On the Existence and uniqueness for solution of system Fractional Differential Equations

Transcription

1 OSR Jourl o Mhms OSR-JM SSN: Volum 4 ssu 3 Nov. - D. PP -5 O h Es d uquss or soluo o ssm rol Drl Equos Mh Ad Al-Wh Dprm o Appld S Uvrs o holog Bghdd- rq Asr: hs ppr w d horm o s d uquss o rol drl quos o ssm o rol drl quos.. roduo rol drl quo s grlzo o ordr drl quos d gros o rrr o gr ordr. or h ps hr urs hs suj hs dl wh mhms d h ls rs hs ssl o udr sd h soluo o m pplo vrous lds o ss l phss d grg [] hmsr d ompur hrd ds orol [34] lso ulr rg s [56] d dm ssms [7]. hs sud s dls wh s d uquss o soluos or ssm o rol drl ordr quo o h orm wh s som os d <p - os hoosg λ suh h d ouous uo o [ < γ posv os g[ suh h g + [.. Prlmrs hs so w rodu oo dos d prlmr s whh r usd hrough ou hs ppr. Do - [8]: L A={::[ R R ouous uo} Suh h = whr = d s h rspos o. L h orm o A dd Sup [ whr provdd h hs orm ss or som γ>. Do. [8]: - L lsgu- msurl uo d o []. > h w d d provdd h grl Lsgu ss. - R s d o [] w d d d or ll [] provdd h - ss. Lmm.[9]: L < d g ouous uo o whr R d suh h Sup{ g : }=M<. D g d or ll > d s som os. h. Pg

2 O h Es d uquss or soluo o ssm rol Drl Equos Pg Lmm. [8]: Suppos h G s h sp d l LG suh h <. h - s rgulr d whr h srs ovrg LG. Lmm.3[9]: L us d =- - o whr dd Lmm. d <. h [. Lmm.4 [8]: L γ R γ>-. > h gr gv gr gv Lmm.5 [9]: < d s ouous o ] M or ll ] whr MR + M>. h or ll ].. M Rsuls hs so w prov h s d uquss soluo o ssm o rol drl quos. horm 3: L < d γ posv os. L g=g g g [ whr g r ouous o [ = d g = g d g + 3. whr s posv os. L = suh h [ d Sup{ : [ }=M<. Choos λ suh h. h hr ss ouous vor uo = suh h =λ wh - = whr = s som os vor d sss <p - os. Proo L A h sp dd do. [ +h] omp survl o [. Cosdr 3. d whr ollows rom Lmm. h. h d whr L = whr gv 3. d d =- - << 3.4 h rom Lmm.3 w hv [. Now d opror o A s

3 O h Es d uquss or soluo o ssm rol Drl Equos 3 Pg 3.5 ] [ h d h d h d ] [ d so rom do. w hv d hus usg 3. w hv 3.6 d d h rom [] w hv d 3.7 h rom 3.6 d 3.7 w hv hus 3.8 H d so do. w g d h 3.9 Now or λ < - w o λ = λ < - = d hs mpls h o usg Lmm. {-λ} - ss. Also rom 3.3 w hv d 3. whr Now 3.5 d 3. mpl h =+λ d so {-λ}= d hror ={-λ} - whr s h d opror d h ss d s h uqu soluo o d Now smpl lulo o show h

4 O h Es d uquss or soluo o ssm rol Drl Equos 4 Pg m whr γ=. S rom 3.9 hr or m hus or h soluo o 3. ss d sss = + λ 3. Ag rom 3.3 w hv d or ll +h] d usg 3.3 ollows h d or ll +h] d hror do. w g +h] 3. Bu rom Lmm.4 w hv d Lmm.5 w g or ll +h] hus +h] h usg do. w g +h] urhr mor rom 3. w hv ollows rom Lmm.4 h

5 O h Es d uquss or soluo o ssm rol Drl Equos d d so ss or ll +h] s do. hror Now rom 3. w hv + λ d rom 3.8 g hus usg 3.3 w o s h h d so h soluo uo sss p os V. Coluso hs sud w prov h s d uquss o soluo or ssm o rol drl quos usg horm d horm [9]. h soluo od usd o solv m prolms h mhm d ohr ss suh s mhs grg hmsr phss. Rrs [] Sr J. ;Agrwl O.P.; vro Mhdo J. A; Advs rol Clulus: h orl Dvlopms d pplo phss d Egrg sprgr 7. [] Hms N.; Podlu.; Phsl rpro o l Codos o rol Drl Equos wh Rm- Louvll rol Drvvs. RhoLAl [3] Ju Bsqur Alr omp. hor o Elrohml mpd o Amous Duso J o Elroll hmsr 499. [4] Brl Dsr Corol or Compur Hrd Ds rol Ordr Hold Dvs. Uvrs o psvs sp. [5] Sh Ds B. B. Bsws ol Erg Vzo rom Nulr Sours POR. 6 Nulr Erg or Nw Europ Slov 6. [6] Sh Ds l. Ro Corol wh Logrhm Logs w P d P orol lgorhm or ru ul- ror.. J. Nulr Erg s d holog. Vol.3 No. 7. [7] V. Lshmhm S. Ll d J. vsudhr; hor o rol Dm Ssms Cmrdg Adm pulshs Cmrdg 9. [8] Al-hl M. A. o h s d uquss or rol drl Equos Ms. hss Uvrs o Musol. [9] Dr. Ahmd Z-Ad Mh Ahd-Wh horm o Cr rol uos d Drvv Eg. d h. Jourl. Vol.3 No.. [] Spr Kh B. Oldhm d Jrom h rol Clulus hor d Applo o Dro o Arrr Ordr Muurd h Ud S Courr Corporo: SBN-3: Pg