MARKUS POMPER
Received 27 February 2005 and in revised form 5 July 2005
LetKbe a compact Hausdorffspace andC(K) the Banach space of all real-valued contin- uous functions onK, with the sup-norm. Types overC(K) (in the sense of Krivine and Maurey) can be uniquely represented by pairs (,u) of bounded real-valued functions on K, whereis lower semicontinuous,uis upper semicontinuous,≤u, and(x)=u(x) for all isolated pointsxofK. A condition that characterizes the pairs (,u) that represent double-dual types overC(K) is given.
1. Statement of the main theorem
The concept oftype over a Banach spaceEwas first introduced by Krivine and Maurey [7]
in the context of separable Banach spaces. The reader is referred to Garling’s monograph [4] for more details. We consider general, not necessarily separable Banach spaces. LetE be a Banach space. For everyx∈E, we define a functionτx:E→Rby lettingτx(y)= x+yfor ally∈E.
Definition 1.1. A functionτ:E→Ris atype overEifτis in the closure (with respect to the topology of pointwise convergence) of the set{τx:x∈E}.
The definition given here is equivalent to the definition given in [1]. That is,τis a type overEif and only if there exists an ultrafilterᐁover an infinite index setλand a bounded family of elements (xα)α∈λinEsuch thatτ(y)=limα∈ᐁxα+yfor ally∈E. The reader is referred to [5] for more details regarding the choice of the ultrafilter.
Throughout, we letKbe a compact Hausdorfftopological space. The topology onK is denoted byΩ. We let∞(K) denote the Banach lattice of bounded real-valued func- tions onKequipped with the sup-norm. Forf,g∈∞(K), the lattice ordering is defined pointwise.
An scpair (semicontinuous pair)is a pair of functions (,u) from∞(K) such thatis lower semicontinuous (lsc),uis upper semicontinuous (usc),≤u, and(x)=u(x) for all isolated pointsx∈K.
The Banach space of continuous real-valued functions onKwith sup-norm is denoted byC(K). The constant function with value 1 is denoted by1.
Copyright©2005 Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences 2005:16 (2005) 2533–2545 DOI:10.1155/IJMMS.2005.2533
The following theorem gives a concrete representation of types overC(K) in terms of sc pairs [9,10].
Theorem1.2. Letτ:C(K)→Rbe a function. Then the following are equivalent:
(i)τis a type overC(K);
(ii)there exists an sc pair(,u)such thatτ(g)=max{+g,u+g}for allg∈C(K).
The correspondence between types overC(K) and sc pairs (,u) is one-to-one.
The following proposition is immediate fromDefinition 1.1; see [9] for more equiva- lent conditions and a detailed proof.
Proposition1.3. LetEbe a Banach space andτ:E→Ra function. Then the following are equivalent:
(i)τis a type overE;
(ii)for every finite subsetα⊆Eand everyε >0, there exists an elementx=x(α,ε)∈E such that|τ(y)− x+y|< εfor ally∈α;
(iii)there exists a bounded net(xα)α∈IinEsuch that
limα,I xα+y=τ(y) (1.1)
for ally∈E.
Ifτis a type overEand (xα)α∈I is as in (iii) above, we say that (xα)α∈I generatesthe typeτ. A net (xα)α∈IinEdoubly generatesτif for everyλ∈[0, 1] and everyy∈E,
limβ,Ilim
α,I y+λxα+ (1−λ)xβ=τ(y). (1.2) LetEbe a Banach space and letEbe its second dual. Throughout, we considerE as a subspace ofE. For every fixedg∈E, define the functionτg:E→Rby letting τg(x)= x+gfor allx∈E. It is immediate from the principle of local reflexivity that τgis a type overE.
Ifτis a type overEthat can be represented in this way, we callτadouble-dual type overE.
Maurey [8] and Rosenthal [11] have given a characterization of double-dual types over separable Banach spaces. The author [9] has generalized this characterization to not necessarily separable Banach spaces as follows.
Theorem1.4. LetEbe a Banach space andτ:E→Ra type overE. Then the following are equivalent:
(i)τis a double-dual type overE;
(ii)there exists a net(xα)α∈IinEthat doubly generatesτ.
This paper is devoted to proving the following characterization of double-dual types overC(K) in terms of the representation using the sc pairs.
Theorem1.5. Letτbe a type overC(K), represented by the sc pair(,u)as inTheorem 1.2.
Let
Y=
x∈K:xis not isolated and(x)<lim infy
→x (y), Yu=
x∈K:xis not isolated andu(x)>lim sup
y→x u(y). (1.3) The following are equivalent:
(i)τis a double-dual type overC(K);
(ii)Y∩Yu= ∅;
(iii)there exists a net(fα)α∈Iwhich doubly generatesτ.
The next section will include a discussion of generating nets. In Section 3, several properties of singular points of sc pairs will be proved. The mainTheorem 1.5will then be proved inSection 4.
2. Generating nets inC(K)
In this section, we introduce concepts that are needed to prove the main theorem.
We use the standard notion for convergence of nets in topological spaces according to [3, Section 1.6]. We recall the basic definitions for the convenience of the reader.
Definition 2.1. (i) A partially ordered set (I,≤) is adirected setif for anyα,β∈I there existsγ∈Isuch thatγ≥αandγ≥β. Such an elementγis called asuccessorofα(andβ).
(ii) Let (I,≤) be a directed set. For every elementα0∈I, define|α0| =card ({α∈I: α≤α0}), the number of predecessors ofα0.
(iii) Let (I,≤) and (J,≤) be directed sets. A functionk:I→J isorder-preserving if α≤β∈Iimpliesk(α)≤k(β). A functionk:I→Jiscofinalif for everyγ∈Jthere exists α∈Isuch thatγ≤k(α).
(iv) Let (I,≤) be a directed set andKa topological space. We say that (xα)α∈Iis anet inKindexed byIifxα∈Kfor allα∈I. IfK is a normed space, then (xα)α∈Iis bounded if{xα:α∈I}is bounded inR.
(v) Let (I,≤) be a directed set,Ka topological space, and (xα)α∈Ia net inKindexed by I. Ifj:I→Iis a cofinal order-preserving function, then (xj(α))α∈Iis asubnetof (xα)α∈I.
(vi) Let (I,≤) be a directed set,K a topological space, and (xα)α∈Ia net inKindexed byI. Letx∈K. Then limα,Ixα=xif and only if for every neighborhoodUofxinKthere existsα∈Isuch thatxβ∈Ufor allβ≥α.
(vii) Let (I,≤) be a directed set and (rα)α∈Ia bounded net of real numbers. Then define lim sup
α,I rα=inf
α∈Isuprβ:β∈Iandβ≥α, lim inf
α,I rα=sup
α∈Iinfrβ:β∈Iandβ≥α. (2.1) Observe that lim supα,Irαand lim infα,Irαexist for every bounded net (rα)α∈IinR. We now consider the Banach lattice∞(K) of bounded real-valued functions onK, equipped with the sup-norm.
A subsetH⊆∞is calledboundedif sup{f:f ∈H}<∞. LetHbe such a set. The pointwise supremum ofHis the real-valued functionLdefined byL(x)=sup{h(x) :h∈ H}for everyx∈K. We writeL=Hfor this function. Similarly, the pointwise infimum ofH is the real-valued functionUdefined byU(x)=inf{h(x) :h∈H}for everyx∈K.
This function is denoted by H. Note that bothHand Hare again in∞(K).
IfH⊆∞(K) is a bounded set of usc functions, then the pointwise infimum H is usc. Similarly, the pointwise supremum of a bounded set of lsc functions is lsc. Finally, it is clear that f ∈C(K) is continuous if and only if f is usc and lsc. Therefore, ifH is a bounded set of continuous functions onK, then His usc andHis lsc.
Letτbe a type overC(K) and let (fα)α∈Igenerateτas inProposition 1.3(iii) above.
We construct the sc pair (,u) ofTheorem 1.2as follows.
For everyα∈I, define a lower semicontinuous functionsαand an upper semicontin- uous functionuαonKby setting
α=
f ∈C(K) : f ≤fβ∀β≥α, uα=
f ∈C(K) : f ≥ fβ∀β≥α. (2.2) Then set
u=
α uα, =
α α. (2.3)
Here are some basic properties of the functionsandudefined in (2.3). See [10] for details.
Remark 2.2. Let (fα)α∈Ibe a bounded net of functions and letα,,uα, andube as in (2.3) above.
(i) Ifα1,α2∈Iandα1≤α2, thenα1≤α2≤anduα1≥uα2≥u.
(ii) If x∈K andε >0, then there exists an α0=α(x,ε)∈I such that for all indices α > α0,
α(x)≥(x)−ε, uα(x)≤u(x) +ε. (2.4) (iii) For everyβ∈I, everyx∈K, everyδ >0, and every neighborhoodUofx, there
existsy∈Uandγ≥βsuch that fγ(y)≤β(x) +δ.
(iv) For everyβ∈I, everyx∈K, everyδ >0, and every neighborhoodUofx, there existsy∈Uandγ≥βsuch that fγ(y)≥uβ(x)−δ.
Proof. (i) and (ii) are trivial. To prove (iii) letβ∈I, letx∈K,δ >0, andUa neighbor- hood ofx. Suppose that for everyy∈Uand allγ≥βwe have fγ(y)> β(x) +δ. Then we may choose a functiong∈C(K) such thatg≤ fγfor allγ≥βandg(x)=β(x) +δ. This would imply thatβ(x)=
{f ∈C(K) : f ≤ fγfor allγ≥β} ≥g(x)=β(x) +δ. This is a contradiction. The proof of (iv) is dual to the proof of (iii).
Let (fα)α∈Ibe a bounded net of functions inC(K) that generates a typeτoverC(K).
Chooseanduas in (2.3) and assumex∈Kandu(x)=r. It can be shown that for every neighborhoodU of x and for everyε >0, there exists an indexα0 such that for every
α≥α0, there existsy∈Usuch that fα(y)> r−ε. IfU,ε, andrare fixed, then we define for everyα∈I,
Vα:=
y∈U:fα(y)> r−ε. (2.5) Hence, for everyx∈K, every neighborhoodU ofx, and everyε >0, there exists an indexα0such thatVα= ∅for allα > α0.
The following definition introduces stronger conditions.
Definition 2.3. Let (fα)α∈Ibe a bounded net of functions inC(K). Letandube as in (2.3).
(i) (fα)α∈I generatesu atx withinΩ if for every ε >0 and every neighborhoodU of x, there exists an indexα0 such that for all α > α0, there exists β0 such that Vα∩Vβ= ∅for allβ > β0.
(ii) The net (fα)α∈IgeneratesuwithinΩif it generatesuatxwithinΩfor everyx∈K.
(iii) (fα)α∈IgeneratesatxwithinΩif (−fα)α∈Igenerates−atxwithinΩ.
(iv) The net (fα)α∈IgenerateswithinΩif it generatesatxwithinΩfor everyx∈K.
Proposition2.4. Let(fα)α∈Ibe a bounded net of functions inC(K)that generates a type τ. Letube as in (2.3).
(i)Ifuis continuous atx, then(fα)α∈IgeneratesuatxwithinΩ. (ii)Iflimα,Ifα(x)=u(x), then(fα)α∈IgeneratesuatxwithinΩ.
(iii)If (xβ)β∈I is a net in K that converges to x and if limβ,Iu(xβ)=u(x)and limα,I
fα(xβ)=u(xβ)for allβ, then(fα)α∈IgeneratesuatxwithinΩ. The statement is also true ifuis replaced with.
Proof. To show (i) letε >0 andU a neighborhood ofx. We may assume that|u(y)− u(x)|< ε/2 for ally∈U. ByRemark 2.2(ii) there existsα0∈Isuch that for allα > α0,
Vα=y∈U: fα(y)> u(x)−ε= ∅. (2.6) Now fix such anαand choosey∈Vα. Then (usingRemark 2.2(iii)) there existsβ0such that for everyβ≥β0,
fβ zβ
> u(y)−ε
2 (2.7)
for somezβ∈Vα. Therefore,zβ∈Vα∩Vβ, which shows that the net (fα)α∈Igeneratesu atxwithinΩ. Statement (ii) is immediate from the definition.
To show (iii) letU be a neighborhood of xand ε >0. There existsβ∈I such that xβ∈Uand|u(x)−u(xβ)|< ε/2. Fix such aβ∈I and chooseα0∈Isuch that|fα(xβ)− u(xβ)|< ε/2 for allα > α0. Thenxβ∈Vα= {y∈U: fα(y)> u(x)−ε}for allα > α0; that
is,Vα∩Vα= ∅for allα,α> α0.
3. Singular points of semicontinuous pairs
Our next goal is to find necessary and sufficient conditions onandufor the existence of a single net that generates bothanduwithinΩ.
Definition 3.1. Letube a usc function andx∈K. We callxasingularpoint ofu, ifxis not an isolated point ofKand
u(x)>lim sup
y→x u(y). (3.1)
Similarly, we callxasingularpoint of an lsc functionifxis not isolated and (x)<lim infy
→x (y). (3.2)
We callxaregular pointofu(resp.,) ifxis not isolated and not a singular point ofu (resp.,).
It is immediate from the definition thatxis a singular point ofuif and only if there exists an open neighborhoodUofxsuch that
u(x)>supu(y) :y∈U\ {x}. (3.3) IfUis such a neighborhood andV⊆Uis another neighborhood ofx, then
u(x)>supu(y) :y∈V\ {x}
. (3.4)
Ifxis a regular point ofu, then there exists a net (xβ)β∈I inK which converges tox such thatu(x)=limβ,Iu(xβ) andxβ=xfor allβ∈I.
Proposition3.2. Let(,u)be an sc pair in∞(K). Letx∈K be a nonisolated point and (fα)α∈Ia net which generates bothanduwithinΩatx.
(i)Ifxis a singular point ofu, thenxis a regular point ofandlimα,Ifα(x)=u(x).
(ii)Ifxis a singular point of, thenxis a regular point ofuandlimα,Ifα(x)=(x).
Proof. First we prove the following claim, which is the second statement of (i).
Ifxis a singular point ofu, then lim
α,I fα(x)=u(x). (3.5) Proof of the claim. Letxbe a singular point ofuand suppose limα,Ifα(x)=u(x). Choose ε >0 and an open neighborhoodUofxsuch that
u(x)−ε >supu(y) :y∈U\ {x}
(3.6) and such that
lim sup
α,I fα(x)< u(x)−2ε. (3.7) There exists a further open neighborhoodU ofx such thatx∈U⊆U⊆UandU is compact. We may fixα0such that for allα > α0,
fα(x)< u(x)−ε, Vα=
y∈U:fα(y)> u(x)−ε 3
= ∅. (3.8)
Letα > α0. Then
Wα=
y∈U:fα(y)< u(x)−2ε 3
(3.9) is an open neighborhood ofx which is disjoint fromVα. Since (fα)α∈I generatesuatx withinΩ, there existsβ0such that for allβ > β0, we may choose
yβ∈
y∈Vα: fβ(y)> u(x)−ε 3
. (3.10)
By passing to a subnet if necessary, we may assume that limβ,Iyβ=yfor somey∈U. We obtainuβ(y)≥u(x)−ε/3 for allβ∈Iwithβ≥β0and hence
u(y)≥u(x)−ε
3, (3.11)
which contradicts (3.6). So limα,Ifα(x)=u(x) and the claim is established.
The dual statement of claim (3.5) reads as follows:
ifxis a singular point of, then limα,I fα(x)=(x). (3.12) It is proved using an argument dual to the proof of claim (3.5). This shows the second part of (ii).
To prove the first part of (i) observe thatx singular foruimplies(x)< u(x), and therefore limα,Ifα(x)=u(x)=(x). Using the contrapositive of statement (3.12) above shows thatxis not a singular point of; that is,xis a regular point of.
Likewise, (3.5) can be used to show that ifxa singular point of, thenxis a regular
point ofu.
Let (,u) be an sc pair andYandYuthe sets of singular points ofandu, respectively.
If (fα)α∈I is a net that generates bothanduwithinΩ, thenY andYuare disjoint by Proposition 3.2. The following proposition proves the existence of such a net, provided thatYandYuare disjoint.
Proposition 3.3. Let K be a compact Hausdorffspace and (,u)an sc pair in ∞(K).
Consider the setsY,Yuof singular points of,u, respectively. Suppose thatY∩Yu= ∅. Then there exists a net(fα)α∈Iof continuous functions which generatesanduwithinΩ.
The proof of this proposition requires the following theorem.
Theorem3.4 (Edwards [2]). LetUbe a usc function andLan lsc function on a compact HausdorffspaceK such thatU≤L. Then there exists a continuous functionF such that U≤F≤L.
A proof of this theorem can be found in Kaplan [6, (48.5)].
Proof ofProposition 3.3. Letᐁbe a base for the topologyΩsuch thatᐁdoes not contain the empty set and the only finite sets inᐁare singletons. LetI=P<∞(ᐁ)\ {∅}, the set of finite subsets ofᐁ, be partially ordered by inclusion.
By induction on|α|construct an increasing net of integers (kα)α∈Iand for every 1≤ k≤kαconstruct functionsgα(1),gα(2) and fα∈C(K) and finite collections of nonempty open setsBα= {Vα,1,...,Vα,kα⊇αand elementszi,α,k∈Vα,kfori=1, 2 and all 1≤k≤kα, such that the following conditions hold for everyα∈Iand everyk=1,...,kα:
uz1,α,k
≥supu(y) :y∈Vα,k
− 1
|α|, (3.13)
z2,α,k
≤inf(y) :y∈Vα,k + 1
|α|, (3.14)
gα(1)zj,α,k
=uzj,α,k
forj=1, 2, (3.15)
gα(2)zj,α,k
=zj,α,k
forj=1, 2, (3.16)
u≤gα(1)≤
β<α
gβ(1)≤ u1, (3.17)
≥gα(2)≥
β<α
gβ(2)≥ −1, (3.18)
fα z1,α,k
=uz1,α,k
, fα(z2,α,k)=z2,α,k
. (3.19)
Furthermore, for everyβ < αand every 1≤k≤kβ, the following nonempty open sets are required to be among the elements ofBα:
V1,β,k(α) =
y∈Vβ,k: fβ(y)> uz1,β,k
− 1
|α|
, (3.20)
V2,β,k(α) =
y∈Vβ,k: fβ(y)< z2,β,k + 1
|α|
. (3.21)
We use induction on |α|. Ifα= ∅, let f∅=g∅(1)= u1andg∅(2)= −1and set B∅= ∅. With this choice, conditions (3.13)–(3.21) are either trivial or vacuously true.
Ifα∈I andα= ∅, suppose as inductive hypothesis that the construction has been completed for everyβ∈Iwithβ < α. Let
Bα=
Vi,β,k(α) :i=1, 2;β < α; 1≤k≤kβ
∪α∪
β<α
Bβ, (3.22)
whereV1,β,kandV2,β,kare as in (3.20) for all 1≤k≤kα. Observe thatBαis a finite col- lection of nonempty open sets. Say
Bα=
Vα,1,...,Vα,kα
, (3.23)
where (Vα,k)kkα=1 are pairwise distinct. Fori=1, 2 and 1≤k≤kα, we choosezi,α,k∈Vα,k
satisfying (3.13) and (3.14), and such that for all 1≤k,j≤kα, andi1,i2∈ {1, 2}, we have zi1,α,k=zi2,α,jif and only if eitherj=kandi1=i2orj=kandVα,kis a singleton.
Note that such a choice is possible, since the singular points ofanduare disjoint and the only finite sets inᐁare singletons.
We now constructgα(1)andgα(2)satisfying (3.15) through (3.18).
By inductive hypothesis in (3.17),u≤ β<αgβ(1)≤ u1. We define an lsc functionL onKby setting
L(x)=
u(x) ifx=zj,α,k for somej=1, 2; 1≤k≤kα,
β<αgβ(1)(x) otherwise. (3.24)
Becauseu≤L, we may applyTheorem 3.4and obtaingα(1)∈C(K) withu≤gα(1)≤L. This choice ofgα(1)satisfies (3.15) and (3.17). We use a dual construction to definegα(2)satisfy- ing conditions (3.16) and (3.18).
In order to construct fαdefine a usc functionUand an lsc functionLonKby setting for everyx∈K,
U(x)=
gα(1)(x) ifx=z1,α,k for some 1≤k≤k(α), gα(2)(x) otherwise,
L(x)=
gα(2)(x) ifx=z2,α,k for some 1≤k≤k(α), gα(1)(x) otherwise.
(3.25)
Observe thatU≤L; byTheorem 3.4there exists a continuous function fαwithU≤ fα≤L. By construction ofUandLand (3.15), we have
Ux1,α,k
=Lx1,α,k
=gα(1)z1,α,k
=uz1,α,k
(3.26) for all 1≤k≤kα. Hence, fα(z1,α,k)=u(z1,α,k). Furthermore,
Ux2,α,k
=Lx2,α,k
=gα(2)z2,α,k
=z2,α,k
(3.27) for all 1≤k≤kα. Thus,fα(z2,α,k)=(z2,α,k). Condition (3.19) follows from these last two observations.
This completes the construction and we now proceed to show that the net (fα)α∈I
generatesuandwithinΩ.
Fixx∈K,ε >0, andU∈Ω. Choosen∈Nsuch that 1/n < ε/2. Fixβ∈Iwith|β|> n, such that for someV∈βwe havex∈V ⊆U. Choose 1≤k≤kβ such thatV =Vβ,k∈ Bβ. Applying (3.13) yields
uz1,β,k
≥supu(y) :y∈Vβ,k
− 1
|β|≥u(x)− 1
|β|. (3.28)
So by (3.19), fβ(z1,β,k)=u(z1,β,k). Now letα > β. By (3.20) there exists 1≤j≤kαsuch that
Vα,j=V1,β,k(α) =
y∈Vβ,k:fβ(y)> uz1,β,k
− 1
|α|
. (3.29)
In particular,
z1,β,k∈Vα,j. (3.30)
Observe that by (3.13)
uz1,α,j
≥supu(y) :y∈Vα,j
− 1
|α| (3.31)
andz1,α,j∈Vα,j. Thus, fβ
z1,α,j
> uz1,β,k
− 1
|α| by (3.29)
≥u(x)− 1
|β|− 1
|α| by (3.28)
> u(x)−ε.
(3.32)
On the other hand,
fα(z1,α,j)=uz1,α,j
by (3.19)
≥supu(y) :y∈Vα,j
− 1
|α| by (3.31)
≥uz1,β,k
− 1
|α| by (3.30)
≥u(x)− 1
|β|− 1
|α| by (3.28)
> u(x)−ε.
(3.33)
Therefore, z1,α,j∈
y∈U:fα(y)> u(x)−ε∩
y∈U: fβ(y)> u(x)−ε= ∅ (3.34) for allα≥β. This shows that the net (fα)α∈I generates u withinΩ. The proof that it
generateswithinΩfollows from a similar argument.
Letτbe a type overC(K) that is represented by the sc pair (,u) as inTheorem 1.2.
Propositions3.2and3.3prove that the setsY andYuof singular points ofanduare disjoint if and only if there exists a net (fα)α∈Ithat generates bothanduwithinΩ.
4. Proof of the main theorem
We now consider a net (fα)α∈Ithat generates a typeτoverC(K). As before, let this type be represented by the sc pair (,u).
To establish the main theorem, we will now prove that the net doubly generates the typeτif and only if the net generates bothanduwithinΩ. This is accomplished in the following two lemmas.
Lemma4.1. LetKbe a compact Hausdorffspace andτa type overC(K). Let(fα)α∈I be a net that doubly generatesτ. Let(,u)be the sc pair such thatτ(g)=max{+g,u+g} for allg∈C(K). Then(fα)α∈IgeneratesanduwithinΩ.
Proof. Assume the conclusion does not hold. Then either (fα)α∈I does not generate u withinΩat somex∈K, or it does not generatewithinΩat somex∈K. We distinguish between these two cases.
Case 1. (fα)α∈I does not generate uat x within Ω. Letλ=1/2. There exists an open neighborhoodUofxandε >0 such that for allα0∈Iandβ0∈Ithere existα > α0and β > β0, for which
y∈U: fα(y)> u(x)−ε∩
y∈U:fβ(y)> u(x)−ε= ∅. (4.1) LetU0= {y∈U:u(y)< u(x) +ε/2} and choose an open neighborhoodU1 ofx such thatU1⊆U1⊆U0⊆U. We claim that there existsα0∈Isuch thatfα ≤ τ+ε/2 and
fαU1≤u(x) +ε/2 for allα≥α0. (Here,τ =τ(0).)
First observe that there existsα1 such that for allα≥α1 we havefα ≤ τ+ε/2.
Suppose there does not exist α0≥α1 such that fαU1≤u(x) +ε/2 for allα≥α0. Then there exist a cofinal order-preserving mapi:I→Isuch thatfi(α)(yi(α))> u(x) +ε/2, where yi(α)∈U1for allα∈I. We may assume that (yi(α))α∈Iconverges toy0∈U1. Thus,u(y0)≥ u(x) +ε/2, which contradicts the choice ofU0and establishes the claim.
Fix a functiong∈C(K) such thatgK\U1=0 andg(x)=3τand 0≤g≤3τ. Ob- serve thatu+g ≥g(x) +u(x)=3τ+u(x).
Further, for eachα≥α0, there exists α2≥αand a cofinal order-preserving function j=jα2:I→Isuch that
y∈U1:1
2fα2(y)>1
2u(x)−1 2ε∩
y∈U1:1
2fj(β)(y)>1
2u(x)−1
2ε= ∅. (4.2) Fix suchα2andj=jα2. Ify∈U1,
g(y) +1
2fα2(y) +1
2fj(β)(y)≤3τ+u(x)−ε
4 (4.3)
for allβ∈I. Ify∈K\U1, we have g(y) +1
2fα2(y) +1
2fj(β)(y)≤ τ+ ε
2. (4.4)
Observe that limβ,Ig+ 1/2fα2+ 1/2fβexists. Thus, limβ,I
g+1 2fα2+1
2fβ =lim
β,I
g+1 2fα2+1
2fj(β)
≤3τ+u(x)−ε
4. (4.5) Hence,
lim inf
α,I lim
β,I
g+1 2fα+1
2fβ
≤3τ+u(x)−ε 4<lim
α,I g+fα. (4.6) This contradicts the assumption that (fα)α∈Idoubly generatesτ.
Case 2. (fα)α∈Idoes not generateatxwithinΩ. This case is handled with an argument
dual to the one inCase 1.
Lemma 4.2. Let K be a compact Hausdorffspace and τ a type overC(K). Let(,u)be the sc pair such thatτ(g)=max{+g,u+g}for allg∈C(K). Assume that(fα)α∈I
generatesanduwithinΩ. Then(fα)α∈Idoubly generatesτ.
Proof. Fixg∈C(K). Becauseτ(g)=max{+g,u+g}, we distinguish between two cases.
Case 1. Suppose thatτ(g)= u+g. Choosex∈K such thatu+g =u(x) +g(x). Let ε >0 and choose a neighborhoodU of x such that |g(y)−g(x)|< ε/2 for all y∈U.
Chooseα0∈Isuch that for allα > α0, there existsβ0∈Isuch that for allβ > β0, we have fα(z)> u(x)−ε/2 and fβ(z)> u(x)−ε/2 for somez∈U. Then
g+λ fα+ (1−λ)fβ≥g(z) +λ fα(z) + (1−λ)fβ(z)> u(x) +g(x)−ε= u+g −ε.
(4.7) Therefore,
lim inf
α,I lim
β,I g+λ fβ+ (1−λ)fα≥ u+g −ε. (4.8) On the other hand,
lim sup
α,I lim
β,I g+λ fα+ (1−λ)fβ
≤lim sup
α,I λg+fα+ lim
β,I(1−λ)fβ+g
≤u+g.
(4.9)
Becauseεwas arbitrary, this shows that limα,I lim
β,I g+λ fα+ (1−λ)fβ (4.10) exists and equalsτ(g).
Case 2. Ifτ(g)= +g, consider the net (−fα)α∈I, which generates−uand−within Ωand the function−g∈C(K). We infer fromCase 1that
limβ,I lim
α,I g+λ fα+ (1−λ)fβ=lim
β,I lim
α,I −g+λ(−fα) + (1−λ)(−fβ)= +g. (4.11) Therefore, limβ,Ilimα,Ig+λ fα+ (1−λ)fβ =τ(g) for allg∈C(K).
Proof ofTheorem 1.5. The equivalence between (i) and (iii) is Theorem 1.4above. The implication (ii)⇒(iii) follows fromProposition 3.2andLemma 4.1and (iii)⇒(ii) follows
fromProposition 3.3andLemma 4.2.
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Markus Pomper: Department of Mathematics, Indiana University East, Richmond, IN 47374, USA E-mail address:[email protected]
Special Issue on
Modeling Experimental Nonlinear Dynamics and Chaotic Scenarios
Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system. Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision. In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from “Qualitative Theory of Differential Equations,”
allowing more precise analysis and synthesis, in order to produce new vital products and services. Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Prob- lems in Engineering aims to provide a picture of the impor- tance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.
Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophis- ticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System athttp://
mts.hindawi.com/according to the following timetable:
Manuscript Due December 1, 2008 First Round of Reviews March 1, 2009 Publication Date June 1, 2009
Guest Editors
José Roberto Castilho Piqueira,Telecommunication and Control Engineering Department, Polytechnic School, The University of São Paulo, 05508-970 São Paulo, Brazil;
Elbert E. Neher Macau,Laboratório Associado de Matemática Aplicada e Computação (LAC), Instituto Nacional de Pesquisas Espaciais (INPE), São Josè dos Campos, 12227-010 São Paulo, Brazil ; [email protected] Celso Grebogi,Center for Applied Dynamics Research, King’s College, University of Aberdeen, Aberdeen AB24 3UE, UK; [email protected]
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