Cauchy type problems of linear Riemann-Liouville fractional differential equations with variable coefficients

Transcription

1 Cuch tpe proes of er Re-ouve frcto dfferet equtos wth vre coeffcets Mo-H K u-cho R u-so Choe Ho Cho O* Fcut of thetcs K Su Uverst Po PRK Correspod uthor E: ro@hooco Astrct: The estece of soutos to Cuch tpe proes of er Re-ouve frcto dfferet equtos wth vre coeffcets s cosdered spce of tere fuctos Frst we cosder the estece d uqueess of souto for Cuch tpe proe wth spec t codtos the spce of tere fuctos The we provde epe of the proe tht hs o souto the spce of tere fuctos We ve sov ethod d represetto of soutos for the Cuch tpe proe st we ve soe epes Kewords: vre coeffcets Re-ouve frcto dervtves frcto dfferet equtos Cuch tpe proes Mthetcs Suect Cssfcto: J A troducto recet ers the reserch o the frcto ccuus s eco terest d usefu topc o pped scetsts[877] M phsc d chec processes re descred frcto dfferet equtos[8 ] The estece d uqueess of the souto to Cuch tpe proe for the frcto dfferet equtos were studed ppers[9] reserches for the sov ethods d represettos of the soutos to Cuch tpe proes the er frcto dfferet equtos wth costt coeffcets were cosdered whe those wth vre coeffcets were prt cosdered for soe spec fors of equtos[ 9 9 ] [ ] er frcto dfferet equtos wth costt coeffcets were cosdered us pce trsfor d [ 7 9 9] cosdered us operto ethod Espec [ ] represetto of ree s fucto for er frcto dfferet opertor wth costt coeffcets ters of ut-vre Mtt-effer fucto ws provded us pce trsfor [ ] souto represetto of er Cputo frcto dfferet equto wth costt coeffcets ws provded us operto ccuus of Muss's tpe [9 9] souto represetto of er eerzed Re-ouve frcto dfferet equto wth costt coeffcets ws provded us operto ccuus of Muss's tpe Sov ethods d souto represettos of er frcto dfferet equtos wth costt coeffcets were provded us the ethod of dstruto theor [7 8] Neu seres ethod [7] d Ado decoposto ethod [] [] the sov ethod ree s fucto of the sste of er frcto dfferet equtos wth costt coeffcets ws provded d [] represetto of ree s fucto for er frcto dfferet opertor wth vre coeffcets ws ve [] the tc souto of er frcto dfferet opertor wth vre coeffcets ws ve power seres ethod [] the tc souto of css of frcto dfferet equtos wth vre coeffcets ws provded us propertes of uerre dervtves d Cputo frcto

2 dervtves Ths rtce cocers wth the estece d represetto of soutos to Cuch tpe proes of er Re - ouve frcto dfferet equtos wth vre coeffcets spce of tere fuctos Frst we cosder the estece d uqueess of souto for Cuch tpe proe wth spec t codtos the spce of tere fuctos The we provde epe of the proe tht hs o souto the spce of tere fuctos We ve sov ethod d represetto of soutos for such Cuch tpe proes st we ve soe epes whch provde couter epes of coror t pe 8 of [] Our ethod s ust the se ethod of [] We o too otce tht soe ters re ot tere Preres Here the deftos d propertes of frcto dervtves re descred sed o [ 7 ] For re uers wth < we use ottos R : [ ] R : [ R We use ottos Z d N to deote the sets of teers d tur uers For Z wth Z s the set of teers stsf d Z the set of teers stsf Thus N Z Sr we use ottos N N for N wth We deote C γ γ [ ] the sets of fuctos stsf f C[ ] for f : R d γ < Whe we deote C γ [ ] : C γ [ ] et Ω [ ] < e fte or fte terv of the re s R We deote the set of those eesue cope-vued esure fuctos f for whch p f p < where Here esssup f f f p p / p : p f d p < : ess sup f p s the esset u of the fucto f Whe p we deote : et [ ] < < < e fte terv d et AC [ ] e the spce of soute cotuous fuctos f o [ ] t s ow tht AC [ ] cocdes wth the spce of prtves of eesue sue fuctos tht s f AC[ ] f c ϕ t dt ϕ Therefore soute cotuous fucto f hs sue dervtve f ϕ ost everwhere o [ ] d c f

3 For N we deote AC [ ] the spce of cope-vued fuctos f whch hve cotuous dervtves of order o [ ] such tht f AC[ ] : d AC [ ]: f : [ ] C f AC[ ] d AC [ ]: AC [ ] Ths spce s chrcterzed the foow sserto c AC [ ] f f ϕ Here ϕ d c re rtrr costts d Fro 7 we hve ϕ ϕ t t dt 7 f ϕ f c 8 et C e the set of cope uers The Re-ouve frcto ters f of order C Re > re defed : f t dt f > Re > 9 t t Here s the fucto d t e These ters re ced the eft-sded frcto ters Whe N the defto 9 cocdes wth the th terted ter t t f dt dt f t dt t f t dt N The Re-ouve frcto dervtves of order C Re re defed d d t dt : d d t [Re] > Here [Re] es the ter prt of Re prtcur whe Z the Here s the usu dervtve of of order f < Re < the d t dt < Re < > d [Re ] t Whe R the te the foow for: d t dt [ ] > d t

4 d s ve d t dt d [ ] t f Re the eds frcto dervtves of pure r order: θ d t dt θ R \ {} > θ d θ t For C Re > p < the spces of fuctos s defed p ϕ p : { f : f ϕ } 7 For C Re > the spces of fuctos s defed : { : } 8 e f C Re > the p t Re > 9 t Re prtcur f d Re the Re-ouve frcto dervtves of costt fuctos re ot equ to zero d we hve O the other hd for [Re] we hve t Fro we hve the foow resut e et Re > [Re] < Re < The equt s vd f d o f c Here R re rtrr costts prtcur whe < Re < the reto c hods f d o f c c R e f Re [Re] AC [ ] the the frcto dervtves est ost everwhere o [ ] d we hve t dt t e f Re > Re > d f p p the we hve f f

5 t ost ever pot [ ] f > the s true for pot of [ ] e f Re > f p p < the for ost [ ] we hve f f 7 Fro e e d e we hve the foow e e f Re > Re > f p the we hve p f f e [ ] 8 prtcur f N d Re > the we hve f f e [ ] 9 d e 7 et Re N f the frcto dervtves d est the we hve e [ ] e 8 et Re > [Re] f : f f p d f the we hve p f f f f f AC [ ] < Re the we hve f N < f f f the we hve f f f e e [ ] [ ] Rer wht foows we w rerd tht the equtes hod ost everwhere o the terv Cuch tpe proes of er equtos For cope uer C wth Re we defe the tur uer et [Re] N : N N d ssue tht cope uers C stsf < Re < < Re < Re We cosder the foow Cuch tpe proe of the er frcto dfferet equto of order C wth vre coeffcets where the fuctos d re cope vued fuctos wth re vre t

6 stsf C[ ] - d C - We use the tur uer defed N : Z t A spec Cuch tpe proe We cosder spec Cuch tpe proe fd the fucto stsf the equto d the t codto H Here H 7 < We cosder the foow Voterr ter equto of the secod d correspod to the Cuch tpe proe H 8 Frst we cosder the equvece of the Cuch tpe proe d wth the Voterr ter equto 8 Theore Assue tht for d - - d hod The s the souto to the Cuch tpe proe d f d o f 9 stsfes the ter equto 8 Proof Frst we prove the ecesst et e souto to the proe The we hve H d Fro we hve d 8 ed B 9 d the defto of the frcto dervtve we hve d fro we hve AC [ ]

7 The fct tht < < Re d e 8 ve Sce < Re Re we hve Therefore we hve App the frcto ter opertor the oth sdes of 9 uder cosderto of d we hve H Thus fro 9 d we ot the reto Coverse we c es ot the retos 9 d fro Thus the retos 9 d the reto re equvet Fro cosder d 8 we hve H 7 Therefore we hve H 8 Here the frst ter of the rht sde of 8 s tere the defto of d the eft sde s tere The secod ter of the rht sde s tere the ssuptos o d Therefore the equto 8 hods Susttut 9 d 8 to we hve 8 e 9 stsfes the ter equto 8 Now we prove the suffcec Assue tht stsfes the equto 8 The equto 8 c e rewrtte s H 9 For stsf 8 the equt d the equtes 9 d re equvet d thus we hve 7 Therefore for stsf 8 we hve H H H

8 Susttut d to 9 the we hve d Thus f stsfes 8 the ve stsfes d d Thus the suffcec s proved whch copetes the proof of theore Theore we proved tht the Cuch tpe proe d the ter equto 8 re equvet Therefore order to prove the estece d uqueess of souto to the t ve proe s suffcet to prove the estece of uque souto the ter equto 8 Theore Uder the codtos of Theore there ests uque souto to the Cuch tpe proe Proof B Theore t s suffcet to estsh the estece of uque souto to the ter equto efe H : T : H The the ter equto 8 c e rewrtte s T 7 et A 8 Choose postve uer δ such tht Z δ ω A < 9 The we c prove the estece of uque souto to the equto 8 wth { δ } B we hve T : f the fro the oudedess of frcto terto opertor we hve T T A [ ] ω d < ω < there ests the uque souto to the ter equto 8 o * the terv ] Ad the souto s the t of the sequece { T } e [

9 T * Here s fucto f for t est oe { } the t codto the we c te T s wrtte the recurso forus T T f we deote : T the we hve c e wrtte s foows: f the the proof s copeted f < the et Note tht : The 8 s rewrtte s foows Fro 9 d the ove cosderto we c prove tht the equto hs the uque souto wth { δ } Repet these processes we cocude tht the equto 8 hs uque souto Thus there ests uque souto to the Cuch tpe proe Theore B we hve H Ths copetes the proof of Theore 7 A Note for eer Cuch tpe proes We proved tht the spec Cuch tpe proe hs the uque souto But the cse whe the tur uer defed s ess th [Re ] the Cuch tpe proe 8 9

10 ht ot hve souto Uder the ssupto of Theore f we ssue tht there ests the souto of the proe 8 8 the we hve fro the defto of The equto 8 c e rewrtte s Sce we hve B the defto of frcto dervtves d thus fro we hve ] [ AC Fro < < Re d e 8 we hve Fro < Re Re d the defto of frcto dervtves we hve T the opertor to the oth sdes of d cosder 9 we hve Opert the frcto dervtve opertor the oth sdes d cosder d 8 we hve Therefore we hve B the defto of the cotut of the tert of d the eft sde d the frst d thrd ters of rht sde of re tere o f soe } { re ot zero the soe ters re ot tere the secod ter of rht sde of Becuse of these ters ths cse there ht ot est the souto of

11 8 d 9 Epe For C cosder the proe < < T f we ssue tht the proe d hs souto T the fro we hve Here ters ecept for the secod thrd d fourth ters of rht sde re tere d thus f the ove equt hods the the su of these secod thrd d fourth ters of rht sde s tere Ths s o posse whe { } re zero So the proe of d hs o souto f d o f soe of { } re ot zero Ths epe show us tht f cost the the proe 8 d 9 wth costt coeffcets hs souto f d o f { } Sov Method of the Cuch tpe proes secto we sw tht there ests uque souto to the Cuch tpe proe d the souto s represeted us the souto of the ter equto 8 where s oted successve pprotos: H ths secto us ths successve pprotos we et represetto of the souto to the proe coeffcets d t vue of the equto Represetto of the souto to the hooeeous equto We fd represetto of the souto stsf the er hooeeous equto > 7

12 d hooeeous t codto efto et e the souto of the proe > 8 δ 9 The sste of } { s ced the coc fudet sste of soutos of the er hooeeous equto 7 Theore Uder the codtos of Theore for the coc fudet sste of soutos of the hooeeous equto 7 s ve s foows: Proof B theore the Cuch tpe proe 89 hs uque souto The ter equto 8 correspod to the Cuch tpe proe 8 9 s For the souto of the equto the souto of the proe 8 d 9 s represeted f we pp the successve pprotos to sove the souto of the equto the we hve Ccut we hve Ccut we hve

13 Ccut sr we hve Therefore we hve Here s the coposto opertor of tes of the opertor d cse wth s ut opertor Susttut to we hve The proof of the theore s copeted Coror Uder the codtos of Theore for the souto to the Cuch tpe proe s represeted s foows: Here s the coc fudet sste of the souto to the hooeeous equto 7 Rer wth cope uers C s the eer souto to the hooeeous equto Represetto of the souto to the hooeeous equto We fd represetto of the souto to the proe

14 > 7 Theore Uder the codtos of Theore for the souto to the proe -7 s represeted 8 Proof B Theore ths proe hs uque souto Ths souto s oted 8 d The ter equto 8 correspod to the proe d 7 s 9 The correspod s The correspod successve pproto s ve the sr w wth theore derv of 8 s derved 9 d The proof of the theore s copeted efto The souto > to the Cuch tpe proe > s ced the ree s fucto of the hooeeous Cuch tpe proe 7 Rer Accord to the defto the ree s fucto s ust the frst fucto of the coc fudet sste of the souto to the hooeeous equto whe d thus there ests the ree s fucto of the Cuch tpe proe 7 d t s represeted s foows: Theore Uder the codtos of Theore for the souto to the hooeeous Cuch tpe proe 7 s

15 represeted s foows: d Ad s equ to the souto ve 8 Proof Theore we proved tht the uque souto to the hooeeous Cuch tpe proe 7 s represeted 8 Therefore f we prove tht s equ to 8 the proof s copeted B the represetto of the ree s fucto d d d B the defto of frcto ter opertors d the eerzed Fu s theore we hve Theore 7 Uder the codtos of Theore for C the uque souto to the proe s represeted s foows d Here { } s the coc fudet sste of the soutos to the hooeeous equto 7 d the fucto s the ree s fucto of the hooeeous Cuch tpe proe 7 Note o er frcto dfferet equtos wth costt coeffcets We cosder the er frcto dfferet equto wth costt coeffcets > 7 H 8 Here re the cope costts The hooeeous equto correspod to the hooeeous equto 7 s

16 9 Now we cosder the represetto of the coc fudet sste of the soutos to the hooeeous equto 9 As spec cse of the resuts of Theore we hve the foow theore Theore 8 Uder the codtos of Theore for C C the coc fudet sste of the soutos to the hooeeous equto 9 s represeted s foows: 7 Here Z d Rer The ree s fucto for the Cuch tpe proe 7 8 s represeted s foows: 7 Rer f we et 7 the we hve 7 Ths s equ to the resut of [9] See the equto of [9] Rer Uder the codtos of Theore for C C C N the uque souto to the proe 7-8 s represeted s foows: d 7 Here d re equ to 7 d 7 respectve Ths resut s oted [9] us Neu s seres ethod the cse of See d theore [9] Soe epes Epe Cosder the foow er hooeeous frcto dfferet equto wth costt coeffcets > < < T T 7 Ths s the cse whe T We c ot the the equto Sce

17 T T we hve < There ests uque souto T to the equto 7 stsf the foow t codtos 7 The the coc fudet sste of the souto s provded us the foru 7 d the souto s represet s But there does t est the souto T to the equto 7 stsf the t codtos 7 wth C d Epe Cosder the foow er hooeeous frcto dfferet equto wth vre coeffcets < < T 77 Ths s the cse whe T We c ot the the equto Sce T we hve B the foru the coc fudet sste T of the soutos to 77 s represeted s foows: 78 The uque souto T to 77 stsf the t codtos 79 for C s represeted s foows: 8 Epe Cosder the foow er hooeeous frcto dfferet equto wth costt coeffcets Ths s the cse whe 9 7 8

18 7 9 We use the foow te to decde us 9 7 T T - T - - T B ths te < B the foru 7 the coc fudet sste of the soutos s provded s foows The the uque souto T to 8 stsf the t codtos 8 wth C d s represeted s foows:

19 But there does t est the souto T to 8 stsf the t codtos 8 for C d Epe Cosder the foow er hooeeous frcto dfferet equto of cope order wth costt coeffcets < < T 8 ths cse d the coc fudet sste of the soutos s represeted 8 The uque souto T to 8 stsf the foow t codto 8 wth cope uer C s represeted But Theore there does t est the souto T to the equto 8 stsf the t codtos 87 wth C such tht t est oe of the re ot equ to zero Cocuso Our epes d re couter epes of coror t pe 8 of [] whch sserts tht the proe 8 d 9 wth hs uque souto [ ] Accord to our resuts the Cuch tpe proe 8 d 9 ht ot hve souto the cse whe soe of C re ot zero We provded sov of ethod for the Cuch tpe proe wth d the souto represettos Our ethod s ust the se ethod of [] We o too otce tht soe ters re ot tere Refereces [] BoBKsAA d TruoJJ Sstes of oer frcto dfferet equtos the spce of sue fuctostrstmtms 8- [] BoBRveroM d TruoJJO sstes er frcto dfferet equtos wth costt coeffcetsapped Mthetcs d Coputto [] CpoettoRooFortuPetrs Frcto order sstes:mode d Cotro Appcto : Word Scetfc Seres o Noer SceceVo7Word Scetfc [] FuuM d ShzuN Atc d uerc Soutos for Frcto Vscoestc Equtos JSME t J Seres C7-9 [] rr R Atc souto of css of frcto dfferet equtos wth vre coeffcets opertor ethods Cou Noer Sc Nuer Sut 7 9 [] oreforuchoyu Operto ethod for sov eerzed Ae ter equtos of secod d ter trsfors d spec fuctos [7] HddSBuchoYuF A Operto ethod for sov frcto dfferet equtos of rtrr re orderpermthj997-

20 [8] HferRAppctos of Frcto Ccuus Phscs Word Scetfc Spore [9] HferHuchoY d ToovsZ Operto ethod for the souto of frcto dfferet equtos wth eerzed Re-ouve frcto dervtves FrctCcAppA999-8 [] HuYJusho Atc Souto of the er frcto dfferet equto Ado decoposto ethodcoput d pp Mth7do:/ c [] KsAA BoB d TruoJJ Estece d uqueess theores for oer frcto dfferet equtos eostrmth8- [] KsAA BoB d TruoJJFrcto ters d dervtves d dfferet equtos of frcto order wehted spces of cotuous fuctosruss ontsadnu Berus 8- [] KsAA BoB d TruoJJ Noer dfferet equtos of frcto order spce of tere fuctosrussoadnu7-9trsted o Mth - [] KsAARveroMRodrez-er Truo JJ -Atc soutos of soe er frcto dfferet equtos wth vre coeffcetsapp Mth d Coput [] KsAA SoM O Mtt-effer tpe fuctosfrcto Ccuus Opertos d Souto of ter equtoster Trsfors d Spec Fuctos99-7 [] KsAASrvstvHM d TruoJJ Theor d Appctos of Frcto fferet EqutosEseverAsterd-Too [7] V Krov eerzed Frcto Ccuus d Appctos o- & J We Hrow & N Yor 99 [8] uchoy Operto ethod frcto ccuus FrctCcAppA [9] uchoy d orefor A operto ethod for sov frcto dfferet equtosact Mthetc Vetc [] uchoyuf d SrvstvHM The Ect Souto of Cert fferet Equtos of Frcto Order Us Operto CcuusCoputers MthAppc [] MchdoJJKrovVMrdF Recet hstor of frcto ccuus CouNoer Sc Nuer Su- [] MrdFFrcto reto d frcto dffuso equtosthetc spects:proceeds of the -th MACS Word CoressEdWFAeseor Tech AttVo999- [] MerKS d RossB A troducto to the Frcto Ccuus d Frcto fferet EqutosWe d SosNew Yor99 [] MoS d OdtZ Nuerc coprso of ethods for sov er dfferet equtos of frcto orderchossotos d Frcts78- [] MortT Souto of the t-vue Proe of Frcto fferet EqutotJAppMth99- [] MortT Souto of the t-vue proe of Frcto fferet Equto Srt verstedssfuthhrmorto d YOurNov Scece PushescNew Yor-7 [7] MortT d StoKNeu-Seres Souto of Frcto fferet Equto terdscpr forto Sceces 7-7 [8] MortT d StoK Souto of Frcto fferet Equto Teros of struto Theor terdsc f Sc 7-8 [9] Mo-H K u-cho R d Ho-Cho O Operto Method for Sov Mut-ter frcto fferet Equtos wth the eerzed frcto ervtves Frct Cc App A

21 [] Mo-H K Ho-Cho O Epct Represetto of ree s Fucto for er Frcto fferet opertor wth vre CoeffcetsJour of frcto Ccuus d Appcto -8 [] Podu Frcto fferet EqutosAcdec PressS eo999 [] Podu Souto of er frcto dfferet equtos wth costt coeffcets:trsfor Methods d Spec Fuctos ES PRusev ovs V Krov SporeScece Cuture Techoo PushSCTP997-7 [] Podu The pce Trsfor Method for er fferet Equtos of the Frcto ordersov Acde of Scecessttute of Eperet Phscs rv:fuct-/97Ⅳv Oct997 [] Podu The pce Trsfor Method for er fferet Equtos of the Frcto order stepphssov AcdScNo UEF--999Kosce- [] SoSKsAAMrchevO Frcto trs d ervtves:theor d AppctosNew Yor d odoordo d Brech Scece Pushers99 [] SdevTMetzerRToovsZ Frcto dffuso equto wth eerzed Re-ouve te frcto dervtve J Phs A:MthTheor Artce p [7] ToovsZ eerzed Cuch tpe proes for oer frcto dfferet equtos wth coposte frcto dervtve opertornoer Ass7-8