Existence Of Solutions For Nonlinear Fractional Differential Equation With Integral Boundary Conditions

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1 Reserch Ivey: Ieriol Jourl Of Egieerig Ad Sciece Vol., Issue (April 3), Pp 8- Iss(e): 78-47, Iss(p): , Exisece Of Soluios For Nolier Frciol Differeil Equio Wih Iegrl Boudry Codiios, Azhr H. Sllo,, Afrh S. Hs,, Deprme of Mhemics, Fculy of Sciece, Uiversiy of Duhok, Kurdis Regio, Irq. Absrc. I his pper we discuss he exisece of soluios defied i C[, ] for boudry vlue problems for olier frciol differeil equio wih iegrl codiio. he resuls re derived by usig he Ascoli-Arzel heorem d Schuder-ychooff fixed poi heorem. Keywords: Riem_Liouville frciol derivive d iegrl, boudry vlue problem, olier frciol differeil equio, iegrl codiio, Ascoli-Arzel heorem d Schuder-ychooff fixed poi heorem. I. INRODUCION Frciol boudry vlue problem occur i mechics d my reled fields of egieerig d mhemicl physics, see Ahmd d Nouys [], Drwish d Nouys [4], Hmi, Bechohr d Gref [6], Kilbs, Srivsv d rujillo [7] d refereces herei. Vrious problems hs fced i differe fields such s populio dymics, blood flow models, chemicl egieerig d cellulr sysems h c be modeled o olier frciol differeil equio wih iegrl boudry codiios. Recely, my uhors focused o boudry vlue problems for frciol differeil equios, see Ahmd d Nieo [], Drwish d Nouys [4] d he refereces herei. Some works hs bee published by my uhors o exisece d uiqueess of soluios for olocl d iegrl boudry vlue problems such s Ahmd d Nouys [] d Hmi, Bechohr d Gref [6]. I his pper we prove he exisece of he soluios of olier frciol differeil equio wih iegrl boudry codiio he righ ed poi of [, ] i C[, ], where C[, ] is he spce of ll coiuous fucios over [, ],which resuls re bsed o Ascoli-Arzel heorem d Schuder- ychooff fixed poi heorem. II. PRELIMINARIES I his secio we iroduce defiiios, lemms d heorems which re used hroughou his pper. For refereces see Brre [3], Kilbs, Srivsv d rujillo [7] d refereces herei. Defiiio.. Le f be fucio which is defied lmos everywhere o[ b, ]. For b D f b f () b d ( ) provided h his iegrl exiss i Lebesgue sese, where is he gmm fucio. Lemm.. Assume h f C(,) L(,) wih frciol derivive of order o C(,) L(,), he for some Ci Lemm.3. Le D D f f C C C ( ) ( )... R ; i =,,,, where is he smlles ieger greer h or equl o., R,. If x, he x ( x ) ( ) ; egive ieger x I ( ) ( ) ; egive ieger Lemm.4. he followig relio x x x ( ) D D f D f 8 holds if, we defie: h belogs

2 Exisece Of Soluios For Nolier Frciol Differeil., d he fucio f ( x ) C o closed iervl [ b, ]. ( ) or, d he fucio f ( x ) C o closed iervl [ b, ]. b. x Lemm.5. If d f( x) is coiuous o [ b, ], he D f ( x ) wih respec o x o [ b, ]. exiss d i is coiuous heorem.6. (he Arzel Ascoli heorem) Le F be equicoiuous, uiformly bouded fmily of rel vlued fucios f o iervl I (fiie or ifiie).he F cois uiformly coverge sequece of fucio f, covergig o fucio f C ( I ) where C( I) deoes he spce of ll coiuous bouded fucios o I. hus y sequece i F cois uiformly bouded coverge subsequece o I d cosequely F hs compc closure i C( I ). heorem.7. (Shuder-ychoff Fixed Poi heorem) Le B be loclly covex, opologicl vecor spce. Le Y be compc, covex subse of B d coiuous mp of Y io iself. he hs fixed poi y Y. III. MAIN RESUL he semes d proofs for he mi resuls re crried ou i his secio. Lemm 3.. Le, h, x belog o C[, ], d, he he soluio of Where (, ) Where f x d D x f, x, h(,, x ( )) d x (3.), (, ) (3.) x x() d (3.3) d is cos, is give by x f (, x( ), h(,, x( )) d ) d ( ) f (, x( ), ( ) ( ) ( ) h(, s, x( s)) ds) d ( ) f, x, h(, s, x( s)) ds d ( ) d ( ). Proof. Opere boh sides of equio (3.) by he operor D, o obi D D x D f, x, h(,, x ( )) d From Lemm (.), we ge x C C D f, x, h(,, x ( )) d (3.4) Now, opere boh sides of equio (3.4) by he operor D, o hve D x D C D C D D f, x, h(,, x ( )) d (3.5) From Lemm (.) d Lemm (.3), D C C ( ), D C d 9

3 Exisece Of Soluios For Nolier Frciol Differeil D D f, x, h(,, x( )) d D f, x, h(,, x( )) d herefore equio (3.5) c be wrie s D x C( ) D f, x, h(,, x ( )) d (3.6) D Now, operig o boh sides of equio (3.4) by he operor d usig gi Lemm (.) d Lemm (.3), equio (3.4) kes he form D x C( ) C ( ) D f, x, h(,, x ( )) d (3.7) Now from he codiio (3.) d equio (3.7), i follows h C, d from he codiio (3.3) d equio (3.6) we ge x( ) d C( ) f, x, h(,, x( )) d d C d ( s) f s, xs, h( s, z, x( z)) dz dsd C( ) f, x, h(,, x( )) d d ( ) C f, x, h(,, x( )) d d ( ) f, x, h(, s, x( s)) ds d ( ) ( ) Where d ( ), herefore he soluio of he give boudry vlue problem kes he form x f (, x( ), h(,, x( )) d ) d ( ) f (, x( ), ( ) ( ) ( ) heorem: Assume h h(, s, x( s)) ds) d ( ) f, x, h(, s, x( s)) ds d. f, x d, problem (3.-3) hs uique soluio o [, ]. Proof: Le X { x ( ); x ( ) C [, ]} ( ) h x belog o C[, ], he he frciol boudry vlue d he mppig : C, C, defied by x( ) f (, x( ), h(,, x( )) d ) d ( ) ( ) ( ) f (, x( ), h(, s, x( s)) ds) d ( ) f (, x( ), h(, s, x( s)) ds) d ( ) (3.8) i order o pply he Schuder-ychooff fixed poi heorem, we should prove he followig seps Sep: mps X io iself. Le x X, sice f is coiuous o [, ] mps Y io iself Sep: is coiuous mppig o X. x () be sequece i X such h lim x ( ) x ( ) Le x ( ) x ( ) ( ), i gurees h ll he erms o (3.8) re coiuous. hus where x ( ) C, ( f (, x ( ), h(,, x ( )) d f (, x( ), h(,, x( )) d d, cosider

4 Exisece Of Soluios For Nolier Frciol Differeil ( ) [ f (, x ( ), h(, s, x ( s)) ds) f (, x( ), (,, ( )) ) ] h s x s ds d ( ) f x h s x s ds f x h s x s ds d (3.9) ( ) ( ) [ (, ( ), (,, ( )) ) (, ( ), (,, ( )) )] he righ hd side of he equio (3.9) eds o zero s eds o ifiiy, sice f is coiuous fucio x () d he sequece coverges o x (), h is x ( ) x ( ) s Also sice f is bouded hece by Lebesgue's domied covergece heorem we hve x ( ) x ( ) s herefore is coiuous mppig o X. Sep3: he closure of X { x ( ) ; x ( ) X } is compc. o prove sep3 we will prove h he fmily X is uiformly bouded d equicoiuous. X is uiformly, (, ], he bouded s show i sep, for provig he equicoiuiy, le such h x( ) x( ) f (, x( ), h(,, x( )) d ) d ( ) ( )( ( )) ( ) f (, x( ), h(, s, x( s)) ds) d ( ) ( ) f (, x( ), h(, s, x( s)) ds) d f (, x( ), h(,, x( )) d ) d ( ) ( )( ( )) ( ) f (, x( ), h(, s, x( s)) ds) d ( ) f (, x( ), h(, s, x( s)) ds) d ( ) By coiuiy of f o[, ] here exiss posiive cos M such h x( ) x( ) Md ( ) Md ( ) ( )( ( )) ( ) ( ) ( ) Md Md ( )( ( )) ( ) M ( ) Md ( ) Md M d = ( ) ( ) ( ) ( ) ( )( ( )) M [ ( ) d ( ) d ] ( ) M M ( ) ( ) ( ) ( )( ( ))

5 Exisece Of Soluios For Nolier Frciol Differeil x( ) x( ) M [ [( ) ( ) ] d ( ) d ] ( ) M M ( ) ( ) ( ) ( )( ( )) M ( ) ( ) ( ) M M M ( ) ( ) ( ) ( ) ( )( ( )) ( ) whe eds o, wih, we hve x( ) x( ), which proves h he fmily X is equicoiuous. hus by Ascoli-Arzel heorem, X hs compc closure. I view of sep, sep d sep3, he Schuder-ychooff fixed poi heorem gurees h hs les oe fixed poi x X, h is x( ) x( ). REFERENCES [] Ahmd B d Nieo J.J., Riem-Liouville frciol iegro-differeil equios wih frciol olocl iegrl boudry codiios. Boudry Vlue Problems,, :36. [] Ahmd B. d Nouys S. K., Boudry Vlue Problems for Frciol Differeil Iclusios wih Four-Poi Iegrl Boudry Codiios, Surveys i Mhemics d is Applicios, 6 (), [3] Brre, J.H., Differeil equio of o-ieger order, Cd. J. Mh., 6 (4) (954), [4] Drwish M.A. d Nouys S.K., O iiil d boudry vlue problems for frciol order mixed ype fuciol differeil iclusios, Compu. Mh. Appl. 59 (), [5] Goldberg, R.R., Mehods of Rel Alysis, Joh Wiley d Sos, Ic, USA (976). [6] Hmi S., Bechohr M. d Gref J.R., Exisece resuls for boudry vlue problems wih olier frciol differeil iclusios d iegrl codiios, Elecro. J. Differeil Equios, No., 6 pp. [7] Kilbs, A.A., Srivsv, H.M. d rujillo, J.J., heory d Applicios of Frciol Differeil Equios. Elsevier (Norh- Holld), Mh. Sudies, 4, Amserdm (6).