©2015 Tomás Ca aballo e al., licensee De G uy e Open.
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Nonau on. Dyn. Sys . 2015; 2:31–51
Resea ch A icle Open Access
Tomás Ca aballo, F ancisco Mo illas, and José Vale o*
A ac o s o non-au onomous e a ded
la ice dynamical sys ems
DOI 10.1515/msds-2015-0003
Recei ed Janua y 22, 2015; accep ed May 9, 2015
Abs ac : In his pape we s udy a non-au onomous la ice dynamical sys em wi h delay. Unde a he gene al
g ow h and dissipa i e condi ions on he nonlinea e m, we de ine a non-au onomous dynamical sys em and
p o e he exis ence o a pullback a ac o o such sys em as well. Bo h mul i alued and single- alued cases
a e conside ed.
Keywo ds: la ice dynamical sys ems, non-au onomous sys ems, di e en ial equa ions wi h delay, se - alued
dynamical sys ems, pullback a ac o
MSC: 34K05, 34K31, 35B40, 35B41, 35K55, 35K40, 37L30, 58C06
1In oduc ion
La ice dynamical sys ems o en a ise as an app oxima i e sys em o in ini e di e en ial equa ions o a pa ial
di e en ial equa ion in an unbounded domain, al hough hey also appea as models o a a ie y o phenom-
ena such as image p ocessing, pa e n ecogni ion, b ain science, among o he s.
In he las yea s many au ho s ha e been in e es ed in he asymp o ic beha iou o solu ions o such sys-
ems. As a esul , a shee numbe o pape s ha e been published conce ning he exis ence and p ope ies o
global a ac o s in he au onomous, nonau onomous and s ochas ic cases; wi h o wi hou uniqueness; in
weigh ed o unweigh ed spaces. Usually, he models unde conside a ion a e ob ained by a spa ial disc e iza-
ion o a pa abolic o a hype bolic equa ion (see e.g. [1], [2], [4], [5], [8], [11] [12], [15], [16], [19], [20], [22], [23],
[26], [28], [29]).
The addi ion o a delay in he sys em, which appea s na u ally in eal models, gi es ise o new di icul ies.
Re a ded au onomous la ice dynamical sys ems we e s udied om he poin o iew o dynamical sys ems
in [25], [27], [24]. These esul s we e imp o ed la e on by Ca aballo e . al. [13].
Ou main aim in his pape is o analyze he asymp o ic beha io o he ollowing nonau onomous e-
a ded la ice di e en ial equa ion







dui
d −(ui−1−2ui+ui+1)+λui+ i( ,ui )= 0, >τ,i∈Z,
ui(s)=ψi(s),∀s∈[τ−h,τ],
(1)
Tomás Ca aballo: Dp o. Ecuaciones Di e enciales y Análisis Numé ico, Uni e sidad de Se illa, Apdo. de Co eos 1160, 41080-
Se illa, Spain, E-mail: ca [email protected]
F ancisco Mo illas: Depa men d’Economia Aplicada, Facul a d’Economia, Uni e si a de Valéncia, Campus del Ta onge s
s/n, 46022-Valéncia, Spain, E-mail: F ancisco.Mo illas@u .es
*Co esponding Au ho : José Vale o: Cen o de In es igación Ope a i a, Uni e sidad Miguel He nández, A da. de la Uni e si-
dad, s/n, 03202-Elche, Spain, E-mail: j ale [email protected]
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32 |Tomás Ca aballo, F ancisco Mo illas, and José Vale o
whe e λ∈R. This model is ob ained a e a spa ial disc e iza ion o he scala e a ded eac ion-di usion
equa ion: 






du
d −∂2u
∂x2+λu + ( ,u ) = 0, >τ,x∈R,
u(s)=ψ(s),∀s∈[τ−h,τ].
He e u= (ui)i∈Z∈`2,Zdeno es he in ege s se and o a con inuous unc ion u: [τ−h,T]→Y(whe e Y
is some space), u deno es he segmen o he solu ion, i.e., he elemen in C[−h,0],Yde ined by u (s)=
u( +s),s∈[−h,0].
The exis ence and uniqueness o solu ions o p oblem (1) we e add essed in [13]. I is wo h poin ing
ou ha a he gene al assump ions on he nonlinea unc ions i(jus con inui y and g ow h condi ions) a e
imposed, no ensu ing any kind o compac ness p ope ies in he space `2 o he co esponding Nemy skii
ope a o , which a e necessa y in o de o apply he sol abili y esul s s a ed in o he pape s (see [14], [17], [21]).
Also, in he au onomous case, when does no depend explici ely on , he exis ence o global a ac o s was
es ablished in bo h he mul i alued and single- alued se ings o a pa icula ype o unc ions i.
In he p esen pape we ex end he esul s ca ied ou in [13] o he nonau onomous case. Fo his aim we
apply he well-known heo y o pullback a ac o s [7], [9].
The pape is o ganized in wo pa s. In Sec ion 2 we ecall b ie ly he gene al sol abili y heo ems p o ed
in [13] and apply hem o p oblem (1) unde a he gene al assump ions on he nonlinea e m . In Sec ion 3
we conside he pa icula case o a la ice dynamical sys em wi h a nonlinea e m o he o m
i( ,ui ) = F0,i(ui( )) +F1,iui −ρ( )+
0
Z
−h
bi( ,s,ui( +s)) ds,
wi h ρ(·)∈C1(R)and ρ( )∈[0,h] o all ∈R.Unde some dissipa i e and sublinea g ow h condi ions
on he maps F0,i,F1,i,bi, we de ine o his p oblem a mul i alued p ocess Uand p o e he exis ence o a
pullback a ac o . Addi ionally, wi h ex a Lipschi z condi ions we ob ain uniqueness o he Cauchy p oblem,
so ha Uis in ac a single- alued p ocess.
2Exis ence o solu ions o a la ice di e en ial equa ion wi h delay
2.1 Some esul s on he exis ence o solu ions o di e en ial equa ions wi h delay
in Banach spaces
Le us i s ecall some abs ac esul s which we e p o ed in [13] and which will be use ul in he p esen case.
Le Ebe a eal Banach space wi h dual E*,and le E0=C([−h,0],E), wi h no ms k·k,k·k*and k·kE0,
espec i ely, whe e kφkE0= max ∈[−h,0] kφ( )k. Also,
BX(y0, ) = {y∈X:ky−y0kX≤ },
whe e X=Eo E0, and (·,·)will deno e he pai ing be ween Eand E*.
Le us conside he ollowing Cauchy p oblem o a unc ional di e en ial equa ion in a Banach space:







du
d =F( ,u ),
uτ=ψ∈E0,
(2)
whe e F:R×E0→E.Also, o any u∈C([τ−h,+∞),E), he unc ion u ∈E0, ≥τ,is de ined by
u (s)=u( +s),s∈[−h,0].
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A ac o s o non-au onomous e a ded la ice dynamical sys ems |33
Le Ewbe he space Eendowed wi h he weak opology. We conside he space E0,w=C([−h,0],Ew). Le
un,u∈E0,w. We say ha un→uin E0,wi
un(sn)→u(s)in Ew o all sn→s∈[−h,0].
We will say ha he unc ion Fis sequen ially weakly con inuous in bounded se s i n→ ,un→uin E0,w
and kunkE0≤M, o all n, imply F( n,un)→F( ,u)in Ew.
On he o he hand, we will say ha he unc ion Fis bounded i i maps bounded subse s o R×E0on o
bounded subse s o E.
De ini ion 1. The map u: [τ−h,T]→Eis called a solu ion o p oblem (2) i uτ=ψ,u(·)is con inuous, once
weakly con inuously di e en iable in [τ,T]and sa is ies
u( )=u(τ)+
Zτ
(s,us)ds, o all ∈[τ,T].
Rema k 2. I ollows om his de ini ion ha o any solu ion uo (2), he map 7→ u ∈E0is con inuous.
Rema k 3. We no e ha i F:R×E0→Eis sequen ially weakly con inuous in bounded se s and he map
7→ u ∈E0is con inuous, hen 7→ F( ,u )is weakly con inuous, hence weakly measu able. I Eis sepa able,
we ob ain ha 7→ F( ,u )is s ongly measu able. I we assume, mo eo e , ha he map Fis bounded, hen we
ha e ha F(·,u·)∈L1(τ,T;E).
I F:R×E0→Eand 7→ u ∈E0a e con inuous, hen he map 7→ F( ,u )is con inuous, hence s ongly
measu able. I we assume, mo eo e , ha he map Fis bounded, hen we ha e ha F(·,u·)∈L1(τ,T;E).
Then, we ecall now some esul s ensu ing he exis ence and uniqueness o solu ions o p oblem (2), which
we e p o ed in [13].
Theo em 4. Assume ha Eis e lexi e and sepa able. Le :R×E0→Ebe sequen ially weakly con inuous
in bounded se s, and le Fbe a bounded map. Then, o each >0, he e exis s a( )>0such ha i ψ∈E0and
kψkE0≤ , p oblem (2) possesses a leas one solu ion de ined on [0,a( )].Mo eo e , u(·)is a.e. di e en iable
and du
d = ( ,u ) o a.a. ∈0,a( ).
I we assume addi ionally ha :R×E0→Eis con inuous, hen u∈C1[0,a]; Eand he sepa abili y o
Eis no needed.
Theo em 5. Assume he condi ions o Theo em 4. I a solu ion u(·)o (2) has a maximal in e al o exis ence
[0,b)and he e exis s K>0such ha ku( )k≤K, o all ∈[0,b), hen b= +∞, ha is, u(·)is a globally
de ined solu ion.
Le J:E→2E*be he duali y map, i.e. J(y) = {ξ∈E*: (y,ξ) = kyk2=kξk2
*},∀y∈E. We s a e a esul
conce ning uniqueness o solu ions.
Theo em 6. Assume he hypo heses o Theo em 4. Also, suppose ha , o any M>0, he e exis s β(·,M)∈
L1
loc (R)such ha β( ,M)≥0 o a.a. ∈Rand he ollowing inequali y holds:
( ( , )− ( ,w),j)≤β( ,M)k −wk2
E0,(3)
o all j∈J( (0)−w(0)), all ,w∈E0wi h k kE0,kukE0≤M, and a.a. ∈R. Then, o each >0, he e exis s
a( )>0such ha i ψ∈E0and kψkE0≤ , p oblem (2) has a unique solu ion de ined on [0,a( )].
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34 |Tomás Ca aballo, F ancisco Mo illas, and José Vale o
2.2 La ice dynamical sys ems wi h delay: se ing o he p oblem
Fo a gi en τ∈R, conside he ollowing i s o de la ice dynamical sys em wi h ini e delay







dui
d −(ui−1−2ui+ui+1)+λui+ i( ,ui )= 0, >τ,i∈Z,
ui(s)=ψi(s−τ),∀s∈[τ−h,τ],
(4)
whe e λ∈R.
We conside he sepa able Hilbe space `2={ =( i)i∈Z:Pi∈Z 2
i<∞}wi h no m k k=qPi∈Z 2
i
and scala p oduc (w, )=Pi∈Zwi i,and also he Banach space `∞={ =( i)i∈Z: supi∈Z| i|<∞}wi h
no m k k∞= supi∈Z| i|.
Fu he , we shall use he no a ion E=`2,E0=C[−h,0],`2,E1=C([−h,0],R), wi h he no ms
kukE0= maxs∈[−h,0] ku(s)k,kukE1= maxs∈[−h,0] |u(s)|. Also, pu E∞=C[−h,0],`∞wi h no m kukE∞=
maxs∈[−h,0] ku(s)k∞. We no e ha E0⊂E∞, as
ku( )−u(s)k∞= sup
i∈Z
|ui( )−ui(s)|≤sX
i∈Z
|ui( )−ui(s)|2=ku( )−u(s)k,∀ ,s∈[−h,0],
and
kukE∞= max
s∈[−h,0] sup
i∈Z
|ui|≤max
s∈[−h,0] sX
i∈Z
|ui|2=kukE0.
We conside he ollowing condi ions:
(H1)The ope a o :R×E0→Egi en by he ule ( ( , ))i= i( , i),i∈Z, is well de ined and bounded.
(H2)The maps i:R×C([−h,0],R)→Ra e con inuous.
We shall i s p o e he exis ence o solu ions o p oblem (4). Fo his aim we shall ew i e i in an abs ac
o m. We de ine he ope a o A:E→Eby
(A )i:= − i−1+ 2 i− i+1,i∈Z.
Also, we de ine he ope a o s B,B*:E→Eby
(B )i:= i+1 − i,B* i:= i−1− i.
I is easy o check ha
A=B*B=BB*,
B*w, =(w,B ).
Then he ope a o F:R×E0→Eis de ined by
F( , )=−A (0)− ( , )−λ (0)
and (4) can be ew i en as







du
d =F( ,u ), >τ,
uτ=ψ,i.e. u(s)=ψ(s−τ),∀s∈[τ−h,τ].
(5)
Lemma 7. Le (H1)-(H2) hold. Then he map :R×E0→Eis sequen ially weakly con inuous in bounded se s.
Also, he map A:E→Eis weakly con inuous.
P oo . Le n→ in R, and n→ ∈E0,w,wi h  nE0≤M1 o all n, and le w∈`2be a bi a y. Fo
any ε>0we ake K0(ε)>0such ha P|i|≥K0|wi|2<ε. Since is bounded, he e exis s M2>0such ha
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A ac o s o non-au onomous e a ded la ice dynamical sys ems |35
  n, n≤M2,k ( , )k≤M2, o all n. Also, as n→ and n
i→ iin C([−h,0],R), o all i, (H2) implies
he exis ence o N(K0,ε)such ha P|i|<K0 i n, n
i− i( , i)2<ε2i n≥N.Hence,
  n, n− ( , ),w≤sX
|i|<K0 i n, n
i− i( , i)2kwk+k ( , )k+  n, nsX
|i|≥K0
|wi|2
≤εkwk+ 2M2ε.
The esul o he ope a o Acan be p o ed simila ly. This comple es he p oo .
Theo em 8. Le (H1)-(H2) hold. Fo each >0 he e exis s a( )>0such ha i ψ∈E0and kψkE0≤ ,
hen p oblem (4) has a leas one solu ion de ined on [τ,τ+a( )].Mo eo e , u(·)is a.e. di e en iable and
du
d =F( ,u ) o a.a. ∈τ,τ+a( ).
P oo . Lemma 7 implies ha he ope a o Fis sequen ially weakly con inuous in bounded se s. Since is
bounded, Fis also bounded. The esul ollows om Theo em 4.
In o de o ob ain ha he map is con inuous, we need an assump ion which is s onge han (H1).
(H3)The ope a o :R×E0→Egi en by ( ( , ))i= i( , i),i∈Z, is well de ined, and o any ( , )∈R×E0,
we ha e
X
|i|≥K
| i( , i)|2≤Ck kE0
max
s∈[−h,0] X
|i|≥K
2
i(s)+bK( )
, o all K∈Z+,
whe e bK( )→0+as K→∞uni o mly in compac se s, and C(·)≥0is a con inuous non-dec easing
unc ion.
Rema k 9. Condi ion (H3) implies ha he map is bounded.
Lemma 10. Le (H2)-(H3) hold. Then, he map :R×E0→Eis con inuous.
P oo . Le n→ in R, and n→ in E0. Then o any ε>0 he e exis s K(ε)such ha
max
s∈[−h,0] X
|i|≥K n
i(s)2<ε,max
s∈[−h,0] X
|i|≥K
| i(s)|2<ε.
Then by (H3) one can choose K1(ε)≥K(ε)such ha
X
|i|≥K1 i n, n
i2≤Rε,X
|i|≥K1
| i( , i)|2≤Rε,
o some R>0.On he o he hand, by (H2) we ob ain he exis ence o N(ε,K)such ha
X
|i|<K1 i n, n
i− i( , i)2<εi n≥N.
Thus,
X
i∈Z i n, n
i− i( , i)2≤X
|i|<K1 i n, n
i− i( , i)2+ 2 X
|i|≥K1 i n, n
i2+ 2 X
|i|≥K1
| i( , i)|2
≤ε+ 2Rε, i n≥N.
Co olla y 11. Unde condi ions(H2)-(H3), hesolu iongi eninTheo em 8 belongs o he space C1[τ,τ+a]; E.
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36 |Tomás Ca aballo, F ancisco Mo illas, and José Vale o
In o de o ob ain he uniqueness o solu ions we need an addi ional Lipschi z assump ion.
(H4)Fo any M>0 he e exis s β( ,M)≥0such ha β(·,M)∈L1(R)and
 ( ,z)− ( , ),z(0) − (0)≥ −β( ,M)kz− k2
E0,
i kzkE0,k kE0≤M, ∈R.
Theo em 12. Assume (H1)-(H2) and (H4). Then he solu ion gi en in Theo em 8 is unique.
P oo . Le z, ∈E0,kzkE0,k kE0≤M,and w=z− . I ollows om (H4) and Aw(0),w(0)=Bw(0),Bw(0)≥
0 ha
F( ,z)−F( , ),z(0) − (0)=−Aw(0),w(0)−λkwkE0− ( ,z)− ( , ),w(0)
≤β( ,M)kwkE0.
Then he esul ollows om Theo em 6.
We now aim o s udy he asymp o ic beha iou o solu ions o p oblem (4). In pa icula , we will show he
exis ence o a non-au onomous a ac o . When condi ions (H1)-(H2), (H4) hold, i we assume ha e e y so-
lu ion is global ( his is ue i we ob ain an es ima e o he solu ions by Theo em 5), hen we can de ine he
map U:Rd×E0→E0,R2
d={( ,τ)∈R2: ≥τ}by
U( ,τ,ψ)=u ,
whe e u(·)is he unique solu ion o (4) wi h uτ=ψ.Mo eo e , i is easy o p o e, using (3) and G onwall’s
lemma, ha he map ψ7→ U( ,τ,ψ)is con inuous o any τ≤ .The map Uis a p ocess, ha is, U(τ,τ,ψ) = ψ
and
U( ,τ,ψ) = U( , ,U( ,τ,ψ)) o all τ≤ ≤ and ψ∈E0.(6)
On he o he hand, i we assume only (H1)-(H2) and ha e e y solu ion is global, hen we can de ine a
mul i alued semi low by U:R2
d×E0→P(E0)(P(E0)is he se o all non-emp y subse s o E0) by
U( ,τ,ψ)={u :u(·)is a solu ion o (4) wi h uτ=ψ}.(7)
Since we do no ha e uniqueness o he Cauchy p oblem, his map is in gene al mul i alued. In a simila way
o he au onomous case [19, Lemma 13] one can p o e ha i is a mul i alued p ocess, ha is:
1. U(τ,τ,·)=Id ( he iden i y map);
2. U( ,τ,ψ)⊂U( , ,U( ,τ,ψ)) o all ψ∈E0,τ≤ ≤ .
Mo eo e , i is s ic , ha is, U( ,τ,ψ)=U( , ,U( ,τ,ψ)) o all ψ∈E0,τ≤ ≤ .
Now, we will ecall he main esul s om he heo y o pullback a ac o s. Fi s , le us conside he case
o a single- alued p ocess [9], [10] (see also [18]).
Le Xbe a comple e me ic space. Suppose ha Dis a nonemp y class o pa ame e ized se s b
D={D( ); ∈
R} ⊂ P(X),whe e P(X)deno es he amily o all nonemp y subse s o X.
De ini ion 13. The p ocess Uis said o be pullback D-asymp o ically compac i o any ∈R, any b
D∈D,
any sequence τn→−∞,and any sequence yn∈U( ,τn,D(τn))}is ela i ely compac in X.
De ini ion 14. I is said ha b
B∈Dis pullback D-abso bing o he p ocess Ui o any ∈Rand any b
D∈D,
he e exis s a τ0( ,b
D)≤ such ha
U( ,τ,D(τ)) ⊂B( ) o all τ≤τ0( ,b
D).
De ini ion 15. The amily b
A={A( ); ∈R} ⊂ P(X)is said o be a pullback D-a ac o o U(·,·)i :
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A ac o s o non-au onomous e a ded la ice dynamical sys ems |37
1. A( )is compac o all ∈R,
2. b
Ais pullback D-a ac ing, i.e.,
lim
τ→−∞ dis (U( ,τ,D(τ)),A( )) = 0,
o all b
D∈D, and all ∈R,
3. b
Ais in a ian , i.e.,
U( ,τ,A(τ)) = A( ), o − ∞ <τ≤ <+∞.
We ha e he ollowing esul .
Theo em 16. Suppose ha he map ψ7→ U( ,τ,ψ)is con inuous o any τ≤ and ha he p ocess Uis
pullback D-asymp o ically compac . Le b
B∈Dbe a amily o pullback D-abso bing se s o U(·,·). Then, he
amily b
A={A( ); ∈R} ⊂ P(X)de ined by A( ) = Λ(b
B, ), ∈R,whe e
Λ(b
D, ) =
s≤ [
τ≤s
U( ,τ,D(τ))!, o each b
D∈D,
is a pullback D-a ac o o U(·,·)which sa is ies in addi ion ha
A( ) = [
b
D∈D
Λ(b
D, ), o ∈R.
Fu hemo e, b
Ais minimal in he sense ha i b
C={C( ); ∈R} ⊂ P(X)is a amily o closed se s such ha
limτ→−∞ dis (U( ,τ,B(τ)),C( )) = 0, hen A( )⊂C( ).
The amily Dis said o be inclusion-closed i b
D∈Dand ∅=B( )⊂D( ), o all ∈R, implies b
B∈D.I he
amily is inclusion-closed and he abso bing se b
B∈Dsa is ies ha he se s B( )a e closed, hen A( )⊂B( )
implies ha he a ac o b
Abelongs o D.
Le us conside now he case o a mul i alued p ocess. The ollowing esul is p o ed in [7] (see also [6]
o a mo e gene al non-au onomous and andom amewo k).
The de ini ions o pullback D-asymp o ically compac ness, pullback D-abso bing amily and pullback
D-a ac ion a e he same as in he single- alued case. Fo ixed τ≤ he mapping U( ,τ,·)is said o be
uppe -semicon inuous i o any x0∈Xand o e e y neighbo hood Nin Xo he se U( ,τ,x0), he e exis s
δ>0such ha U( ,τ,y)⊂Nwhene e dX(x0,y)<δ.
De ini ion 17. A amily b
A=A( ) : ∈R⊂P(X)is said o be a global pullback D-a ac o o he MNDS U
i A( )is compac o any ∈R,b
Ais pullback D-a ac ing, and b
Ais nega i ely in a ian , i.e.,
A( )⊂U( ,τ,A(τ)), o any ( ,τ)∈R2
d.
b
Ais said o be a s ic global pullback D-a ac o i he in a iance p ope y in he hi d i em is s ic , i.e.,
A( ) = U( ,τ,A(τ)), o ( ,τ)∈R2
d.
Theo em 18. Assume ha he map ψ7→ U( ,τ,ψ)is uppe -semicon inuous and possesses closed alues. Le
b
B=B( ) : ∈R∈Dbe pullback D-abso bing and such ha Uis asymp o ically compac wi h espec o b
B.
Then, he se b
Agi en by
A( ) := Λb
B, =
s≤ [
τ≤s
U( ,τ,B(τ)) ∈R,(8)
is a pullback D-a ac o o he MNDS U.
Mo eo e , suppose ha Dis inclusion-closed and ha B( )is closed in X o any ∈R. Then he amily b
A
de ined by (8) belongs o D, and is he unique pullback D-a ac o wi h his p ope y. In addi ion, in his case,
i Uis a s ic MNDS, hen b
Ais s ic ly in a ian .
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38 |Tomás Ca aballo, F ancisco Mo illas, and José Vale o
3A la ice sys em wi h sublinea non-au onomous e a ded e ms
We shall conside a unc ion :R×E0→Egi en by he ule ( ( , ))i= i( , i)and
i( , i)=F0,i( , i(0)) +F1,i , i−ρ( )+
0
Z
−h
bi( ,s, i(s)) ds,
whe e ρ(·)∈C1(R)and ρ( )∈[0,h] o all ∈R, ha is, pu ing =u =u( +·), p oblem (4) can be
ew i en as 






dui
d −(ui−1−2ui+ui+1)+λui+F0,i( ,ui( )) +F1,i ,ui −ρ( )
+R0
−hbi( ,s,ui( +s)) ds = 0, >τ,i∈Z,
ui(s)=ψi(s−τ),∀s∈[τ−h,τ].
(9)
We conside he ollowing condi ions:
(C1) λ>0.
(C2) F0,i:R2→Ra e con inuous and sa is y ha F0,i(x)x≥ −C0,i( ),C0∈C(R;`1)and
Z
−∞
kC0(s)k`1eδsds <∞, o all ∈Rand δ>0.
(C3) F0,i( ,x)≤H(|x|)|x|+C1,i( ), o all x∈R, whe e C1∈C(R;`2), and H(·)≥0is a con inuous and
non-dec easing unc ion.
(C4) F1,i:R2→Ra e con inuous and sa is y ha F1,i( ,x)≤K1|x|+C2,i( ), o all x∈R,whe e C2∈
C(R;`2),K1>0and
Z
−∞
kC2(s)k2
`2eδsds <∞, o all ∈Rand δ>0.
(C5) |bi( ,s,x)|≤m0,i( ,s)+m1,i(s)|x|, o all x∈Rand a.a. s∈(−h,0), whe e bia e Ca a heodo y in
he sense ha i is measu able in sand con inuous in ( ,x).
Also, m0,i( ,·),m1,i(·)∈L1(−h,0),m0,i( ,s),m1,i(s)≥0and de ining M0,i( ) = R0
−hm0,i( ,s)ds
and M1,i=R0
−hm1,i(s)ds we assume ha M1:= qPi∈ZM2
1,i<∞,M0( ) := qPi∈ZM2
0,i( )<∞,
M0∈C(R;R+)and
Z
−∞
(M0(s))2eδsds <∞, o all ∈Rand δ>0.
(C6) ρ∈C1R,[0,h]and ρ′( )≤ρ*<1.
Le us check condi ions (H1)-(H3). Fi s , in o de o ob ain (H1) we p o e ha is well de ined and
bounded. We no e ha
| i( , i)|≤F0,i( , i(0))+F1,i , i−ρ( )+
0
Z
−h
|bi( ,s, i(s))|ds.(10)
Fo he i s e m we ha e by (C3) ha
F0,i( , i(0))2≤2H2(| i(0)|)| i(0)|2+C2
1,i( )(11)
≤2χ(k kE0)| i(0)|2+ 2C2
1,i( ),
whe e χ(k kE0) = maxi∈ZH2(| i(0)|), which exis s because H(·)is non-dec easing and ∈E0.Then,
X
i∈ZF0,i( , i(0))2≤2χ(k kE0)k k2
E0+ 2 kC1( )k2. (12)
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A ac o s o non-au onomous e a ded la ice dynamical sys ems |39
As o he second e m we ob ain hanks o (C4) ha
X
i∈ZF1,i , i−ρ( )2≤2K2
1X
i∈Z i−ρ( )2+ 2 kC2( )k2
≤2K2
1k k2
E0+ 2 kC2( )k2. (13)
Now, o he e m wi h he in eg al delay, aking in o accoun (C5), we p oceed as ollows:
0
Z
−h
|bi( ,s, i(s))|ds ≤
0
Z
−hm0,i(s)+m1,i(s)| i(s)|ds
≤M0,i( ) + k kE∞M1,i.
Then
X
i∈Z

0
Z
−h
|bi( ,s, i(s))|ds

2
≤2X
i∈Z
M2
0,i+ 2 k k2
E∞X
i∈Z
M2
1,i
≤2M2
0( ) + 2 k k2
E0M2
1. (14)
Using (12)-(14) in (10) we ob ain ha is well de ined and bounded.
Now, we check (H2), i.e., ha he maps i:R×C([−h,0],R)→Ra e con inuous. We conside n∈
R, nn∈N⊂C([−h,0],R)and 0∈R, 0∈C([−h,0],R)such ha n→ 0, n→ 0in C([−h,0],R). Now,
we ha e  i n, n− i 0, 0≤F0,i n, n(0)−F0,i 0, 0(0)
+F1,i n, n−ρ( n)−F1,i 0, 0−ρ( 0)
+
0
Z
−h
bi n,s, n(s)ds −
0
Z
−h
bi 0,s, 0(s)ds
.
F om (C2) and (C4), F0,iand F1,ia e con inuous unc ions. Also, om (C5) and Lebesgue’s heo em, he las
e m con e ges o 0. Thus, he con inui y o i ollows.
To check (H3) we obse e ha
X
|i|≥K

0
Z
−h
|bi( ,s, i(s))|ds

2
≤2X
|i|≥K

0
Z
−h
m0,i(s)ds

2
+ 2 X
|i|≥K

0
Z
−h
m1,i(s)| i(s)|ds

2
≤2X
|i|≥K
M2
0,i( ) + 2 k k2
E0X
|i|≥K
M2
1,i.
Also, by (10), (11) and (C4) we ha e
X
|i|≥K
| i( , i)|2≤R
χ(k kE0)X
|i|≥K
| i(0)|2+X
|i|≥K
C2
1,i( ) + K2
1X
|i|≥K i−ρ( )2
+X
|i|≥K
C2
2,i( )+X
|i|≥K
M2
0,i( ) + k k2
E0X
|i|≥K
M2
1,i

≤Ck kE0
max
s∈[−h,0] X
|i|≥K
2
i(s)+bK( )
,
whe e bK→0+as K→∞uni o mly in compac se s, and C(·)≥0is a con inuous non-dec easing unc ion.
Thus, (H3) holds.
Then Theo em 8 and Co olla y 11 imply ha o any ψ∈E0 he e exis s, a leas , one solu ion u(·)∈
C1[τ,α),Ein a maximal in e al [τ,α). In o de o ob ain ha e e y solu ion is globally de ined we need
o p o e some es ima es. This will be done in he nex sec ion.
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46 |Tomás Ca aballo, F ancisco Mo illas, and José Vale o
Taking ϵ=λand using condi ion (16), we ha e
eη X
i∈Z
ρK,i|ui( )|2≤eητ X
i∈Z
ρK,i|ui(τ)|2+2
ηeη −eητC( 2)
K
+
Zτ
eηs 


2ρ1
2
KC2(s)
2
λ+Pi∈ZρK,iM2
0,i(s)
ˆ
ϵ+ 2 kρKC0(s)k`1

ds
+2K2
1
λ
eηh
η1−ρ*ρ1
2
Kψ
2
E0eητ −eη(τ−h)+ 2M1
Zτ
eηs ρ1
2
Kus
2
E0
ds.(40)
Le θ∈[−h,0]. We eplace by +θin (40), and use ha ρ1
2
Ku( +θ)=ρ1
2
Kψ( +θ)≤ρ1
2
KψE0
i +θ<τ;
mul iplying by e−η( +θ)we ob ain
X
i∈Z
ρK,i|ui( +θ)|2≤eητe−η( +θ)X
i∈Z
ρK,i|ui(τ)|2
+e−η( +θ)
+θ
Zτ
eηs 


2ρ1
2
KC2(s)
2
λ+ρ1
2
KM0(s)
2
ˆ
ϵ+ 2 kρKC0(s)k`1

ds
+2
ηeη( +θ)−eητC( 2)
Ke−η( +θ)
+2K2
1
λ
eηhe−η( +θ)
η1−ρ*ρ1
2
Kψ
2
E0eητ −eη(τ−h)+ 2M1e−η( +θ)
+θ
Zτ
eηs ρ1
2
Kus
2
E0
ds,
and
eη ρ1
2
Ku 
2
E0
≤ eηh +2K2
1e2ηh
λη 1−ρ*!ρ1
2
Kψ
2
E0
eητ
+eηh
Zτ
eηs 


2ρ1
2
KC2(s)
2
λ+Pi∈ZρK,iM2
0,i(s)
ˆ
ϵ+ 2 kρKC0(s)k`1

ds
+2
η
C( 2)
Keη + 2M1eηh
Zτ
eηs ρ1
2
Kus
2
E0
ds.
We can ew i e his exp ession as
eη ρ1
2
Ku 
2
E0
≤2
η
C( 2)
Keη +ˆ
Cρ1
2
Kψ
2
E0
eητ +
Zτ
eηsβρK(s)ds +L
Zτ
eηs ρ1
2
Kus
2
E0
ds, (41)
whe e we ha e used he no a ion
ˆ
C:= eηh +2K2
1
λη 1−ρ*e2ηh,
βρK( ):= eηh 


2ρ1
2
KC2( )
2
λ+Pi∈ZρK,iM2
0,i(s)
ˆ
ϵ+ 2 kρKC0( )k`1

,
L:= 2M1eηh.
Now, p oceeding in a simila way o (32) and using η−L>0(see Rema k 20) we ob ain
ρ1
2
Ku 
2
E0
≤2e(L−η) e(η−L)τˆ
Cρ1
2
Kψ
2
E0
+2C( 2)
K(η−L)+e−η
Zτ
eηsβρK(s)ds +e(L−η)
Zτ
e(η−L)sβρK(s)ds. (42)
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A ac o s o non-au onomous e a ded la ice dynamical sys ems |47
Now, i is con enien o keep in mind he de ini ion o βρk, and i s dependence on C0,M0and C2. I ollows
ha
βρK(s)→0as K→∞ o any s.
Hence, Lebesgue’s Domina ed Con e gence Theo em implies ha
Z
−∞
eδsβρK(s)ds →0,as K→∞ o any ∈[ 1, 2],δ>0.
Thus, he e exis Tϵ, 1, 2,b
Bη≤ 1,Kϵ, 1, 2,b
Bη≥1such ha
max
s∈[−h,0] sX
|i|≥2K
(ui( +s))2≤max
s∈[−h,0] sX
i∈Z
ρK,i(ui( +s))2
=ρ1
2
Ku E0
≤ϵ,i τ≤T, ∈[ 1, 2].
3.3 Exis ence o he pullback a ac o : gene al case
We know ha unde he assump ions o P oposi ion 19, he map Ugi en by (7) is a s ic mul i alued p ocess.
Fo any ini ial da a ψ∈E0we deno e
Dτ(ψ)=u(·)is a global solu ion o (9) wi h ini ial da a uτ=ψ.
We will p o e ha he map ψ7→ U( ,τ,ψ)is uppe -semicon inuous and has closed alues, and also ha
Uis asymp o ically compac wi h espec o he pullback Dη-abso bing amily b
Bηde ined in Co olla y 22.
Fi s , we ob ain an auxilia y lemma.
Lemma 26. We assume he condi ions o Lemma 23. Le ψn→ψin E0. Then:
1. Fo a bi a y ϵ>0,τ≤T he e exis s K(ϵ,τ,T)such ha o any un(·)∈Dτψn,
max
s∈[−h,0] sX
|i|≥2Kun
i( +s)2≤ϵ,∀ ∈[τ,T].(43)
2. Le un(·)∈Dτψn. Then he e exis s u(·)∈Dτ(ψ)and a subsequence unko unsuch ha
unk→uin C([τ,T],E) o all T>τ.(44)
P oo . I ollows om ψn→ψin E0 he exis ence o K1(ϵ)>0such ha
X
i∈Z
ρK,iψn
i(s)2<ϵ,∀n,s∈[−h,0]
X
i∈Z
ρK,iψ0
i(s)
2<ϵ,∀s∈[−h,0],
i K≥K1.Now, om (42) we ob ain he exis ence o K(ϵ,τ,T)≥K1such ha
ρ1
2
Kun

2
E0
≤2e(L−η) e(η−L)τˆ
Cρ1
2
Kψn
2
E0
+2C(T)
K(η−L)+e−η
Zτ
eηsβρK(s)ds +e(L−η)
Zτ
e(η−L)sβρK(s)ds ≤ϵ,
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48 |Tomás Ca aballo, F ancisco Mo illas, and José Vale o
o all ∈[τ,T], whe e we ha e used βρK(s)→0as K→∞ o any sand he Lebesgue Domina ed Con e -
gence Theo em. The e o e,
max
s∈[−h,0] sX
|i|≥2Kun
i( +s)2≤max
s∈[−h,0] sX
i∈Z
ρK,iun
i( +s)2
=ρ1
2
Kun
E0
≤ϵ,
which p o es (43). Nex , om P oposi ion 19 we ha e ha un
is bounded in E0. Then, using (43) one can p o e
in a s anda d way (see [13, p.71] o he de ails) ha un( )is p ecompac in E o any ∈[τ,T]. A e ha ,
ollowing he same lines as in [13, p.71], we can ob ain he exis ence o u(·)∈Dτ(ψ)and a subsequence such
ha un(·)→u(·)in C([τ,T],E) o all T>τ.
As a di ec consequence we ha e he ollowing esul . The p oo is a he simila o ha in [13, p.72].
Co olla y 27. Assume he condi ions o Lemma 23. Then, he mul i alued map ψ7→ G( ,τ,ψ)possesses closed
g aph and is uppe semicon inous. Mo eo e , i has compac alues.
Lemma 28. Assume hecondi ionso Lemma23. Then, hemul i alued p ocess UispullbackDη-asymp o ically
compac . In pa icula , i is pullback asymp o ically compac wi h espec o he pullback Dη-abso bing amily
b
Bη.
P oo . We conside ξn=un
∈U( ,τn,ψn), whe e un(·)∈Dτnψn,ψn∈D(τn), and b
D={D( )} ∈ Dη. In
iew o Co olla y 22, o nla ge enough we ha e un
∈Bη( ). Hence,
un
(s)≤C,∀s∈[−h,0],
o some C>0. Fo ixed s∈[−h,0]we can ind a subsequence (deno ed again as un) such ha
un( +s)→ωsin Ew.
Using a simila a gumen as in [13, p.71] (wi h he help o Lemma 25) we ob ain ha un( n+s)→ωsin E.
The e o e, un
(s)is a p ecompac sequence o any s∈[−h,0]. In o de o apply he Ascoli-A zelà heo-
em, we need o ob ain he equicon inui y p ope y. Using P oposi ion 19, he boundedness o he sequence
ψn2
E0e(η−L)τn, he ac ha he ope a o Fis bounded and he in eg al ep esen a ion o solu ion we can
ob ain ha
un( +s2)−un( +s1)≤
+s2
Z
+s1F ,un
d
≤K(s2−s1),i −h≤s1<s2≤0.
Then, he Ascoli-A zelà heo em implies ha ξnis ela i ely compac in E0.Since by Lemma 23 we ha e ha
b
Bη∈Dη,Uis pullback asymp o ically compac wi h espec o his amily as well.
The exis ence o he pullback a ac o ollows now om P oposi ion 19, Lemma 28, Co olla ies 22, 27 and
Theo em 18.
Theo em 29. Assume he condi ions o Lemma 23. Then, he mul i alued p ocess Upossesses a unique pull-
back Dη-a ac o b
A, which belongs o Dη. Mo eo e , i is s ic ly in a ian .
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A ac o s o non-au onomous e a ded la ice dynamical sys ems |49
3.4 Exis ence o he pullback a ac o : case o uniqueness
We can p o e uniqueness o he solu ion o he Cauchy p oblem (9) i we assume he ollowing ex a assump-
ion:
(C6) Fo any x,y∈Rand s∈[−h,0] we ha e
F0,i( ,x)−F0,i( ,y)≤k0( )C3(|x|,|y|)|x−y|,
F1,i( ,x)−F1,i( ,y)≤k1( )C4(|x|,|y|)|x−y|,
|bi( ,s,x)−bi( ,s,y)|≤k2( )k3(s)C5(|x|,|y|)|x−y|,
whe e Cj(·,·)≥0a e con inuous and non-dec easing unc ions in bo h a iables and ki(·)∈L2
loc (R)
o i= 0,1,2,k3(·)∈L2(−h,0).
Lemma 30. I (C6) holds, he map :R×E0→Esa is ies he local Lipschi z assump ion (H4).
P oo . Le ,w∈E0be such ha k kE0,kwkE0≤M. On he one hand, we ha e ha
X
i∈ZF0,i( , i(0)) −F0,i( ,wi(0))2≤k2
0( )max
i∈Z(C3(| i(0)|,|wi(0)|))2X
i∈Z
| i(0)−wi(0)|2
≤k2
0( )χ2
3k kE0,kwkE0k −wk2
E0,
X
i∈ZF1,i( i(−h1)) −F1,i(wi(−h1))2≤k2
1( )max
i∈Z(C4(| i(−h1)|,|wi(−h1)|))2X
i∈Z
| i(−h1)−wi(−h1)|2
≤k2
1( )χ2
4k kE0,kwkE0k −wk2
E0,
whe e χjk kE0,kwkE0= maxi∈Z,s∈[−h,0](Cj(| i(s)|,|wi(s)|)). On he o he hand,
X
i∈Z

0
Z
−h
|bi( ,s, i(s)) −bi( ,s,wi(s))|ds

2
≤k2
2( )max
i∈Z,s∈[−h,0](C5(| i(s)|,|wi(s)|))2X
i∈Z

0
Z
−h
k3(s)| i(s)−wi(s)|ds

2
≤k2
2( )χ2
5k kE0,kwkE0X
i∈Z
0
Z
−h
k2
3(s)ds
0
Z
−h
| i(s)−wi(s)|2ds
=k2
2( )χ2
5k kE0,kwkE00
Z
−h
k2
3(s)ds
0
Z
−hX
i∈Z
| i(s)−wi(s)|2ds
≤k2
2( )χ2
5k kE0,kwkE0h
0
Z
−h
k2
3(s)ds k −wk2
E0,
whe e χ5k kE0,kwkE0= maxi∈Z,s∈[−h,0](C5(| i(s)|,|wi(s)|)). The ac ha he sum and he in eg al can be
exchanged ollows easily using Lebesgue’s heo em. Thus, he e exis K(M),β(·)∈L1
loc (R)such ha
k ( , )− ( ,w)k2≤β( )K(M)k −wk2
E0,
which p o es he esul .
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50 |Tomás Ca aballo, F ancisco Mo illas, and José Vale o
Then, i we assume condi ions (C1)-(C6) and (15)-(16), Theo ems 5, 8, 12, Co olla y 11 and P oposi ion 19 imply
ha o any ψ∈E0 he e exis s a unique global solu ion u(·)∈C1[τ,∞),Ewi h u(τ)=ψ.
Hence, as shown in Sec ion 2.2, we can de ine he p ocess Uby pu ing U( ,τ,ψ)=u ,whe e u(·)is he
unique solu ion o (9) wi h ψ=u0.Mo eo e , his map is con inuous wi h espec o he ini ial da a ψ.
We ob ain now he exis ence o a pullback a ac o .
Theo em 31. Assume condi ions (C1)-(C6) and (15)-(16), (34). Then, he p ocess Upossesses a pullback Dη-
a ac o b
A, which belongs o Dη.
P oo . P oposi ion 19, Lemma 28, Co olla y 22 and Theo em 16 imply he exis ence o he pullback Dη-
a ac o b
A. Since he se s Bη( )o he abso bing amily a e closed and Bη∈Dη, we ob ain ha b
A∈Dη.
Acknowledgmen s
Pa ially suppo ed by FEDER and Minis e io de Economía y Compe i i idad (Spain) unde g an s
MTM2011-22411 and MTM2012-31698, and by Jun a de Andalucía unde P oyec o de Excelencia P12-FQM-1492.
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