R E S E A R C H
Open Access
Dislocated metric and fixed point
theorems
Lech Pasicki
**Correspondence:
Faculty of Applied Mathematics, AGH University of Science and Technology, Al. Mickiewicza 30, Kraków, 30-059, Poland
Abstract
The paper deals with the here defined dislocated strong quasi-metric (ifp(y,x) = 0, thenx=y; 0≤p(z,x)≤p(z,y) +p(y,x)) and with the well-known notion of the dislocated metric (in addition,p(y,x) =p(x,y)). In particular, the partial metric is a kind of dislocated metric. Our basic results on general contractions (also for cyclic
mappings) and results of variational type can be treated as a starting point for further development.
MSC: 47H10; 54H25; 49J45; 54E99
Keywords: dislocated metric; dislocated strong quasi-metric; partial metric; fixed point; generalized contraction; cyclic mapping; variational principle
1 Introduction
Recent years have witnessed the appearance of many papers devoted to fixed point theo-rems for partial metric spaces. The aim of the present paper is to show that the dislocated (strong quasi-)metric as presented here (Definition .) has a great potential. Each partial metric is a dislocated metric and examples preceding Definition . show that the dislo-cated metric is more general. The paper is divided into three sections.
In Section the definitions of a dislocated (strong quasi-)metric and of a partial metric are presented. This section contains some examples and a comparison between the two notions. Also some additional ideas (-completeness,Kerp) are included.
Section is devoted to fixed point theorems for general contractions. The simplest re-quirement is condition (.):p(f(y),f(x))≤ϕ(p(y,x)), for allx,y∈X, wherepis a dislocated metric onX,f :X→Xis a mapping, and the comparison functionϕ: [,∞)→[,∞) belongs to a wide class of mappings defined in [] and here. The main classical results are Theorem . (a direct extension of the celebrated theorems of Matkowski [], Theo-rem ., and of Boyd-Wong [], TheoTheo-rem ), and a more general TheoTheo-rem .. The most sophisticated ones are the theorems for cyclic mappings (see Definition .): Theorem ., and a result of a new type, Theorem ., which is proved with the use of cross mappings defined in []. Our theorems extend also some general results of Karapinar and Salimi [], Theorems ., ..
of the smallest element in a maximal chain. Typical are Theorems . and ., and an extension of the Ekeland principle (Theorem .). The fixed point results are Theorem . and Theorem . (an extension of the celebrated theorems of Caristi and Takahashi).
2 Dislocated metric and dislocated strong quasi-metric
The notion of dislocated metric was introduced by Hitzler and Seda in [].
Definition . LetXbe a nonempty set, andp:X×X→[,∞) a mapping satisfying
ifp(y,x) = thenx=y, x,y∈X, (.)
p(z,x)≤p(z,y) +p(y,x), x,y,z∈X. (.)
Thenpis called adislocated strong quasi-metric(briefly a dsq-metric), and (X,p) is called adislocated strong quasi-metric space(briefly a dsq-metric space). Thekernelofpis the set
Ker p=x∈X: lim
n→∞p(x,xn) = for a sequence (xn)n∈NinX
.
If, in addition,
p(y,x) =p(x,y), x,y∈X, (.)
holds, thenpis called adislocated metric(briefly ad-metric), and (X,p) is called a dislo-cated metric space(briefly ad-metric space).
In the previous version of the present paper (entitled ‘Near (quasi-)metric and fixed point theorems’) dislocated metric was called near metric because the author was not aware of Hitzler’s definition, and a dislocated strong metric was called a near quasi-metric.
The nonalphabetical order ofx,y,zin conditions (.), (.), (.) is better suited to the results of Section (y=f(x)xcorresponds withψ(y)≤ψ(x)).
The topology of ad-metric (or dsq-metric) space (X,p) is generated by ballsB(x,r) = {y∈X:p(x,y) <r}. Clearly,x∈B(x,r) does not necessarily hold, but the family of all balls generates the respective smallest topology forX={B(x,r) :x∈X,r> }[], Theorem , p.. IfZ=Kerpis nonempty, then (Z,p|Z×Z) is a metric (or quasi-metric) subspace of (X,p).
Recently, Amini-Harandi has defined metric-like mappingσ (identic with the idea of d-metricp) and metric-like space (X,σ) [], Definition .. The topology of his space gen-erated byσ-ballsBσ(x,) ={y∈X:|σ(x,y) –σ(x,x)|<}usually differs from the topology
for ad-metric space.
It should be noted that also Karapinar and Salimi [] follow the ideas of Amini-Harandi, which are better suited to partial metric spaces (see Definition .).
For thed-metricpthe following condition is satisfied:
lim
n→∞p(x,xn) =nlim→∞p(y,xn) = yieldsx=y∈Kerp=
Indeed, from ≤p(y,x)≤p(y,xn) +p(x,xn) it follows thatp(y,x) = ,x=y(see (.)), and p(x,x) = means thatx∈Kerp.
Proposition . Let(X,p)be a d-metric(or dsq-metric)space.Then p(·,y)is lower semi-continuous at points ofKerp,y∈X.
Proof Let (xn)n∈N be such that limn→∞p(x,xn) = . Then the inequality p(x,y) ≤
lim infn→∞p(xn,y) is a consequence ofp(x,y)≤p(x,xn) +p(xn,y).
Definition . Ad-metric (or dsq-metric) space (X,p) is called -completeif the follow-ing condition is satisfied:
for every sequence (xn)n∈NinXsuch that lim
n>m→∞p(xn,xm) =
there exists anx∈Xsuch that lim
n→∞p(x,xn) = ;
(.)
a nonempty setA⊂Xis -complete if (A,p|A×A) is -complete.
In view of (.) a pointxas in (.) is unique ifpis ad-metric, and thenp(x,x) = . To present a simple example of a -completed-metric space let us considerX= [–,∞) andp(x,y) =x+y+ . An easy computation shows thatp(y,x) = yieldsx=y= –. In addition,
p(z,x) =z+x+ ≤z+x+ + y+ =p(z,y) +p(y,x)
yields (.). Clearly,limm,n→∞p(xn,xm) = means thatlimn→∞p(–,xn) = , and (X,p) is -complete. ForX= (–,∞) and the samepwe obtain a non--completed-metric space. Another example is X={(x,x)∈R :x≥–,x ≥} withp defined by p((x,x),
(y,y)) =x+x+y+y+ .
Let us recall the notions of a partial metric due to Matthews [], and of a dualistic partial metric due to Oltra and Valero [] and O’Neill [].
Definition . Adualistic partial metricis a mappingp:X×X→Rsuch that
y=x iff p(y,y) =p(y,x) =p(x,x), x,y∈X, (.)
p(y,y)≤p(y,x), x,y∈X, (.)
p(y,x) =p(x,y), x,y∈X, (.)
p(z,x)≤p(z,y) +p(y,x) –p(y,y), x,y,z∈X. (.)
Ifpis nonnegative, then it is called apartial metric.
It is well known (see,e.g., []) that a metricdcan be defined by a (dualistic) partial metricpas follows:
d(y,x) =maxp(y,x) –p(y,y),p(x,y) –p(x,x), x,y∈X.
A dualistic partial metric space (X,p) is -complete (see [], Definition ., [], Corol-lary ) if for every sequence such thatlimm,n→∞p(xn,xm) = there exists anx∈Xsuch thatlimn→∞d(x,xn) = andp(x,x) = .
Now, it is clear that if a partial metric space (X,p) is -complete, then (X,p) treated as a dislocated metric space is also -complete.
Lemma .(cf.[], Lemma .) Let(X,p)be a d-metric space with a nonempty kernel Z. Then(Z,p|Z×Z)is a metric subspace of(X,p);if(X,p)is-complete,then(Z,p|Z×Z)is com-plete.
Proof Clearly,p|Z×Zis a metric onZ. Fromlimn→∞p(x,xn) = and
≤p(x,x)≤p(x,xn) +p(xn,x) = p(x,xn)
it follows thatp(x,x) = ,i.e. x∈Z. Therefore, if (X,p) is -complete, then (Z,p|Z×Z) is
complete.
3 General contractions
In the present section we are interested in mappingsf :X→Xsatisfying
pf(y),f(x)≤ϕp(y,x) (.)
or
pf(y),f(x)≤ϕmf(y,x)
(.)
for
mf(y,x) =max
p(y,x),pf(y),y,pf(x),x, (.)
where (X,p) is ad-metric space, andϕis a comparison function.
It should be noted that thed-metricpdefines the metricδin the following way:δ(x,x) = , andδ(x,y) =p(x,y) forx=y. One can see that ad-metric space (X,p) is -complete (Definition .) iff the metric space (X,δ) is complete.
Let us note that forf(y) =yandy=x=f(x)
mf(y,x) =max
p(y,x),p(y,y),pf(x),x
does not necessarily equal
(e.g.forp(y,y) = andp≡ elsewhere). Therefore, it is better to apply thed-metricpthan the metricδ.
Our theorems in the present section are new also ifpis a metric. The special features of the mappingϕwill enable one to give proofs of these theorems using metricδbut such full proofs would be unnecessarily complex (see also []).
According to the notations from []is a class of mappingsϕ: [,∞)→[,∞) such thatϕ(α) <α,α> ; andϕ∈iffϕ∈andϕ() = . In turn,Pconsists of mappingsϕ: [,∞)→[,∞) for which every sequence (an)n∈Nsuch thatan+≤ϕ(an),n∈Nconverges to zero. It appears [], Proposition , thatP⊂; ifϕ∈satisfies
lim sup β→α+
ϕ(β) <α, α> , (.)
thenϕ∈P. In particular (see []), ifϕ∈is upper semicontinuous from the right (see
[]), thenϕ∈P; also, ifϕ∈is nondecreasing andlimn→∞ϕn(α) = ,α> (see []), thenϕ∈P.
Let us considerP(P⊂P⊂) consisting of mappingsϕfor which every sequence (an)n∈N such that <an+≤ϕ(an),n∈Nconverges to zero. Ifϕ∈satisfies (.), then ϕ∈P(see the proof of [], Proposition ). It was noted in [] that Theorems and of that paper are valid also forPreplaced byP, asϕ() is meaningless.
Example Let us consider linear mappingsgn:R→R,gn(x) = –nx,n∈N. One can see thatgn(/(n+ )) = /(n+ ), andgn(/n) = . Therefore,ϕ∈defined byϕ() = =ϕ(x),
x> ,ϕ(x) =gn(x),x∈(/(n+ ), /n],n∈N, has the following properties:
(i) ϕ(x) < /,x∈R, (ii) ϕ(x) < /(n+ ),x≤/n,
(iii) lim supβ→(/n)+ϕ(β) = /n, <n∈N.
Clearly,ϕis not monotone, and (iii) means that (.) is not satisfied. Ifan+≤ϕ(an),n∈N, holds, then (i) yieldsa≤ϕ(a) < /, and from (ii) it follows (by induction) thatan+≤
ϕ(an) < /(n+ ),n∈N. Consequently, we obtainϕ∈P.
The subsequent two lemmas for (.), (.) are partial extensions of [], Lemmas , , proved for partial metric spaces (see also [], Lemmas , ).
Lemma . Let X be a nonempty set,and let p:X×X→[,∞),f :X→X be mappings satisfying condition(.)or(.),for all x,y∈X and aϕ∈.Then the condition
pf(x),f(x)≤ϕpf(x),x, x∈X, (.)
holds,and ifϕ∈P,thenlimn→∞p(fn+(x),fn(x)) = ,x∈X.
Proof For notational simplicity let us adoptxn=fn(x),n∈N. We have mf(x,x) =max
p(x,x),p(x,x)
.
Supposep(x,x) <p(x,x). Then (.) yields
<α=p(x,x)≤ϕ
mf(x,x)
=ϕp(x,x)
a contradiction (ϕ∈). Now,mf(x,x) =p(x,x) holds, and we obtain (.) (which for (.)
is trivial). Now, foran=p(xn+,xn),n∈N, andϕ∈Pwe getlimn→∞an= .
Lemma . Let (X,p)be a -complete d-metric space, and let f :X→X be a map-ping satisfying condition (.)or(.), for all x,y∈X and a ϕ ∈. If for xn=fn(x), limm,n→∞p(xn,xm) = holds,then there exists a unique fixed point x of f,limn→∞p(x,xn) = (i.e. x=limn→∞xnin(X,p)),and p(x,x) = .
Proof Letx∈Xbe such thatlimn→∞p(x,xn) = . For condition (.) we have
pf(x),x≤pxn+,f(x)
+p(xn+,x)≤ϕ
p(xn,x)+p(xn+,x).
Supposep(f(x),x) =α> . Thenϕ(p(xn,x))≥α/ holds for largen, asp(xn+,x)→.
Con-sequently,p(x,xn) = for largen, and
≤p(xn+,xn)≤p(xn+,x) +p(xn,x) =
means thatf(xn) =xn(see (.)) andx=xnis a fixed point off. For condition (.),p(f(x),x) > , and largenwe have
<pf(x),x≤pxn+,f(x)
+p(xn+,x)≤ϕ
mf(xn,x)
+p(xn+,x)
=ϕmaxp(xn,x),p(xn+,xn),p
f(x),x+p(xn+,x)
=ϕpf(x),x+p(xn+,x).
Consequently, <p(f(x),x)≤ϕ(p(f(x),x)) holds, a contradiction (ϕ∈). In view of (.), p(f(x),x) = yieldsx=f(x).
Ifyis a fixed point off, then
≤pf(y),y=p(y,y) =pf(y),f(y)≤ϕp(y,y)
=ϕmaxp(y,y),pf(y),y=ϕmf(y,y)
means thatp(y,y) = ,i.e. y∈Kerp.
Supposex,yare two fixed points off. Then
p(y,x) =pf(y),f(x)≤ϕp(y,x)
=ϕmaxp(y,x),pf(y),y,pf(x),x=ϕmf(y,x)
means thatp(y,x) = , andx=y.
Now, we are ready to prove the following extension of [], Theorem (the proof is almost the same as in []).
Theorem . Let(X,p)be a-complete d-metric space,and let f :X→X be a mapping satisfying condition(.)or(.),for all x,y∈X and aϕ∈having property(.)or a ϕ∈Psuch that
lim sup
β→α– ϕ(β) <α,
(e.g. ifϕis nondecreasing)holds.Then f has a unique fixed point;if x=f(x),then p(x,x) = andlimn→∞p(x,fn(x)) = ,x∈X.
Proof It is sufficient to prove thatlimm,n→∞p(xn,xm) = holds forxn=fn(x),n∈N(see
Lemma .). Suppose that there are infinitely manyk,n∈Nsuch thatp(fn++k(x),fk(x))≥
> . Letn=n(k) > be the smallest numbers satisfying this inequality for infinitely many largek. For simplicity let us adoptx=fk(x
) andxn=fn(x),n∈N. We have ≤p(xn+,x)≤p(xn+,xn) +p(xn,x) <p(xn+,xn) +,
which forn=n(k) means that
lim
k→∞p(xn+,x) =klim→∞p(xn,x) =,
asϕ∈P andlimk→∞p(xn+,xn) =limk→∞p(x,x) = (see Lemma .). Now fory=xn condition (.) yields
≤p(xn+,x)≤p(xn+,x) +p(x,x)≤ϕ
mf(xn,x)
+p(x,x)
=ϕmaxp(xn,x),p(xn+,xn),p(x,x)
+p(x,x),
and we obtain (from (.) as well)
≤ϕp(xn,x)+p(x,x)
for largek. Now,p(xn,x) <,limk→∞p(xn,x) =, and condition (.) yield
≤lim sup
k→∞ ϕ
p(xn,x)
<,
a contradiction. Similarly,p(xn+,x)≥,
p(xn+,x) –p(xn+,xn+) –p(x,x)≤p(xn+,x)≤ϕ
p(xn+,x)
,
and condition (.) yield
≤lim sup
k→∞
ϕp(xn+,x)
<,
a contradiction. Therefore,limm,n→∞p(xn,xm) = holds.
Let us recall the following.
Lemma .[], Lemma Let f :X→X be a mapping such that ftfor a t∈Nhas a unique fixed point,say x.Then x is the unique fixed point of f.If,in addition,x∈limn→∞(ft)n(x
),
x∈X,then x∈limn→∞fn(x),x∈X holds.
Theorem . Let(X,p)be a-complete d-metric space,and let f :X→X be a mapping satisfying condition(.)or(.),for all x,y∈X with f replaced by ft for a t∈N,and a ϕ∈having property(.)or aϕ∈Psuch that(.)holds.Then f has a unique fixed point;if x=f(x),then x satisfies p(x,x) = andlimn→∞p(x,fn(x)) = ,x∈X.
Kirket al.[] suggested the idea of cyclic mappings which was later formalized by Rus in [] as the cyclic representation ofX=X∪· · ·∪Xtwith respect tof. The next definition means the same, but it is more compact.
Definition . A mappingf :X→Xis calledcycliconX, . . . ,Xt (for at> ) if∅ =X= X∪ · · · ∪Xt, andf(Xj)⊂Xj++,j= , . . . ,t, wherej+ + =j+ forj= , . . . ,t– , andt+ + = .
Clearly,Xj=∅for ajin Definition ., and henceXj=∅,j= , . . . ,t. The proof of Lemma . works also for the following.
Lemma . Let p:X×X→[,∞)be a mapping,and let f :X→X be cyclic on X, . . . ,Xt.
Assume that(.)or(.)is satisfied for all x∈Xj,y∈Xj++,j= , . . . ,t,and aϕ∈.Then
condition(.)holds,and ifϕ∈P,thenlimn→∞p(fn+(x),fn(x)) = ,x∈X.
If we consider nsuch thatx∈Xj andxn∈Xj++ for aj∈ {, . . . ,t}, then the proof of
Lemma . yields the following.
Lemma . Let(X,p)be a-complete d-metric space, and let f :X→X be cyclic on X, . . . ,Xt.Assume that(.)or(.)is satisfied for all x∈Xj,y∈Xj++,j= , . . . ,t,and a
ϕ∈.If for xn=fn(x),limm,n→∞p(xn,xm) = holds,then there exists a unique fixed point x of f,limn→∞p(x,xn) = (i.e. x=limn→∞xnin(X,p)),and p(x,x) = .
Lemmas . and . yield the following extension of Theorem ..
Theorem . Let(X,p)be a-complete d-metric space,and let f :X→X be cyclic on X, . . . ,Xt.Assume that(.)or(.)is satisfied for all x∈Xj,y∈Xj++,j= , . . . ,t,and a
ϕ∈having property(.)or aϕ∈Psuch that(.)holds.Then f has a unique fixed point;if x=f(x),then p(x,x) = andlimn→∞p(x,fn(x)) = ,x∈X.
Proof It is sufficient to prove thatlimm,n→∞p(xn,xm) = holds forxn=fn(x),n∈N(see
Lemma .). Suppose that there are infinitely manyk,n∈Nsuch thatp(x(n+)t+k+,xk)≥
> . Letn=n(k) > be the smallest numbers satisfying this inequality for infinitely many largek. For simplicity let us adoptx=xk=fk(x), andxn=fn(x),n∈N. Clearly,x∈Xj yieldsxnt+,x(n+)t+∈Xj++. We have
≤p(x(n+)t+,x)≤p(x(n+)t+,xnt+) +p(xnt+,x)
<p(x(n+)t+,xnt+) +≤p(x(n+)t+,x(n+)t) +· · ·+p(xnt+,xnt+) +,
which forn=n(k) means that
lim
asϕ∈Pandlimm→∞p(xm+,xm) = (see Lemma .). Now fory=xnt+condition (.)
yields
≤p(x(n+)t+,x)≤p(x(n+)t+,x(n+)t) +· · ·+p(xnt+,x) +p(x,x)
≤p(x(n+)t+,x(n+)t) +· · ·+ϕ
mf(xnt+,x)
+p(x,x)
=p(x(n+)t+,x(n+)t) +· · ·
+ϕmaxp(xnt+,x),p(xnt+,xnt+),p(x,x)
+p(x,x),
and we obtain (from (.) as well)
≤p(x(n+)t+,x(n+)t) +· · ·+ϕ
p(xnt+,x)
+p(x,x)
for largek. Now,p(xnt+,x) <,limk→∞p(xnt+,x) =, and condition (.) yield
≤lim sup
k→∞ ϕ
p(xnt+,x)
<,
a contradiction. Similarly,p(x(n+)t+,x)≥,
p(x(n+)t+,x) –p(x(n+)t+,x(n+)t+) –p(x,x)
≤p(x(n+)t+,x)≤ϕ
p(x(n+)t+,x)
,
and condition (.) yield
≤lim sup
k→∞
ϕp(x(n+)t+,x)
<,
a contradiction. Now, it is clear thatlimm,n→∞p(xm+nt+,xm) = . Consequently,
lim
m,n→∞p(xm+nt+s,xm)
≤ lim
m,n→∞
p(xm+nt+s,xm+nt+s–) +· · ·
+p(xm+nt+,xm+nt+) +p(xm+nt+,xm)
=
for anys∈ {, . . . ,t},i.e.limm,n→∞p(xn,xm) = .
Karapinar and Salimi [], Definition ., have defined the notion of a generalized φ-ψ-contractive mapping. It appears thatϕ= (id+ψ)–◦(id+ψ–φ)∈Pbecause condi-tion (.) is satisfied. Therefore, [], Theorem ., is a particular case of Theorem ., and [], Theorem ., is a consequence of Theorem . (see also the example preceding Lemma .).
Let us present cyclic mappings of the second type,i.e.those for (.) or (.) withx,y∈ Xj,j= , . . . ,t. It is convenient to apply the idea of cross mappings introduced in [].
LetFj:Xj→Ej++,j= , . . . ,t(fort> ), be multivalued mappings. Then forY=X×
· · · ×Xt,E=E× · · · ×Etwe define across mapping F:Y→Eas follows [], (.):
We can see that forEj⊂Xj,j= , . . . ,tthe compositionFt◦Ft–◦ · · · ◦Fhas a fixed point
inXiffFhas a fixed point. This concept is very efficient for multivalued mappings (see
[], Section ). Let us apply cross mappings to prove the following.
Theorem . Let(X,p)be a-complete d-metric space,and let f :X→X be cyclic on -complete sets X, . . . ,Xt.Assume that(.)or(.)is satisfied for all x,y∈Xj,j= , . . . ,t, and a nondecreasingϕ∈P.Then f has a unique fixed point;if x=f(x),then p(x,x) = andlimn→∞p(x,fn(x)) = ,x∈X.
Proof Let us considerY=X× · · · ×Xtand
q(y,x) =maxp(y,x), . . . ,p(yt,xt)
, x,y∈Y.
Then (Y,q) is ad-metric space, and it is -complete. Ifϕis nondecreasing and (.) or (.) is satisfied forp, then it is also satisfied forq, ase.g.max{ϕ(a),ϕ(b)}=ϕ(max{a,b}). In view of Theorem . the cross mappinghdefined by
h(x) =f(xt),f(x), . . . ,f(xt–)
, x∈Y,
has a unique fixed point. This means that ft has a unique fixed point. Now we apply
Lemma ..
There exist many papers concerning cyclic mappings (see,e.g., the references of []), and it is very likely that Theorems . and . are just a starting point for further devel-opment.
4 Variational results
In this sectionpis a dislocated strong quasi-metric (i.e.a dsq-metric).
To present a simple example of a -complete dsq-metric space let us consider X= [–,∞) andp(x,y) =x+ y+ . An easy computation shows thatp(y,x) = yieldsx=y= –. Clearly,
p(z,x) =z+ x+ ≤z+ x+ + y+ =z+ y+ +y+ x+ =p(z,y) +p(y,x)
yields (.). Moreover, (.) is not satisfied and therefore,pis not ad-metric. In addition, (X,p) is -complete, aslimn>m→∞p(xn,xm) = means thatlimn→∞p(–,xn) = .
Let us prove the following analog of [], Theorem .
Theorem . Let(X,p)be a-complete dsq-metric space andψ:X→Ra mapping lower semicontinuous at points ofKerp.Let us adopt yx iffψ(y) +p(y,x) –ψ(x)≤,x,y∈X, and assume thatψ is bounded below on each chain and the following holds:
for each x∈X\B there exists a y∈X\ {x}such that yx. (.)
Then for any x∈X\B,each maximal chain A⊂X containing xhas a unique smallest
element x,in addition satisfying
(ii) ψ(x) +p(x,x) –ψ(x) =inf{ψ(z) +p(z,x) –ψ(x) :z∈A} ≤,
(iii) <ψ(y) +p(y,x) –ψ(x),for eachy∈X\ {x}, (iv) x∈Kerp(i.e.p(x,x) = ifpis ad-metric).
Proof The relationis transitive and in view of Kuratowski’s lemma [], p., for any x∈X\Bthere exists a maximal chainAcontainingx. Let us adoptα=inf{ψ(z) :z∈A}
and supposeα<ψ(x), for eachx∈A. Then there exists a sequence (xn)n∈N inA such
that (ψ(xn))n∈Ndecreases toα. For anyy∈Asuch thatψ(xn) <ψ(y) the conditionyxn cannot hold, as then
≤p(y,xn)≤ψ(xn) –ψ(y) < .
Therefore, ify∈Aandψ(xn) <ψ(y), thenxny. In particular,m<nyieldsxnxm,i.e.,
≤p(xn,xm)≤ψ(xm) –ψ(xn),
and consequently,limn>m→∞p(xn,xm) = , asψis bounded below onA. Therefore, there exists anxsuch thatlimn→∞p(x,xn) = ,i.e. x∈Kerp(see Definition .), andp(x,x) = ifpis ad-metric (see (.)). Clearly,
ψ(·) +p(·,xm) –ψ(xm)
is lower semicontinuous atx(see Proposition .). Now, we obtain
ψ(x) +p(x,xm) –ψ(xm)≤ lim
n→∞ψ(xn) +lim supn→∞ p(xn,xm) –ψ(xm)≤,
i.e. xxm, andψ(x)≤limn→∞ψ(xn) =α. For anyy∈Aandψ(xm) <ψ(y) we getx xmy. Consequently,x∈Aand (iv) is satisfied. Suppose ay∈A\ {x}such thatψ(y) =α. Then ≤min{p(x,y),p(y,x)} ≤ |ψ(y) –ψ(x)|= yieldsx=y. Therefore,xis the unique smallest element ofAand conditions (ii), (iii) follow. In view of (.)x∈B, and we get (i).
The ‘order’ reasoning fails for a quasi-metric (x=yiff q(x,y) =q(y,x) = , q(z,x)≤ q(z,y) +q(y,x)), asq(y,x) = does not necessarily yieldq(x,y) = .
A reasoning similar to the one presented in the above proof yields the following analog of [], Theorem .
Theorem . Let(X,p)be a-complete dsq-metric space andψ:X→Ra mapping lower semicontinuous at points ofKerp.Let us adopt yx iffψ(y) +p(y,x) –ψ(x)≤,x,y∈X, and assume that x∈X belongs to a chain andψis bounded below on each chain
contain-ing x.Then each maximal chain A⊂X containing xhas a unique smallest element x,in
addition satisfying
(i) ψ(x) =inf{ψ(z) :z∈A},
(ii) ψ(x) +p(x,x) –ψ(x) =inf{ψ(z) +p(z,x) –ψ(x) :z∈A} ≤,
Proof In view of Kuratowski’s lemma [], p., there exists a maximal chainAcontaining x. Now, forAwe follow the proof of Theorem ., omitting the last sentence.
We also have the following analog of [], Theorem .
Theorem . Let(X,p)be a-complete dsq-metric space andψ:X→Ra mapping lower semicontinuous at points ofKerp.Let us adopt yx iffψ(y) +p(y,x) –ψ(x)≤,x,y∈X, and assume thatψis bounded below on each chain.If X⊂Y and F:X→Yis a mapping satisfying
for each x∈X\F(x)there exists a y∈X\ {x}such that yx. (.)
Then for any x∈X,x∈F(x)holds or each maximal chain A⊂X containing xhas a
unique smallest element x,which in addition satisfies conditions(i), . . . , (iv)of Theorem. and is such that x∈F(x).
Proof Ifx∈/F(x), then Theorem . applies and (iii) contradicts (.) forx∈/F(x).
The subsequent theorem extends the theorems of Caristi [], Theorem (.), and Taka-hashi [], Theorem .
Theorem . Let(X,p)be a-complete dsq-metric space andψ:X→Ra mapping lower semicontinuous at points ofKerp.Let us adopt yx iffψ(y) +p(y,x) –ψ(x)≤,x,y∈X, and assume thatψis bounded below on each chain.If X⊂Y and F:X→Yis a mapping satisfying
for each x∈X there exists a y∈F(x)such that yx. (.)
Then for any x∈X,each maximal chain A⊂X containing xhas a unique smallest
ele-ment x,which in addition satisfies conditions(i), . . . , (iv)of Theorem.and is such that x∈F(x).
Proof In view of (.) Theorem . works. Now condition (.) and (iii) mean thatx∈
F(x).
The subsequent theorem extends Ekeland’s variational principle [], Theorem , (cf. [], Theorem , and [], Theorem ).
Theorem . Let(X,p)be a-complete dsq-metric space andψ:X→Ra bounded below mapping lower semicontinuous at points ofKerp.Let us adopt yx iffψ(y) +p(y,x) – ψ(x)≤,x,y∈X,and assume that x∈X belongs to a chain(e.g. if p(x,x) = ).Then
the following conditions are satisfied:
(a) there exists anx∈Xsuch thatψ(x)≤ψ(x)andψ(x) –p(y,x) <ψ(y),for each
y∈X\ {x},
(b) for any> and eachx∈Xwithp(x,x) = there exists anx∈Xsuch that
ψ(x)≤ψ(x),andψ(x) –p(y,x) <ψ(y),for eachy∈X\ {x};if,in addition,
Proof Clearly, Theorem . (i), (iii) yield (a). If we considerpin place ofpforp(x,x) = ,
then for the smallest element xin any maximal chainAcontainingxwe haveψ(x)≤
ψ(x), and <ψ(y) +p(y,x) –ψ(x),y∈X\ {x}(Theorem .(i), (iii)). Fromp(x,x)≤
ψ(x) –ψ(x) (Theorem .(ii)) and the second assumption of (b) we obtain
p(x,x)≤ ψ(x) –ψ(x)
/≤ ψ(x) –inf
ψ(z) :z∈X/≤.
Kirk and Saliga [], p., say that for a Hausdorff space (X,τ) a mappingψ:X→R isτ-lower semicontinuous from above if given any sequence (xn)n∈NinX, the conditions: (ψ(xn))n∈Ndecreases toαandlimn→∞xn=x, yieldψ(x)≤α. The proof of Theorem . works without any change if we relax in such a way the assumption of lower semicontinuity ofψat points ofKerp(we may then say thatψis lower semicontinuous from above at the points ofKerp). Consequently, all results of the present section stay valid with this weaker assumption.
It should be noted that the dislocated (strong quasi-)metricpdefines the (quasi-)metric δin the following way:δ(x,x) = , andδ(x,y) =p(x,y) forx=y. One can see that a dislocated (strong quasi-)metric space (X,p) is -complete (Definition .) iff the (quasi-)metric space (X,δ) is complete (for the quasi-metric, see [], Definitions , ).
Consequently, if a proof of a fixed point theorem for complete metric spaces is based on a sequence (xn)n∈Nthat converges to a fixed point,δ(xn,xn+)= ,n∈N(andδ(xn,xn) = can be disregarded), then the same proof works for -completed-metric spaces, and further, for -complete partial metric spaces.
Another method is to prove that Z =Kerpis nonempty, f|Z :Z→Z, and to apply Lemma . (see comments on Theorems . and . []).
Competing interests
The author declares that he has no competing interests.
Acknowledgements
The author is obliged to Pascal Hitzler and Anthony Karel Seda for their old idea of dislocated metric (thankfully known to the referee), which is identical to the ‘new’ concept of near metric. That is why the former version of this manuscript entitled ‘Near (quasi-)metric and fixed point theorems’ (sent to FPTAppl in December 2014) had to be rearranged to the present form). The work has been supported by the Polish Ministry of Science and Higher Education.
Received: 21 December 2014 Accepted: 17 May 2015 References
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