# Common Fixed Point Theorems in a New Fuzzy Metric Space

Authors: Weiquan Zhang; Dong Qiu; Zhifeng Li; Gangqiang Xiong
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We generalize the Hausdorff fuzzy metric in the sense of Rodríguez-López and Romaguera, and we introduce a newM∞-fuzzy metric, where M∞-fuzzy
metric can be thought of as the degree of nearness between two fuzzy sets with respect to any positive real number. Moreover, under ϕ-contraction condition, in the fuzzy metric space, we give some common fixed point theorems for fuzzy mappings.


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## Body

## 1. Introduction

The concept of fuzzy sets was introduced initially by Zadeh [1] in 1965. After that, to use this concept in topology and analysis, many authors have expansively developed the theory of fuzzy sets and application [2, 3]. In the theory of fuzzy topological spaces, one of the main problems is to obtain an appropriate and consistent notion of fuzzy metric space. This problem was investigated by many authors [4–13] from different points of view. George and Veeramani’s fuzzy metric space [6] has been widely accepted as an appropriate notion of metric fuzziness in the sense that it provides rich topological structures which can be obtained, in many cases, from classical theorems. Further, it is necessary to mention that this fuzzy metric space has very important application in studying fixed point theorems for contraction-type mappings [7, 14–16]. Besides that, a number of metrics are used on subspaces of fuzzy sets. For example, the sendograph metric [17–19] and the d∞-metric for fuzzy sets [20–25] induced by the Hausdorff-Pompeiu metric have been studied most frequently, where d∞-metric is an ordinary metric between two fuzzy sets. Combining fuzzy metric (in the sense of George and Veeramani) and Hausdorff-Pompeiu metric, Rodríguez-Lópezand Romaguera [26] construct a Hausdorff fuzzy metric, where Hausdorff fuzzy metric can be thought of as the degree of nearness between two crisp nonempty compact sets with respect to any positive real number.In this present investigation, considering the Hausdorff-Pompeiu metric and theories on fuzzy metric spaces (in the sense of George and Veeramani) together, we study the degree of nearness between two fuzzy sets as a natural generalization of the degree of nearness between two crisp sets, in turn, it helps in studying new problems in fuzzy topology. Based on the Hausdorff fuzzy metricHM, we introduce a suitable notion for the M∞-fuzzy metric on the fuzzy sets whose λ-cut are nonempty compact for each λ∈[0,1]. In particular, we explore several properties of M∞-fuzzy metric. Then, under ϕ-contraction condition, we give some common fixed point theorems in the fuzzy metric space on fuzzy sets.


## 2. Preliminaries

According to [27], a binary operation *:[0,1]×[0,1]→[0,1] is called a continuous t-norm if ([0,1],*) is an Abelian topological semigroups with unit 1 such that a*b≤c*d whenever a≤c and b≤d for all a,b,c,d∈[0,1].Definition 2.1 (see [6]).
The 3-tuple(X,M,*) is said to be a fuzzy metric space if X is an arbitrary set, * is a continuous t-norm, and M is a fuzzy set on X2×(0,∞) satisfying the following conditions, for all x,y,z∈X,t,s>0:(i)
M(x,y,t)>0;(ii)
M(x,y,t)=1 if and only if x=y;(iii)
M(x,y,t)=M(y,x,t);(iv)
M(x,z,t+s)≥M(x,y,t)*M(z,y,s);(v)
M(x,y,-):(0,∞)→[0,1] is continuous.
If(X,M,*) is a fuzzy metric space, it will be said that (M,*) is a fuzzy metric on X.A simply but useful fact [7] is that M(x,y,-) is nondecreasing for all x,y∈X. Let (X,d) be a metric space. Denote by a·b the usual multiplication for all a,b∈[0,1], and let Md be the fuzzy set defined on X×X×(0,∞) by(2.1)Md(x,y,t)=tt+d(x,y).
Then, (X,Md,·) is a fuzzy metric space, and (Md,·) is called the standard fuzzy metric induced by d [8].George and Veeramani [6] proved that every fuzzy metric (M,*) on X generates a topology τM on X which has a base the family of open sets of the form:(2.2){BM(x,ε,t):x∈X,0<ε<1,t>0},
where BM(x,ε,t)={y∈X:M(x,y,t)>1-ε} for all ε∈(0,1) and t>0. They proved that (X,τM) is a Hausdorff first countable topological space. Moreover, if (X,d) is a metric space, then the topology generated by d coincides with the topology τMd generated by the induced fuzzy metric (Md,*) (see [8]).Lemma 2.2 (see [6]).
Let(X,M,*) be a fuzzy metric space and let τ be the topology induced by the fuzzy metric. Then, for a sequence {xn}n∈ℕ in X, xn→x if and only if M(xn,x,t)→1 as n→∞ for all t>0.Definition 2.3 (see [6]).
A sequence{xn}n∈ℕ in a fuzzy metric space (X,M,*) is called a Cauchy sequence if and only if for each 1>ε>0,t>0, there exists n0∈ℕ such that M(xn,xm,t)>1-ε for all n,m≥n0. A fuzzy metric space is said to be complete if and only if every Cauchy sequence is convergent.Definition 2.4 (see [13]).
LetA be a nonempty subset of a fuzzy metric space (X,M,*). For a∈X and t>0, M(a,A,t)=sup{M(a,y,t)∣y∈A,t>0}.Lemma 2.5 (see [28]).
LetG be a set and let {Gα:α∈[0,1]} be a family of subsets of G such that(1)
G0=G;(2)
α≤β implies Gβ⊆Gα;(3)
α1≤α2≤⋯, limn→∞αn=α implies Gα=⋂n=1∞Gαn.Then, the function φ:G→[0,1] defined by φ(x)=sup{α∈[0,1]:x∈Gα} has the property that {x∈G:φ(x)≥α}=Gα  for every α∈[0,1].Next, we recall some pertinent concepts on Hausdorff fuzzy metric. Denote byC0(X) the set of nonempty closed and bounded subsets of a metric space (X,d). It is well known (see, e.g., [29]) that the function Hd defined on C0(X)×C0(X) by(2.3)Hd(A,B)=max⁡{sup⁡a∈A⁡d(a,B),sup⁡b∈B⁡d(A,b)},
for all A,B∈C0(X), is a metric on C0(X) called the Hausdorff-Pompeiu metric. In [30], it is proved that the metric (C(X),H) is complete provided X is complete.LetC(X) be the set of all nonempty compact subsets of a fuzzy metric space (X,M,*), A,B∈C(X), t>0, according to [26], the Hausdorff fuzzy metric HM on C(X)×C(X)×(0,∞) is defined as(2.4)HM(A,B,t)=min⁡{inf⁡a∈AM(a,B,t),inf⁡b∈BM(A,b,t)}=min⁡{ρ(A,B,t),ρ(B,A,t)},
where ρ(A,B,t)=infa∈AM(a,B,t), and (HM,*) is a fuzzy metric on C(X). It is shown that ρ(A,B,t)=1 if and only if A⊆B, and HM(A,B,t)=1 if and only if A=B.Lemma 2.6 (see [26]).
Let(X,M,*) be a fuzzy metric space. Then, (C(X),HM,*) is complete if and only if (X,M,*) is complete.Lemma 2.7 (see [26]).
Let(X,d) be a metric space. Then, the Hausdorff fuzzy metric (HMd,·) of the standard fuzzy metric (Md,·) coincides with standard fuzzy metric (MHd,·) of the Hausdorff metric Hd on C(X).


## 3. OnM∞-Fuzzy Metric

Let(X,M,*) be a fuzzy metric space. Denote by 𝒞(X) the totality of fuzzy sets:(3.1)μ:X⟶[0,1]=I,
which satisfy that, for each λ∈I, the λ-cut of μ,(3.2)[μ]λ={x∈X:μ(x)≥λ},
is nonempty compact in X.Definition 3.1.
Let(X,M,*) be a fuzzy metric space. The M∞-fuzzy metric between two fuzzy sets is induced by the Hausdorff fuzzy metric HM as
(3.3)M∞(μ1,μ2,t)=min⁡{ρ∞(μ1,μ2,t),ρ∞(μ2,μ1,t)},
where μ1,μ2∈𝒞(X),t>0, and
(3.4)ρ∞(μ1,μ2,t)=inf⁡0≤λ≤1ρ([μ1]λ,[μ2]λ,t)
is the fuzzy separation of μ1 from μ2.Lemma 3.2.
Let(X,M,*) be a fuzzy metric space, μ1,μ2,μ3∈𝒞(X), s,t>0. Then one has(1)
M∞(μ1,μ2,t)∈(0,1],(2)
M∞(μ1,μ2,t)=M∞(μ2,μ1,t),(3)
ρ∞(μ1,μ2,t)=1 if and only if μ1⊆μ2,(4)
M∞(μ1,μ2,t)=1 if and only if μ1=μ2,(5)
ifμ1⊆μ2, then ρ∞(μ1,μ3,t+s)≥M∞(μ2,μ3,t),(6)
ρ∞(μ1,μ3,t+s)≥M∞(μ1,μ2,t)*ρ∞(μ2,μ3,s),(7)
M∞(μ1,μ3,t+s)≥M∞(μ1,μ2,t)*M∞(μ2,μ3,s),(8)
M∞(μ1,μ2,-):(0,∞)→[0,1] is continuous.Proof.
For (1), by the definition of theλ-cut [μ1]λ, for every λ∈I, [μ1]λ is nonempty compact in X. By the theorem of nested intervals, there exists a point a0 in [μ1]λ for every λ∈I, likewise, there exists a points b0 in [μ2]λ for every λ∈I. Thus, M∞(μ1,μ2,t)>0. Moreover, it is clear that A=B⇔HM(A,B,t)=1⇔M∞(μ1,μ2,t)=1.
For (2), it is clear thatM∞(μ1,μ2,t)=M∞(μ2,μ1,t).
For (3), sinceρ∞(μ1,μ2,t)=1 if and only if ρ([μ1]λ,[μ2]λ,t)=1 for all λ∈I, which implies [μ1]λ⊆[μ2]λ for all λ∈I, we have that ρ∞(μ1,μ2,t)=1 if and only if μ1⊆μ2.
For (4), it follows from (3).
For (5), for everyλ∈I, any x∈[μ1]λ, y∈[μ2]λ and z∈[μ3]λ, by the proof of Theorem  1 in [26], we have
(3.5)M(x,z,t+s)≥M(x,y,t)*M(y,z,s)
with all x,y,z∈X, which implies
(3.6)M(x,[μ3]λ,t+s)≥M(x,y,t)*M(y,[μ3]λ,s)
for all x∈[μ1]λ and all y∈[μ2]λ. Since μ1⊆μ2, then ρ([μ1]λ,[μ2]λ,s)=1. By  (iv) of Definition 2.1 and the arbitrariness of x and y, we have
(3.7)ρ([μ1]λ,[μ3]λ,t+s)=inf⁡x∈[μ1]λM(x,[μ3]λ,t+s)≥inf⁡x∈[μ1]λM(x,[μ2]λ,s)*inf⁡y∈[μ2]λM(y,[μ3]λ,t)=ρ([μ1]λ,[μ2]λ,s)*ρ([μ2]λ,[μ3]λ,t)=ρ([μ2]λ,[μ3]λ,t)≥HM([μ2]λ,[μ3]λ,t),
which implies
(3.8)inf⁡0≤λ≤1ρ([μ1]λ,[μ3]λ,t+s)≥inf⁡0≤λ≤1HM([μ2]λ,[μ3]λ,t).
Consequently, ρ∞(μ1,μ3,t+s)≥M∞(μ2,μ3,t).
For (6), for everyλ∈I, by the proof of (5) and (iv) of Definition 2.1, we have
(3.9)inf⁡0≤λ≤1ρ([μ1]λ,[μ3]λ,t+s)≥inf⁡0≤λ≤1{ρ([μ1]λ,[μ2]λ,t)*ρ([μ2]λ,[μ3]λ,s)}≥inf⁡0≤λ≤1{HM([μ1]λ,[μ2]λ,t)*ρ([μ2]λ,[μ3]λ,s)}.
Consequently, ρ∞(μ1,μ3,t+s)≥M∞(μ1,μ2,t)*ρ∞(μ2,μ3,s).
For (7), for everyλ∈I, by the proof of (6), we have
(3.10)inf⁡0≤λ≤1ρ([μ1]λ,[μ3]λ,t+s)≥inf⁡0≤λ≤1{ρ([μ1]λ,[μ2]λ,t)*ρ([μ2]λ,[μ3]λ,s)}≥{inf⁡0≤λ≤1ρ([μ1]λ,[μ2]λ,t)}*{inf⁡0≤λ≤1ρ([μ2]λ,[μ3]λ,s)}.
Similarly, it can be shown that
(3.11)inf⁡0≤λ≤1ρ([μ3]λ,[μ1]λ,t+s)≥{inf⁡0≤λ≤1ρ([μ3]λ,[μ2]λ,s)}*{inf⁡0≤λ≤1ρ([μ2]λ,[μ1]λ,t)}.
Hence, M∞(μ1,μ3,t+s)≥M∞(μ1,μ2,t)*M∞(μ2,μ3,s).
For (8), by the continuity on(0,∞) of the function t↦HM(A,B,t), it is clear that M∞(μ1,μ2,-):(0,∞)→[0,1] is continuous.Theorem 3.3.
Let(X,M,*) be a fuzzy metric space. Then, (𝒞(X),M∞,*) is a fuzzy metric space, where M∞ is a fuzzy set on the 𝒞(X)×𝒞(X)×(0,+∞).Proof.
It is easily proved by Lemma3.2.Example 3.4.
Letd be the Euclidean metric on ℝ, and let A=[a1,a2] and let B=[b1,b2] be two compact intervals. Then, Hd(A,B)=max{|a1-b1|,|a2-b2|}. Let (ℝ,Md,*) be a fuzzy metric space, where a*b the usual multiplication for all a,b∈[0,1], and Md is defined on ℝ×ℝ×(0,∞) by
(3.12)Md(x,y,t)=tt+d(x,y).
Denote by 𝒞(ℝ) the totality of fuzzy sets μ:ℝ→[0,1] which satisfy that for each λ∈I, the λ-cut of μ[μ]λ={x∈ℝ:μ(x)≥λ} is a nonempty compact interval. For any λ-cuts of fuzzy sets μ1,μ2∈𝒞(ℝ) and for all t>0, by a simple calculation, we have
(3.13)HM([μ1]λ,[μ2]λ,t)=tt+Hd([μ1]λ,[μ2]λ).
So by Definition 3.1, we get
(3.14)M∞(μ1,μ2,t)=inf⁡0≤λ≤1tt+Hd([μ1]λ,[μ2]λ).


## 4. Properties of theM∞-Fuzzy Metric

Definition 4.1.
Let(𝒞(X),M∞,*) be a fuzzy metric space. For t∈(0,+∞), define B(μ,r,t) with center a fuzzy set μ∈𝒞(X) and radius r,0<r<1,t>0 as
(4.1)B(μ,r,t)={γ∈C(X)∣M∞(μ,γ,t)>1-r}.Proposition 4.2.
EveryB(μ,r,t) is an open set.Proof.
It is identical with the proof in [6].Proposition 4.3.
Let(𝒞(X),M∞,*) be a fuzzy metric space. Define τM∞={𝒜⊂𝒞(X)∣μ∈𝒜ifandonlyifthereexistt>0andr,0<r<1suchthatB(μ,r,t)⊂𝒜}.
Then,τM∞ is a topology on 𝒞(X).Proof.
It is identical with the proof in [6].Definition 4.4.
A sequence{μn} in a fuzzy metric space (𝒞(X),M∞,*) is a Cauchy sequence if and only if for each ε>0, t>0, there exists n0∈ℕ such that M∞(μn,μm,t)>1-ε for all n,m≥n0.Lemma 4.5.
Let(𝒞(X),M∞,*) be a fuzzy metric space on fuzzy metric M∞ and let τ be the topology induced by the fuzzy metric M∞. Then, for a sequence {μn} in 𝒞(X), μn→μ if and only if M∞(μ,μn,t)→1 as n→∞.Proof.
It is identical with the proof of Theorem  3.11 in [6].Theorem 4.6.
The fuzzy metric space(𝒞(X),M∞,*) is complete provided (X,M,*) is complete.Proof.
Let(X,M,*) be a complete fuzzy metric space and let a sequence {μn,n≥1} be a Cauchy sequence in 𝒞(X). Consider a fixed 0<λ<1. Then, {[μn]λ,n≥1} is a Cauchy sequence in (C(X),HM,*), where C(X) denotes all nonempty compact subsets of (X,M,*).
Since(C(X),HM,*) is complete by Lemma 2.6, it follows that [μn]λ→μλ∈C(X). Actually, from the definition of M∞ and the continuity of HM, it is easy to see that [μn]λ→μλ, uniformly in λ∈[0,1].
Now, consider the family{μλ:λ∈[0,1]}, where μ0=X. Take λ≤β, we have
(4.2)ρ(μβ,μλ,t)≥ρ(μβ,[μn]β,t3)*ρ([μn]β,[μn]λ,t3)*ρ([μn]λ,μλ,t3).
Since [μn]β⊆[μn]λ, it follows that ρ([μn]β,[μn]λ,t/3)=1. Thus, for each 0<ε<1, ρ(μβ,μλ,t)≥ρ(μβ,[μn]β,t/3)*ρ([μn]λ,μλ,t/3) if n is large enough. Hence, ρ(μβ,μλ,t)=1, and by Lemma 3.2, we have μβ⊆μλ.
Now, takeλn↑ and limn→∞λn=λ. We have to show that μλ=⋂n=1∞μλn. It is clear that
(*)μλ⊆⋂n=1∞μλn.
On the other hand, we have
(4.3)ρ(⋂n=1∞μλn,μλ,t)≥ρ(⋂n=1∞μλn,⋂n=1∞[μj]λn,t3)*ρ(⋂n=1∞[μj]λn,[μj]λ,t3)*ρ([μj]λ,μλ,t3),
for fixed j. However,
(4.4)ρ(⋂n=1∞[μj]λn,[μj]λ,t3)=1.
Consequently, for every 0<ε<1, there exists 0<ε0<ε<1 such that (1-ε0)*(1-ε0)*(1-ε0)>1-ε. For given ε0, since [μj]λ→μλ, there exists jε0 such that
(4.5)ρ(⋂n=1∞μλn,μλ,t)≥ρ(⋂n=1∞μλn,⋂n=1∞[μj]λn,t3)*(1-ε0),
for j≥jε0. Now,
(4.6)ρ(⋂n=1∞μλn,⋂n=1∞[μj]λn,t3)≥ρ(⋂n=1∞μλn,μλp,t9)*ρ(μλp,[μj]λn,t9)*ρ([μj]λp,⋂n=1∞[μj]λn,t9),
for any p≥1. Since ⋂n=1∞μλn⊆μλp, we obtain
(4.7)ρ(⋂n=1∞μλn,⋂n=1∞[μj]λn,t)≥ρ(μλp,[μj]λp,t2)*ρ([μj]λp,⋂n=1∞[μj]λn,t2).
Now, ρ(μλp,[μj]λp,t/2)>1-ε0 for j≥j0 and all t>0. Note that (since the convergence [μj]λ→μλ is uniform in λ) j0 does not depend on p. Since {[μj]λp,p≥1} decreases to ⋂n=1∞[μj]λn, if follows that ρ([μj]λp0,⋂n=1∞[μj]λn,t/2)>1-ε0 for some p0 (depending on j).
Thus,ρ(⋂n=1∞μλn,⋂n=1∞[μj]λn,t/3)≥(1-ε0)*(1-ε0), if j is large.
Finally, by takingj large enough, we obtain
(4.8)ρ(⋂n=1∞μλn,μλ,t)≥(1-ε0)*(1-ε0)*(1-ε0)≥1-ε,
that is,
(**)⋂n=1∞μλn⊆μλ.
From (4.3) and (4.9), it yields ⋂n=1∞μλn=μλ. Thus, Lemma 2.5 is applicable and there exists μ∈𝒞(X) for every λ∈[0,1] such that [μn]λ→μλ. It remains to show that μn→μ in (𝒞(X),M∞,*).
Letε>0. Then, since {μn} is a Cauchy sequence, there exists nε such that n,m>nε implies M∞(μn,μm,t)>1-ε.
Letn(>nε) be fixed. Then,
(4.9)HM([μn]λ,[μ]λ,t)=lim⁡m→∞HM([μn]λ,[μm]λ,t)≥lim⁡¯m→∞inf⁡0≤λ≤1HM([μn]λ,[μm]λ,t)=lim⁡¯m→∞M∞(μn,μm,t)>1-ε.
Thus, μn→μ in the M∞-fuzzy metric. The proof is completed.Lemma 4.7.
Let(X,M,*) be a compact fuzzy metric space and compact subsets A,B∈C(X). Then, for each x∈A and t>0, there exists a y∈B such that M(x,y,t)≥HM(A,B,t).Proof.
Suppose there exists ax0∈A such that M(x0,y,t)<HM(A,B,t) for any y∈B and t>0. Then,
(4.10)sup⁡y∈BM(x0,y,t)<HM(A,B,t),
that is,
(4.11)sup⁡y∈BM(x0,y,t)<min⁡{inf⁡x∈Bsup⁡y∈AM(x,y,t),inf⁡x∈Asup⁡y∈BM(x,y,t)}.
So,
(4.12)sup⁡y∈BM(x0,y,t)<inf⁡x∈Asup⁡y∈BM(x,y,t).
This is a contradiction with x∈A.Lemma 4.8.
Let(X,M,*) be a compact fuzzy metric space, t>0 and A,B∈C(X). Then, for any compact set A1⊆A, there exists a compact set B1⊆B such that HM(A1,B1,t)≥HM(A,B,t).Proof.
LetC={y: there exists a x∈A1 such that M(x,y,t)≥HM(A,B,t)} and let B1=C⋂B. For any x∈A1⊆A,t>0, by Lemma 4.7, there exists a y∈B such that
(4.13)M(x,y,t)≥HM(A,B,t).
Thus, B1≠∅, moreover, B1 is compact since it is closed in X and B1⊆B.
Now, for anyx∈A1, t>0, there exists a y∈B1 such that
(4.14)M(x,y,t)≥HM(A,B,t).
Thus, we have M(x,B1,t)≥HM(A,B,t), which implies that
(4.15)ρ(A1,B1,t)=inf⁡x∈A1M(x,B1,t)≥HM(A,B,t).
Similarly, it can be shown that ρ(A1,B1,t)≥HM(A,B,t).
Hence,HM(A1,B1,t)≥HM(A,B,t). This completes the proof.Theorem 4.9.
Let(X,M,*) be a compact fuzzy metric space and μ1,μ2∈𝒞(X), t>0. Then, for any μ3∈𝒞(X) satisfying μ3⊆μ1, there exists a μ4∈𝒞(X) such that μ4⊆μ2 and
(4.16)M∞(μ3,μ4,t)≥M∞(μ1,μ2,t).Proof.
Sinceμ1,μ2, and μ3 are normal, we have ∅≠[μ3]λ⊆[μ1]λ and ∅≠[μ2]λ for all λ∈I. Let
(4.17)Cλ={y:thereexistsax∈[μ3]λsuchthatM(x,y,t)≥M∞(μ1,μ2,t)},
and let Bλ=Cλ⋂[μ2]λ. For any x∈[μ3]λ⊆[μ1]λ, by Lemma 4.7, there exists a y∈[μ2]λ such that
(4.18)M(x,y,t)≥HM([μ1]λ,[μ2]λ,t)≥M∞(μ1,μ2,t).
Thus, Bλ is nonempty compact in X, moreover, Bλ⊆Bγ if 0≤γ≤λ≤1.
From the proof of Lemma4.8, we have
(4.19)HM([μ3]λ,Bλ,t)≥M∞(μ1,μ2,t).
By Lemma  3.1 in [28], there exists a fuzzy set μ4 with the property that [μ4]λ=Bλ for λ∈I. Since Bλ are nonempty compact for all λ∈I, we have μ4∈𝒞(X). Consequently,
(4.20)M∞(μ3,μ4,t)≥M∞(μ1,μ2,t).
This completes the proof.Definition 4.10 (see [24]).
LetX,Y be any fuzzy metric space. ℱ is said to be a fuzzy mapping if and only if ℱ is a mapping from the space 𝒞(X) into 𝒞(Y), that is, ℱ(μ)∈𝒞(Y) for each μ∈𝒞(X).


## 5. Common Fixed Point Theorems in the Fuzzy Metric Space on Fuzzy Sets

Theorem 5.1.
Let(X,M,*) be a compact fuzzy metric space and let {ℱi}i=1∞ be a sequence of fuzzy self-mappings of 𝒞(X). Let ϕ:[0,1]→[0,1] be a nondecreasing function satisfying the following condition: ϕ is continuous from the left and
(5.1)ϕ(h)*ϕ2(h)*⋯*ϕn(h)⟶1asn⟶∞,∀h∈(0,1],
where ϕn denote the nth iterative function of ϕ. Suppose that for each μ1,μ2∈𝒞(X), and for arbitrary positive integers i and j,i≠j,t>0,
(5.2)M∞(Fi(μ1),Fj(μ2),t)≥ϕ(12inf⁡{12M∞(μ1,μ2,t),ρ∞(μ1,Fi(μ1),2t),ρ∞(μ2,Fj(μ2),2t),12[ρ∞(μ2,Fi(μ1),4t)+ρ∞(μ1,Fj(μ2),4t)]}),
then there exists μ*∈𝒞(X) such that μ*⊆ℱi(μ*) for all i∈Z+.Proof.
Letμ0∈𝒞(X) and μ1⊆ℱ1(μ0). By Theorem 4.9, for any t>0, there exists μ2∈𝒞(X) such that μ2⊆ℱ2(μ1) and
(5.3)M∞(μ1,μ2,t)≥M∞(F1(μ0),F2(μ1),t).
Again by Theorem 4.9, for any t>0, we can find μ3∈𝒞(X) such that μ3⊆ℱ3(μ2) and
(5.4)M∞(μ2,μ3,t)≥M∞(F2(μ1),F3(μ2),t).
By induction, we produce a sequence {μn} of points of 𝒞(X) such that
(5.5)μn+1⊆Fn+1(μn),n=0,1,2,…;M∞(μn,μn+1,t)≥M∞(Fn(μn-1),Fn+1(μn),t).
Now, we prove that {μn} is a Cauchy sequence in 𝒞(X). In fact, for arbitrary positive integer n, by the inequality (5.2), Lemma 3.2, and the formula (5.5), we have
(5.6)M∞(μn,μn+1,t)≥M∞(Fn(μn-1),Fn+1(μn),t)≥ϕ(12inf⁡{12M∞(μn-1,μn,t),ρ∞(μn-1,Fn(μn-1),2t),ρ∞(μn,Fn+1(μ2),2t),12[ρ∞(μn-1,Fn+1(μn),4t)+ρ∞(μn,Fn(μn-1),4t)]}),≥ϕ(inf⁡{12M∞(μn-1,μn,t),M∞(μn-1,μn,t),M∞(μn,μn+1,2t),12[M∞(μn-1,μn+1,2t)+1]}),≥ϕ(inf⁡{12M∞(μn-1,μn,t),M∞(μn,μn+1,t),M∞(μn,μn+1,2t),12[M∞(μn-1,μn,t)+1*M∞(μn,μn+1,t)+1]}),
where μn⊆ℱn(μn-1) implies ρ∞(μn,ℱn(μn-1),2t)=1, by (3) of Lemma 3.2. In addition, it is easy to get that ϕ(h)>h for all h∈(0,1). In fact, suppose that there exists some t0∈(0,1) such that ϕ(h0)≤h0. Since ϕ is nondecreasing, we have
(5.7)ϕn(h0)≤ϕn-1(h0)≤⋯≤ϕ(h0)≤h0.
Since ϕ(h)*ϕ2(h)*⋯*ϕn(h)→1 as n→∞, for all h∈(0,1), then we have ϕn(h0)→1 as n→∞. From the inequality (5.7), we have 1≤h0. This is a contradiction which implies ϕ(h)>h for all h∈(0,1). We can prove that M∞(μn-1,μn,t)≤M∞(μn,μn+1,t). In fact, if M∞(μn-1,μn,t)>M∞(μn,μn+1,t), then from the inequality (5.6), we get
(5.8)M∞(μn,μn+1,t)≥ϕ(M∞(μn,μn+1,t))>M∞(μn,μn+1,t),
which is a contradiction. Thus, from the inequality (5.6), we have
(5.9)M∞(μn,μn+1,t)≥ϕ(M∞(μn-1,μn,t))≥⋯≥ϕn(M∞(μ0,μ1,t)).
Furthermore, for arbitrary positive integers m and k, we have
(5.10)1≥M∞(μk,μk+m,t)≥M∞(μk,μk+1,tm)*M∞(μk+1,μk+2,tm)*⋯*M∞(μk+m-1,μk+m,tm)≥ϕk(M∞(μ0,μ1,tm))*ϕk+1(M∞(μ0,μ1,tm))*⋯*ϕk+m-1(M∞(μ0,μ1,tm)),
and ϕ(h)*ϕ2(h)*⋯*ϕn(h)→1 as n→∞, for all h∈(0,1), it follows that
(5.11)ϕk(M∞(μ0,μ1,tm))*ϕk+1(M∞(μ0,μ1,tm))*⋯*ϕk+m-1(M∞(μ0,μ1,tm))
is convergent, which implies that {μn} is a Cauchy sequence in 𝒞(X). Since X is a compact fuzzy metric space, it follows X is complete. By Theorem 4.6, 𝒞(X) is complete. Let μn→μ*. Next, we show that μ*⊆ℱi(μ*) for all i∈Z+. In fact, for arbitrary positive integers i and j, i≠j, by Theorem 4.9, we have
(5.12)ρ∞(μ*,Fi(μ*),t)≥M∞(μ*,μj,t4)*ρ∞(μj,Fi(μ*),3t4)≥M∞(μ*,μj,t4)*M∞(Fj(μj-1),Fi(μ*),t2)≥M∞(μ*,μj,t4)*ϕ(inf⁡{M∞(μj-1,μ*,t2),ρ∞(μj-1,Fj(μj-1),t),ρ∞(μ*,Fi(μ*),t),12[ρ∞(μ*,Fj(μj-1),2t)+ρ∞(μj-1,Fi(μ*),2t)]})≥M∞(μ*,μj,t4)*ϕ(inf⁡{M∞(μj-1,μ*,t2),M∞(μj-1,μj,t2),ρ∞(μ*,Fi(μ*),t),12[M∞(μ*,μj,t)+M∞(μ*,μj-1,t)*ρ∞(μ*,Fi(μ*),t)]}),
where μj⊆ℱj(μj-1) implies ρ∞(μj,ℱj(μj-1),t)=1. Letting n→∞,M∞(μn,μ*,t)=1, and using the left continuity of ϕ, we have
(5.13)ρ∞(μ*,Fi(μ*),t)≥ϕ(ρ∞(μ*,Fi(μ*),t)),
which implies ρ∞(μ*,ℱi(μ*),t)=1. Hence, by Lemma 3.2, it follows that μ*⊆ℱi(μ*). Then, the proof is completed.Theorem 5.2.
Let(X,M,*) be a compact fuzzy metric space and let {ℱi}i=1∞ be a sequence of fuzzy self-mappings of 𝒞(X). Suppose that for each μ1,μ2∈𝒞(X), and for arbitrary positive integers i and j, i≠j, t>0,
(5.14)M∞(Fi(μ1),Fj(μ2),t)≥ϕ(M∞(μ1,μ2,t),ρ∞(μ1,Fi(μ1),2t),ρ∞(μ2,Fj(μ2),2t),ρ∞(μ1,Fj(μ2),4t),ρ∞(μ2,Fi(μ1),t)),
where ϕ(h1,h2,h3,h4,h5):(0,1]5→[0,1] is nondecreasing and continuous from the left for each variable. Denote γ(h)=ϕ(h,h,h,a,b), where (a,b)∈{(h*h,1),(1,h*h)}. If
(5.15)γ(h)*γ2(h)*⋯*γn(h)⟶1asn⟶∞,∀h∈(0,1],
where γn denote the nth iterative function of γ, then there exists μ*∈𝒞(X) such that μ*⊆ℱi(μ*) for all i∈Z+.Proof.
Letμ0∈𝒞(X) and μ1⊆ℱ1(μ0). By Theorem 4.9, for any t>0, there exists μ2∈𝒞(X) such that μ2⊆ℱ2(μ1) and
(5.16)M∞(μ1,μ2,t)≥M∞(F1(μ0),F2(μ1),t).
Again by Theorem 4.9, for any t>0, we can find μ3∈𝒞(X) such that μ3⊆ℱ3(μ2) and
(5.17)M∞(μ2,μ3,t)≥M∞(F2(μ1),F3(μ2),t).
By induction, we produce a sequence {μn} of points of 𝒞(X) such that
(5.18)μn+1⊆Fn+1(μn),n=0,1,2,…;M∞(μn,μn+1,t)≥M∞(Fn(μn-1),Fn+1(μn),t).
Now, we prove that {μn} is a Cauchy sequence in 𝒞(X). In fact, for arbitrary positive integer n, by the inequality (5.14), Lemma 3.2, and the formula (5.18), we have
(5.19)M∞(μn,μn+1,t)≥M∞(Fn(μn-1),Fn+1(μn),t)≥ϕ(M∞(μn-1,μn,t),ρ∞(μn-1,Fn(μn-1),2t),ρ∞(μn,Fn+1(μn),2t),ρ∞(μn-1,Fn+1(μn),4t),ρ∞(μn,Fn(μn-1),t))≥ϕ(M∞(μn-1,μn,t),M∞(μn-1,μn,t),M∞(μn,μn+1,t),M∞(μn-1,μn+1,2t),1)≥ϕ(M∞(μn-1,μn,t),M∞(μn-1,μn,t),M∞(μn,μn+1,t),M∞(μn-1,μn,t)*M∞(μn,μn+1,t),1),
where μn⊆ℱn(μn-1) implies ρ∞(μn,ℱn(μn-1),2t)=1 by (3) in Lemma 3.2 Likewise, we have γ(h)>h for all h∈(0,1), t>0. If M∞(μn-1,μn,t)>M∞(μn,μn+1,t), then from the inequality (5.19), we obtain
(5.20)M∞(μn,μn+1,t)≥γ(M∞(μn,μn+1,t))>M∞(μn,μn+1,t),
which is a contradiction. Thus, from the inequality (5.19), we have
(5.21)M∞(μn,μn+1,t)≥γ(M∞(μn-1,μn,t))≥⋯≥γn(M∞(μ0,μ1,t)).
Furthermore, for arbitrary positive integers m and k, we have
(5.22)M∞(μn,μn+1,t)≥ϕ(M∞(μn-1,μn,t),M∞(μn-1,μn,t),M∞(μn,μn+1,t),M∞(μn-1,μn,t)*M∞(μn,μn+1,t),1)≥ϕ(M∞(μn-1,μn,t),M∞(μn-1,μn,t),M∞(μn-1,μn,t),M∞(μn-1,μn,t)*M∞(μn-1,μn,t),1)=γ(M∞(μn-1,μn,t)),M∞(μn,μn+1,t)≥γ(M∞(μn-1,μn,t))≥⋯≥γ(M∞(μ0,μ1,t)).
Furthermore, for arbitrary positive integers m and k, we have
(5.23)1≥M∞(μk,μk+m,t)≥M∞(μk,μk+1,tm)*M∞(μk+1,μk+2,tm)*⋯*M∞(μk+m-1,μk+m,tm)≥γk(M∞(μ0,μ1,tm))*γk+1(M∞(μ0,μ1,tm))*⋯*γk+m-1(M∞(μ0,μ1,tm)).
Since ϕ(h)*ϕ2(h)*⋯*ϕn(h)→1 as n→∞, for all h∈(0,1), it follows that
(5.24)γk(M∞(μ0,μ1,tm))*γk+1(M∞(μ0,μ1,tm))*⋯*γk+m-1(M∞(μ0,μ1,tm))
is convergent, this implies that {μn} is a Cauchy sequence in 𝒞(X). Since X is a compact fuzzy metric space, it follows that X is complete. By Theorem 4.6, 𝒞(X) is complete. Let μn→μ*. Now, we show that μ*⊆ℱi(μ*) for all i∈Z+. In fact, for arbitrary positive integers i and j, i≠j, by Theorem 4.9, we have
(5.25)ρ∞(μ*,Fi(μ*),t)≥M∞(μ*,μj,t4)*ρ∞(μj,Fi(μ*),3t4)≥M∞(μ*,μj,t4)*M∞(Fj(μj-1),Fi(μ*),t2)≥M∞(μ*,μj,t4)*ϕ(M∞(μj-1,μ*,t2),ρ∞(μj-1,Fj(μj-1),t),ρ∞(μ*,Fi(μ*),t),ρ∞(μj-1,Fi(μ*),2t),ρ∞(μ*,Fj(μj-1),t)12)≥M∞(μ*,μj,t4)*ϕ(M∞(μj-1,μ*,t2),M∞(μj-1,μj,t2),ρ∞(μ*,Fi(μ*),t),M∞(μj-1,μ*,t)*ρ∞(μ*,Fi(μ*),t),M∞(μ*,μj,t2)),
where μj⊆ℱj(μj-1) implies ρ∞(μj,ℱj(μj-1),t)=1. Letting n→∞,M∞(μn,μ*,t)=1, and using the left continuity of ϕ, we have
(5.26)ρ∞(μ*,Fi(μ*),t)≥ϕ(1,1,ρ∞(μ*,Fi(μ*),t),ρ∞(μ*,Fi(μ*),t),1)≥γ(ρ∞(μ*,Fi(μ*),t)),
which implies ρ∞(μ*,ℱi(μ*),t)=1. Hence, by Lemma 3.2, it follows that μ*⊆ℱi(μ*), then the proof is completed.Now, we give an example to illustrate the validity of the results in fixed point theory. For simplicity, we only exemplify Theorem5.1, while the example may be similarly constructed for Theorem 5.2.Example 5.3.
Let(𝒞(X),M∞,*) be a fuzzy metric space, where X=[-1,1], Md,HM, and M∞ are the same as in Example 3.4. Then, (𝒞(X),M∞,*) is a compact metric space.
Now, defineϕ:[0,1]→[0,1] as ϕ(x)=x, and define {ℱi}i=1∞ a sequence of fuzzy self-mappings of 𝒞(X) as
(5.27)Fi(μ)=12iμ,foranyμ∈C(X).
For arbitrary positive integersi and j, without loss of generality, suppose i<j. For each μ1,μ2∈𝒞(X), by a routine calculation, we have
(5.28)M∞(Fi(μ1),Fj(μ2),t)=M∞(12iμ1,12jμ2,t)=M∞(μ1,12j-iμ2,2it)≥M∞(μ1,μ2,2it)≥ϕ(M∞(μ1,μ2,t))≥ϕ(12inf⁡{12M∞(μ1,μ2),ρ∞(μ1,Fi(μ1),2t),ρ∞(μ2,Fj(μ2),2t),12[ρ∞(μ2,Fi(μ1),4t)+ρ∞(μ1,Fj(μ2),4t)]}).
Therefore, by Theorem 5.1, we assert that the sequence of fuzzy self-mappings {ℱi}i=1∞ has a common fixed point μ* in 𝒞(X). In fact, it is easy to check that
(5.29)μ*(x)={1,ifx=(0,0,…),0,otherwise.


## 6. Conclusion

So far many authors have made a great deal of work in the Hausdorff-Pompeiu metric [20–25]. To describe the degree of nearness between two crisp sets, Rodrguez-López and Romaguera have defined Hausdorff fuzzy metric. In this paper, we define a new M∞-fuzzy metric, which describes the degree of nearness between two fuzzy sets. Then, some properties on M∞-fuzzy metric are discussed. In addition, in this new circumstances, we give some fixed point theorems which are the important generalizations of contraction mapping principle in functional analysis.The results of the present paper may be applied in different settings. In terms of topology, one can make use of topology in data analysis and knowledge acquisition [31]. For another, topologies corresponding to fuzzy sets are used to detect dependencies of attributes in information systems with respect to gradual rules as in [32]. Furthermore, fuzzy fixed point theory can be used in existence and continuity theorems for dynamical systems with some vague parameters [33, 34]. In addition, this work offers a new tool for the description and analysis of fuzzy metric spaces. It would be possible to obtain more topological properties on the new fuzzy metric space. So, we hope our results contribute to dealing with some problems in practical applications for future study.



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