ,ς=DU,V,ς⟹ξDu,v,ς,ψDϑ,θ,ς≥0,for all u,v,ϑ,θ∈U and ς>0.Remark 2.
Note that Definition14 cannot be reduced to Definition 13 since ψt=t does not belong to Ψ.Definition 15.
LetU and V be nonempty subsets of fuzzy metric space Λ,D,∗ and α:Λ×Λ×0,∞⟶0,∞. 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(16)αϑ,θ,ς≥1,Du,ℒϑ,ς=DU,V,ς,Dv,ℒθ,ς=DU,V,ς⟹ξDu,v,ς,ℛϑ,θ,ς≥0,for all u,v,ϑ,θ∈U and ς>0, where(17)ℛϑ,θ,ς=minDϑ,θ,ς,Dϑ,u,ςDθ,v,ςDϑ,θ,ς.
Next, we give our first main result.Theorem 1.
LetU and V be nonempty subsets of a complete fuzzy metric space Λ,D,∗, α:Λ×Λ×0,∞⟶0,∞, ψ∈Ψ, and ξ∈FZ is nonincreasing in its second argument. Assume that ℒ:U⟶V is an α−ψ−FZ-contraction with respect to ξ and(i)
ℒ is triangular weak-α-admissible(ii)
U is closed(iii)
ℒU0⊆V0(iv)
There existsϑ0,ϑ1∈U such that Dϑ0,ℒϑ0,ς=DU,V,ς and αϑ0,ϑ1,ς≥1 for all ς>0(v)
ℒ is continuous.
Then, there existsω∈U such that Dω,ℒω,ς=DU,V,ς for all ς>0; that is, ℒ has a best proximity point ω∈U.Proof.
Due to condition (iv), there existsϑ0,ϑ1∈U such that αϑ0,ϑ1,ς≥1 and(18)Dϑ1,ℒϑ0,ς=DU,V,ς.
Regarding (iii), we deduce thatℒϑ1∈V0; hence, there exists ϑ2∈U such that(19)Dϑ2,ℒϑ1,ς=DU,V,ς.
Sinceαϑ0,ϑ1,ς≥1 and ℒ is an α-proximal admissible, consequently, αϑ1,ϑ2,ς≥1. Recursively, a sequence ϑn⊂U0 can be defined as follows:(20)αϑn,ϑn+1,ς≥1,for alln∈ℕ,(21)Dϑn+1,ℒϑn,ς=DU,V,ς,for alln∈ℕ.
If there existsn0∈ℕ such that ϑn0+1=ϑn0, we obtain(22)Dϑn0,ℒϑn0,ς=Dϑn0+1,ℒϑn0,ς=DU,V,ς,which means that ϑn0 is a best proximity point of ℒ. Thus, to continue our proof, we suppose that ϑn≠ϑn+1 for all n∈ℕ. Making use of (20) and (21), we obtain(23)Dϑn,ℒϑn−1,ς=Dϑn+1,ℒϑn,ς=DU,V,ς,for alln∈ℕ.
Regarding thatℒ is an α−ψ−FZ-contraction with respect to ξ∈FZ, together with (20), (21), and ξ2, we obtain(24)0≤ξDϑn,ϑn+1,ς,ψDϑn−1,ϑn,ς<1ψDϑn−1,ϑn,ς−1Dϑn,ϑn+1,ς.
Consequently, we have(25)Dϑn−1,ϑn,ς<ψDϑn−1,ϑn,ς<Dϑn,ϑn+1,ς,which means that Dϑn,ϑn+1,ς is a nondecreasing sequence of positive real numbers in 0,1. Then, there exists cς≤1 such that limn⟶∞Dϑn,ϑn+1,ς=cς≥1 for all ς>0. We shall prove that cς=1. Reasoning by contradiction, suppose that cς0<1 for some ς0>0. Now, if we take the sequences τn=Dϑn,ϑn+1,ς0 and δn=Dϑn−1,ϑn,ς0 and considering ψ2 and ξ3 and that ξ is nonincreasing with respect to its second argument, we obtain(26)0≤limn⟶∞supξDϑn,ϑn+1,ς0,ψDϑn−1,ϑn,ς0≤limn⟶∞supξDϑn,ϑn+1,ς0,Dϑn−1,ϑn,ς0<0,which is a contradiction and yields(27)limn⟶∞Dϑn,ϑn+1,ς=1,for allς>0.
Next, we show that the sequenceϑn is Cauchy. Reasoning by contradiction, suppose that ϑn is not a Cauchy sequence. Thus, there exists ε∈0,1, ς0>0, and two subsequences ϑnk and ϑmk of ϑn with nk>mk≥k for all k∈ℕ such that(28)Dϑmk,ϑnk,ς0≤1−ε.
Taking into account Lemma1, we derive(29)Dϑmk,ϑnk,ς02≤1−ε.
By choosingmk as the smallest index satisfying (29), we have(30)Dϑmk,ϑnk−1,ς02>1−ε.
On account of (28), (30), and ℳS4, we have(31)1−ε≥Dϑmk,ϑnk,ς0≥Dϑmk,ϑnk−1,ς02∗Dϑnk−1,ϑnk,ς02>1−ε∗Dϑnk−1,ϑnk,ς02.
Taking limit ask⟶∞ and employing (27), we derive(32)limn⟶∞Dϑmk,ϑnk,ς0=1−ε.
On the other hand, we have(33)Dϑmk−1,ϑnk−1,ς0≥Dϑmk−1,ϑmk,ς03∗Dϑmk,ϑnk,ς03∗Dϑnk,ϑnk−1,ς03,Dϑmk,ϑnk,ς0≥Dϑmk,ϑmk−1,ς03∗Dϑmk−1,ϑnk−1,ς03∗Dϑnk−1,ϑnk,ς03,which imply that(34)limn⟶∞Dϑmk−1,ϑnk−1,ς0=1−ε.
Furthermore, given thatℒ is triangular weak-α-admissible and taking into account (20), we deduce that(35)αϑn,ϑm,ς≥1,for alln,m∈ℕwithn>m.
So that(36)αϑmk,ϑnk,ς0≥1,(37)Dϑmk,ℒϑmk−1,ς0=Dϑnk,ℒϑnk−1,ς0=DU,V,ς0,for allk∈ℕ.
Regarding the fact thatℒ is an α−ψ−FZ-contraction with respect to ξ∈FZ and making use of (35) and (36), we have(38)0≤ξDϑmk,ϑnk,ς0,ψDϑmk−1,ϑnk−1,ς0,for allk∈ℕ.
From (32) and (34), we see that the sequences μk=Dϑmk,ϑnk,ς0 and νk=Dϑmk−1,ϑnk−1,ς0 have the same limit 1−ε<1, taking into consideration that ξ is nonincreasing with respect to its second argument; by the property ξ3, we conclude that(39)0≤limn⟶∞supξμk,ψνk≤limn⟶∞supξμk,νk<0,which is a contradiction. So that ϑn 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(40)limn⟶∞Dϑn,ω,ς=1.
Asℒ is continuous, we conclude that ℒϑn converges to ℒω; thus,(41)limn⟶∞Dℒϑn,ℒω,ς=1.
Due to the continuity ofD, we have Dϑn+1,ℒϑn,ς⟶Dω,ℒω,ς. From (21), we deduce(42)DU,V,ς=limn⟶∞Dϑn+1,ℒϑn,ς=Dω,ℒω,ς,which means that ω∈U is a best proximity point of ℒ.
In the next theorem, we substitute the continuity ofℒ in Theorem 1 with the following condition.
C: if ϑn is a sequence in U such that αϑn,ϑn+1,ς≥1 for all n∈ℕ, ς>0, and ϑn⟶ϑ∈U as n⟶∞, then there exists a subsequence ϑnk of ϑn such that αϑnk,ϑ,ς≥1 for all k∈ℕ and ς>0.Theorem 2.
LetU and V be nonempty subsets of a complete fuzzy metric space Λ,D,∗ and α:Λ×Λ×0,∞⟶0,∞, ψ∈Ψ, and ξ∈FZ is nonincreasing in its second argument. Assume that ℒ:U⟶V is an α−ψ−FZ-contraction with respect to ξ∈FZ and(i)
ℒ is triangular weak-α-admissible(ii)
U is closed(iii)
ℒU0⊆V0(iv)
There existsϑ0,ϑ1∈U such that Dϑ1,ℒϑ0,ς=DU,V,ς and αϑ0,ϑ1,ς≥1 for all ς>0(v)
Ifϑn is a sequence in U such that αϑn,ϑn+1,ς≥1 for all n∈ℕ, ς>0, and ϑn⟶ϑ∈U as n⟶∞, then there exists a subsequence ϑnk of ϑn such that αϑnk,ϑ,ς≥1 for all k∈ℕ and ς>0
Then, there existsω∈U such that Dω,ℒω,ς=DU,V,ς for all ς>0.Proof.
Following the lines of the proof of Theorem1, we deduce that there exists a Cauchy sequence ϑn in U0 which converges to ω∈U0. Since ℒU0⊆V0, we have ℒω∈V0, and then(43)Da1,ℒω,ς=DU,V,ς,for somea1∈U0.
By condition (v), there exists a subsequenceϑnk of ϑn such that(44)αϑnk,ω,ς≥1,for allk∈ℕ,ς>0.
Regarding thatℒ is an α-proximal admissible and(45)Da1,ℒω,ς=Dϑnk+1,ℒϑnk,ς=DU,V,ς,we obtain that αϑnk+1,a1,ς≥1. Hence,(46)0≤ξDa1,ϑnk+1,ς,ψDω,ϑnk,ς,for allk∈ℕ.
Applying the propertyξ2, it follows that(47)Dω,ϑnk,ς<ψDω,ϑnk,ς<Da1,ϑnk+1,ς,which yields limk⟶∞Da1,ϑnk+1,ς=1. Then, a1=ω; from (45), we derive that Dω,ℒω,ς=DU,V,ς.Theorem 3.
LetU and V be nonempty subsets of a complete fuzzy metric space Λ,D,∗ and α:Λ×Λ×0,∞⟶0,∞. Assume that ℒ:U⟶V is a generalized α−FZ-contraction with respect to ξ∈FZ and(i)
ℒ is triangular weak-α-admissible(ii)
U is closed(iii)
ℒU0⊆V0(iv)
There existsϑ0,ϑ1∈U such that Dϑ1,ℒϑ0,ς=DU,V,ς and αϑ0,ℒϑ0,ς≥1 for all ς>0(v)
ℒ is continuous
Then, there existsω∈U such that Dω,ℒω,ς=DU,V,ς for all ς>0.Proof.
Using condition (iv), there existsϑ0,ϑ1∈U such that αϑ0,ϑ1,ς≥1 and Dϑ1,ℒϑ0,ς=DU,V,ς. Regarding (iii), we have ℒϑ1∈V0 which yields that there exists ϑ2∈U such that(48)Dϑ2,ℒϑ1,ς=DU,V,ς.
Sinceαϑ0,ϑ1,ς≥1 and ℒ is an α-proximal admissible, it therefore follows that αϑ1,ϑ2,ς≥1. We recursively construct the sequence ϑn⊂U0 as follows:(49)αϑn,ϑn+1,ς≥1,Dϑn+1,ℒϑn,ς=DU,V,ς,for alln∈ℕ.
Suppose that there exists certainm0∈ℕ such that ϑm0+1=ϑm0. Hence,(50)Dϑm0,ℒϑm0,ς=Dϑm0+1,ℒϑm0,ς=DU,V,ς,which means that ϑm0 is a best proximity point of ℒ; thus, the proof is finished. For this reason, to continue our proof, we assume that ϑn≠ϑn+1 for all n∈ℕ. Making use of (56) and (60), we obtain(51)Dϑn,ℒϑn−1,ς=Dϑn+1,ℒϑn,ς=DU,V,ς,for alln∈ℕ.
Sinceℒ is a generalized α−FZ-contraction with respect to ξ∈FZ, we have(52)0≤ξDϑn,ϑn+1,ς,ℛϑn−1,ϑn,ς,where(53)ℛϑn−1,ϑn,ς=minDϑn−1,ϑn,ς,Dϑn−1,ϑn,ςDϑn,ϑn+1,ςDϑn−1,ϑn,ς=minDϑn−1,ϑn,ς,Dϑn,ϑn+1,ς.
Now, if(54)minDϑn,ϑn+1,ς,Dϑn−1,ϑn,ς=Dϑn,ϑn+1,ς.
applyingξ2, we get that(55)0≤ξDϑn,ϑn+1,ς,ℛϑn−1,ϑn,ς<1ℛϑn−1,ϑn,ς−1Dϑn,ϑn+1,ς.
Thus,(56)ℛϑn−1,ϑn,ς=Dϑn,ϑn+1,ς<Dϑn,ϑn+1,ς,which is a contradiction. Consequently,(57)ℛϑn−1,ϑn,ς=minDϑn,ϑn+1,ς,Dϑn−1,ϑn,ς=Dϑn−1,ϑn,ς.
By (55), we obtain that(58)Dϑn−1,ϑn,ς<Dϑn,ϑn+1,ς,for alln∈ℕ.
Hence, we deduce thatDϑn,ϑn+1,ς is a nondecreasing sequence in 0,1. Thus, there exists sς≤1 such that limn⟶∞Dϑn,ϑn+1,ς=sς for all ς>0. We shall prove that sς=1. Reasoning by contradiction, suppose that sς0<1 for some ς0>0. Now, if we take the sequences Tn=Dϑn,ϑn+1,ς0 and Sn=Dϑn−1,ϑn,ς0 and consider ξ3, we obtain(59)0≤limn⟶∞supξTn,Sn<0,which is a contradiction. Therefore,(60)limn⟶∞Dϑn,ϑn+1,ς=1,for allς>0.
Next, we show thatϑn is Cauchy sequence. On the contrary, assume that ϑn is not a Cauchy. Hence, there exist ε∈0,1, ς0>0, and two subsequences ϑnk and ϑmk of ϑn with nk>mk≥k for all k∈ℕ such that(61)Dϑmk,ϑnk,ς0≤1−ε.
Taking into account Lemma1, we derive that(62)Dϑmk,ϑnk,ς02≤1−ε.
By choosingmk as the smallest index satisfying (29), we have(63)Dϑmk,ϑnk−1,ς02>1−ε.
Making use of (61) and (63) and the triangular inequality, we get(64)1−ε≥Dϑmk,ϑnk,ς0≥Dϑmk,ϑnk−1,ς02∗Dϑnk−1,ϑnk,ς02>1−ε∗Dϑnk−1,ϑnk,ς02.
Passing to the limitk⟶∞ and using (60), we derive that(65)limn⟶∞Dϑmk,ϑnk,ς0=1−ε.
On the other hand,(66)Dϑmk−1,ϑnk−1,ς0≥Dϑmk−1,ϑmk,ς03∗Dϑmk,ϑnk,ς03∗Dϑnk,ϑnk−1,ς03,Dϑmk,ϑnk,ς0≥Dϑmk,ϑmk−1,ς03∗Dϑmk−1,ϑnk−1,ς03∗Dϑnk−1,ϑnk,ς03,which imply that(67)limn⟶∞Dϑmk−1,ϑnk−1,ς0=1−ε.
Furthermore, sinceℒ is triangular weak-α-admissible, we deduce that(68)αϑn,ϑm,ς≥1,for alln,m∈ℕwithn>m.
Thus,(69)αϑmk,ϑnk,ς0≥1,(70)Dϑmk,ℒϑmk−1,ς0=Dϑnk,ℒϑnk−1,ς0=DU,V,ς0,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(71)0≤ξDϑmk,ϑnk,ς0,ℛϑmk−1,ϑnk−1,ς0,for allk∈ℕ,where(72)ℛϑmk−1,ϑnk−1,ς0=minDϑmk−1,ϑnk−1,ς0,Dϑmk−1,ϑmk,ς0Dϑnk−1,ϑnk,ς0Dϑmk−1,ϑnk−1,ς0.
Lettingk⟶∞ in equality (72) and using (60), we derive(73)limk⟶∞ℛϑmk−1,ϑnk−1,ς0=min11−ε,1−ε=1−ε.
Take the sequencesμk=Dϑmk,ϑnk,ς0 and νk=ℛϑmk−1,ϑnk−1,ς0 for all k∈ℕ. Applying ξ3, we derive that(74)0≤limn⟶∞supξμk,νk<0,which is a contradiction. Then, ϑn 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(75)limn⟶∞Dϑn,ω,ς=1.
Asℒ is continuous, we obtain that ℒϑn converges to ℒω; thus,(76)limn⟶∞Dℒϑn,ℒω,ς=1.
As the metric functionD is continuous, we have Dϑn+1,ℒϑn,ς⟶Dω,ℒω,ς. In view of (51), we get(77)DU,V,ς=limn⟶∞Dϑn+1,ℒϑn,ς=Dω,ℒω,ς.
Thus,ω∈U is a best proximity point of ℒ.Theorem 4.
LetU and V be nonempty subsets of a complete fuzzy metric space Λ,D,∗ and α:Λ×Λ×0,∞⟶0,∞. Assuming that ℒ:U⟶V is an α−FZ-contraction with respect to ξ∈FZ,(i)
ℒ is triangular weak-α-admissible(ii)
U is closed(iii)
ℒU0⊆V0(iv)
There existsϑ0,ϑ1∈U such that Dϑ0,ℒϑ0,ς=DU,V,ς and αϑ0,ϑ1,ς≥1 for all ς>0(v)
ℒ is continuous or C holds
Then, there existsω∈U such that Dω,ℒω,ς=DU,V,ς for all ς>0.Proof.
Pursuant to the same arguments as those given in the proof of Theorem3, we know that there exists a Cauchy sequence ϑn in U which converges to ω∈U. Further,(78)limk⟶∞Dϑn,ω,ς=1,for alln∈ℕ,ς>0.
Ifℒ is continuous, then(79)limk⟶∞Dℒϑn,ℒω,ς=1,for alln∈ℕ,ς>0.
Taking into account (21), we deduce(80)DU,V,ς=limn⟶∞Dϑn+1,ℒϑn,ς=Dω,ℒω,ς,which means that ω∈U is a best proximity point of ℒ.
Now, suppose thatC holds. Since ℒU0⊆V0, we have ℒω∈V0 and then(81)Da1,ℒω,ς=DU,V,ς,for somea1∈U0.
By conditionC, there exists a subsequence ϑnk of ϑn such that(82)αϑnk,ω,ς≥1,for allk∈ℕ,ς>0.
Regarding thatℒ is an α-proximal admissible and(83)Da1,ℒω,ς=Dϑnk+1,ℒϑnk,ς=DU,V,ς,we obtain that αϑnk+1,a1,ς≥1. Hence,(84)0≤ξDa1,ϑnk+1,ς,Dω,ϑnk,ς,for allk∈ℕ.
Applying the propertyξ2, it follows that(85)Dω,ϑnk,ς<Da1,ϑnk+1,ς,which yields limk⟶∞Da1,ϑnk+1,ς=1. Then, a1=ω; from (45), we derive that Dω,ℒω,ς=DU,V,ς. This completes the proof.
Note that Theorem4 cannot be deduced by combining Theorems 1 and 2 since the function ψc=c does not belong to Ψ. Moreover, in Theorems 1 and 2, we have an added condition that ξ is nonincreasing in its second argument.


## 4. Consequences

Now, we shall clarify that diverse consequences of the existence results can be easily derived and developed from our main results.Corollary 1.
LetU and V be nonempty subsets of a complete fuzzy metric space Λ,D,∗, α:Λ×Λ×0,∞⟶0,∞, ψ∈Ψ. Assume that ℒ:U⟶V is an α-admissible proximal mapping such that(86)αϑ,θ,ς≥1,Du,ℒϑ,ς=DU,V,ς,Dv,ℒθ,ς=DU,V,ς⟹Du,v,ς≥ψDϑ,θ,ς,for all u,v,ϑ,θ∈U and ς>0. Assume also that(i)
ℒ is triangular weak-α-admissible(ii)
U is closed(iii)
ℒU0⊆V0(iv)
There existsϑ0,ϑ1∈U such that Dϑ0,ℒϑ0,ς=DU,V,ς and αϑ0,ϑ1,ς≥1 for all ς>0(v)
ℒ is continuous or C holds
Then, there existsω∈U such that Dω,ℒω,ς=DU,V,ς for all ς>0.Proof.
Defineξ:0,1×0,1⟶ℝ by(87)ξμ,ν=1ψν−1μ,for allμ,ν∈0,1.
Sinceξ∈FZ, Theorem 4 leads to the desired results.Corollary 2.
LetU and V be nonempty subsets of a complete fuzzy metric space Λ,D,∗, α:Λ×Λ×0,∞⟶0,∞, η∈ℋ. Assume that ℒ:U⟶V is an α-admissible proximal mapping such that(88)αϑ,θ,ς≥1,Du,ℒϑ,ς=DU,V,ς,Dv,ℒθ,ς=DU,V,ς⟹ηDu,v,ς≤aηDϑ,θ,ς,for all u,v,ϑ,θ∈U and ς>0. Assume also that(i)
ℒ is triangular weak-α-admissible(ii)
U is closed(iii)
ℒU0⊆V0(iv)
There existsϑ0,ϑ1∈U such that Dϑ0,ℒϑ0,ς=DU,V,ς and αϑ0,ϑ1,ς≥1 for all ς>0(v)
ℒ is continuous or C holds
Then, there existsω∈U such that Dω,ℒω,ς=DU,V,ς for all ς>0.Proof.
It follows from Theorem4 using the FZ-simulation function ξμ,ν=1/η−1a.ην−1/μ, for all μ,ν∈0,1.Corollary 3.
LetU and V be nonempty subsets of a complete fuzzy metric space Λ,D,∗, α:Λ×Λ×0,∞⟶0,∞. Assume that ℒ:U⟶V is an α-admissible proximal mapping such that(89)αϑ,θ,ς≥1,Du,ℒϑ,ς=DU,V,ς,Dv,ℒθ,ς=DU,V,ς⟹1Du,v,ς−1≤ϕ1Dϑ,θ,ς−1,for all u,v,ϑ,θ∈U and ς>0, where ϕ:0,∞⟶0,∞ with ϕt<t for all t>0 and ϕ0=0. Assume also(i)
ℒ is triangular weak-α-admissible(ii)
U is closed(iii)
ℒU0⊆V0(iv)
There existsϑ0,ϑ1∈U such that Dϑ0,ℒϑ0,ς=DU,V,ς and αϑ0,ϑ1,ς≥1 for all ς>0(v)
ℒ is continuous or C holds.
Then, there existsω∈U such that Dω,ℒω,ς=DU,V,ς for all ς>0, i.e., ℒ has a best proximity point ω∈U.Proof.
It follows from Theorem4 by taking ξμ,ν=ϕ1/ν−1−1/μ+1 for all μ,ν∈0,1.Example 5.
LetΛ=U=V=0,∞ and D be a fuzzy set on Λ2×0,∞ given by Dϑ,θ,ς=minϑ,θ/maxϑ,θ, ς>0, and ∗ is a t-norm given by J∗ı=J.ı for all J,ı∈0,1. Λ,D,∗ is a complete fuzzy metric space. Consider the mappings ℒ:Λ⟶Λ by ℒϑ=ϑ. Define αϑ,θ,ς=1 for all ϑ,θ∈Λ, ς>0, and ξ:0,1×0,1⟶ℝ by(90)ξμ,ν=1ψν−1μ,for allμ,ν∈0,1,whereψc=c.
It is easy to see thatξ∈FZ and ξDℒϑ,ℒθ,ς,Dϑ,θ,ς≥0. Therefore, all conditions of Theorem 4 are satisfied, and x=1 is a fixed point of ℒ.
We must point to the fact that, by defining the control functionξ and the admissible mapping αϑ,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.


## 5. Conclusion

This paper has dealt with aFZ-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.



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