ON APPLICATION OF CONTRACTION PRINCIPLE TO SOLVE TWO-TERM FRACTIONAL DIFFERENTIAL EQUATIONS
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1 men e uom vol5 no ON PPCON OF CONRCON PRNCP O SOV WO-R FRCON FFRN QUONS re BŁSK* łorz KK* * nsue o ems Czesoow Unvers o enolo ul ąrowseo Częsoow mrels@mlom mlme@mpzpl sr We solve wo-erm ronl derenl equons w le-sded Cpuo dervves sene-unqueness eorems re proved usn newl-nrodued equvlen norms/mer on e spe o onnuous unons e mers re moded n su w e spe o onnuous unons s omplee nd e Bn eorem on ed pon n e ppled ppers e enerl soluon s enered e sonr unon o e es order dervve nd ess n n rrr nervl NROUCON Fronl derenl equons F emered n ppled mems s n mporn ool o desre mn proesses nd penomen n pss mens eonoms onrol eor enneern nd oenneern ompre monorps nd revew ppers rwl e l 4; Hler ; Kls e l 6; n 6; ezler nd Kler 4; Wes e l 3 nd e reerenes ven eren urn e ls dedes e eor o ronl derenl equons s eome n neresn nd mennul eld o mems e resuls nd reerenes re summrzed n monorps eelm ; Kls e l 6; Kls nd rullo ; Klme 9; smnm e l 9; ller nd Ross 993; Podlun 999 Sne e F eor lso ws lssed n e SC ssem under 348 or ordnr ronl derenl equons 34K37 or unonl ronl derenl equons nd under 35R or prl ronl derenl equons Sll n e eor o ronl derenl equons mn prolems remn open ven n se o s esene-unqueness resuls ere s n re or nvesons onernn e een provn meods orrespondn spe o soluons oe nd e eenson o resuls rom s o more enerl equons onnn mn erms w dervves o le- nd r-sded e presen pper s devoed o e sud o wo-erm ronl derenl equons w le-sded Cpuo dervves sene-unqueness resuls re oned or equons o rrr ronl order wen e es order dervve s ven s sequenl operor e s omposon o wo Cpuo dervves e us pon ou e equons rom s lss o sequenl F were ppled n e eor o vsoels Wn nd n drodnms Kn e l 9; Sn e l 9; n e l 6 e proposed meod o dervn e soluon s n eenson o e Bele meod nown rom derenl equons eor Bele 956 He ppled equvlen norms/mers moded usn e eponenl unon nd e Bn eorem o solve derenl equons o neer order n e pper l Reem 3 smple ronl derenl equon o e order n e rne ws solved usn e sme ppro en smnm nd s ollorors smnm e l 8 9 ppled n e modon o norms/mers e one-prmeer -eler unon s llowed em o prove e esene nd unqueness resul or equons w e Cpuo dervve o order n e rne Furer resuls or smlr equons re ven n e pper Blenu nd us n e ppers Klme Klme e were eended o e mul-erm F dependen on s Remnn-ouvlle Cpuo or Hdmrd dervve Here we sud equons w sequenl powers o Cpuo dervves o n rrr ronl order ssumn e nonlner erm oes e psz ondon we on onnuous soluons n n rrrl lon nervl e pper s ornzed s ollows n Seon we rell e s denons rom ronl lulus nd nrodue ml o norms n e spe o unons onnuous n ne nervl e re equvlen o e remum norm us wenever we endow e spe o onnuous unons w e mer enered e new norm we on omplee mer spe We lso ormule e lemm on e properes o ronl neron nd urer ppl o solve ronl derenl equons Seon 3 s devoed o e esene-unqueness resul or wo-erm ronl derenl equon w e omposed Cpuo dervves e nloous resuls or enerl sequenl equon w e omposed Cpuo dervves re nluded n Seon 4 e pper s losed sor dsusson o e presened meod o solvn nd s possle urer pplons 5
2 re Błs łorz Klme On pplon o e Conron Prnple o Solve e wo-erm Fronl erenl quons PRNRS n s seon we rell e s denon rom ronl lulus nd nrodue lss o norms equvlen o e remum norm on e spe o onnuous unons e Remnn-ouvlle nerl nd Cpuo dervve re dened s ollows Smo e l 993 Kls e l 6 enon e le-sded Remnn-ouvlle nerl o order denoed s s ven e ollown ormul or Re > Γ u du u enon e Re e le-sded Cpuo dervve o order denoed s s ven e ormul Γ n n u du n u e denon elow desres enerlzed eponenl unon - e -eler unon enon 3 e γ > > e wo-prmeer -eler unon s ven s e ollown seres z γ δ z 3 Γ γ δ enon 4 We denoe s e spe o unons onnuous n nervl s spe w remum norm nd e respeve enered mer d s omplee mer spe We sll ppl e -eler unons n e onsruon o lss o norms n spe ndeed posve prmeer enon 5 e ollown ormul denes oneprmeer lss o norms nd mers on spe 4 γ γ d 5 Proper 6 Supremum norm nd e norms ven ormul 4 re equvlen Proo s proper resuls rom e se o nequles γ γ w re ullled or n nd Now we quoe n mporn resul onernn e ronl nerls o e -eler unon nd er remum emm 7 e ollown neron ormul s vld or n 6 e > > en onsn ess so e ollown nequl s vld or prmeer e rs pr o e ove lemm s srorwrd orollr o relon e proo o e seond pr s rer lon nd enl so we om ese lulons n e presen pper 3 SOUON OF WO-R FRCON FFRN QUON CS We sll onsder wo-erm ronl derenl equon n n rrr ne nervl nludn le -sded Cpuo dervves n sequenl orm w > e prelmnr resuls or equons o s pe re dsussed n pper Klme Błs n e presen pper we sll ve ull proo o e esene-unqueness resuls or e enerl soluon o equon 8 nd or e nl vlue prolem n se n e rnsormon o e ove equon we sll ppl e ollown omposon rule or e Cpuo dervve nd Remnn-ouvlle nerl s rule olds or n unon onnuous n nervl nd we quoe er e monorps Smo e l 993; Kls e l 6 Proper 3 e R nd > e ollown equles old or n pon C ssumn e nonlner erm o e onnuous unon o wo vrles nd usn e ove proper we n reormule equon 8 s ollows 3 provded Now we oserve e unon n e res elons o e ernel o sequenl dervve e us denoe s unon s nd wre e orrespondn equon or sonr unon 6
3 men e uom vol5 no 7 4 w leds o e epl ormul Γ Γ n n d 5 wen e respeve orders ulll e ondons n n nd n n quon 3 rewren usn sonr unon 5 eomes e ronl nerl equon 6 w n urn ondes w ed pon ondon 7 or mppn enered sonr unon 8 ssumn unon R R nd oservn sonr unon s onnuous n nervl we onlude e ove mppn rnsorms n onnuous unon no s onnuous me e dsussed rnsormon o e F ven n 8 no ed pon ondon 7 llows us o ormule e ollown resul on e esene o soluon o equon 8 Proposon 3 e > nd unon R R ulll e psz ondon 9 R en e sonr unon o dervve ven n 5 elds unque soluon o equon 8 n e spe o unons onnuous n nervl Su soluon s lm o e erons o mppn lm χ were s n rrr onnuous unon Proo We reormule equon 8 s ed pon ondon 7 w mppn dened n 8 We sll onsder e properes o s mppn on e spe o unons onnuous n nervl endowed w mer enered e norm onsrued ordn o enon 5 For n wo onnuous unons nd e dsne o er mes nd mesured usn e mer deermned e ove norm n e esmed s ollows n e ove lulons we ve ppled emm 7 on e ronl neron o -eler unons e n e summrzed n e orm o e ollown nequl w onsn ven ormul 3 Relon s ullled or n wo onnuous unons nd sonr unon nd e vlue o prmeer ordn o emm 7 e us noe e numeror o e ron denn onsn does no depend on e vlue o us or prmeer lre enou nd mppn s onron n e spe o onnuous unons C endowed w e mer
4 re Błs łorz Klme On pplon o e Conron Prnple o Solve e wo-erm Fronl erenl quons enered norm s s spe s omplee en usn e Bn eorem on ed pon we onlude ed pon ess nd ullls ondon 7 ordn o e menoned eorem e ed pon s lm o erons o mppn s desred n e ess o Proposon 3 nd ns o e omposon rule rom Proper 3 lso solves nl F 8 e us pon ou e presened onsruon wors or n sonr unon o dervve us e oned soluon onneed o s n nloue o enerl soluon rom e lssl eor o derenl equons nd onns rrr oeens o e oeens we dd se o nl ondons e ne proposon ves e esene-unqueness resul or e se wen orders o dervves re n e rne o Proposon 33 e e ssumpons o Proposon 3 e ullled nd en e unque soluon o equon 8 oen e nl ondons w w ess n e C spe Su soluon s lm o e erons o mppn enered e ollown sonr unon w w w Proo From Proposon 3 ollows e sonr unon eneres unque onnuous soluon o equon 8 We sll prove soluon enered unon solves e nl vlue prolem ven n Proposon 33 Frs rom 6 we on e relon w w quon 6 nd omposon rule eld ormul Ψ d n we on w d Solvn e ove equons we rrve e ollown vlues o oeens nd n enerl ormul 5 w d w w nd s ends e proo 4 SOUON OF WO-R SQUN FRCON FFRN QUON CS n s seon we sll solve e enerl wo-erm sequenl F n ne nervl s equon loos s ollows Ψ 4 were e sequenl dervve on e le-nd sde s undersood derenl n n equon 8 e onsdered F n e rewren n e veor orm 5 Ψ 6 were we denoed Su rnsormon s pl or e ove lss o equons ompre monorp eelm eelm e reerenes ven eren nd ppers Kls e l e novel o our pper s new meod o proo w leds o e esene resul or e soluon n n rrrl lon nervl Smlr o e lulons n e prevous seon we rnsorm e ove ssem o F no e ssem o ronl nerl equons on e spe o onnuous unons 7 Ψ 8 Funons nd re e orrespondn sonr unons o dervves nd 9 3 e epl orm o e ove sonr unons depends on e order o e dervves e n n nd n n en unons nd loo s ollows n 3 Γ n 3 Γ w oeens nd en rrr rel numers We noe e ssem o ronl nerl equons n e reormuled s ed pon ondon or e woomponen rel-vlued unon 33 were e omponens o e mppn re dened s ollows or n wo-omponen onnuous unon 34 Ψ 35 Solvn e ssems o equons 5 6 nd 7 8 we sll prove e mppn ven ove s onron on e spe o onnuous unons 8
5 men e uom vol5 no 9 endowed w e respeve norm nd mer rom e lss ndeed posve prmeer e us oserve or e posve vlue o prmeer e norm nd e respeve mer dened ormuls eld omplee mer spe s e re equvlen o e sndrd norm n Proposon 4 e > nd unon R R ulll e psz ondon Ψ Ψ 4 R en e pr o sonr unons o e dervves ven n 3 3 elds unque soluon o equon 4 n e spe o unons onnuous n nervl Su soluon s lm o e erons o mppn lm χ χ 4 4 were re rrr onnuous unons deermned n nervl Proo e us ssume > nd esme e dsne eween e mes o n rrr pr o wo -omponen unons nd We on e ollown nequles vld or e rs omponen o e me e presened lulons n e summrzed e ollown relon 43 Respevel or e seond omponens o e mes we on Ψ Ψ { } m e ove lulons eld e ollown nequl or e seond omponens o mes 44 { } m Now usn derved relons we re red o esme e dsne eween e mes nd were onsn s nversel proporonl o e vlue o prmeer { } m
6 re Błs łorz Klme On pplon o e Conron Prnple o Solve e wo-erm Fronl erenl quons s prmeers do no depend on e vlue o we onlude or lre enou mppn s onron on spe R us we n ppl e Bn eorem on ed pon nd ner unon R ess so e ed pon ondon s ullled Su unon n e onsrued usn e eron lm s desred n ormuls 4 4 e rs omponen o e ed pon solves F 4 ordn o relon 4 e proo n se s nloous 5 FN RRKS We developed n een meod o provn e esene-unqueness resuls or wo-erm ronl derenl equons For equons onnn le-sded Cpuo dervves nd er omposon we sowed enerl soluon ess n n rrrl lon nervl nd s enered e sonr unon o e es order dervve e ppled meod o equvlen norms/mers eends e de ven Bele Bele 956 or derenl equons o neer order s ws sown n ormer ppers l-reem 3; smnm e l 8 9; Blenu nd us ; Klme n F eor we sould mod e mers usn one- or wo-prmeer -eler unons n e presen pper e respeve verson o e meod s ven or wo-erm ronl derenl equons onsdered on e C spe However reul nlss o e oned resuls nd er proo mples usn emm 7 e n e eended o mul-erm F e us lso pon ou s ppro seems ppropre o sud nd solve ronl derenl equons on oer unon spes or nsne on e spe o derenle unons RFRNCS rwl O P enrero-do J Ser J 4 ds Fronl ervves nd er pplon Nonlner nms Sprner-Verl Berln vol 38 Blenu us O G On e lol esene o soluons o lss o ronl derenl equons Comp ppl Vol Bele 956 Une remrque sur l meode de Bn- Copol-onov dns l eore des equons derenelles ordnres Bull d Polon S Cl - V eelm K e nlss o Fronl erenl quons Sprner-Verl Berln 5 l-reem Z F 3 odon o e pplon o onron mppn meod on lss o ronl derenl equon ppl & Compu Vol Hler R d pplons o Fronl Clulus n Pss World Sen Snpore 7 Kn Hder l S Feeu C Ho Q 9 e o poenl vore or vsoles lud w r-onl well model ppl Compu Vol Kls Srvsv H rullo J J 6 eor nd pplons o Fronl erenl quons lsever mserdm 9 Kls rullo J J erenl equon o ronl order meods resuls nd prolems ppl nl Vol Kls rullo J J erenl equon o ronl order meods resuls nd prolems ppl nl Vol Klme 9 On Soluons o ner Fronl erenl quons o Vronl pe e Pulsn Oe o e Czesoow Unvers o enolo Czesoow Klme On onron prnple ppled o nonlner ronl derenl equons w dervves o order Bn Cener Pul o pper 3 Klme Sequenl ronl derenl equons w Hdmrd dervve Commun Nonlner S Numer Smul do 6/nsns8 4 Klme Błs sene-unqueness resul or nonlner wo-erm sequenl F Proeedns o e 7 uropen Nonlner nms Conerene NOC Rome o pper 5 smnm V eel S Vsundr ev J 9 eor o Fronl nm Ssems Cmrde Sen Pulsers Cmrde 6 smnm V Vsundr ev J 8 eor o ronl derenl equons n Bn spe uropen J Pure nd ppl Vol n R 6 Fronl Clulus n Boenneern Reddn Beell House Pulser 8 ezler R Kler J 4 e resurn e end o e rndom wl reen developmens n e desrpon o nomlous rnspor ronl dnms J Ps Vol 37 R6-R8 9 ller K S Ross B 993 n nroduon o e Fronl Clulus nd Fronl erenl quons Wle nd Sons New Yor Podlun 999 Fronl erenl quons dem Press Sn eo Smo S G Kls rev O 993 Fronl nerls nd ervves Gordon & Bre mserdm Sn on Xue 9 Unsed low o non -Newonn vso-els lud n dul poros med w e ronl dervve J Hdrodn B Vol n J on 6 e low nlss o luds n rl reservor w e ronl dervve J Hdrodn Vol Wn ZH Wn X Generl soluon o e Ble -orv equons w ronl order dervve Commun Nonlner S Numer SmulVol Wes B J Bolon Groln P 3 Pss o Fronl Operors Sprner-Verl Berln
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