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
4and 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)whenevera≤candb≤dfor 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×X→D(Fx,ydenotes the value ofFat the pair (x,y)) satisfying the following conditions:
(PM-) Fx,y(t) = for allx,y∈Xandt> if and onlyx=y; (PM-) Fx,y(t) =Fy,x(t)for allx,y∈Xandt> ;
(PM-) Fx,z(t+s)≥T(Fx,y(t),Fy,z(s))for allx,y,z∈Xandt,s≥.
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,z∈X; (G) <G(x,x,y)for allx,y∈Xwithx=y;
(G) G(x,x,y)≤G(x,y,z)for allx,y,z∈Xwithz=y; (G) G(x,y,z) =G(x,z,y) =G(y,z,x) =· · · for allx,y,z∈X; (G) G(x,y,z)≤G(x,a,a) +G(a,y,z)for allx,y,z,a∈X.
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(G∗x,y,zdenotes the value ofG∗at the point (x,y,z)) satisfying the following conditions:
(PGM-) G∗x,y,z(t) = for allx,y,z∈Xandt> if and only ifx=y=z; (PGM-) G∗x,x,y(t)≥G∗x,y,z(t)for allx,y∈Xwithz=yandt> ;
(PGM-) G∗x,y,z(t) =G∗x,z,y(t) =G∗y,x,z(t) =· · · (symmetry in all three variables); (PGM-) G∗x,y,z(t+s)≥T(G∗x,a,a(s),Ga∗,y,z(t))for allx,y,z,a∈Xands,t≥.
Remark . Golet introduced a concept of probabilistic -metric (or -Menger space)
Example . LetHdenote the specific distribution function defined by
H(t) =
⎧ ⎨ ⎩
, t≤,
, t> ,
andDbe a distribution function defined by
D(t) =
⎧ ⎨ ⎩
, t≤,
–e–t, t> .
For anyt> , define a functionG∗:X×X×X→R+by
G∗x,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,a∈Xand 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
G∗x,y,z(t) =min Fx,y(t),Fy,z(t),Fx,z(t)
Proof It is obvious thatG∗satisfies (PGM-), (PGM-), and (PGM-). To prove thatG∗
satisfies (PGM-), we need to show that, for allx,y,z,a∈Xand alls,t≥,
G∗x,y,z(s+t)≥TG∗x,a,a(s),G∗a,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
G∗x,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
G∗x,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 G∗x,a,a(t),G∗a,y,z(s).
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 G∗x,y,y(t),G∗y,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,z∈X, we have
Fx,y(s+t) =min G∗x,y,y(s+t),G∗y,x,x(s+t)
,
TFx,z(s),Fz,y(t)
=Tmin G∗x,z,z(s),G∗z,x,x(s),min G∗z,y,y(t),G∗y,z,z(t).
It follows from (PGM-) that
min G∗x,y,y(s+t),G∗y,x,x(s+t)
≥min TG∗x,z,z(s),G∗z,y,y(t),TG∗y,z,z(t),Gz∗,x,x(s).
Since
G∗x,z,z(s)≥min G∗x,z,z(s),G∗z,x,x(s), G∗z,y,y(t)≥min G∗z,y,y(t),G∗y,z,z(t)
and
G∗y,z,z(t)≥min G∗z,y,y(t),G∗y,z,z(t), G∗z,x,x(s)≥min G∗x,z,z(s),G∗z,x,x(s),
it follows from (PGM-) that
G∗x,y,y(s+t)≥TG∗x,z,z(s),Gz∗,y,y(t)
≥Tmin G∗x,z,z(s),G∗z,x,x(s),min G∗z,y,y(t),Gy∗,z,z(t)
=TFz,x(s),Fz,y(t)
and
G∗y,x,x(s+t)≥TG∗y,z,z(t),Gz∗,x,x(s)
≥Tmin Gz∗,y,y(t),G∗y,z,z(t),min G∗x,z,z(s),G∗z,x,x(s)
=TFz,y(t),Fz,x(s)
.
Therefore, we have
Fx,y(s+t) =min G∗x,y,y(s+t),G∗y,x,x(s+t)
≥TFx,z(s),Fz,y(t)
.
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 andxbe any point inX. For any> andδwith <δ< , an (,δ)-neighborhoodofxis the set of all pointsyinXfor which G∗x,y,y() > –δandG∗y,x,x() > –δ. We write
Nx(,δ) = y∈X:G
∗
x,y,y() > –δ,G∗y,x,x() > –δ
.
This means thatNx(,δ) is the set of all pointsyinXfor which the probability of the
distance fromxtoybeing less thanis greater than –δ.
Lemma . If≤andδ≤δ,then Nx(,δ)⊂Nx(,δ).
Proof Suppose thatz∈Nx(,δ), soG∗x,z,z() > –δandGz∗,x,x() > –δ. SinceFis
monotone, we have
G∗x,z,z()≥Gx∗,z,z()≥ –δ≥ –δ
and
G∗z,x,x()≥G∗z,x,x()≥ –δ≥ –δ.
Therefore, by the definition,z∈Nx(,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, ofxand every
neighborhood ofxcontainsx. (B) IfN
xandN
xare neighborhoods ofx, then there exists a neighborhood ofx,N
x,
such thatN
x⊂N
x∩N
x.
(C) IfNxis a neighborhood ofxandy∈Nx, then there exists a neighborhood ofy, Ny, such thatNy⊂Nx.
(D) Ifx=y, then there exist disjoint neighborhoods,NxandNy, such thatx∈Nx
andy∈Ny.
Now, we prove that (A)-(D) hold.
(A) For any> and <δ< ,x∈Nx(,δ) sinceG∗x,x,x() = for any> .
(B) For any,> and <δ,δ< , let
Nx(,δ) = y∈X:G∗x,y,y() > –δ,G∗y,x,x() > –δ
and
be the neighborhoods ofx. Consider
Nx= y∈X:G∗x,y,ymin{,}
> –min{δ,δ},
andG∗y,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={z∈X:G∗x,z,z() > –δ,G∗z,x,x() > –δ}be the neighborhood ofx.
Sincey∈Nx,
G∗x,y,y() > –δ, G∗y,x,x() > –δ.
Now,Gx∗,y,yis left-continuous at, so there exist<andδ<δsuch that
G∗x,y,y() > –δ> –δ, G∗y,x,x() > –δ> –δ.
LetNy={z∈X:G∗y,z,z() > –δ,Gz∗,y,y() > –δ}, where <<–andδis chosen
such that
T( –δ, –δ) > –δ.
Such aδexists sinceTis continuous,T(a, ) =afor alla∈[, ] and –δ> –δ. Now, suppose thats∈Ny, so that
G∗y,s,s() > –δ, G∗s,y,y() > –δ.
Then, sinceG∗is monotone, it follows from (PGM-) that
G∗x,s,s()≥TGx∗,y,y(),G∗y,s,s(–)≥TG∗x,y,y(),G∗y,s,s()
≥T( –δ, –δ) > –δ.
Similarly, we also haveG∗s,x,x() > –δ. This showss∈Nxand henceNy⊂Nx.
(D) Lety=x. Then there exist> anda,awith ≤a,a< such thatG∗x,y,y() =a
andG∗y,x,x() =a. Let
Nx= z:G
∗
x,z,z(/) >b,G∗z,x,x(/) >b
and
Ny= z:Gy∗,z,z(/) >b,G∗z,y,y(/) >b
,
Now, suppose that there exists a points∈Nx∩Nysuch that
G∗x,s,s(/) >b, G∗s,x,x(/) >b, G∗y,s,s(/) >b, G∗s,y,y(/) >b.
Then, by (PGM-), we have
a=G∗x,y,y()≥TG∗x,s,s(/),G∗s,y,y(/)≥T(b,b) >a≥a
and
a=Gy∗,x,x()≥TG∗y,s,s(/),G∗s,x,x(/)≥T(b,b) >a≥a,
which are contradictions. Therefore,NxandNyare 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 pointx∈X (writexn→x) if, for any> and <δ< , there exists a positive integerM,δ such
thatxn∈Nx(,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 thatG∗xn,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,z∈X.If xn→x,yn→y and zn→z as n→ ∞,then,for any t> ,Gxn∗,yn,zn(t)→
G∗x,y,z(t)as n→ ∞.
Proof For anyt> , there existsδ> such thatt> δ. Then, by (PGM-), we have
G∗xn,yn,zn(t)≥G∗xn,yn,zn(t–δ)
≥TG∗xn,x,x(δ/),G∗x,yn,zn(t– δ/)
≥TG∗xn,x,x(δ/),TG∗yn,y,y(δ/),G∗y,x,zn(t– δ/)
≥TG∗xn,x,x(δ/),TG∗yn,y,y(δ/),TG∗z,z,zn(δ/),G∗x,y,z(t– δ)
and
G∗x,y,z(t)≥G∗x,y,z(t–δ)
≥TG∗x,xn,xn(δ/),G∗xn,y,z(t– δ/)
≥TG∗x,xn,xn(δ/),TG∗y,yn,yn(δ/),G∗yn,xn,z(t– δ/)
Lettingn→ ∞in the above two inequalities and noting thatT is continuous, we have
lim
n→∞G
∗
xn,yn,zn(t)≥G∗x,y,z(t– δ)
and
G∗x,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) =G∗x,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 :X→Xis said to be contrac-tiveif there exists a constantλ∈(, ) such that
G∗fx,fy,fz(t)≥G∗x,y,z(t/λ) (.)
for allx,y,z∈Xandt> .
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> –δ ⇒ Tn(t) > –
for eachn≥.
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
G∗xn,xn+,xn+(t)≥G∗xn–,xn,xn(t/λ)
for any n≥and t> .Then{xn}is a Cauchy sequence in X.
Proof SinceG∗xn,xn+,xn+(t)≥G∗xn–,xn,xn(t/λ), by induction, we have
G∗xn,xn+,xn+(t)≥G∗x,x,xt/λn.
SinceXis a Menger PGM-space, we haveG∗x,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≥,
G∗xn,xn
+k,xn+k(t)≥T kG∗
xn,xn+,xn+(t–λt)
. (.)
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+)(t–λt), which implies that (.) holds fork= . Assume that (.) holds
for somek≥. Then, sinceTis monotone, it follows from (PGM-) that
G∗xn,xn+k+,xn+k+(t) =Gxn∗,xn+k+,xn+k+(t–λt+λt)
≥TG∗xn,xn+,xn+(t–λt),G∗xn+,xn+k+,xn+k+(λt)
≥TG∗xn,xn+,xn+(t–λt),G∗xn,xn+k,xn+k(t)
≥TG∗xn,xn+,xn+(t–λt),TkG∗xn,xn+,xn+(t–λt)
=Tk+G∗xn,xn+,xn+(t–λt),
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→∞G∗xn,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) > – (.)
anyt> . By (GPM-), we have
G∗xn,xm,x
l(t)≥T
G∗xn,xn,xm(t/),G∗xn,xn,x
l(t/)
,
G∗xn,xn,xm(t/)≥TG∗xn,xm,xm(t/),Gxn∗,xm,xm(t/),
G∗xn,xn,xl(t/)≥TG∗xn,xl,xl(t/),G∗xn,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 DefineG∗x,y,z(t) =min{Fx,y(t),Fy,z(t),Fx,z(t)}for allx,y,z∈Xand allt> . Example . shows that (X,G∗,T) is a PGM-space. SinceG∗xn,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 thatG∗xn,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 :X→X 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+=fnxfor all n≥. By (.), for anyt> , we have
G∗xn,xn+,xn+(t) =G∗fxn–,fxn,fxn(t)
≥G∗xn–,xn,xn(t/λ).
Lemma . shows that{xn}is a Cauchy sequence inX. SinceXis complete, there exists a pointx∈Xsuch thatxn→xasn→ ∞. By (.), it follows that
Lettingn→ ∞, sincexn→xandfxn→xasn→ ∞, we have
G∗fx,x,x(t) =
for anyt> . Hencex=fx.
Next, suppose thatyis another fixed point off. Then, by (.), we have
G∗x,y,y(t) =Gfx∗,fy,fy(t)≥G∗x,y,y(t/λ)≥ · · · ≥Gx∗,y,yt/λn.
Lettingn→ ∞, sinceXis a Menger PGM-space,G∗x,y,y(t/λn)→ asn→ ∞, so
G∗x,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 :X→X be a mapping satisfying
G∗fx,fy,fz(λt)≥
G∗x,fx,fx(t) +G∗y,fy,fy(t) +G∗z,fz,fz(t) (.)
for all x,y,z∈X,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+=fnxfor all n≥. By (.), for anyt> , we have
G∗xn,xn+,xn+(λt) =G∗fxn–,fxn,fxn(λt)
≥
G∗xn–,fxn–,fxn–(t) + G∗xn,fxn,fxn(t)
≥
G∗xn–,fxn–,fxn–(t) + G∗xn,fxn,fxn(λt)
=
G∗xn–,xn,xn(t) + G∗xn,xn+,xn+(λt).
This shows that
G∗xn,xn+,xn+(t)≥G∗xn–,xn,xn(t/λ).
Lemma . shows that{xn}is a Cauchy sequence inX. SinceXis complete, there exists a pointx∈Xsuch thatxn→xasn→ ∞. By (.), it follows that
G∗fx,fx,fxn(t)≥
G∗x,fx,fx(t/λ) +G∗xn,fxn,fxn(t/λ).
Lettingn→ ∞, sincexn→xandfxn→xasn→ ∞, we have, for anyt> ,
G∗fx,fx,x(t)≥
G∗x,fx,fx(t/λ) +G∗x,x,x(t/λ)≥
i.e.,
G∗fx,fx,x(t)≥G∗x,x,x(t/λ) = .
Hencex=Tx.
Next, suppose thatyis another fixed point off. Then, by (.), we have, for anyy> ,
G∗x,y,y(t) =G∗fx,fy,fy(t)
≥
G∗x,fx,fx(t/λ) + G∗y,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
G∗x,y,z(t) = t
t+G(x,y,z)
for allx,y,z∈X, whereG(x,y,z) =|x–y|+|y–z|+|z–x|. 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,a∈X, 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 :X→Xbyfx=λxfor allx∈X. For anyt> , we have
G∗fx,fy,fz(t) = t
t+λ(|x–y|+|y–z|+|z–x|)
and
G∗x,y,z(t/λ) = t/λ
t/λ+ (|x–y|+|y–z|+|z–x|).
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 :X→Xbyfx= for allx∈X. For anyt> and all
x,y,z∈X, since
and
G∗x,fx,fx(λt) +G∗y,fy,fy(λt) +G∗z,fz,fz(λt)≤,
we conclude that
G∗fx,fy,fz(t)≥
G∗x,fx,fx(λt) +G∗y,fy,fy(λt) +G∗z,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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