Generalized probabilistic G-contractions

(1)

R E S E A R C H

Open Access

Generalized probabilistic

G

-contractions

Seong-Hoon Cho

*

*_{Correspondence:}

[email protected] Department of Mathematics, Hanseo University, Seosan, Chungnam 356-706, South Korea

Abstract

In this paper, the notion of generalized probabilisticG-contractions in Menger probabilistic metric spaces endowed with a directed graphGis introduced and some new ﬁxed point theorems for such mappings are established.

MSC: Primary 47H10; secondary 54H25

Keywords: ﬁxed point; coincidence point; directed graph; Menger probabilistic

metric space

1 Introduction and preliminaries

Ran and Reurings [] gave a generalization of Banach contraction principle to partially ordered metric spaces. Since then, many authors obtained generalization and extension of the results of [–].

In particular, Ćirićet al. [] extended the results of [, , ] to partially ordered Menger probabilistic metric spaces.

Sametet al. [] introduced the notion ofα-ψ-contractive type mappings and established some ﬁxed point theorems for such mappings in complete metric spaces.

Cho [] obtained a generalization of the results of [] by introducing the concept of

α-contractive type mappings in Menger probabilistic metric spaces.

Recently, Wu [] obtained a generalization of the results of [], and improved and ex-tended the ﬁxed point results of [, , ]. Also, Kamranet al. [] introduced the notion of probabilisticG-contractions in Menger PM-spaces endowed with a graphGand ob-tained some ﬁxed point results. Especially, they obob-tained the following result.

Theorem . Let(X,F,)be a complete Menger PM-space,whereis of Hadžić-type.

Let G= (V(G),E(G))be a directed graph such that V(G) =X and⊂E(G).Suppose that a map f :X→X satisﬁes f preserves edges and there exists k∈(, )such that, for all x,y∈X with(x,y)∈E(G),

Ffx,fy(kt)≥Fx,y(t).

Assume that there exists x∈X such that (x,fx)∈E(G).If either f is orbitally

G-continuous or G is a C-graph,then f has a ﬁxed point in[x]G.

Further if(x,y)∈E(G)for any x,y∈M,where M={x∈X: (x,fx)∈E(G)},then f has a unique ﬁxed point.

(2)

In this paper, we give some new ﬁxed point theorems which are generalizations of the re-sults of [, , , ], by introducing a concept of generalized probabilisticG-contractions in Menger PM-spaces with a directed graphG= (V(G),E(G)) such that V(G) =Xand

⊂E(G).

We recall some deﬁnitions and results which will be needed in the sequel. A mappingf :R→[,∞) is called adistributionif the following conditions hold:

() f is nondecreasing and left-continuous; () sup{f(t) :t∈R}= ;

() inf{f(t) :t∈R}= .

We denote byDthe set of all distribution functions. Let:R→[,∞) be a function deﬁned by

(t) =

⎧ ⎨ ⎩

 (t≤),

 (t> ).

Then∈D.

Let: [, ]×[, ]→[, ] be a mapping such that

() (a,b) =(b,a)for alla,b∈[, ];

() ((a,b),c) =(a,(b,c))for alla,b,c∈[, ]; () (a, ) =afor alla∈[, ];

() (a,b)≥(c,d), whenevera≥candb≥dfor alla,b,c,d∈[, ].

Thenis called atriangular norm(for shortt-norm). We denoteNby the set of all natural numbers. For at-norm, we consider the following notation:

(t) =(t,t), n(t) =t,n–(t) for alln∈Nandt∈[, ].

A t-norm is said to be of Hadžić-type [] whenever the family of {n(t)}∞_n_= is equicontinuous att= .

For example, the minimumt-normmdeﬁned by

m(a,b) =min{a,b}, ∀a,b∈[, ],

is of Hadžić-type.

It is easy to see that the following are equivalent (see []):

() for at-norm,

it is of Hadžić-type; (.)

() given∈(, ), there is aδ∈(, )such thatn₍_x_{) >  –}_{for all}_n_∈_N_{, whenever}

x>  –δ.

Also, it is well known that ifsatisﬁes condition(a,a)≥afor alla∈[, ], then=

m(see []). Hence we have

(3)

LetXbe a nonempty set, and letbe at-norm. Suppose that a mappingF:X×X→D

(forx,y∈X, we denoteF(x,y) byFx,y) satisﬁes the following conditions:

(PM) Fx,y(t) =(t)for allt∈Rif and only ifx=y;

(PM) Fx,y=Fy,xfor allx,y∈X;

(PM) Fx,y(t+s)≥(Fx,z(t),Fz,y(s))for allx,y,z∈Xand allt,s≥.

Then a -tuple (X,F,) is called aMenger probabilistic metric space(brieﬂy,Menger PM-space) [, ].

Let (X,F,) be a Menger PM-space and∈X, and let>  andλ∈(, ].

Schweizer and Sklar [] brought in the notion of neighborhood Ux(,λ) ofx, where

Ux(,λ) is deﬁned as follows:

Ux(,λ) =

y∈X:Fx,y() >  –λ .

The family

Ux(,λ) :x∈X,> ,λ∈(, ] (.)

does not necessarily determine a topology onX(see [, ]). It is well known that ifsatisﬁes condition

sup(t,t) :  <t<  =  (.)

then (.) determines a Hausdorﬀ topology onX, and it is called (,λ)-topology.

So if (.) holds, then Menger space (X,F,) is a Hausdorﬀ topological space in the (,λ)-topology (see [, ]).

Remark . The following are satisﬁed:

() condition (.) is the weakest condition which ensure the existence of the

(,λ)-topology (see []);

() condition (.)⇒condition (.) (see []).

Let (X,F,) be a Menger PM-space, and let{xn}be a sequence inXandx∈X. Then we say that

() {xn}isconvergenttox(we writelimn→∞xn=x) if and only if, given> and

λ∈(, ), there existsn∈Nsuch thatFxn,x() >  –λ, for alln≥n.

() {xn}is aCauchy sequenceif and only if, given> andλ∈(, ), there existsn∈N

such thatFxn,xm() >  –λ, for allm>n≥n.

() (X,F,)iscompleteif and only if each Cauchy sequence inXis convergent to some point inX.

Example . LetDbe a distribution function deﬁned by

D(t) = ⎧ ⎨ ⎩

 (t≤),

(4)

Let

Fx,y(t) =

⎧ ⎨ ⎩

(t) (x=y),

D(_d₍_xt_,_y₎) (x=y),

for allx,y∈Xandt> , wheredis a metric on a nonempty setX. Then (X,F,m) is a Menger PM-space (see []).

Remark . If (X,d) is complete, then (X,F,m) is complete. In fact, let {xn} be any Cauchy sequence in (X,F,m).

Then

lim

n,m→∞D

t d(xn,xm)

= lim

n,m→∞Fxn,xm(t) = 

for allt> , which implieslimn,m→∞d(xn,xm) = .

Hence,{xn}is a Cauchy sequence in (X,d). Since (X,d) is complete, there existsx∗∈X such thatlimn→∞d(xn,x∗) = .

Thus, we have

lim

n→∞Fxn,x∗(t) =nlim→∞D

t d(xn,x∗)

= 

for allt> . Hence, (X,F,m) is complete.

From now on, let

=φ: [,∞)→[,∞)| lim

n→∞φ

n₍_t_{) = ,}_∀_t_{> }

and let

w=

φ: [,∞)→[,∞)| ∀t> ,∃r≥ts.t. lim

n→∞φ

n₍_r_{) = }_.

Note that ⊂ w.

Fang [] gave the corrected version of Theorem  of [] by introducing the notion of right-locally monotone functions as follows:φ: [,∞)→[,∞) is right-locally monotone if and only if∀t≥,∃δ>  s.t. it is monotone on [t,t+δ).

Lemma .[] The following are satisﬁed:

() If a right-locally monotone functionφ: [,∞)→[,∞)satisﬁes

φ() = , φ(t) <t and lim

r→t+infφ(r) <t for allt> ,

thenφ∈ .

() If a functionφ: [,∞)→[,∞)satisﬁes

φ(t) <t and lim

r→t+supφ(r) <t for allt> ,

(5)

() If a functionα: [,∞)→[, )is piecewise monotone and

φ(t) =α(t)t for allt≥,

thenφ∈ .

Lemma .[] Ifφ∈ w,then∀t> ,∃r≥t s.t.φ(r) <t.

Lemma .[] Let(X,F,)be a Menger PM-space,and let x,y∈X.If

Fx,y

φ(t)≥Fx,y(t)

for all t> ,whereφ∈ w,then x=y.

Lemma .[] Let(X,F,)be a Menger PM-space and x,y∈X,whereis continuous.

Suppose that{xn}is a sequence of points in X.Iflimn→∞xn=x,thenlimn→∞infFxn,y(t) =

Fx,y(t)for all t> .

Lemma . Let(X,F,)be a Menger PM-space,whereis of Hadžić-type.Let{xn}be a

sequence of points in X such that xn–=xnfor all n∈N.If there existsφ∈ wsuch that

Fxn,xm

φ(s)≥minFxn–,xm–(s),Fxn–,xn(s),Fxm–,xm(s) (.)

for all s> and all n,m∈N,then for each t> there exists r≥t such that

Fxn,xm(t)≥m–n

Fxn,xn+

t–φ(r) for all m≥n+ . (.)

Proof It is easy to see that (.) implies thatφ(t) >  for allt> . In fact, if there exists

t>  such thatφ(t) = , then we obtain

 =Fxn,xn

φ(t)

≥Fxn–,xn(t) > 

which is a contradiction. We claim that

Fxn,xn+(u)≥Fxn–,xn(u) for allu>  andn∈N.

From (.) we have

Fxn,xn+

φ(s)≥minFxn–,xn(s),Fxn,xn+(s)

for alls>  and alln∈N.

If there exists n∈Nsuch thatFxn–,xn(s)≥Fxn,xn+(s) for alls> , thenFxn,xn+(φ(s))≥ Fxn,xn+(s) for alls> . Thus,xn=xn+, which is a contradiction. Hence we haveFxn–,xn(s) <

Fxn,xn+(s) for alls>  andn∈N, and so

Fxn,xn+

φ(s)≥Fxn–,xn(s)

(6)

Sinceφ∈ w, for eachu> , there existsv≥usuch that

φ(v) <u.

Hence,

Fxn,xn+(u)≥Fxn,xn+

φ(v)≥Fxn–,xn(v)≥Fxn–,xn(u)

for allu>  andn∈N. So the claim is proved.

Lett>  be given. By Lemma ., there existsr≥tsuch that

φ(r) <t. (.)

By induction, we show that (.) holds. Letm=n+ .

Then

Fxn,xn+(t) ≥Fxn,xn+

t–φ(r)

=Fxn,xn+

t–φ(r), 

≥Fxn,xn+

t–φ(r).

Thus, (.) holds form=n+ .

Assume that (.) holds for some ﬁxedm>n+ . That is,

Fxn,xm(t)≥m–n

Fxn,xn+

t–φ(r) holds for somem>n+ . (.)

Then

Fxn,xm+(t)

=Fxn,xm+

t–φ(r) +φ(r)

≥Fxn,xn+

t–φ(r),Fxn+,xm+

φ(r). (.)

From (.) we obtain

Fxn+,xm+

φ(r)

≥minFxn,xm(r),Fxn,xn+(r),Fxm,xm+(r) .

By the above claim, sinceFxm,xm+(t)≥Fxn,xn+(t), from (.) and (.) we obtain

Fxn+,xm+

φ(r)

≥minFxn,xm(t),Fxn,xn+(t) ≥minm–nFxn,xn+

t–φ(r),Fxn,xn+

t–φ(r)

=m–nFxn,xn+

(7)

Thus, from (.) and (.) we have

Fxn,xm+(t) ≥Fxn,xn+

t–φ(r),m–nFxn,xn+

t–φ(r)

=m–n+Fxn,xn+

t–φ(r).

Hence, (.) holds for allm≥n+ .

Lemma .[] Let(X,d)be a metric space.Suppose that F:X×X→D is a mapping deﬁned by

F(x,y)(t) =Fx,y(t) =

t–d(x,y)

for all x,y∈X and all t> .

Then(X,F,m)is a Menger PM-space,which is called a Menger PM-space induced by

the metric d.

Remark . Let (X,d) be a metric space. Suppose that (X,F,m) is a Menger PM-space induced byd.

Then we have the following.

() Iff :X→Xis continuous in(X,d), then it is continuous in(X,F,m).

() If a sequence{xn}is convergent to a pointxin(X,d), then it is convergent toxin (X,F,m).

() If(X,d)is complete, then(X,F,m)is complete.

Lemma .[] If X is a nonempty set and h:X→X is a function,then there exists Y⊂X

such that h(Y) =h(X)and h:Y→X is one-to-one.

LetXbe a nonempty set, and let={(x,x) :x∈X}the diagonal of the Cartesian product

X×X.

LetGbe a directed graph such that the following conditions are satisﬁed:

() the setV(G)of its vertices coincides withX,i.e.V(G) =X; () the setE(G)of its edges contains all loops,i.e.⊂E(G).

IfGhas no parallel edges, then we can identifyGwith the pair (V(G),E(G)). LetG= (V(G),E(G)) be a directed graph.

Then theconversionof the graphG(denoted byG–_{) is an ordered pair (}_V₍_G–_),_E₍_G–₎₎

consisting of a setV(G–_{) of vertices and a set}_E₍_G–_{) of edges, where}

VG–=V(G) and EG–=(x,y)∈X×X: (y,x)∈E(G) .

Note thatG–_{= (}_V₍_G_),_E₍_G–_)).

Given a directed graphG= (V(G),E(G)), letG= (V(G),E(G)) be a directed graph such that

(8)

Forx,y∈V(G), letp= (x=x,x,x, . . . ,xN=y) be a ﬁnite sequence such that

(xn–,xn)∈E(G) forn= , , . . . ,N.

Thenpis called a path inGfromxtoyof lengthN. Denote(G) by the family of all path inG.

If, for any x,y∈V(G), there is a pathp∈(G) fromxtoy, then the graphGcalled

connected. A graphGis calledweakly connected, wheneverGis connected. LetGbe a graph such thatE(G) is symmetric andx∈V(G).

Then the subgraphGx= (V(Gx),E(Gx)) is calledcomponentofGcontainingxif and only if there is a pathp∈(G) beginning atxsuch that

v∈p for allv∈V(Gx) and e⊂p for alle∈E(Gx).

Deﬁne a relationonV(G) as follows:

(y,z)∈ ⇐⇒ there is ap∈(G) fromytoz.

Then the relationis an equivalence relation onV(G), and [x]G=V(Gx), where [x]Gis the equivalence class ofx∈V(G).

Note that the componentGxofGcontainingxis connected. For the details of the graph theory, we refer to [].

Let (X,F,) be a Menger PM-space, and letG= (V(G),E(G)) be a directed graph such thatV(G) =Xand⊂E(G).

Then the graph Gis said to be a C-graphif and only if, for any sequence{xn} ⊂X with limn→∞xn=x∗ ∈X, there exist a subsequence {xnk} of {xn} and an N∈N such that (xnk,x∗)∈E(G) (resp. (x∗,xnk)∈E(G)) for allk≥Nwhenever (xn,xn+)∈E(G) (resp. (xn+,xn)∈E(G)) for alln∈N.

The following deﬁnitions are in [].

Let (X,F,) be a Menger PM-space, and letG= (V(G),E(G)) be a directed graph such thatV(G) =Xand⊂E(G). Letf:X→Xbe a map. Then we say that:

() f iscontinuousif and only if, for anyx∈Xand a sequence{xn} ⊂Xwith

limn→∞xn=x,

lim

n→∞fxn=fx.

() f isG-continuousif and only if, for anyx∈Xand a sequence{xn} ⊂Xwith

limn→∞xn=xand(xn,xn+)∈E(G)for alln∈N,

lim

n→∞fxn=fx.

() f isorbitally continuousif and only if, for allx,y∈Xand any sequence{kn} ⊂Nwith

lim_n_→∞fkn_x₌_y_,

lim

(9)

() f isorbitallyG-continuousif and only if, for allx,y∈Xand any sequence{kn} ⊂N

withlimn→∞fknx=yand(fknx,fkn+x)∈E(G)for allk∈N,

lim

n→∞ﬀ kn_x₌_fy_.

2 Main results

From now on, let (X,F,) be a Menger PM-space, whereis at-norm of Hadžić-type. LetG= (V(G),E(G)) be a directed graph satisfying conditions

V(G) =X and ⊂E(G).

A mapf :X→Xis said to be ageneralized probabilistic G-contractionif and only if the following conditions are satisﬁed:

() f preserves edges ofG,i.e.(x,y)∈E(G)⇒(fx,fy)∈E(G); () there existsφ∈ wsuch that

Ffx,fy

φ(t)≥minFx,y(t),Fx,fx(t),Fy,fy(t) (.)

for allx,y∈Xwith(x,y)∈E(G)and allt> .

Theorem . Let(X,F,)be complete.Suppose that a map f :X→X is a generalized probabilistic G-contraction.Assume that there exists x∈X such that(x,fx)∈E(G).If

either f is orbitally G-continuous oris a continuous t-norm and G is a C-graph,then f has a ﬁxed point in[x]G.

Proof Letx∈Xbe such that (x,fx)∈E(G). Letxn=fnxfor alln∈N∪ {}.

If there existsn∈Nsuch thatxn=xn+, thenxn=xn+=fxn, and soxn is a ﬁxed

point off.

Consider the pathpinGfromxtoxn+:

p= (x,x,x, . . . ,xn=xn+)∈(G).

Then the above path is inG. Hence,xn=xn+∈[x]G. Hence, the proof is ﬁnished.

Assume thatxn–=xnfor alln∈N.

As in the proof of Lemma ., we haveφ(t) >  for allt> .

Sincefis a generalized probabilisticG-contraction, (xn,xn+)∈E(G) for alln= , , , . . . ,

and from (.) withx=xn–,y=xnwe have

Fxn,xn+

φ(t)=Ffxn–,fxn

φ(t)

≥minFxn–,xn(t),Fxn–,fxn–(t),Fxn,fxn(t)

=minFxn–,xn(t),Fxn,xn+(t)

(10)

If there existsn∈Nsuch thatFxn–,xn(t)≥Fxn,xn+(t) for allt> , then

Fxn,xn+

φ(t)≥Fxn,xn+(t)

for allt> .

By Lemma .,xn=xn+, which is a contradiction. Thus, we haveFxn–,xn(t) <Fxn,xn+(t)

for allt>  andn∈N, and so

Fxn,xn+

φ(t)≥Fxn–,xn(t)

for allt>  andn∈N. Thus, we have

Fxn,xn+

φn(t)≥Fx,x(t)

for allt>  andn∈N.

We now show that

lim

n→∞Fxn,xn+(t) =  (.)

for allt> . Sincelimt→∞Fx,x(t) = , for any∈(, ) there existst>  such that

Fx,x(t) >  –.

Becauseφ∈ w, there existst≥tsuch that

lim

t→∞φ n₍_t

) = .

Thus, for eacht> , there existsNsuch thatφn(t) <tfor alln>N. Hence, we have

Fxn,xn+(t)≥Fxn,xn+

φn(t)

≥Fx,x(t)≥Fx,x(t) >  –

for alln>N. Thus,limn→∞Fxn,xn+(t) =  for allt> .

Next, we show that{xn}is a Cauchy sequence. Let∈(, ) be given.

Sinceis of Hadžić-type, there existsλ∈(, ) such that

n(s) >  – for alln= , , . . . , whenevers>  –λ. (.)

Sinceφ∈ w, for eacht> , there existsr≥tsuch thatφ(r) <t. From (.) we have

lim

n→∞Fxn,xn+

t–φ(r)= .

Thus, there existsNsuch that

Fxn,xn+

t–φ(r)>  –λ (.)

(11)

Since (.) is satisﬁed,

Fxn,xm(t)≥m–n

Fxn,xn+

t–φ(r) (.)

holds for allm≥n+  by Lemma .. By applying (.) with (.) and (.),

Fxn,xm(t) >  –

for allm>n>N.

Thus,{xn}is a Cauchy sequence inX. It follows from the completeness ofXthat there existsx∗∈Xsuch that

lim

n→∞xn=x∗.

Iff is orbitallyG-continuous, thenlimn→∞xn=fx∗. Hence,x∗=fx∗. Suppose thatis continuous andGisC-graph.

Then there exist a subsequence{xnk}of{xn}and anN∈Nsuch that

(xnk,x∗)∈E(G)

for allk≥N. Sincef is a generalized probabilisticG-contraction and (xnk,x∗)∈E(G) for allk≥N, from (.) withx=xnkandy=x∗we have

Fx_nk+,fx∗

φ(t)

=Ffx_nk,fx∗

φ(t)

≥minFx_nk,x∗(t),Fx_nk,fx_nk(t),Fx∗,fx∗(t)

=minFx_nk,x∗(t),Fx_nk,x_nk+(t),Fx∗,fx∗(t)

for allt> .

By Lemma ., we obtain

Fx∗,fx∗

φ(t)

= lim

k→∞infFxnk+,fx∗

φ(t)

≥ lim

k→∞inf min

Fx_nk,x∗(t),Fx_nk,fx_nk(t),Fx∗,fx∗(t)

=min, ,Fx∗,fx∗(t)

=Fx∗,fx∗(t)

for allt> . By Lemma .,x∗=fx∗. Consider the pathqinGfromxtox∗:

q= (x,x,x, . . . ,xnN,x∗)∈(G).

(12)

Suppose that (x,y)∈E(G) for anyx,y∈M. Letx∗andy∗be two ﬁxed point off.

Thenx∗,y∗∈M. By assumption, (x∗,y∗)∈E(G). From (.) withx=x∗,y=y∗we have

Fx∗,y∗

φ(t)=Ffx∗,fy∗

φ(t)

≥minFx∗,y∗(t),Fx∗,fx∗(t),Fy∗,fy∗(t)

=minFx∗,y∗(t), , 

=Fy∗,x∗(t)

for allt> . By Lemma .,x∗=y∗. Thus,f has a unique ﬁxed point. Example . LetX= [,∞), and letd(x,y) =|x–y|for allx,y∈X.

Let

Fx,y(t) =

⎧ ⎨ ⎩

(t) (x=y),

D(_d₍_xt_,_y₎) (x=y),

for allx,y∈Xandt> , whereDis a distribution function deﬁned by

D(t) = ⎧ ⎨ ⎩

 (t≤),

 –e–t ₍_t_{> ).}

Then (X,F,m) is a complete Menger PM-space. Letfx=

xfor allx∈X, and let

φ(t) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩



t (≤t< ),

–_t+_ (≤t≤_),

t–_ (_<t<∞).

Thenφ∈ wandφ(t)≥_tfor allt≥.

Further assume thatXis endowed with a graphGconsisting ofV(G) =XandE(G) =

{(x,y)∈X×X:yx}.

Obviously,f preserves edges, and it is orbitallyG-continuous. Ifx= , then (x,fx) =

(, )∈E(G). We have

Ffx,fy

φ(t)=D

φ(t)

|fx–fy|

≥D



t  t|x–y|

=D

t t|x–y|

=Fx,y(t)

≥minFx,y(t),Fx,fx(t),Fy,fy(t)

(13)

Thus, (.) is satisfied. Hence, all the conditions of Theorem . are satisfied andf has a fixed pointx∗= ∈[]G. Furthermore,M={}and the fixed point is unique.

Remark . Note that in Theorem . the assumption of orbitallyG-continuity can be replaced by orbitally continuity,G-continuity or continuity.

Remark . Theorem . is a generalization of Theorem . in [] to the case of a Menger PM-space endowed with a graph.

Corollary . Let(X,F,)be complete,and let f :X→X be a map.Suppose that the following are satisﬁed:

() f preserves edges ofG; () there existsφ∈ such that

Ffx,fy

φ(t)≥minFx,y(t),Fx,fx(t),Fy,fy(t)

for allx,y∈Xwith(x,y)∈E(G)and allt> .

Assume that there exists x∈X such that (x,fx)∈ E(G). If either f is orbitally

G-continuous oris a continuous t-norm and G is a C-graph,then f has a ﬁxed point in

[x]G.

Remark .

() Corollary ., in part, is a generalization of Theorem . and Theorem . of []. () In Corollary ., letφ(s) =ksfor alls≥, wherek∈(, ). IfGis a graph such that

V(G) =XandE(G) ={(x,y)∈X×X:α(x,y)≥}, whereα:X×X→[,∞)is a function, then Corollary . reduces to Theorem . of [].

() IfGis a graph such thatV(G) =XandE(G) ={(x,y)∈X×X:xy}, whereis a partial order onX, then Corollary . become to Theorem . of [].

Corollary . Let(X,F,)be complete.Suppose that a map f :X→X is generalized probabilistic G-contraction.Assume that either f is continuous oris a continuous t-norm and G is a C-graph.

Then f has a ﬁxed point in[x]Gfor some x∈Q if and only if Q=∅,where Q={x∈X:

(x,fx)∈E(G)}.Further if,for any x,y∈Q, (x,y)∈E(G)then f has a unique ﬁxed point.

Proof Iffhas a ﬁxed point in [x]G, sayx∗, then (x∗,fx∗) = (x∗,x∗)∈⊂E(G). Thus,Q=∅. Suppose thatQ=∅.

Then there existsx∈Xsuch that (x,fx)∈E(G).

We have two cases: (x,fx)∈E(G) or (x,fx)∈E(G–).

If (x,fx)∈E(G), then following Theorem .f has a ﬁxed point in [x]G. Assume that (x,fx)∈E(G–).

Then (fx,x)∈E(G). Sincef is preserves edges ofG, (fn+x,fnx)∈E(G) for alln∈

N∪ {}.

In the same way as the proof of Theorem . with condition (PM), we deduce thatf

has a ﬁxed point in [x]G.

Suppose that, for anyx,y∈Q, (x,y)∈E(G). Letx∗andy∗be two ﬁxed points off.

(14)

If (x∗,y∗)∈E(G), then

Fx∗,y∗

φ(t)≥minFx∗,y∗(t),Fx∗,x∗(t),Fy∗,y∗(t) =Fx∗,y∗(t)

for allt> . By Lemma .,x∗=y∗. Let (x∗,y∗)∈E(G–), then (y∗,x∗)∈E(G). Then

Fy∗,x∗

φ(t)≥Fy∗,x∗(t)

for allt> . Hence,y∗=x∗. Thus,f has a unique ﬁxed point. Remark . Ifφ∈ andGis a graph such thatV(G) =XandE(G) ={(x,y)∈X×X:

xy}, whereis a partial order onX, then Corollary . reduces to Theorem . of [].

In the following result, we can drop continuity of thet-norm.

Corollary . Let(X,F,)be complete.Suppose that a map f :X→X satisﬁes

Ffx,fy

φ(t)≥Fx,y(t) (.)

for all x,y∈X with(x,y)∈E(G)and all t> ,whereφ∈ w.

Assume that there exists x∈X such that (x,fx)∈E(G). If either f is orbitally

G-continuous or G is a C-graph,then f has a ﬁxed point in[x]G.

Proof Letx∈Xbe such that (x,fx)∈E(G), and letxn=fnxfor alln∈N∪ {}.

Note that (.) to be satisﬁed implies that (.) is satisﬁed.

As in the proof of Theorem .,xn–=xnand (xn–,xn)∈E(G) for alln∈Nand there

exists

lim

n→∞xn=x∗∈X.

Iff is orbitallyG-continuous, thenlimn→∞xn=fx∗, and sox∗=fx∗. Assume thatGis aC-graph.

Then there exist a subsequence{xnk}of{xn}and anN∈Nsuch that

(xnk,x∗)∈E(G)

for allk≥N.

Sinceφ∈ w, for eacht> , there existsr≥tsuch thatφ(r) <t. We have

Fx∗,fx∗(t) ≥Fx∗,x_nk+

t–φ(r),Ffx_nk,fx∗

(15)

≥Fx∗,x_nk+

t–φ(r),Fx_nk,x∗(r)

≥Fx∗,x_nk+

t–φ(r),Fx_nk,x∗(t)

≥(an,an) (.)

for allt> , wherean=min{Fx∗,x_nk+(t–φ(r)),Fxnk,x∗(t)}.

Sincelimn→∞an=  and(t,t) is continuous att= , limn→∞(an,an) =(, ) = . Hence, from (.) we haveFx∗,fx∗(t) =  for allt> , and sox∗=fx∗.

Remark . Corollary . is a generalization of Theorem . in [] to the case of a Menger PM-space endowed with a graph.

Theorem . Let(X,F,)be complete such thatis continuous. Let f,h:X→X be maps,and let G be a directed graph satisfying V(G) =h(X)and{(hx,hx) :x∈X} ⊂E(G).

Suppose that the following are satisﬁed:

() f(X)⊂h(X); () h(X)is closed;

() (hx,hy)∈E(G)implies(fx,fy)∈E(G); () there existsx∈Xsuch that(hx,fx)∈E(G);

() there existsφ∈ wsuch that

Ffx,fy

φ(t)≥minFhx,hy(t),Fhx,fx(t),Fhy,fy(t) (.)

for allx,y∈Xwith(hx,hy)∈E(G)and allt> ;

() if{xn}is a sequence inXsuch that(hxn,hxn+)∈E(G)for alln∈N∪ {}and

limn→∞hxn=hufor someu∈X,then(hxn,hu)∈E(G)for alln∈N∪ {}.

Then f and h have a coincidence point in X.Further if f and h commute at their coinci-dence points and(hu,hhu)∈E(G),then f and h have a common ﬁxed point in X.

Proof By Lemma ., there existsY⊂Xsuch thath(Y) =h(X) andh:Y→Xis one-to-one. Deﬁne a mappingU:h(Y)→h(Y) byU(hx) =fx. Sinceh:Y→Xis one-to-one,U

is well deﬁned.

By (), (hx,hy)∈E(G) implies (U(hx),U(hy))∈E(G). By (), (hx,U(hx))∈E(G) for somex∈X. We have

FU(hx),U(hy)

φ(t)

=Ffx,fy

φ(t)

≥minFhx,hy(t),Fhx,fx(t),Fhy,fy(t)

=minFhx,hy(t),Fhx,U(hx)(t),Fhy,U(hy)(t)

for allhx,hy∈h(Y) with (hx,hy)∈E(G). Sinceh(Y) =h(X) is complete, by applying Theo-rem ., there existsu∈Xsuch thatU(hu) =hu, and sohu=fu. Hence,uis a coincidence point off andh.

Suppose thatf andhcommute at their coincidence points and (hu,hhu)∈E(G). Let

(16)

Applying inequality (.) withx=u,y=w, we have

Fw,fw

φ(t)

=Ffu,fw

φ(t)

≥minFhu,hw(t),Fhu,fu(t),Fhw,fw(t)

=minFw,fw(t),Fw,w(t),Ffw,fw(t)

=minFw,fw(t), , 

=Ffw,w(t)

for allt> .

By Lemma ., w=fw. Hence w=fw=hw. Thus, w is a common ﬁxed point of f

andh.

Remark . Theorem . is a generalization of Theorem . of []. If we haveφ(s) =ks

for alls≥, wherek∈(, ), andV(G) =XandE(G) ={(x,y) :x≤y}, where≤is a partial order onX, then Theorem . reduces to Theorem . of [].

Theorem . Let(X,F,)be complete.Suppose that maps f,f:X→X satisfy the

fol-lowing:

Ffx,fy

φ(t)≥Fx,y(t), (.)

whereφ∈ wand

Ffx,fy(t)≥min

Fx,y(t),Fx,fx(t),Fy,fy(t) (.)

for all x,y∈X with(x,y)∈E(G)and all t> .

Suppose that f preserves edges,and assume that there exists x∈X such that(x,fx)∈

E(G),where f =ff.If either f is orbitally G-continuous oris a continuous t-norm and

G is a C-graph,then f has a ﬁxed point in[x]G.

Further if(x,y)∈E(G)for any x,y∈M,where M={x∈X: (x,fx)∈E(G)},then fand f

have a common ﬁxed point whenever fis commutative with f.

Proof From (.) and (.) we have

Ffx,fy

for allx,y∈Xwith (x,y)∈E(G) and allt> . By Theorem .,f has a ﬁxed point in [x]G, sayx∗.

Suppose that (x,y)∈E(G) for anyx,y∈M.

Then from Theorem .f has a unique ﬁxed point.

Sincefis commutative withf andfx∗=x∗,ffx∗=f(ffx∗) =f(ffx∗) =ffx∗=fx∗. Similarly, we obtainffx∗=fx∗. From the uniqueness of fixed point of f, we havex∗=

(17)

Example . LetX= [,∞), and letFx,y(t) =_t₊_dt₍_x_,_y₎for allx,y∈Xand allt> , where

d(x,y) = ⎧ ⎨ ⎩

max{x,y} (x=y),

 (otherwise).

Then (X,F,m) is a complete Menger PM-space. Let

φ(t) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 

t (≤t< ),

–_t+_ (≤t≤_),

t–_ (_<t<∞).

Thenφ∈ wandφ(t)≥tfor allt≥.

Further assume thatXis endowed with a graphGconsisting ofV(G) =XandE(G) =

{(x,y)∈X×X:yx}. Obviously,Gis aC-graph.

Letf:X→Xbe a map deﬁned byfx=_xfor allx≥, and deﬁne a mapf:X→Xby

fx=

⎧ ⎨ ⎩

x

(+x) (≤x≤), 

x (x> ).

Then

fx=ffx=

⎧ ⎨ ⎩

x

(+x) (≤x≤), 

x (x> ).

Obviously,f preserves edges. Let (x,y)∈E(G).

Thenyx, and we obtain

Ffx,fy

φ(t)= φ(t)

φ(t) +d(_x,_y)

≥  t  t+

 x

= t

t+x

= t

t+max{x,y}=Fx,y(t)

for allt> . Hence, (.) is satisﬁed. We consider the following three cases: Case . ≤y<x≤:

Ffx,fy(t) =

t

t+d(_(+x_x₎,_(+y_y₎)

= t

t+_(+x_x₎ ≥

(18)

= t

t+max{x,y}= t

t+d(x,y)=Fx,y(t)

for allt> . Case .  <y<x:

Ffx,fy(t) = t t+d(x

,

y

)

= t

t+_x ≥

t t+x=

t t+max{x,y}

= t

t+d(x,y)=Fx,y(t)

for allt> .

Case . ≤y≤ and  <x:

Ffx,fy(t) =

t t+d(_x,_(+y_y₎)

= t

t+ x

 ≥ t

t+x= t t+max{x,y}

= t

t+d(x,y)=Fx,y(t)

for allt> .

Thus, (.) is satisﬁed.

Forx= , (x,fx) = (,_)∈E(G). Hence, all the conditions of Theorem . are satisﬁed

andf has a ﬁxed pointx∗= ∈[x]G.

Corollary . Let(X,F,)be complete.Suppose that maps f,f:X→X satisfy the

fol-lowing:

Ffx,fy

φ(t)≥Fx,y(t), (.)

whereφ∈ wand

Ffx,fy(t)≥Fx,y(t) (.)

for all x,y∈X with(x,y)∈E(G)and all t> .

Suppose that f preserves edges,and assume that there exists x∈X such that(x,fx)∈

E(G),where f =ff.If f is orbitally G-continuous or G is a C-graph,then f has a ﬁxed

point in[x]G.

Further if(x,y)∈E(G)for any x,y∈M,where M={x∈X: (x,fx)∈E(G)},then fand f

(19)

Proof From (.) and (.) we have

Ffx,fy

φ(t)≥Fx,y(t)

for allx,y∈Xwith (x,y)∈E(G) and allt> . By Corollary .,f has a ﬁxed point in [x]G, sayx∗.

Suppose that (x,y)∈E(G) for anyx,y∈M.

Then from Corollary .f has a unique ﬁxed point.

Sincef is commutative withf, as in the proof of Theorem . we havex∗ =fx∗=

fx∗.

Remark . Corollary . is a generalization of Corollary . of [] to the case of Menger PM-space endowed with a graph.

Corollary . Let(X,d)be a complete metric space,and let G= (V(G),E(G))be a directed graph satisfying V(G) =X and⊂E(G).Let f :X→X be a map.Suppose that the follow-ing are satisﬁed:

() (x,y)∈E(G)implies(fx,fy)∈E(G); () there existsφ∈ wsuch that

d(fx,fy)

≤φmaxd(x,y),d(x,fx),d(y,fy) (.)

for allx,y∈Xwith(x,y)∈E(G),whereφis nondecreasing; () there existsx∈Xsuch that(x,fx)∈E(G);

(a) f is continuous,or

(b) if{xn}is a sequence inXsuch thatlimn→∞xn=x∗∈Xand(xn,xn+)∈E(G)for all

n∈N,then there exists a subsequence{xnk}of{xn}such that(xnk,x∗)∈E(G)for all

k∈N.

Then f has a ﬁxed point in[x]G.

Proof Suppose that equality holds in (.) andx=fxfor allx∈X. Letx∈Xbe ﬁxed. Then (x,x)∈E(G), and from (.) we have

 =d(fx,fx)

=φmaxd(x,x),d(x,fx),d(x,fx)

=φd(x,fx)

,

which impliesd(x,fx) =  and sox=fx, which is a contradiction.

Thus, if equality holds in (.), thenf has a ﬁxed point. Assume that equality is not satisﬁed in (.).

Let (X,F,m) be the induced Menger PM-space by (X,d).

By Lemma ., (X,F,m) is complete. By Remark ., (a) impliesf is continuous in (X,F,m), and (b) impliesGisC-graph.

(20)

We know that the values of each distribution functionFu,v(·),u,v∈X, in the induced

Menger PM-space only can equal  or . Hence, without loss of generality, we may assume that

minFx,y(t),Fx,fx(t),Fy,fy(t) = 

for allx,y∈E(G) andt> . Then

t>d(x,y), t>d(x,fx) and t>d(y,fy).

Thus,

t>maxd(x,y),d(x,fx),d(y,fy) .

Sinceφis nondecreasing,

φmaxd(x,y),d(x,fx),d(y,fy) ≤φ(t).

By assumption, we have

d(fx,fy) <φ(t).

Hence,φ(t) –d(fx,fy) > . SoFfx,fy(φ(t)) = . Thus we have

Ffx,fy

for allx,y∈Xwith (x,y)∈E(G) and allt> .

Hence, (.) is satisﬁed. By Theorem . and Remark ., f has a ﬁxed point in

[x]G.

Corollary . Let(X,d)be a complete metric space,and let G= (V(G),E(G))be a directed graph satisfying V(G) =X and⊂E(G).Let f :X→X be a map.

Suppose that the following are satisﬁed:

() (x,y)∈E(G)implies(fx,fy)∈E(G); () there existsφ∈ wsuch that

d(fx,fy)≤φd(x,y)

for allx,y∈Xwith(x,y)∈E(G),whereφis nondecreasing; () there existsx∈Xsuch that(x,fx)∈E(G);

() eitherf is continuous or if{xn}is a sequence inXsuch thatlimn→∞xn=x∗∈Xand (xn,xn+)∈E(G)for alln∈N,then there exists a subsequence{xnk}of{xn}such that (xnk,x∗)∈E(G)for allk∈N.

Then f has a ﬁxed point in[x]G.

Remark . Corollary . is a generalization of the results of []. If we have a graphG

(21)

Competing interests

The author declares that he has no competing interests.

Author’s contributions

The author completed the paper himself. The author read and approved the ﬁnal manuscript.

Acknowledgement

The author thanks the editor and the referees for their useful comments and suggestions.

Received: 10 August 2015 Accepted: 1 April 2016

References

1. Ran, ACM, Reurings, MCB: A ﬁxed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc.132, 1435-1443 (2004)

2. Agarwal, RP, El-Gebeily, MA, O’Regan, D: Generalized contractions in partially ordered metric spaces. Appl. Anal.87, 109-116 (2008)

3. ´Ciri´c, L, Mihe¸t, D, Saadati, R: Monotone generalized contractions in partially ordered probabilistic metric spaces. Topol. Appl.156, 2838-2844 (2009)

4. ´Ciri´c, L, Agawal, RP, Samet, B: Mixed monotone generalized contractions in partially ordered probabilistic metric spaces. Fixed Point Theory Appl.2011, 56 (2011)

5. Nieto, JJ, Lopez, RR: Contractive mapping theorems in partially ordered sets and applications to ordinary diﬀerential equations. Order22, 223-239 (2005)

6. Nieto, JJ, Lopez, RR: Existence and uniqueness of ﬁxed point in partially ordered sets and applications to ordinary diﬀerential equations. Acta Math. Sin. Engl. Ser.23, 2205-2212 (2007)

7. Petru¸sel, A, Rus, AI: Fixed point theorems in orderedL-spaces. Proc. Am. Math. Soc.134, 411-418 (2006)

8. Samet, B, Vetro, C, Vetro, P: Fixed point theorems forα-ψ-contractive type mappings. Nonlinear Anal.75, 2154-2165 (2012)

9. Cho, SH: Fixed point theorems forα-contractive type mappings in Menger probabilistic metric spaces. Int. J. Math. Anal.7, 2981-2993 (2013)

10. Wu, J: Some ﬁxed-point theorems for mixed monotone operators in partially ordered probabilistic metric spaces. Fixed Point Theory Appl.2014, 49 (2014)

11. Ćirić, L: Solving the Banach fixed point principle for nonlinear contractions in probabilistic metric spaces. Nonlinear Anal.72, 2009-2018 (2010)

12. Jachymski, J: On probabilisticϕ-contractions on Menger spaces. Nonlinear Anal.73, 2199-2203 (2010) 13. Kamran, T, Samreen, M, Shahzad, N: ProbabilisticG-contractions. Fixed Point Theory Appl.2013, 223 (2013) 14. Hadzic, O: Fixed point theorems for multi-valued mappings in probabilistic metric space. Mat. Vesn.3, 125-133 (1979) 15. Kelement, EP, Mesiar, R, Pap, E: Triangular Norm. Kluwer Academic, Dordrecht (2001)

16. Menger, K: Statistical metrics. Proc. Natl. Acad. Sci. USA28, 535-537 (1942) 17. Schweizer, B, Sklar, A: Probabilistic Metric Space. North-Holland, Amsterdam (1983) 18. Schweizer, B, Sklar, A: Statistical metric spaces. Pac. J. Math.10, 313-334 (1960)

19. Morrel, B, Nagata, J: Statistical metric spaces as related to topological spaces. Gen. Topol. Appl.9, 233-237 (1978) 20. Throp, E: Generalized topologies for statistical metric spaces. Fundam. Math.51, 9-21 (1962)

21. Schweizer, B, Sklar, A, Pap, E: The metrization of statistical metric spaces. Pac. J. Math.10, 673-675 (1960) 22. Mihet, D: Multivalued generalizations of probabilistic contraction. J. Math. Anal. Appl.304, 464-472 (2005) 23. Fang, JX: Onϕ-contractions in probabilistic and fuzzy metric spaces. Fuzzy Sets Syst.267, 86-99 (2015) 24. Sehgal, VM, Bharucha-Reid, AT: Fixed points of contraction mappings in PM-spaces. Math. Syst. Theory6, 97-102

(1972)

25. Haghi, RH, Rezapour, S, Shahzad, N: Some ﬁxed point generalizations are not real generalizations. Nonlinear Anal.74, 1799-1803 (2011)