A ac o s o Di e en ial Equa ions wi h Va iable Delays
Tom´as Ca aballo, Jos´e A. Langa
Dp o. 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)
E-mail: ca aball@nume .us.es ; [email p o ec ed]
and
James C. Robinson
Ma hema ics Ins i u e. Uni e si y o Wa wick. Co en y, CV4 7AL. UK
E-mail: [email p o ec ed]a wick.ac.uk
Using he ela i ely new concep o a pullback a ac o , we p esen some
esul s on he exis ence o a ac o s o di e en ial equa ions wi h a iable
delay. We gi e a a ie y o examples o which ou esul applies.
1. INTRODUCTION
The heo y o global a ac o s o au onomous sys ems as de eloped
by Hale in [7] owes much o examples a ising in he s udy o e a ded
unc ional di e en ial equa ions [8] ( o sligh ly di e en app oaches see
Babin and Vishik [1], Ladyzhenskaya [12], o Temam [15]). Al hough he
classical heo y can be ex ended in a ela i ely s aigh o wa d manne o
deal wi h ime-pe iodic equa ions, gene al non-au onomous equa ions such
as
˙x( ) = F( , x( ), x( −ρ( ))) (1)
all ou side i s scope.
Recen ly, a heo y o ‘pullback a ac o s’ has been de eloped (see sec ion
2) which allows many o he ideas o he au onomous heo y o be ex ended
o deal wi h such examples. Howe e , un il now his has only been applied
o o dina y and pa ial di e en ial equa ions.
I is ou in en ion he e o show how pullback a ac o s can be used o
in es iga e he beha iou o non-au onomous delay equa ions. In pa icu-
la , we a e able o compa e he dynamics o sys ems o o dina y di e en ial
equa ions wi h ha o he same sys em wi h a small delay, and show ha
hese a e ‘close’ in some global sense.
1
2TOM´
AS CARABALLO, JOS´
E A. LANGA AND JAMES C. ROBINSON
2. DELAY DIFFERENTIAL EQUATIONS AS DYNAMICAL
SYSTEMS
We ake as ou canonical example o a non-au onomous delay equa ion
a sys em wi h one, ime- a ying delay, ρ( ) whe e ρ:R→[0, h] is a con-
inuous unc ion and h > 0,
d
d x( ) = F( , x( ), x( −ρ( ))) xs=ψ, ψ ∈ C.(2)
The ini ial condi ion xsis speci ied in C, he space C0([−h, 0]; Rn) o con in-
uous unc ions om [−h, 0] in o Rn, and, o a unc ion x∈C0([−h, T]; Rn),
he no a ion xsdeno es he unc ion in Cgi en by
xs(θ) = x(s+θ) o all θ∈[−h, 0]
(and so makes sense o any 0 ≤s≤T).
This equa ion can be w i en in a mo e gene al amewo k, which allows
one o conside a la ge se o p oblems in a uni ied way. Ra he han make
he delay explici , we w i e
( , x ) = F( , x( ), x( −ρ( ))),
and so can ew i e (2) as
˙x( ) = ( , x )xs=ψ, ψ ∈ C.(3)
In wha ollows we concen a e on his o m o he equa ion, assuming ha
:R×C → Rnis con inuous and ‘a bounded map’ (i.e. maps bounded se s
in o bounded se s).
We no e he e ha his o mula ion immedia ely includes examples o he
han he single, ime- a ying delay o (2). Fo example, he in eg o-
di e en ial equa ion (see Kuang [11] o mo e de ails)
˙x( ) = Z0
−h
g( , s, x( +s)) ds
also i s in o his amewo k, al hough we do no de elop his heo y he e.
I is known (Hale [7]) ha o any (s, ψ)∈R×C he e exis s a unique so-
lu ion x( ;s, ψ) o (3) de ined on [s−h, αs,ψ). We assume ha αs,ψ = +∞,
o all s∈R,since we a e in e es ed in long- ime beha iou o solu ions.
We de ine a solu ion ope a o φ( , s) which gi es he solu ion (in C) a ime
when xs=ψ, ia
φ( , s)ψ=x (·;s, ψ).(4)
ATTRACTORS FOR DELAY DIFFERENTIAL EQUATIONS 3
3. PULLBACK ATTRACTORS
We now discuss he heo y o pullback a ac o s, as de eloped in Kloe-
den and S onie [9], Kloeden and Schmal uss [10], and C auel e al. [5]). As
is clea abo e, in he case o non-au onomous di e en ial equa ions he ini-
ial ime is jus as impo an as he inal ime, and he classical semig oup
p ope y o au onomous dynamical sys ems is no longe a ailable.
Ins ead o a amily o one ime-dependen maps S( ) we need o use a
wo-pa ame e p ocess φ( , s), as in oduced abo e in (4) (c . Sell [14]);
φ( , s)ψdeno es he solu ion a ime which was equal o ψa ime s.
The semig oup p ope y is eplaced by he p ocess composi ion p ope y
φ( , s)φ(s, ) = φ( , ) o all ≥s≥ ,
and, ob iously, he ini ial condi ion implies φ(s, s) =Id. As wi h he semi-
g oup composi ion S( )S(s) = S( +s), his jus exp esses he uniqueness
o solu ions.
[I is possible o p esen he heo y wi hin he mo e gene al amewo k
o cocycle dynamical sys ems. In his case he second componen o φis
iewed as an elemen o some pa ame e space J, so ha he solu ion can
be w i en as φ( , p)ψ, and a shi map θ :J→Jis de ined so ha he
p ocess composi ion becomes he cocycle p ope y,
φ( +τ, p) = φ( , θτp)φ(τ, p).
We do no pu sue his app oach he e, bu no e ha i has p o ed ex emely
ui ul, pa icula ly in he case o andom dynamical sys ems. Fo a ious
examples using his gene al se ing, see Kloeden and Schmal uss [10], o Sell
[14]. Fo his eason, pullback a ac o s a e o en e e ed o as ‘cocycle
a ac o s’].
As in he s anda d heo y o a ac o s, we seek an in a ian a ac ing
se . Howe e , since he equa ion is non-au onomous his se also depends
on ime.
De ini ion 3.1. Le φbe a p ocess on a comple e me ic space X. A
amily o compac se s {A( )} ∈Ris said o be a (global) pullback a ac o
o φi , o all s∈R, i sa is ies
i) φ( , s)A(s) = A( ) o all ≥s, and
ii) lims→∞ dis (φ( , −s)D,
A( ))=0, o allboundedsubse sDo X.
In he de ini ion, dis (A, B) is he Hausdo semidis ance be ween Aand
B, de ined as
dis (A, B) = sup
a∈A
in
b∈Bd(a, b), o A, B ⊆X.
4TOM´
AS CARABALLO, JOS´
E A. LANGA AND JAMES C. ROBINSON
P ope y i) is a gene aliza ion o he in a iance p ope y o au onomous
dynamical sys ems. The pullback a ac ing p ope y ii) conside s he s a e
o he sys em a ime when he ini ial ime −sgoes o −∞ (c . Chepyzho
and Vishik [3])
The no ion o an a ac o is closely ela ed o ha o an abso bing se .
De ini ion 3.2. {B( )} ∈Ris said o be abso bing wi h espec o he
p ocess φi , o all ∈Rand all D⊂Xbounded, he e exis s TD( )>0
such ha o all τ≥TD( )
φ( , −τ)D⊂B( ).
Indeed, jus as in he au onomous case, he exis ence o compac abso b-
ing se s is he c ucial p ope y in o de o ob ain pullback a ac o s. Fo
he ollowing esul see C auel and Flandoli [4] o Schmal uss [13].
Theo em 3.1. Le φ( , s)be a wo-pa ame e p ocess, and suppose φ( , s) :
X→Xis con inuous o all ≥s. I he e exis s a amily o compac ab-
so bing se s {B( )} ∈R, hen he e exis s a pullback a ac o {A( )} ∈R,
and
A( )⊂B( ) o all ∈R. Fu he mo e,
A( ) =
S
SS
D⊂X
boundedΛD( ),whe e
ΛD( ) =
n∈[
s≥n
φ(s, −s)D.
4. ATTRACTORS FOR NON-AUTONOMOUS DELAY
DIFFERENTIAL EQUATIONS
We showed abo e how o de ine he p ocess associa ed wi h he solu ion
o he delay di e en ial equa ion
˙x( ) = ( , x )xs=ψ(5)
ia
φ( , s)ψ=x (·;s, ψ).
ATTRACTORS FOR DELAY DIFFERENTIAL EQUATIONS 5
We now p o e a simple gene al esul on he exis ence o pullback a -
ac o s o his p oblem. P oo ha he condi ion o he heo em holds is
signi ican ly mo e one ous han he p oo o he heo em i sel .
Theo em 4.1. Suppose ha φ( , s)maps bounded se s in o bounded se s,
and ha he e exis s a amily {B( 0)} ∈Ro bounded abso bing se s o φ.
Then he e exis s a pullback a ac o o p oblem (5).
P oo . Using heo em 3.1 i su ices o p o e ha he e exis s a amily
o compac abso bing se s o φ. Fo each 0∈R, de ine
K( 0) = φ( 0, 0−h)B( 0−h).
K( 0) is clea ly abso bing, since o any bounded D⊂ C we ha e, o
≥TD( 0) + h, (he e TD( 0) deno es he abso p ion ime co esponding o
he amily {B( 0)} ∈R)
φ( 0, 0− )D=φ( 0, 0−h)φ( 0−h, ( 0−h)−( −h))D
⊂φ( 0, 0−h)B( 0−h) = K( 0).
Also, K( 0) is bounded, since φmaps bounded se s in o bounded se s.
Finally, K( 0) is a compac subse o C. This ollows using he A zel`a-
Ascoli heo em, since we ha e jus shown ha K( 0) is bounded, and he
equicon inui y ollows since, o ψ∈B( 0−h) and θ∈[−h, 0],

d
dθφ( 0, 0−h)ψ(θ)=
d
dθx( 0+θ; 0−h, ψ)=| ( 0+θ, x 0+θ(·; 0−h, ψ)|
which is bounded, using he assump ion on .
4.1. The case o s ong dissipa i i y
In his sec ion we suppose a dissipa i e p ope y o he nonlinea e m
o he di e en ial equa ion which will lead us o he exis ence o a uni o m
(o e ∈R) bounded abso bing se o he p ocess φand hence a pullback
a ac o .
We will suppose in his sec ion ha o some α > 0, β≥0
h ( , ψ), ψ(0)i≤−α|ψ(0)|2+β o all ψ∈Φ(h)C(6)
whe e h·,·i deno es he scala p oduc in Rnand
Φ(h)C={χ∈ C :χ=φ(s+h, s)ψ, some s∈R, ψ ∈ C}.
(No e ha Φ(h)Cis essen ially he se o all hose unc ions in Cwhich a e
ealisable as solu ions o he equa ion a e a ime h).

6TOM´
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E A. LANGA AND JAMES C. ROBINSON
Al hough his seems s ange a a i s iew, no e ha (6) is a consequence
o a mo e na u al se o assump ions in a ious pa icula examples. Indeed,
i we conside (2) wi h F:Rn→Rnuni o mly bounded and uni o mly
con inuous, i.e., o some k≥0 and some unc ion ω:R+→R+,
|F(x)| ≤ kand |F(x)−F(y)| ≤ ω(|x−y|),
and dissipa i e in a simila sense o (6), so ha , o some α0>0 and
β0≥0
hF(x), xi≤−α0|x|2+β0,
we eco e (6). Obse e ha , in his case, we a e assuming ha ( , x ) =
F(x( −ρ( )) o , mo e gene ally, ( , ψ) = F(ψ(−ρ( )), o all ψ∈ C, ∈R.
Indeed, we ha e
hF(x( −ρ( )), x( )i ≤ hF(x( )), x( )i+hF(x( −ρ( )) −F(x( )), x( )i
≤ −α0|x( )|2+β0+|x( )| |F(x( −ρ( )) −F(x( ))|
≤ −α0|x( )|2+β0+|x( )|ω(|x( −ρ( )) −x( )|)
≤ −α0|x( )|2+β0+|x( )|ω(kh)
≤ −α|x( )|2+β
o all ≥h, since hen x( ) is a solu ion o (2).
We now show ha (6) ensu es he exis ence o a pullback a ac o .
Theo em 4.2. Suppose ha (6) holds. Then he e exis s a amily o
bounded abso bing se s {B( 0)} 0∈R o (3), and hus we can conclude he
exis ence o a pullback a ac o o his p oblem.
P oo . We will p o e mo e han he exis ence o a amily o bounded
abso bing se s: in ac , he e exis s a uni o m (in 0) bounded abso bing
se o (3). Indeed, gi en D⊂ C bounded, he e exis s d≥0 such ha o
all ψ∈D, kψkC≤d, i.e.
kψkC= sup
θ∈[−h,0]
|ψ(θ)| ≤ d.
Take now ψ∈Dand conside
|φ( 0, 0− )ψ|= sup
θ∈[−h,0]
|x( 0+θ; 0− , ψ)|
= sup
τ∈[ 0−h, 0]
|x(τ; 0− , ψ)|.
Le us w i e x(τ) = x(τ; 0− , ψ), τ∈[ 0− , 0]. Then, mul iplying (5)
by x(τ) we ge
d
dτ|x(τ)|2= 2 hx(τ), (τ, xτ)i ≤ 2β−2α|x(τ)|2,
ATTRACTORS FOR DELAY DIFFERENTIAL EQUATIONS 7
o all τ≥ 0− .
Then, by G onwall’s lemma
|x(τ)|2≤ |x( 0− )|2e−2α(τ− 0+ )+β
α(1 −e−2α(τ− 0+ ))
≤ |ψ|2e−2α(τ− 0+ )+β
α
≤ |ψ|2e−2α(θ+ )+β
α
≤de2αhe−2α +β
α.
Thus we ob ain
sup
θ∈[−h,0]
|x( 0+θ)|2≤de2αhe−2α +β
α≤1 + β
α
i we ake ≥1
2αlog(de2αh) = TD. No e ha his ime TDdoes no depend
on 0.
4.2. A mo e gene al case
In he p e ious sec ion we conside ed di e en ial equa ions which only
depended on he delay e m, and had no explici dependence o he cu en
s a e (in o he wo ds, ( , x ) = F(x( ), x( −ρ( ))) = F(x( −ρ( )))).
Howe e , a dependence on bo h he cu en and e a ded s a e is mo e usual
in applica ions, and o en he equa ion can be in e p e ed as a pe u ba ion
o an o dina y di e en ial equa ion.
In his sec ion we will assume ha Fcan be w i en as he ollowing
sum:
F(x( ), x( −ρ( ))) = F0(x( )) + F1(x( −ρ( ))).
In his si ua ion, we shall show ha i a dissipa i e hypo hesis o he e m
F0holds, hen he assump ions on he o he e m can be elaxed.
Le us assume ha F0:Rn→Rnis a con inuous unc ions sa is ying
he dissipa i e assump ion as abo e:
hF0(x), xi ≤ −α0|x|2+β0, o all x∈Rn.(7)
Fi s ly, i we suppose ha F1:Rn→Rnis a con inuous and bounded
unc ion, i.e. he e exis s k≥0 such ha
|F1(x)| ≤ k, o all x∈Rn,
8TOM´
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E A. LANGA AND JAMES C. ROBINSON
hen i is easy o p o e ha (6) holds. Indeed, o e e y ψ∈Φ(h)Cand a
ixed ε<α0,
h ( , ψ), ψ(0)i=hF0(ψ(0)), ψ(0)i+hF1(ψ(−ρ( ))), ψ(0)i
≤ −α0|ψ(0)|2+β0+k|ψ(0)|
≤ −(α0−ε)|ψ(0)|2+β0+k2
4ε.
Secondly, i is s ill possible o weaken his boundedness on F1,al hough
now i is necessa y o assume mo e egula i y o he delay unc ion. Ins ead
o p o ing ha (6) holds, we p o e he exis ence o a amily o bounded
abso bing se s di ec ly.
Theo em 4.3. Assume F0sa is ies (7). Assume ha F1is sublinea ,
i.e. he e exis s k > 0such ha
|F1(x)|2≤k2(1 + |x|2), o all x∈Rn,
and suppose ha he delay unc ion ρis con inuously di e en iable wi h
ρ0( )≤ρ∗<1.Then, i k2< α2
0(1 −ρ∗), he e exis s a amily o bounded
abso bing se s, {B( 0)} 0∈R o (3), and consequen ly he e exis s a pullback
a ac o o his p oblem.
P oo . Choose a posi i e λ(small enough) and ano he posi i e εwhich
will be ixed la e . As in he las heo em, le us w i e x(τ) = x(τ; 0− , ψ),
τ∈[ 0− , 0], o ψin a gi en bounded se D⊂ C,i.e. kψkC≤d, o all
ψ∈D. Then, i ollows
d
dτeλτ |x(τ)|2=λeλτ |x(τ)|2+ 2eλτ hx(τ), (τ, xτ)i
=λeλτ |x(τ)|2+ 2eλτ hx(τ), F0(x(τ))i
+2eλτ hx(τ), F1(x(τ−ρ(τ)))i
≤(λ−2α0)eλτ |x(τ)|2+ 2β0eλτ
+εeλτ |x(τ)|2+eλτ ε−1|F1(x(τ−ρ(τ)))|2
≤(λ−2α0+ε)eλτ |x(τ)|2+ (2β0+k2ε−1)eλτ
+k2ε−1eλτ |x(τ−ρ(τ))|2.
By in eg a ion on he in e al [ 0− , τ],
eλτ |x(τ)|2−eλ( 0− )|x( 0− )|2≤2β0+k2ε−1
λeλτ −eλ( 0− )
+(λ−2α0+ε)Rτ
0− eλs|x(s)|2ds
+k2
εRτ
0− eλs|x(s−ρ(s))|2ds.
(8)
ATTRACTORS FOR DELAY DIFFERENTIAL EQUATIONS 9
E alua ing he e m con aining he delay unc ion by making he change
o a iable s−ρ(s) = uin he in eg al, we ob ain
Rτ
0− eλs|x(s−ρ(s))|2ds ≤1
1−ρ∗Rτ
0− −heλu+λh|x(u)|2du
≤eλh
1−ρ∗hR 0−
0− −heλu|x(u)|2du +Rτ
0− eλu|x(u)|2dui
≤eλh
1−ρ∗hR 0−
0− −heλu|ψ(u)|2du +Rτ
0− eλu|x(u)|2dui
≤eλh
1−ρ∗Rτ
0− eλu|x(u)|2du
+d2eλh
λ(1−ρ∗)eλ( 0− )−eλ( 0− −h),
and, consequen ly,
eλτ |x(τ)|2≤eλ( 0− )d2+2β0+k2ε−1λ−1eλτ −eλ( 0− )
+d2eλhk2ε−1
λ(1−ρ∗)eλ( 0− )−eλ( 0− −h)
+hλ−2α0+ε+eλhk2ε−1
(1−ρ∗)iRτ
0− eλs|x(s)|2ds.
Now, aking ε=α0and no icing ha o λsmall enough we can assu e
ha λ−2α0+ε+eλhk2ε−1
(1−ρ∗)is nega i e, i immedia ely ollows ha
|x(τ)|2≤d2h1 + eλh k2ε−1
λ(1−ρ∗)ieλ( 0− −τ)+2β0+k2ε−1λ−1,
and se ing τ= 0+θ, o θ∈[−h, 0],
|x( 0+θ)|2≤d21 + eλhk2ε−1
λ(1 −ρ∗)e−λ( +θ)+2β0+k2ε−1λ−1,
and, hus
sup
θ∈[−h,0]
|x( 0+θ)|2≤d2h1 + eλh k2ε−1
λ(1−ρ∗)ie−λ +λh +2β0+k2ε−1λ−1
≤1 + 2β0+k2ε−1λ−1
i
≥TD=λ−1log d21 + eλhk2ε−1
λ(1 −ρ∗)eλh.
16 TOM´
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E A. LANGA AND JAMES C. ROBINSON
Gi en an ini ial condi ion xs=ψ, wi h kψkC0≤M, conside
d
d (x( )−y( )) = F(x( )) −F(y( −ρ( ))).
Taking he inne p oduc wi h x( )−y( ) gi es
1
2
d
d |x( )−y( )|2=hF(x( )) −F(y( )), x( )−y( )i
+hF(y( )) −F(y( −ρ( ))), x( )−y( )i
≤L|x( )−y( )|2+L|y( )−y( −ρ( ))||x( )−y( )|
≤2L|x( )−y( )|2+2M0≤ ≤
kL|ρ( )| ≥
≤2L|x( )−y( )|2+2M0≤ ≤
kL ≥.
On [0, ] we can deduce ha
|x( )−y( )|2≤(eL −1)2M
L,
and so, in pa icula ,
|x( )−y( )|2≤(eL −1)2M
L o all ∈[0, ].
Now, s a ing om =we ha e
|x( )−y( )|2≤1
L[2M(eL −1) + kL]eL( −),
and, he e o e,
|x( )−y( )|2≤C(, ),
whe e C(, )→0 as ↓0+uni o mly on bounded ime in e als. In
pa icula , i ollows ha
sup
s∈R
kφ( +s, s)ψ− S( )ψkC0→0 o all ≥0.(20)
3. This ollows immedia ely om (15), since he p oo o he exis ence
o an abso bing se in Rnunde his condi ion is simple. The e is hen a
global a ac o A ⊂Rn, and we can de ine Aas in (19).
4. Finally, no e ha he adius o he abso bing se Bin heo em 4.2
depends on αand β, and using he calcula ions om sec ion 4.1.1 i ollows

ATTRACTORS FOR DELAY DIFFERENTIAL EQUATIONS 17
ha αand βcan be aken uni o m o e ∈(0, 0]. I hen ollows ha he
compac abso bing se in heo em 4.2 is gi en by
φ( 0, 0−0)B.
I ollows om (20) ha o e e y ∈(0, 0] his is a subse o a ixed
compac se K. Since A( )⊂B( ) (see heo em 3.1) he inal condi ion
is sa is ied, and an applica ion o heo em 6.1 gi es he esul as s a ed.
Rema k 6.1. No e ha one could also compa e he a ac o s by con-
side ing he subse s o Rn
A( ) = {y∈Rn:y=x( ), ∈[−, 0], x ∈A( )}.
I hen ollows ha
dis (A( ),A)→0
as →0, whe e now he dis ance is measu ed in Rn.
7. CONCLUSIONS
By using he cocycle a ac o we ha e ex ended he classical ea men
o a ac o s o delay di e en ial equa ions o he gene al nonau onomous
case, and in pa icula we ha e eco e ed esul s on pe iodic equa ions.
Fu he mo e, by using he uppe semicon inui y esul om Ca ballo and
Langa [2], we ha e shown ha he in oduc ion o a small delay has li le
e ec on he asymp o ic dynamics.
Finally, we no e ha hese ideas should be applicable o a wide class o
equa ions. In pa icula , equa ions wi h dis ibu ed delays, such as
˙x( ) = Z0
−h
( , s, x( +s)) ds,
and e en equa ions wi h in ini e delays, such as
˙x( ) = Z0
−∞
( , s, x( +s)) ds.
Fo he second o hese he phase space needs o be chosen much mo e
ca e ully han abo e, and we ha e no p esen ed esul s o his sys em
he e o a oid oo much no a ion. Fo de ails o he s anda d heo y see
Kuang [11].
18 TOM´
AS CARABALLO, JOS´
E A. LANGA AND JAMES C. ROBINSON
ACKNOWLEDGMENT
This wo k has been pa ially suppo ed by P oyec o DGICYT PB98-1134. JCR is a
Royal Socie y Uni e si y Resea ch Fellow, and would like o hank he Socie y o all
hei suppo . He would also like o hank EDAN o hei hospi ali y, and Ibe d ola o
hei gene osi y.
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