R E S E A R C H
Open Access
Some fixed point theorems in Menger
probabilistic metric-like spaces
Antonio Francisco Roldán López de Hierro
1*and Manuel de la Sen
2*Correspondence: aroldan@ugr.es 1Department of Quantitative Methods for Economics and Business, Faculty of Economics and Business Sciences, University of Granada, Campus Universitario de Cartuja, s/n, Granada, 18011, Spain Full list of author information is available at the end of the article
Abstract
In this manuscript we extend very recent fixed point theorems in the setting of Menger spaces in three senses: on the one hand, we introduce the notion ofMenger probabilistic metric-like spaceby avoiding a non-necessary constraint (at least, for our purposes) in the properties that define a Menger space; on the other hand, we consider a more general class of auxiliary functions in the contractivity condition; finally, we show that the functiont→1/t– 1 (which appears in many fixed point theorems in the fuzzy context) can be replaced by more appropriate and general functions. We illustrate our main statements with an example in which previous results cannot be applied.
Keywords: probabilistic metric space; Menger space; fixed point; contractive mapping
1 Introduction
There are two well known extensions of the notion ofmetric spaceto frameworks in which imprecise models are considered:fuzzy metric spaces(see [–]) andprobabilistic metric spaces[–]. The two concepts are very similar, but they are different in nature.
In a metric space, the distance between two points is, necessarily, a non-negative real number. In , Menger extended the notion of metric space replacing non-negative numbers by a random variable that only took non-negative real values. Such random vari-able was characterized by itsdistribution function. Hence, in this kind of spaces (which Menger originally calledstatistical metric spaces), the distance between two pointsxand yis given by a distribution functionFxywhich can be seen as follows:afor allt∈[,∞),
the valueFxy(t) gives the probability of the event that occurs when the distance betweenx
andyis less thant, that is,
Fxy(t) =P
d(x,y) <t for allt∈[,∞). ()
Thus, statistical metric spaces depend on the notion ofdistribution function. Although there is a great agreement about the global properties that a distribution function must satisfy, in practice, we can find several papers involving different notions. Such different definitions lead to different properties and, in some cases, to incomparable results. For instance, Schweizer and Sklar [], Definition .., established that adistribution function (briefly,d.f.) is a nondecreasing functionF:R→[, ] such thatF(–∞) = andF(∞) =
(whereR= [–∞,∞] is the extended real line). Note that not all such functions can be interpreted as cumulative distribution functions for a random variable. However, when a random experiment is considered, additional properties are required. For instance, if F is given as in (), it is not difficult to show thatF is left-continuous inR= (–∞,∞). Furthermore, ifdonly takes non-negative values, thenF(t) = for allt∈[–∞, ]. Thus, Schweizer and Sklar denoted by the family of all d.f.’s that are left-continuous onR, and by+the family of allF∈such thatF(t) = for allt∈[–∞, ]. Functions in+
were calleddistance distribution functions(briefly,d.d.f.) because they represent random variables that only take non-negative values (as in (), where a distance is involved). For convenience, d.d.f.’s are only considered in [,∞], whereF() = andF(∞) = .
The set+of all d.d.f.’s is partially ordered by the binary relation≤, beingF≤Gwhen
F(t)≤G(t) for allt∈(,∞). The minimal and the maximal elements of+are∞and,
respectively, where
∞(t) =
, if ≤t<∞,
, ift=∞, and (t) =
, ift= , , if <t≤ ∞.
On the other hand, many authors calldistribution function(see, for instance, []) to a nondecreasing, left-continuous function F :R→[, ] such that inft∈RF(t) = and supt∈RF(t) = . AsFis nondecreasing, it follows that
lim
t→–∞F(t) =tinf∈RF(t) = and tlim→∞F(t) =supt∈RF(t) = .
This notion does not coincide with Schweizer and Sklar’s view-point. For instance,ε∞is a d.d.f. in Schweizer and Sklar’s sense, but it does not satisfy the conditionlimt→∞F(t) = (because ift∈[,∞), thenF(t) = ).
In recent times, many fixed point theorems have been presented in the setting of proba-bilistic metric spaces. Many of them were inspired by their corresponding results on met-ric spaces. One of the most attractive, effective ways to introduce contractivity conditions in the probabilistic framework is based on considering some terms like in the following expression:
Fxy(t)
– , wherex,y∈Xandt>
(see [–]). For instance, in [], Kutbiet al.stated the following result (whereand
are appropriate collections of auxiliary functions that we will describe in Section ).
Theorem (Kutbiet al.[], Theorem .) Let(X,F,T)be a G-complete Menger space and let f :X→X be a mapping.Assume that there exist a constant c∈(, )and two functionsφ∈andψ∈satisfying the inequality
Ffx,fy(φ(ct))
– ≤ψ
Fx,y(φ(t))
–
We must highlight that the uniqueness of the fixed point was not proved in such result (it was studied in Theorem . in []). The main aims of the present manuscript are the following ones:
() to extend the previous result to a wider class of Menger spaces, that we shall call
Menger probabilistic metric-like spaces;
() to show that some conditions in the functions of the familyare not necessary; () to introduce more general contractivity conditions replacing the function
t→/t– by an appropriate functionh.
For the sake of clarity, we advise the reader that we shall use the terminology introduced in [].
2 Preliminaries
In this section, we recall some definitions and basic results. Throughout this manuscript, let N={, , , . . .} be the set of all non-negative integer numbers and letN∗=N{}. Afixed pointof a self-mappingT:X→Xis a pointx∈Xsuch thatTx=x. We will denote byFix(T) the family of all fixed points ofT.
Definition At-norm(or atriangular norm) is a function∗: [, ]×[, ]→[, ] sat-isfying, for alla,b,c,d∈[, ], the following properties:
() Associativity:(a∗b)∗c=a∗(b∗c). () Commutativity:a∗b=b∗a. () Existence of unity:a∗ =a.
() Monotonicity:ifa≤bandc≤d, thena∗c≤b∗d.
At-norm iscontinuousif it is continuous as a function.
It can be proved thata∗ = for alla∈[, ]. Examples oft-norms are:
Minimalt-norm:a∗Mb=min(a,b),
Productt-norm:a∗Pb=ab,
Lukasiewiczt-norm:a∗Lb=max(a+b– , ),
Drastict-norm:a∗Db=
min(a,b), if max(a,b) = , , otherwise.
The three firstt-norms are continuous, but the drastict-norm is not continuous. Givena∈[,∞), leta: [,∞]→[, ] be the function given by
a(t) =
, if ≤t≤a,
, ifa<t≤ ∞. ()
Functions{a:a∈[,∞)} ∪ {∞}are calledstep functionsand they are examples of d.d.f.’s.
[].
(PM) Fxx(t) = for allt> ;
(PM) identity: ifFxy(t) = for allt> , thenx=y; (PM) symmetry:Fxy(t) =Fyx(t);
(PM) triangular inequality:Fxz(t+s)≥Fxy(t)∗Fyz(s).
This definition of a Menger PMS is taken from Schweizer and Sklar’s notion given in []. However, it is different from the one we find in [], Definition ., because the authors of this reference assumed thatFxyis a distribution function in their own sense (in particular,
the random variable can take values on intervals of negative real numbers).
Remark The reader can observe the following subtle details.
() SinceF(X×X)⊆+, each d.d.f.F
xyin a Menger PMS is a nondecreasing function on[,∞]. Furthermore, it always satisfiesFxy() = . In the following text, we will often use the monotonicity ofFxy. However, we will never use the fact that
Fxy() = . In this sense, it is possible to consider a similar definition of Menger space where the probabilistic metricFtake values inrather than in+. However, notice that conditions (PM)-(PM) only refer to the interval[,∞).
() In the field of fixed point theory, a fifth assumption is usually considered.
(PM) limt→∞Fxy(t) = for allx,y∈X.
A discussion about the necessity of this condition can be found in []. Our main result will not need such property, but it will appear in some corollaries.
The following families of auxiliary functions were considered in [].
Definition Letbe the family of all functionsφ: [,∞)→[,∞) satisfying:
() φ(t) = if, and only if,t= ;
() φis strictly increasing andlimt→∞φ(t) =∞;
() φis left-continuous in(,∞); () φis continuous att= .
Definition Letbe the class of all functionsψ: [,∞)→[,∞) satisfying: () ψis nondecreasing;
() ψis continuous att= ; () ψ() = ;
() if{an} ⊂[,∞)is a sequence such that{an} →, then{ψn(an)} →(whereψn denotes thenth-iterate ofψ).
First of all, we show that we do not need to assume thatψ is continuous att= for functions inunder the rest of the assumptions.
Proposition Letψ: [,∞)→[,∞)be a nondecreasing function such thatψ() = .
() Ifψis not continuous att= ,then there existsε> such thatψ(t)≥εfor allt> . () Ifψ satisfies ‘{ψn(a
n)} →whenever{an} →asn→ ∞’,thenψis continuous at
Proof () Taking into account thatψ() = , then the continuity condition att= is ∀ε> ,∃δ> :|t– |<δ⇒ψ(t) –ψ()<ε.
Assume thatψ is not continuous att= . Then there existsε> verifying
for allδ> , there existst∈(,δ) such thatψ(t)≥ε.
By takingδn= /n> for alln∈N, we can find a strictly decreasing sequence{an} ⊂(,∞)
such that
<an+<an<
n and ψ(an)≥ε for alln∈N.
Next, we show thatψ(t)≥εfor allt> . Lett> be arbitrary. As{an} →, there exists
n∈Nsuch that <an<t. Asψis nondecreasing,
ψ(t)≥ψ(an)≥ε. Thenψ(t)≥ε.
() We reason by contradiction. Assume thatψ is not continuous att= . By the first item, there existsε> such thatψ(t)≥εfor allt> . Letan= /nfor alln∈N. Then
{an} →. By hypothesis,{ψn(an)} →. However, for alln∈N,
an=
n> ⇒ ψ(an)≥ε> ,
ψ(an) > ⇒ ψ(an)≥ε> ,
.. .
ψn–(an) > ⇒ ψn(an) =ψ
ψn–(an)
≥ε.
Sinceψn(a
n)≥εfor alln∈N,{ψn(an)} → is a contradiction.
As a consequence of the previous properties, we can express
=
ψ: [,∞)→[,∞)
ψnondecreasing,ψ() = , if{an} →, then{ψn(an)} →.
3 Menger probabilistic metric-like spaces
In this section, we introduce the notion ofMenger probabilistic metric-like spaceas a nat-ural way to extend Menger PM-spaces.
Definition AMenger probabilistic metric-like space(briefly, aMenger PMLS) is a triple (X,F,∗) whereXis a nonempty set,∗is a continuoust-norm andF:X×X→+is a
mapping satisfying (PM), (PM), and (PM).
Example LetX= [,∞) be the interval of non-negative real numbers and letF:X× X→+be the mapping:
Fxy=max{x,y} for allx,y∈X,
where{a:a∈[,∞)}are the step functions given in (). Let us show that (X,F,∗P) is a
Menger PMLS that also verifies (PM), but it is not a Menger PMS. Indeed, sinceF() =
() = , thenFdoes not satisfy axiom (PM), so (X,F,∗P) is not a Menger PMS. However,
conditions (PM), (PM), and (PM) are obvious. Let us prove condition (PM). Letx,y,z∈
Xandt,s> . SinceFxz(t+s)∈ {, }, (PM) trivially holds ifFxz(t+s) = . Assume that
Fxz(t+s) = . Hencemax{x,z}(t+s) = implies thatt+s≤max{x,z}. Thent+s≤x or
t+s≤z. In the first case,t≤t+s≤x, soFxy(t) =max{x,y}(t) = . In the second case,s≤ t+s≤z, soFyz(s) =max{y,z}(s) = . In any case, ∈ {Fxy(t),Fyz(s)}. Then
Fxy(t)∗PFyz(s) =Fxy(t)·Fyz(s) = =Fxz(t+s),
and this proves that (PM) also holds. Thus, (X,F,∗P) is a Menger PMLS that also verifies
(PM).
Example Using X andF as in the previous example, and taking into account that a∗Pb≤a∗Mbfor alla,b∈[, ], we deduce that (X,F,∗M) is also a Menger PMLS that,
additionally, verifies (PM), but it is not a Menger PMS.
Definition Let{xn}be a sequence in a Menger PMLS (X,F,∗). We will say that: • {xn}converges toxif for allε> and allλ∈(, ), there existsn∈Nsuch that
Fxnx(ε) > –λfor alln≥n(in such a case, we will write{xn} →x);
• {xn}is aCauchy sequenceif for allε> and allλ∈(, ), there existsn∈Nsuch that
Fxnxm(ε) > –λfor allm>n≥n;
• (X,F,∗)isM-completeif every Cauchy sequence is convergent; • {xn}is aG-Cauchy sequenceif for allε> and allp∈N∗, we have
limn→∞Fxnxn+p(ε) = ;
• (X,F,∗)isG-completeif everyG-Cauchy sequence is convergent.
Notice that thet-norm∗has not a role in the previous definition. It is clear that ev-ery Cauchy sequence is also aG-Cauchy sequence. As a consequence, everyG-complete Menger PMLS is anM-complete Menger PMLS, but the converse is false.
Example Let us show that the Menger PMLS (X,F,∗P) introduced in Example is
G-complete. Let{xn}be aG-Cauchy sequence in (X,F,∗P). We claim that{xn} d
→ using the Euclidean metricd(x,y) =|x–y|for allx,y∈X. Letε> be arbitrary. Usingp= , we have limn→∞Fxnxn+(ε) = . AsFxnxn+(ε)∈ {, }for alln∈N, we infer that there existsn∈N such thatFxnxn+(ε) = for alln≥n. Hence
Thenxn≤max{xn,xn+}<εfor alln≥n. This proves that{xn} d
→. Next, we are going to prove that{xn}converges to in (X,F,∗P). Indeed, notice that
Fxn,(ε) =max{xn,}(ε) =xn(ε) =
, ifε≤xn,
, ifxn<ε.
As{xn} d
→, givenε> , there existsn∈Nsuch thatxn<εfor alln≥n. In particular,
Fxn,(ε) =xn(ε) = > –λ,
for alln≥n, whateverλ∈(, ). Thus,{xn}converges to in (X,F,∗P) and this space is
G-complete.
In a Menger PMLS (X,F,∗), the subsets
Nε,λ(x) =
y∈X:Fxy(ε) > –λ
do not determine a topology onXbecause we cannot ensure thatx∈Nε,λ(x) since axiom
(PM) was avoided. However, we are going to show that the limit of a convergent sequence
in a Menger PMLS is unique.
Lemma If{xn} is a sequence in a Menger PMLS(X,F,∗)and x,y∈X are such that
{xn} →x and{xn} →y,then x=y.
Proof Lett> be arbitrary. As{xn} →xand{xn} →y, usingε=t/ > andλk=k∈(, )
for allk∈N,k≥, there existsxn(k)such that
Fx,xn(k)
t
> –
k+ and Fxn(k),y
t
> –
k.
Therefore,
Fxy(t)≥Fx,xn(k)
t
∗Fxn(k),y
t
≥
–
k
∗
– k
.
Lettingk→ ∞and taking into account that∗is continuous, we observe that
Fxy(t)≥ lim k→∞
–
k
∗
– k
= ∗ = .
As a consequence,Fxy(t) = for allt> , sox=yby virtue of (PM).
The following result characterizes the Menger PMS as a particular subclass of Menger PMLS.
Lemma If(X,F,∗)is a Menger PMLS,then the following conditions are equivalent.
() (X,F,∗)is a Menger PMS.
Proof Assume that (X,F,∗) is a Menger PMS and letxn=z∈Xfor alln∈N. Then, for all
ε> and allλ∈(, ),
Fxn,z(ε) =Fz,z(ε) = > –λ for alln∈N.
Conversely, assume the second condition and letx∈Xbe arbitrary. Let definexn=xfor
all n∈N. By hypothesis,{xn} →x. Letε> and letλk= /k for allk∈N. Then there
existsxn(k)such thatFxx(ε) =Fxn(k),x(ε) > –λk= – /k. Lettingk→ ∞, we deduce that Fxx(ε) = for allε> , so assumption (PM) holds.
Lemma If{xn}is a sequence in a Menger PMLS(X,F,∗)such that
lim
n→∞Fxn,xn+(t) = for all t> ,
then{xn}is a G-Cauchy sequence in(X,F,∗).
Proof Letn,p∈N∗be arbitrary and letε> . We observe that
Fxn,xn+p(ε)≥Fxn,xn+
ε
p
∗Fxn+,xn+
ε
p
∗Fxn+,xn+
ε
p
∗ · · · ∗Fxn+p–,xn+p
ε
p
.
Ifεandpare fixed, butn→ ∞, we deduce, taking into account that∗is a continuous t-norm, that
lim
n→∞Fxn,xn+p(ε)≥
lim
n→∞Fxn,xn+
ε
p
∗
lim
n→∞Fxn+,xn+
ε
p
∗ · · ·
∗
lim
n→∞Fxn+p–,xn+p
ε
p
= ∗∗ · · · ∗ = .
We have just proved that
lim
n→∞Fxn,xn+p(ε) = for allε> and allp∈N∗,
which means that{xn}is aG-Cauchy sequence in (X,F,∗).
4 Fixed point theorems in Menger probabilistic metric-like spaces
In this section we present an extension of Theorem in several ways: the probabilistic metric is more general, the contractivity condition is better and the involved auxiliary functions form a wider class.
Definition We shall denote byHthe family of all functionsh: (, ]→[,∞) satisfy-ing:
The previous conditions are guaranteed whenh: (, ]→[,∞) is a strictly decreasing bijection between (, ] and [,∞) such thathandh–are continuous (in a broad sense, it
is sufficient to assume the continuities ofhandh–on the extremes of the respective
do-mains). For instance, this is the case of the functionh(t) = /t– for allt∈(, ]. However, the functions inHneed not be continuous nor monotone.
Proposition If h∈H,then h() = .Furthermore,h(t) = if,and only if,t= .
Proof Using the sequencean= for alln∈Nin (H), we deduce that{h() =h(an)} →,
soh() = . On the other hand, assume that there existst∈(, ] such thath(t) = . If we definean=tfor alln∈N, then{h(an) = } →. By condition (H),{t=an} →, sot= .
The main result of the present manuscript is the following one.
Theorem Let(X,F,∗)be a G-complete Menger PMLS and let T:X→X be a mapping. Suppose that there exist c∈(, ),φ∈,ψ∈,and h∈Hsuch that
hFTx,Ty
φ(ct)≤ψhFx,y
φ(t) ()
for all x,y∈X and all t> for which Fx,y(φ(t)) > . If there exists x∈X such that
limt→∞Fx,Tx(t) = ,then T has at least one fixed point.
Additionally,assume that for all x,y∈Fix(T)with x=y,we havelimt→∞Fxy(t) = .Then
T has a unique fixed point.
Notice that we shall not use the left-continuity ofφ neither its strictly monotony. We shall only use thatφin nondecreasing. Hence,φbelongs to a more general class of auxiliary functions.
Proof Notice that condition () implies that ifFx,y(φ(t)) > , thenhmust be applicable to
FTx,Ty(φ(ct)). HenceFTx,Ty(φ(ct))∈domh= (, ], which means that
Fx,y
φ(t)> ⇒ FTx,Ty
φ(ct)> . ()
Letx∈X be the point such thatFx,Tx(tx) > , and let{xn} be the Picard sequence ofT based onx, that is,xn+=Txnfor alln∈N. If there exists somen∈Nsuch that
xn=xn+, thenxnis a fixed point ofT, and the existence part of the proof is finished. On the contrary case, assume thatxn=xn+for alln∈N.
Sincelimt→∞Fx,Tx(t) = , there existst> such thatFx,x(t) =Fx,Tx(t) > . More-over, aslimt→∞φ(t) =∞, it follows that there existss∈[,∞) (we can suppose, without
loss of generality, thats≥t) such thatφ(s) >t. Hence
Fxx
φ(s)
≥Fxx(t) > .
It follows from () that
Fxx
φ(cs)
=FTxTx
φ(cs)
and, by induction, it can be proved that
Fxnxn+
φcns
> for alln∈N.
Ifn,m,r∈Nandr≤n, thencns≤crs≤s≤csm. SinceφandFxnxn+are nondecreasing functions, it follows that
ifn,m,r∈Nandr≤n, then
<Fxnxn+
φcns
≤Fxnxn+
φcrs
≤Fxnxn+
φ(s)
≤Fxnxn+ φ s cm . ()
We claim that
lim
n→∞Fxn,xn+(s) = for alls> . ()
In order to prove it, lets> be arbitrary. Aslimr→∞(crs) = andφis continuous att= ,
thenlimr→∞φ(crs) =φ() = . Sinces> , there existsr∈Nsuch that
φcrs
≤s. ()
Letn∈Nbe such thatn>r. Applying the contractivity condition () tox=xnandy=xn+,
it follows that
hFxn,xn+
φcrs
=hFTxn–,Txn
φcrs
≤ψhFxn–,xn
φcr–s
, ()
where we have usedFxn–,xn(φ(c r–s
)) > by (). Repeating this argument, we find that
hFxn–,xn
φcr–s
=hFTxn–,Txn–
φcr–s
≤ψhFxn–,xn–
φcr–s
,
where we have usedFxn–,xn–(φ(c
r–s
)) > by (). Asψis nondecreasing, then
ψhFxn–,xn
φcr–s
≤ψhFxn–,xn–
φcr–s
. ()
Combining inequalities () and (), we deduce that
hFxn,xn+
φcrs
≤ψhFxn–,xn
φcr–s
≤ψhFxn–,xn–
φcr–s
.
Inequalities () permit us to repeat this argumentntimes, and it follows that
hFxn,xn+
φcrs
≤ψnhFx,x
φcr–ns
=ψn
h
Fx,x
for alln>r. As a consequence,
lim
n→∞ s
cn–r =∞ ⇒ nlim→∞φ
s
cn–r
=∞
⇒ lim
n→∞Fx,x
φ
s
cn–r
=
⇒ lim
n→∞h
Fx,x
φ
s
cn–r
= .
As the sequence{an=h(Fx,x(φ(
s
cn–r)))} → andh∈H, we have{ψn(an)} →. By (),
we deduce that
lim
n→∞h
Fxn,xn+
φcrs
= .
In particular, ash∈H, condition (H) implies that
lim
n→∞Fxn,xn+
φcrs
= .
Taking into account (), we observe that
Fxn,xn+
φcrs
≤Fxn,xn+(s)≤.
Therefore,
lim
n→∞Fxn,xn+(s) = ,
which means that () holds. Lemma guarantees that{xn}is aG-Cauchy sequence in
(X,F,∗). As it isG-complete, there existsz∈Xsuch that{xn} →z, that is,
∀ε> ,∀λ∈(, ),∃n∈N:
Fxnz(ε) > –λ,∀n≥n
. ()
We claim thatzis a fixed point ofT. To prove it, observe that, for allt> and alln∈N,
Fz,Tz(t)≥Fz,xn+
t
∗Fxn+,Tz
t
=Fz,xn+
t
∗FTxn,Tz
t
. ()
By (), it is clear that
lim
n→∞Fz,xn(s) = for alls> . ()
In particular,
lim
n→∞Fz,xn+
t
Let us show that the second factor in () also converges to whenntends to∞. Taking into account thatφ is continuous att= , we havelims→φ(s) =φ() = . Sincet/ > ,
there existsδ> such thatφ(δ) <t/. Sinceδ/c> ,φ(δ/c) > . By (),
lim
n→∞Fxn,z
φ δ c = .
Hence, there existsn∈Nsuch that
Fxn,z
φ δ c
> for alln≥n.
Applying the contractivity condition () tox=zandy=xn+forn≥n, we observe that
hFxn+,Tz
φ(δ)=hFTxn,Tz
φ(δ)≤ψ
h
Fxn,z
φ δ c . Therefore lim
n→∞Fxn,z
φ δ c = ⇒ lim
n→∞h
Fxn,z
φ δ c = ⇒ lim
n→∞ψ
h
Fxn,z
φ δ c = ⇒ lim
n→∞h
Fxn+,Tz
φ(δ)=
⇒ lim
n→∞Fxn+,Tz
φ(δ)= . Taking into account that
Fxn+,Tz
φ(δ)≤Fxn+,Tz
t
≤,
we deduce that
lim
n→∞FTxn,Tz
t
= lim
n→∞Fxn+,Tz
t
= . ()
Lettingn→ ∞in () and using () and (), we deduce that
Fz,Tz(t)≥ lim n→∞
Fz,xn+
t
∗FTxn,Tz
t = lim
n→∞Fz,xn+ t ∗ lim
n→∞FTxn,Tz
t
= ∗ = .
We have just proved thatFz,Tz(t) = for allt> , and axiom (PM) guarantees thatTz=z,
that is,zis a fixed point ofT.
limt→∞Fxy(t) = . Then there existst> such thatFxy(t) > . Moreover, there exists
s> such that
φ(s) >t.
Consequently, asφandFxyare nondecreasing functions,
Fxy
φ(s)
≥Fxy(t) > .
By (), we have
Fxy
φ(s)
> ⇒ Fxy
φ(cs)
=FTx,Ty
φ(cs)
> .
By induction,
Fxy
φcns
> for alln∈N.
We claim that
Fxy
φcrs
= for allr∈N. ()
To prove it, letr∈Nbe arbitrary and letn,m∈Nbe such thatn>r. Ascns
≤crs≤s≤ s
cm, andφandFxyare nondecreasing functions, it follows that
ifn,m∈Nandr≤n, then
<Fxy
φcns
≤Fxy
φcrs
≤Fxy
φ(s)
≤Fxy
φ s cm . ()
Applying the contractivity condition () toxandy, it follows that
hFxy
φcrs
=hFTx,Ty
φcrs
≤ψhFxy
φcr–s
, ()
where we have usedFxy(φ(cr–s)) > by (). Repeating this argument, we find that
hFxy
φcr–s
=hFTx,Ty
φcr–s
≤ψhFxy
φcr–s
,
where we have usedFxy(φ(cr–s)) > by (). Asψis nondecreasing, we have
ψhFxy
φcr–s
≤ψhFxy
φcr–s
. ()
Combining inequalities () and (), we deduce that
hFxy
φcrs
≤ψhFxy
φcr–s
≤ψhFxy
φcr–s
.
Inequalities () permit us to repeat this argumentntimes, and it follows that
hFxy
φcrs
≤ψnhFxy
φcr–ns
=ψn
for alln>r. As a consequence,
lim
n→∞
s
cn–r =∞ ⇒ nlim→∞φ
s
cn–r
=∞
⇒ lim
n→∞Fxy
φ
s
cn–r
=
⇒ lim
n→∞h
Fxy
φ
s
cn–r
= .
As the sequence{an=h(Fxy(φ(cns–r)))} → andh∈H, we have{ψn(an)} →. By (), we
deduce that
hFxy
φcrs
= .
In particular, ash∈H, Proposition implies that
Fxy
φcrs
= ,
which means that () holds. Next, let us show thatFxy(t) = for allt> . Lett>
arbi-trary. Sincelimn→∞(cns) = andlimn→∞φ(cns) =φ() = , there existsr∈Nsuch that
φ(crs
) <t. Hence
=Fxy
φcrs
≤Fxy(t) = ,
soFxy(t) = . Varyingt> , we conclude thatx=yby virtue of (PM), which contradicts
the fact thatx=y. As a result,Tcan only have a unique fixed point.
In the following result, condition (PM) guarantees that any initial conditionx∈X
yields a fixed point.
Corollary Let(X,F,∗)be a G-complete Menger PMLS verifying(PM)and let T:X→
X be a mapping.Suppose that there exist c∈(, ),φ∈,ψ∈,and h∈Hsuch that
hFTx,Ty
φ(ct)≤ψhFx,y
φ(t)
for all x,y∈X and all t> for which Fx,y(φ(t)) > .Then T has a unique fixed point.
The previous results are also valid for Menger PM-spaces.
Corollary Let(X,F,∗)be a G-complete Menger PMS and let T:X→X be a mapping. Suppose that there exist c∈(, ),φ∈,ψ∈,and h∈Hsuch that
hFTx,Ty
φ(ct)≤ψhFx,y
φ(t)
for all x,y∈X and all t> for which Fx,y(φ(t)) > . If there exists x∈X such that
limt→∞Fx,Tx(t) = ,then T has at least one fixed point.
Additionally,assume that for all x,y∈Fix(T)with x=y,we havelimt→∞Fxy(t) = .Then
Corollary Let(X,F,∗)be a G-complete Menger PMS verifying(PM)and let T:X→X
be a mapping.Suppose that there exist c∈(, ),φ∈,ψ∈,and h∈Hsuch that
hFTx,Ty
φ(ct)≤ψhFx,y
φ(t)
for all x,y∈X and all t> for which Fx,y(φ(t)) > .Then T has a unique fixed point.
The following statement trivially follows from Theorem using h(t) = /t– for all t∈(, ].
Corollary Theoremimmediately follows from Theorem.
Example Let (X,F,∗P) be the Menger PMLS introduced in Example and letT:X→
Xbe the self-mapping defined byTx=x/ for allx∈X. We know that (X,F,∗P) is a Menger
PMLS that also verifies (PM), but it is not a Menger PMS. In Example we showed
that (X,F,∗P) isG-complete. Assume that ψ(t) =φ(t) =t for allt∈[,∞), and leth:
(, ]→[,∞) be whatever strictly decreasing bijection between (, ] and [,∞) such thathandh–are continuous (for instance,h(t) = /t– for allt∈(, ], but any other
function verifying these properties yields the same result). In this context, the contractivity condition () is equivalent to
hFTx,Ty
φ(ct)≤ψhFx,y
φ(t) ⇔ hFTx,Ty(ct)
≤hFx,y(t)
⇔ FTx,Ty(ct)≥Fx,y(t).
Ifc= /, then, for allt> ,
FTx,Ty(ct) =Fx/,y/
t
=Fx/,y/
t
=max{x/,y/}
t
=
, if ≤ t ≤max{
x ,
y },
, if t >max{x,y}
=
, if ≤t≤max{x,y}, , ift>max{x,y}
=max{x,y}(t) =Fx,y(t).
As a result, the contractivity condition is verified. Hence, Corollary guarantees that T has a unique fixed point (which isz= ). Notice that other previous results are not applicable to this example because (X,F,∗P) is not a Menger PMS.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Author details
1Department of Quantitative Methods for Economics and Business, Faculty of Economics and Business Sciences, University of Granada, Campus Universitario de Cartuja, s/n, Granada, 18011, Spain.2Institute of Research and Development of Processes, Faculty of Science and Technology, University of Basque Country, Campus of Leioa, Barrio Sarriena, Leioa (Bizkaia), 48940, Spain.
Acknowledgements
The authors are very thankful to the anonymous reviewers for their careful reading of this manuscript and for their constructive reports, which have been very useful to improve the paper. M de la Sen is grateful to the Spanish Government for its support of this research through Grant DPI2012-30651 and to the Basque Government for its support of this research through Grants IT378-10 and SAIOTEK S-PE12UN015. He is also grateful to the University of Basque Country for its financial support through Grant UFI 2011/07. A-F Roldán-López-de-Hierro has been partially supported by Junta de Andalucía by project FQM-268 of the Andalusian CICYE. He is also grateful to the Department of Quantitative Methods for Economics and Business of the University of Granada for its support.
Endnote
a At present, a cumulative distribution function is usually defined as probability of a variable which is less than or
equal tot, not less thant; since Menger’s work in probabilistic spaces, it is commonly used the strict inequality.
Received: 18 May 2015 Accepted: 8 September 2015
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