Attractors for non-autonomous retarded lattice dynamical systems

©2015 Tomás Ca aballo e al., licensee De G uy e Open. This wo k is licensed unde he C ea i e Commons A ibu ion-NonComme cial-NoDe i s 3.0 License. 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] B ough o you by | Biblio eca de la Uni e sidad de Se illa Au hen ica ed Download Da e | 9/29/15 11:23 AM 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]. B ough o you by | Biblio eca de la Uni e sidad de Se illa Au hen ica ed Download Da e | 9/29/15 11:23 AM 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( )]. B ough o you by | Biblio eca de la Uni e sidad de Se illa Au hen ica ed Download Da e | 9/29/15 11:23 AM 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 B ough o you by | Biblio eca de la Uni e sidad de Se illa Au hen ica ed Download Da e | 9/29/15 11:23 AM 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. B ough o you by | Biblio eca de la Uni e sidad de Se illa Au hen ica ed Download Da e | 9/29/15 11:23 AM 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 : B ough o you by | Biblio eca de la Uni e sidad de Se illa Au hen ica ed Download Da e | 9/29/15 11:23 AM 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 . B ough o you by | Biblio eca de la Uni e sidad de Se illa Au hen ica ed Download Da e | 9/29/15 11:23 AM 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) B ough o you by | Biblio eca de la Uni e sidad de Se illa Au hen ica ed Download Da e | 9/29/15 11:23 AM 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. B ough o you by | Biblio eca de la Uni e sidad de Se illa Au hen ica ed Download Da e | 9/29/15 11:23 AM 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) B ough o you by | Biblio eca de la Uni e sidad de Se illa Au hen ica ed Download Da e | 9/29/15 11:23 AM 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, B ough o you by | Biblio eca de la Uni e sidad de Se illa Au hen ica ed Download Da e | 9/29/15 11:23 AM 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 . B ough o you by | Biblio eca de la Uni e sidad de Se illa Au hen ica ed Download Da e | 9/29/15 11:23 AM 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 . B ough o you by | Biblio eca de la Uni e sidad de Se illa Au hen ica ed Download Da e | 9/29/15 11:23 AM 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. 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