EXISTENCE OF A SOLUTION FOR THE FRACTIONAL FORCED PENDULUM

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1 Jourl of Alied Mhemics d Comuiol Mechics 4, 3(), 5-4 EXISENCE OF A SOUION FOR HE FRACIONA FORCED PENDUUM Césr orres Dermeo de Igeierí Memáic, Cero de Modelmieo Memáico Uiversidd de Chile, Sigo, Chile [email protected] Absrc. I his work we sudy he frciol forced edulum equio wih combied frciol derivives ( ) = u = D D u + g u = f,, (.) u. Usig miimizio echiques form vriiol clculus we show h (.) hs orivil soluio. where < <, g C( R, R ), bouded d f C[, ] Keywords: frciol clculus, frciol boudry vlue roblem, moui ss heorem, frciol sce. Iroducio Frciol order models c be foud o be more deque h ieger order models i some rel world roblems s frciol derivives rovide excelle ool for he descriio of memory d herediry roeries of vrious merils d rocesses. he mhemicl modelig of sysems d rocesses i he fields of hysics, chemisry, erodymics, elecrodymics of comlex medium, olymer rheology, ec. ivolves derivives of frciol order. As cosequece, he subjec of frciol differeil equios is giig more imorce d eio. here hs bee sigific develome i ordiry d ril differeil equios ivolvig boh Riem-iouville d Cuo frciol derivives. For deils d exmles, oe c see he moogrhs [-3] d he ers [4-3]. Recely, lso equios icludig boh - lef d righ frciol derivives, re discussed. Equios of his ye re kow i lierure s he frciol Euler- -grge equio d re obied by modifyig he ricile of les cio d lyig he rule of frciol iegrio by rs. he firs resuls were derived by Riewe [4, 5]. he he grgi d he Hmiloi formulio of frciol mechics were develoed for models wih symmeric d isymmeric frciol derivive [6], subsequely for models wih sequeil frciol derivives [7] d for models wih cosris [8].

2 6 C. orres he chrcerisic feure of hese equios of moio is he mixig of lef- d righ-sided Riem-iouville frciol derivives. herefore, hese ew clsses of frciol differeil equios become ieresig re of ivesigio. Ceri frciol equios of his ye were sudied i ers [9-]. I he soluio of frciol equios of vriiol ye he comosiio rules of frciol clculus ogeher wih fixed oi heorems were lied. Uforuely, his soluio is rereseed by series of lerely lef d righ frciol iegrls d herefore is difficul i y rcicl clculios. he he Melli rsform ws roosed s mehod of solvig some equios icludig he comosiio of lef- d righ-sided derivives [3], bu his soluio is rereseed by comliced series of secil fucios. his grely limis rcicl imlemeios, for exmle someimes i is very difficul o illusre he soluio i grhicl form. We oe, i riculr, h he frciol oscillor equio wih mixed derivives d he olier versio b D D u = λu, [, b] ( D D λ) u + V '( u ) =, [, b] were sudied i he works meioed bove. his ye of equios hs bee foud very useful ools for modellig my heome i he cosrucio idusry [4, 5]. Moived by hese revious works, i his er we del wih he frciol forced edulum equio ( ) = u = D D u + g u = f,, u (.) where < <, g C(, ) R R, bouded d f C [, ]. I riculr, if =, boudry vlue roblem (.) reduces o he sdrd secod order forced edulum equio ( ) u" + g u = f (.) Hmel ws he firs resercher who cosidered his roblem i riculr cse: g(u()) = si(u()) d f() = bsi(). Hmel s er srs by exisece resul for π-eriodic soluio of equio (.) by usig he direc mehod of he clculus of vriios were mde rigorously by Hilber he begiig of he ceury. Afer Hmel s work hd bee gre ieres i sudy exisece of eriodic soluio o (.) d is geerlizio see [6] d [7].

3 Exisece of soluio for he frciol forced edulum 7 Before sig our mi resuls, le us iroduce he mi igredies ivolved i our roch. We defie he frciol sce E = { u [, ]: D u [, ] d u() = u( ) = } d we sy h u is wek soluio of (.) if ( ) D u D v + g u v d = f v d, v E Moreover, we oe h u is wek soluio of (.3) if d oly if u is criicl oi of he fuciol D u I( u) = G( u ) f u + d which is C, wekly-lower semicoiuous d sisfies he ( PS ) C codiio. Now we re i osiio o se our mi exisece heorem. heorem.. e g C R, R, bouded d f C [, ]. he roblem (.3) hs les oe soluio. Our roch o rove heorem. is vriiol. We used o clssicl resul from he clculus of vriio. We recll his heorem for he reder s coveiece. < <, heorem.. [8] e φ be wekly lower semi-coiuous fuciol bouded from below o he reflexive Bch sce X. If φ is coercive, he c= ifϕ X is ied oi x X. heorem.3. [8] e C ( X, ) ϕ R be bouded below d c= ifϕ X. Assume h φ sisfies ( PS ) C codiio. he c is chieved oi x X d Where fuciol ϕ C ( X, ) ϕ ( x ) =. R sisfies he Plis-Smle (PS) codiio if every sequece x X such h ϕ ( x ) is bouded d lim ϕ ( x ) = i * X hs coverge subsequece. A vri of (PS) codiio, oed s ( PS ws iro- duced by Brézis, Coro d Niremberg [8]: e c R, ϕ C ( X, ) ( PS ) C codiio if every sequece x X such h lim ϕ( x ) = c d lim ϕ ( x ) = i * X, ) C R sisfies he hs coverge subsequece. I is cler h (PS) codiio imlies he ( PS ) C codiio for every c R. his ricle is orgized s follows. I we rese relimiries o frciol clculus d we iroduce he fuciol seig of he roblem (.3). I 3 we rove he heorem..

4 8 C. orres. Remider bou frciol clculus.. Some sces of fucios For y, : = (, b) deoes he clssicl ebesgue sce of -iegr- ble fucios edowed wih is usul orm. e us give some usul oios of sces of coiuous fucios defied o [ b, ] wih vlues i R: AC : = AC[, b] he sce of bsoluely coiuous fucios; C : = C [, b] he sce of ifiiely differeible fucios; C : = C [, b] he sce of ifiiely differeible fucios d comcly suored i ( b, ). We remid h fucio f AC if d oly if f d he followig equliy holds:, b, f = f + f ' s ds (.) [ ] where f deoes he derivive of f. Filly, we deoe by C (res. AC or C ) he sce of fucios f C (res. AC or C ) such h f ( ) =. I riculr, C C AC.. Frciol clculus oerors e > d u be fucio defied. e. o ( b, ) wih vlues i R. he lef (res. righ) frciol iegrl i he sese of Riem-iouville wih iferior limi (res. suerior limi ) of order of u is give by: resecively: I u = ( s) u( s) ds, (, b] Γ (.) b Ib u = ( s ) u( s) ds, [, b) Γ (.3) where Γ deoes Euler s Gmm fucio. If u, he I u d I b u re defied. e. o ( b., ) Now, le us cosider < <. he lef (res. righ) frciol derivive i he sese of Riem-iouville wih iferior limi (res. suerior limi b) of order of u is give by: d D u = I u, (, b] (.4) d

5 Exisece of soluio for he frciol forced edulum 9 resecively: d D u = I u, [, b) (.5) d b b From [], if u AC, he D u d D b u re defied. e. o ( b, ) d sisfy: d D u = I u ' + b b D u = I u ' + u ( ) Γ( ) u( b) ( b ) Γ( ) (.6) (.7) I riculr, if u AC, he D u() = I u(). So i his cse we hve he equliy of Riem-iouville frciol derivive d Cuo derivive defied by d c D u = I u ' (.8) c b b So wih his defiiio (.6) d (.7) c be rewrie d c D u = I u ' (.9) D u = D u + c b b D u = D u + u ( ) Γ( ) u( b) ( b ) Γ( ).3. Some roeries of frciol clculus oerors I his secio we rovide some roeries cocerig he lef frciol oerors of Riem-iouville. Oe c esily derive he logous versio for he righ oes. he firs resul yields he semi-grou roery of he lef Riem- -iouville frciol iegrl: Proery.. For y, β> d y fucio u, he followig equliy holds: β + = β I I u I u (.)

6 3 C. orres From Proery. d he equios (.6) d (.7), oe c esily deduce he followig resuls cocerig he comosiio bewee frciol iegrl d frciol derivive. For y < <, he followig equliies hold: d, u D I u = u (.) u AC, I D u = u (.) Aoher clssicl resul is he boudedess of he lef frciol iegrl from o : Proery.. For y > d y, I u is lier d coiuous from o. Precisely, he followig iequliy holds: ( b ) ( ) u, I u u (.3) Γ + Proery.3. (Iegrio by rs) e < <. e u, v q, where or, q d + < + q, q d + = + q he, he followig equliy holds: b b b I u v d = u I v d (.4) I he discussio o follow, we will lso eed he followig formule for frciol iegrio by rs Proery.4. e < <, he b = b c = + u D v d v I u v D u d (.5) b b = Moreover, if v is fucio such h v = v( b) =, we hve simler formule: b b b c u D v d= v Db u d (.6)

7 Exisece of soluio for he frciol forced edulum 3 he followig roery comlees Proery. i he cse ideed, i his cse, I u is ddiiolly bouded from o C : Proery.5. [9] e < < < d q=. he, for y hve: I u is Hölder coiuous o ( b, ] wih exoe > ; lim I u =. < < < : u, we Cosequely, I u c be coiuously exeded by i =. Filly, for y u, we hve I u C. Moreover, he followig iequliy holds: ( b ) ( ) / / q u, I u u Γ + ( q ) (.7).4. Frciol derivive sce I order o rove he exisece of wek soluio of (.3) usig vriiol mehod, we eed he iroducio of rorie sce of fucios. his sce hs o rese some roeries like reflexiviy, see [3]. hroughou his er, we deoe by he orm of he sce [, ] for + s u u d = d u = mx [, ] u. Defiiio.. e < < d < <. he frciol derivive sce is defied by { :[, ] : R [, ] }, c E = u u is bsoluely coiuous d D u E, For every u E,, we defie / c (.8) u = u d + D u d, Defiiio.. e < d < <. he frciol derivive sce is defied by he closure of C [, ] wih resec o he orm (.8), h is E,, =,, [ ] E C

8 3 C. orres Remrk.. i. I is obvious h his frciol derivive sce, E is equl o { [, ]: [, ] d }, c E = u D u u = u = ii. I follows from he boudry codiio u() = u( ) = h we see he fc h c D u= D u, c D u= D u, [, ]. his mes h he lef d righ Riem-iouville frciol derivives of order re equivle o he lef d righ Cuo frciol derivives of order. he roeries of he frciol derivive sces he followig lemm:, E d, E re lised s emm.. e < d < <., ) Boh he frciol sces E, d E re reflexive d serble Bch sces., ) For y u E we hve c [ ] D u = D u, for y, 3), E [, ] is coiuous d u Γ ( + ) D u 4) Assume h > d he sequece { u } coverges wekly o u i u. he { u } coverges srogly o u i C[, ], i.e. u, E, i.e. Moreover, if + =, he q u u, s / u Γ( ) + ( q ) / q D u, By he roery (3) i emm., we observe h he equivle orm i E /, u = D u d, c is defied by, E u.

9 Exisece of soluio for he frciol forced edulum 33, I his er, he work sce for roblem (.3) is E = E wih <. he sce E is Hilber sce wih he ier roduc d he corresodig orm defied by c c c u, v = D u D v d d u = u = D, u d 3. Frciol forced edulum / I his secio we del wih he frciol boudry vlue roblem ( ) = u = D D u + g u = f,, u (3.) Where < <, g C( R, R ), bouded d f C [, ]. We recll he oio of soluio for (3.). Defiiio 3.. A fucio u : [, ] R is clled soluio of (3.) if. I u( D u ) d I u re derivble i (, ) d. u sisfies (3.). Moreover, ssocied o (3.) we hve he fuciol I : E R defied by where D u I( u) = G( u ) f u + d G = g s ds d remember h u is wek soluio of (3.) if u is criicl oi of he fuciol I. We recll our mi heorem. heorem 3.. e < <, g C( R, R ), bouded d f C [, ]. he roblem (.3) hs les oe soluio. he roof of heorem 3. is divided io wo rs. I he firs r we rove he exisece of u E such h = mi I( v) I u E

10 34 C. orres for his urose we use he heorem.. O he secod r usig heorem.3, we jus roved h I ( u) =. Firs, we cosider he followig frciol boudry vlue roblem: ( ) = u = D D u = f, u u (3.) Where f :[, ] R R is coiuous d whe here exiss K> such h is idefiie iegrl sisfies he codiio y F x, y f x, s ds = (, ) for ll (, ) [, ] F x y K x y R (3.3) Wek soluios of (3.) re he criicl ois of he fuciol ϕ : E R We hve he followig heorem. D u u ϕ( u) = F(, u ) d heorem 3.. e <, f C([, ] R, R ) such h is idefiie iegrl F sisfies codiio (3.3). he he roblem (3.) hs les oe wek soluio. Proof. By codiio (3.3) D u u ϕ( u) = F(, u ) d K (3.4) d he fuciol ϕ is coercive. We ow show h i is wekly lower semicoiuous. e ( u) E such h u u. he, by he comc embeddig of E io C[, ], u u uiformly o [, ]. Furhermore, sice D u D u d

11 Exisece of soluio for he frciol forced edulum 35 we deduce d hece ( ) ( ) D u d D u D u d D u d lim if ( D u ) d lim if D u D u d ( D u ) d his imlies ( D u ) d ( u ) ( u) lim ifϕ ϕ he ϕ is wekly lower-semicoiuous o E d he exisece of miimum for ϕ follows from heorem.. We ly his heorem o (3.). Firs, x-milgrm heorem shows h he lier roblem u D D u = f u = = (3.5) hs uique soluio U( x ). eig u= U+ v, he roblem (3.) is reduced o he equivle oe v ( ) D D v + g U + v = v = = Moreover, le M= mx G( u), he u R, G u M u R d he corresodig fucio F give by (, ) F u = G U + v M (3.6) sisfies ssumio (3.3). hus Problem (3.) lwys hs les oe soluio by heorem 3.. h is, here exiss u E such h ϕ ( u) = ifϕ( v) E

12 36 C. orres where D u ϕ( u) = G( U u + ) d Filly we rove h u is criicl oi of ϕ. For his fc, firs we rove some lemms. emm 3.. I C ( E, R ) d (3.7) I ' u v = D u D v g u v + f v d Proof. We oly show h he fuciol is φ : E R defied by ( u) Φ = C ( E, R ). We re goig o rove h G u d where By he FC we hve he herefore v r v lim = v r v = Φ u + v Φ u g u vd (3.8) d G( u+ v) G( u) = G( u+ xv) dx dx = ( + ) r v G u v G u d g u vd = ( g( u + xv) g( u) ) vdx d r( v) g( u + xv) g( u) v dx d (3.9)

13 Now, sice g K, R we hve Exisece of soluio for he frciol forced edulum 37 ( + ) 4 [, ] g u xv g u K So by Fubii heorem, Hölder iequliy d emm. we ge he r( v) g( u + xv) g( u) v d dx g u + xv g u v dx ( ) g( u + xv) g( u) v dx Γ + r v v ( ) g( u + xv) g( u) dx Γ + Now for ech N, le θ herefore For ech fixed x, we ge [ ] :, R x θ x = g u + xv g u r v v his imlies, for ech fixed x ( x) ( ) θ Γ + x dx lim g u+ xv g u d=, θ, x [,]. O he oher hd ( x) g( u xv ) g( u) / [ ] θ + + K, By he ebesgue s domied covergece heorem ( x) dx = θ [ ] lim θ i,

14 38 C. orres Now we show h φ is coiuous. e { v} such h v. By emm. By he coiuiy of g his imlies O he oher hd E [ ] [ ] v i, d v x. e. o, (( + )).e. i [, ] g u v g u ().e. i [, ] g u + v g u () ( ) [, ] g u + v g u K By ebesgue s domied covergece heorem () lim g u + v g u d = Now by Hölder iequliy d emm., we ge his imlies ( ) ( + ) Φ, = ( + ) Φ u v u v g u v g u vd / / g( u v) g( u) d v d + g( u + v) g( u) v Γ ( + ) ( + ) ( = ) ( + ) Φ u v Φ u su Φ u v Φ u, v s. herefore Φ is coiuous. E * v g( u + v) g( u) Γ ( + )

15 Exisece of soluio for he frciol forced edulum 39 emm 3.. I sisfies he ( PS ) c - codiio. Proof. For his fc, we use he equivle roblem (3.6), ssocied o his roblem, we hve he fuciol D u ϕ( u) = G( U u) + d ϕ C ( E, R ) d is bouded below. If u E is sequece such h ϕ u c d ϕ u s We clim u is bouded i E. For lrge eough d (3.4) we ge c + ϕ( u) D u M herefore u is bouded i E d hece, u o subsequece, wekly coverges o some u E d uiformly o u o [, ]. Cosequely Bu herefore lim ϕ u ϕ u, u u = ϕ u ϕ u, u u = ϕ u, u u ϕ u, u u ( ) ( + )( ) D u D u u G U u u u d D u D u u G U + u u u d ( ) = D u d G U + u G U u u u d d ( PS ) c codiio is roved. u u s

16 4 C. orres Proof of heorem 3.. By he revious resul we c ly heorem.3 o ge ϕ '( u) =. Hece we rove h here is u E wek soluio of (3.). Now followig he ides of [3], sice u is wek soluio of (3.) we c show h here exiss cos C R such h We oice h for ( ) I D u() = g U + v d+ C (3.) u E, I u is derivble i (, ) d Moreover, by (3.) we hve ( ) I u = D u [, ] d ( I D u ) [, ] D D u() = I D u() = g U + u d sice u imlies h u () = u =. he roof is comlee. E Ackowledgemes C.. ws rilly suored by MECESUP 67. Refereces [] Kilbs A., Srivsv H., rujillo J., heory d Alicios of Frciol Differeil Equios, Norh-Holld Mhemics Sudies, Amserdm 6. [] Miller K., Ross B., A Iroducio o he Frciol Clculus d Frciol Differeil Equios, Wiley d Sos, New York 993. [3] Podluby I., Frciol Differeil Equios, Acdemic Press, New York 999. [4] Agrwl R., Bechohr M., Hmi S., Boudry vlue roblems for frciol differeil equios, Georg. Mh. J. 9, 6, 3, 4-4. [5] Agrwl R., Belmekki M., Bechohr M., A survey o semilier differeil equios d iclusios ivolvig Riem-iouville frciol derivive, Adv. Differece Equ. 9, 9. [6] Agrwl O., ereiro Mchdo J., Sbier J., Frciol Derivives d heir Alicio: Nolier Dymics, Sriger-Verlg, Berli 4. [7] Ah V., Mcviish R., Frciol differeil equios drive by évy oise, J. Al. Mh. d Soch. Al. 3, 6,, [8] Bechohr M., Hederso J., Nouys S., Ouhb A., Exisece resuls for frciol order fuciol differeil equios wih ifiie dely, J. of Mh. Al. Al. 8, 338,,

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