R E S E A R C H
Open Access
A fixed point theorem for preordered
complete fuzzy quasi-metric spaces and an
application
Francisco Castro-Company
1, Salvador Romaguera
2*and Pedro Tirado
2*Correspondence:
2Instituto Universitario de
Matemática Pura y Aplicada, Universitat Politècnica de València, Valencia, 46022, Spain
Full list of author information is available at the end of the article
Abstract
We obtain a fixed point theorem for a type of generalized contractions on preordered complete fuzzy quasi-metric spaces which is applied to deduce, among other results, a procedure to show in a direct and easy fashion the existence of solution for the recurrence equations that are typically associated to Quicksort and Divide and Conquer algorithms, respectively.
MSC: 47H10; 54H25; 06A06; 68Q25
Keywords: fixed point; preordered fuzzy quasi-metric space; recurrence equation; algorithm
1 Introduction
In , Matkowski [, Theorem .] proved the following distinguished generalization of Banach’s contraction principle.
Theorem [] Let(X,d)be a complete fuzzy metric space and f :X→X a self-map such that
d(fx,fy)≤ϕd(x,y)
for all x,y ∈ X, where ϕ : [,∞) → [,∞) is a nondecreasing function satisfying
limn→∞ϕn(t) = for all t> .Then f has a unique fixed point.
Remark It is well known and easy to check that ifϕ: [,∞)→[,∞) is a nondecreasing function such thatlimn→∞ϕn(t) = for allt> , thenϕ(t) <tfor allt> .
Matkowski’s theorem has been generalized or extended in several directions (seee.g.[– ]). In particular, Jachymski obtained in [, Theorem ] the following nice fuzzy version of it.
Theorem [] Let(X,M,∗)be a complete fuzzy metric space,with∗a continuous t-norm of Hadžić type,and let f :X→X be a self-map such that
Mfx,fy,ϕ(t)≥M(x,y,t)
for all x,y∈X and t> ,whereϕ: [,∞)→[,∞)is a function satisfying <ϕ(t) <t and
limn→∞ϕn(t) = for all t> .Then f has a unique fixed point.
Remark See [] or [] for the notion of at-norm of Hadžić type (or ofh-type).
Recently, Ricarte and Romaguera [, Theorem .] established the following new fuzzy version of Matowski’s theorem by using a type of contraction introduced in the fuzzy in-tuitionistic context by Huanget al.[], and that generalizesC-contractions as defined by Hicks in [].
Theorem [] Let(X,M,∗)be a complete fuzzy metric space and f :X→X a self-map such that
M(x,y,t) > –t ⇒ Mfx,fy,ϕ(t)> –ϕ(t)
for all x,y∈X and t> ,whereϕ: [,∞)→[,∞)is a nondecreasing function satisfying
limn→∞ϕn(t) = for all t> .Then f has a unique fixed point.
In this paper we obtain a generalization of Theorem to preordered fuzzy quasi-metric spaces which is applied to deduce, among other results, a procedure to show in a direct and easy way the existence of solution for the recurrence equations that are typically as-sociated to Quicksort and Divide and Conquer algorithms, respectively. The key for this application is the nice fact that, for the specialization order of a fuzzy quasi-metric space, the contraction condition of Theorem is automatically satisfied whenever the self-map
f is nondecreasing for the specialization order andϕ: [,∞)→[,∞) is any function verifyingϕ(t) > for allt> (see Theorem in Section ).
2 Background
In this section we recall several notions and properties which will be useful in the rest of the paper. Our basic reference for quasi-metric spaces is [] and for fuzzy (quasi-)metric spaces they are [, ].
The lettersR,Nandωwill denote the set of real numbers, the set of positive integer numbers and the set of non-negative integer numbers, respectively.
A preorder on a nonempty setXis a reflexive and transitive binary relation onX. The preorder is called a partial order, or simply an order, if it is antisymmetric (i.e., conditionx yandy x, impliesx=y).
Note that for any nonempty setX, the binary relation tdefined byx tyif and only if x,y∈X, is obviously a preorder onX, the so-called trivial preorder onX.
A quasi-metric on a setXis a functiond:X×X→[,∞) such that for allx,y,z∈X: (i)d(x,y) =d(y,x) = if and only ifx=y, and (ii)d(x,y)≤d(x,z) +d(z,y).
A quasi-metric space is a pair (X,d) such thatXis a nonempty set anddis a quasi-metric onX.
Given a quasi-metricdonXthe functiondsdefined byds(x,y) =max{d(x,y),d(y,x)}for
allx,y∈X, is a metric onX.
Ifdis a quasi-metric onX, then the relation≤donXgiven by
x≤dy ⇔ d(x,y) = ,
According to [, ], a binary operation∗: [, ]×[, ]→[, ] is a continuoust-norm if∗satisfies the following conditions: (i)∗is associative and commutative; (ii)∗is contin-uous; (iii)a∗ =afor everya∈[, ]; (iv)a∗b≤c∗dwhenevera≤candb≤d, with
a,b,c,d∈[, ].
Two interesting examples are∧, and∗L, where, for alla,b∈[, ],a∧b=min{a,b}, and
∗Lis the well-known Lukasiewiczt-norm defined bya∗Lb=max{a+b– , }.
It seems appropriate to point out that∗ ≤ ∧for any continuoust-norm∗.
Definition [, ] A KM-fuzzy quasi-metric on a setXis a pair (M,∗) such that∗is a continuoust-norm andMis a fuzzy set inX×X×[,∞) (i.e., a function fromX×X×
[,∞) into [, ]) such that for allx,y,z∈X:
(KM) M(x,y, ) = ;
(KM) x=yif and only ifM(x,y,t) =M(y,x,t) = for allt> ; (KM) M(x,z,t+s)≥M(x,y,t)∗M(y,z,s)for allt,s≥; (KM) M(x,y,_) : [,∞)→[, ]is left continuous.
A KM-fuzzy quasi-metric (M,∗) onXsuch that for eachx,y∈X:
(KM) M(x,y,t) =M(y,x,t)for allt>
is a fuzzy metric onX(in the sense of Kramosil and Michalek []).
A simple but useful fact (seee.g.[, ]) is that for each KM-fuzzy quasi-metric (M,∗) on a setXand eachx,y∈X, the functionM(x,y, _) is nondecreasing.
In the following, KM-fuzzy quasi-metrics will be simply called fuzzy quasi-metrics. If (M,∗) is a fuzzy quasi-metric on a setX, then the pair (Mi,∗) is a fuzzy metric onX
whereMiis the fuzzy set inX×X×[,∞) defined byMi(x,y,t) =min{M(x,y,t),M(y,x,t)}
for allx,y∈Xandt≥.
As in the fuzzy metric case, each fuzzy quasi-metric (M,∗) on a setXinduces a topology
τM onXwhich has as a base the family of open balls{BM(x,ε,t) :x∈X, <ε< ,t> },
whereBM(x,ε,t) :={y∈X:M(x,y,t) > –ε}.
It immediately follows that a sequence (xn)n∈ωin a fuzzy quasi-metric space (X,M,∗) converges to a pointx∈Xwith respect toτMif and only iflimn→∞M(x,xn,t) = for all t> .
Definition A fuzzy (quasi-)metric space is a triple (X,M,∗) such thatXis a set and (M,∗) is a fuzzy (quasi-)metric onX.
Definition A preordered fuzzy (quasi-)metric space is -tuple (X,M, ,∗) such that
(X,M,∗) is a fuzzy (quasi-)metric space and is a preorder onX.
The notion of an ordered fuzzy (quasi-)metric space is defined in the obvious man-ner.
Remark If (M,∗) is a fuzzy quasi-metric onX, then the relation≤MonXgiven by
x≤My ⇔ M(x,y,t) = for allt> ,
It is interesting to note that, however, the relation≤Mpgiven by
x≤Mpy ⇔ M(x,y,t) = for somet> ,
is a preorder onX(seee.g.[, Remark ]).
We conclude this section with two typical examples of fuzzy quasi-metrics induced by a given quasi-metric space.
Example Let (X,d) be a quasi-metric space. Then the pair (M,∗) is a fuzzy quasi-metric onXwhere∗is any continuoust-norm andMis the fuzzy set onX×X×[,∞) given byM(x,y,t) = ift> andd(x,y) <t, andM(x,y,t) = otherwise.
Example [, ] (see also [, Example .]) Let (X,d) be a quasi-metric space. Then the pair (Md,∗) is a fuzzy quasi-metric onXwhere∗is any continuoust-norm∗andMd
is the fuzzy set onX×X×[,∞) given byMd(x,y, ) = for allx,y∈Xand
Md(x,y,t) = t t+d(x,y)
for allx,y∈Xandt> .
Remark Let (X,d) be a quasi-metric space. It is clear that the specialization orders≤d,
≤Mand≤Md coincide, while the preorder≤(M)pcoincides with the trivial preorder t
onX, and the preorder≤(Md)pcoincides with the specialization order≤Md.
3 The fixed point theorem and some of its consequences
We start this section with the notions of fuzzy quasi-metric completeness and continuity of self-maps which will be used in our main result (Theorem below).
A leftK-Cauchy sequence in a fuzzy quasi-metric space (X,M,∗) is a sequence (xn)n∈ω
inXsuch that for eacht> and eachε∈(, ) there isnε∈Nsuch thatM(xn,xm,t) > –ε
wheneverm≥n≥nε.
A preordered fuzzy quasi-metric space (X,M, ,∗) will be called -complete if for each nondecreasing leftK-Cauchy sequence (xn)n∈ωthere isx∈Xsuch that (xn)n∈ωconverges toxwith respect toτMi, andxn xfor alln∈ω.
Observe that if (X,M, t,∗) is a preordered fuzzy metric space which is t-complete, then (X,M,∗) is a complete fuzzy metric space in the usual sense (seee.g.[]).
Let (X,M, ,∗) be a preordered fuzzy quasi-metric space. A self-mapf:X→Xis said to be -nondecreasing if conditionx yimpliesfx fyfor allx,y∈X, and it will be called -continuous if whenever (xn)n∈ω is a nondecreasing sequence for , which converges with respect toτMito somex∈Xsuch thatxn xfor alln∈ω, then the sequence (fxn)n∈ω converges tofxwith respect toτMi.
Theorem Let(X,M, ,∗)be a preordered -complete fuzzy quasi-metric space and f :
X→X a -continuous nondecreasing self-map such that
for all x,y∈X with x y,and t> ,whereϕ: [,∞)→[,∞)satisfieslimn→∞ϕn(t) =
for all t> .If there is x∈X such that x fx,thenFix(f)=∅.
If,in addition,condition()is satisfied for each x,y∈Fix(f),then f has a unique fixed point.
Proof Taket> . Letx,y∈Xwithx y. SinceM(x,y,t) > –t, it follows that
Mfx,fy,ϕ(t)> –ϕ(t). ()
Sincef is nondecreasing, we deduce thatfnx fnyfor alln∈ω, so, by () and (), we
immediately deduce that
Mfnx,fny,ϕn(t)> –ϕn(t) ()
for allx,y∈Xwithx yandn∈ω.
Now letx∈Xsuch thatx fx. Putxn=fnxfor alln∈ω. Sincef is nondecreasing it follows that (xn)n∈ωis a nondecreasing sequence for .
Similarly to the proof of [, Theorem .], we show that (xn)n∈ω is a leftK-Cauchy
sequence in (X,M,∗). Choose ε∈(, ) and t> . Then there exists nε∈Nsuch that
ϕn(t) <min{ε,t}for alln≥nε. Letm>n≥nε. Thenm=n+kfor somek∈N, so, by ()
M(xn,xm,t) =M
fnx,fnfkx,t≥Mfnx,fnfkx,ϕn(t) > –ϕn(t) > –ε.
Therefore (xn)n∈ω is a nondecreasing leftK-Cauchy sequence in (X,M,∗). Since (X,M, ,∗) is -complete there existsz∈Xsuch thatlimn→∞Mi(x
n,z,t) = for allt> , and xn zfor alln∈ω. Solimn→∞Mi(xn,fz,t) = for allt> , by -continuity off. Hence z=fz.
Finally suppose thatu=fufor someu∈X. From our hypothesis it follows, exactly as in the first part of the proof, that
Mfnz,fnu,ϕn(t)> –ϕn(t)
for alln∈ω. Sincefnz=z andfnu=uwe deduce, fort> ,ε∈(,t) andn∈Nwith
ϕn(t) <ε, that
M(z,u,t)≥M(z,u,ε)≥Mz,u,ϕn(t)
> –ϕn(t) > –ε.
Sinceεis arbitrary, we conclude thatM(z,u,t) = for allt> . SimilarlyM(u,z,t) = for
allt> , sou=z. This completes the proof.
Corollary Let(X,M, ,∗)be a preordered -complete fuzzy quasi-metric space and f :X→X a -nondecreasing self-map such that
for all x,y∈X with x y,and t> ,whereϕ: [,∞)→[,∞)satisfies <ϕ(t) <t and
limn→∞ϕn(t) = for all t> .If there is x∈X such that x fx,thenFix(f)=∅.
If,in addition,condition()is satisfied for each x,y∈Fix(f),then f has a unique fixed point.
Proof We shall show thatfis -continuous in (X,M, ,∗). Let (xn)n∈ωbe a nondecreasing
sequence for , convergent with respect toτMi to somex∈X such thatxn xfor all n∈ω. Givenε∈(, ) there isnε∈Nsuch thatMi(x,xn,ε) > –ε. By condition () and
the fact that <ϕ(ε) <εit follows thatMi(fx,fxn,ε) > –εfor alln≥nε. Thereforef is
-continuous. The conclusions follow from Theorem .
As an immediate consequence of Corollary we obtain the following improvement of Theorem .
Corollary Let(X,M,∗)be a complete fuzzy metric space and f :X→X a self-map such that
M(x,y,t) > –t ⇒ Mfx,fy,ϕ(t)> –ϕ(t)
for all x,y∈X and t> ,whereϕ: [,∞)→[,∞)satisfies <ϕ(t) <t andlimn→∞ϕn(t) = for all t> .Then f has a unique fixed point.
Proof It is clear that (X,M, t,∗) is a preordered t-complete fuzzy metric space. Now
the result follows from Corollary .
The contraction () is almost trivially satisfied in the case that the preorder is the spe-cialization order≤Mandf is nondecreasing for≤M. This situation, which will be crucial
in our application in Section , is described in the following result.
Theorem If the ordered fuzzy quasi-metric space(X,M,≤M,∗)is≤M-complete and f : X→X is a≤M-nondecreasing self-map such that there is x∈X satisfying x≤Mfx,then f has a fixed point.
Proof Letϕ: [,∞)→[,∞) be any function satisfying <ϕ(t) <tandlimn→∞ϕn(t) =
for allt> . Then, for eachx,y∈Xsuch thatx≤My, we havefx≤Mfy, and thus
Mfx,fy,ϕ(t)= > –ϕ(t)
for allt> . This shows that condition () is satisfied, and thus f has a fixed point by
Corollary .
We conclude this section with two examples illustrating the obtained results.
Example LetX= [,∞) and letMbe the fuzzy set inX×X×[,∞) defined as
M(x,y, ) = for allx,y∈X,
M(x,y,t) = ifx≤yandt> and
It is routine to check that (M,∧) is a fuzzy quasi-metric. Note that the specialization or-der≤Mcoincides with the usual order≤onX. Moreover, a sequence inXis leftK-Cauchy
in (X,M,∧) if and only it is eventually constant, so (X,M,≤,∧) is an ordered≤-complete fuzzy quasi-metric space.
Now letϕ: [,∞)→[,∞) defined as
ϕ(t) =t/ if ≤t≤/,
ϕ(t) = ift> / witht= and
ϕ() = /.
Clearlylimn→∞ϕn(t) = for allt> . We show that iff :X→Xis any≤-nondecreasing
self-map, the contraction condition () is satisfied forx,y∈Xwithx<yandt> . (Note thatf is automatically continuous and hence≤-continuous, becauseτMi is the discrete
topology onX.)
Letx≤yandt> . Thenfx≤fyandϕ(t) > , so
Mfx,fy,ϕ(t)= > –ϕ(t),
so condition () is trivially satisfied. If, in addition, there existsx∈Xsuch thatx≤fx, thenf has a fixed point by Theorem or Theorem . Observe that in this example we cannot apply Corollary becauseϕ(t) >tfor / <t< ; in fact, it is not nondecreasing.
Example LetX={a,b,c}and let (M,∧) be the fuzzy metric onXdefined asM(x,x,t) = for allx∈X,M(a,b,t) =M(b,a,t) = / for allt∈(, ],M(a,b,t) =M(b,a,t) = for allt> , andM(x,y,t) = otherwise. Now define a preorder onXas follows:x xfor allx∈X,
a b, andb a. Obviously is not an order onX. Clearly (X,M, ,∗) is -complete. Let
f :X→Xbe such thatfa=a,fb=aandfc=c. Thenf is a -nondecreasing self-map witha fa(alsoc fc). Letϕ: [,∞)→[,∞) be any function such that <ϕ(t) <t
andlimn→∞ϕn(t) = for allt> . SinceM(fa,fb,ϕ(t)) =M(a,a,ϕ(t)) = for allt> , we deduce that the conditions of Corollary , and hence of Theorem , forx,y∈Xwithx y, are satisfied. However, we cannot apply Corollary to this example. Indeed, suppose that there exists a functionϕ: [,∞)→[,∞) satisfying <ϕ(t) <tandlimn→∞ϕn(t) = for allt> , and such that
M(a,c,t) > –t ⇒ Mfa,fc,ϕ(t)> –ϕ(t)
for allt> . Since conditionM(a,c,t) > –tholds fort> , we deduce that, fort> ,
M(fa,fc,ϕ(t)) > –ϕ(t), soϕ(t) > becauseM(fa,fc,ϕ(t)) =M(a,c,ϕ(t)) = . Repeating this argument, we deduce that ϕn(t) > for all n∈ω, which contradicts the fact that limn→∞ϕn(t) = .
4 An application
Let us recall that Schellekens introduced in [] the so-called complexity quasi-metric space in order to construct a topological foundation for the complexity analysis of pro-grams and algorithms. In that paper, he also applied his theory to show that the existence and uniqueness of solution for the recurrence associated to Divide and Conquer algo-rithms. Further contributions to the study of these spaces and of other related ones may be found in [, –],etc.
The complexity (quasi-metric) space (see []) consists of the pair (C,dC), where
C=
f :N→(,∞] : ∞
n= –n
f(n)<∞
,
anddCis the quasi-metric onCgiven by
dC(f,g) = ∞
n= –n
g(n)–
f(n)
∨
for allf,g∈C. (We adopt the convention that ∞ = .) The elements ofCare called com-plexity functions anddC is said to be the complexity quasi-metric. Observe that
dC(f,g) = ⇔ f(n)≤g(n) for alln∈N,
and thus conditiondC(f,g) = , can be computationally interpreted asf to be ‘more effi-cient’ thangon all inputs (see [, Section ]).
In our context we shall work on the subsetCofCdefined as
C= f ∈C:f(n)≥ for alln∈N
,
and we shall use the functionQCintroduced in [] and defined, for eachf,g∈Candt> , as
QC(f,g,t) = ∞
k=n
–k
g(k)–
f(k)
∨
,
wheret∈(n– ,n],n∈N.
The following well-known facts will be useful.
Remark [, Remark ] dC(f,g) =QC(f,g,t) for allf,g∈Candt∈(, ].
Remark QC(f,g,t) < wheneverf,g∈Candf=g.
Remark [, Lemma ] Let (fn)k∈ωbe a sequence inCsuch thatfn≤fn+ for alln∈ω, and letF∈Cdefined asF(k) =sup{fn(k) :n∈ω}for allk∈N. Thenlimn→∞(dC)s(F,f
n) = .
In the following, for anyf,g∈C, byf ≤gwe mean thatf(n)≤g(n) for alln∈N. Now we construct a fuzzy setMinC×C×[,∞) as
M(f,g, ) = for allf,g∈C and
Note thatM(f,g,t) = for allt> if and only iff ≤g. Then we have:
Lemma (C,M,≤M,∗L)is a≤M-complete fuzzy quasi-metric space.
Proof We first see that (M,∗L) is a fuzzy quasi-metric onC(recall thata∗Lb=max{a+ b– , }for alla,b∈[, ]). Indeed, conditions (KM), (KM) and (KM) of Definition are almost trivially satisfied, while condition (KM) follows immediately from the fact showed in [, Lemma ] that
QC(f,g,t+s)≤QC(f,h,t) +QC(h,g,s)
for allf,g∈Candt,s> . Hence (C,M,∗L) is a fuzzy quasi-metric space.
Now let (fn)n∈ωbe a nondecreasing leftK-Cauchy sequence in (C,M,≤M,∗L). Then
fn≤fn+for alln∈ω, so, by Remark , lim
n→∞dC(F,fn) =nlim→∞dC(fn,F) = . ()
Sincefn≤Ffor alln∈ω(note thatF∈C), we deduce that
M(fn,F,t) = ()
for alln∈ωandt> . On the other hand, and assuming thatF=fnfor alln∈ω, we deduce
from Remark and () that
lim
n→∞M(F,fn,t) =nlim→∞
–QC(F,fn,t)
= lim
n→∞
–dC(F,fn,t)
= ()
for allt∈(, ], and thus for allt> . By () and () we deduce that (fn)n∈ω converges toF with respect toτMi. We have shown that (C,M,≤M,∗L) is a≤M-complete fuzzy
quasi-metric space.
Taking into account the preceding constructions and results, we immediately obtain the following consequence of Theorem .
Theorem If:C →C is a≤M-nondecreasing map and there is f∈C such that
f≤Mf,thenhas a fixed point.
We finish the paper by applying Theorem to show the existence of solution for the recurrence equations associated to Quicksort and Divide and Conquer algorithm, respec-tively.
Example Consider the recurrence equationT given byT() = , and
T(n) =(n– )
n +
n+
n T(n– )
In this case, we define a functional:C→Cas
f() =∞, f() = ,
f(n) = (n– )
n +
n+
n f(n– )
for alln> , wheref∈C(see for instance [, ]).
It is clear that iff ≤gthenf ≤g, so by the definition ofM,is nondecreasing for ≤M. Moreover, the complexity functionf∈Cdefined asf(n) = for alln∈N, satisfies
f≤f,i.e.,f≤Mf. Therefore, we can apply Theorem and thushas a fixed point
g∈C. Consequently, the functionh:N→[,∞) given byh() = andh(n) =g(n) for alln> is solution of the recurrence equationT.
Example It is well known that Divide and Conquer algorithms solve a problem by re-cursively splitting it into subproblems each of which is solved separately by the same algo-rithm, after which the results are combined into a solution of the original problem (seee.g.
[, ]). Thus, the complexity of a Divide and Conquer algorithm typically is the solution of the recurrence equation given by
T() =c and
T(n) =aT
n b
+h(n),
wherea,b,c∈Nwitha,b≥,nranges over the set{bp:p= , , , . . .}andh(n) <∞if n∈ {bp:p= , , , . . .}.
This recurrence equation induces, in a natural way the associated functional:C→C defined by f() =c,f(n) =af(nb) +h(n), ifn∈ {bp :p= , , , . . .}andf(n) =∞ if n∈ {/ bp:p= , , , . . .}.
As in Example ,is≤M-nondecreasing. Since the complexity functionf∈Cdefined asf(n) = for alln∈N, satisfiesf≤Mf, we can apply Theorem and thushas a fixed pointg∈Cwhich is solution of the recurrence equationT.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally in writing this article. They read and approved the final manuscript.
Author details
1Gilmation SL, Calle 232, 66 La Cañada, Paterna, 46182, Spain.2Instituto Universitario de Matemática Pura y Aplicada,
Universitat Politècnica de València, Valencia, 46022, Spain.
Acknowledgements
The second and third named authors thank the supports of the Universitat Politècnica de València, grant
PAID-06-12-SP20120471, and the Ministry of Economy and Competitiveness of Spain, grant MTM2012-37894-C02-01.
Received: 18 November 2013 Accepted: 19 February 2014 Published:27 Mar 2014
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