R E S E A R C H
Open Access
Some fixed point theorems in Menger
PbM-spaces with an application
Firoozeh Hasanvand and Mahnaz Khanehgir
**Correspondence:
[email protected] Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran
Abstract
In this paper, we establish the structure of Menger PbM-spaces as a generalization of Menger PM-spaces. We present some fixed point theorems for a new class of contractive mappings in the framework of Menger PbM-spaces. We also provide examples to illustrate the results presented herein. Then we utilize our main result to obtain the existence and uniqueness of a solution for a Volterra type integral equation.
MSC: Primary 47H10; secondary 37C25; 54H25; 55M20
Keywords: contractive mapping; fixed point;b-metric space; Menger probabilistic metric space; Volterra integral equation
1 Introduction and preliminaries
The concept of a Menger probabilistic metric space (briefly, Menger PM-space) was initi-ated by Menger []. The idea of Menger was to use a distribution function instead of a non-negative number for the value of a metric. The notion of a probabilistic metric space cor-responds to the situation when we do not know exactly the distance between two points. Thus, one thinks of the distance between two pointsxandyas being probabilistic with
Fx,y(t) representing the probability that the distance betweenxandyis less thant. In , Sehgal and Bharucha-Reid [] obtained a generalization of the Banach con-traction principle on a complete Menger PM-space, which is a milestone in developing fixed point theory in a Menger PM-space. After that, Schweizer and Sklar [] studied the properties of Menger PM-spaces and gave some basic results on these spaces.
In recent times, the study on existence of fixed points for mappings satisfying general-ized contractive type conditions in Menger PM-spaces has attracted much attention (see [–]). This study was initiated by Ćirić in []; more details in []. Also a nice overview of this research can be found in the book of Hadžić and Pap [].
On the other hand, the notion of ab-metric space was studied by Czerwik [, ] and many fixed point results were obtained for single and multivalued mappings by Czerwik and many other authors (see [–] and references cited therein).
In this paper, motivated by [, ], we establish the structure of Menger PbM-spaces and obtain fixed point results for classes of mappings that extend the notion of generalized
β-type contractive mappings introduced by Gopalet al.[] in Menger PbM-spaces. We also give some examples to show that our fixed point theorems for the new type of
tractive mappings are independent. Then we use our main results to obtain the existence and uniqueness of a solution for a Volterra type integral equation.
In the following, we provide some notations, definitions and auxiliary facts will be used later in this paper. Throughout this paper,R+denotes the set of nonnegative real numbers.
Definition .[, ] LetXbe a nonempty set, and let the functionald:X×X→[,∞) satisfy:
(b) d(x,y) = if and only ifx=y, (b) d(x,y) =d(y,x)for allx,y∈X,
(b) there exists a real numbers≥such thatd(x,z)≤s[d(x,y) +d(y,z)]for all x,y,z∈X.
Thendis called ab-metric onXand a pair (X,d) is called ab-metric space with coeffi-cients.
Definition .[] Let (X,d) be ab-metric space. Then a sequence{xn}inXis called:
(i) convergent if and only if there existsx∈Xsuch thatd(xn,x)→asn→ ∞; in this
case, we writelimn→∞xn=x;
(ii) Cauchy if and only ifd(xn,xm)→asm,n→ ∞. Theb-metric space(X,d)is
complete if every Cauchy sequence inXconverges inX.
Remark .[] Notice that in ab-metric space (X,d) the following assertions hold:
(i) a convergent sequence has a unique limit; (ii) each convergent sequence is Cauchy; (iii) (X,d
–
→)is anL-space (see [, ]);
(iv) in general, ab-metric is not continuous;
(v) in general, ab-metric does not induce a topology onX.
Example .[] LetX= [,∞) and defined:X×X→[,∞) as
d(x,y) =|x–y| for allx,y∈X.
Then (X,d) is a completeb-metric space with coefficients= > , but it is not a usual metric space.
Definition .[] Let (X,d) and (X,d) be twob-metric spaces with coefficientsands, respectively. A mappingT :X→Xis called continuous if for each sequence{xn}inX, which converges tox∈Xwith respect tod, then{Txn}converges toTxwith respect tod.
We recall the following definitions in the class of Menger PM-spaces.
Definition .[] A binary operationT: [, ]×[, ]→[, ] is a continuoust-norm if the following conditions hold:
(i) Tis commutative and associative, (ii) Tis continuous,
(iii) T(a, ) =afor alla∈[, ],
The following are three basic continuoust-norms:
() The minimumt-norm, sayTM, defined byTM(a,b) =min{a,b}.
() The productt-norm, sayTP, defined byTP(a,b) =ab.
() The Lukasiewiczt-norm, sayTL, defined byTL(a,b) =max{a+b– , }. Theset-norms are related in the following way:TL≤TP≤TM.
Definition .[] A functionF: (–∞, +∞)→[, ] is called a distribution function if it is non-decreasing and left-continuous withlimt→–∞F(t) = . If in additionF() = , then Fis called a distance distribution function.
Definition .[] A distance distribution functionFsatisfyinglimt→+∞F(t) = is called
a Menger distance distribution function. The set of all Menger distance distribution func-tions is denoted byD+. A special Menger distance distribution function is given by
H(t) =
⎧ ⎨ ⎩
, t≤,
, t> .
Definition .[] A Menger probabilistic metric space (briefly, Menger PM-space) is a triple (X,F,T) whereXis a nonempty set,T is a continuoust-norm, andFis a mapping fromX×XintoD+such that, ifF
x,ydenotes the value ofFat the pair (x,y), the following conditions hold:
(PM) Fx,y(t) =H(t)if and only ifx=y,
(PM) Fx,y(t) =Fy,x(t),
(PM) Fx,y(t+s)≥T(Fx,z(t),Fz,y(s))for allx,y,z∈Xands,t≥.
Definition .[] Let (X,F,T) be a Menger PM-space. Then:
(i) A sequence{xn}inXis said to be convergent toxinXif, for everyε> andλ>
there exists a positive integerNsuch thatFxn,x(ε) > –λ, whenevern≥N. (ii) A sequence{xn}inXis called a Cauchy sequence if, for everyε> andλ> there
exists a positive integerNsuch thatFxn,xm(ε) > –λ, whenevern,m≥N. (iii) A Menger PM-space is said to be complete if every Cauchy sequence inXis
convergent to a point inX.
According to [], the (ε,λ)-topology in a Menger PM-space (X,F,T) is introduced by the family of neighborhoodsNxof a pointx∈Xgiven byNx={Nx(ε,λ) :ε> ,λ∈(, )}, whereNx(ε,λ) ={y∈X:Fx,y(ε) > –λ}.
The (ε,λ)-topology is a Hausdorff topology. In this topology a functionf is continuous inx∈Xif and only iff(xn)→f(x), for every sequencexn→x.
Example .[] Let (X,d) be a metric space. Define a mappingF:X×X→D+by
F(x,y)(t) =Fx,y(t) =H
t–d(x,y), ∀x,y∈X,t∈R.
Definition .[] A functionφ:R+→R+ is said to be a-function if it satisfies the following conditions:
(i) φ(t) = if and only ift= ,
(ii) φ(t)is strictly monotone increasing andφ(t)→ ∞ast→ ∞, (iii) φis left-continuous in(,∞),
(iv) φis continuous at.
In the sequel, the class of all-functions will be denoted by.
We conclude this section recalling the following fixed point theorem of Gopalet al., see []. Before this, we quote some definitions.
Definition .[] Let (X,F,T) be a Menger PM-space andf :X→Xbe a given map-ping. We say thatf is a generalizedβ-type contractive mapping if there exists a function
β:X×X×(,∞)→(,∞) such that
β(x,y,t)Ffx,fy
φ(t)≥min
Fx,y
φ
t c
,Fx,fx
φ
t c
,Fy,fy
φ
t c
,
Fx,fy
φ
t c
,Fy,fx
φ
t c
,
for allx,y∈Xand for allt> , whereφ∈andc∈(, ).
Definition .[] Let (X,F,T) be a Menger PM-space,f :X→Xbe a given mapping andβ:X×X×(,∞)→(,∞) be a function. We say thatf isβ-admissible if
x,y∈X, for allt> , β(x,y,t)≤ ⇒ β(fx,fy,t)≤.
Theorem .[] Let(X,F,T)be a complete Menger PM-space with continuous t-norm T which satisfies T(a,a)≥a with a∈[, ].Let f :X→X be a generalizedβ-type contractive mapping satisfying the following conditions:
(i) f isβ-admissible,
(ii) there existsx∈Xsuch thatβ(x,fx,t)≤for allt> ,
(iii) if{xn}is a sequence inXsuch thatβ(xn,xn+,t)≤for alln∈Nand for allt>
andxn→xasn→ ∞,thenβ(xn,x,t)≤for alln∈Nand for allt> .
Then f has a fixed point.
We denote byFix(f) the set of fixed points off. Consider the following condition:
(J) For allu,v∈Fix(f)and for allt> there existsz∈Xsuch thatβ(z,fz,t)≤with β(u,z,t)≤andβ(v,z,t)≤.
Theorem .[] Adding condition(J)to the hypotheses of Theorem.,we find that f has a unique fixed point.
2 Main results
Definition . A Menger probabilisticb-metric space (briefly, Menger PbM-space) with coefficient αis a triple (X,F,T) whereX is a nonempty set,T is a continuoust-norm,
Fis a mapping fromX×XintoD+(forx,y∈X, we denoteF(x,y) byF
x,y), andαis a real number in (, ] such that the following conditions hold:
(PbM) Fx,y(t) =H(t)for allt∈R, if and only ifx=y,
(PbM) Fx,y(t) =Fy,x(t)for allx,y∈Xandt∈R,
(PbM) Fx,y(t+s)≥T(Fx,z(αt),Fz,y(αs))for allx,y,z∈X, andt,s≥.
It should be noted that the class of Menger PbM-spaces is larger than the class of Menger
PM-spaces, since a Menger PbM-space is a Menger PM-space whenα= .
Definition . Let (X,F,T) be a Menger PbM-space. Then a sequence{xn}inXis called:
(i) convergent toxinX(often denoted byxn→x) if for any givenε> andλ∈(, ),
there exists a positive integerN=N(ε,λ)such thatFxn,x(ε) > –λwhenevern≥N, which is equivalent tolimn→∞Fxn,x(t) = for allt> ;
(ii) Cauchy if for any givenε> andλ∈(, ), there exists a positive integerN=N(ε,λ)
such thatFxn,xm(ε) > –λwhenevern,m≥N.
The Menger PbM-space (X,F,T) is said to be complete if every Cauchy sequence inXis convergent inX.
Remark . In a Menger PbM-space (X,F,T) the following assertions hold:
(i) a convergent sequence has a unique limit,
(ii) in general, a Menger PbM-space is not a topological space.
In the following we present examples which show that introducing a Menger PbM-space instead of Menger PM-space is meaningful.
Example . LetX=R+. DefineF:X×X→D+by
Fx,y(t) =
⎧ ⎨ ⎩
t
t+|x–y|, ift> ,
, ift≤,
for all x,y∈X. It is easy to show that (X,F,TM) is a complete Menger PbM-space with
α= . However, it is not a Menger PM-space. We show that (PM) does not hold. To prove this, letx= ,y= ,z= ,t= , andt= . ThenFx,y(t+t) =,Fx,z(t) =, and
Fz,y(t) =, henceFx,y(t+t) = =TM(Fx,z(t),Fz,y(t)).
Example . Let (X,d) be ab-metric space with coefficients≥. Define a mappingF:
X×X→D+as in Example .. Then (X,F,TM) is a Menger PbM-space withα=s. We know thatt+t–d(x,y)≥t–sd(x,z) +t–sd(z,y), for eachx,y,z∈X,t,t∈R, and hence by the properties ofH, we get
Ht+t–d(x,y)
≥Ht–sd(x,z) +t–sd(z,y)
≥minHt–sd(x,z)
,Ht–sd(z,y)
=min
H
t
s –d(x,z) ,H
t
It gives (PbM). Furthermore, a straightforward computation shows that if (X,d) is com-plete, then (X,F,TM) is complete.
Now assume that (X,d) is as in Example .. Then by the above comment, (X,F,TM) is a complete Menger PbM-space withα= . We claim that (X,F,TM) is not a Menger PM-space. Indeed, (PM) does not hold. To see this, let x= , y= , z= , t = , andt= . Then Fx,y(t+t) = and Fx,z(t) =Fz,y(t) = , henceFx,y(t+t) = =
TM(Fx,z(t),Fz,y(t)).
Note that the above examples are Menger PbM-spaces (but are not Menger PM-spaces) ifTMsubstitutes withTPorTL.
The following result is used in our next considerations. It is a generalization of [], Lemma . in Menger PbM-spaces.
Lemma . Let (X,F,T) be a Menger PbM-space with coefficient α. Then the func-tion F is a lower semi-continuous funcfunc-tion of points,i.e.,for every fixed t> and every two convergent sequences {xn}, {yn} in X such that xn→x and yn→y it follows that limn→∞infFxn,yn(t) =Fx,y(t).
Proof Lett> andε> be given. SinceFx,yis left-continuous attso there existsδsuch that <δ<t andFx,y(t) –Fx,y(t–δ) <ε. Supposehis an arbitrary fixed real number with < h<δ, thenFx,y(t) –Fx,y(t– h) <ε. Using again left-continuity ofFx,yatt–δ, there existsδ> such thatFx,y(t–δ) –Fx,y(t–δ) <ε. By repeating this argument we can findk∈N,δi,δi+> (i= , . . . ,k) in whichFx,y(t–δi) –Fx,y(t–δi+) <εandαt– αh∈ (t–δk,t–δk+). We deduce that
Fx,y(t) –Fx,y
αt– αh=Fx,y(t) –Fx,y(t–δ)
+Fx,y(t–δ) –Fx,y(t–δ)
+· · · +Fx,y(t–δk) –Fx,y
αt– αh
< (k+ )ε. ()
SetFx,y(αt– αh) =a. Taking into account continuity ofTandT(a, ) =a, there is a real numberlin (, ), fulfills
T(a,l) >a–ε
and T
a–ε
,l >a– ε
. ()
On the other hand, sincexn→xandyn→y, there exists an integerMh,lsuch that
Fxn,x
αh>l and Fyn,y(αh) >l, ()
whenevern>Mh,l. Now, by (PbM)
Fxn,yn(t)≥T
Fxn,y(αt–αh),Fy,yn(αh)
()
and
Fxn,y(αt–αh)≥T
Fxn,x
αh,Fx,y
From (), (), and (), we obtain
Fxn,y(αt–αh)≥T(a,l) >a– ε
. ()
Thus, on combining (), (), (), (), and (), we get
Fxn,yn(t)≥T
a–ε
,l >a– ε
>Fx,y(t) –
(k+ )ε
.
This completes the proof.
3 Generalized
β
-γ
-type contractive mappingsIn this section, we generalize the results obtained by Gopalet al.[] for the wider class of generalizedβ-γ-type contractive mappings in Menger PbM-spaces.
Definition . Let (X,F,T) be a Menger PbM-space with coefficientαandf :X→Xbe a given mapping. We say thatf is a generalizedβ-γ-type contractive mapping of degreek
(k∈N), if there exist two functionsβ:X×X×(,∞)→(,∞) andγ :X×X×(,∞)→ (,∞) such that
βx,y,αktFfx,fy
αkφ(t)≥γ
fx,fy,αk–t c min
Fx,y
αk–φ
t c
,
Fx,fx
αk–φ
t c
,Fy,fy
αk–φ
t c
,
Fx,fy
αk–φ
t c
,Fy,fx
αk–φ
t c
, ()
for allx,y∈Xand for allt> , whereφ∈andc∈(, ). Further, the mappingf is called a generalizedβ-γ-type contractive mapping if it is a generalized β-γ-type contractive mapping of degreekfor eachk∈N.
Definition . Let (X,F,T) be a Menger PbM-space,f :X→Xbe a mapping, andβ :
X×X×(,∞)→(,∞) andγ :X×X×(,∞)→(,∞) be two functions. We say that
f is (β,γ)-admissible ifx,y∈X, for allt> ,β(x,y,t)≤⇒β(fx,fy,t)≤ andγ(x,y,t)≥ ⇒γ(fx,fy,t)≥.
Imitating the proof of [], Lemma ., we can easily obtain the following lemma.
Lemma . Let(X,F,T)be a Menger PbM-space with coefficientα.Letφbe a-function.
Then the following statement holds:
If forx,y∈X,c∈(, ),andk∈Nwe haveFx,y(αkφ(t))≥Fx,y(αk–φ(tc))for allt> ,
thenx=y.
Our first main result is the following.
Theorem . Let(X,F,T)be a complete Menger PbM-space with coefficientα,which sat-isfies T(a,a)≥a with a∈[, ].Let f :X→X be a generalizedβ-γ-type contractive map-ping satisfying the following conditions:
(ii) there existsx∈Xsuch thatβ(x,fx,t)≤andγ(x,fx,t)≥for allt> ,
(iii) if{xn}is a sequence inXsuch thatβ(xn–,xn,t)≤andγ(xn,xn+,t)≥for all
n∈N,and for allt> andxn→xasn→ ∞,thenβ(xn–,x,t)≤and
γ(xn,fx,t)≥for alln∈Nand for allt> .
Then f has a fixed point.
Proof SinceT(a,a)≥afor alla∈[, ],T ≥TM. Letx∈Xbe such that (ii) holds and define a sequence{xn}inXso thatxn+=fxn, for alln= , , . . . . We supposexn+=xnfor alln= , , . . . , otherwisef has trivially a fixed point. From (i), (ii), and by induction, we getβ(xn–,xn,t)≤ andγ(xn,xn+,t)≥ for alln∈Nand allt> . Taking into account the continuity ofφat zero, we can findr> such thatt>φ(r) and therefore we have
Fxn,xn+(t)≥β
xn–,xn,αkr
Ffxn–,fxn
αkφ(r)
≥γ
xn,xn+,αk–
r c min
Fxn–,fxn–
αk–φ
r c
,Fxn–,xn
αk–φ
r c
,
Fxn,fxn
αk–φ
r c
,Fxn–,fxn
αk–φ
r c
,Fxn,fxn–
αk–φ r c ≥min Fxn–,xn
αk–φ
r c
,Fxn,xn+
αk–φ
r c
.
We will show that
Fxn,xn+
αkφ(r)≥Fxn–,xn
αk–φ
r c
. ()
If we assume thatFxn,xn+(αk–φ(
r
c)) is the minimum, then from Lemma ., we getxn=
xn+, which is a contradiction with the assumptionxn=xn+and soFxn–,xn(αk–φ(
r c)) is the minimumi.e., inequality () holds. Now from (), one obtains that
Fxn,xn+
αkt≥Fxn,xn+
αkφ(r)≥Fxn–,xn
αk–φ
r c
≥ · · · ≥Fx,x
αk–nφ r cn , that is,
Fxn,xn+
αkt≥Fx,x
αk–nφ
r cn
,
for arbitraryn∈N. Next, letm,n∈Nwithm>n, then by (PbM) and strictly increasing ofφwe have
Fxn,xm
(m–n)t
≥minFxn,xn+(αt), . . . ,Fxm–,xm–
αm–n–t,Fxm–,xm
αm–n–t
≥min
Fx,x
α–nφ
r cn
, . . . ,Fx,x
α–nφ
r cm–
,Fx,x
α–nφ r cm– ≥min Fx,x
α–nφ
r cn
, . . . ,Fx,x
α–nφ
r cm–
,Fx,x
α–nφ
r cm–
=Fx,x
α–nφ
r cn
Since α–nφ(r
cn)→ ∞ as n→ ∞, for fixed ε∈ (, ) there exists n ∈N such that
Fx,x(α–nφ(
r
cn)) > –ε, whenevern≥n. This implies that, for everym>n≥n,
Fxn,xm
(m–n)t> –ε.
Sincet> andε∈(, ) are arbitrary, we deduce that{xn}is a Cauchy sequence in the complete Menger PbM-space (X,F,T). Then there existsu∈Xsuch thatxn→uasn→
∞. We are going to show thatuis a fixed point off. Using (PbM), we have
Ffu,u(t)≥T
Ffu,xn
αφ(r),Fxn,u
αt–αφ(r)
≥minFfu,xn
αφ(r),Fxn,u
αt–αφ(r).
Note that, ifxn=fufor infinitely many values ofn, thenu=fu, and hence the proof is finished. Therefore, we assume thatxn=fufor alln∈N. Now, sincexn→u, then, for any arbitraryε∈(, ) andnlarge enough, we getFxn,u(αt–αφ(r)) > –ε. Hence,Ffu,u(t)≥ min{Ffu,xn(αφ(r)), –ε}. Sinceε> is arbitrary, we haveFfu,u(t)≥Ffu,xn(αφ(r)). Next, using
(iii) we get
Fu,fu(t)≥Fxn,fu
αφ(r)
=Ffxn–,fu
αφ(r)
≥β(xn–,u,αr)Ffxn–,fu
αφ(r)
≥γ
fxn–,fu,
r c min
Fxn–,u
φ r c
,Fxn–,xn φ r c , Fu,fu φ r c
,Fxn–,fu αφ r c
,Fu,xn αφ r c ≥min Fxn–,u
φ r c
,Fu,fu
φ r c
,Fxn–,xn φ r c .
It follows that
Fu,fu(t)≥ lim
n→∞infFxn,fu
αφ(r)
≥ lim n→∞inf min
Fxn–,u
φ r c
,Fu,fu
φ r c
,Fxn–,xn φ r c ≥min
–ε,Fu,fu
φ r c
, –ε
.
Finally, sinceε∈(, ) is arbitrary, then Ffu,u(αφ(r))≥Fu,fu(φ(rc)). From Lemma ., we
conclude thatu=fuand so we achieve our goal.
In the following we present an example of a generalizedβ-γ-type contractive mapping, which is not a generalizedβ-type contractive mapping.
Example . LetX= [
fromX×X×(,∞) into (,∞) as follows:
fx=
⎧ ⎨ ⎩
, ifx∈[, ], , otherwise,
β(x,y,t) = ,
γ(x,y,t) =
⎧ ⎨ ⎩
, ifx,y∈[
, ], orx,y∈/[ , ],
, otherwise,
for allt> . Now, we considerφ:R+→R+ defined byφ(t) =tand letc=
. To prove thatf is a generalizedβ-γ-type contractive mapping, it suffices to check the following condition:
βx,y,αktFfx,fy
αkφ(t)
≥γ
fx,fy,αk–t c min
Fx,y
αk–φ
t c
,Fx,fx
αk–φ
t c
,
Fy,fy
αk–φ
t c
,Fx,fy
αk–φ
t c
,Fy,fx
αk–φ
t c
.
We distinguish three cases:
CaseI. Ifx,y∈[
, ] orx,y∈/[
, ], then the left-hand side of above inequality is equal to
andγ(fx,fy,αk– t
c) =γ(, , t
k–) =γ(, ,kt–) =. Hence, the inequality obviously true. CaseII. Ifx∈/[
, ] andy∈[
, ], then
βx,y,αktFfx,fy
αkφ(t)= t t+ k+ ≥γ
fx,fy,αk–t c Ffx,y
αk–φ
t c
= t
t+ k+|y– | and hence the inequality is again true.
CaseIII. Ifx∈[
, ] andy∈/[
, ], then
βx,y,αktFfx,fy
αkφ(t)= t t+ k+ ≥γ
fx,fy,αk–t c Fx,fy
αk–φ
t c
= t
t+ k+|x– | and hence the inequality is again true.
Also, if we takeγ(x,y,t) = for allx,y∈Xand allt> , thenf is not a generalizedβ-type contractive mapping. Indeed, forx= ,y= , andt= kwe have
≥min
+c, , ,
+c,
+c
=
+c.
Example . LetX,F,f be as in Example .. Define the functionsβ:X×X×(,∞)→ (,∞) andγ :X×X×(,∞)→(,∞) as follows:
β(x,y,t) =
⎧ ⎨ ⎩
, ifx,y∈[, ], t+
t+|x–y|, otherwise,
γ(x,y,t) =
⎧ ⎨ ⎩
, ifx=y= ,
t+
t+|x–y|, otherwise,
for allt> .
Now, we considerφ:R+→R+defined byφ(t) =tand letc=. Thenf is a generalized
β-γ-type contractive mapping. We distinguish three cases:
Case I. If x,y∈[, ], then the left-hand side of above inequality is equal to and
γ(fx,fy,αk–ct) =γ(, ,kt–) = . Hence, the inequality is obviously true. Case II. If x,y∈/ [
, ], then β(x,y,α kt)F
fx,fy(αkφ(t)) =γ(fx,fy,αk–tc)Fx,y(αk–φ(tc)) = t+k
t+k–|x–y| and hence the inequality is again true.
CaseIII. Ifx∈[, ] andy∈/[, ] orx∈/[, ] andy∈[, ], then we have the same result as case II.
On the other hand, f does not satisfy inequality () if we assume that β(x,y,t) =
γ(x,y,t) = for allx,y∈Xand allt> . Indeed, forx= andy= we get
t
t+ k ≥min
t
t+ k–c, , ,
t t+ k–c,
t t+ k–c
= t
t+ k–c,
which givesc≥, a contradiction.
Under an additional hypothesis onf, from Theorem ., we obtain the uniqueness of the fixed point.
(J) For allu,v∈Fix(f)and for allt> there existsz∈X such thatβ(z,fz,t)≤ with β(u,z,t)≤, andβ(v,z,t)≤andγ(z,fz,t)≥withγ(u,z,t)≥andγ(v,z,t)≥.
Theorem . Adding condition(J)to the hypotheses of Theorem.,we find that f has a unique fixed point.
Proof Letu,v∈Xbe such thatu=fuandv=fv. From condition (J), there existsz∈Xsuch thatβ(z,fz,t)≤ withβ(u,z,t)≤ andβ(v,z,t)≤, andγ(z,fz,t)≥ withγ(u,z,t)≥ andγ(v,z,t)≥. By virtue of the fact thatf is (β,γ)-admissible, we deduce that
βfz,fz,t≤, β(u,fz,t)≤, β(v,fz,t)≤,
and
By induction, we derive
β(zn,zn+,t)≤, β(u,zn,t)≤, β(v,zn,t)≤,
γ(zn+,zn+,t)≥, γ(u,zn+,t)≥, γ(v,zn+,t)≥,
for allt> , wherezn=fnz(n∈N). By continuity ofφ, there existsr> such thatt>φ(r) and therefore by (PbM) and (PbM) we have
Fu,zn+(t)≥Fu,zn+
αkφ(r)
=Ffu,fzn
αkφ(r)
≥βu,zn,αkr
Ffu,fzn
αkφ(r)
≥γ
fu,fzn,αk–
r c min
Fu,zn
αk–φ
r c
,Fu,fu
αk–φ
r c
,
Fzn,fzn
αk–φ
r c
,Fu,fzn
αk–φ
r c
,Fzn,fu
αk–φ r c ≥min Fu,zn
αk–φ
r c
,Fzn,zn+
αk–φ
r c
,
wherek∈N. Now, we consider following cases:
CaseI. IfFzn,zn+(α
k–φ(r
c)) is the minimum, then by (), (PbM), and (PbM), it follows that
Fu,zn+
αkφ(r)
≥Fzn,zn+
αk–φ
r c
=Ffzn–,fzn
αk–φ r c ≥β
zn–,zn,αk–
r
c Ffzn–,fzn
αk–φ r c ≥γ
zn,zn+,αk–
r c min
Fzn–,zn
αk–φ
r c
,Fzn–,fzn–
αk–φ
r c
,
Fzn,fzn
αk–φ
r c
,Fzn–,fzn
αk–φ
r c
,Fzn,fzn–
αk–φ r c ≥min Fzn–,zn
αk–φ
r c
,Fzn,zn+
αk–φ
r c
.
Now, ifFzn,zn+(αk–φ(
r
c)) is the minimum for somen∈N, then by Lemma ., we de-duce that zn=zn+. SinceFu,zn+(αkφ(r))≥Fzn,zn+(αk–φ(
r
c)) = ,u=zn+. Consequently
β(v,u,t)≤ andγ(fv,fu,t)≥ for allt> and so by (), (PbM), and (PbM) we have
Fv,u
αkφ(t)≥βv,u,αktFfv,fu
αkφ(t)
≥γ
fv,fu,αk–t c min
Fv,u
αk–φ
t c
,Fv,fv
αk–φ
t c
Fu,fu
αk–φ
t c
,Fv,fu
αk–φ
t c
,Fu,fv
αk–φ
t c
≥Fv,u
αk–φ
t c
.
Again, by Lemma ., we conclude thatu=v.
On the other hand, ifFzn–,zn(αk–φ(cr)) is the minimum, then
Fzn,zn+
αk–φ
r c
≥Fzn–,zn
αk–φ
r c
≥ · · · ≥Fz,z
αk–(n+)φ
r cn+
,
and, lettingn→ ∞, we getFzn,zn+(αk–φ(
r
c))→. Thereforelimn→∞Fu,zn+(t) = , which implies thatzn+→uasn→ ∞. A similar method shows thatzn+→v, forn→ ∞. Since the limit is unique,u=v.
CaseII. Suppose thatFu,zn(α
k–φ(r
c)) is the minimum, then we get
Fu,zn+
αkφ(r)≥Fu,zn
αk–φ
r c
≥Fu,zn–
αk–φ
r c
≥ · · ·
≥Fu,z
αk–(n+)φ
r cn+
.
Lettingn→ ∞, we obtainlimn→∞Fu,zn+(αkφ(r)) = , that is,zn+→uasn→ ∞. A similar argument shows thatzn+→v, forn→ ∞. Now, uniqueness of the limit gives usu=v,
and the proof is complete.
Our last existence theorem is a version of [], Theorem ., for generalizedβ-γ-type contractive mappings in Menger PbM-spaces.
Theorem . Let(X,F,T)be a complete Menger PbM-space with coefficientα,and f :
X→X be a mapping.Assume that there existβ:X×X×(,∞)→(,∞)andγ :X× X×(,∞)→(,∞)such that the following conditions hold:
(i)
βx,y,αktFfx,fy
αkφ(t)
≥γ
fx,fy,αk–t c min
Fx,y
αk–φ
t c
,Fx,fx
αk–φ
t c
,
Fy,fy
αk–φ
t c
,Fy,fx
αk–φ
t c
,
for allx,y∈X,for allt> and for allk∈N,wherec∈(, )andφ∈; (ii) f is(β,γ)-admissible;
(iii) there existsx∈Xsuch thatβ(x,fx,t)≤andγ(x,fx,t)≥for allt> ;
(iv) for each sequence{xn}inXsuch thatβ(xn–,xn,t)≤andγ(xn,xn+,t)≥,for all
n∈Nand for allt> ,there existsk∈Nsuch thatβ(xm–,xn–,t)≤and
γ(xm,xn,t)≥,for allm,n∈Nwithm>n≥kand for allt> ;
(v) if{xn}is a sequence inXsuch thatβ(xn–,xn,t)≤andγ(xn,xn+,t)≥for all
n∈Nand for allt> andxn→xasn→ ∞,thenβ(xn–,x,t)≤and
Then f has a fixed point. If in addition,condition(J)holds,then f has a unique fixed point.
Proof Letx∈Xbe such that (iii) holds. Define a sequence{xn}inXsuch thatxn+=fxnfor alln= , , . . . . We suppose thatxn+=xnfor alln= , , . . . , otherwisef has trivially a fixed point. By (ii) and (iii), and applying induction, we getβ(xn–,xn,t)≤ andγ(xn,xn+,t)≥ for alln∈Nand for allt> . By continuity ofφ at zero, we can findr> such that
t>φ(r), thusβ(xn–,xn,αkr)≤ andγ(xn,xn+,αk–rc)≥, wherek∈N. It follows from conditions (i) and (PbM) that
Fxn,xn+(t)≥Fxn,xn+
αkφ(r)
≥βxn–,xn,αkr
Ffxn–,fxn
αkφ(r)
≥γ
xn,xn+,αk–
r c min
Fxn–,xn
αk–φ
r c
,Fxn–,xn
αk–φ
r c
,
Fxn,xn+
αk–φ
r c
,Fxn,xn
αk–φ
r c
≥γ
xn,xn+,αk–
r c min
Fxn–,xn
αk–φ
r c
,Fxn,xn+
αk–φ
r c
≥min
Fxn–,xn
αk–φ
r c
,Fxn,xn+
αk–φ
r c
.
Next, if Fxn,xn+(α
k–φ(r
c)) is the minimum, then Fxn,xn+(α
kφ(r)) ≥ F xn,xn+(α
k–φ(r c)) and so by Lemma ., xn=xn+, which contradicts the assumption xn=xn+. Now if
Fxn–,xn(αk–φ(rc)) is the minimum, then
Fxn,xn+(t)≥Fxn,xn+
αkφ(r)≥Fxn–,xn
αk–φ
r c
≥ · · · ≥Fx,x
αk–nφ
r cn
.
Lettingn→ ∞, then
Fxn,xn+(t)→. ()
We claim that{xn}is a Cauchy sequence. Suppose the contrary. Then there existε> ,
λ∈(, ) for which we can find subsequences{xm(s)}and{xn(s)}of{xn}such thatn(s) is the smallest index for which
s<m(s) <n(s), Fxm(s),xn(s)(ε)≤ –λ, Fxm(s),xn(s)–(ε) > –λ. () By the properties ofφthere existsε> such that
φ(ε) <ε. ()
From () and (), we deduce thatFxm(s),xn(s)(αφ(ε))≤ –λ, so{xn}is not Cauchy sequence with respect toαφ(ε) andλ. Thus there exist increasing sequences of integersm(s) and
n(s), such thatn(s) is the smallest index for which
s<m(s) <n(s), Fxm(s),xn(s)
αφ(ε)
≤ –λ, Fxm(s),xn(s)–
αφ(ε)
Take a real numberηsuch that <η<φ(ε
c) –φ(ε). From () it follows that
Fxm(s),xn(s)–
αφ
ε
c –αη > –λ.
Then, for any <λ<λ< , by () it is possible to find a positive integerNsuch that for alls>N, we have
Fxm(s)–,xm(s)(αη) > –λ, Fxn(s)–,xn(s)(αη) > –λ. ()
By () and also applying (PbM), we have
Fxm(s)–,xn(s)–
φ
ε
c
≥T
Fxm(s)–,xm(s)(αη),Fxm(s),xn(s)–
αφ
ε
c –αη
>T( –λ, –λ).
Sinceλis arbitrary andTis continuous, it follows that
Fxm(s)–,xn(s)–
φ
ε
c
> –λ. ()
A direct consequence of () is
Fxm(s)–,xm(s)
φ
ε
c
≥Fxm(s)–,xm(s)
αφ
ε
c
≥Fxm(s)–,xm(s)(αη) > –λ> –λ. ()
A similar relation holds when one substitutesxm(s)–andxm(s)withxn(s)–andxn(s), respec-tively. On the other hand, we observe that
Fxm(s),xn(s)–
φ
ε
c
≥Fxm(s),xn(s)–
αφ
ε
c
≥Fxm(s),xn(s)–
αφ
ε
c –αη > –λ. ()
Applying assumptions (i), (iv), and (), (), (), () we get
–λ≥Fxm(s),xn(s)
αφ(ε)
=Ffxm(s)–,fxn(s)–
αφ(ε)
≥β(xm(s)–,xn(s)–,αε)Ffxm(s)–,fxn(s)–
αφ(ε)
≥γ
fxm(s)–,fxn(s)–,
ε
c min
Fxm(s)–,xn(s)–
φ
ε
c
,Fxm(s)–,xm(s)
φ
ε
c
,
Fxn(s)–,xn(s)
φ
ε
c
,Fxn(s)–,xm(s)
φ
ε
c
>γ
fxm(s)–,fxn(s)–,
ε
c { –λ, –λ, –λ, –λ}
This is a contradiction; therefore{xn}is a Cauchy sequence in the complete Menger PbM-space. Thusxn→uasn→ ∞for someu∈X.
Now, we show thatuis a fixed point off. We have
Ffu,u(t)≥T
Ffu,xn
αφ(r),Fxn,u
αt–αφ(r). ()
Sinceφ is continuous, there existsr> such thatt>φ(r). Further, sinceu=limn→∞xn, then, for arbitraryδ∈(, ), there existsn∈Nsuch that for alln≥n, we get
Fxn,u
αt–αφ(r)> –δ. ()
Hence, from () and (), we find that
Ffu,u(t)≥T
Ffu,xn
αφ(r), –δ.
Sinceδ> is arbitrary andTis continuous, we can writeFfu,u(t)≥Ffu,xn(αφ(r)). Without
loss of generality we may assume thatxn=fufor alln∈N, otherwise if for infinitely many values ofn,xn=fu, thenu=fu, and hence the proof is finished. Applying (i) and (v), we derive
Fu,fu(t)≥Fxn,fu
αφ(r)
≥β(xn–,u,αr)Ffxn–,fu
αφ(r)
≥γ
fxn–,fu,
r c min
Fxn–,u
φ
r c
,Fxn–,fxn–
φ
r c
,
Fu,fu
φ
r c
,Fu,fxn–
φ
r c
≥min
Fxn–,u
φ
r c
,Fxn–,fxn–
φ
r c
,Fu,fu
φ
r c
,Fu,fxn–
φ
r c
.
Lettingn→ ∞in the above inequality, we getFfu,u(αφ(r))≥Fu,fu(φ(rc)). Thusu=fuby Lemma .. Hencef has a fixed point. Furthermore, if (J) holds, then by using a similar technique as in the proof of Theorem . one can see thatuis a unique fixed point off.
4 Application to integral equation
As an application of our results, we will consider the following Volterra type integral equa-tion:
x(t) =g(t) +
t
t,s,x(s)ds, ()
for allt∈[,k], wherek> .
LetC([,k],R) be the space of all continuous functions defined on [,k] endowed with theb-metric
d(x,y) = max t∈[,k]
Alternatively the spaceC([,k],R) can be endowed with theb-metric
dB(x,y) = max t∈[,k]
x(t) –y(t)
e–Lt, x,y∈C,k,R,L> .
One can see thatd anddB are completeb-metrics withs= . We define the mapping
F:C([,k],R)×C([,k],R)→D+by
Fx,y(t) =H
t–dB(x,y)
, t> ,x,y∈C,k,R. ()
We know that (C([,k],R),F,TM) is a complete Menger PbM-space with coefficientα=. Now we discuss the existence of a solution for the Volterra type integral equation ().
Theorem . Let (C([,k],R),F,TM) be the Menger PbM-space and ∈C([,k]× [,k]×R,R)be an operator satisfying the following conditions:
(i) ∞=supt,s∈[,k],x∈C([,k],R)|(t,s,x(s))|<∞,
(ii) there existsL> such that for allx,y∈C([,k],R)and allt,s∈[,k]we obtain
t,s,fx(s)–t,s,fy(s)
≤√L
max
x(s) –y(s),x(s) –fx(s),y(s) –fy(s),y(s) –fx(s),
wheref :C([,k],R)→C([,k],R)is defined by
fx(t) =g(t) +
t
t,s,fx(s)ds, g∈C,k,R.
Then the Volterra type integral equation()has a unique solution x∗∈C([,k],R).
Proof For each x,y∈ C([,k],R) we consider dB(x,y) =maxt∈[,k](|x(t) –y(t)|e–Lt), whereLsatisfies condition (ii). As we mentioned above (C([,k],R),F,TM) is a complete Menger PbM-space with coefficientα=. Therefore, for allx,y∈C([,k],R), we get
dB(fx,fy) = max t∈[,k]
fx(t) –fy(t)e–Lt
= max
t∈[,k]
t
t,s,fx(s)–t,s,fy(s)ds
e–Lt
≤L
max
dB(x,y),dB(x,fx),dB(y,fy),dB(y,fx)
max t∈[,k]
t
eL(s–t)ds
=
–e–LkmaxdB(x,y),dB(x,fx),dB(y,fy),dB(y,fx)
.
Puttingc= ( –e–Lk), by using (), for anyr> andk∈Nwe derive
Ffx,fy
r
k =H
r
k–dB(fx,fy)
≥H
r
k–
c
max
=H
r
k–c–max
dB(x,y),dB(x,fx),dB(y,fy),dB(y,fx)
=min
Fx,y
r
k–c ,Fx,fx
r
k–c ,Fy,fy
r
k–c ,Fy,fx
r
k–c ,
for allx,y∈C([,k],R). Therefore by Theorem . withφ(r) =rfor allr> andβ(x,y,t) =
γ(x,y,t) = for allx,y∈C([,k],R) andt> , we deduce that the operatorf has a unique fixed pointx∗∈C([,k],R), which is the unique solution of the integral equation ().
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Acknowledgements
The authors gratefully acknowledge the anonymous reviewers for carefully reading of the paper and giving helpful suggestions.
Received: 11 March 2015 Accepted: 14 May 2015
References
1. Menger, K: Statistical metrics. Proc. Natl. Acad. Sci. USA28, 535-537 (1942)
2. Sehgal, VM, Bharucha-Reid, AT: Fixed point of contraction mappings in PM-spaces. Math. Syst. Theory6, 97-102 (1972)
3. Schweizer, B, Sklar, A: Probabilistic Metric Spaces. North-Holland, Amsterdam (1983)
4. Babaˆcev, NA: Nonlinear generalized contraction on Menger PM-spaces. Appl. Anal. Discrete Math.6, 257-264 (2012) 5. Gopal, D, Abbas, M, Vetro, C: Some new fixed point theorems in Menger PM-spaces with application to Volterra type
integral equation. Appl. Math. Comput.232, 955-967 (2014)
6. Su, Y, Zhang, J: Fixed point and best proximity point theorems for contractions in new class of probabilistic metric spaces. Fixed Point Theory Appl.2014, 170 (2014)
7. Zhu, C, Xu, W, Wu, Z: Coincidence point theorems for generalized cyclic (kh,φL)S-weak contractions in partially ordered Menger PM-spaces. Fixed Point Theory Appl.2014, 214 (2014)
8. ´Ciri´c, LB: On fixed point of generalized contractions on probabilistic metric spaces. Publ. Inst. Math. (Belgr.)18, 71-78 (1975)
9. Gaji´c, L, Rakoˆcevi´c, V: Pair of non self mappings and common fixed points. Appl. Math. Comput.187, 999-1006 (2007) 10. Hadži´c, O, Pap, E: The Fixed Point Theory in Probabilistic Metric Spaces. Kluwer Academic, Dordrecht (2001) 11. Czerwik, S: Contraction mappings inb-metric spaces. Acta Math. Inform. Univ. Ostrav.1, 5-11 (1993)
12. Czerwik, S: Nonlinear set-valued contraction mappings inb-metric spaces. Atti Semin. Mat. Fis. Univ. Modena46, 263-276 (1998)
13. Allahyari, R, Arab, R, Shole Haghighi, A: A generalization on weak contractions in partially orderedb-metric spaces and its application to quadratic integral equations. J. Inequal. Appl.2014, 355 (2014)
14. Allahyari, R, Arab, R, Shole Haghighi, A: Fixed points of admissible almost contractive type mappings onb-metric spaces with an application to quadratic integral equations. J. Inequal. Appl.2015, 32 (2015)
15. Parvaneh, V, Roshan, JR, Radenovi´c, S: Existence of tripled coincidence point in orderedb-metric spaces and application to a system of integral equation. Fixed Point Theory Appl.2013, 130 (2013)
16. Phiangsungnoen, S, Sintunavarat, W, Kumam, P: Fixed point results, generalized Ulam-Hyers stability and well-posedness viaα-admissible mappings inb-metric spaces. Fixed Point Theory Appl.2014, 188 (2014) 17. Bakhtin, IA: The contraction mapping principle in almost metric spaces. In: Functional Analysis, vol. 30, pp. 26-37.
Ul’yanovsk. Gos. Ped. Inst., Ul’yanovsk (1989)
18. Bota, M, Molnár, A, Varga, C: On Ekeland’s variational principle inb-metric spaces. Fixed Point Theory12(2), 21-28 (2011)
19. Blumenthal, LM: Theory and Applications of Distance Geometry. Oxford University Press, Oxford (1953) 20. Fréchet, M: Les espaces abstraits. Gauthier-Villars, Paris (1928)
21. Shatanawi, W, Pitea, A, Lazovi´c, R: Contraction conditions using comparison functions onb-metric spaces. Fixed Point Theory Appl.2014, 135 (2014)
