On the Existence and Uniqueness of Solutions for. Q-Fractional Boundary Value Problem

Transcription

1 I Joural of ah Aalysis, Vol 5, 2, o 33, O he Eisee ad Uiueess of Soluios for Q-Fraioal Boudary Value Prolem ousafa El-Shahed Deparme of ahemais, College of Eduaio Qassim Uiversiy PO Bo 377 Uizah, Qassim, Saudi Araia elshahedm@yahooom aryam Al-Yami Deparme of ahemais, College of Eduaio Kig Adulaziz Uiversiy POBo 229 Jedda, Saudi Araia mohad22@homailom Asra We disuss i his paper he eisee ad uiueess of soluios for oudary value prolem Du( ) f, u( ), a u() u( ), i a Baah spae Uder erai odiios o f, he eisee of soluios is oaied y applyig Baah fied poi heorem ad Shaefer's fied poi heorem Keywords: Q-differeial euaio; Capuo fraioal -derivaive; Fraioal - iegral; Eisee soluio; Fied poi heorem

2 62 El-Shahed ad Al-Yami Iroduio Fraioal alulus is a disiplie o whih may researhers are dediaig heir ime, perhaps eause of is demosraed appliaios i various fields of siee ad egieerig [6] I pariular, he eisee of soluios o fraioal oudary value prolems is urrely uder srog researh[3] he -differee alulus or uaum alulus is a old suje ha was iiially developed y Jakso [9,], Basi defiiios ad properies of - differee alulus a e foud i he ook [] he fraioal -differee alulus had is origi i he works y Al-Salam [2] ad Agarwal [] ore reely, maye due o he eplosio i researh wihi he fraioal differeial alulus seig, ew developmes i his heory of fraioal -differee alulus were made, eg, -aalogues of he iegral ad differeial fraioal operaors properies suh as iage-leffler fuio [7], jus o meio some Very reely some asi heory for he iiial value prolems of fraioal differeial euaios ivolvig Riema-Liouville differeial operaor has ee disussed y Lakshmikaham ad Vasala [2,3] Some eisee resuls were give for he prolem ()-(2) wih y [4] ad, y isdell i [9] I his paper, we prese eisee resuls for he prolem for eah Du( ) f, u( ), I,,,, () a u() u( ), (2) where f :,, is a oiuous fuio, a,,, are real osas wih a I Seio 3, we give wo resuls, oe ased o Baah fied poi heorem (heorem 3) ad aoher oe ased o Shaefer's fied poi heorem (heorem 32) D is he Capuo fraioal -derivaive, 2 Prelimiaries I his seio, we irodue oaios, defiiios, ad prelimiary fas whih C I, we deoe he Baah spae of all are used hroughou his paper By oiuous fuios from I io wih he orm u : sup u ( ) : I

3 Eisee ad uiueess of soluios 62 Le, defied y [] a a, a a he -aalogue of he power fuio a wih is k ( a ), ( a ) ( a ), a,, k ore geerally, if, he ( a ) ( ) a i ( a ( a i ) i ) ) Noe ha, if he a ( a he -gamma fuio is defied y ( ) ( ) ( ), \,, 2,,, ( ) ad saisfies ( ) ( ) he -derivaive of a fuio f ( ) is here defied y df ( ) f ( ) f ( ) Df ( ), d ( ) ad -derivaives of higher order y ( ), f if D f ( ) DD f ( ) if he -iegral of a fuio f defied i he ierval, is give y f ( ) d ( ) f ( ),,, If a, ad f defied i he ierval,, is iegral from a o is defied y

4 622 El-Shahed ad Al-Yami a f ( ) d f ( ) d a f ( ) d Similarly as doe for derivaives, i a e defied a operaor I, amely, ( I f )( ) f ( ) ad ( I f )( ) I ( I f )( ), he fudameal heorem of alulus applies o hese operaors I ad ( D I f )( ) f ( ), ad if f is oiuous a, he ( I D f )( ) f ( ) f () D, ie, Basi properies of he wo operaors a e foud i he ook [] We ow poi ou hree formulas ha will e used laer ( i D deoes he derivaive wih respe o variale i ) [6] ( ) ( ) a ( s) a ( s), D ( s) ( ) ( s), ( ) D f (, ) d ( ) D f (, ) d f (, ) Remark 2 [6] We oe ha if ad a Defiiio 2[8] Le, he ( ( ) ( ) a) ( ) ad f e a fuio defied o, fraioal -iegral of he Riema Liouville ype is ( RL I f )( ) f ( ) ad ( ) ( RLI f )( ) ( ) f ( ) d,,, ( ) a he Defiiio 22[8] he fraioal -derivaive of he Riema Liouville ype of order is defied y ( RL D f )( ) f ( ) ad ( D f )( ) ( D I f )( ), RL, where is he smalles ieger greaer ha or eual o

5 Eisee ad uiueess of soluios 623 Defiiio 23[8] he fraioal -derivaive of he Capuo ype of order is defied y ( D f )( ) ( I D f )( ), C, where is he smalles ieger greaer ha or eual o Lemma 2[8] Le, e formulas hold: ( I I f )( ) ( I f )( ), 2 ( D I f )( ) f ( ) C ad f e a fuio defied o, he, he heorem 2[8] Le ad p e a posiive ieger he, he followig eualiy holds: ( I D f )( ) ( D I f )( ) ( D f )() p pk p p k RL RL k ( k p) heorem 22[8] Le ad \ he, he followig eualiy holds: ( I C D f )( ) f ( ) k k ( D ( k ) k f )() 3 Eisee of soluios Le us sar y defiig wha we mea y a soluio of he prolem ()-(2) Defiiio 3[4] A fuio u C,, (2) if u saisfies he euaio D u( ) f, u( ) a u() u( ) is said o e a soluio of ()- o I, ad he odiio For he eisee of soluios for he prolem ()-(2), we eed he followig auiliary lemma Lemma 3[4] Le, ad le :, A fuio u is a soluio of fraioal -iegral euaio y e oiuous

6 624 El-Shahed ad Al-Yami ( u ( ) u ) s y ( s ) d s ( ) if ad oly if u is a soluio of he iiial value prolem for he fraioal - differeial euaio Du( ) y ( ),,, u() u As a oseuee of lemma 3 we have he followig resul whih is useful i wha follows Lemma 32[4] Le, ad le :, A fuio u is a soluio of he fraioal -iegral euaio y e oiuous ( ( ) ) u s y ( s ) d s ( ) ( ) s y ( s ) d s a ( ) if ad oly if u is a soluio of he fraioal BVP Du( ) y ( ),,, a u() u( ) Our firs resul is ased o Baah fied poi heorem heorem 3[8] Assume ha: (H) here eiss a osa K suh ha f, u f, u 2 K u u 2, for eah I, ad all u, u2 K a If, he he BVP ()-(2) has a uiue soluio o, Proof rasform he prolem ()-(2) io a fied poi prolem Cosider he operaor defied y F : C,, C,, ( ) F ( u )( ) s f s, u ( s ) d s ( ) (3)

7 Eisee ad uiueess of soluios 625 ( ) s f s, u ( s ) d s a ( ) (4) Clearly, he fied poi of he operaor F are soluio of he prolem ()-(2) We shall use he Baah oraio priiple o prove ha F defied y (4) has a fied poi We shall show ha F is a oraio Le 2 C,,, he, for eah I we have ( ) F ( )( ) F ( )( ) s f s, ( s ) f s, ( s ) d s ( ) 2 2 ( ) s f s, ( s ) f s, 2( s ) d s ( ) a K 2 ( ) K 2 ( ) a ( ) s d s ( ) s d s hus K a 2 K a F ( ) F ( ) 2 2

8 626 El-Shahed ad Al-Yami Coseuely y (3) F is a oraio As a oseuee of Baah fied poi heorem, we dedue ha F has a fied poi whih is a soluio of he prolem ()-(2) he seod resul is ased o Shaefer's fied poi heorem heorem 32 Assume ha: (H2) he fuio f :, is oiuous (H3) here eiss a osa suh ha f, u for eah I ad all u he he BVP ()-(2) has a leas oe soluio o, Proof We shall use Shaefer's fied poi heorem o prove ha F defied y (4) has a fied poi he proof will e give i several seps Sep F is oiuous Le u e a seuee suh ha u, u i C,, he for eah ( ) F ( u )( ) F ( u )( ) s f s, u ( s ) f s, u ( s ) d s ( ) ( ) s f s, u ( s ) f s, u ( s ) d s ( ) a ( ) s sup f s, u ( s ) f s, u ( s ) d s ( ) s, ( ) s sup f s, u ( s ) f s, u ( s ) d s ( ) a s,, ( ), ( ) f u f u ( ) ( ) s ds s d s ( ) a f, u ( ) f, u ( ) a ( )

9 Eisee ad uiueess of soluios 627 Sie f is a oiuous fuio, we have f, u ( ) f, u ( ) a F u F u ( ) as Sep 2: F maps ouded ses io ouded ses i Ideed, i is eough o show ha for ay r suh ha for eah F u r By (H3) we have for eah, C,,, here eis a posiive osa u B u C,, : u, we have, ( ) F ( u )( ) s f s, u ( s ) d s ( ) ( ) s f s, u ( s ) d s ( ) a a ( ) ( ) s d s s d s ( ) ( ) a a hus ( ) ( ) a a F ( u ) : r a a Sep 3 F maps ouded ses io euioiuous ses of Le,,,, e ouded se of ad le u 2 2 B B he 2 2 C,, C,, as i sep 2, ( ) ( ) F ( u)( ) F ( u )( ) s s f s, u ( s ) d s 2 ( ) 2 s f s, u ( s ) d s ( )

10 628 El-Shahed ad Al-Yami ( ) ( ) 2 s s d s ( ) 2 ( ) ( 2 ) 2 ( ) s d s As 2, he righ-had side of he aove ieualiy eds o zero As a oseuee of Sep o 3 ogeher wih he Arzelá-Asoli heorem, we a F : C,, C,, is oiuous ad ompleely olude ha oiuous Sep 4 A priori ouds Now i remais o show ha he se u C I, : u F u for some is ouded Le u, he u F u for some hus, for eah I we have ( ) u ( ) s f s, u ( s ) d s ( ) ( ) s f s, u( s) ds a ( ) his implies y (H3) ha for I we have ( ) F ( u )( ) s f s, u ( s ) d s ( ) ( ) s f s, u ( s ) d s ( ) a a ( s ) d s ( ) ( s ) d s ( ) a a

11 Eisee ad uiueess of soluios 629 hus for every, a a, we have F ( u ) : a a his shows ha he se is ouded As a oseuee of Shaefer's fied poi heorem, we dedue ha F has a fied poi whih is a soluio of he prolem ()-(2) Referees [] R P Agarwal, Cerai fraioal -iegrals ad -derivaives, Pro Camridge Philos So 66 (969) [2] W A Al-Salam, Some fraioal -iegrals ad -derivaives, Pro Edi ah So 5 ( ), No 2, 35 4 [3] El-Shahed, Eisee of soluio for a oudary value prolem of fraioal order, Adv Appl ah Aal, 2 (27), No, -8 [4] El-Shahed, Posiive soluios of oudary value prolems for h order ordiary differeial euaios, Eleroi Joural of Qualiaive heory of Differeial Euaios, (28) ), No, -9 [5] El-Shahed ad H A Hassa, Posiive soluios of -differee euaio, Pro Amer ah So 38 (2), No 5, [6] R A C Ferreira, Norivial soluios for fraioal -differee oudary Value Prolems, Elero J Qual heory Differ Eu(2), No 7, - [7] R A C Ferreria, Posiive soluios for a lass of oudary value prolems wih fraioal -differees, Compu ah Appl, ( I press) [8] H Gauhma, Iegral ieualiies i -alulus, Compu ah Appl 47 (2-3) (24) 28-3 [9] F H Jakso, O -fuios ad a erai differee operaor ras Roy So Ediurgh 46 (98) [] F H Jakso, O -defiie iegrals, Quar J Pure Appl ah 4 (9) 93 23

12 63 El-Shahed ad Al-Yami [] V Ka ad P Cheug, Quaum Calulus, Spriger-Verlag, New York, 22 [2] V Lakshmikaham, ad A S Vasala, heory of fraioal differeial ieualiies ad appliaios, Commu Appl Aal (3&4) (27), [3] V Lakshmikaham ad A S Vasala, Geeral uiueess ad moooe ieraive ehiue for fraioal differeial euaios, Appl ah Leers, o appear [4] Behohra, S Hamai ad S K Nouyas, Boudary value prolems for differeial euaios wih fraioal order, Surveys i ahemais ad is Appliaios, Volume 3 (28), -2 [5] V A Ploikov, A V Ploikov ad A N Viyuk, Differeial Euaios wih a ulivalued Righ-Had Side, Asympoi ehods AsroPri", Odessa, 999 R (2k:3422) [6] I Podluy, Fraioal Differeial Euaios, Sa Diego Aademi Press, 999 [7] P Rajkovi, S D arikovi ad S Sakovi, O -aalogues of Capuo derivaive ad iag-leffler fuio, Fra Cal Appl Aal ( 27 ), No 4, [8] S Saković, P Rajković ad S D ariković, O -fraioal derivaives of Riema-Liouville ad Capuo ype, arxiv:99 [mah CA] 2 sep 29 [9] C C isdell, O he solvailiy of oliear firs-order oudary-value prolems, Elero J Differeial Euaios 26, No 8, 8 pp R224828(27e:344) Zl 7342 Reeived: arh, 2