Probabilistic G-contractions

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R E S E A R C H

Open Access

Probabilistic

G-contractions

Tayyab Kamran

1

, Maria Samreen

2

and Naseer Shahzad

3*

*Correspondence:

nshahzad@kau.edu.sa

3Department of Mathematics, King

Abdulaziz University, Jeddah, Saudi Arabia

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

Abstract

In this paper we introduce the notion of probabilisticG-contraction and establish some fixed point theorems in such settings. Our results generalize/extend some recent results of Jachymski and Sehgal and Bharucha-Reid. Consequently, we obtain fixed point results for (,δ)-chainable PM-spaces and for cyclic operators.

MSC: 47H10; 54H25

Keywords: fixed point; Menger PM-space; directed graph; Picard operator

1 Introduction

In recent years the Banach contraction principle has been widely used to study the exis-tence of solutions for the nonlinear Volterra integral equations, nonlinear integrodiffer-ential equations in Banach spaces and to prove the convergence of algorithms in compu-tational mathematics. It has been extended in many different directions for single- and multi-valued mappings. Recently, Nieto and Rodríguez-López [], Ran and Reurings [], Petruşl and Rus [] established some new results for contractions in partially ordered met-ric spaces. The following is the main result due to Nieto and Rodríguez-López [, ], Ran and Reurings [].

Theorem . Let(S,d)be a complete metric space endowed with the partial order ‘’.

Assume that the mapping f:SS is nondecreasing(or nonincreasing)with respect to the partial order ‘’ on S and there exists a real numberα,  <α< ,such that

d(fx,fy)≤αd(x,y) for all x,yS,xy. (.)

Also suppose that either

(i) f is continuous;or

(ii) for every nondecreasing sequence{xn}inSsuch thatxnxinS,we havexnxfor alln≥.

If there exists x∈S with xfx(or xfx),then f has a fixed point.Furthermore,if (S,)is such that every pair of elements of S has an upper or lower bound,then f is a Picard operator(PO).

Many authors undertook further investigations in this direction to obtain some general-izations and extensions of the above main result (see,e.g., [–]). In this context, Jachymski [] established a generalized and novel version of Theorem . by utilizing graph theoretic

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approach. From then on, investigations have been carried out to obtain better and gener-alized versions by weakening contraction condition and analyzing connectivity of a graph (see [–]).

Motivated by the work of Jachymski, we can pose a very natural question: Is it possible to establish a probabilistic version of the result of Jachymski [] (see Corollary .)? In this paper, we give an affirmative answer to this question. Our results are substantial gen-eralizations and improvements of the corresponding results of Jachymski [] and Sehgal [] and others (see,e.g., [, , ]). Subsequently, we apply our main results to the setting of cyclical contractions and to that of (,δ)-contractions as well.

2 Preliminaries

In  Menger introduced the notion of probabilistic metric space (briefly, PM space), and since then enormous developments in the theory of probabilistic metric space have been made in many directions [–]. The fundamental idea of Menger was to replace real numbers with distribution functions as values of a metric.

A mapping F:R→[, ] is called a distribution function if it is nondecreasing, left continuous and inftRF(t) = , suptRF(t) = . In addition, if F() = , thenF is called a distance distribution function. LetD+denote the set of all distance distribution func-tions satisfyinglimt→∞F(t) = . The spaceD+is partially ordered with respect to the usual

pointwise ordering of functions,i.e.,FGif and only ifF(t)≤G(t) for allt∈R. The ele-ment∈D+acts as the maximal element in the space and is defined by

(t) = ⎧ ⎨ ⎩

 ift≤,

 ift> . (.)

Definition . A mapping: [, ]×[, ]→[, ] is called a triangular norm (briefly

t-norm) if the following conditions hold: (i) is associative and commutative, (ii) (a, ) =afor alla∈[, ],

(iii) (a,b)≤(c,d)for alla,b,c,d∈[, ]withacandbd.

Typical examples oft-norms areM(a,b) =min{a,b}andP(a,b) =ab.

Definition .(Hadzić [], Hadzić and Pap []) At-normis said to be ofH-type if the family of functions{n(t)}n∈Nis equicontinuous att= , wheren: [, ]→[, ] is

recursively defined by

(t) =(t,t), and n(t) =n–(t),t; t∈[, ], n= , , . . . .

A trivial example of at-norm ofH-type isM:=min, but there existt-norms ofH-type

with=M(see,e.g., []).

Definition . A probabilistic metric space (briefly, PM-space) is an ordered pair (S,F), whereSis a nonempty set andF:S×SD+ if the following conditions are satisfied (F(p,q) =Fp,q,∀(p,q)∈S×S):

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(PM) Fx,y(t) =Fy,x(t)for allx,ySandt∈R;

(PM) ifFx,y(t) = andFy,z(s) = , thenFx,z(t+s) = for allx,y,zSand for every t,s≥.

Definition . A Menger probabilistic metric space (briefly, Menger PM-space) is a triple (S,F,), where (S,F) is a PM-space,is at-norm and instead of (PM) in Definition . it satisfies the following triangle inequality:

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

Remark .(Sehgal []) Let (S,d) be a metric space. DefineFxy(t) =(td(x,y)) for all

x,ySandt> . Then the triple (S,F,M) is a Menger PM-space induced by the metricd.

Furthermore, (S,F,M) is complete iffdis complete.

Schweizeret al.[] introduced the concept of neighborhood in PM-spaces. Forε>  andδ∈(, ], the (ε,δ)-neighborhood ofxSis denoted byNx(ε,δ) and is defined by

Nx(ε,δ) =

yS:Fx,y(ε) >  –δ

.

Furthermore, if (S,F,) is a Menger PM-space withsup<a<(a,a) = , then the family of neighborhoods{Nx(ε,δ) :xS,ε> ,δ∈(, ]}determines a Hausdorff topology forS.

Definition . Let (S,F,) be a Menger PM-space.

() A sequence{xn}inSconverges to an elementxinS(we writexnxor

limn→∞xn=x) if for everyε> andδ> there exists a natural numberN(ε,δ)such

thatFxn,x(ε) >  –δ, whenevernN.

() A sequence{xn}inSis a Cauchy sequence if for everyε> andδ> there exists a

natural numberN(ε,δ)such thatFxn,xm(ε) >  –δ, whenevern,mN.

() A Menger PM-space is complete if and only if every Cauchy sequence inS

converges to a point inS.

Now we recall some basic notions from graph theory which we need subsequently. Let (S,d) be a metric space, letbe the diagonal of the Cartesian productS×S, and letGbe a directed graph such that the setV(G) of its vertices coincides withSand the setE(G) of its edges contains all loops,i.e.,E(G)⊇. Assume thatGhas no parallel edges. Let

G= (V(G),E(G)) be a directed graph. By letterGwe denote the undirected graph obtained fromGby ignoring the direction of edges and byG–we denote the graph obtained by reversing the direction of edges. Equivalently, the graphGcan be treated as a directed graph havingE(G) :=E(G)∪E(G–). Ifxandyare vertices in a graphG, then a path in

Gfromxtoyof lengthlis a sequence (xi)li=ofl+  vertices such thatx=x,xl=yand

(xi–,xi)∈E(G) fori= , . . . ,l. A graphGis called connected if there is a path between

any two vertices.Gis weakly connected ifGis connected. For a graphGsuch thatE(G) is symmetric andxis a vertex inG, the subgraphGxconsisting of all edges and vertices which

are contained in some path beginning atxis called the component ofGcontainingx. In this caseV(Gx) = [x]G, where [x]Gis the equivalence class of a relationRdefined onV(G)

by the rule:yRzif there is a path inGfromytoz. Clearly,Gxis connected. A mapping f :SSis called a BanachG-contraction [] if∀x,yS; (x,y)∈E(G)⇒ (fx,fy)∈E(G),

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3 Main results

We start with the following definition.

Definition . A mappingf :SSis said to be a probabilisticG-contraction iff

pre-serves edges and there existsα∈(, ) such that

x,yS, (x,y)∈E(G) ⇒ Ffx,fy(αt)≥Fx,y(t). (.)

Example . Let (S,d) be a metric space endowed with a graphG, and let the mapping

f :SSbe a BanachG-contraction. Then the induced Menger PM space (S,F,M) is a

probabilisticG-contraction.

To see this, let (x,y)∈E(G), then (fx,fy)∈E(G) and there exists α∈(, ) such that

d(fx,fy)≤αd(x,y). Now, fort> , we have

Ffx,fy(αt) =

αtd(fx,fy)

αtαd(x,y)=Fx,y(t).

Thusf satisfies (.).

From Example . it is inferred that every Banach G-contraction is a probabilistic

G-contraction with the same contraction constant.

Proposition . Let f :SS be a probabilistic G-contraction with contraction constant

α∈(, ).Then

(i) f is both a probabilisticG-contraction and a probabilisticG–-contraction with the

same contraction constantα.

(ii) [x]Gisf-invariant andf|[x]Gis a probabilisticGx-contraction provided that

x∈Sis such thatfx∈[x]G.

Proof

(i) It follows from the symmetry ofFx,y.

(ii) Letx∈[x]G˜. Then there is a pathx=z,z, . . . ,zl=xbetweenxandx. Sincef is a probabilisticG-contraction,(fzi–,fzi)∈E(G)∀i= , , . . . ,l. Thusfx∈[fx]G˜ = [x]G˜.

Suppose (x,y)∈E(G˜x). Then (fx,fy)∈E(G) sincef is a probabilisticG-contraction. But [x]G˜ isf invariant, so we conclude that (fx,fy)∈E(G˜x). Condition (.) is satisfied

auto-matically, sinceG˜xis a subgraph ofG.

Lemma . Let (S,F,) be a Menger PM-space under a t-norm satisfying supa<

(a,a) = . Assume that the mapping f :SS is a probabilistic G-contraction. Let

y∈[x]G,then Ffnx,fny(t)→as n→ ∞(t> ).Moreover,for zS,fnxz(n→ ∞)if and only if fnyz(n→ ∞).

Proof LetxS andy∈[x]G, then there exists a path (xi),i= , , , . . . ,l, inGfrom x

to ywith x=x, xl =yand (xi–,xi)∈E(G). From Proposition ., f is a probabilistic G-contraction. By induction, fort> , we have (fnx

i–,fnxi)∈E(G) andFfnx

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Ffn–x

i–,fn–yi(t) for alln∈Nandi= , . . . ,l. Thus we obtain

Ffnx

i–,fnxi(t) ≥ Ffn–x

i–,fn–xi

t α

Ffn–x

i–,fn–xi

t α

· · ·

Fxi–,xi

t αn

→ (asn→ ∞).

Lett>  andδ>  be given. Sincesupa<(a,a) = , then there existsλ(δ)∈(, ) such that( –λ,  –λ) >  –δ. Choose a natural numbernsuch that for allnnwe have

Ffnx

,fnx(

t

) >  –λandFfnx,fnx(

t

) >  –λ. We get, for allnn,

Ffnx,fnx(t)≥

Ffnx,fnx

t

,Ffnx,fnx

t

( –λ,  –λ) >  –δ,

so thatFfnx

,fnx(t)→ asn→ ∞(t> ). Continuing recursively, one can easily show that

Ffnx

,fnxl(t)→ asn→ ∞(t> ).

LetfnxzS. Lett>  andδ>  be given. Sincesup

a<(a,a) = , then there exists

λ(δ)∈(, ) such that( –λ,  –λ) >  –δ. Choose a natural numbernsuch that for all

nnwe haveFfnx,fny(t

) >  –λandFz,fnx(

t

) >  –λ. So that for allnn, we have

Fz,fny(t)≥

Fz,fnx

t

,Ffnx,fny

t

( –λ,  –λ) >  –δ.

Hence,fnyzasn→ ∞.

Everyt-norm can be extended in a unique way to ann-ary as follows:i=xi= ,ni=xi= (ni=–xi,xn) forn= , , . . . . Let (xi)li=be a path between two verticesxandyin a graphG. Let us denote withLx,y(t) =li=Fxi–,xi(t) for allt. Clearly the functionLx,yis monotone

nondecreasing.

Definition . Let (S,F,) be a PM-space andf :SS. Suppose that there exists a sequence{fnx}inSsuch thatfnx−→xand (fnx,fn+x)E(G) fornN. We say that:

(i) Gis a(Cf)-graph inSif there exist a subsequence{fnkx}of{fnx}and a natural

numberNsuch that(fnkx,x)E(G)forkN;

(ii) Gis an(Hf)-graph inSiffnx∈[x∗]Gforn≥and the sequence of functions

{Lfnx,x∗(t)}converges to(t)uniformly asn→ ∞(t> ).

Example . Let (S,F,) be a Menger PM-space induced by the metricd(x,y) =|xy|

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Consider the graphGconsisting ofV(G) =Sand

E(G) =

n,

n+ 

,

n+ ,n

, (n, ),

 n, 

;n∈N

.

We note thatxn=n → asn→ ∞. Also, it is easy to see thatGis a (CI)-graph. But since (a,b) =min{a,b}, then

Lxn,(t) =

t–

n

n+  ,

t– 

n+ –n

,(tn)

=(tn)(t) asn→ ∞.

ThusGis not an (HI)-graph.

Example . Let (S,F,) be a Menger PM space induced by the metricd(x,y) =|xy|

onS={n:n∈N} ∪ {

n+ :n∈N} ∪ {}, and letIbe an identity map onS. Consider the graphGconsisting ofV(G) =Sand

E(G) =

n,

n+  ,  n, √ 

n+ 

,

n+ , 

;n∈N

.

Sincexn=n → asn→ ∞. Clearly,Gis not a (CI)-graph. ButLxn,(t) =(t

n+)→

(t) asn→ ∞(t> ). ThusGis an (HI)-graph.

From the above examples, we note that the notions of (Cf)-graph and (Hf)-graph are

independent even iff is an identity map.

The following lemma is essential to prove our fixed point results.

Lemma .(Miheţ []) Let(S,F,)be a Menger PM-space under a t-normofH-type.

Let{xn}be a sequence in S,and let there existα∈(, )such that

Fxn,xn+(αt)≥Fxn–,xn(t) for all n∈N,t> .

Then{xn}is a Cauchy sequence.

Theorem . Let(S,F,)be a complete Menger PM-space under a t-normofH-type.

Assume that the mapping f :SS is a probabilistic G-contraction and there exists x∈S

such that(x,fx)∈E(G),then the following assertions hold.

(i) IfGis a(Cf)-graph,thenf has a unique fixed point ∈[x]Gand for anyy∈[x]G, fny→ .Moreover,ifGis weakly connected,thenf is a Picard operator.

(ii) IfGis a weakly connected(Hf)-graph,thenf is a Picard operator.

Proof Sincef is a probabilisticG-contraction and there existsx∈Ssuch that (x,fx)∈

E(G). By induction (fnx

,fn+x)∈E(G) for alln≥ and

Ffnx

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(i) Since thet-normis ofH-type, then from Lemma . it can be inferred that{fnx}

is a Cauchy sequence inS. From completeness of the Menger PM-spaceS, there exists ∈Ssuch that

lim

n→∞f nx

= . (.)

Now we prove that is a fixed point off. LetGbe a (Cf)-graph. Then there exists a

sub-sequence{fnkx}of{fnx}andN∈Nsuch that (fnkx

, )∈E(G) for allkN. Note that (x,fx,fx, . . . ,fnx, . . . ,fnNx, ) is a path inGand so inGfromxto , thus ∈[x]G.

Sincef is a probabilisticG-contraction and (fnkx

, )∈E(G) for allkN. Fort>  and

kN, we get

Ffnk+

x,f (t)≥Ffnk+x,f (αt)

Ffnkx, (t)→ ask→ ∞.

We obtain

lim

k→∞f nk+x

=f . (.)

Hence, we conclude thatf = . Now, lety∈[x]G, then from Lemma . we get

lim

n→∞f

ny= . (.)

Next to prove the uniqueness of a fixed point, suppose ∗∈[x]G= [ ]Gsuch thatf ∗= ∗.

Then from Lemma ., fort> , we have

F , ∗(t) =Ffn ,fn ∗(t)→, n→ ∞. (.)

Hence, ∗= . Moreover, if G is weakly connected, thenf is a Picard operator as [x]G=S.

(ii) LetGbe a weakly connected (Hf)-graph. By using the same arguments as in the first

part of the proof, we obtainlimn→∞fnx= . For eachn∈Nlet (zni);i= , . . . ,Mnbe a

path inGfromfnx

to withzn=fnx,znM= and (zin–,zni)∈E(G).

F ,f (t)≥F ,f (αt)

F ,fn+x

αt

,Ffn+x ,f

αt  ≥

F ,fn+x

αt

,Mn i=Ffzn

i–,fzni

αt

Mn

F ,fn+x

αt

,Mn i=Fzni–,zni

t

Mn

F ,fn+x

αt

,Lfnx

, tM , (.)

whereM=max{Mn:n∈N}.

SinceGis an (Hf)-graph and (fnx,fn+x)∈E(G) forn∈Nwithlimn→∞fnx= ∈S, then the sequence of functions{Lfnx

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andδ>  be given. Since the family{p(t)}

p∈Nis equicontinuous at pointt= , there

existsλ(δ)∈(, ) such thatp( –λ) >  –δfor everypN. Choosen

∈Nsuch that for allnnwe haveF ,fn+x

(

αt

) >  –λandLfnx, (

t

M) >  –λ. So that in view of (.), for

allnn, we have

F ,f (t)≥( –λ,  –λ)

=( –λ) >  –δ. (.)

Hence, we deducef = . Finally, let yS= [x]G be arbitrary, then from Lemma .,

limn→∞fny= .

Corollary . Let(S,F,)be a complete Menger PM-space under a t-normofH-type.

Assume that S is endowed with a graph G which is either(Cf)-graph or(Hf)-graph.Then the following statements are equivalent:

(i) Gis weakly connected.

(ii) For every probabilisticG-contractionf onS,if there existsx∈Ssuch that (x,fx)∈E(G),thenf is a Picard operator.

Proof (i)⇒(ii): It is immediate from Theorem ..

(ii)⇒(i): Suppose thatGis not weakly connected. ThenGis disconnected,i.e., there existsx∗∈Ssuch that [x∗]G=∅andS\[x∗]G=∅. Lety∗∈S\[x∗]G, we construct a self-mappingf by

fx= ⎧ ⎨ ⎩

x∗ ifx∈[x∗]G,

y∗ ifxS\[x∗]G.

Let (x,y)∈E(G), then [x]G:= [y]G, which impliesfx=fy. Hence (fx,fy)∈E(G), sinceG

contains all loops. Thus the mappingf preserves edges. Also, fort>  andα∈(, ), we haveFfx,fy(αt) = ≥Fx,y(t); thus (.) is trivially satisfied. Butx∗andy∗are two fixed points

off contradicting the fact thatf is a Picard operator.

Remark . TakingG= (S,S×S), Theorem . improves and extends the result of Sehgal [, Theorem ] to all Menger PM-spaces witht-norms ofH-type. Theorem . general-izes claim of [, Theorem .], and thus we have the following consequence.

Corollary .(Jachymski [, Theorem .]) Let(S,d)be a complete metric space en-dowed with the graph G.Assume that the mapping f :SS is a Banach G-contraction and the following property is satisfied:

(P) For any sequence{xn}inS,ifxnxinSand(xn,xn+)∈E(G)for alln≥,then there

exists a subsequence{xnk}with(xnk,x)∈E(G)for allk≥.

If there exists x∈S with(x,fx)∈E(G),then f|[x]Gis a Picard operator.Furthermore,if G is weakly connected,then f is a Picard operator.

Proof Let (S,F,M) be the Menger PM-space induced by the metricd. Since the

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and property (P) invokes thatGis a (Cf)-graph. Hence the conclusion follows from

The-orem .(i).

Example . Let (X,F,M) be a Menger PM-space whereX= [,∞) andFx,y(t) = t+|xty|

fort> . Then (X,F,M) is complete. Define a self-mappingf onXby

fx= ⎧ ⎨ ⎩

xp ifx=

nandp≥ is a fixed integer,

 otherwise. (.)

Further assume thatXis endowed with a graphGconsisting ofV(G) :=X andE(G) :=

∪ {(n,m) :n,m∈Nandn|m} ∪ {(x, ) :x=n}. It can be seen thatf is a probabilistic

G-contraction withα=

p and satisfies all the conditions of Theorem .(i).

Note that forx=  andy=and for eachα∈(, ), we can easily set  <t<(–α) such that

αt αt+|– |<

t t+| –|,

or

Ff,f

(αt) <F,(t) for  <t<  ( –α).

Hence, one cannot invoke [, Theorem ].

Definition . Let (S,F,) be a Menger PM-space under at-normofH-type. A map-pingf :SSis said to be: (i) continuous at pointxSwheneverxnxinSimplies fxnfxasn→ ∞; (ii) orbitally continuous if for allx,ySand any sequence{kn}n∈Nof positive integers,fknxyimpliesf(fknx)fyasn→ ∞; (iii) orbitallyG-continuous if

for allx,ySand any sequence{kn}n∈Nof positive integers,fknxyand (fknx,fkn+x)∈

E(G)∀n∈Nimplyf(fknx)fy(see []).

Theorem . Let(S,F,)be a complete Menger PM-space under a t-normofH-type.

Assume that the mapping f :SS is a probabilistic G-contraction such that f is orbitally G-continuous,and let there exist x∈S such that(x,fx)∈E(G).Then f has a unique

fixed pointS and for every y∈[x]G,fny→ .Moreover,if G is weakly connected,then f is a Picard operator.

Proof Let (x,fx)∈E(G), by induction (fnx,fn+x)∈E(G) for alln∈N. By using Lemma ., it follows thatfnx

→ ∈S. Sincefis orbitallyG-continuous, thenf(fnx)→f . This

gives =f . From Lemma . for anyy∈[x]G,fny→ .

Remark . We note that in Theorem . the assumption that f is orbitally G -continuous can be replaced by orbital continuity or continuity off.

Remark . Theorem . generalizes and extends claims and [, Theorem .]

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As a consequence of Theorems . and ., we obtain the following corollary, which is actually a probabilistic version of Theorem . and thus generalizes and extends the re-sults of Nieto and Rodríguez-López [, Theorems . and .], Petruşel and Rus [, The-orem .] and Ran and Reurings [, TheThe-orem .].

Corollary . Let(S,)be a partially ordered set,and let(S,F,)be a complete Menger PM-space under a t-normofH-type.Assume that the mapping f:SS is nondecreas-ing(nonincreasing)with respect to the order ‘’ on S and there existsα∈(, )such that

Ffx,fy(αt)≥Fx,y(t) for all x,yS,xy(t> ). (.)

Also suppose that either

(i) f is continuous,or

(ii) for every nondecreasing sequence{xn}inSsuch thatxnxinS,we havexnxfor alln≥.

If there exists x∈S with xfx,then f has a fixed point.Furthermore,if(S,)is such

that every pair of elements of S has an upper or lower bound,then f is a Picard operator.

Proof Consider a graphG consisting ofV(G) =SandE(G) ={(x,y)∈S×S:xy}. If

f is nondecreasing, then it preserves edges w.r.t. graphG and condition (.) becomes equivalent to (.). Thus f is a probabilisticG-contraction. In casef is nonincreasing, considerGwithE(G) ={(x,y)∈S×S:xyorxy}and a vertex set coincides withS. Actually, G:=G and from Proposition . iff is a probabilistic G-contraction, then it is a probabilisticGcontraction. Now iff is continuous, then the conclusion follows from Theorem .. On the other hand, if (ii) holds, thenGandGare (Cf)-graphs and

conclusions follow from the first part of Theorem ..

By relaxingH-type condition on at-norm, our next result deals with a compact Menger PM-space using the following class of graphs as the fixed point property is closely related to the connectivity of a graph.

Definition . Let (S,F) be a PM-space endowed with a graphGandf :SS. Assume the sequence{fnx}inSwith (fnx,fn+x)E(G) fornNandF

fnx,fn+x(t)→ (t> ), we say that the graphGis (Ef)-graph if for any subsequencefnkxzS, there exists a natural

numberNsuch that (fnkx,z)E(G) for allkN.

Theorem . Let(S,F,)be a compact Menger PM-space under a t-normsatisfying

supa<(a,a) = .Assume that the mapping f :SS is a probabilistic G-contraction,and let there exist x∈S such that(x,fx)∈E(G).If G is an(Ef)-graph,then f has a unique fixed point ∈[x]G.

Proof Since (x,fx)∈E(G), then (fnx,fn+x)∈E(G) forn∈Nand

Ffnx

,fn+x(t) ≥ Ffn–x,fnx

t α

· · ·

Fx,fx

t αn

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From compactness, let{fnkx}be a subsequence such thatfnkx

→ ∈S. Lett>  andδ>  be given. Sincesupa<(a,a) = , then there existsλ(δ)∈(, ) such that( –λ,  –λ) >  –δchoosen∈Nsuch that for allknwe haveFfnkx, (

t

) >  –λandFfnkx,fnk+x(

t

) >  –λ. Then we obtain

Ffnk+x

, (t)≥

Ffnk+x ,fnkx

t

,Ffnkx,

t

( –λ,  –λ) >  –δ.

Thus,fnk+x

→ .

Choosen∈Nsuch that for allknwe haveFfnk+x, (

t

) >  –λandFfnkx, (

t

α) >  –λ.

SinceGis an (Ef)-graph, there existsn∈Nsuch that (fnkx, )∈E(G) for allkn. Let

n=max{n,n}, then forknwe get

Ff , (t)≥

Ffnk+x ,f

t

,Ffnk+x ,

t

Ffnkx,

t

α

,Ffnk+x

,

t

( –λ,  –λ) >  –δ.

Hence,f = . Note that{x,fx, . . . ,fnx, . . . ,fnNx, }is a path inG, so that ∈[x]G.

So far it remains to investigate whether Theorem . can be extended to a complete PM-space?

Definition .[] Let (S,F) be a PM-space, and letε>  and  <δ<  be fixed real numbers. A mappingf :SSis said to be (ε,δ)-contraction if there exists a constant

α∈(, ) such that forxSandyNx(ε,δ) we have

Ffx,fy(αt)≥Fx,y(t) for allt> . (.)

The PM space (S,F) is said to be (ε,δ)-chainable if for eachx,ySthere exists a finite sequence (xn)Nn=of elements inSwithx=xandxN =ysuch thatxi+∈Nxi(ε,δ) fori=

, , . . . ,N– .

It is important to note that every (ε,δ)-contraction mapping is continuous. Letxnx

inS, then there exists a natural numberN(ε,δ) such thatxnNx(ε,δ) for allnN. Thus,

fort>  and for allnN, we obtain

Ffxn,fx(t)≥Ffxn,fx(αt)

Fxn,x(t)→ asn→ ∞.

Hence,fxnfx.

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Proof Consider the graphGconsisting ofE(G) ={(x,y)∈S×S:Fx,y(ε) >  –δ}andV(G)

coinciding withS. Letx,yS. Since the PM-space (S,F) is (ε,δ)-chainable, there ex-ists a finite sequence (xi)Ni= inS withx=xandxN =ysuch thatFxi,xi+(ε) >  –δ for

i= , , . . .N– . Hence, (xi,xi+)∈E(G) fori= , , . . . ,N– , which yields thatGis con-nected. Let (x,y)∈E(G), thenyNx(ε,δ). Since the mappingf is an (ε,δ)-contraction,

thus (.) is satisfied. Finally we have

Ffx,fy(ε)≥Ffx,fy(αε)

Fx,y(ε) >  –δ.

Thus, (fx,fy)∈E(G). Hence,f is a probabilisticG-contraction and the conclusion follows

from Theorem ..

Remark . Theorem . has an advantage over Theorem  of Sehgal and Bharucha-Reid [] which is only restricted to continuoust-norms satisfying(t,t)≥t. Moreover, the proof of our result is rather simple and easy, which invokes the novelty of Theo-rem ..

Definition .(Edelstein [, ]) The metric space (S,d) isε-chainbale for someε>  if for everyx,yS, there exists a finite sequence (xi)Nn=of elements inSwithx=x,xN =y

andd(xi,xi+) <εfori= , , . . . ,N– .

Remark .[] If (S,d) is anε-chainable metric space, then the induced Menger PM-space (S,F,M) is an (ε,δ)-chainable space.

Corollary .(Edelstein [, ]) Let(S,d)be a completeε-chainable metric space.Let f :SS and let there existα∈(, )such that

x,yS

d(x,y) <εd(fx,fy)≤αd(x,y). (.)

Then f is a Picard operator.

Proof Since the metric space (S,d) isε-chainable, then the induced Menger PM-space (S,F,) is (ε,δ)-chainable for each  <δ< . We only need to show that the self-mapping

f onSis an (ε,δ)-contraction. Letx,ySbe such thatyNx(ε,δ),i.e.,Fx,y(ε) >  –δ or (εd(x,y)) >  –δ. The definition ofimpliesd(x,y) <εand thusd(fx,fy)≤αd(x,y). Now, fort> , we get

Ffx,fy(αt) =

αtd(fx,fy)

td(x,y)

=Fx,y(t).

Hence the conclusion follows from Theorem ..

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LetSbe a nonempty set, letmbe a positive integer, let{Ai}mi=be nonempty closed subsets ofS, and letf :mi=Aiim=Aibe an operator. ThenS:=mi=Aiis known as a cyclic

representation ofSw.r.t.f if

f(A)⊂A, . . . ,f(Am–)⊂Am,f(Am)⊂A (.)

and the operatorf is known as a cyclic operator [].

In the following, we present the probabilistic version of the main result of [], as a last consequence of Theorem ..

Theorem . Let(S,F,)be a complete Menger PM-space under a t-norm ofH -type.Let m be a positive integer,let{Ai}mi= be nonempty closed subsets of S,Y :=

m

i=Ai and f :YY.Assume that

(i) mi=Aiis a cyclic representation ofYw.r.t.f;

(ii) ∃α∈(, )such thatd(fx,fy)≤αd(x,y)wheneverxAi,yAi+,whereAm+=A.

Then f has a unique fixed pointmi=Aiand fnyfor any y

m

i=Ai.

Proof Sincemi=Aiis closed, then (Y,F,) is complete. Let us consider a graphG

con-sisting ofV(G) :=YandE(G) :=∪ {(x,y)∈Y×Y:xAi,yAi+;i= , . . . ,m}. By (i) it follows thatf preserves edges. Now, letfnxx∗inYsuch that (fnx,fn+x)∈E(G) for all

n∈N. Then by (.) it is inferred that the sequence{fnx}has infinitely many terms in

eachAi;i∈ {, , . . . ,m}. So that one can easily identify a subsequence of{fnx}converging

tox∗ in eachAi; and sinceAi’s are closed, thenx∗∈

m

i=Ai. Thus, we can easily form a

subsequence{fnkx}in someA

j,j∈ {, . . . ,m}such that (fnkx,x∗)∈E(G) fork≥. It elicits

thatGis a weakly connected (Cf)-graph. Hence, by Theorem . conclusion follows.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Author details

1Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan.2Centre for Advanced Mathematics and

Physics, National University of Sciences and Technology, H-12, Islamabad, Pakistan. 3Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia.

Acknowledgements

The research of N. Shahzad was partially supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.

Received: 31 May 2013 Accepted: 5 August 2013 Published: 22 August 2013

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doi:10.1186/1687-1812-2013-223

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