PROBABILISTIC METRIC SPACES. PART II
DOREL MIHET¸Received 7 June 2004 and in revised form 3 December 2004
A fixed point theorem concerning probabilistic contractions satisfying an implicit rela-tion, which generalizes a well-known result of Hadˇzi´c, is proved.
1. Preliminaries
In this section we recall some useful facts from the probabilistic metric spaces theory. For more details concerning this problematic we refer the reader to the books [1,3,9].
1.1.t-norms. Atriangular norm(shortlyt-norm) is a binary operationT: [0, 1]×[0, 1]→ [0, 1] :=I which is commutative, associative, monotone in each place, and has 1 as the unit element.
Basic examples areTL:I×I→I,TL(a,b)=Max(a+b−1, 0) (Łukasiewiczt-norm),
TP(a,b)=ab, andTM(a,b)=Min{a,b}. We also mention the following families oft -norms:
(i)Sugeno-Weber family(TSW
λ )λ∈(−1,∞), defined byTλSW=max(0, (x+y−1 +λxy)/
(1 +λ)),
(ii)Domby family (TλD)λ∈(0,∞), defined by TλD =(1 + (((1−x)/x)λ+ ((1− y)/ y)λ)1/λ)−1,
(iii)Aczel-Alsina family(TλAA)λ∈(0,∞), defined byTλAA=e−(|logx|
λ+|logy|λ)1/λ .
Definition 1.1[2,3]. It is said that thet-normTisof Hadˇzi´c-type(H-typefor short) and
T∈Ᏼif the family{Tn}
n∈Nof its iterates defined, for eachxin [0, 1], by
T0(x)=1, Tn+1(x)=TTn(x),x, ∀n≥0, (1.1)
is equicontinuous atx=1, that is,
∀ε∈(0, 1)∃δ∈(0, 1) such thatx >1−δ=⇒Tn(x)>1−ε, ∀n≥1. (1.2)
There is a nice characterization of continuoust-norms Tof the classᏴ[8].
Copyright©2005 Hindawi Publishing Corporation
(i) If there exists a strictly increasing sequence (bn)n∈Nin [0, 1] such that limn→∞bn=
1 andT(bn,bn)=bn∀n∈N, thenTis of Hadˇzi´c-type.
(ii) IfTis continuous andT∈Ᏼ, then there exists a sequence (bn)n∈Nas in (i).
Thet-normTM is an trivial example of at-norm ofH-type, but there aret-normsT
of Hadˇzi´c-type withT=TM(see, e.g., [3]).
Definition 1.2[3]. IfT is at-norm and (x1,x2,...,xn)∈[0, 1]n (n∈N), thenTin=1xi is
defined recurrently by 1, ifn=0 andTin=1xi=T(Tin=−11xi,xn) for alln≥1. If (xi)i∈Nis a
sequence of numbers from [0, 1], thenTi∞=1xiis defined as limn→∞Tin=1xi(this limit always exists) andTi∞=nxiasTi∞=1xn+i. In fixed point theory in probabilistic metric spaces there are
of particular interest thet-normsT and sequences (xn)⊂[0, 1] such that limn→∞xn=1
and limn→∞Ti∞=1xn+i=1. Some examples oft-norms with the above property are given in
the following proposition.
Proposition1.3 [3]. (i)ForT≥TLthe following implication holds:
lim
n→∞T ∞
i=1xn+i=1⇐⇒ ∞
n=1
1−xn<∞. (1.3)
(ii)(1.3) also holds forT=TλSW.
(iii)IfT ∈Ᏼ, then for every sequence (xn)n∈N in I such thatlimn→∞xn=1, one has
limn→∞Ti∞=1xn+i=1.
(iv)IfT∈ {TD
λ,TλAA}, thenlimn→∞Ti∞=1xn+i=1⇔∞n=1(1−xn)λ<∞.
Note [4, Remark 13] that ifTis at-norm for which there exists a sequence (xn)⊂[0, 1]
such that limn→∞xn=1 and limn→∞Ti∞=1xn+i=1, then supt<1T(t,t)=1.
1.2. Menger spaces and generalized Menger spaces. Probabilistic contractions of Sehgal type. Let ∆+be the class of distance distribution functions[9], that is, the class of all functionsF: [0,∞)→[0, 1] with the properties
(a)F(0)=0;
(b)Fis nondecreasing;
(c)Fis left continuous on (0,∞).
D+ is the subset of∆+ containing the functions F which also satisfy the condition limx→∞F(x)=1.
A special element ofD+is the functionε0, defined by
ε0(t)=
0, ift=0,
1, ift >0. (1.4)
A sequence (Fn) in∆+is said to beweakly convergent toF∈∆+(shortlyFn−−→w F) if limn→∞Fn(x)=F(x) for every continuity pointxofF.
IfXis a nonempty set, a mappingF:X×X→∆+is calleda probabilistic distance onX andF(x,y) is denoted byFxy.
Schweizer and Sklar) if the following conditions hold:
Fxy=ε0⇐⇒x=y, (1.5)
Fxy=Fyx, ∀x,y∈X, (1.6)
Fxy(t+s)≥T
Fxz(t),Fzy(s)
, ∀x,y,z∈X,∀t,s >0. (1.7)
AMenger spaceis a generalized Menger space with the property Range (F)⊂D+. If (X,F,T) is a generalized Menger space with supt<1T(t,t)=1, then the family
Uε,λ ε>0,λ∈(0,1), Uε,λ=
(x,y)∈X×X:Fxy(ε)>1−λ (1.8)
is a base for a metrizable uniformity onX, named theF-uniformityand denoted byᐁF.
ᐁFnaturally determines a topology onX, calledtheF-topology:
O∈᐀F⇐⇒ ∀x∈O∃ε >0, ∃λ∈(0, 1) such thatUε,λ(x)⊂O. (1.9)
ᐁFis also generated by the family{Vδ}δ>0whereVδ:=Uδ,δ. In what follows the
topo-logical notions refer to theF-topology. Thus, a sequence (xn)n∈NisF-convergent tox∈X
if for allε >0,λ∈(0, 1) there existsk∈Nsuch thatFxxn(ε)>1−λfor alln≥k.
Definition 1.4. A sequence (xn)n∈N in X is calledF-Cauchyif for eachε >0,λ∈(0, 1)
there existsk∈Nsuch thatFxrxs(ε)>1−λfor alls≥r≥k.
Probabilistic contractions were first defined and studied byV. M. Sehgalin his doctoral dissertation at Wayne State University.
Definition 1.5[10]. LetSbe a nonempty set and letFbe a probabilistic distance onS. A mapping f :S→Sis calleda probabilistic contraction(orB-contraction) if there exists
k∈(0, 1) such that
Ff(p)f(q)(kt)≥Fpq(t), ∀p,q∈S,∀t >0. (1.10)
In [10] it is showed that any contraction map on a complete Menger space in which the triangle inequality is formulated under the strongest triangular normTM has a unique fixed point. In [11]Sherwoodshowed that one can construct a complete Menger space underTLand a fixed-point-free contraction map on that space.Hadˇzi´c[2] introduced the
classᏴwhich have the property that Sehgal’s result can be extended to any continuous triangular norm in that class. Completing the result ofHadˇzi´c, Radusolved the problem of the existence of fixed points for probabilistic contractions in complete Menger spaces (S,F,T) withTcontinuous. Namely, the following theorem holds.
Theorem1.6 [7]. EveryB-contraction in a complete Menger space(S,F,T)withT contin-uous has a (unique) fixed point if and only ifTis of Hadˇzi´c-type.
2. Main results
The main result of this paper isTheorem 2.4concerning contractive mappings satisfying an implicit relation similar to that in [6,12]. This theorem generalizes the mentioned result of Hadˇzi´c (seeCorollary 2.7). Note that we work in generalized Menger spaces.
We begin with an auxiliary result, which is formulated as follows.
Lemma2.1. Let(X,F,T)be a generalized Menger space and let(xn)n∈Nbe a sequence inX
such that, for somek∈(0, 1),
Fxnxn+1(kt)≥Fxn−1xn(t), ∀n≥1,∀t >0. (2.1)
If there existsγ >1such that
lim
n→∞T ∞ i=nFx0x1
γi=1, (2.2)
then(xn)n∈Nis anF-Cauchy sequence.
Proof. First note [4] that if the condition limn→∞Ti∞=nFx0x1(γi)=1 holds for someγ= γ0>1, then it is satisfied for allγ >1. Indeed, if limn→∞Ti∞=nFx0x1(γi0)=1 andγ≥γ0, then
limn→∞Ti∞=nFx0x1(γi)≥limn→∞Ti∞=nFx0x1(γ0)i =1 and therefore limn→∞Ti∞=nFx0x1(γi)=1,
while if γ < γ0, then γs> γ0, for some s∈N, and now lim
n→∞Ti∞=n+sFx0x1(γi)≥
limn→∞Ti∞=nFx0x1(γi0)=1. We will prove that
∀ε >0,∃n0=n0(ε) :Fxnxn+m(ε)>1−ε, ∀n≥n0,∀m∈N. (2.3)
Letµ∈(k, 1) and letδ=k/µ. From the above remark it follows that
lim
n→∞T ∞ i=nFx0x1
1
µi
=1. (2.4)
Letε >0 be given andyi:=Fx0x1(1/µi). From limn→∞Ti∞=1yn+i=1 it follows that there
existsn1∈Nsuch thatTim=1yn+i−1>1−ε, for alln≥n1, for allm∈N. Since the series∞n=1δnis convergent, there existsn2∈Nsuch that
∞
n=n2δn< ε.
Letn0=max{n1,n2}. Then, for alln≥n0andm∈N, we have
Fxnxn+m(ε)≥Fxnxn+m
n+m−1
i=n δi
≥Tim=−01Fxn+ixn+i+1
δn+i≥Tim=−01yn+i>1−ε,
(2.5)
where the last “≥” inequality follows fromFxsxs+1(δs)=Fxsxs+1(k/µ)s≥Fx0x1(1/µs) for all
s≥1, which immediately can be proved by induction.
In the following we deal with the classΦof all continuous functionsϕ: [0, 1]4→R with the property:
ϕ(u,v,v,u)≥0=⇒u≥v. (2.6)
Example 2.2. Ifa,b,c,d∈Randa+b+c+d=0, thenϕ(t1,t2,t3,t4) :=at1+bt2+ct3+
dt4∈Φif and only ifa+d >0.
Indeed,a+d≤0⇒b+c≥0. Choosingu=0,v=1 we haveu < vandϕ(u,v,v,u)= (a+d)u+ (b+c)v=b+c≥0.
Conversely, if a+d >0 andϕ(u,v,v,u)≥0, then (a+d)u≥ −(b+c)v, that is (a+
d)u≥(a+d)v, which implies thatu≥v. Thus, the functionsϕ1,ϕ2,
ϕ1t1,t2,t3,t4=t1−t2,
ϕ2
t1,t2,t3,t4
=t1−t3,
(2.7)
are inΦ.
Also, the functionϕdefined byϕ(t1,t2,t3,t4)=t2
1−t2t3and, more generally,ϕ(t1,t2,
t3,t4)=t21−(at22+bt32)−t2t3witha+b=0 are inΦ.
In the proof of Theorem 2.4we need the following lemma, which is the analog of uniform continuity of a metric (note that ([0, 1],T) is rather a semigroup than a group). Lemma2.3. Let(S,F,T)be a generalized Menger space withTcontinuous in(a, 1)for all
a∈(0, 1), that is,
lim
n→∞an=a, nlim→∞bn=1=⇒nlim→∞T
an,bn
=a. (2.8)
Ifp,q∈Sand(pn)is a sequence inSsuch thatpn→p, thenFpnq
w
−−→Fpq.
Proof. Letp,q∈S,pn→pandtbe a continuity point ofFpq. By (1.7) it follows that for all 0< ε < t,
Fpnq(t)≥T
Fpnp(ε),Fpq(t−ε)
,
Fpq(t+ε)≥TFpnp(ε),Fpnq(t)
. (2.9)
Therefore, limninfFpnq(t)≥Fpq(t−ε) andFpq(t+ε)≥limnsupFpnq(t). Lettingε→0 we obtain limnsupFpnq(t)≤Fpq(t)≤limninfFpnq(t), and thus limn→∞Fpnq(t)=Fpq(t).
Theorem2.4. Let (X,F,T)be anF-complete generalized Menger space under a t-norm
T which is continuous in(a, 1)for alla∈(0, 1),k∈(0, 1), andϕ∈Φ. If f :X→X is a mapping such that
ϕf:ϕFf(x)f(y)(kt),Fxy(t),Fx f(x)(t),Fy f(y)(kt)≥0, ∀x,y∈X,∀t >0 (2.10)
and there existx0∈Xandγ >1for whichlimn→∞Ti∞=nFx0f(x0)(γi)=1, then f has a fixed point.
Proof. Letx0∈Xbe such that limn→∞Ti∞=nFx0f(x0)(γi)=1 and, for alln≥1,xn=f(xn−1).
Note that (ϕf) implies that
On taking in this relationx=xnwe obtain
ϕFxn+1xn+2(kt),Fxnxn+1(t),Fxnxn+1(t),Fxn+1xn+2(kt)
≥0, ∀n∈N,∀t >0. (2.12)
It follows that Fxn+1xn+2(kt)≥Fxnxn+1(t), for all n∈N, for all t >0 and therefore, by Lemma 2.1, (xn) is a Cauchy sequence.
By theF-completeness ofXit follows that there existsu∈Xsuch that limn→∞Fuxn(t)= 1, for allt >0.
Notice that from Fxn+1xn+2(kt)≥Fxnxn+1(t), for all n∈N, for all t >0 it follows that
limn→∞Fxnxn+1(t)= 1, for all t > 0, for limn→∞Ti∞=nFx0f(x0)(γi) = 1 implies that
limn→∞Fx0f(x0)(γn)=1 (thereforeFx0f(x0)∈D+) andFxnxn+1(t)≥Fx0x1(t/kn), for alln∈N,
for allt >0.
Next, on takingx=xn,y=uin (ϕf) one obtains
ϕFxn+1f(u)(kt),Fxnu(t),Fxnxn+1(t),Fu f(u)(kt)
≥0, ∀n∈N,∀t >0. (2.13)
Ifktis a continuity point ofFu f(u), then, on takingn→ ∞in the above inequality and
usingLemma 2.3, we get
ϕFu f(u)(kt), 1, 1,Fu f(u)(kt)
≥0. (2.14)
ThusFu f(u)(kt)=1. SinceFu f(u)is increasing, the set of its discontinuity points is at most countable. HenceFu f(u)(kt)=1 for allt >0, from which (using (1.5)) we obtainu= f(u).
This completes the proof.
Corollary2.5 [5, Theorem 2.1]. Let(X,F,T)be anF-complete generalized Menger space under a continuoust-normT∈Ᏼ,k∈(0, 1), andϕ∈Φ. If f :X→Xis a mapping such that
ϕFf(x)f(y)(kt),Fxy(t),Fx f(x)(t),Fy f(y)(kt)≥0, ∀x,y∈X,∀t >0 (2.15)
and there existsx0∈Xfor whichFx0f(x0)∈D+, thenf has a fixed point.
Proof. Choose a µ > 1. Since limn→∞µn = ∞ and Fx0x1 ∈ D+, it follows that
limn→∞Fx0f(x0)(µn)=1. Therefore, byProposition 1.3(iii),
lim
n→∞T ∞ i=nFx0f(x0)
µi=1. (2.16)
Now applyTheorem 2.4.
Corollary2.6. Let(X,F,TL)be anF-complete generalized Menger space andϕ∈Φ. If f :X→Xis a mapping such that
ϕFf(x)f(y)(kt),Fxy(t),Fx f(x)(t),Fy f(y)(kt)≥0, ∀x,y∈X,∀t >0, (2.17)
and∞n=1(1−Fx0f(x0)(γn))<∞for somex0∈Xandγ >1, then f has a fixed point.
Corollary 2.7. Let (X,F,T) be an F-complete generalized Menger space under T∈ {TD
λ,TλAA},k∈(0, 1), andϕ∈Φ. If f :X→Xis a mapping such that
ϕFf(x)f(y)(kt),Fxy(t),Fx f(x)(t),Fy f(y)(kt)≥0, ∀x,y∈X,∀t >0 (2.18)
and∞n=1(1−Fx0f(x0)(γn))λ<∞for somex0∈Xandγ >1, thenf has a fixed point. Corollary2.8. Let(X,F,T)be anF-complete generalized Menger space under a continu-oust-normT∈Ᏼandk∈(0, 1). If f :X→Xis a mapping satisfying one of the following conditions:
Ff(x)f(y)(kt)≥Fxy(t), ∀x,y∈X,∀t >0, (2.19) F2
f(x)f(y)(kt)≥Fxy(t)Fx f(x)(t), ∀x,y∈X,∀t >0, (2.20) Ff(x)f(y)(kt)≥2Fxy(t)−Fx f(x)(t), ∀x,y∈X,∀t >0 (2.21)
and there existsx0∈Xfor whichFx0f(x0)∈D+, thenf has a fixed point.
As a final result for this section, we consider an example to see the generality of Theorem 2.4.
Example 2.9. LetX be a set containing at least two elements and the mappingF from
X×Xto∆+, defined by
Fxy(t)=
0, ift≤1 1
2, ift >1
forx,y∈X,x=y, Fxx=ε0, ∀x∈X. (2.22)
It is easy to show (see [14]) that (X,F,TM) is a complete Menger space.
We are going to prove that the mapping f :X→X, f(x)=x satisfies the contrac-tivity condition (2.21) from the above corollary withb=2,c= −1, however it is not a
B-contraction (here we took advantage of working in∆+rather than inD+). First, we show that
Fxy(kt) + 1≥2Fxy(t), ∀x,y∈X,∀t >0. (2.23)
Indeed, the above inequality holds with equality ifx=y, while ifx=ythen the right-hand member is at most 1.
Next, for everyt∈(1, 1/k],Fxy(kt)=0, whileFxy(t)=1/2, which means that f is not a Sehgal contraction.
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Dorel Mihet¸: Faculty of Mathematics and Computer Science, West University of Timisoara, Bd. V. Parvan 4, 300223 Timisoara, Romania
