# On Common Fixed Point Theorems in the Stationary Fuzzy Metric Space of the Bounded Closed Sets

Authors: Dong Qiu; Chongxia Lu; Wei Zhang; Yuzhong Mo
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

Under theϕ-contraction conditions, we prove common fixed point theorems for self-mappings in the space 𝒞ℬ(X) of the bounded closed sets in the complete stationary fuzzy metric space with the HM-fuzzy metric for the bounded closed sets.


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## Body

## 1. Introduction

It is well known that not only the Hausdorff metric is very important concept in general topology and analysis, but also many authors have expansively developed the theory of fuzzy sets and application (see [1–11]). As a natural generalization of the concept of set, fuzzy sets were introduced initially by Zadeh [12] in 1965. Various concepts of the fuzzy metrics on ordinary set were considered in [13–19].In [20], Rodríguez-López and Romaguera introduced and discussed a suitable notion for the Hausdorff fuzzy metric of a given fuzzy metric space (in the sense of George and Veeramani) on the set of its nonempty compact subsets. It is necessary to note that such fuzzy metric space has very important application in studying fixed point theorems for contraction-type mappings [21–30]. In fuzzy functional analysis, many researches have been done on the fixed point theory in the space of compact fuzzy sets equipped with the supremum metric [1, 16, 31–38].In this paper, we will establish the completeness of(𝒞ℬ(X),HM,*) with respect to the completeness of the stationary fuzzy metric space (X,M,*), where 𝒞ℬ(X) is the class of sets with nonempty bounded closed subsets of X, and HM is the stationary Hausdorff fuzzy metric on 𝒞ℬ(X) induced by M. Finally, we will prove some common fixed point theorems for self-mappings in the space 𝒞ℬ(X).


## 2. Preliminaries

We start this section by recalling some pertinent concepts.Definition 1 (see [39]).
A triangular norm (ort-norm for short) is a binary operation * on the unit interval [0,1], that is, a function *:[0,1]2→[0,1], such that for all a,b,c,d∈[0,1] the following four axioms are satisfied:(1)
a*1=a (boundary condition);(2)
a*b≤c*d whenever a≤c and b≤d (monotonicity);(3)
a*b=b*a (commutativity);(4)
a*(b*c)=(a*b)*c (associativity). A t-norm * is said to be continuous if it is a continuous function in [0,1]2; a t-norm * is said to be positive if a*b>0 whenever a,b∈(0,1]. The following are examples of t-norms: a*Pb=a·b; a∧b=min(a,b), where a·b denotes the usual multiplication for all a,b∈[0,1].Definition 2 (see [40]).
A stationary fuzzy metric space is an ordered triple(X,M,*) such that X is an arbitrary nonempty set, * is a continuous t-norm, and M is a fuzzy set of X×X satisfying the following conditions, for all x,y,z∈X:(1)
M(x,y)>0,(2)
M(x,y)=1 if and only if x=y,(3)
M(x,y)=M(y,x),(4)
M(x,y)≥M(x,z)*M(z,y).If(X,M,*) is a stationary fuzzy metric space, we will say that (M,*) is a stationary fuzzy metric on X. Since a stationary fuzzy metric is a special fuzzy metric, just like fuzzy metrics in [14], we can prove that every stationary fuzzy metric (M,*) on X generates a topology τM on X which has as a base the family of sets of the form {BM(x,ε):x∈X,0<ε<1}, where BM(x,ε)={y∈X:M(x,y)>1-ε} for all ε∈(0,1). A sequence (xi)i∈ℕ in a stationary fuzzy metric space (X,M) is said to be Cauchy if limi,j→∞M(xi,xj)=1; a sequence (xi)i∈ℕ in X converges to x if limi→∞M(xi,x)=1 [40].Example 3 (see [14]).
Let(X,d) be a metric space. Denote by a·b the usual multiplication for all a,b∈[0,1], and define Md on X×X by
(1)Md(x,y)=11+d(x,y)
for all x,y∈X. Then (Md,·) is a stationary fuzzy metric on X which will be called a standard stationary fuzzy metric.Definition 4 (see [41]).
A stationary fuzzy pseudometric space is an ordered triple(X,M,*) such that X is an arbitrary nonempty set, * is a continuous t-norm, and M is a fuzzy set of X×X satisfying the following conditions, for all x,y,z∈X:(1)
M(x,y)=1 if and only if x=y,(2)
M(x,y)=M(y,x),(3)
M(x,y)≥M(x,z)*M(z,y).If(X,M,*) is a stationary fuzzy pseudometric space, we will say that (M,*) is a stationary fuzzy pseudometric on X. In the following we always suppose that the t-norm * is positive.Definition 5 (see [42]).
Let(X,M,*) be a stationary fuzzy metric space and A⊂X. If, for all ε∈(0,1), BM(x,ε)⋂‍(A-{x})≠∅, then x is an accumulation point of A; the set of all accumulation points of A is called the derived set of A, denoted by d(A); the union of A and d(A) is called the closure of A, denoted by A¯. If d(A)⊂A, then A is a closed set of X.Definition 6 (see [42]).
Let(X,M,*) be a stationary fuzzy metric space, and A⊂X. If there exists r∈(0,1) such that for all x,y∈A we have M(x,y)>1-r, then we say that A is a bounded subset of X; if X itself is a bounded set we will say that (X,M,*) is a bounded stationary fuzzy metric space.Given a stationary fuzzy metric space(X,M,*), we will denote by 𝒫(X), 𝒫0(X), and 𝒞ℬ(X), the powerset, the set of nonempty subsets, and the set of nonempty bounded closed subsets of (X,M,*), respectively.LetB be a nonempty subset of a stationary fuzzy metric space (X,M,*). For all x∈X, let
(2)M(x,B)=supy∈BM(x,y)=M(B,x).For the empty index set∅, we will make the convention that, for ax∈[0,1],
(3)supx∈∅ax=0,infx∈∅ax=1.
It follows that M(x,∅)=M(∅,x)=0.Definition 7 (see [43]).
Let(X,M,*) be a stationary fuzzy metric space. For all A,B∈𝒫(X), we define a function HM on 𝒫(X)×𝒫(X) by
(4)HM(A,B)=infx∈AM(x,B)∧infy∈BM(y,A)=ρ(A,B)∧ρ(B,A),
where ρ(A,B)=infx∈AM(x,B).Definition 8.
Let(X,M,*) be any stationary metric space. A0∈𝒞ℬ(X) is said to be a fixed point of a self-mapping f of 𝒞ℬ(X) if and only if A0⊆f(A0).


## 3. Main Results

Now we will establish our main theorems.Proposition 9.
Let(X,M,*) be a stationary fuzzy metric space. Then, for all A,B,C∈𝒫(X),(1)
ρ(A,B)=1if and only ifA⊂B¯if and only ifA¯⊂B¯if and only ifρ(A¯,B¯)=1,(2)
for allx,x′∈X,M(x,B)≥M(x,x′)*M(x′,B),(3)
ρ(A,B)=ρ(A,B¯)=ρ(A¯,B¯),(4)
ρ(A,C)≥ρ(A,B)*ρ(B,C),(5)
ifA⊂B¯, thenρ(A,C)≥HM(B,C),(6)
ρ(A,C)≥HM(A,B)*ρ(B,C),(7)
HM(A,B)=1if and only ifA¯=B¯.Proof.
In fact, we can prove this proposition by a similar proof of Proposition 1 in [42].Theorem 10.
Let(X,M,*) be a stationary fuzzy metric space, and then (𝒫(X),HM,*) is a stationary fuzzy pseudometric space.Proof.
LetA,B,C∈𝒫(X); by the definition of HM, (1) of Proposition 9, and the commutativity of *, it is clear that HM(A,B)=1 if and only if A=B and HM(A,B)=HM(B,A).
In addition, by (4) of Proposition9 and the commutativity of *, we obtain
(5)ρ(A,C)∧ρ(C,A)≥(ρ(A,B)*ρ(B,C))∧(ρ(C,B)*ρ(B,A))≥(ρ(A,B)∧ρ(B,A))*(ρ(B,C)∧ρ(C,B)).
Consequently, by the definition of HM, we get
(6)HM(A,C)≥HM(A,B)*HM(B,C).
We conclude that (𝒫(X),HM,*) is a stationary fuzzy pseudometric space.Proposition 11 (see [42]).
Let(X,M,*) be a stationary fuzzy metric space. If A,B⊂X are any two bounded subsets of X, then A∪B is a bounded subset of X.Theorem 12.
Let(X,M,*) be a stationary fuzzy metric space. Then (𝒞ℬ(X),HM,*) is a stationary fuzzy metric space.Proof.
LetA,B,C∈𝒞ℬ(X). By Proposition 11, we have A∪B∈𝒞ℬ(X), which means there exists r∈(0,1) such that M(x,y)>1-r, for all x∈A, y∈B. Hence, for any x∈A, we can get that
(7)M(x,B)=supy∈BM(x,y)>1-r>0.
Thus we obtain
(8)ρ(A,B)=infx∈AM(x,B)≥(1-r)>0.
Similarly, we can get
(9)ρ(B,A)=infy∈AM(y,A)≥(1-r)>0.
Consequently, by the positivity of *, we have HM(A,B)=ρ(A,B)*ρ(B,A)>0.
By the definition ofHM, (7) of Proposition 9, and the commutativity of *, it is clear that HM(A,B)=1 if and only if A=B and HM(A,B)=HM(B,A). In addition, by (4) of Proposition 9 and the commutativity of *, we obtain
(10)ρ(A,C)∧ρ(C,A)≥(ρ(A,B)*ρ(B,C))∧(ρ(C,B)*ρ(B,A))≥(ρ(A,B)∧ρ(B,A))*(ρ(B,C)∧ρ(C,B)).
Consequently, by the definition of HM, we get
(11)HM(A,C)≥HM(A,B)*HM(B,C).
We conclude that (𝒞ℬ(X),HM,*) is a stationary fuzzy metric space.Example 13.
Let(X,d) be a metric space. Then the Hausdorff stationary fuzzy metric (HMd,·) of the standard fuzzy metric (Md,·) coincides with the standard fuzzy metric (MHd,·) of the Hausdorff fuzzy metric Hd on 𝒞ℬ(X).
In fact, letA,B∈𝒞ℬ(X); for each x0∈A, we have
(12)Md(x0,B)=supy∈BMd(x0,y)=supy∈B11+d(x0,y)=11+infy∈Bd(x0,y)=11+d(x0,B).
Consequently, we obtain
(13)HMd(A,B)=infx∈AMd(x,B)∧infy∈BMd(y,A)=11+supx∈Ad(x,B)∧11+supy∈Bd(y,A)=11+supx∈Ad(x,B)∨supy∈Bd(y,A)=11+Hd(A,B)=MHd(A,B).
We conclude that HMd=MHd on 𝒞ℬ(X).Let us recall that if(X,𝒰) is a uniform space, then the Hausdorff-Bourbaki uniformity H𝒰 (of 𝒰) on 𝒫(X) has as a base the family of sets of the form
(14)HU={(A,B)∈𝒫(X)×𝒫(X):B⊆U(A),A⊆U(B)},
where U∈𝒰 [44].The restriction ofH𝒰 to 𝒞ℬ(X)×𝒞ℬ(X) will be also denoted by H𝒰. In addition, if (X,M,*) is a stationary fuzzy metric space, then {Un:n∈ℕ+} is a (countable) base for the uniformity 𝒰M on X compatible with τM, where
(15)Un={(x,y)∈X×X:M(x,y)>1-1n}
for all n∈ℕ+. 𝒰M is called the uniformity induced by (M,*). In particular, 𝒰HM is the uniformity induced by the Hausdorff stationary fuzzy metric of (M,*). We have the following useful result.Theorem 14.
Let(X,M,*) be a stationary fuzzy metric space. Then the Hausdorff-Bourbaki uniformity H𝒰M coincides with the uniformity 𝒰HM on 𝒞ℬ(X).Proof.
If(A,B)∈{(A,B)∈𝒞ℬ(X)×𝒞ℬ(X):B⊆Un+1(A),A⊆Un+1(B)}, then, for any y∈B⊆Un+1(A), there exists xy∈A⊆Un+1(B), such that M(xy,y)>1-(1/(n+1)). Thus we obtain
(16)M(A,y)=supx∈AM(x,y)≥M(xy,y)>1-1n+1,
for all y∈B⊆Un+1(A). Consequently we have
(17)infy∈BM(A,y)≥1-1n+1.
Similarly, we can get
(18)infx∈AM(x,B)≥1-1n+1,
for all A⊆Un+1(B). Thus we conclude that
(19)HM(A,B)=infx∈AM(x,B)∧infy∈BM(A,y)≥1-1n+1.
If(A,B)∈{(A,B)∈𝒞ℬ(X)×𝒞ℬ(X):HM(A,B)>1-(1/n)}, then, for each x∈A, we have
(20)M(x,B)=supy∈BM(x,y)>1-1n.
Let α=M(x,B), and then, for each x∈A, there exists xy∈B such that
(21)M(x,xy)≥M(x,B)-α+1/n-12=α-1/n+12>1-1n.
Thus we obtain A⊆Un(B).
Similarly, by(22)M(A,y)=supx∈AM(x,y)>1-1n,
we can get B⊆Un(A). It follows that
(23)(A,B)∈{(A,B)∈𝒞ℬ(X)×𝒞ℬ(X):B⊆Un(A),A⊆Un(B)}.
Hence we obtain the following relations:
(24)(A,B)∈{(A,B)∈𝒞ℬ(X)×𝒞ℬ(X):B⊆Un+1(A),fffA⊆Un+1(B)}⊆{(A,B)∈𝒞ℬ(X)×𝒞ℬ(X):HM(A,B)≥1-1n+1}⊆{(A,B)∈𝒞ℬ(X)×𝒞ℬ(X):HM(A,B)>1-1n}⊆(A,B)∈{(A,B)∈𝒞ℬ(X)×𝒞ℬ(X):B⊆Un(A),A⊆Un(B)}.
We conclude thatH𝒰M=𝒰HM on 𝒞ℬ(X).Theorem 15.
Let(X,M,*) be a stationary fuzzy metric space. Then (𝒞ℬ(X),HM,*) is complete if and only if (X,M,*) is complete.Proof.
In fact, we can prove it by a similar proof of Theorem 3 in [20].Another type of convergence for a sequence of sets was defined by Kuratowski.We say that a sequence of sets{Ck}k=1∞, Ck⊆X, converges to a set C⊆X, denoted by limk→∞Ck=C, if
(25)C=liminfCk=limsupCk,
where liminfCk={x∈X:x=limk→∞xk,xk∈Ck}, limsupCk=⋂k=1∞(⋃i=k∞Ci¯).We mention that, for sequence of closed sets, convergence in Hausdorff metric implies convergence in the sense of Kuratoski. But for sequence of bounded closed sets, both types of convergence are equivalent provided that the limit set is nonempty [37].Lemma 16.
Let(X,M,*) be a stationary fuzzy metric space and A,B∈𝒞ℬ(X). Then(1)
for arbitrarilyε∈(0,1) and any x∈A, there exists y∈B such that M(x,y)≥HM(A,B)-ε;(2)
for anyx∈A and any β∈[0,1), there exists y∈B such that M(x,y)≥βHM(A,B).Proof.
We only prove (1) since it is equivalent to (2).
For eachx∈A, there exists y∈B such that, for any ε∈(0,1),
(26)M(x,y)≥supy∈BM(x,y)-ε≥infx∈Asupy∈BM(x,y)-ε=ρ(A,B)-ε≥ρ(A,B)∧ρ(B,A)-ε=HM(A,B)-ε.
This completes the proof.In fact, we can get a more general result.Lemma 17.
Let(X,M,*) be a stationary fuzzy metric space and A,B∈𝒞ℬ(X). Then(1)
for arbitrarilyε∈(0,1) and any closed subset A1⊆A, there exists closed subset B1⊆B such that HM(A1,B1)≥HM(A,B)-ε;(2)
for any closed subsetA1⊆A and any β∈[0,1), there exists closed subset B1⊆B such that HM(A1,B1)≥βHM(A,B).Proof.
We only prove (1) since it is equivalent to (2). Letε∈(0,1) and let
(27)C0={y:thereexistsanx∈A1GsuchthatM(x,y)≥HM(A,B)-ε}.
Assume z∈C0¯. For any δ>0, there exists a y∈C0 such that M(z,y)>1-δ. By (2) of Proposition 9, we have
(28)M(z,A1)≥M(z,y)*M(y,A1)≥(1-δ)*(HM(A,B)-ε).
By the arbitrariness of δ, we have M(z,A1)≥HM(A,B)-ε. Then we get
(29)C0¯⊆D={z:M(z,A1)≥HM(A,B)-ε}.
Conversely, supposez∈D. Take a descending positive number sequence {δn}n=1∞⊆(0,1) such that δn→0 as n→∞. For each δn, there exists a x∈A1 such that
(30)M(z,x)≥(1-δn)*(HM(A,B)-ε).
LetCδn={y:thereexistsanx∈A1suchthatM(x,y)≥(1-δn)*(HM(A,B)-ε)}. Then we have z∈Cδn⊆Cδn¯,n=1,2,…, and Cδi¯⊆Cδj¯(i≥j). Hence we can get
(31)z∈limn→∞Cδn=⋂n=1∞‍Cδn¯=C0¯.
Thus, we obtain C0¯=D. Let B1=B∩D. For each x∈A1⊆A, by Lemma 16, there exists a y∈B such that
(32)M(x,y)≥HM(A,B)-ε.
Consequently, B1 is a nonempty closed subset of B.
For anyx∈A1, there exists y∈B1 such that M(x,y)≥HM(A,B)-ε, which implies that
(33)M(x,B1)=supy∈B1M(x,y)≥HM(A,B)-ε.
Thus we obtain
(34)ρ(A1,B1)=infx∈A1M(x,B1)≥HM(A,B)-ε.
By the definition ofB1, we can get
(35)M(A1,y)=supx∈A1M(x,y)≥HM(A,B)-ε,
for all y∈B1. Thus we obtain
(36)ρ(B1,A1)=infy∈B1M(A1,y)≥HM(A,B)-ε.
Consequently, we easily obtain the following inequality:
(37)HM(A1,B1)=ρ(A1,B1)∧ρ(B1,A1)≥HM(A,B)-ε.
This completes the proof.Lemma 18 (see [29]).
Letϕ:[0,1]→[0,1] be a nondecreasing function satisfying the following conditions:(i)
ϕ is continuous from the left,(ii)
ϕn(h)→1(n→∞)forallh∈(0,1],where ϕn denote the nth iterative function of ϕ. Then(1)
for eachh∈(0,1), such that ϕ(h)>h,(2)
ϕ(1)=1.Theorem 19.
Let(X,M,*) be a complete stationary fuzzy metric space and let {fn}n=1∞ be a sequence of self-mappings of 𝒞ℬ(X). If there exists a constant q∈(1,+∞), such that, for each A1,A2∈𝒞ℬ(X), and for arbitrary positive integers i and j, i≠j,
(38)HM(fi(A1),fj(A2))≥qϕ([ρ(A2,fi(A1))+ρ(A1,fj(A2))]1/2min{[ρ(A2,fi(A1))+ρ(A1,fj(A2))]1/2HM(A1,A2),ρ(A1,fi(A1)),fffffffffffffρ(A2,fj(A2)),fffffffffffff[ρ(A2,fi(A1))+ρ(A1,fj(A2))]1/2}),
where ϕ satisfies the conditions of Lemma 18. Then there exists an A*∈𝒞ℬ(X) such that A*⊆fi(A*), for all i∈ℕ+.Proof.
LetA0,A1∈𝒞ℬ(X) and A1⊆f1(A0), and β=(1/q)∈(0,1). By Lemma 17, there exists A2∈𝒞ℬ(X), such that A2⊆f2(A1) and
(39)HM(A1,A2)≥βHM(f1(A0),f2(A1)).
Again by Lemma 17, we can find A3∈𝒞ℬ(X) such that A3⊆f3(A2) and
(40)HM(A2,A3)≥βHM(f2(A1),f3(A2)).
By induction, we produce a sequence {An} of points of 𝒞ℬ(X) such that
(41)An+1⊆fn+1(An)n=0,1,2,…HM(An+1,An)≥βHM(fn+1(An),fn(An-1)).
Now we prove that {An} is a Cauchy sequence in 𝒞ℬ(X). In fact, for arbitrary positive integer n, by inequality (38) and formula (41), we have
(42)HM(An,An+1)≥βHM(fn(An-1),fn+1(An))≥βqϕ([ρ(An,fn(An-1))+ρ(An-1,fn+1(An))]1/2min{[ρ(An,fn(An-1))+ρ(An-1,fn+1(An))]1/2HM(An-1,An),ffffffffffffddρ(An-1,fn(An-1)),ρ(An,fn+1(An)),ffffffffffffdd[ρ(An,fn(An-1))ffffffffffddddd+ρ(An-1,fn+1(An))]1/2})=ϕ([ρ(An,fn(An-1))+ρ(An-1,fn+1(An))]1/2min{[ρ(An,fn(An-1))+ρ(An-1,fn+1(An))]1/2HM(An-1,An),ρ(An-1,fn(An-1)),ffffffffffffρ(An,fn+1(An)),fffffffffff[ρ(An,fn(An-1))+ρ(An-1,fn+1(An))]1/2})≥ϕ([1+HM(An-1,An+1)]1/2min{[1+HM(An-1,An+1)]1/2HM(An-1,An),HM(An-1,An),ffffffffffffHM(An,An+1),fffffffffff[1+HM(An-1,An+1)]1/2})=ϕ(min{HM(An-1,An),HM(An,An+1)}),
where An⊆fn(An-1), which implies that ρ(An,fn(An-1))=1.
IfHM(An-1,An)∧HM(An,An+1)=HM(An,An+1), then
(43)HM(An,An+1)≥ϕ(HM(An,An+1)).
From An+1⊆fn+1(An)={y:thereexistsx∈Ansuchthatfn+1(x)=y}, it follows that HM(An,An+1)∈(0,1]. Hence, there are two cases.
Case
1. If HM(An,An+1)=1, by (2) of Lemma 18 we can get
(44)HM(An,An+1)=HM(An-1,An)=1,
that is, HM(An,An+1)≥ϕ(HM(An-1,An)).
Case
2. If HM(An,An+1)∈(0,1), by (1) of Lemma 18, we can get
(45)HM(An,An+1)<ϕ(HM(An,An+1)).
Obviously, (43) and (45) are contradictory. Hence, we have
(46)HM(An-1,An)∧HM(An,An+1)=HM(An-1,An);
that is, HM(An,An+1)≥ϕ(HM(An-1,An)).
Consequently, we easily obtain the following inequalities:(47)HM(An+1,An)≥ϕ(HM(An,An-1))≥ϕ2(HM(An-1,An-2))≥⋯≥ϕn(HM(A1,A0)).
Thus, for arbitrary positive integer p, we have
(48)HM(An+p,An)≥HM(An+p,An+p-1)*HM(An+p-1,An+p-2)*⋯*HM(An+1,An)≥(ϕn+p-1*ϕn+p-2*⋯*ϕn)(HM(A1,A0)).
Sinceϕn(h)→1(n→∞), for all h∈(0,1], we get
(49)limn→∞HM(An+p,An)≥limn→∞(ϕn+p-1*ϕn+p-2*⋯*ϕn)(HM(A1,A0))=1,
which implies that limn→∞HM(An+p,An)=1. Hence, {An} is a Cauchy sequence. In addition, since (X,M,*) is a complete stationary fuzzy metric space, by Theorem 15, we get that (𝒞ℬ(X),HM,*) is complete. Thus there exists an A*∈𝒞ℬ(X) such that An→A* as n→∞; that is, limn→∞HM(An,A*)=1.
Next, we show thatA*⊆fi(A*), that is, ρ(A*,fi(A*))=1, for all i∈ℕ+. In fact, for arbitrary positive integers i and j, i≠j, by Proposition 9, we have
(50)ρ(A*,fi(A*))≥HM(A*,Aj)*ρ(Aj,fi(A*))≥HM(A*,Aj)*ρ(Aj,fj(Aj-1))*ρ(fj(Aj-1),fi(A*))=HM(A*,Aj)*ρ(fj(Aj-1),fi(A*))≥HM(A*,Aj)*HM(fj(Aj-1),fi(A*)).
Moreover, we have
(51)HM(fj(Aj-1),fi(A*))≥qϕ([ρ(A*,fj(Aj-1))+ρ(Aj-1,fi(A*))]1/2min{[ρ(A*,fj(Aj-1))+ρ(Aj-1,fi(A*))]1/2HM(Aj-1,A*),ρ(Aj-1,fj(Aj-1)),gggggggffggρ(A*,fi(A*)),ggggggffggg[ρ(A*,fj(Aj-1))+ρ(Aj-1,fi(A*))]1/2})>ϕ(*ρ(A*,fi(A*))]1/2min{ffgggfggff*ρ(A*,fi(A*))HM(A*,Aj)+HM(Aj-1,A*)]1/2HM(Aj-1,A*),HM(Aj-1,Aj),dddggggggdρ(A*,fi(A*)),ffffggggfff[HM(A*,Aj)+HM(Aj-1,A*)ffgggfggggff*ρ(A*,fi(A*))HM(A*,Aj)+HM(Aj-1,A*)]1/2}).
Since ϕ is continuous from the left and * is a continuous positive t-norm, we can obtain
(52)limj→∞HM(fj(Aj-1),fi(A*))≥limj→∞ϕ(*ρ(A*,fi(A*))HM(A*,Aj)+HM(Aj-1,A*)]1/2min{*ρ(A*,fi(A*))HM(A*,Aj)+HM(Aj-1,A*)]1/2HM(Aj-1,A*),ggggggfffffffggHM(Aj-1,Aj),ρ(A*,fi(A*)),ggggffffffffggg[HM(A*,Aj)+HM(Aj-1,A*)ggggfffffgffgggg*ρ(A*,fi(A*))HM(A*,Aj)+HM(Aj-1,A*)]1/2})=ϕ([1+1*ρ(A*,fi(A*))]1/2min{[1+1*ρ(A*,fi(A*))]1/21,1,ρ(A*,fi(A*)),ddddddddd[1+1*ρ(A*,fi(A*))]1/2})=ϕ(ρ(A*,fi(A*))).
Consequently, we conclude that
(53)ρ(A*,fi(A*))≥limj→∞HM(A*,Aj)*limj→∞HM(fj(Aj-1),fi(A*))≥1*ϕ(ρ(A*,fi(A*)))=ϕ(ρ(A*,fi(A*)));
that is, ρ(A*,fi(A*))=1. By (1) of Proposition 9, we obtain A*⊆fi(A*), for all i∈ℕ+.Corollary 20.
Let(X,M,*) be a complete stationary fuzzy metric space and let f be a self-mapping of 𝒞ℬ(X). If there exists a constant q∈(1,+∞), such that, for each A1,A2∈𝒞ℬ(X),
(54)HM(f(A1),f(A2))≥qϕ([ρ(A2,f(A1))+ρ(A1,f(A2))]1/2min{[ρ(A2,f(A1))+ρ(A1,f(A2))]1/2HM(A1,A2),ρ(A1,f(A1)),ρ(A2,f(A2)),gggggggggg[ρ(A2,f(A1))+ρ(A1,f(A2))]1/2}),
where ϕ satisfies the conditions of Lemma 18, then there exists an A*∈𝒞ℬ(X) such that A*⊆f(A*).Proof.
In fact, we can define a sequence of fuzzy self-mappings of𝒞𝒞(X) as fi=f, for i=1,2,…. Thus, this result is a special case of Theorem 19.Theorem 21.
Let(X,M,*) be a complete stationary fuzzy metric space and let {fn}n=1∞ be a sequence of self-mappings of 𝒞ℬ(X). If there exists a constant q∈(1,+∞), such that, for each A1,A2∈𝒞ℬ(X), and for arbitrary positive integers i and j, i≠j,
(55)HM(fi(A1),fj(A2))≥qϕ(HM(A1,A2),ρ(A1,fi(A1)),ρ(A2,fj(A2)),ffffffρ(A2,fi(A1)),ρ(A1,fj(A2))),
where ϕ(h1,h2,h3,h4,h5):(0,1]5→[0,1] is nondecreasing and continuous from the left for each variable, let rn(h)=ϕ(h,h,h,a,b), where (a,b)∈{(h*h,1),(1,h*h)}. If
(56)rn(h)⟶1asn⟶∞,∀h∈(0,1],
where rn denotes the nth iterative function of r, then there exists an A*∈𝒞ℬ(X) such that A*⊆fi(A*), for all i∈ℕ+.Proof.
LetA0,A1∈𝒞ℬ(X) and A1⊆f1(A0), and β=(1/q)∈(0,1). By Lemma 17, there exists A2∈𝒞ℬ(X), such that A2⊆f2(A1) and
(57)HM(A1,A2)≥βHM(f1(A0),f2(A1)).
Again by Lemma 17, we can find A3∈𝒞ℬ(X) such that A3⊆f3(A2) and
(58)HM(A2,A3)≥βHM(f2(A1),f3(A2)).
By induction, we produce a sequence {An} of points of 𝒞ℬ(X) such that
(59)An+1⊆fn+1(An),n=0,1,2,…HM(An+1,An)≥βHM(fn+1(An),fn(An-1)).
Now we prove that {An} is a Cauchy sequence in 𝒞ℬ(X). In fact, for arbitrary positive integer n, by inequality (55) and formula (59), we have
(60)HM(An,An+1)≥βHM(fn(An-1),fn+1(An))≥βqϕ(HM(An-1,An),ρ(An-1,fn(An-1)),ffffffffρ(An,fn+1(An)),ρ(An-1,fn+1(An)),fffffffρ(An,fn(An-1)))=ϕ(HM(An-1,An),ρ(An-1,fn(An-1)),fffffρ(An,fn+1(An)),ρ(An-1,fn+1(An)),fffffρ(An,fn(An-1)))≥ϕ(HM(An-1,An),HM(An-1,An),HM(An,An+1),ffffffHM(An-1,An)*HM(An,An+1),1),
where An⊆fn(An-1), which implies that ρ(An,fn(An-1))=1.
IfHM(An-1,An)∧HM(An,An+1)=HM(An,An+1), then
(61)HM(An,An+1)≥r(HM(An,An+1)).
From
(62)An+1⊆fn+1(An)={y:thereexistsanx∈Ansuchthatfn+1(x)=y},
we get HM(An,An+1)∈(0,1]. Hence, there are two cases.
Case
1. If HM(An,An+1)=1, by (2) of Lemma 18, we can get
(63)HM(An,An+1)=HM(An-1,An)=1;
that is, HM(An,An+1)≥r(HM(An-1,An)).
Case
2. If HM(An,An+1)∈(0,1), by (1) of Lemma 18, we can get
(64)HM(An,An+1)<r(HM(An,An+1)).
Obviously, (61) and (64) are contradictory. Hence, we have
(65)HM(An-1,An)∧HM(An,An+1)=HM(An-1,An);
that is, HM(An,An+1)≥r(HM(An-1,An)).
Consequently, we easily obtain the following relations:(66)HM(An+1,An)≥r(HM(An,An-1))≥r2(HM(An-1,An-2))≥⋯≥rn(HM(A1,A0)).
Thus, for arbitrary positive integer p, we have
(67)HM(An+p,An)≥HM(An+p,An+p-1)*HM(An+p-1,An+p-2)*⋯*HM(An+1,An)≥(rn+p-1*rn+p-2*⋯*rn)×(HM(A1,A0)).
Byrn(h)→1  (n→∞), for all h∈(0,1], and continuity of *, we can get
(68)limn→∞HM(An+p,An)≥limn→∞(rn+p-1*rn+p-2*⋯*rn)(HM(A1,A0))=1,
which implies that limn→∞HM(An+p,An)=1. Hence, we get {An}⊆𝒞ℬ(X) that is a Cauchy sequence. In addition, since (X,M,*) is a complete stationary fuzzy metric space, by Theorem 15, we have that (𝒞ℬ(X),HM,*) is complete. Thus there exists an A*∈𝒞ℬ(X) such that An→A* as n→∞; that is, limn→∞HM(An,A*)=1.
Next, we show thatA*⊆fi(A*), that is, ρ(A*,fi(A*))=1, for all i∈ℕ+. In fact, for arbitrary positive integers i and j, i≠j, by Proposition 9, we have
(69)ρ(A*,fi(A*))≥HM(A*,Aj)*ρ(Aj,fi(A*))≥HM(A*,Aj)*ρ(Aj,fj(Aj-1))*ρ(fj(Aj-1),fi(A*))=HM(A*,Aj)*ρ(fj(Aj-1),fi(A*))≥HM(A*,Aj)*HM(fj(Aj-1),fi(A*)).
Moreover, we have
(70)HM(fj(Aj-1),fi(A*))≥qϕ((A*,fj(Aj-1))HM(Aj-1,A*),ρ(Aj-1,fj(Aj-1)),ggggggρ(A*,fi(A*)),ρ(Aj-1,fi(A*)),ggggggρ(A*,fj(Aj-1)))>ϕ((A*,fj(Aj-1))HM(Aj-1,A*),ρ(Aj-1,fj(Aj-1)),ρ(A*,fi(A*)),ssssssρ(Aj-1,fi(A*)),ρ(A*,fj(Aj-1)))≥ϕ(HM(Aj-1,A*),HM(Aj-1,Aj),ρ(A*,fi(A*)),ffffffHM(Aj-1,A*)*ρ(A*,fi(A*)),HM(A*,Aj)).
Since ϕ is continuous from the left and * is a continuous positive t-norm, hence, we can obtain
(71)limj→∞HM(fj(Aj-1),fi(A*))≥limj→∞ϕ(HM(Aj-1,A*)*ρ(A*,fi(A*)),HM(A*,Aj)HM(Aj-1,A*),HM(Aj-1,Aj),ρ(A*,fi(A*)),fffffffffHM(Aj-1,A*)*ρ(A*,fi(A*)),HM(A*,Aj))=ϕ(1,1,ρ(A*,fi(A*)),ρ(A*,fi(A*)),1)≥ϕ(ρ(A*,fi(A*)),ρ(A*,fi(A*)),ρ(A*,fi(A*)),2222ρ(A*,fi(A*))*ρ(A*,fi(A*)),1)=r(ρ(A*,fi(A*))).
Consequently, we conclude that
(72)ρ(A*,fi(A*))≥limj→∞HM(A*,Aj)*limj→∞HM(fj(Aj-1),fi(A*))≥1*r(ρ(A*,fi(A*)))=r(ρ(A*,fi(A*)));
that is, ρ(A*,fi(A*))=1. By (1) of Proposition 9, we obtain A*⊆fi(A*), for all i∈ℕ+.Corollary 22.
Let(X,M,*) be a complete stationary fuzzy metric space and let f be a self-mapping of 𝒞ℬ(X). If there exists a constant q∈(1,+∞), such that, for each A1,A2∈𝒞ℬ(X),
(73)HM(f(A1),f(A2))≥qϕ(HM(A1,A2),ρ(A1,f(A1)),ρ(A2,f(A2)),ffffgfffρ(A2,f(A1)),ρ(A1,f(A2))),
where ϕ(h1,h2,h3,h4,h5):(0,1]5→[0,1] is nondecreasing and continuous from the left for each variable, let rn(h)=ϕ(h,h,h,a,b), where (a,b)∈{(h*h,1),(1,h*h)}. If
(74)rn(h)⟶1asn⟶∞,∀h∈(0,1],
where rn denotes the nth iterative function of r, then there exists an A*∈𝒞ℬ(X) such that A*⊆f(A*).Proof.
In fact, we can define a sequence of fuzzy self-mappings of𝒞𝒞(X) as fi=f, for i=1,2,…. Thus, this result is a special case of Theorem 21.Theorem 23.
Let(X,M,*) be a complete stationary fuzzy metric space and let {fn}n=1∞ be a sequence of self-mappings of 𝒞ℬ(X). If there exists a constant q∈(1,+∞), such that, for each A1,A2∈𝒞ℬ(X), and, for arbitrary positive integers i and j, i≠j,
(75)HM(fi(A1),fj(A2))≥qϕ(HM(A1,A2)),
where ϕ satisfies the conditions of Lemma 18, then there exists an A*∈𝒞ℬ(X) such that A*⊆fi(A*), for all i∈ℕ+.Proof.
LetA0,A1∈𝒞ℬ(X) and A1⊆f1(A0), and β=(1/q)∈(0,1). By Lemma 17, there exists A2∈𝒞ℬ(X), such that A2⊆f2(A1) and
(76)HM(A1,A2)≥βHM(f1(A0),f2(A1)).
Again by Lemma 17, we can find A3∈𝒞ℬ(X) such that A3⊆f3(A2) and
(77)HM(A2,A3)≥βHM(f2(A1),f3(A2)).
By induction, we produce a sequence {An} of points of 𝒞ℬ(X) such that
(78)An+1⊆fn+1(An),n=0,1,2,…,HM(An+1,An)≥βHM(fn+1(An),fn(An-1)).
Now we prove that {An} is a Cauchy sequence in 𝒞ℬ(X). In fact, for arbitrary positive integer n, by inequality (75) and formula (78), we have
(79)HM(An+1,An)≥βHM(fn+1(An),fn(An-1))≥βqϕ(HM(An,An-1))=ϕ(HM(An,An-1)).
Thus, from inequality (79), we easily obtain the following relations:
(80)HM(An+1,An)≥ϕ(HM(An,An-1))≥ϕ2(HM(An,An-1))≥⋯≥ϕn(HM(A1,A0)).
Furthermore, for arbitrary positive integers n and p, we get that
(81)HM(An+p,An)≥HM(An+p,An+p-1)*HM(An+p-1,An+p-2)*⋯*HM(An+1,An)≥(ϕn+p-1*ϕn+p-2*⋯*ϕn)×(HM(A1,A0)).
Since for arbitrary h∈(0,1), ϕn(h)→1(n→∞), and by continuity of *, we have
(82)HM(An+p,An)⟶1(n⟶∞);
that is, {An} is a Cauchy sequence in 𝒞ℬ(X). By Theorem 15, 𝒞ℬ(X) is complete since X is complete. Consequently, there exists A*∈𝒞ℬ(X) such that An→A*(n→∞); that is, limn→∞HM(An,A*)=1.
Next, we show thatA*⊆fi(A*); that is, ρ(A*,fi(A*))=1, for all i∈ℕ+. In fact, for arbitrary positive integers i and j, i≠j, by Proposition 9 we have
(83)ρ(A*,fi(A*))≥HM(A*,Aj)*ρ(Aj,fi(A*))≥HM(A*,Aj)*ρ(Aj,fj(Aj-1))*ρ(fj(Aj-1),fi(A*))=HM(A*,Aj)*ρ(fj(Aj-1),fi(A*))≥HM(A*,Aj)*HM(fj(Aj-1),fi(A*)).
Moreover, we have
(84)HM(fj(Aj-1),fi(A*))≥qϕ(HM(Aj-1,A*))>ϕ(HM(Aj-1,A*)).
Consequently, we get
(85)ρ(A*,fi(A*))>HM(A*,Aj)*ϕ(HM(Aj-1,A*)).
Since ϕ is continuous from the left and * is a continuous positive t-norm, hence, we can obtain
(86)ρ(A*,fi(A*))≥limj→∞HM(A*,Aj)*ϕ(limj→∞HM(Aj-1,A*))=1*ϕ(1)=1;
that is, ρ(A*,fi(A*))=1. By (1) of Proposition 9, we obtain A*⊆fi(A*), for all i∈ℕ+.Corollary 24.
Let(X,M,*) be a complete stationary fuzzy metric space and let f be a self-mapping of 𝒞ℬ(X). If there exists a constant q∈(1,+∞), such that, for each A1,A2∈𝒞ℬ(X),
(87)HM(f(A1),f(A2))≥qϕ(HM(A1,A2)),
where ϕ satisfies the conditions of Lemma 18, then there exists an A*∈𝒞ℬ(X) such that A*⊆f(A*).Proof.
In fact, we can define a sequence of fuzzy self-mappings of𝒞ℬ(X) as fi=f, for i=1,2,…. Thus, this result is a special case of Theorem 23.Example 25.
Let(𝒞ℬ(X),HMd,·) be a stationary fuzzy metric space, where X=[-1,1], and HMd is the same as in Example 13. Then, (𝒞ℬ(X),HMd,·) is a complete stationary fuzzy metric space.
Now, defineϕ:[0,1]→[0,1] as ϕ(x)=x, and let {fn}n=1∞ be a sequence of self-mappings of 𝒞ℬ(X) as
(88)fn(A)=12nA,foranyA∈𝒞ℬ(X).
For arbitrary positive integers i and j, without loss of generality, suppose i<j. For each A1,A2∈𝒞ℬ(X), by a routine calculation, we have
(89)HMd(fi(A1),fj(A2))=HMd(12iA1,12jA2)=2iHMd(A1,12j-iA2)≥2iHMd(A1,A2)≥2ϕ(HMd(A1,A2)).
Therefore, by Theorem 23, we assert that the sequence of self-mappings {fn}n=1∞ has a common fixed point A* in 𝒞ℬ(X). In fact, it is easy to check that A*={0}.


## 4. Conclusions

In this paper, we have established the completeness of𝒞ℬ(X) with respect to the completeness of the stationary fuzzy metric space X. We also present some common fixed point theorems for the self-mapping of stationary fuzzy metric space 𝒞ℬ(X) under some ϕ-contraction conditions.Several possible applications of our results may be suggested. We briefly mention some of them. Fuzzy fixed point theory can be used in existence and continuity theorems for dynamical systems with some vague parameters [45–47]. More specifically in the field of qualitative behavior, these may be used demonstrating the existence of solutions of the fuzzy differential equation [25] and fuzzy integral equation, and so forth. In addition, the recent work of Veeramani [48] on best approximation in fuzzy metric spaces could constitute the start of a theory for which the Hausdorff fuzzy metric must play an important role. So we hope our results would provide a mathematical background to ongoing work in the problems of those related fields.



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