R E S E A R C H
Open Access
Fixed points of set-valued maps in locally
complete spaces
Carlos Bosch
1*, César L García
1, Thomas Gilsdorf
2, Claudia Gómez-Wulschner
1and Rigoberto Vera
1*Correspondence: [email protected] 1Departamento de Matemáticas,
ITAM, Río Hondo 1, Ciudad de México, 01080, México Full list of author information is available at the end of the article
Abstract
We prove an extension of the Pareto optimization criterion to locally complete locally convex vector spaces to guarantee the existence of fixed points of set-valued maps.
MSC: Primary 47H10; 47H04; secondary 46N10; 46A03
Keywords: fixed points; set-valued map; Pareto optimization
1 Introduction
Ekeland’s variational principle, a milestone in the theory of nonlinear optimization, fo-cuses on solving an optimization problem via a perturbed optimization problem. Since its appearance many extensions and equivalent formulations have been shown. Some of them, related to our discussion below, are contained in [–]. In [] Azé and Corvellec gen-eralized, in the setting of metric spaces, a result due to Lim ([]) on the existence of fixed points for weakly inward multivalued contractions, defined on a nonempty closed subset of a Banach space. Their argument is remarkably simple: on the one hand, it avoids the use of transfinite induction (as in Lim’s paper) and, on the other hand, it uses Ekeland’s variational principle as a main tool to guarantee the existence of fixed points. Ekeland’s principle has been shown equivalent to other optimization statements, in particular, and of interest in this paper, is the equivalence to an optimization criterion of Pareto (see []) for which the existence of optima (critical points for dynamical systems) has been shown in complete metric spaces (see []). Our main result (Theorem .) is a modification of that optimization criterion of Pareto, which shows how the existence of fixed points for set-valued maps can be extended to the setting of locally complete spaces. Among the consequences of Theorem ., we provide a simple argument of Azé and Corvellec’s the-orem ([], Thethe-orem .). Although our results are set in the context of locally complete spaces, let us say that related to Ekeland’s type variational principles, some other forms of completeness have been use in the literature (e.g., quasi-metric spaces withQ-functions, fuzzy metric spaces, or sequentially lower complete spaces). Some of these include equiv-alences to Ekeland’s variational principle, for instance, to Caristi fixed point theorem or Takahashi minimization theorem, all of them with their natural connection to solution to equilibrium or fixed point theorems for set-valued maps (see, for instance [, –]).
2 Preliminaries
Throughout this paper (X,τ) will denote a Hausdorff locally convex space where the topol-ogyτ is generated by a saturated family of seminorms{ρj}j∈J. IfBis a subset ofXwhich
is balanced and convex, we will callBadisk. LetXB be the linear span ofB, endowed with the topology generated by the Minkowski gauge ofB,ρB. WhenBis boundedρBis
a norm, and the norm topology is finer than the topology inherited fromX. If (XB,ρB) is
a Banach space we say thatBis aBanach disk. We say thatXis alocally completespace if each closed, bounded disk is a Banach disk. Local complete spaces are also known asc∞or
convenient spaces. The notion of local completeness has become important in all sorts of applications, for instance, in nonlinear distribution theory (see [, ] and the references therein), or as a context where existence and uniqueness for nonlinear integro-differential equations can be shown (see []). Bylscwe refer to a lower semicontinuous functions,
f:X→R∪ {∞}, which areproper, that is, their effective domain,dom(f) :={x:f(x) <∞}, is nonempty.
For a closed subsetA⊂Xwe will consider set-valued mapsT:A→A. Such mapsT
are known in the literature asdynamical systemsand, in our discussion, points of interest arex∗∈Asuch thatx∗∈Tx∗(known asfixed pointsofT) and, especially,critical points
ofT(pointsx∗∈Asuch thatTx∗={x∗}). Let us start with the definition and properties of some functions that will be used in the sequel. For eachj∈J,φj,φj: X→[,∞) will be
the functions defined by
φj(x) =ρj-diam(Tx) =sup
ρj
y–y:y,y∈Tx ()
and
φj(x) =ρj-dist(x,Tx) =inf
ρj(x–y) :y∈Tx
. ()
Also, for eachj∈J, and for every pair of subsetsA,B⊂X, we define
ej(A,B) =supρj(x–B) :x∈A
, ()
whereρj(x–B) =inf{ρj(x–b) :b∈B}, is theρj-distance fromxtoB. In general, we define
ρj(A–B) =inf
ρj(a–B) :a∈A
=infρj(b–A) :b∈B
=ρj(B–A),
whereρj(a–B) denotes theinf{ρj(a–b) :b∈B}andρj(b–A) =inf{ρj(b–a) :a∈A}.
It is important to note that, in general,ej(Tx,Tz)=ej(Tz,Tx). For instance, takeX=R
equipped with the usual Euclidean normdand consider the subsetsA={(a, ) : –≤a≤
}andB={(x,y) :x+y≤}. An easy computation shows thated(A,B) =√ – while
ed(B,A) = .
Now we provide sufficient conditions on the functionsejto get lower semicontinuity and continuity properties for the functionsφjandφj. We start with the following.
Proposition . If Mj∈R+ is such that ej(Tx,Tz)≤Mjρj(x–z)for every x,z∈X then
Proof Letx,z∈Xandy,y∈Tx. Then, for eachn∈Nchoosewn∈Tzsuch that
ρj(y–Tz) <ρj(y–wn) <ρj(y–Tz) +
n,
and similarly forychoose the correspondingwn∈Tz. Then
ρj
y–y≤ρj(y–wn) +ρj
wn–wn+ρj
wn–y
<ρj(y–Tz) +ρj
wn–wn
+ρj
y–Tz+
n
≤φj(z) + ej(Tx,Tz) +
n.
Thus,ρj(y–y)≤φj(z) + ej(Tx,Tz). By taking the supremum over ally,y, we obtain
φj(x)≤φj(z) + ej(Tx,Tz)≤φj(z) + Mjρj(x–z).
A similar argument shows that
φj(z) –φj(x)≤Mjρj(z–x)
also holds. Henceφjisρj-uniformly continuous.
For the functionφj, the result corresponding to Proposition . is as follows.
Proposition . If Mj∈R+is such that ej(Tx,Tz)≤Mjρj(x–z)for all x,z∈X thenφjis
ρj-continuous henceτ-continuous.
Proof Letx∈Xand{xλ}be a net such thatxλ ρj
→x. Forz∈Txλwe have
φj(x) =ρj(x–Tx)≤ρj(x–xλ) +ρj(xλ–Tx)
≤ρj(x–xλ) +ρj(xλ–z) +ej(Txλ,Tx)
≤( +Mj)ρj(x–xλ) +ρj(xλ–z).
By taking the infimum with respect toz∈Txλwe get
φj(x)≤( +Mj)ρj(x–xλ) +ρj(xλ–Txλ) = ( +Mj)ρj(x–xλ) +φj(xλ).
Thus
φj(x) –φj(xλ)≤( +Mj)ρj(xλ–x).
Similarly,
φj(xλ) –φj(x)≤( +Mj)ρj(xλ–x),
Definition . Forj∈J,φj: X→[,∞) will be the function defined byφj(x) =φj(x) +
φj(x).
Lemma . Let A⊂X be any closed subset,and suppose thatφjandφjareρj-lsc.Suppose that T: A→Ais such that Tx isρj-sequentially compact and for each x∈A and each n∈Nthere exists yn∈Tx with the following properties:φj(yn)≤φj(x) +n andρj(x–yn) <
φj(x) +n.Then there exists y∈Tx such thatφj(y) +ρj(x–y)≤φj(x).
Proof Let{yn} ⊂Txbe a sequence such that eachynsatisfies the conditions as in the state-ment of Lemma .. SinceTxisρj-sequentially compact, there is a subsequence{ynk}k∈N,
ρj-convergent to some, not necessarily unique,y∈Tx. Then
φj(ynk) +ρj(x–ynk)≤φj(x) +
nk +φj(x) +
nk =φj(x) +
nk.
By takinglim infoverkon both sides of the inequality we have
φj(y) +ρj(x–y)≤φj(x)
as desired.
To close this section we define the sets that we will consider as target values for the dynamical systems in Theorem .. Note that Lemma . is tailored to provide conditions for these sets to be nonempty.
Definition . For eachx∈Xand each indexj∈Jlet
Cxj=y∈Tx:φj(y) +ρj(x–y)≤φj(x)
.
Lemma . Ifφjis lsc then the set Cjxisρj-closed.
Proof Let (yn) be a sequence inCjxsuch thatyn
ρj
→y∈X. Note thatφj(yn) +ρj(x–yn)≤φj(x) implies that
φj(y) +ρj(x–y)≤lim infφj(yn) +limρj(x–yn)≤φj(x).
Hencey∈Cjx.
Corollary . Ifφjis lsc and for each y∈Cxjand y+Zρj⊂C
j
x,where Zρj={x∈X:ρj(x) = }
is the zero set ofρj,then Cjxis aρj-closed linear subspace of X.
Definition . LetA⊂Xbe any closed subset. IfT:A→Ais a dynamical system and
ρj is one of the seminorms defining the topologyτ,x∗∈Ais aρj-critical point ofT if Tx∗⊂x∗+Zρj.
3 Modified Pareto case
Recall that the topologyτ in the lcsX is generated by a saturated family of seminorms {ρj:j∈J}. Theorem of [] provides with sufficient conditions for a given dynamical
systemT:A→A, over a locally complete subsetAofX, to have critical points. In [],
Theorem , one essentially sees that if for every x∈Aand everyu∈Tx, cjρj(x–u)≤
(x) –(u) for everyj∈J and for some function: A→Rlsc and bounded below (cj
are positive scalars such thatj∈J{x∈X:cjρj(x)≤}is a nonzero Banach disk) thenT
has a critical point. In our setting, if we suppose that, for a fixedj∈J,Aisρj-complete
and: A→Risρj-lsc, that is,ρj(xn–x)→ implies(x)≤lim infn(xn) (which in turn
impliesτ-lsc of) then we have the following proposition.
Proposition . If for each x∈A and each u∈Tx we haveρj(x–u)≤(x) –(u),then T has aρj-critical point,x∗j ∈A.
The proof goes along the same lines as that of Theorem in [] and we omit it.
Theorem .(Main Result) Let T: A→A be a dynamical system and for each j∈J let Tj:A→Abe the dynamical system defined by Tj(x) =Cjx.Suppose that the following conditions are satisfied:
. The seminormρj,from the family of seminorms defining the topologyτ,is such that the subsetAisρj-sequentially complete.
. The functionφj(see definition.)satisfies the conditionφj(x)≤lim infφj(xn)
wheneverρj(x–xn)→. . Cxj=∅for allx∈A.
Then there exists x∗j ∈A such that Tjx∗j ⊂x∗j +Zρj,that is,x∗j is aρj-critical point of the function Tj.
Proof The hypotheses of this theorem are essentially the same as those of Proposition ., withT=Tj. The fact that the functionφjisρj-lsc implies that the setTjx=Cxjisρj-closed
for each vectorx∈A. Then, for ally∈Tjx=Cjx,
y+Zρj={y}
ρj
⊂Tjx.
On the other hand, the fact thatAisρj-sequentially complete implies that the setTjx= Cxj is alsoρj-sequentially complete. Hence, by Proposition ., we see that there exists x∗j ∈Asuch that
Tjx∗j ⊂x∗j +Zρj
as desired.
Corollary . If x∗j ∈Tjx∗j then Tjxj∗=x∗j +Zρj.
A weak version (in the sense of weak topology) of Theorem . can be obtained as fol-lows. Consider X= (X,τ), the topological dual ofX, and the saturated family of semi-norms{ρf :f ∈X}, whereρf(x) =|f(x)|andZf =ker(f). Assume as before thatA⊂Xis
ρf-complete for some fixed, but arbitrary,f∈X, and let: A→Rbe a lsc function. Then
Proposition . If each x∈A and each u∈Tx satisfies|f(x–u)| ≤(x) –(u),then the dynamical system T has aρf-critical point x∗f ∈A,that is,
Tx∗f ⊂x∗f +Zρf =x
∗ f +Zf.
We also have the following restatement of Theorem ., where the continuous linear functionsf∈Xtake the place of the indicesj∈J.
Theorem . For each f ∈X,let Tf: A→Adenote the function given by Tf(x) =Cxf. Suppose there exists a seminorm,ρf(x) =|f(x)|with f ∈X,such that:
. The functionφf =|f|satifiesφf(x)≤lim infφf(xn) =lim infn|f(xn)|(lower
semicontinuity). . Cxf =∅for allx∈A.
Then there exists x∗f ∈A such that Tfx∗j ⊂x∗f +Zf.
Proof Just notice that if we consider the weak topology in place ofτ onX, then we have
the same hypotheses as in Theorem . above.
4 Applications to fixed point theory
With the tools we have developed so far we can present a couple of interesting applications to fixed point theory. Let (X,τ) be a complete locally convex topological vector space and takef ∈X,A⊂Xaτ-closed subset, and <M< . Leth: A→Abe any function such that
fh(x)–fh(z)≤Mf(x) –f(z) for allx,z∈A. ()
Theorem . Under the above assumptions there exists x∗∈A such that,for all k∈N,
fx∗=fhx∗=fhkx∗. ()
Equivalently,for all k∈N,ρf(x∗–hk(x∗)) = .Moreover,if x∗∗∈A also satisfies()then f(x∗) =f(x∗∗).
First note that Theorem . fails forM= . Indeed, in [] Khamsi provides an example [], Example , to answer Kirk’s problem (in the negative) on the existence of fixed points for a mapT:X→Xsuch that, for allx∈Xand for some positive functionη,η(d(x,Tx))≤ φ(x) –T(φ(x)) (φ is a non-negative lsc function). Here we see that for M= andA= {x,x, . . .} ⊂X=Rwithxn= ++· · ·+n; if we takef =I∈(R)we have the inequality
h(xn) –h(xm)≤ |xn–xm| for alln,m∈N
and then the functionh(xn) =xn+does not have a fixed point. Note that the setAis closed
but it is not bounded.
Also note that ifA=Xthen as a consequence of inequality () we see thathˆ: X/Zf → X/Zf, such thathˆ([x]) = [h(x)] is a function. Furthermore, ifA=Xand the identities in ()
Proof of Theorem.. LetT:A→Adefined byTx={x,h(x),h(x), . . .}. Then we have
fh(x) –h(z)=fh(x)–fh(z)≤Mf(x) –f(z)=Mf(x–z)
for everyx,z∈A.
Consideref:{Tx:x∈X} × {Tz:z∈X} →[,∞) given by
ef(Tx,Tz) =supf(y–Tz):y∈Tz
=supρf(y–Tz) :y∈Tx
.
Observe that sinceTx⊃Th(x)⊃Th(x)⊃ · · · we haveef(Thk+m(x),Thk(x)) = for all k,m∈N. NowTxis not necessarily closed or bounded, however, ifk≤m,
fhk(x)–fhm(z)≤Mkf(x) –fhm–k(z).
Thus,
inf m≥k
fhkx–fhmz≤ inf m≥k
Mkf(x) –fhm–kz
≤Mkf(x) –f(z)
=Mkf(x–z).
As a consequence we see that the function ef satisfiesef(Tx,Tz)≤M|f(x–z)|for all x,z∈A.
Note that for the given functionhand the seminormρfwe can use Proposition . to see
that the functionφf(x) =diamfTxis Lipschitz and thusρf-uniformly continuous. Also,
sincex∈Tx,φf(x) = andφf(x) =φf(x). Thereforeφf isρf uniformly continuous and
satisfies condition () in Theorem .. Observe that sincex∈Txwe have trivially from Lemma . thatφf(x) +ρf(x–x) =φf(x). Alsox∈Cxf={y∈Tx|φf(y) +ρf(x–y)≤φf(x)},
that is,Cfx=∅, which is condition () in Theorem ..
By Lemma ., and since ρf is continuous, we see that Cxf is ρf-closed, thus ρf
-sequentially complete sinceXisρf-sequentially complete. By Theorem . we conclude
that, for the map Tf: A→A, defined via Tf(x) =Cf
x, there exists x∗ ∈A such that Tf(x∗)⊂x∗+Zf.
Now, sincex∗∈Tf(x∗), by Corollary . we see thatTf(x∗) =x∗+Zf. Hence, for eachy∈ Tx∗there existsz∈Zf(f(z) = ) such thaty=x∗+z. From which we obtaindiamfTx∗= .
That is,f(x∗) =f(hk(x∗)) for allk∈N. Equivalentlyρf(x∗–hk(x∗)) = for allk∈N.
As for the uniqueness, ifx∗∗∈Xis such thatf(x∗∗) =f(hk(x∗∗)) for allk∈N, we would
have
fx∗–x∗∗=fhx∗–fhx∗∗≤Mfx∗–fx∗∗=Mfx∗–x∗∗.
Thus ≤M, which contradicts our hypothesis. This concludes Theorem ..
f(x–u) andf(u–hx) are both positive numbers (or both negative) wherehis a function satisfying the conditions of Theorem ..
All we did in the previous paragraphs can be repeated if we change the weak topology for a generic Hausdorff topology for a lcs (X,τ). Indeed, if we use a fixed seminormρjamong
those that generate the topologyτ, then we have the following result corresponding to Proposition ..
Proposition . Let(X,τ)be a complete locally convex topological vector space,takeρj a seminorm, from those that generate the topologyτ.Let A⊂X a closed subset and let
<M< .Take h: A→A defined as a function not necessarily linear or continuous and such that forρjwe have
ρj
h(x) –h(z)≤Mρj(x–z) for all x,z∈A. ()
Then there exists x∗∈A such that
x∗–hkx∗∈Z
ρj for all k∈N. ()
Equivalently
ρj
x∗–hx∗=ρj
x∗–hkx∗= for all k∈N.
If x∗∗∈A also satisfies()we have,ρj(x∗–x∗∗) = .
A consequence of () forA=Xis that the relationhˆ:X/Zρj→X/Zρjdefined byhˆ([x]) =
[h(x)] is a function. A consequence of () forA=X, is that the functionhˆhas a unique fixed point.
Proof of Proposition. To prove Proposition . let us define the functionT:X→X
viaTx={x,h(x),h(x), . . .}.
Then we have ρj(h(x) –h(z))≤Mρj(x–z) for all x,z∈A. Note that Tx⊃Th(x)⊃ Th(x)⊃ · · ·, which means thate
f(Thk+m(x),Thk(x)) = for allk,m∈N. Recall thatTx
is not necessarily closed but it is bounded. Furthermore, ifk≤m,
ρj
hk(x) –hm(z)≤Mkρj
x–hm–k(z),
and then
inf m≥k
ρj
hkx–hmz≤inf m≥k
Mkρ j
x–hm–kz≤Mkρ j(x–z).
The trick now is to follow the argument in Theorem . but replacing the seminorm ρf by the seminormρjin order to getx∗∈Xsuch that, by Theorems ., ., and
Corol-lary .,Tx∗ρj=x∗+Zρj. Finally, sinceˆ∈Zρj we obtainx∗∈Tx∗. Uniqueness is proved in
the same way as in Theorem ..
Note that in Theorem . if we hadx∗∈Tx∗ andx∗∗∈Tx∗∗, that is,x∗=hk(x∗) and
would have, for eachn∈N,x∗=hnk(x∗), andx∗∗=hnj(x∗∗). Thus, forj≤k,
fx∗–x∗∗=f(hnkx∗–hnjx∗∗≤Mnjfhnk–njx∗–x∗∗.
Since{hn(k–j)(x∗)|n∈N}is bounded, since it is finite with at mostkpoints, and <M< ,
the sequence{Mnj|f(hn(k–j)(x∗) –x∗∗)|}converges to asn→ ∞; in other words,|f(x∗–
x∗∗)|= hence|f(h((x∗)) –h(x∗∗))|= . We conclude thatρf(x∗–x∗∗) = =ρf(h(x∗) – h(x∗∗)). In general,|f(hnj(x∗) –hnj(x∗∗))|= =|f(hnk(x∗) –hnk(x∗∗))|.
As another application of our results, we show how the Azé and Corvellec theorem ([], Theorem .) follows easily from Theorem .. Azé and Corvellec defined the set of fixed points forT asFT={x∈X:x∈Tx}. For the seminormsρjthe set of fixed points
of T is described asFTj ={x∈X|ρj(x–Tx) = }or in the case ofρf, the set of fixed
point forTisFTf ={x∈X| |f(x–Tx)|= }. Note that this sets areρj-closed (respectively,
ρf-closed). In [], Azé and Corvellec proved that for allx∈X, ρj(x–FTj)≤ρj(x–Tx)
(|f(x–FTf)| ≤ |f(x–Tx)|). We discuss the case forρf (the case forρjis analogous). Let
Af =x∈X|fx–FTf>f(x–Tx)=φf(x)
.
We can prove, under the conditions of Propositions . and ., which we assume, that
Af =∅. Indeed, define the functionψf: (X,d)→Rby
ψf(x) =f
x–FTf–f(x–Tx)=fx–FTf–φf(x).
Then we have
Af =x∈X:ψf(x) >
=ψf–(,∞).
Thus if the functionψf is lsc thenAf is an open set, andAf ∩FTf =∅. Consider the
functionγf(x) =|f(x–FTf)|=inf{|f(x–z)|:z∈F f
T}, which is always continuous. Then,
under the conditions of Propositions . and ., the functionφf is continuous. Thus,
we see that the functionψf(x) =γf(x) –φf(x) is continuous, giving us thatAf is open. If Af =∅, takex∈Af andr> such that
Dfr(x) =z∈X:f(z–x)≤r⊂Af.
Note thatDfr(x) isρf-closed andρf-bounded.
If we defineT:Dfr(x)→Xto beTx=Tfx,T=Tf|Dfr(x), thenT
fsatisfies the
hypothe-ses of Theorem .. Thus, there existsz∗∈Dfr(x)⊂Afsuch thatz∗∈Tz∗, in other words, z∗∈Af∩FTf =∅and the setAf must be empty.
As as additional consequence (and easy example) of Proposition . one can get well known results such as the contraction mapping theorem. Indeed, if (X,ρ) is a normed space andAis a nonempty compact subset ofX then any contractionf: A→Ahas a fixed point. This follows from Proposition . by lettingT:A→AasTx={f(x)}and
(x) =
∞
k=
ρfk(x) –fk+(x)= –cρ
where <c< is the contraction constant forf. It is easy to show thatis lsc (in fact continuous) and thatρ(x–f(x)) =(x) –(f(x)). HenceTmust have critical points which are, by definition ofT, fixed points forf.
5 Conclusions
The paper can be considered as an extension of the Pareto optimization criterion to lo-cally complete lolo-cally convex vector spaces [] with some applications to fixed points. Local completeness is a very weak completeness property. This type of spaces is becom-ing the convenient settbecom-ing for several applications. Here in this settbecom-ing we get a fixed point theorem and as an application we obtain the results of Azé and Corvellec’s [].
Acknowledgements
The authors were partially supported by the Asociación Mexicana de Cultura, A.C. Research by T. Gilsdorf was done while on the faculty at the ITAM. The authors gratefully acknowledge all the remarks and suggestions made by the anonymous referees.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Author details
1Departamento de Matemáticas, ITAM, Río Hondo 1, Ciudad de México, 01080, México.2Department of Mathematics,
Central Michigan University, Mt Pleasant, MI 48858, USA.
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Received: 9 March 2017 Accepted: 6 July 2017 References
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