# Fuzzy Meir-Keeler’s Contraction and Characterization

Authors: Kianoush Fathi Vajargah; Hamid Mottaghi Golshan
Journal: Advances in Mathematical Physics
 (2021)

---

## Abstract

In this study, a fuzzy Meir-Keeler’s contraction theorem for complete FMS based on George and Veeramani idea is established. Then, we characterize fuzzy Meir-Keeler’s contractions as contractive types induced by functions called fuzzyL-function. Moreover, we show that the converse of it is true. Finally, we bring some examples and corollaries certify our results and new improvement.


---

## Body

## 1. Introduction

Fixed point theory and related topics are an active research field with a wide range of applications in mathematics, engineering, chemistry, physics, economics, and computer sciences. Many authors have been studied this theory in hyperstructure spaces alongside the classical metric spaces and normed spaces. Among them, it could be cited probabilistic (and fuzzy) metric spaces. The phrase of fuzzy metric space (FMS), introduced by Kramosil and Michalek [1], then George and Veeramani [2], modified this idea which has applications in quantum particle physics [3] and in the two-slit experiment [4, 5]. Also, the theory of FMS is, in this framework, very disparate from the usual theory of metric best approximation and completion, e.g., see [6] and [7–9], respectively. Grabiec [10] developed and extended fixed point theory to probabilistic metric space. Later on, several authors have participated to the growth of this theory (see [10–17]).In 2006, Lim [18] introduced L-functions (LF) and characterized Meir-Keeler’s contractive as a self-map T on M that satisfies dTp,Tq<φdp,q,p≠q∈X for some LF φ (see also [19]). This characterization prepares it easy to compare such maps with those satisfying Boyd-Wong’s condition (see [20]). Then, Meir-Keeler [21] developed Boyd-Wong’s result as follows
(1)∀ε>0;∃δ>0,ε≤dp,q<ε+δ⇒dTp,Tq<ε.In 2005, Razani ([22], Theorem 2.2) introduced a contraction theorem in FMS. Our main result in this paper is to extend this Theorem to fuzzy Meir-Keeler’s contraction. We assert that if X,M,∗ is a FMS and T on X be a fuzzy Meir-Keeler’s contractive self-mapping, then, T has a unique fixed point in X. Our works are an extension of some recent results that we notice them. Then, we characterize fuzzy Meir-Keeler’s contractive map as a map so that
(2)1MTp,Tq,t−1<φ1Mp,q,t−1,∀p,q∈X,p≠q,t>0,where φ is a fuzzy LF and ∗ is the minimum t-norm, i.e., r∗s=minr,s,r,s∈I≔0,1.


## 2. Preliminaries

In what follows, we mention some reported results, definitions, and examples related to the theory of FMS which are needed. More details and explanations can be followed in [2, 6–9, 23, 24].Definition 1 (see [24]).
At-norm is a function ∗:I×I⟶I such that ∗ is continuous, commutative, associative, s∗1=s,s∈I, and p∗q≤t∗s, where p≤t and q≤s, and p,q,t,s∈I.In the study of probabilistic and FM spaces, the presentation oft-norms was raised by the requirement to assign the triangle inequality (condition (iv) below) in the setting of metric spaces to that of fuzzy metric spaces. Numerous examples of this concept have been cited by various researchers, for instance, one may be found in [14, 15].Definition 2 (George and Veeramani [2]).
X,M,∗ is said FMS where X≠∅ is a set, ∗ is a continuous t-norm, and M on X2×0,+∞⟶I is a function with
(1)
Mp,q,t>0, for all t>0(2)
Mp,q,t=1, for all t>0 iff p=q(3)
Mp,q,t=Mq,p,t(4)
Mp,q,t∗Mq,r,s≤Mp,r,t+s,(5)
Mp,q,⋅:0,+∞⟶I be a continuous fuzzy setwhere p,q,r∈X and t,s>0.
Now, we bring two theorems and definitions that play as key roles in this paper, and we continue the next sections based on these concepts to reach our aims.Theorem 3 (see [2]).
In a FMSX,M,∗, pn⟶p (i.e., pn converges to p) iff,∀t>0,Mpn,p,t⟶1 as n⟶∞.Definition 4 (see [2]).
pnn≥1 is said a Cauchy sequence in X,M,∗ if for all t>0 and ε∈0,1, ∃n0=n0ε,t∈ℕ so that Mpn,pm,t>1−ε whenever n,m≥n0. Also, we call it complete if every Cauchy sequence is convergent.Theorem 5 (see [9, 17, 23]).
In a FMSX,M,∗, M:X×X×0,+∞⟶I is a continuous function.Definition 6 (see [22]).
In a FMSX,M,∗, T on X is said a fuzzy contractive self-mapping (FCM), if
(3)1MTp,Tq,t−1<1Mp,q,t−1,∀p,q∈X,p≠q,t>0.


## 3. Main Results

In this section, we discuss concerning fuzzy Meir-Keeler’s contractive self-mapping. We give a proof of Meir-Keeler’s fixed point theorem in FMS. Here, we consider fuzzy Meir-Keeler’s contraction (FMK) to state our main results.Definition 7.
In a FMSX,M,∗, T on X is a (FMK), if for all ε∈0,1, ∃δ>0 such that for all p≠q∈X,t>0,
(4)Mp,q,t>1ε+δ+1impliesMTp,Tq,t>1ε+1.Remark 8.
Each FMK is a FCM but the inverse is not necessarily true. To prove this claim, we bring the flowing example.Example 1.
It is precise that FMK implies fuzzy contraction but note the inverse is not true because assume that(5)Tp=3p>1,4p=0.and Md defined as Example 2.9 of [2] with ∗=min. If ∣p∣>1 and q=0 then Tp=3 and Tq=4. Thus, ∣Tp−Tq∣=1<∣p∣=∣p−q∣. Therefore, we get ∣Tp−Tq∣<∣p−q∣, for all p,q,p≠q, i.e., T is a FCM. But if ∣p∣>1, q=0, and ε=1/2, then, there exists δ>0 so that
(6)Mp,q,t=tt+∣p∣>11+1/2+δ,holds. Hence, if T be a FMK, we easily have that
(7)MTp,Tq,t=tt+∣Tp−Tq∣=tt+1>11+1/2,for all p≠q∈X. It means that t>2, which is a contradiction. Thus, T does not satisfy FMK, and this proves our claim.Theorem 9.
IfX,M,∗ is a complete FMS, where t-norm is defined as ∗=min. Suppose self-mapping T on X is a FMK. Then, T has a unique fixed point.Proof.
For allp≠q∈X and t>0, we get MTp,Tq,t>Mp,q,t. Since if there are p0,q0∈X and t0>0 so that MTp0,Tq0,t0≤Mp0,q0,t0. For simplicity, we put M2=MTp0,Tq0,t0 and M1=Mp0,q0,t0. Hence, for each δ>0(8)M1≥M2≥M21+δM2=11/M2−1+δ+1.
Then, by (4), M1>M2, which is a contradiction. Let p0∈X. Set pn+1=Tpn,n∈ℕ and suppose that pn+1≠pn,∀n∈ℕ, since otherwise T has a fixed point. Assume that t>0 be arbitrary and fixed after choosing. Now, let cn=1/Mpn+1,pn,t−1. cn is a nonincreasing sequence. So, cnconverges to η. Thus, ∃N∈ℕ such that, ∣cn−η∣<r, for all n≥N. We claim that η=0. If 0<η, there is d>0 such that, for all p,q∈X and t>0, where Mp,q,t>1/1+η+d, we have MTp,Tq,t>1/1+η. Select r>0 such that d/2−r,d/2+r⊂0,+∞. We have
(9)∣d/2−cn−η−d/2∣<r,thus
(10)d2−cn−η∈d2−r,d2+r,so 1/Mpn+1,pn,t−1−η<d. It means that Mpn+1,pn,t>1/1+η+d. Hence by 2, Mpn+2,pn+1,t>1/1+η, which is a contradiction, since cj=1/Mpj+1,pj,t−1≥η, for all j∈ℕ.
Now, we prove thatpn is a Cauchy sequence. If pn is not, then there is 0<ε0<1, so that, for any m∈ℕ, there are im,jm>m such that Mpim,pjm,t≤1−ε0. For each 0<e<ε0, there is d>0 such that for any p,q∈X and t>0, if Mp,q,t>1/1+e+d then MTp,Tq,t>1/1+e. There exists M>0 such that for all i≥M and t>0, Mpi,pi+1,t≥1−e+d/1+e+d. Take jM>iM≥M such that
(11)MpjM,piM,t≤1−ε0.
Then, we get(12)MpiM−1,piM+1,t>MpiM−1,piM,t2∗MpiM,piM+1,t2≥1−e+d1+e+d∗1−e+d1+e+d=11+e+d.
It presents that1/MpiM−1,piM+1,t−1<e+d. Thus, by (4), 1/MpiM,piM+2,t−1<e. Also,
(13)MpiM−1,piM+2,t>MpiM−1,piM,t2∗MpiM,piM+2,t2≥1−e+d1+e+d∗11+e=11+e+d.
In other words,MpiM−1,piM+2,t>1/1+e+d. Thus, by (4), MpiM,piM+3,t>1/1+e. By induction, MpiM,pjM,t>1/1+ε0. By (11), 1−ε0>1/1+ε0, i.e., 1−ε02>1, which is a contradiction. Thus, pn is convergent to p∗∈X. We get
(14)Mpn+1,Tp∗,t=MTpn,Tp∗,t>Mpn,p∗,t⟶1.
Then,Mpn+1,Tp∗,t⟶1. So, Tp∗=p∗.
For proving the uniqueness ofp∗, suppose there is a q∗≠p∗ with Tq∗=q∗, it follows that
(15)Mp∗,q∗,t=MTp∗,Tq∗,t>Mp∗,q∗,t.
This is a contradiction, then, the uniqueness is proved.
Note that Theorem9 is a generalization of ([22], Theorem 2.2); when we consider t-norm ∗=min, this shows one of the most reason of improvement of [22].☐


## 4. Characterization of Fuzzy Meir-Keeler’s Contractions

In this part of the paper, we characterize FMK maps. In Theorem11, we provided a sufficient and necessary condition for FMK maps by tools of fuzzy L-function. Also, this generalizes Theorem 1 of [21]. More precisely, we show that if a self-mapping T on X be a FMK then there is a fuzzy L-function ϕ from 0,+∞ into itself such that, for all t>0,p≠q∈X, 1/MTp,Tq,t−1<ϕ1/Mp,q,t−1. Also, we show that the converse of it is true. In some sense, our work is very close to Suzuki [19], and Lim [18].Definition 10 (see [18]).
Functionφ from 0,+∞ into itself is said to be a fuzzy LF if φ−10=0 and ∀s∈0,+∞,∃δ>0 so that φt≤s,∀t∈s,s+δ.
In the following, we bring a theorem to show the condition of reaching a self-mapT on X in a FMS to FMK.Theorem 11.
SupposeX,M,∗ is a FMS, and the t-norm is defined as ∗=min. Then, a self-map T on X is FMK iff there exists a (nondecreasing) fuzzy LF φ as (2) is satisfied.Proof.
Assume thatT is a FMK. By Definition 7, let a function η from 0,1 into 0,+∞ is defined such that
(16)1Mp,q,t−1<ε+2ηεimplies1MTp,Tq,t−1<ε,for ε∈0,1. With such η, let present a nondecreasing function κ from 0,1 into 0,+∞ by
(17)κt=infε>0:t≤ηε+ε,for any t∈0,1,p,q∈X,p≠q. Since t≤t+ηt, we get κt≤t for t∈0,+∞. Suppose that the function φ¯:0,+∞⟶0,+∞ is defined by
(18)φ¯t=κtifminε>0:t≤ηε+εexists,wheret>0,0ift=0,t+κt2otherwise.
It is obvious thatφ¯0=0. 0<φ¯s≤s for s∈0,+∞. Let s∈0,+∞ be fixed. If φ¯t≤s, where t∈s,s+ηs, one can put δ=κs. Otherwise, ∃σ∈s,s+ηs with s<φ¯σ. But σ≤s+ηs, then, we get κσ≤s, and if κσ=s then
(19)φ¯σ=κσ=s.
Thus, we gets=φ¯σ. This is a contradiction. Thus, it is concluded that
(20)κσ<s<φ¯σ=σ+κσ2.
Now, we shall selectu∈κσ,s with σ≤u+ηu, and let δ=s−u>0. Consider t∈s,s+δ. Since
(21)u+ηu≥σ=2σ+κσ2−κσ>2s−u=δ+s≥t,we obtain κt≤u. Therefore
(22)s=u+s+δ2≥κt+t2≥φ¯t.
Hence,φ¯ is a fuzzy LF. If we consider p,q∈X with p≠q and fixed. The definition of φ¯ implies that ∀t>0 there is ε∈0,φ¯t in which t≤ηε+ε. Thus, ∃ε∈0,φ1/Mp,q,l−1 where 1/Mp,q,l−1≤ε+ηε. Therefore,
(23)1MTp,Tq,l−1≤ε≤φ¯1Mp,q,l−1,holds. Therefore, φ¯ satisfies (2). We define function φ¯¯ as
(24)φ¯¯t=supφ¯s:s≤t,for any t∈0,1, we get
(25)0<φ¯t≤φ¯¯t≤t,∀t∈0,1.
Hence,φ¯¯ also satisfies (2). Easily can be verified that φ¯¯ is a nondecreasing fuzzy LF. This completes the proof.☐Considering the following example, we briefly explain the Theorems9 and 11.Example 2.
LetX=I∪3n,3n+1n∈ℕ with usual distance on ℝ and T on X be defined as follows:
(26)Tp=p20≤p≤1,1−1n+2p=3n+1,0p=3n.
Let we setM as Example 2.9 of [2] and ∗=min then,
(27)δε=2ε0<ε≤12,112<ε<1.
Therefore,T satisfies the conditions of Theorem 9. Hence, T has a (unique) fixed point in X. Also, if we consider
(28)φt=23t0≤t≤34,2t−134<t≤1,11≤t<∞.
Thus,φ is a fuzzy LF and
(29)1MTp,Tq,t−1=∣Tp−Tq∣t,≤φ∣p−q∣t,=φ1Mp,q,t−1.
In this step, we can easily reach to the following corollaries.Corollary 12.
SupposeX,M,∗ is a FMS, where ∗=min and T are a self-mapping on X. If there exists a fuzzy L-function φ:0,+∞⟶0,+∞ where (2) is satisfied, then, T has a unique fixed point.Proof.
Letε>0 has given. By Definition 10 there is δ>0 such that for all t∈ε,ε+δ, φt≤ε. It means that if
(30)ε≤1Mp,q,t−1<ε+δ,then by (2)
(31)1MTp,Tq,t−1<φ1Mp,q,t−1<ε.
It means that (4) holds. Thus, T has a (unique) fixed point.☐Based on theorems in Section3 and 4, we have the following result:Corollary 13.
Suppose thatX,M,∗ be a FMS where ∗=min and T is a self-mapping on X. The followings statements are equivalent:
(1)
T is a FMK(2)
There is a fuzzy LFφ such that (2) satisfiesProof.
By Theorem11 and Corollary 12, we easily obtain the desired results.☐


## 5. Conclusion

In this paper motivated by the results of Razani [22], a new class of FMK contractions in a complete FMS was introduced by reducing the contractive condition of the so-called Meir-Keeler’s contractive maps. In Theorems 9, we established a fixed point theorem; and in Theorem 11, we provided a sufficient and necessary condition for fuzzy FMK maps. Our work generalizes Theorem 1.1 of [22] and Theorem 1 of [21].



---

*Source: 9971505-2021-07-28.xml*