Common fixed points for self-mappings on partial metric spaces

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R E S E A R C H

Open Access

Common ﬁxed points for self-mappings on

partial metric spaces

Cristina Di Bari

1

_{, Marija Milojevi´c}

2

_{, Stojan Radenovi´c}

3*

_{and Pasquale Vetro}

1

*_{Correspondence:}

[email protected]

3_{Faculty of Mechanical Engineering,}

University of Belgrade, Kraljice Marije 16, Beograd, 11120, Serbia Full list of author information is available at the end of the article

Abstract

In this paper, we prove some results of a common fixed point for two self-mappings on partial metric spaces. Our results generalize some interesting results of Ilićet al. (Appl. Math. Lett. 24:1326-1330, 2011). We conclude with a result of the existence of a fixed point for set-valued mappings in the context of 0-complete partial metric spaces.

MSC: 54H25; 47H10

Keywords: points of coincidence; common ﬁxed points; 0-complete partial metric space;

ψ

-contractions

1 Introduction

In the mathematical ﬁeld of domain theory, attempts were made to equip semantics do-main with a notion of distance. In particular, Matthews [] introduced the notion of a partial metric space as a part of the study of denotational semantics of data for networks, showing that the contraction mapping principle can be generalized to the partial met-ric context for applications in program veriﬁcation. Moreover, the existence of several connections between partial metrics and topological aspects of domain theory have been lately pointed out by other authors such as O’Neill [], Bukatin and Scott [], Bukatin and Shorina [], Romaguera and Schellekens [] and others.

After the definition of the concept of a partial metric space, Matthews [] obtained a Ba-nach type fixed point theorem on complete partial metric spaces. This result was recently generalized by Ilićet al. []. In this paper, we prove some results of a common fixed point for two self-mappings on partial metric spaces. Our results generalize some interesting results of Ilićet al. We conclude this paper with a new existence result of a fixed point for set-valued mappings in a partial metric space.

2 Preliminaries

First, we recall some deﬁnitions and some properties of partial metric spaces that can be found in [, –]. A partial metric on a nonempty setXis a functionp:X×X→[, +∞[ such that for allx,y,z∈X,

(p) x=y⇔p(x,x) =p(x,y) =p(y,y),

(p) p(x,x)≤p(x,y),

(p) p(x,y) =p(y,x),

(p) p(x,y)≤p(x,z) +p(z,y) –p(z,z).

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It is clear that (p) implies the triangular inequality. A partial metric space is a pair (X,p)

such thatXis a nonempty set andpis a partial metric onX. It is clear that ifp(x,y) = , then from (p) and (p), it follows thatx=y. But ifx=y,p(x,y) may not be . A basic example of

a partial metric space is the pair ([, +∞[,p), wherep(x,y) =max{x,y}for allx,y∈[, +∞[. Other examples of partial metric spaces, which are interesting from a computational point of view, can be found in [].

Each partial metric ponXgenerates a Ttopologyτp onX which has as a base the family of openp-balls{Bp(x,ε) :x∈X,ε> }, where

Bp(x,ε) =

y∈X:p(x,y) <p(x,x) +ε

for allx∈Xandε> .

Ifpis a partial metric onX, then the functionps_:_X_×_X_→_{[, +}_∞_{[ given by}

ps(x,y) = p(x,y) –p(x,x) –p(y,y)

is a metric onX.

Deﬁnition  Let (X,p) be a partial metric space.

(i) A sequence{xn}in(X,p)converges to a pointx∈Xif and only if

p(x,x) =limn→+∞p(x,xn).

(ii) A sequence{xn}in(X,p)is called a Cauchy sequence if there exists (and is ﬁnite) lim_n_,_m_→₊_∞p(xn,xm).

(iii) A partial metric space(X,p)is said to be complete if every Cauchy sequence{xn}in

Xconverges, with respect toτp, to a pointx∈Xsuch that

p(x,x) =limn,m→+∞p(xn,xm).

(iv) A sequence{xn}in(X,p)is called -Cauchy iflimn,m→+∞p(xn,xm) = .

We say that (X,p) is -complete if every -Cauchy sequence inXconverges, with respect toτp, to a pointx∈Xsuch thatp(x,x) = .

On the other hand, the partial metric space (Q∩[, +∞[,p), whereQdenotes the set of rational numbers and the partial metric pis given byp(x,y) =max{x,y}, provides an example of a -complete partial metric space which is not complete.

It is easy to see that every closed subset of a complete partial metric space is complete.

Lemma  ([]) Let (X,p)be a partial metric space and {xn} ⊂X. If xn→x∈X and

p(x,x) = , thenlimn→+∞p(xn,z) =p(x,z)for all z∈X.

Deﬁnep(x,A) =inf{p(x,a) :a∈A}. Thena∈A⇔p(a,A) =p(a,a), whereAdenotes the closure ofA(for details see [], Lemma ).

LetXbe a non-empty set andT,f :X→X. The mappingsT,f are said to be weakly compatible if they commute at their coincidence point (i.e.,Tfx=fTxwheneverTx=fx). A pointy∈Xis called a point of coincidence ofTandf if there exists a pointx∈Xsuch thaty=Tx=fx.

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3 Main results

Let (X,p) be a partial metric space andT,f :X→Xbe such thatTX⊂fX. For every

x∈X, we consider a sequence{xn} ⊂Xdeﬁned byfxn=Txn–for alln∈Nand we say

that{Txn}is aT-f-sequence of the initial pointx(see []). Set

ρf :=inf

p(fx,fx) :x∈X and Xf=

x∈X:p(fx,fx) =ρf

.

It is not alwaysXf =∅. This is true if and only ifρf=min{p(fx,fx) :x∈X}. IfX= [, +∞[,

p(x,y) =max{x,y},fx=x,x=  andf > , thenρf=  andXf =∅.

Let (X,p) be a partial metric space. We denote with F the family of pairs (T,f) such that:

(i) Tandf are self-mappings onXwithTX⊂fX; (ii) for eachx,y∈Xthe following condition holds:

p(Tx,Ty)≤maxkp(fx,fy),p(fx,fx),p(fy,fy), (.)

wherek∈[, [.

Remark  If (T,f)∈F, then for eachx∈X, we have

ρf ≤p(Tx,Tx)≤p(fx,fx).

Indeed, becauseTX⊂fX, we have that{p(Tx,Tx) :x∈X}=B⊂A={p(fx,fx) :x∈X}

from which followsinfA≤infB, that is,ρf ≤ρT≤p(Tx,Tx)≤p(fx,fx).

Lemma  Let(X,p)be a partial metric space with(T,f)∈Fand Xf =∅. If u,v∈Xf are

coincidence points for T and f , then fu=fv.

Proof From

p(fu,fv) =p(Tu,Tv)≤maxkp(fu,fv),p(fu,fu),p(fv,fv)

it follows that either ( –k)p(fu,fv) =  orp(fu,fv)≤p(fu,fu) =p(fv,fv) =ρf. Now, from (p), it followsp(fu,fv) =ρf=min{p(fx,fx) :x∈X}, that is,fu=fv.

Lemma  Let(X,p)be a partial metric space and(T,f)∈F. If fX is a complete subspace

of X, then for each s∈N, there is x∈X such that

p(fx,fx) =p(fx,Tx) <ρf+ 

s.

Proof We ﬁxx∈Xand prove that eachT-f-sequence{Txn}of the initial pointxis a

Cauchy sequence inTX.

From Remark , we deduce that p(Txn+,Txn+)≤p(fxn+,fxn+) =p(Txn,Txn). Hence,

{p(Txn,Txn)}is a nonincreasing sequence. Set

r:=infp(Txn,Txn)

= lim

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and

M:= 

 –kp(fx,Tx) +p(fx,fx).

We prove that

p(fx,Txn)≤M, for alln∈N. (.)

Obviously, (.) is true forn= , . Assume that (.) is true for somen≥, then

p(fx,Txn+)≤p(fx,Tx) +p(Tx,Txn+)

≤p(fx,Tx) +max

kp(fx,Txn),p(fx,fx)

≤p(fx,Tx) +

k

 –kp(fx,Tx) +p(fx,fx)

=M.

We ﬁxε>  and choosen∈Nsuch thatp(Txn,Txn) <r+εfor alln≥nand knM<

r+ε. For eachm≥n≥n, we have

p(Txn,Txm)≤max

kp(Txn–,Txm–),p(Txn–,Txn–)

≤maxkp(Txn–,Txm–),p(Txn–,Txn–)

. . .

≤maxkn_p₍_Tx

n–n,Txm–n),p(Txn–n,Txn–n)

≤maxkn_p₍_Tx

n–n,fx) +p(fx,Txm–n)

,

p(Txn–n,Txn–n)

<r+ε.

Now, from

r≤p(Txn,Txn)≤p(Txn,Txm) <r+ε

for eachm≥n≥n, we obtain that

lim

n,m→+∞p(Txn,Txm) =r=n,mlim→+∞p(fxn,fxm).

SincefXis a complete subspace ofX, there isz∈fXsuch that

p(z,z) = lim

n→+∞p(z,Txn) =n,mlim→+∞p(Txn,Txm) =r.

Letx∈Xsuch thatfx=z. We show thatp(fx,fx) =p(fx,Tx) =r. Now, from (.), we deduce that there existI,I,I⊂Nsuch that

(i) p(Txn,Tx)≤kp(fxn,fx)for alln∈I;

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(iii) p(Txn,Tx)≤p(fx,fx)for alln∈I.

Clearly, at least one of the setsI,I,Iis inﬁnite.

Then, from

p(fx,Tx)≤p(fx,Txn) +p(Txn,Tx) –p(Txn,Txn),

ifIi(i= , , ) is inﬁnite, taking the limit asn→+∞andn∈Ii, it follows thatp(fx,Tx)≤

p(fx,fx) and sop(fx,Tx) =p(fx,fx).

Now, if we choose x∈X such that p(fx,fx) <ρf + /s, we deduce thatp(fx,fx)≤

p(fx,fx) <ρf + /s. Ifx∈Xf, alsox∈Xf.

Lemma  Let(X,p)be a partial metric space and(T,f)∈F. If fX is a complete subspace of X, then there exists u∈Xf such that fu=Tu.

Proof By Lemma , there exists a sequence{xn} ⊂Xsuch that

p(fxn,fxn) =p(fxn,Txn) <ρf + 

n, (.)

for alln∈N. First, we prove that

lim

n,m→+∞p(fxn,fxm) =ρf. (.)

Forε>  ﬁxed, we choosen> /ε( –k). From Remark , it follows

ρf ≤p(Txn,Txn)≤p(fxn,fxn) =p(fxn,Txn) <ρf+ 

n<ρf+

ε

( –k),

which implies

p(fxn,Txn) –p(Txn,Txn) <

ε

( –k).

Now, for alln,m> /ε( –k), we have

p(fxn,fxm)≤p(fxn,Txn) +p(Txn,Txm) +p(Txm,fxm)

–p(Txn,Txn) –p(Txm,Txm)

=p(fxn,Txn) –p(Txn,Txn) +p(Txm,fxm)

–p(Txm,Txm) +p(Txn,Txm)

< 

ε( –k) +max

kp(fxn,fxm),p(fxn,fxn),p(fxm,fxm)

.

Ifmax{kp(fxn,fxm),p(fxn,fxn),p(fxm,fxm)}=kp(fxn,fxm), then

p(fxn,fxm) < 

ε( –k) +kp(fxn,fxm),

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If

maxkp(fxn,fxm),p(fxn,fxn),p(fxm,fxm)

=maxp(fxn,fxn),p(fxm,fxm)

<ρf+ 

ε( –k),

then

ρf ≤p(fxn,fxm) < 

ε( –k) +ρf+ 

ε( –k).

This implies that

lim

n,m→+∞p(fxn,fxm) =ρf.

In both cases, (.) holds. SincefXis a complete subspace ofX, there isz∈fXsuch that

p(z,z) = lim

n→+∞p(z,fxn) =n,mlim→+∞p(fxn,fxm) =ρf.

Letu∈Xsuch thatfu=z. Fromp(fu,fu) =p(z,z) it follows thatu∈Xf. We prove that

p(fu,fu) =p(fu,Tu). Asp(fxn,fxn) =p(fxn,Txn) andρf ≤p(Txn,Txn) for alln∈N, we get that

p(fu,Tu)≤p(fu,fxn) +p(fxn,Txn) +p(Txn,Tu)

–p(fxn,fxn) –p(Txn,Txn)

=p(fu,fxn) +p(Txn,Tu) –p(Txn,Txn)

≤p(fu,fxn) –ρf +p(Txn,Tu)

≤p(fu,fxn) –ρf +max

kp(fxn,fu),p(fxn,fxn),p(fu,fu)

.

Asn→+∞, we obtainp(fu,Tu) =p(fu,fu) =p(Tu,Tu) =ρf and sofu=Tu.

The following theorem of a common ﬁxed point in a partial metric space is one of our main results.

Theorem  Let(X,p)be a partial metric space with(T,f)∈F. If fX is a complete subspace of X, then T and f have a unique point of coincidence z=fu=Tu with u∈Xf. Moreover, if

T and f are weakly compatible and fXf⊂Xf, then T and f have a unique common ﬁxed

point z∈Xf.

Proof By Lemma  and Lemma , there existsu∈Xf such thatz=fu=Tuis a unique point of coincidence forT andf withp(z,z) =ρf. IfT andf are weakly compatible and

fXf ⊂Xf, by Lemma , fromTz=Tfu=fTu=fz∈Xf, that isp(fz,fz) =ρf, it follows that

fz=fu. By Lemma ,zis a unique common ﬁxed point forTandf belonging toXf.

Theorem  Let(X,p)be a partial metric space and T,f:X→X. Suppose that the follow-ing condition holds:

p(Tx,Ty)≤max

kp(fx,fy),p(fx,fx) +p(fy,fy) 

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for each x,y∈X, where k∈[, [. If fX is a complete subspace of X, and T and f are weakly compatible, then T and f have a unique common ﬁxed point z∈X.

Proof Clearly, (T,f)∈F, by Lemma , there existsu∈Xf such thatz=fu=Tu. Ifv∈Xis such thatfv=Tvandp(fv,fv) =p(fu,fu), thenfu=fvby Lemma . Ifp(fv,fv) >p(fu,fu), from (.) we obtain either p(fu,fv) = p(Tu,Tv) ≤kp(fu,fv) or p(fu,fv) =p(Tu,Tv) <

p(fv,fv). In each of these cases, we deduce thatfu=fv, and soTandf have a unique point of coincidence. By Lemma , sinceTandf are weakly compatible,T andf have a unique

common ﬁxed pointz∈X.

If in Theorems  and  we choosef(x) =x, we obtain Theorems . and . of Ilićet al.

Example  LetX= [, ] andp:X×X→Rbe deﬁned byp(x,y) =max{x,y}. Then (X,p) is a complete partial metric space. LetT,f :X→Xbe deﬁned by

Tx=

⎧ ⎨ ⎩

x–  ifx∈[, ],

 ifx∈], ]

and

fx=

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

 ifx= ,

 ifx∈], /[,

x ifx∈[/, ],

 ifx∈], ],

respectively.

In order to show thatTandf satisfy the contractive condition (.) in Theorem  with

k= /, we consider the following cases. Case .x=y= . We have

p(T,T) =  =p(f,f) +p(f,f) 

=max

kp(f,f),p(f,f) +p(f,f) 

.

Case .x,y∈[, /[ andx<y. We have

p(Tx,Ty) = y– ≤

≤max

.

Case .x∈[, ],y∈[/, ] andx<y. We have

p(Tx,Ty) = y– ≤

y≤max

.

Case .x∈[, ] andy∈], ]. We have

p(Tx,Ty) =max{, x– } ≤

≤max

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Case .x,y∈], ]. We have

p(Tx,Ty) = ≤

≤max

.

SincefXis a complete subspace ofX, andTandf are weakly compatible, by Theorem ,

T andf have a unique common ﬁxed pointz= .

4 Fixed points for set-valued mappings

Investigations of the existence of ﬁxed points of set-valued contractions in metric spaces were initiated by Nadler []. The following theorem is motivated by Nadler’s results and also generalizes the well-known Banach contraction theorem in several ways.

Denote with the family of nondecreasing functionsψ: [, +∞[→[, +∞[ such that

+∞

n=ψn(t) < +∞for eacht> , whereψnis thenth iterate ofψ.

Lemma  For every functionψ∈, the following holds:ψ(t) <t for each t> .

Deﬁnition  Let (X,p) be a partial metric space and letT:X→C(X), whereC(X) is the family of nonempty closed subsets ofX.Tisψ-contractive if there existsψ∈such that, for anyx,x∈Xandy∈Tx, there isy∈Txwith

p(y,y)≤ψ

p(x,x)

.

Theorem  Let(X,p)be a-complete partial metric space and let T:X→C(X)be aψ -contractive mapping. Then there exists z∈X such that z∈Tz, i.e., z is a ﬁxed point of T , and p(z,z) = .

Proof Fixx∈X and letx∈Tx. Ifp(x,x) = , thenx=xandxis a ﬁxed point ofT.

Assume, hence,p(x,x) > ; then there existsx∈Tx such thatp(x,x)≤ψ(p(x,x)).

Proceeding in this way, we have a sequence{xn}inXsuch thatxn+∈Txnandp(xn,xn+)≤

ψ(p(xn–,xn)) for everyn∈N. Consequently,

p(xn,xn+)≤ψn

p(x,x)

, for alln∈N.

Since, the series+_n_=∞ψn(p(x,x)) converges, we get that

+∞

n=p(xn,xn+) converges too.

It follows, form>n,

p(xm,xn)≤ m–

k=n

p(xk,xk+)≤ +∞

k=n

p(xk,xk+)→, asn→+∞. (.)

Now, from (.), we deduce thatlimn,m→+∞p(xn,xm) =  and hence{xn}is a -Cauchy sequence inX. SinceXis a -complete space, there existsz∈Xsuch thatxn→zand

p(z,z) = . Forxn∈Txn–there isyn∈Tzsuch thatp(xn,yn)≤ψ(p(xn–,z)). From

p(z,Tz)≤p(z,yn)≤p(z,xn) +p(xn,yn)

≤p(z,xn) +ψ

p(xn–,z)

≤p(z,xn) +p(xn–,z)

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Example  LetX=Q∩[, +∞[ andp:X×X→Rbe deﬁned byp(x,y) =_|x–y|+



max{x,y}. Then (X,p) is a -complete partial metric space. LetT:X→C(X) be deﬁned

by

Tx=Q∩

x

,

x



.

Note thatTxis closed and bounded for allx∈Xunder the given partial metricp. We show thatT is aψ-contractive mapping with respect toψ: [, +∞[→[, +∞[ de-ﬁned byψ(t) =t/ for allt∈[, +∞[. In fact, for allx,x∈Xandy∈Tx, ify=kx/,

k∈Q∩[/, ], then we choosey=kx/. It implies that

p(y,y) =k



|x–x|+  |x–x|

=kψp(x,x)

≤ψp(x,x)

.

By Theorem ,Thas a ﬁxed pointz= .

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

1_{Dipartimento di Matematica e Informatica, Università di Palermo, Via Archiraﬁ, 34, Palermo, 90123, Italy.}2_Mathematical

Institute of Serbian Academy of Sciences and Arts, Kneza Mihaila 36, Belgrade, 11000, Serbia.3_{Faculty of Mechanical} Engineering, University of Belgrade, Kraljice Marije 16, Beograd, 11120, Serbia.

Acknowledgements

The authors thank the referees for their valuable comments that helped us to improve the text. The ﬁrst and fourth authors are supported by Università degli Studi di Palermo (Local University Project ex 60%). The second and third authors are thankful to the Ministry of Science and Technological Development of Serbia (project III44006).

Received: 20 February 2012 Accepted: 20 August 2012 Published: 4 September 2012 References

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