Modified proof of Caristi’s fixed point theorem on partial metric spaces

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R E S E A R C H

Open Access

Modiﬁed proof of Caristi’s ﬁxed point

theorem on partial metric spaces

Chakkrid Klin-eam

*

See related article: http://www.journaloﬁnequalitiesandapplications.com/content/2013/1/355.

*_{Correspondence:}

[email protected] Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok, 65000, Thailand

PERDO National Centre of Excellence in Mathematics, Faculty of Science, Mahidol University, Bangkok, 10400, Thailand

Abstract

In this paper, lower semi-continuous functions are used to modify the proof of Caristi’s ﬁxed point theorems on partial metric spaces. We prove a new type of ﬁxed point theorems in complete partial metric spaces, and then generalize them to metric spaces. Some more general results are also obtained on partial metric spaces.

Keywords: lower semi-continuous functions; Caristi’s ﬁxed point theorem; partial metric spaces

1 Introduction

The number of extensions of the Banach contraction principle have appeared in literature. One of its most important extensions is known as Caristi’s ﬁxed point theorem because Caristi’s theorem is a variety of Ekeland’s-variational principle []. In , Caristi [] proved the following ﬁxed point theorem.

Theorem . Let(X,d)be a complete metric space and f :X→X,and letφbe a lower semicontinuous function from X into[,∞).Assume that d(x,f(x))≤φ(x) –φ(f(x))for all x∈X.Then f has a ﬁxed point in X.

Caristi’s ﬁxed point theorem was generalized by several authors. For example, Bae [] generalized Caristi’s theorem to prove the ﬁxed point theorem for weakly contractive set-valued mappings. Downing and Kirk [] generalized Caristi’s theorem to prove the sur-jectivity theorem for a nonlinear closed mapping. See also [] and others [, ].

In , Siegel [] found that based on the work of Brondsted [] (see also []), Caristi [] had given a significant generalization of the contraction theorem. However, Caristi’s proof, as well as other more recent ones [], lacked the constructive aspect of the original proof. Therefore, he presented a version of Caristi’s theorem which offered a construction of the fixed point as a countable iteration of application of suitable operators in a complete metric space.

In recent years many works on domain theory have been made in order to equip seman-tics domain with a notion of distance, see []. In particular, Matthews [] introduced the notion of a partial metric space as a part of the study of denotational semantics of dataﬂow network, showing that the Banach contraction mapping theorem can be generalized to the partial metric context for applications in program veriﬁcation.

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In this paper we present a version of Caristi’s theorem which oﬀers a construction of the ﬁxed point as a countable iteration of application of suitable operators in a complete partial metric space. The theorem we present is proved by a very simple argument which is in the spirit of the original contraction theorem.

2 Preliminaries

First, we start with some preliminaries on partial metric spaces. For more details, we refer the reader to [].

Deﬁnition . LetXbe a nonempty set. The mappingp:X×X→R+satisﬁes: (i) p(x,x)≤p(x,y)for allx,y∈X.

(ii) x=yif and only ifp(x,x) =p(y,y) =p(x,y). (iii) p(x,y) =p(y,x)for allx,y∈X.

(iv) p(x,y)≤p(x,z) +p(z,y) –p(z,z)for allx,y,z∈X.

Thenpis called apartial metriconXand (X,p) is called apartial metric space.

Note that the self-distance of any point need not be zero, hence the idea of generalizing metrics so that a metric on a non-empty setXis precisely a partial metricponXsuch thatp(x,x) =  for anyx∈X.

Deﬁnition . Let (X,p) be a partial metric space. For anyx∈Xandε> , we deﬁne respectively theopenandclosed ballfor the partial metricpby setting

Bε(x) =

y∈X:p(x,y) <ε, B¯ε(x) =

y∈X:p(x,y)≤ε.

Contrary to the metric space case, some open balls may be empty. As an example, in a partial metric space (X,p), the open ballsBp(x,x)(x) are empty for anyx∈X. For example,

consider the functionp:R–_×_R–_→_R+_{deﬁned by}_p₍_x_,_y_{) = –}_min_{_x_,_y_} _{for any}_x_,_y_∈_X_.

Then the pair (R–,p) is a partial metric space. In a partial metric space (X,p), the set of open balls is the basis of aTtopology onX, calledthe partial metric topologyand denoted byT[p].

Lemma . Let(X,p)be a partial metric space,and let ps:X×X→R+∪ {}be deﬁned by

ps(x,y) = p(x,y) –p(x,x) –p(y,y), ∀x,y∈X.

Then(X,ps₎_{is a metric space}_.

Deﬁnition . A sequence{xn}in a partial metric space (X,p) converges tox∈X, and

we writelimn→∞xn=xif, for anyε>  such thatx∈Bε(x), there existsN≥ so that for

anyn≥N,xn∈Bε(x).

Deﬁnition . If{xn}is a sequence in a partial metric space (X,p), thenx∈Xis a proper

limit of{xn}writtenxn→x(properly) ifxn→xin (X,ps). If a sequence has a proper limit,

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Deﬁnition . A sequence{xn}in a partial metric space (X,p) is called aCauchy sequence

iflimn,m→∞p(xn,xm) exists and is ﬁnite. A partial metric space (X,p) is said to becomplete

if every Cauchy sequence{xn}inXconverges with respect toT[p]to a pointx∈Xsuch

thatp(x,x) =limn,m→∞p(xn,xm).

The following lemma on a partial metric space can be derived easily (see,e.g., [, ]).

Lemma . Let(X,p)be a partial metric space.Then

(i) a sequence{xn}in a partial metric space(X,p)converges to a pointx∈Xif and only

ifp(x,x) =limn→∞p(x,xn),

(ii) a sequence{xn}in a partial metric space(X,p)converges properly to a pointx∈Xif

and only ifp(x,x) =limn→∞p(xn,xn) =limn→∞p(x,xn),

(iii) a sequence{xn}in a partial metric space(X,p)is a Cauchy sequence if and only if it

is a Cauchy sequence in the metric(X,ps_),

(iv) a partial metric space(X,p)is complete if and only if the metric space(X,ps₎_is

complete.Moreover,

p(x,x) = lim

n→∞p(xn,xn) =nlim→∞p(x,xn) ⇔ nlim→∞p s₍_x_,_x

n) = .

Deﬁnition . A functionf :X→Ris calledlower semi-continuousif for any sequence

{xn}inXandx∈Xsuch that{xn}converges tox, we havef(x)≤lim infn→∞f(xn).

3 Main results

Let (X,p) be a complete partial metric space. Letφ:X→R+_and_g_:_X_→_X_{be not}

neces-sarily continuous functions such that

px,g(x)–p(x,x) –pg(x),g(x)≤φ(x) –φg(x), x∈X. Given a sequence of functionsfi, ≤i<∞,

∞

i=

fi(x) = lim

i→∞fifi–· · ·f(x).

If it exists, call it thecountable compositionof thefi.

Now, we let={f |f :X→Xand p(x,f(x)) –p(x,x) –p(f(x),f(x))≤φ(x) –φ(f(x))},

g={f |f∈andφ(f)≤φ(g)}, and then we prove a simple lemma as follows.

Lemma . Bothandg are closed under compositions.Moreover,ifφis lower semi-continuous,thenandgare closed under countable compositions.

Proof We ﬁrst show thatis closed under compositions. Letfandfbe in, then we have

px,ff(x)

–p(x,x) –pff(x),ff(x)

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= px,f(x)+ pf(x),ff(x)– pf(x),f(x)

= px,f(x)+ pf(x),ff(x)–pf(x),f(x)

–pf(x),f(x)

=px,f(x)–p(x,x) –pf(x),f(x)

+pf(x),ff(x)

–pf(x),f(x)

–pff(x),ff(x)

≤φ(x) –φf(x)+φf(x)–φff(x) =φ(x) –φff(x)

implies thatff∈.

Furthermore, we havegis closed under compositions.

Indeed, we letf∈g, which gives thatφ(f(x))≤φ(g(x)), and we now consider

φf(x)–φff(x)≥pff(x),f(x)–pff(x),ff(x)–pf(x),f(x) =pff(x),f(x)+pff(x),f(x)–pff(x),ff(x)

–pf(x),f(x)

≥pff(x),f(x)+pff(x),f(x)–pff(x),f(x) –pf(x),ff(x)

= .

So, we obtain from the above inequalities that

≤φf(x)

–φff(x)

≤φg(x)–φff(x)

,

and thenφ(ff)≤φ(g). This implies thatff∈g.

Thereforeandgare closed under compositions.

To show the classes are closed under countable composition, we use the following lemma.

Lemma . Let{xn}be a sequence in a partial metric space(X,p)such that

p(xn+,xn) –p(xn+,xn+) –p(xn,xn)≤φ(xn) –φ(xn+), ∀n∈N,

thenlimn→∞xn=x and¯ p(x¯,xn) –p(x¯,x¯) –p(xn,xn)≤φ(xn) –φ(x¯),for each n,whereφis a lower semi-continuous function.

Proof We will prove thatlimn−→∞xn=x¯and

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We note that

p(xn+,xn) –p(xn+,xn+) –p(xn,xn)≤φ(xn) –φ(xn+), ∀n∈N,

and we see that

p(xn+,xn) –p(xn+,xn+) –p(xn,xn)≥p(xn+,xn) –p(xn+,xn) –p(xn,xn+)

= p(xn+,xn) –p(xn+,xn) –p(xn+,xn)

= .

It follows that

≤φ(xn) –φ(xn+), ∀n∈N.

Thus

φ(x)≥φ(x)≥ · · · ≥φ(xn)≥φ(xn+)≥ · · ·> ,

and then the sequence of valueφ(xn) is decreasing and bounded below, which implies that

{φ(xn)}is a convergent sequence inR, that is, a Cauchy sequence. Thus, forε> , there

existsn∈Nsuch that for allm>n>n, we have

φ(xn) –φ(xm) <ε.

By the triangle inequality, we have

p(xn,xm) –p(xn,xn) –p(xm,xm)≤φ(xn) –φ(xm).

Therefore

p(xn,xm) –p(xn,xn) –p(xm,xm)≤ φ(xn) –φ(xm) <ε.

It follows thatps₍_x

n,xm) <ε. Thus{xn} is a Cauchy sequence inX, and the

complete-ness of the space (X,ps_{) implies that}_{_x

n} converges and so there existsx¯∈Xsuch that

lim_n_→∞xn=x¯. Moreover, we have

p(xn,x¯) –p(xn,xn) –p(x¯,x¯) =ps(xn,x¯) = lim m→∞p

s₍_x n,xm)

≤ lim

m→∞

φ(xn) –φ(xm)

= lim

m→∞φ(xn) –mlim→∞φ(xm)

=φ(xn) – lim m→∞φ(xm)

≤φ(xn) –lim inf m→∞φ(xm)

≤φ(xn) –φ(x¯).

This last inequality obtained byφis lower semi-continuous.

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The remainder of the proof of Lemma . amounts to the observation that for eachx∈X, the sequencexi=fifi–· · ·f(x) satisﬁes the conditions of Lemma .. Before starting the

main theorem of this paper, we also need the following.

Deﬁnition . Let (X,p) be a partial metric space. () ForA⊆X, deﬁnethe diameter of A, writtenD(A), by

D(A) = sup

xi,xj∈A

p(xi,xj) –p(xi,xi) –p(xj,xj)

.

() Letr(A) =infx∈A(φ(x)); noteB⊆Aimpliesr(B)≥r(A).

() Let⊆. For eachx∈X, deﬁneSx={f(x)|f∈}. Lemma . D(Sx)≤(φ(x) –r(Sx)).

Proof Letf,fbe inSx, then we have

pf(x),f(x)–pf(x),f(x)–pf(x),f(x)

≤pf(x),x+ px,f(x)– p(x,x) –pf(x),f(x)

–pf(x),f(x)

= pf(x),x

+ px,f(x)

–p(x,x) –p(x,x)

–pf(x),f(x)

–pf(x),f(x)

=pf(x),x–pf(x),f(x)–p(x,x)

+px,f(x)–pf(x),f(x)–p(x,x)

=px,f(x)–p(x,x) –pf(x),f(x)

+px,f(x)–p(x,x) –pf(x),f(x)

≤φ(x) –φf(x)+φ(x) –φf(x) = φ(x) –φf(x)–φf(x)

= φ(x) –r(Sx)

.

ThusD(Sx)≤(φ(x) –r(Sx)).

Theorem . Let⊆be closed under compositions.Let x∈X.

. Letbe closed under countable compositions.Then there exists anf¯∈such that

¯

x=f¯(x)andg(x¯) =x¯for allg∈.

. Let the elements ofbe continuous functions.Then there exists a sequence of

functionsfi∈andx¯=limi→∞fifi–· · ·f(x)such thatg(x¯) =x¯for allg∈. Proof Letεibe a positive sequence converging to . Choosef∈such that

φf(x)–r(Sx) <

ε

.

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Sinceis closed under compositions, we haveSx⊆Sxand

D(Sx)≤

φ(x) –r(Sx)

≤φf(x)–r(Sx)

< 

ε



=ε.

Again, choosef∈such that

φf(x)–r(Sx) <

ε

.

Setx=f(x).

Sinceis closed under compositions, we haveSx⊆Sxand

D(Sx)≤

φ(x) –r(Sx)

≤φf(x)–r(Sx)

< 

ε



=ε.

Continuing this procedure, we obtain a sequence offisuch that

xi+=fi+(xi),Sxi+⊆Sxi and D(Sxi+) <εi+.

Next, to show that g(x¯) =x¯ under hypothesis . Letf¯=_i∞_=fi andx¯=f¯(x). Since

is closed under compositions, thenf¯∈. Sincex¯ =∞_j₌_i_+fj(xi), it implies thatx¯∈Sxi

for eachi. On the other hand, sincelim_i_→∞D(Sxi) = , we havex¯=

∞

i=Sxi. Sinceg(x¯) = g(∞_j₌_i_+fj(xi)), we obtain thatg(x¯)∈Sxi. Thusg(x¯) =x¯. Finally, to show thatg(x¯) =x¯under

hypothesis . Letx¯=limi→∞fifi–· · ·f(x) =limi→∞xi. First, since{xj}j>i⊆Si, for eachi

we have thatx¯∈ ¯Si, the closure ofSi. SinceD(S¯i) =D(Si), we have thatx¯=

∞

i=Sxi.

To verify that g(x¯) =x¯, observe thatg(xi)∈Sxi for eachi. Hence, for any ε_ > , there

existsisuch thatBε

(g(x¯))∩Sxi=∅,i>i(here we needgto be continuous). Therefore, fori>i,

px¯,g(x¯)–p(x¯,x¯) –pg(x¯),g(x¯)

≤px¯,g(xi)

+pg(xi),g(x¯)

–pg(xi),g(xi)

–pg(x¯),g(x¯)–p(x¯,x¯) = px¯,g(xi)

+ pg(xi),g(x¯)

– pg(xi),g(xi)

–pg(x¯),g(x¯)–p(x¯,x¯) = px¯,g(xi)

+ pg(xi),g(x¯)

–p(x¯,x¯) –pg(xi),g(xi)

–pg(xi),g(xi)

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< pg(x¯),g(xi)

+px¯,g(xi)

–p(x¯,x¯) –pg(xi),g(xi)

< 

ε



+εi

=ε+εi.

And soεi→ implies that

pg(x¯),x¯–pg(x¯),g(x¯)–p(x¯,x¯)≤ε.

Thereforeg(x¯) =x¯. The proof is completed.

Remark In Theorem .() one may choose={gn_}_{, the set consisting of}_g_{and its ﬁnite}

iterates. For this choice of, one hasx¯=limn→∞gn(x) as in the contraction theorem.

Competing interests

The author declares that they have no competing interests.

Acknowledgements

The author is grateful to the referees for precise remarks allowing us to improve the presentation of the paper and would like to thank the faculty of Science, Naresuan University, Phitsanulok for the ﬁnancial support.

Received: 25 December 2012 Accepted: 12 April 2013 Published: 26 April 2013 References

1. Ekeland, I: Sur les problemes variationnels. C. R. Acad. Sci. Paris275, 1057-1059 (1972)

2. Caristi, J: Fixed point theorems for mappings satisfying inwardness conditions. Trans. Am. Math. Soc.215, 241-251 (1976)

3. Bae, JS: Fixed point theorems for weakly contractive multivalued maps. J. Math. Anal. Appl.284, 690-697 (2003) 4. Downing, D, Kirk, WA: A generalization of Caristi’s theorem with applications to nonlinear mapping theory. Pac. J.

Math.69, 339-346 (1977)

5. Bae, JS, Cho, EW, Yeom, SH: A generalization of the Caristi-Kirk ﬁxed point theorem and its applications to mapping theorems. J. Korean Math. Soc.31, 29-48 (1994)

6. Reich, S: Approximate selections best approximations ﬁxed points and invariant set. J. Math. Anal. Appl.62, 104-113 (1978)

7. De Blasi, FS, Myjak, J, Reich, S, Zaslavski, AJ: Generic existence and approximation of ﬁxed points for nonexpansive set-valued maps. Set-Valued Var. Anal.17, 97-112 (2009)

8. Siegel, J: A new proof of Caristi’s ﬁxed point theorem. Proc. Am. Math. Soc.66, 54-56 (1977) 9. Brondsted, A: On a lemma of Bishop and Phelps. Pac. J. Math.55, 335-341 (1974)

10. Wong, CS: On a ﬁxed point theorem of contractive type. Proc. Am. Math. Soc.57, 283-284 (1976)

11. Matthews, SG: Partial metric topology. In: Proc. 8th Summer Conference on General Topology and Applications. Ann. New York Acad. Sci., vol. 728, pp. 183-197 (1994)

12. Valero, O: On Banach ﬁxed point theorems for partial metric spaces. Appl. Gen. Topol.6, 229-240 (2005)

doi:10.1186/1029-242X-2013-210