PROBABILISTIC METRIC SPACES. PART II

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(T_λ^SW)_λ∈(−1,∞), defined byT_λ^SW=max(0, (x+y−1 +λxy)/

(1 +λ)),

(ii)Domby family (T_λ^D)_λ∈(0,_∞), defined by T_λ^D =(1 + (((1−x)/x)^λ+ ((1− y)/

y)^λ)^1/λ)⁻¹,

(iii)Aczel-Alsina family(T_λ^AA)_λ∈(0,_∞), defined byT_λ^AA=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

International Journal of Mathematics and Mathematical Sciences 2005:5 (2005) 729–736 DOI:10.1155/IJMMS.2005.729

(i) If there exists a strictly increasing sequence (b_n)_n∈Nin [0, 1] such that lim_n→∞b_n= 1 andT(b_n,b_n)=b_n∀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=T_M(see, e.g., [3]).

Definition 1.2[3]. IfT is at-norm and (x1,x2,...,x_n)∈[0, 1]ⁿ (n∈N), thenT_iⁿ₌1x_i is defined recurrently by 1, ifn=0 andT_iⁿ₌1xi=T(T_iⁿ₌⁻1¹xi,xn) for alln≥1. If (xi)_i∈Nis a sequence of numbers from [0, 1], thenT_i^∞₌1xiis defined as lim_n→∞T_iⁿ₌1xi(this limit always exists) andT_i^∞_=nx_iasT_i^∞₌1x_n+i. In fixed point theory in probabilistic metric spaces there are of particular interest thet-normsT and sequences (xn)⊂[0, 1] such that lim_n→∞xn=1 and lim_n→∞T_i^∞₌1xn+i=1. Some examples oft-norms with the above property are given in the following proposition.

Proposition1.3 [3]. (i)ForT≥T_Lthe following implication holds:

nlim→∞T_i^∞₌1xn+i=1⇐⇒

∞ n=1

1−xn

<∞. (1.3)

(ii)(1.3) also holds forT=T_λ^SW.

(iii)IfT ∈Ᏼ, then for every sequence (xn)_n∈N in I such thatlim_n→∞xn=1, one has lim_n→∞T_i^∞₌1x_n+i=1.

(iv)IfT∈ {T_λ^D,T_λ^AA}, thenlim_n→∞T_i^∞₌1xn+i=1⇔_∞

n=1(1−xn)^λ<∞.

Note [4, Remark 13] that ifTis at-norm for which there exists a sequence (x_n)⊂[0, 1]

such that lim_n→∞xn=1 and lim_n→∞T_i^∞₌1xn+i=1, then sup_t<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 lim_x→∞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 lim_n→∞F_n(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.

The triple (X,F,T), where X is a nonempty set,F is a probabilistic distance onX, andT is at-norm, is calleda generalized Menger space(or aMenger space in the sense of

Schweizer and Sklar) if the following conditions hold:

Fxy=ε0⇐⇒x=y, (1.5)

F_xy=F_yx, ∀x,y∈X, (1.6)

F_xy(t+s)≥TF_xz(t),F_zy(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 sup_t<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{V_δ}δ>0whereV_δ:=U_δ,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 thatF_xrxs(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 underT_Land 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)withTcontin- uous has a (unique) fixed point if and only ifTis of Hadˇzi´c-type.

However, under some additional growth conditions on the probabilistic metricFone may replace the t-norm of H-type in the above theorem, as in Tardiff’s paper [13].

Corollary 2.6in our paper gives another result in this respect.

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

nlim→∞T_i^∞_=nF_x0x¹

γⁱ=1, (2.2)

then(x_n)_n∈Nis anF-Cauchy sequence.

Proof. First note [4] that if the condition lim_n→∞T_i^∞_=nF_x0x¹(γⁱ)=1 holds for someγ= γ0>1, then it is satisfied for allγ >1. Indeed, if lim_n→∞T_i^∞_=nFx0x1(γⁱ0)=1 andγ≥γ0, then lim_n→∞T_i^∞_=nFx0x1(γⁱ)≥lim_n→∞T_i^∞_=nFx0x1(γ0ⁱ)=1 and therefore lim_n→∞T_i^∞_=nFx0x1(γⁱ)=1, while if γ < γ0, then γ^s> γ0, for some s∈N, and now lim_n→∞T_i^∞_=n+sF_x0x¹(γⁱ)≥ lim_n→∞T_i^∞_=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

nlim→∞T_i^∞_=nFx0x1

1 µⁱ

=1. (2.4)

Letε >0 be given andyi:=Fx0x1(1/µⁱ). From lim_n→∞T_i^∞₌1yn+i=1 it follows that there existsn1∈Nsuch thatT_i^m₌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 F_xnxn+m(ε)≥F_xnxn+m

_n+m−1

i=n

δⁱ

≥T_i^m₌⁻0¹Fxn+ixn+i+1

δⁿ⁺ⁱ≥T_i^m₌⁻0¹yn+i>1−ε,

(2.5)

where the last “≥” inequality follows fromFxsxs+1(δ^s)=Fxsxs+1(k/µ)^s≥Fx0x1(1/µ^s) 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)

Next we give some examples of functions inΦ.

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, ϕ1

t1,t2,t3,t4

=t1−t2, ϕ2

t1,t2,t3,t4

=t1−t3, (2.7)

are inΦ.

Also, the functionϕdefined byϕ(t1,t2,t3,t4)=t²1−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,

nlim→∞a_n=a, lim

n→∞b_n=1=⇒lim

n→∞Ta_n,b_n=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)≥TFpnp(ε),Fpq(t−ε),

Fpq(t+ε)≥TFpnp(ε),Fpnq(t). (2.9) Therefore, lim_ninfFpnq(t)≥Fpq(t−ε) andFpq(t+ε)≥lim_nsupFpnq(t). Lettingε→0 we obtain lim_nsupFpnq(t)≤Fpq(t)≤lim_ninfFpnq(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 whichlim_n→∞T_i^∞_=nF_x0f(x⁰)(γⁱ)=1, then f has a fixed point.

Proof. Letx0∈Xbe such that lim_n→∞T_i^∞_=nFx0f(x0)(γⁱ)=1 and, for alln≥1,xn=f(xn−1).

Note that (ϕf) implies that

Ff(x)f²(x)(kt)≥Fx f(x)(t), ∀x∈X,∀t >0. (2.11)

On taking in this relationx=x_nwe obtain

ϕF_xn+1xn+2(kt),F_xnxn+1(t),F_xnxn+1(t),F_xn+1xn+2(kt)≥0, ∀n∈N,∀t >0. (2.12) It follows that F_xn+1xn+2(kt)≥F_xnxn+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 lim_n→∞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 lim_n→∞Fxnxn+1(t)= 1, for all t > 0, for lim_n→∞T_i^∞_=nFx0f(x0)(γⁱ) = 1 implies that lim_n→∞F_x0f(x0)(γⁿ)=1 (thereforeF_x0f(x0)∈D+) andF_xnxn+1(t)≥F_x0x1(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

ϕF_{u f}(u)(kt), 1, 1,F_{u f}(u)(kt)≥0. (2.14) ThusF_{u f}(u)(kt)=1. SinceF_{u 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 lim_n→∞µⁿ = ∞ and Fx0x1 ∈ D+, it follows that lim_n→∞Fx0f(x0)(µⁿ)=1. Therefore, byProposition 1.3(iii),

nlim→∞T_i^∞_=nFx0f(x0)

µⁱ=1. (2.16)

Now applyTheorem 2.4.

Corollary2.6. Let(X,F,T_L)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.

For the proof seeProposition 1.3.

Corollary 2.7. Let (X,F,T) be an F-complete generalized Menger space under T∈ {T_λ^D,T_λ^AA},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

F_xy(t)=







0, ift≤1 1

2, ift >1 forx,y∈X,x=y, F_xx=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 E-mail address:[email protected]