φ-Contraction in generalized probabilistic metric spaces

R E S E A R C H

Open Access

ϕ

-Contraction in generalized probabilistic

metric spaces

Saud M Alsulami

_{, Binayak S Choudhury}

_{and Pradyut Das}

*_{Correspondence:} alsulami@kau.edu.sa

1_{Department of Mathematics, King}

Abdulaziz University, P.O. Box 138381, Jeddah, 21323, Saudi Arabia Full list of author information is available at the end of the article

Abstract

We use the gauge function introduced by Fang to gain a fixed point result in probabilistic G-metric spaces. Our work extends some existing results. Moreover, our result is supported with an example.

MSC: 47H10; 54H25

Keywords: generalized PM-space; Had˘zi´c typet-norm;

ϕ-function; Cauchy

1 Introduction

Probabilistic fixed point theory originated in the work of Sehgal and Bharucha-Reid [] where they introduced a contraction mapping principle in probabilistic metric spaces. After that this line of research has developed through the works of different authors over the years. A comprehensive description of this development is given in the book of Had˘zić and Pap []. Some more recent references are noted in [–].

Metric spaces are generalized in different ways creating spaces like fuzzy metric spaces, -metric spaces,etc.Generalized metric space is one of such spaces in which to every triple of points a nonnegative real number is attached. Such spaces were introduced by Mustafa and Sims []. Fixed point theory and related topics have been developed through a good number of works in recent times, some instances of which are in [–].

In the same vein probabilistic generalized metric spaces have been introduced by Zhou

et al.[] wherein they also proved certain primary fixed point results in these spaces. An interesting class of problems in probabilistic fixed point theory was addressed in re-cent times by use of gauge functions. These are control functions which have been used to extend the Sehgal contraction in probabilistic metric spaces. Some examples of these applications are in [, , , , , ]. One of such gauge functions was introduced by Fang []. Here we use the gauge function used by Fang [] to obtain a fixed point result in probabilistic G-metric spaces. Our result is supported with an example.

It is perceived that the study of fixed points for contractions defined by using control functions is an important category of problems in fixed point theory. One of the causes for this interest is the particularities involved in the proofs. With this motivation we work out the results in this paper.

2 Mathematical preliminaries

In this section we discuss certain definitions and lemmas which will be necessary for es-tablishing the results of the next section.

Definition .[] A mappingF:R→R+is called a distribution function if it is nonde-creasing and left continuous withinft∈RF(t) =  andsupt∈RF(t) = , whereRis the set of

real numbers andR+_{denotes the set of all nonnegative real numbers.}

Definition .[, ] A binary operation: [, ]_{→[, ] is called a continuoust-norm}

if the following properties are satisfied: (i) is associative and commutative, (ii) (a, ) =afor alla∈[, ],

(iii) (a,b)≤(c,d)whenevera≤candb≤dfor alla,b,c,d∈[, ], (iv) is continuous.

Generic examples of continuoust-norm areM(a,b) =min{a,b},P(a,b) =ab,etc.

Definition .[] A Menger space is a triplet (X,F,), whereXis a nonempty set,Fis a function defined onX×Xto the set of distribution functions andis at-norm such that the following are satisfied:

(i) Fx,y() = for allx,y∈X,

(ii) Fx,y(s) = for alls> if and only ifx=y,

(iii) Fx,y(s) =Fy,x(s)for alls> ,x,y∈X,

(iv) Fx,y(u+v)≥(Fx,z(u),Fz,y(v))for allu,v≥andx,y,z∈X.

Definition .[] The -tuple (X,G,) is called a probabilistic G-metric space (shortly PGM-space) ifXis a nonempty set,is a continuoust-norm andF∗is a function from

X_{×(,∞) to the set of distribution functions satisfying the following conditions for each}

x,y,z∈Xandt,s> :

(i) F_x∗_,y,z(t) = if and only ifx=y=z, (ii) F_x∗_,x,y(t)≥F_x∗_,y,z(t)withy=z,

(iii) F_x∗_,y,z(t) =F_p∗_(x,y,z)(t), wherepis a permutation function, (iv) (F_x∗_,a,a(t),F_a∗_,y,z(s))≤F_x∗_,y,z(t+s).

Example .[] Let (X,F∗,) be a probabilistic metric space. Define a functionF∗ :

X×X×X→R+_by

F_x∗_,y,z(t) =minFx,y(t),Fy,z(t),Fz,x(t)

for allx,y,z∈Xandt> .

Then (X,F∗,) is a probabilistic G-metric space.

For more examples of probabilistic G-metric space refer to [].

Definition .[] Let (X,F∗,) be a probabilistic G-metric space andxbe any point

inX. For any>  andλwith  <λ< , an (,λ)-neighborhood ofxis the set of all points

yinXfor whichF_x∗_,y,y() >  –λandF_y∗_,x,x() >  –λ. We write

Nx(,λ) =

This means thatNx(,λ) is the set of all pointsy∈Xfor which the probability of the distance fromxtoybeing less thanis greater than  –λ.

Lemma .[] If≤andλ≤λ,then Nx(,λ)⊆Nx(,λ).

Theorem . [] Let (X,F∗,) be a probabilistic G-metric space. Then (X,F∗,)

is a Hausdorff space in the topology induced by the family {Nx(,λ)} of (,λ) -neigh-borhoods.

Definition .[] Let (X,F∗,) be a probabilistic G-metric space.

(i) A sequence{xn} ⊂Xis said to converge to a pointx∈Xif given> ,λ> we can

find a positive integerN,λsuch that for alln>N,λ,

Fx,xn,xn()≥ –λ.

(ii) A sequence{xn}is said to be a Cauchy sequence inXif given> ,λ> there

exists a positive integerN,λsuch that

Fxn,xm,xl()≥ –λ for allm,n,l>N,λ.

(iii) A probabilistic G-metric space(X,F∗,)is said to be complete if every Cauchy sequence is convergent inX.

Definition .[] At-normis said to be a Had˘zić type (shortly H-type)t-norm if the family{m_}

m>of its iterates defined for eacht∈[, ] by

(t) =(t,t)

and, in general, for allm> ,m(t) =(t,m–(t)) is equi-continuous att= , that is, given

λ>  there existsη(λ)∈(, ) such that

η(λ) <t≤ ⇒ (m)η(λ)≥ –λ for allm> .

Definition .[] Letwdenote the class of all functionsϕ:R+→R+satisfying the

following condition:

for eacht> , there existsr≥tsuch thatlimn→∞ϕn(r) = .

Lemma .[] Letϕ∈w,then for each t> there exists r≥t such thatϕ(r) <t.

3 Main results

Definition . Let (X,F∗,) be a PGM-space with a continuoust-normof H-type. A mappingf :X→Xis said to be a probabilisticϕ-contraction if there exists a function

ϕ∈wsuch that

Lemma . Let{xn}be a sequence in a Menger PGM-space(X,F∗,),whereis a Had˘zić

type t-norm.If there exists a functionϕ∈wsuch that

(i) ϕ(t) >  for all t> ,

(ii) F_x∗_{n,xn+,xn+}ϕ(t)≥F_x∗_{n–,xn,xn}(t) for all n∈Nand t>  (.)

for all t> ,n≥,then{xn}is a Cauchy sequence in X.

Proof The condition (i) implies thatϕn_{(t) >  for allt>  andn≥, and from the condition}

(ii), by induction, we have

F_x∗_{n,xn+,xn+}ϕn(t)≥F_x∗_{,x,x}(t) for allt> ,n≥. (.) We now prove that

lim

n→∞F ∗

xn,xn+,xn+(t) =  for allt> . (.) SinceF_x∗_{,x,x}(t)→ ast→ ∞, for any∈(, ), there existst>  such thatFx∗,x,x(t) >  –. Sinceϕ∈w, there existst≥tsuch thatlimn→∞ϕn(t) = . Thus, for allt> ,

there existsn≥ such thatϕn(t) <tfor alln≥n. Byt>ϕn(t),t≥t, (.) and the

monotonic property of distribution functions, we have

F_x∗_{n,xn+,xn+}(t)≥F_x∗_{n,xn+,xn+}ϕn(t)

≥F_x∗_{,x,x}(t)≥Fx∗,x,x(t) >  – for alln≥n.

Thus (.) holds.

Sinceϕ∈w, by Lemma ., for anyt> , there existsr≥tsuch thatϕ(r) <t. Letn≥

be given. Now we show by induction that, for anyk≥,

F_x∗_n,x

n+k,xn+k(t)≥

k_F∗

xn,xn+,xn+

t–ϕ(r). (.)

Fork= , from (.) we have

F_x∗_{n,xn+,xn+}(t)≥F_x∗_{n,xn+,xn+}t–ϕ(r),F_x∗_{n,xn+,xn+}t–ϕ(r) =F_x∗_{n,xn+,xn+}t–ϕ(r).

Thus (.) holds fork= . Assume that (.) holds for somek≥. Sinceis monotone, from (iv) in Definition ., and then by (.) and (.), it follows that

F_x∗_n,x

n+k+,xn+k+(t) =F ∗

xn,xn+k+,xn+k+

t–ϕ(r) +ϕ(r)

≥F_x∗_{n,xn+,xn+}t–ϕ(r),F_x∗_{n+,xn+k+,xn+k+}ϕ(r)

≥F_x∗_{n,xn+,xn+}t–ϕ(r),F_x∗_n,xn+k,xn+k(r) (by (.))

≥F_x∗_{n,xn+,xn+}t–ϕ(r),F_x∗_n,x

n+k,xn+k(t)

(sincer≥t)

≥F_x∗_{n,xn+,xn+}t–ϕ(r),kF_x∗_{n,xn+,xn+}t–ϕ(r) =k+F_x∗_{n,xn+,xn+}t–ϕ(r),

Next we show that{xn}is a Cauchy sequence inX, that is,limm,n,l→∞Fx∗n,xm,xl(t) =  for

allt> . Lett>  and  << . Since{n_{(t)}is equi-continuous att=  and}n_{() = ,}

then there existsδ>  such that

n(s) >  – for alls∈( –δ, ] andn≥.

We first prove thatlimn,m→∞Fx∗n,xm,xm(t)∈( –δ, ] for alln≥n. Sincet–ϕ(r) > , from

(.) there isn≥ such thatFx∗n,xn+,xn+(t–ϕ(r)) >  –δfor alln≥n. Hence and by (.) we conclude thatF_x∗_{n,xn+k+,xn+k+}(t) >  –for allk≥. Thus we proved that

lim

n,m→∞F ∗

xn,xm,xm(t) =  for allt> . (.a)

By (iv) witha=xmin Definition ., we have, for allt> ,

F_x∗_n,xm,x l(t)≥

F_x∗_n,xm,xm

t



,F_x∗_m,xm,x l

t

 =

F_x∗_n,xm,xm

t



,F_x∗ l,xm,xm

t

 .

From (.a), it follows that

lim

n,m→∞F

∗

xn,xm,xm

t



=  for allt> ,

lim

l,m→∞F

∗

xl,xm,xm

t



=  for allt> .

Thus, by using the continuity of, we have

lim

m,n,l→∞F ∗

xn,xm,xl(t)≥

lim

n,m→∞F ∗

xn,xm,xm

t



, lim

l,m→∞F ∗

xl,xm,xm

t

 =(, ) = .

Therefore, we proved that

lim

m,n,l→∞F ∗

xn,xm,xl(t) =  for allt> .

This shows that{xn}is a Cauchy sequence inX.

Corollary .[] Let{xn}be a sequence in a Menger PM-space(X,F,),whereis a

Had˘zić type t-norm.If there exists a functionϕ∈wsuch that

(i) ϕ(t) >  for all t> ,

(ii) Fxn,xm

ϕ(t)≥Fxn–,xm–(t) (.)

for all t> ,n,m≥,then{xn}is a Cauchy sequence in X.

Proof DefineF_x∗_,y,z(t) =min{Fx,y(t),Fy,z(t),Fx,z(t)}for allx,y,z∈Xandt> . Then (X,F∗,)

is a probabilistic G-metric space.

SinceF_x∗_{n,xn+,xn+}(ϕ(t)) =min{Fxn,xn+(ϕ(t)),Fxn+,xn+(ϕ(t)),Fxn,xn+(ϕ(t))}, it follows F_x∗_{n,xn+,xn+}ϕ(t)=Fxn,xn+

Similarly we have

F_x∗_{n–,xn,xn}(t) =Fxn–,xn(t). (.)

By using (.), (.) and (.), we have, for allt> ,

F_x∗_{n,xn+,xn+}ϕ(t)=Fxn,xn+

ϕ(t)≥Fxn–,xn(t) =Fx∗n–,xn,xn(t).

Hence we conclude that (.) holds. By Lemma ., we conclude that {xn}is a Cauchy

sequence in the sense of PGM-space (X,F∗,), that is, given > ,λ>  there exists a positive integerN,λsuch that

F_x∗_n,xm,xl()≥ –λ for allm,n,l>N,λ. (.)

By definition ofF∗and (.), we have, for allt> ,

minFxn,xm(t),Fxm,xl(t),Fxn,xl(t)

≥ –λ.

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

Lemma . Let(X,F∗,)be a PGM-space.If there exists a functionϕ∈wsuch that

F_x∗_,y,yϕ(t)≥F_x∗_,y,y(t) (.)

for all t> and x,y∈X,then x=y.

Proof SinceF∗is monotonic, it is obvious that from (.) it followsϕ(t) >  for allt> . Therefore we haveϕn(t) >  for allt>  andn≥. By induction, we have from (.) that

F_x∗_,y,yϕn(t)≥F_x∗_,y,y(t) (.) for allt>  andn≥.

To provex=y, it is required thatF_x∗_,y,y(t) =  for allt> . Suppose, to the contrary, that there exists somet>  such thatFx∗,y,y(t) < . Sincelimt→∞Fx∗,y,y(t) = , there existst>t

such that

F_x∗_,y,y(t) >F_x∗_,y,y(t) (.)

for allt≥t.

Sinceϕ∈w, there existst≥tsuch thatlimn→∞ϕn(t) = . Therefore, we can choose

large enoughn≥ such thatϕn(t) <t. By the monotone property ofF∗, using (.)

and (.), we have

F_x∗_,y,y(t)≥Fx∗,y,y

ϕn_(t

)

≥F_x∗_,y,y(t) >Fx∗,y,y(t),

Theorem . Let(X,F∗,)be a PGM-space with a continuous t-normof H-type.If the mapping f is a probabilisticϕ-contraction,then f has a unique fixed point in X.

Proof Letx∈Xbe an arbitrary pointX and the sequence{xn}be defined as follows:

xn+=fnxfor alln≥. Sincef is a probabilisticϕ-contraction, by (.) for allt>  we

have

F_x∗_{n,xn+,xn+}ϕ(t)=F_fx∗_{n–,fxn,fxn}ϕ(t)

≥F_x∗_{n–,xn,xn}(t).

By Lemma . we conclude that the sequence{xn}is a Cauchy sequence inX. SinceXis

complete, there existsx∈Xsuch thatlimn→∞xn=x, that is, for allt> ,

lim

n→∞F

∗

x,xn,xn(t) = . (.)

Now we prove thatxis a fixed point off. By (iv) witha=fxnin Definition ., we have

F_fx∗_,x,x(t)≥F_fx∗_,fxn,fxnϕ(t),F_fx∗_n,x,xt–ϕ(t)=F_fx∗_,fxn,fxnϕ(t),F_x∗_n+,x,xt–ϕ(t). Hence, by (.), we get

F_fx∗_,x,x(t)≥F_x∗_,xn,xn(t),F_x∗_n+,x,xt–ϕ(t). (.) Again by (iii) and (iv) witha=xn+in Definition .,

F_x∗_n+,x,xt–ϕ(t)=F_x∗_,x,xn+t–ϕ(t)≥

F_x∗_{,xn+,xn+}

t–ϕ(t) 

,F_x∗_{n+,x,xn+}

t–ϕ(t) 

.

Hence, by (iii) in Definition ., we have

F_x∗_n+,x,xt–ϕ(t)≥

F_x∗_{,xn+,xn+}

t–ϕ(t) 

,F_x∗_{,xn+,xn+}

t–ϕ(t) 

.

Now from (.) we get

F_fx∗_,x,x(t)≥

F_x∗_,xn,xn(t),

F_x∗_{,xn+,xn+}

t–ϕ(t) 

,F_x∗_{,xn+,xn+}

t–ϕ(t) 

. (.)

Sincelimn→∞xn=limn→∞xn+=xandis continuous, takingn→ ∞in (.) we get,

for allt> ,

F_fx∗_,x,x(t)≥,(, )=(, ) = . Hence

fx=x,

Next suppose thaty=xis another fixed point off. Then, for allt> , we have

F_x∗_,y,yϕ(t)=F_fx∗_,fy,fyϕ(t)

≥F_x∗_,y,y(t),

which implies, by Lemma ., thatx=y. Therefore,f has a unique fixed point inX. This

completes the proof of the theorem.

Example . LetX= [,∞) and(a,b) =min{a,b}for alla,b∈X. Define a function

F∗:X_{×[,∞)→[,∞) by}

F_x∗_,y,z(t) = t

t+G(x,y,z)

for allx,y,z∈XandG(x,y,z) =|x–y|+|y–z|+|z–x|. Then (X,F∗,) is a probabilistic G-metric space (see []).

Letf :X→Xbe a mapping defined byfx=x

andϕ:R

+_→R+_{be defined by}

ϕ(t) =

⎧ ⎪ ⎨ ⎪ ⎩

t

 if ≤t< ,

–_t +_ if ≤t≤_,

t–_ if _<t<∞.

It is easy to verify thatϕ∈wandϕ(t)≥_t for allt≥.

Now we show thatf satisfies (.). We have

F_fx∗_,fy,fzϕ(t)= ϕ(t)

ϕ(t) + [|fx–fy|+|fy–fz|+|fz–fx|]

= ϕ(t)

ϕ(t) + [|x

–

y

|+|

y

–

z

|+|

z

–

x

|]

≥

t



t

+ [|

x

–

y

|+|

y

–

z

|+|

z

–

x

|]

≥ t

t+ [|x–y|+|y–z|+|z–x|]

=F_x∗_,y,z(t).

This shows that (.) holds andf has a fixed point inX. The fixed point is .

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Author details

1_{Department of Mathematics, King Abdulaziz University, P.O. Box 138381, Jeddah, 21323, Saudi Arabia.}2_{Department of}

Acknowledgements

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant No. (319/130/1434). The authors, therefore, acknowledge with thanks DSR technical and financial support. The authors thank the learned referees for suggestions, which helped to improve the paper.

Received: 22 December 2014 Accepted: 25 June 2015

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