Manusc ip submi ed o Websi e: h p://AIMsciences.o g
AIMS’ Jou nals
Volume X, Numbe 0X, XX 200X pp. X–XX
EQUI-ATTRACTION AND THE CONTINUOUS DEPENDENCE
OF ATTRACTORS ON TIME DELAYS
P. E. Kloeden1, P. Ma ´
ın-Rubio2
1FB Ma hema ik,
Johann Wol gang Goe he Uni e si ¨a ,
D-60054 F ank u am Main, Ge many
2Depa amen o de Ecuaciones Di e enciales y An´alisis Num´e ico,
Uni e sidad de Se illa,
Apdo. de Co eos 1160,
41080–Se illa, Spain
(Communica ed by Aim Sciences)
Abs ac . Unde app op ia e egula i y condi ions i is shown ha he con-
inuous dependence o he global a ac o s Aτo semi dynamical sys ems
S(τ)( ) in C([−τ, 0]; Z) wi h Za Banach space and ime delay τ∈[T∗, T ∗],
whe e T∗>0, is equi alen o he equi-a ac ion o he a ac o s. Examples
and coun e examples posed in his igh amewo k a e p o ided.
1. In oduc ion. The uppe semi con inuous dependence o a ac o s on a pa-
ame e is a s anda d esul in dynamical sys ems heo y, see e.g. [5, 11, 13, 15, 16].
In gene al, lowe semi con inuous, and hence con inuous, dependence does no hold
wi hou addi ional assump ions, which usually a e gi en in e ms o he s uc u e
o he a ac o , such as i s being Mo se-Smale. In ano he app oach, Li and Kloe-
den [14] showed ecen ly ha con inuous dependence in a pa ame e is equi alen
o he equi-a ac ion o he pa ame ized a ac o s. These esul s also apply o
a ac o s o delay di e en ial equa ions (DDE) wi h a ixed ime delay.
On he o he hand e y li le has appea ed in he li e a u e abou he dependence
o a ac o s o DDE on he ime delay i sel , a di icul y being ha he a ac o s
belong o di e en s a e spaces. An ea ly pape on uppe semi con inui y o a
conc e e e a ded nonlinea PDE is [1] (see also [2, 4] o he same ques ion abou
ine ial mani olds o de e minis ic and s ochas ic p oblems).
Kloeden [11] showed how he uppe semi con inuous dependence o a ac o s
in he ime delay can be o mula ed by embedding he di e en semi dynamical
sys ems and hei a ac o s in a common s a e space. See also [3, 9] o o he
esul s.
Ou aim in his pape is o ind an analogue o he equi alence o con inuous de-
pendence and equi-a ac ion in [14] (see also [12]) o he dependence o a ac o s
2000 Ma hema ics Subjec Classi ica ion. P ima y: 34D45, 37C70, 34K25.
Key wo ds and ph ases. Semi lows o delay di e en ial equa ions, pa ame ic a ac o s, ex-
ended semi lows and a ac o s, con inui y o a ac o s and equi-a ac ion.
Pa ially suppo ed by Minis e io de Educaci´on y Ciencia (Spain) and FEDER (Eu opean
Communi y) g an MTM2005-01412 and Acci´on in eg ada Hispano-Alemana Re . HA2005-0082.
1
2 P. E. KLOEDEN AND P. MAR´
IN-RUBIO
o semi dynamical sys ems (SDS) gene a ed DDE on he ime delay. Fo his we
i s summa ize he main ing edien s om [14].
Le λbe a pa ame e in a compac me ic space (Λ, DΛ) and le {S(λ)
, ∈
R+}λ∈Λbe a amily o SDS on a comple e me ic space (X, d). De ine d(x, A) =
in y∈Ad(x, y) o any x∈Xand A⊂X, and le BX(a, ) deno e he open ball o
Xwi h cen e aand adius ; and P(X) and C(X) he classes o all nonemp y and
nonemp y and closed subse s o X, espec i ely. In addi ion, deno e he Hausdo
semidis ance and Hausdo dis ance on X, espec i ely, by
H∗
X(A, B) = sup
x∈A
d(x, B), HX(A, B) = max {H∗
X(A, B), H∗
X(B, A)}
o any closed nonemp y subse s Aand Bo X.
De ini ion 1. A nonemp y compac subse Ao Xis called a global a ac o o
an SDS {S , ∈R+}on X(i.e. a semi-g oup o mappings wi h S :X→X
con inuous o each ixed ≥0) i i is in a ian , i.e. S (A) = A o all ∈R+,
and a ac s bounded subse s Bo X, i.e.
H∗(S (B),A)→0as →+∞.
De ini ion 2. Le {S(λ)
, λ ∈Λ}be a amily o SDS on X. I is said o be
(i) equi-dissipa i e on Xi he e exis s a bounded subse Uo Xso ha o
any bounded subse B⊂X, he e exis s a TB∈R+independen o λ∈Λsuch ha
S(λ)
(B)⊂ U, ≥TB;
(ii) e en ually equi-compac (o uni o mly compac o la ge in [14]) i o
any bounded subse Bo X, he e exis s a TB∈R+independen o λ∈Λsuch ha
Sλ∈ΛS(λ)
(B)is ela i ely compac in X o any ≥TB.
Theo em 3. [14, Th.2.9] Suppose ha a amily o SDS {S(λ)
, λ ∈Λ}on Xis
equi-dissipa i e and e en ually equi-compac and ha Aλis he global a ac o o
S(λ)
o λ∈Λ. In addi ion, suppose ha
(A1) o any ∈R+ ixed, S(λ)
(x)is join ly con inuous in (x, λ)on X×Λ.
(A2) S(λ)
(x)is equi-con inuous in λ o ( , x)in any bounded subse o R+×X.
Then {Aλ}is equi-a ac ing i and only i Aλis con inuous in λwi h espec o
he Hausdo dis ance.
Rema k 4. The abo e equi alence also holds i (A2) is eplaced by:
(A2’) S(λ)
(x)is equi-con inuous in λ o in any bounded subse o R+and xin
any bounded subse o Sλ∈ΛAλ.
Theo em 5. [14, Th.2.7] Suppose ha {S(λ)
, λ ∈Λ}is equi-dissipa i e and e en-
ually equi-compac and ha he assump ions (A1) and
(A3) Fo any bounded subse Bo Xand T > 0, S(λ)
xis uni o mly con inuous
in x∈Buni o mly w. . . λ∈Λand ≤T, i.e.
∀ε > 0,∃δ > 0 : x, y ∈B, d(x, y)< δ ⇒dS(λ)
(x), S(λ)
(y)< ε, ∀ ∈[0, T ], λ ∈Λ.
hold.
Then, i Aλis con inuous in λ, he amily {Aλ}is uni o mly Lyapuno s able,
i.e. o any ε > 0, he e exis s δ > 0(independen o λ) such ha o all λ∈Λ,i
d(x, Aλ)< δ, hen d(S(λ)
x, Aλ)< ε o all ∈R+.
EQUI-ATTRACTION AND CONTINUOUS DEPENDENCE FOR DDE 3
In Sec ion 2 we ein e p e he abo e “equi” concep s o SDS gene a ed by DDE
wi h ini e delay, whe e he ini e delay is conside ed as he pa ame e and ex end he
SDS and hei a ac o s (which a e assumed o exis , see [11] o exis ence esul s)
o a common s a e space. In Sec ion 3 we in e p e he abo e “equi” concep s o
he SDS in hei o iginal s a e spaces. Finally, in Sec ion 4 an example is gi en o
a scala DDE wi h a ac o s which a e con inuous and discon inuous in he ime
delay a di e en ime delays.
2. Ex ension o a common s a e space. A delay di e en ial equa ion in a
Banach space (Z, |·|) wi h ime delay τ > 0 gene a es an SDS in he unc ion space
Cτ:= C([−τ, 0]; Z) o con inuous unc ions φ: [−τ, 0] →Z, which is a Banach space
wi h he sup emum no m k· kτ. We deno e his SDS in Cτby S(τ)and conside
a amily o such SDS o di e en , ixed alues o he ime delay τ∈[T∗, T ∗], wi h
0< T∗< T∗<∞. In addi ion, we assume ha each SDS S(τ)possesses a global
a ac o Aτin i s s a e space Cτ. Theo em 3 canno be applied di ec ly o his
amily, bu can be a e we ep esen hen as SDS on he common s a e space CT∗.
In o de o ansla e he di e en SDS o he common space we p ojec a solu ion
S(τ)
φin unc ion space C([−τ, 0]; Z) on o he base space Zand hen econs i u e
i as a ime dependen unc ion aking alues in he unc ion space C([−T∗,0]; Z).
Le φ∈CT∗and le φ|[−τ,0] be i s unca ion in Cτ. Hence S(τ)
φ|[−τ,0] is well
de ined o all ≥0. De ine i s p ojec ion x: [−T∗,∞)×CT∗→Zin Zby
x( , φ) := φ( ) ∈[−T∗,0],
S(τ)
(φ|[−τ,0])(0) > 0,
whe e S(τ)
φ|[−τ,0](0) is he alue ha akes he unc ion S(τ)
(φ|[−τ,0]) in Za ime
0. Finally, de ine b
S(τ)
(φ)∈CT∗ o each ≥0 by
b
S(τ)
(φ)(s) := x( +s, φ), s ∈[−T∗,0].
Theo em 6. I S(τ)be an SDS on Cτ, hen {b
S(τ)
, ∈R+}de ines an SDS on CT∗.
Mo eo e , i S(τ):R+×Cτ→Cτis join ly con inuous in ( , φ)∈R+×Cτ, hen
b
S(τ):R+×CT∗→CT∗is join ly con inuous in ( , φ)∈R+×CT∗.
P oo . The ini ial condi ion p ope y o an SDS ollows di ec ly om he de ini ion
o = 0, speci ically
b
S(τ)
0(φ)(s) = x(s, φ) = φ(s), s ∈[−T∗,0],
so b
S(τ)
0(φ) = φ o all φ∈C([−T∗,0]; Rd).
To check he semi-g oup p ope y, ha is,
b
S(τ)
1+ 2(φ) = b
S(τ)
1b
S(τ)
2(φ), o all 1, 2≥0,and φ∈CT∗,
we conside wo cases:
4 P. E. KLOEDEN AND P. MAR´
IN-RUBIO
Case 1: 1+s > 0. We use he semi-g oup p ope y o he SDS S(τ)se e al
imes:
b
S(τ)
1+ 2(φ)(s) = x( 1+ 2+s, φ)
=S(τ)
1+ 2+s(φ|[−τ,0])(0)
=S(τ)
1+s(S(τ)
2(φ|[−τ,0]))(0)
=x( 1+s, b
S(τ)
2(φ)) = b
S(τ)
1(b
S(τ)
2(φ))(s).
Case 2: 1+s≤0 (since s∈[−T∗,0], his case only holds i T∗> 1). By he
de ini ions we ha e
b
S(τ)
1+ 2(φ)(s) = x( 1+ 2+s, φ),(1)
as well as
b
S(τ)
1(b
S(τ)
2(φ))(s) = x( 1+s, b
S(τ)
2(φ))
=b
S(τ)
2(φ)( 1+s) = x( 1+ 2+s, φ).(2)
Compa ing (2) wi h (1) we ob ain he desi ed semi-g oup p ope y.
The con inui y o b
S(τ)
om CT∗in o CT∗ o each ixed ∈R+and he second
asse ion o he heo em can be p o ed simila ly, so we p o e jus he la e .
Suppose ha φ(n)→¯
φin CT∗and n→ in R+.Then φ(n)|[−τ,0] →¯
φ|[−τ,0] in
Cτand hence S(τ)
n(φ(n)|[−τ,0])→S(τ)
(¯
φ|[−τ,0]) in Cτ o each n→ in R+, which
means ha
x( n+s, φ(n)) = S(τ)
n(φ(n)|[−τ,0])(s)
→S(τ)
n(¯
φ|[−τ,0])(s) = x( +s, ¯
φ)
o all s∈[−τ, 0] (no only punc ually, bu uni o mly in [−τ, 0]). Conca ena ing as
many in e als as necessa y, we ob ain in a ini e numbe o s eps ha
b
S(τ)
n(φ(n))(s) = x( n+s, φ(n))→x( +s, ¯
φ) = b
S(τ)
(¯
φ)(s)
o all s∈[−T∗,0], i.e.
b
S(τ)
n(φ(n))→b
S(τ)
(¯
φ)
in CT∗as n→ in R+and φ(n)→¯
φin CT∗. Hence he mapping ( , φ)7→ b
S(τ)
(φ)
is con inuous.
This comple es he p oo ha b
S(τ)is an SDS on CT∗.
The nex s ep in ou goal is o ex end he a ac o s o he common s a e space
and o ensu e ha he ex ended objec s a e indeed a ac o s o he ex ended SDS.
Obse e ha i is no enough o ha e
H∗
CτS(τ)
−jτ (Bτ),Aτ< ε o j= 0,...,n∗−1,(3)
whe e n∗is he i s in ege wi h n∗τ≥T∗.This does no ensu e ha he e exis s
a co esponding conca ena ed se in CT∗sa is ying he co esponding inequali y
EQUI-ATTRACTION AND CONTINUOUS DEPENDENCE FOR DDE 5
he e. To show his we use he compac ness o he a ac o s and he con inui y o
he SDS.
Theo em 7. Suppose ha an SDS S(τ):R+×Cτ→Cτhas a global a ac o Aτ.
Then, he ex ended SDS b
S(τ)in CT∗gi en in Theo em 6 possesses a global a ac o
b
Aτin CT∗, which is cha ac e ized by
b
Aτ:= ψ∈CT∗:∃en i e ajec o y ¯
Φ(τ)
o S(τ)in Aτ(4)
wi h ψ(s) = ¯
φ(s)∀s∈[−T∗,0],
whe e ¯
φ( )is he p ojec ion in Zo he en i e solu ion ¯
Φ(τ)
de ined by ¯
φ( ) :=
¯
Φ(τ)
(0) o all ∈R.
P oo . ¿F om he s ic in a iance o Aτi is known ha o each φ∈ Aτ he e
exis s a leas one en i e solu ion ¯
Φ(τ)
o he SDS S(τ)in Aτwi h ¯
Φ(τ)
0=φ, so he
se b
Aτis well de ined. The in a iance o b
Aτunde he ex ended SDS b
S(τ) ollows
immedia ely om he de ini ions. The compac ness o Aτin CT∗ ollows om he
de ini ions and he ac ha he backwa d ex ension o an SDS in a compac in a i-
an se gene a es a mul i alued semi-g oup wi h compac a ainabili y se s [10].
I emains o p o e ha b
Aτis he global a ac o o he ex ended SDS b
Sτ.
Le ε > 0 be a bi a y and le n∗be he i s in ege such ha n∗τ≥T∗. Fo
each χ∈ Aτ,de ine δ(χ) := min{δ1(χ),...,δn∗(χ)}, whe e δj(χ) o j= 1,...,n∗
a e such ha he con inuous maps S(τ)
jτ sa is y H∗
Cτ(S(τ)
jτ (χ), S(τ)
jτ (φ)) ≤ε o all
φ∈BCτ(χ, δj(χ)).
Since Aτis compac i has ini e co e o open balls
Aτ⊂
k
[
i=1
BCτ(x(i), δ(x(i))).
The e hus exis s an ρ > 0 wi h
BCτ(Aτ, ρ)⊂
k
[
i=1
BCτ(x(i), δ(x(i))).(5)
Now conside a bounded se Bin CT∗.By he a ac ion o Aτ he e exis s T=
T(ρ, B|[−τ,0])≥0 such ha
H∗
CτS(τ)
(B|[−τ,0]),Aτ≤ρ∀ ≥T=T(ρ, B|[−τ,0]).
Conside an a bi a y elemen ϕ∈Band a x(i0)∈Cτsuch ha , by (5),


S(τ)
jτ S(τ)
T(ϕ)−S(τ)
jτ (x(i0))

τ≤ε o j= 1,...,n∗.
This implies ha
H∗
CT∗b
S(τ)
n∗τ+T(B),b
Aτ≤ε.
Thus b
Aτis he global a ac o o b
S(τ)in CT∗.

6 P. E. KLOEDEN AND P. MAR´
IN-RUBIO
Finally, ollowing [11], we ecall ha he con inuous con e gence o he a ac o s
o di e en ime delays is unde s ood as
HCT∗b
Aτ0,b
Aτ→0 as τ0→τ.
3. The equi-p ope ies o he o iginal SDS. We will now ansla e he con-
cep s o equi-a ac ion, equi-dissipa i e and e en ually equi-compac o he amily
o ex ended SDS {b
S(τ)
, τ ∈[T∗, T ∗]}on he space CT∗in e ms o he o iginal SDS
S(τ)
on hei s a e spaces Cτ. This is impo an as he p ope ies will be e i ied
he e, especially when he SDS a e gene a ed by speci ic DDE.
3.1. Equi-dissipa i i y. The concep o equi-dissipa i i y in De ini ion 2, (i), in
e ms o he ex ended SDS eads: he e exis s a bounded subse Uo CT∗and o
e e y bounded subse Bo CT∗ he e exis s a TB∈R+, which is independen o τ,
such ha
b
S(τ)
(B)⊂ U o all ≥TBand τ∈[T∗, T ∗].(6)
In e ms o he o iginal SDS and s a e space his implies ha
S(τ)
(B|[−τ,0])⊂ U|[−τ,0] o all ≥TBand τ∈[T∗, T∗],
whe e he p e ious no a ion is used o he es ic ed se s, i.e.
B|[−τ,0] ={φ|[−τ,0] :φ∈ B},U|[−τ,0] ={ψ|[−τ,0] :ψ∈ U}.
The de ini ion o equi-dissipa i i y has an equi alen o m in e ms o he unde -
lying base space Z, namely:
Lemma 8. A amily o SDS {b
S(τ), τ ∈[T∗, T∗]}is equi-dissipa i e i and only i
he e exis s a bounded subse Uo Zsuch ha o e e y bounded subse Bo Z
he e exis s a TB∈R+, which is independen o τ, such ha
S(τ)
(B|[−τ,0])(0) ⊂U o all ≥TBand τ∈[T∗, T∗],
whe e
B:= {φ∈CT∗:φ(s)∈B∀s∈[−T∗,0]}.(7)
P oo . S a ing wi h (6), we simply de ine
U={φ(s)∈Z:φ∈ U, s ∈[−T∗,0]}
wi h TB:= TBco esponding o he bounded subse Bo CT∗de ined in (7).
In he o he di ec ion, ollowing (7), we de ine
U:= {φ∈CT∗:φ(s)∈U, s ∈[−T∗,0]}.(8)
Gi en a bounded subse Bo CT∗we de ine TB:= TB+T∗co esponding o he
se
B={φ(s)∈Z:φ∈ B, s ∈[−T∗,0]}.
(No e ha he new se Bde ined by (7) in e ms o his Bwill con ain and in
gene al be la ge han he o iginal se B).
EQUI-ATTRACTION AND CONTINUOUS DEPENDENCE FOR DDE 7
3.2. E en ual equi-compac ness. We i s no e ha he p ope y o e en ual
equi-compac ness in De ini ion 2, (ii), can be ew i en as: o any bounded subse
Bo X, he e exis s a TB∈R+independen o λ∈Λand a amily o compac
subse s {K( ), ≥TB}o Xsuch ha
S(λ)
(B)⊂K( ) o e e y ≥TB.
We simply ake U( ) o be he closu e o Sλ∈ΛS(λ)
(B) in X. The compac se s
U( ) he e need no be uni o mly bounded in – i hey we e hen we would also
ha e equi-dissipa i i y.
In ou si ua ion his de ini ion akes he o m: o e e y bounded subse Bo CT∗
he e exis s a TB∈R+, which is independen o τ, and a amily o compac subse s
{U( ), ≥TB}o CT∗such ha
b
S(τ)
(B)⊂ U( ) o all ≥TBand each τ∈[T∗, T∗].
In e ms o he o iginal dynamical sys ems his ansla es o
S(τ)
(B|[−τ,0])⊂ U( )|[−τ,0] o all ≥TBand each τ∈[T∗, T∗].
Rema k 9. Fo DDE wi h ini e delay, when Zis ini e dimensional, compac ness
ollows om he exis ence o a bounded abso bing amily, hanks o Ascoli-A zel`a
Theo em, i he igh hand side o he DDE is a bounded map (i.e. i maps bounded
se s on o bounded se s).
3.3. Join and equi-con inui y. Theo em 3 equi es ha he amily o SDS sa -
is ies he con inui y p ope ies (A1) and (A2), i.e.
(A1) Fo any ∈R+ ixed, S(λ)
(x) is join ly con inuous in (x, λ) on X×Λ.
(A2) S(λ)
(x) is equi-con inuous in λ o ( , x) in any bounded subse o R+×X.
In ou con ex he join con inui y p ope y (A1) becomes: o any ∈R+ ixed,
b
S(τ)
(φ)is join ly con inuous in (τ, φ) in [T∗, T∗]×CT∗.
Thus, i (τn, φ(n))→(τ, φ) in [T∗, T∗]×CT∗as n→ ∞, hen so oo does
b
S(τn)
(φ(n))→b
S(τ)
(φ) as n→ ∞.
Recalling he p ojec ion no a ion in oduced be o e Theo em 6
x(τ)( +s, φ) := b
S(τ)
(φ)(s), s ∈[−T∗,0],
join con inui y means ha
x(τn)( , φ(n))→x(τ)( , φ) as n→ ∞,
in he base space Zuni o mly on he in e al [ −T∗, ] o each ixed ≥0. Thus i
will also be uni o m on all ini e ime in e als [−T∗, T] wi h T > 0. This uni o m
join con e gence in Zimplies he unc ion space join con inui y o condi ion (A1)
abo e.
Simila ly, he equi-con inui y p ope y (A2) becomes: b
S(τ)
(φ)is equi-con inuous
in τ o ( , φ)in any bounded subse [T1, T2]×B o R+×CT∗,which is essen ially
8 P. E. KLOEDEN AND P. MAR´
IN-RUBIO
uni o m con inui y in τ, i.e. o e e y ε > 0 and bounded subse [T1, T2]× B o
R+×CT∗ he e exis s δ=δ(T1, T2,B, ε)>0 such ha
|τ0−τ|< δ =⇒

b
S(τ0)
(φ)−b
S(τ)
(φ)

T∗< ε ∀( , φ)∈[T1, T2]×B.
In e ms o he p ojec ions in he base space Z his eads as
|τ0−τ|< δ =⇒x(τ0)( , φ)−x(τ)( , φ)< ε ∀( , φ)∈[T1−T∗, T2]×B,
which implies he unc ion space equi-con inui y condi ion (A2) abo e.
3.4. Equi-a ac ion. Suppose ha each SDS S(τ)
on Cτhas global a ac o Aτ
in Cτ o τ∈[T∗, T∗]. Then, by Theo em 7, each ex ended SDS b
S(τ)
on CT∗has
an a ac o b
Aτin CT∗, whe e b
Aτis de ined in e ms o Aτ h ough (4).
These ex ended a ac o s a e equi-a ac ing i o e e y ε > 0 and bounded
subse Bo CT∗ he e exis s Tε,B∈R+independen o τ∈[T∗, T∗] such ha
H∗
CT∗b
S(τ)
(φ),b
Aτ< ε o all ≥Tε,B, φ ∈ B, τ ∈[T∗, T ∗] (9)
which ob iously implies ha
H∗
CτS(τ)
(φ|[−τ,0]),Aτ< ε o all ≥Tε,B, φ ∈ B, τ ∈[T∗, T∗].(10)
O cou se, one would like ha (9) and (10) o be equi alen (pe haps wi h a sligh ly
la ge Tε,B). Howe e , he alue ρappea ing in he p oo o Theo em 7 depends on
τin a no necessa ily uni o m way. We will use p ope y (A3) and bo ow some
ideas om Theo em 5 o ob ain an equi alence.
Rema k 10. Condi ion (A3) in Theo em 5 o he ex ended SDS b
S(τ)is equi alen
o he ollowing condi ion o he o iginal semi dynamical sys ems S(τ):
(A3’) Fo any bounded subse Bo CT∗and T > 0, S(τ)
(χ|[−τ,0])is uni o mly
con inuous in χ|[−τ,0] ∈ B|[−τ,0] uni o mly w. . . τand ≤T, i.e.
∀ε > 0,∃δ > 0 : χ, φ ∈ B,kχ|[−τ,0] −φ|[−τ,0]kτ< δ
⇒ kS(τ)
(χ|[−τ,0])−S(τ)
(φ|[−τ,0])kτ< ε, ∀ ∈[0, T ], τ ∈[T∗, T∗].(11)
Theo em 11. Le S(τ):R+×Cτ→Cτ o τ∈[T∗, T∗]be a amily o SDS wi h
a ac o s Aτ, which is equi-dissipa i e and equi-a ac ing in he sense o (10) and
also sa is ies condi ion (A3’). Then he ex ended a ac o s b
Aτa e equi-a ac ing.
P oo . By he equi-dissipa i eness he e exis s a bounded subse Uo Zsuch ha
Aτ⊂ U|[−τ,0] o all τ, whe e he subse Uo CT∗is de ined om U h ough (8).
Conside any ε > 0 and he bounded se
B={φ∈CT∗:φ(s)∈BZ(U, ε), s ∈[−T∗,0]}.
I is enough o check (9) only wi h his bounded se . By he equi-a ac ion o
{Aτ}τ, he e exis s Tε,Bindependen o τ, such ha (10) holds. In pa icula , his
implies ha B|[−τ,0] is posi i ely in a ian o any S(τ)
wi h ≥Tε,B, i.e.
S(τ)
(B|[−τ,0])⊂ B|[−τ,0] ∀ ≥Tε,B.(12)
EQUI-ATTRACTION AND CONTINUOUS DEPENDENCE FOR DDE 9
Fo he bounded se B,by (A3’), he e exis s δ > 0 depending on εsuch ha
(11) holds o T=n∗T∗,wi h n∗ he i s in ege such ha n∗T∗≥T∗.We will use
(11) o =jT∗wi h j= 1,...,n∗, which ensu es ha we can co e any in e al
o leng h T∗by delays o leng h τ∈[T∗, T ∗].
Le ρ= min(δ, ε).By he equi-a ac ion again, analogously o (10), he e exis s
a ime Tρ,B(which we can ake w.l.o.g. la ge han Tε,B) such ha
H∗
Cτ(S(τ)
(B|[−τ,0]),Aτ)< ρ o all ≥Tρ,B, τ ∈[T∗, T∗].(13)
To inish he p oo , ake ψ∈ B.By (13), o any ≥Tρ,B, he e exis s ξ∈ Aτsuch
ha kS(τ)
(ψ|[−τ,0])−ξkτ< ρ ≤δ. Using (11) o T=n∗T∗we ha e
kS(τ)
jT∗S(τ)
(ψ|[−τ,0])−S(τ)
jT∗(ξ)kτ< ε o j= 1,...,n∗.
This means ha
H∗
CT∗b
S(τ)
n∗T∗+ (ψ),b
Aτ< ε ∀ ≥Tρ,B,
which is he equi-a ac ion p ope y (9) wi h Tε,B eplaced by n∗T∗+Tρ,B.
Rema k 12. The e is an equi alen condi ion o Assump ion (A3’) in he abo e
esul , hough appa en ly is less es ic i e, in which he uni o m con inui y o in
bounded in e als can be subs i u ed by uni o m con inui y a a single ime ins an
∗, namely,
(A3”) The e exis s ∗∈(0, T∗]such ha o any bounded subse Bo CT∗, he
SDS S(τ)
∗(χ)is uni o mly con inuous in χ∈ B|[−τ,0] uni o mly w. . . τ, i.e.
∀ε > 0,∃δ > 0 : χ, φ ∈ B,kχ|[−τ,0] −φ|[−τ,0]kτ< δ (14)
⇒ kS(τ)
T∗(χ|[−τ,0])−S(τ)
T∗(φ|[−τ,0])kτ< ε, ∀τ∈[T∗, T∗].
Ac ually, by (12), he abo e uni o m con inui y o S(τ)
∗in B|[−τ,0] holds o all
S(τ)
j ∗wi h j= 1,...,n∗,whe e n∗now deno es he i s in ege such ha n∗ ∗≥T∗.
Indeed, since S(τ)
∗is uni o mly con inuous in B|[−τ,0], o an a bi a y ε > 0 he e
exis s δ1such ha kS(τ)
∗(χ)−S(τ)
∗(φ)kτ≤εi kχ−φkτ< δ1.Fo S(τ)
2 ∗,choose δ2
associa ed wi h ε2=δ1(p ope y (12) plays an essen ial ole he e). Recu si ely, we
conclude he claim in n∗s eps, wi h δ= min{δ1,...,δn∗}.
4. An example. Li and Kloeden [14, Sec.3,Ex.3.2] ga e he ollowing example o
a scala o dina y di e en ial equa ion o illus a e hei esul s. Le λ0= 2√3/9
and Λ = [0, λ0] and le : Λ ×R→Rbe gi en by
(λ, x) = −x3+x+ 4√3/9−λ,
which is illus a ed in Figu e 1 below. In pa icula , o λ < λ0i has a single ze o
x(λ+)>0 and o λ=λ0a new ze o x(λ−
0)appea s.