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R E S E A R C H

Open Access

Generalized probabilistic metric spaces and

fixed point theorems

Caili Zhou

1,2*

, Shenghua Wang

3

, Ljubomir ´Ciri´c

4

and Saud M Alsulami

5

*Correspondence:

pumpkinlili@163.com

1School of Mathematics and

Statistics, Beijing Institute of Technology, Beijing, 100081, China

2College of Mathematics and

Computer Science, Hebei University, Baoding, 071002, China

Full list of author information is available at the end of the article

Abstract

In this paper, we introduce a new concept of probabilistic metric space, which is a generalization of the Menger probabilistic metric space, and we investigate some topological properties of this space and related examples. Also, we prove some fixed point theorems, which are the probabilistic versions of Banach’s contraction principle. Finally, we present an example to illustrate the main theorems.

MSC: 54E70; 47H25

Keywords: Menger metric space; contraction mapping;G-metric space; fixed point theorem

1 Introduction and preliminaries

LetRbe the set of all real numbers,R+ be the set of all nonnegative real numbers,Δ

denote the set of all probability distribution functions,i.e., Δ={F:R∪ {–∞, +∞} → [, ] :Fis left continuous and nondecreasing onR,F(–∞) =  andF(+∞) = }.

Definition .([]) A mappingT: [, ]×[, ]→[, ] is called acontinuous t-normif

T satisfies the following conditions:

() T is commutative and associative,i.e.,T(a,b) =T(b,a)and

T(a,T(b,c)) =T(T(a,b),c), for alla,b,c∈[, ]; () T is continuous;

() T(a, ) =afor alla∈[, ];

() T(a,b)≤T(c,d)wheneveracandbdfor alla,b,c,d∈[, ].

From the definition ofT it follows thatT(a,b)≤min{a,b}for alla,b∈[, ].

Two simple examples of continuoust-norm areTM(a,b) =min{a,b}andTp(a,b) =ab for alla,b∈[, ].

In , Menger [] developed the theory of metric spaces and proposed a generalization of metric spaces called Menger probabilistic metric spaces (briefly, Menger PM-space).

Definition . AMenger PM-spaceis a triple (X,F,T), whereXis a nonempty set,T is a continuoust-norm andFis a mapping fromX×XD(Fx,ydenotes the value ofFat the pair (x,y)) satisfying the following conditions:

(PM-) Fx,y(t) = for allx,yXandt> if and onlyx=y; (PM-) Fx,y(t) =Fy,x(t)for allx,yXandt> ;

(PM-) Fx,z(t+s)≥T(Fx,y(t),Fy,z(s))for allx,y,zXandt,s≥.

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The idea of Menger was to use distribution functions instead of nonnegative real num-bers as values of the metric. Since Menger, many authors have considered fixed point the-ory in PM-spaces and its applications as a part of probabilistic analysis (see [, –]).

In , Gähler [] investigated the concept of -metric spaces and he claimed that a -metric is a natural generalization of an ordinary metric space (for more detailed results, see the books [, ]). But some authors pointed out that there are no relations between -metric spaces and ordinary -metric spaces []. Later, Dhage [] introduced a new class of generalized metrics calledD-metric spaces. However, as pointed out in [], theD-metric is also not satisfactory.

Recently, Mustafa and Sims [] introduced a new class of metric spaces called general-ized metric spaces orG-metric spaces as follows.

Definition .([]) LetXbe a nonempty set andG:X×X×X:→R+be a function

satisfying the following conditions:

(G) G(x,y,z) = ifx=y=zfor allx,y,zX; (G)  <G(x,x,y)for allx,yXwithx=y;

(G) G(x,x,y)≤G(x,y,z)for allx,y,zXwithz=y; (G) G(x,y,z) =G(x,z,y) =G(y,z,x) =· · · for allx,y,zX; (G) G(x,y,z)≤G(x,a,a) +G(a,y,z)for allx,y,z,aX.

ThenGis called ageneralized metricor aG-metriconXand the pair (X,G) is aG-metric space.

It was proved that theG-metric is a generalization of ordinary metric (see []). Recently, some authors studiedG-metric spaces and obtained fixed point theorems onG-metric spaces [–]. Similar work can be found in [–].

It is well known that the notion of a PM-space corresponds to the situation that we may know probabilities of possible values of the distance although we do not know exactly the distance between two points. This idea leads us to seek a probabilistic version ofG-metric spaces defined by Mustafa and Sims [].

Definition . AMenger probabilistic G-metric space(shortly,PGM-space) is a triple (X,G∗,T), whereX is a nonempty set,T is a continuoust-norm andG∗ is a mapping fromX×X×XintoD(Gx,y,zdenotes the value ofG∗at the point (x,y,z)) satisfying the following conditions:

(PGM-) Gx,y,z(t) = for allx,y,zXandt> if and only ifx=y=z; (PGM-) Gx,x,y(t)≥Gx,y,z(t)for allx,yXwithz=yandt> ;

(PGM-) Gx,y,z(t) =Gx,z,y(t) =Gy,x,z(t) =· · · (symmetry in all three variables); (PGM-) Gx,y,z(t+s)≥T(Gx,a,a(s),Ga,y,z(t))for allx,y,z,aXands,t≥.

Remark . Golet introduced a concept of probabilistic -metric (or -Menger space)

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Example . LetHdenote the specific distribution function defined by

H(t) =

⎧ ⎨ ⎩

, t≤,

, t> ,

andDbe a distribution function defined by

D(t) =

⎧ ⎨ ⎩

, t≤,

 –et, t> .

For anyt> , define a functionG∗:X×X×X→R+by

Gx,y,z(t) =

⎧ ⎨ ⎩

H(t), x=y=z,

D(G(xt,y,z)), otherwise,

whereGis aG-metric as in Definition .. SetT=min. ThenG∗is a probabilisticG-metric.

Proof It is easy to see thatG∗satisfies (PGM-)-(PGM-). Next we showG∗(x,y,z)(s+t)≥

T(G∗(x,a,a)(s),G∗(a,y,y)(t)) for allx,y,z,aXand alls,t> . In fact, we only need show thatDsatisfies

D

t+s G(x,y,z)

≥min D s G(x,a,a)

,D

t D(a,y,z)

. (.)

SinceG(x,y,z)≤G(x,a,a) +G(a,y,z), we have

s+t G(x,y,z)≥

s+t

G(x,a,a) +G(a,y,z). (.)

Furthermore, we have

max

s G(x,a,a),

t G(a,y,z)

s+t

G(x,a,a) +G(a,y,z)

≥min

s G(x,a,a),

t G(a,y,z)

, (.)

which, from (.) and (.), shows that

s+t

G(x,y,z)≥min

s G(x,a,a),

t G(a,y,z)

.

This implies (.) sinceDis nondecreasing.

Example . Let (X,F,T) be a PM-space. Define a functionG∗:X×X×X→R+by

Gx,y,z(t) =min Fx,y(t),Fy,z(t),Fx,z(t)

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Proof It is obvious thatG∗satisfies (PGM-), (PGM-), and (PGM-). To prove thatG

satisfies (PGM-), we need to show that, for allx,y,z,aXand alls,t≥,

Gx,y,z(s+t)≥TGx,a,a(s),Ga,y,z(t), (.)

i.e.,

min Fx,y(s+t),Fy,z(s+t),Fx,z(s+t)

TFx,a(s),min Fa,y(t),Fa,z(t),Fy,z(t)

. (.)

Now, from

Fx,y(s+t)≥T

Fx,a(s),Fa,y(t)

TFx,a(s),min Fa,y(t),Fa,z(t),Fy,z(t)

,

Fx,z(s+t)≥T

Fx,a(s),Fa,z(t)

TFx,a(s),min Fa,y(t),Fa,z(t),Fy,z(t)

and

Fy,z(s+t)≥Fy,z(t)

≥min Fa,y(t),Fa,z(t),Fy,z(t)

TFx,a(s),min Fa,y(t),Fa,z(t),Fy,z(t)

,

we conclude that (.),i.e., (.) holds. Therefore,G∗satisfies (PGM-) and henceG∗is a

probabilisticG-metric.

Example . Let (X,F,TM) be a PM-space. Define a functionG∗:X×X×X→R+by

Gx,y,z(t) =min Fx,y(t/),Fy,z(t/),Fx,z(t/)

for allx,y,z,∈Xandt> . Then (X,G∗,TM) is a PGM-space.

Proof In fact, the proofs of (PGM-)-(PGM-) are immediate. Now, we show thatG∗ sat-isfies (PGM-). It follows that

Gx,y,z(t+s)

=min

Fx,y

t+s

,Fy,z

t+s

,Fx,z

t+s

 ≥min min

Fx,a

t

,Fa,y

s

,Fy,z

s  ,min

Fx,a

t

,Fa,z

s  =min min

Fx,a

t

,Fa,a

t

,Fx,a

t  ,min

Fa,y

s

,Fy,z

s

,Fa,z

s

=min Gx,a,a(t),Ga,y,z(s).

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The following remark shows that the PGM-space is a generalization of the Menger PM-space.

Remark . For any functionG∗:X×X×X→R+, the functionF:X×X→R+defined by

Fx,y(t) =min Gx,y,y(t),Gy,x,x(t)

is a probabilistic metric. It is easy to see thatFsatisfies the conditions (PM-) and (PM-).

Next, we showFsatisfies (PM-). Indeed, for anys,t≥ andx,y,zX, we have

Fx,y(s+t) =min Gx,y,y(s+t),Gy,x,x(s+t)

,

TFx,z(s),Fz,y(t)

=Tmin Gx,z,z(s),Gz,x,x(s),min Gz,y,y(t),Gy,z,z(t).

It follows from (PGM-) that

min Gx,y,y(s+t),Gy,x,x(s+t)

≥min TGx,z,z(s),Gz,y,y(t),TGy,z,z(t),Gz,x,x(s).

Since

Gx,z,z(s)≥min Gx,z,z(s),Gz,x,x(s), Gz,y,y(t)≥min Gz,y,y(t),Gy,z,z(t)

and

Gy,z,z(t)≥min Gz,y,y(t),Gy,z,z(t), Gz,x,x(s)≥min Gx,z,z(s),Gz,x,x(s),

it follows from (PGM-) that

Gx,y,y(s+t)≥TGx,z,z(s),Gz,y,y(t)

Tmin Gx,z,z(s),Gz,x,x(s),min Gz,y,y(t),Gy,z,z(t)

=TFz,x(s),Fz,y(t)

and

Gy,x,x(s+t)≥TGy,z,z(t),Gz,x,x(s)

Tmin Gz,y,y(t),Gy,z,z(t),min Gx,z,z(s),Gz,x,x(s)

=TFz,y(t),Fz,x(s)

.

Therefore, we have

Fx,y(s+t) =min Gx,y,y(s+t),Gy,x,x(s+t)

TFx,z(s),Fz,y(t)

.

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2 Topology, convergence, and completeness

In this section, we first introduce the concept of neighborhoods in the PGM-spaces. For the concept of neighborhoods in PM-spaces, we refer the readers to [, ].

Definition . Let (X,G∗,T) be a PGM-space andxbe any point inX. For any>  andδwith  <δ< , an (,δ)-neighborhoodofxis the set of all pointsyinXfor which Gx,y,y() >  –δandGy,x,x() >  –δ. We write

Nx(,δ) = yX:G

x,y,y() >  –δ,Gy,x,x() >  –δ

.

This means thatNx(,δ) is the set of all pointsyinXfor which the probability of the

distance fromxtoybeing less thanis greater than  –δ.

Lemma . If≤andδ≤δ,then Nx(,δ)⊂Nx(,δ).

Proof Suppose thatzNx(,δ), soGx,z,z() >  –δandGz∗,x,x() >  –δ. SinceFis

monotone, we have

Gx,z,z()≥Gx,z,z()≥ –δ≥ –δ

and

Gz,x,x()≥Gz,x,x()≥ –δ≥ –δ.

Therefore, by the definition,zNx(,δ). This completes the proof.

Theorem . Let(X,G∗,T)be a Menger PGM-space.Then(X,G∗,T)is a Hausdorff space in the topology induced by the family{Nx(,δ)}of(,δ)-neighborhoods.

Proof We show that the following four properties are satisfied:

(A) For anyxX, there exists at least one neighborhood,Nx, ofxand every

neighborhood ofxcontainsx. (B) IfN

xandN

xare neighborhoods ofx, then there exists a neighborhood ofx,N

x,

such thatN

x⊂N

x∩N

x.

(C) IfNxis a neighborhood ofxandyNx, then there exists a neighborhood ofy, Ny, such thatNyNx.

(D) Ifx=y, then there exist disjoint neighborhoods,NxandNy, such thatxNx

andyNy.

Now, we prove that (A)-(D) hold.

(A) For any>  and  <δ< ,x∈Nx(,δ) sinceGx,x,x() =  for any> .

(B) For any,>  and  <δ,δ< , let

Nx(,δ) = yX:Gx,y,y() >  –δ,Gy,x,x() >  –δ

and

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be the neighborhoods ofx. Consider

Nx= yX:Gx,y,ymin{,}

>  –min{δ,δ},

andGy,x,xmin{,}

>  –min{δ,δ}

.

Clearly, xNx and, sincemin{,} ≤ andmin{δ,δ} ≤δ, by Lemma .,Nx⊂ N

x(,δ) andN

x⊂N

x(,δ), so

N

x⊂N

x(,δ)∩N

x(,δ).

(C) LetNx={zX:Gx,z,z() >  –δ,Gz,x,x() >  –δ}be the neighborhood ofx.

SinceyNx,

Gx,y,y() >  –δ, Gy,x,x() >  –δ.

Now,Gx,y,yis left-continuous at, so there exist<andδ<δsuch that

Gx,y,y() >  –δ>  –δ, Gy,x,x() >  –δ>  –δ.

LetNy={zX:Gy,z,z() >  –δ,Gz∗,y,y() >  –δ}, where  <<–andδis chosen

such that

T( –δ,  –δ) >  –δ.

Such aδexists sinceTis continuous,T(a, ) =afor alla∈[, ] and  –δ>  –δ. Now, suppose thatsNy, so that

Gy,s,s() >  –δ, Gs,y,y() >  –δ.

Then, sinceG∗is monotone, it follows from (PGM-) that

Gx,s,s()≥TGx,y,y(),Gy,s,s()≥TGx,y,y(),Gy,s,s()

T( –δ,  –δ) >  –δ.

Similarly, we also haveGs,x,x() >  –δ. This showssNxand henceNyNx.

(D) Lety=x. Then there exist>  anda,awith ≤a,a<  such thatGx,y,y() =a

andGy,x,x() =a. Let

Nx= z:G

x,z,z(/) >b,Gz,x,x(/) >b

and

Ny= z:Gy∗,z,z(/) >b,Gz,y,y(/) >b

,

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Now, suppose that there exists a pointsNx∩Nysuch that

Gx,s,s(/) >b, Gs,x,x(/) >b, Gy,s,s(/) >b, Gs,y,y(/) >b.

Then, by (PGM-), we have

a=Gx,y,y()≥TGx,s,s(/),Gs,y,y(/)≥T(b,b) >aa

and

a=Gy,x,x()≥TGy,s,s(/),Gs,x,x(/)≥T(b,b) >aa,

which are contradictions. Therefore,NxandNyare disjoint. This completes the proof.

Next, we give the definition of convergence of sequences in PGM-spaces.

Definition .

() A sequence{xn}in a PGM-space(X,G∗,T)is said to beconvergentto a pointxX (writexnx) if, for any> and <δ< , there exists a positive integerM,δ such

thatxnNx(,δ)whenevern>M,δ.

() A sequence{xn}in a PGM-space(X,G∗,T)is called aCauchy sequenceif, for any

> and <δ< , there exists a positive integerM,δsuch thatGxn,xm,xl() >  –δ wheneverm,n,l>M,δ.

() A PGM-space(X,G∗,T)is said to becompleteif every Cauchy sequence inX

converges to a point inX.

Theorem . Let(X,G∗,T)be a PGM-space.Let{xn},{yn}and{zn} be sequences in X

and x,y,zX.If xnx,yny and znz as n→ ∞,then,for any t> ,Gxn∗,yn,zn(t)→

Gx,y,z(t)as n→ ∞.

Proof For anyt> , there existsδ>  such thatt> δ. Then, by (PGM-), we have

Gxn,yn,zn(t)≥Gxn,yn,zn(tδ)

TGxn,x,x(δ/),Gx,yn,zn(t– δ/)

TGxn,x,x(δ/),TGyn,y,y(δ/),Gy,x,zn(t– δ/)

TGxn,x,x(δ/),TGyn,y,y(δ/),TGz,z,zn(δ/),Gx,y,z(t– δ)

and

Gx,y,z(t)≥Gx,y,z(tδ)

TGx,xn,xn(δ/),Gxn,y,z(t– δ/)

TGx,xn,xn(δ/),TGy,yn,yn(δ/),Gyn,xn,z(t– δ/)

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Lettingn→ ∞in the above two inequalities and noting thatT is continuous, we have

lim

n→∞G

xn,yn,zn(t)≥Gx,y,z(t– δ)

and

Gx,y,z(t)≥ lim

n→∞G

xn,yn,zn(t– δ).

Lettingδ→ in above two inequalities, sinceG∗is left-continuous, we conclude that

lim

n→∞G

xn,yn,zn(t) =Gx,y,z(t)

for anyt> . This completes the proof.

3 Fixed point theorems

In [], Sehgal extended the notion of a Banach contraction mapping to the setting of Menger PM-spaces. Later on, Sehgal and Bharucha-Raid [] proved a fixed point theorem for a mapping under the contractive condition in a complete Menger PM-space. Before proving our fixed point theorems, we first introduce a new concept of contraction in PGM-spaces, which is a corresponding version of Sehgal’s contraction in PM-spaces.

Definition . Let (X,G∗,T) be a PGM-space. A mappingf :XXis said to be contrac-tiveif there exists a constantλ∈(, ) such that

Gfx,fy,fz(t)≥Gx,y,z(t/λ) (.)

for allx,y,zXandt> .

The mappingf satisfying the condition (.) is called aλ-contraction.

LetTbe a givent-norm. Then (by associativity) a family of mappingsTn: [, ][, ] for eachn≥ is defined as follows:

T(t) =T(t,t), T(t) =Tt,T(t), . . . , Tn(t) =Tt,Tn–(t), . . .

for anyt∈[, ].

Definition .([]) At-normT is said to beof Hadzić-typeif the family of functions

{Tn(t)}∞n=is equicontinuous att= , that is, for any∈(, ), there existsδ∈(, ) such that

t>  –δTn(t) >  –

for eachn≥.

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Lemma . Let(X,G∗,T)be a Menger PGM-space with T of Hadžić-type and{xn}be a

sequence in X.Suppose that there existsλ∈(, )satisfying

Gxn,xn+,xn+(t)≥Gxn–,xn,xn(t/λ)

for any n≥and t> .Then{xn}is a Cauchy sequence in X.

Proof SinceGxn,xn+,xn+(t)≥Gxn–,xn,xn(t/λ), by induction, we have

Gxn,xn+,xn+(t)≥Gx,x,xt/λn.

SinceXis a Menger PGM-space, we haveGx,x,x(t/λn)→ asn→ ∞, so

lim

n→∞G

xn,xn+,xn+(t) =  (.)

for anyt> .

Now, letn≥ andt> . We show, by induction, that, for anyk≥,

Gxn,xn

+k,xn+k(t)≥T kG

xn,xn+,xn+(tλt)

. (.)

Fork= , sinceT(a,b) is a real number,T(a,b) =  for alla,b[, ]. Hence,G

xn,xn,xn(t) =  =T(G

xn,xn+,xn+)(tλt), which implies that (.) holds fork= . Assume that (.) holds

for somek≥. Then, sinceTis monotone, it follows from (PGM-) that

Gxn,xn+k+,xn+k+(t) =Gxn,xn+k+,xn+k+(tλt+λt)

TGxn,xn+,xn+(tλt),Gxn+,xn+k+,xn+k+(λt)

TGxn,xn+,xn+(tλt),Gxn,xn+k,xn+k(t)

TGxn,xn+,xn+(tλt),TkGxn,xn+,xn+(tλt)

=Tk+Gxn,xn+,xn+(tλt),

so we have the conclusion.

Now, we show that{xn}is a Cauchy sequence inX,i.e.,limm,n,l→∞Gxn∗,xm,xl(t) =  for any

t> . To this end, we first prove thatlimn,m→∞Gxn,xm,xm(t) =  for anyt> . Lett>  and

>  be given. By hypothesis,Tn:n is equicontinuous at  andTn() = , so there existsδ>  such that, for anya∈( –δ, ],

Tn(a) >  – (.)

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anyt> . By (GPM-), we have

Gxn,xm,x

l(t)≥T

Gxn,xn,xm(t/),Gxn,xn,x

l(t/)

,

Gxn,xn,xm(t/)≥TGxn,xm,xm(t/),Gxn,xm,xm(t/),

Gxn,xn,xl(t/)≥TGxn,xl,xl(t/),Gxn,xl,xl(t/).

Therefore, by the continuity ofT, we conclude that

lim

m,n,l→∞G

xn,xm,xl(t) = 

for anyt> . This shows that the sequence{xn}is a Cauchy sequence inX. This completes

the proof.

From Example . and Lemma . we get the following corollary.

Corollary .([]) Let(X,F,T)be a PM-space with T of Hadžić-type and{xn} ⊂X be a

sequence.If there exists a constantλ∈(, )such that

Fxn,xn+(t)≥Fxn–,xn(t/λ), n≥,t> ,

then{xn}is a Cauchy sequence.

Proof DefineGx,y,z(t) =min{Fx,y(t),Fy,z(t),Fx,z(t)}for allx,y,zXand allt> . Example . shows that (X,G∗,T) is a PGM-space. SinceGxn,xn+,xn+(t) =Fxn,xn+(t) andGxn∗–,xn,xn(t/λ) =

Fxn–,xn(t/λ),Fxn,xn+(t)≥Fxn–,xn(t/λ) impliesGxn∗,xn+,xn+(t)≥G

xn–,xn,xn(t/λ) for all n≥ andt> . By Lemma . we conclude that{xn}is a Cauchy sequence in the sense of PGM-space (X,G∗,T). That is, for every>  and  <δ< , there exists a positive integerM,δ such thatGxn,xm,x

l() >  –δfor allm,n,l>M,δ. By the definition ofG

, we have

min Fxn,xm(),Fxm,xl(),Fxn,xl()

>  –δ, m,l,n>M,δ.

This shows that{xn}is a Cauchy sequence in the sense of PM-space (X,F,T).

Theorem . Let(X,G∗,T)be a complete Menger PGM-space with T of Hadžić-type.Let λ∈(, )and f :XX be aλ-contraction.Then,for any xX,the sequence {fnx} converges to a unique fixed point of T.

Proof Take an arbitrary pointx inX. Construct a sequence {xn}byxn+=fnxfor all n≥. By (.), for anyt> , we have

Gxn,xn+,xn+(t) =Gfxn–,fxn,fxn(t)

Gxn–,xn,xn(t/λ).

Lemma . shows that{xn}is a Cauchy sequence inX. SinceXis complete, there exists a pointxXsuch thatxnxasn→ ∞. By (.), it follows that

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Lettingn→ ∞, sincexnxandfxnxasn→ ∞, we have

Gfx,x,x(t) = 

for anyt> . Hencex=fx.

Next, suppose thatyis another fixed point off. Then, by (.), we have

Gx,y,y(t) =Gfx,fy,fy(t)≥Gx,y,y(t/λ)≥ · · · ≥Gx,y,yt/λn.

Lettingn→ ∞, sinceXis a Menger PGM-space,Gx,y,y(t/λn) asn→ ∞, so

Gx,y,y(t) = 

for anyt> , which implies thatx=y. Therefore,f has a unique fixed point inX. This

completes the proof.

Theorem . Let(X,G∗,T)be a complete Menger PGM-space with T of Hadžić-type.Let f :XX be a mapping satisfying

Gfx,fy,fz(λt)≥  

Gx,fx,fx(t) +Gy,fy,fy(t) +Gz,fz,fz(t) (.)

for all x,y,zX,whereλ∈(, ).Then,for any xX,the sequence{fnx}converges to a

unique fixed point of f.

Proof Take an arbitrary pointx inX. Construct a sequence {xn}byxn+=fnxfor all n≥. By (.), for anyt> , we have

Gxn,xn+,xn+(λt) =Gfxn–,fxn,fxn(λt)

≥

Gxn–,fxn–,fxn–(t) + Gxn,fxn,fxn(t)

≥

Gxn–,fxn–,fxn–(t) + Gxn,fxn,fxn(λt)

= 

Gxn–,xn,xn(t) + Gxn,xn+,xn+(λt).

This shows that

Gxn,xn+,xn+(t)≥Gxn–,xn,xn(t/λ).

Lemma . shows that{xn}is a Cauchy sequence inX. SinceXis complete, there exists a pointxXsuch thatxnxasn→ ∞. By (.), it follows that

Gfx,fx,fxn(t)≥ 

Gx,fx,fx(t/λ) +Gxn,fxn,fxn(t/λ).

Lettingn→ ∞, sincexnxandfxnxasn→ ∞, we have, for anyt> ,

Gfx,fx,x(t)≥  

Gx,fx,fx(t/λ) +Gx,x,x(t/λ)≥ 

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i.e.,

Gfx,fx,x(t)≥Gx,x,x(t/λ) = .

Hencex=Tx.

Next, suppose thatyis another fixed point off. Then, by (.), we have, for anyy> ,

Gx,y,y(t) =Gfx,fy,fy(t)

≥ 

Gx,fx,fx(t/λ) + Gy,fy,fy(t/λ)

= .

This shows that x=y. Therefore,f has a unique fixed point inX. This completes the

proof.

Finally, we give the following example to illustrate Theorem . and Theorem ..

Example . SetX= [,∞) andT(a,b) =min{a,b}for alla,b∈[, ]. Define a function

G∗:X×[,)[,) by

Gx,y,z(t) = t

t+G(x,y,z)

for allx,y,zX, whereG(x,y,z) =|xy|+|yz|+|zx|. ThenGis aG-metric (see []). It is easy to check thatG∗satisfies (PGM-)-(PGM-). SinceG(x,y,z)≤G(x,a,a) +G(a,y,z) for allx,y,z,aX, we have

t+s s+t+G(x,y,z)≥

t+s

s+t+G(x,a,a) +G(a,y,z)

≥min

s s+G(x,a,a),

t t+G(a,y,z)

.

This shows thatG∗satisfies (PGM-). Hence (X,G∗,min) is a PGM-space.

() Letλ∈(, ). Define a mappingf :XXbyfx=λxfor allxX. For anyt> , we have

Gfx,fy,fz(t) = t

t+λ(|xy|+|yz|+|zx|)

and

Gx,y,z(t/λ) = t/λ

t/λ+ (|xy|+|yz|+|zx|).

Therefore, we conclude thatf is aλ-contraction andf has a fixed point inXby Theo-rem .. In fact, the fixed point isx= .

() Letλ∈(, ). Define a mappingf :XXbyfx=  for allxX. For anyt>  and all

x,y,zX, since

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and

 

Gx,fx,fx(λt) +Gy,fy,fy(λt) +Gz,fz,fz(λt)≤,

we conclude that

Gfx,fy,fz(t)≥  

Gx,fx,fx(λt) +Gy,fy,fy(λt) +Gz,fz,fz(λt)

and hencef has a fixed point inXby Theorem .. In fact, the fixed point isx= .

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Author details

1School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, China.2College of Mathematics

and Computer Science, Hebei University, Baoding, 071002, China.3Department of Mathematics and Physics, North China

Electric Power University, Baoding, 071003, China.4Faculty of Mechanical Engineering, University of Belgrade, Kraljice

Marije 16, Belgrade, 11 000, Serbia.5Department of Mathematics, King Abdulaziz University, Jeddah, 21323, Saudi Arabia.

Acknowledgements

The first author is supported by the National Natural Science Foundation of China (Grant Number: 11371002,41201327) and Specialized Research Fund for the Doctoral Program of Higher Education (Grant Number:20131101110048) and Natural Science Foundation of Hebei Province (Grant Number: A2013201119). The second author is supported by the Fundamental Research Funds for the Central Universities (Grant Number: 13MS109). The authors S. Alsulami and L. Ciric thanks the Deanship of Scientific Research (DSR), King Abdulaziz University for financial support ( Grant Number: 130-037-D1433).

Received: 9 October 2013 Accepted: 26 March 2014 Published:08 Apr 2014

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