EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS WITH TWO-POINT BOUNDARY CONDITIONS

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1 Advaced Mahemaical Models & Applicaios Vol3, No, 8, pp54-6 EXISENCE AND UNIQUENESS OF SOLUIONS FOR NONLINEAR FRACIONAL DIFFERENIAL EQUAIONS WIH WO-OIN BOUNDARY CONDIIONS YA Shariov *, FM Zeyally, SM Zeyally Baku Sae Uiversiy, Baku, Azerbaija Gaja Sae Uiversiy, Gaja, Azerbaija Absrac he suicie codiios are esablished or he eisece o soluios or a class o olocal boudary value problems or racioal diereial equaios ivolvig he Capuo racioal derivaive Keywords: Nolocal boudary codiios, Capuo racioal derivaive, eisece, uiqueess, ied poi AMS Subjec Classiicaio: 6A33 Correspodig Auhor: Yagub Shariov, proessor, Baku Sae Uiversiy, Baku, Azerbaija, shariov@ramblerru Mauscrip received 38; Revised 8; Acceped 38 Iroducio Diereial equaios o racioal order have proved o be valuable ools i he modelig o may pheomea is various ields o sciece ad egieerig Ideed, we ca id umerous applicaios i viscoelasiciy (Bagley, 983; Caaia & Sorreio, 6; Sorreios, 7), dyamical processes i sel-similar srucures (Magi, 4), bioscieces (Lakshmikaham & Vasala, 8), sigal processig (Oldham, ), sysem corol heory (u e al,993; Viagre e al, ), elecrochemisry (Mezler & Klaer, ) ad diusio processes (Maiardi, 997) Furher, racioal calculus has oud may applicaios i classical mechaics (Origueira, 3; Dhaigude & Rizqa, 7) ad he calculus o variaios (Agrawal, ) ad is a very useul ad meas or obaiig soluios o o-homogeous liear ordiary ad parial diereial equaios For more deails we reer he reader o (Sorreios, 7) here are several approaches o racioal derivaives such as Riema- Lowville, Capuo, Weyl, Hadamar ad Gruwald-Leikov, ec Applied problems require hose deiiios o a racioal derivaive ha allow he uilizaio o physically ierpreable iiial ad boudary codiios he Capuo racioal derivaive saisies hese demads, while he Riema-Lowville derivaive is o suiable or mied boudary codiios Recely, he heory o eisece ad uiqueess o soluios o liear ad oliear racioal diereial equaios has araced he aeio o he may auhors, see or eample, (Agarwal e al, ; Ashyralyev & Shariov, ; Shariov,, Ibrahim & Momai, 7, Kha 5, Kha e al,, Bechohra e al, 8; Rabei & Alhalhol, 4) ad reereces herei However, may o he physical sysems ca beer be described by olocal boudary codiios Nolocal boudary 54

2 YA SHARIFOV e al: EXISENCE AND UNIQUENESS OF SOLUIONS FOR NONLINEAR codiios are ecouered i various applicaios such as populaio dyamics, blood low models, chemical egieerig ad cellular sysems Moreover, olocal boudary value problems a very ieresig ad impora class o problems hey iclude wopoi, hree-poi, muli-poi ad olocal boudary value problems as special cases, see (Ahmad & Nieo, 9; Boucheri, 9; Rabei & Alhalhol, 4, Xiwei & Ladog, 7) I his paper, we sudy eisece ad uiqueess o oliear racioal diereial equaios o he ype D,, or,, () subjec o wo-poi boudary codiios A B, () where, D relimiaries C is he Capuo racioal derivaives I his secio, we iroduce oaios, deiiios, ad prelimiary acs ha will be used i he remaider o his paper By C,, R we deoe he Baach space o all coiuous ucios rom, io R wih he orm ma :, Deiiio I g Ca, b ad, he he Riema-Lowville racioal iegral is deied by where I a g a g s s is he Gamma ucio deied or ay comple umber z as z z e d ds, (3) Deiiio he Capuo racioal derivaive o order o a coiuous ucio g : a, b is deied by R c D where, (he oaio a g s s g a ds, (4) sads or he larges ieger o greaer ha ) Remark Uder aural codiios o g, he Capuo racioal derivaive becomes he coveioal ieger order derivaive o he ucio g as Remark Le, > ad, he he ollowig relaios hold: c k D,, ad D, k,,,, Lemma For, g C, L,, he homogeous racioal diereial equaio D g, has a soluio g c c c, i ad where, c i R,,,,, 55

3 ADVANCED MAHEMAICAL MODELS & ALICAIONS, V3, N, 8 Lemma Assume ha g C, L, C, L,, he I D g g c c c where, c i R, i,,,, ad Lemma 3 (Kilbas e al, (6)) Le, q,, wih derivaive o order ha belogs o p L, he q pq q p I II, Moreover, i C, p I I (5) is saisied almos everywhere o, he (5) is rue or all, Lemma 4 (Kilbas e al, (6) I q, C,, he D I or all, 3 Mai Resuls Lemma 5 Le ad, g C,, R he he uique soluio o he boudary value problem or racioal diereial equaio c D y,, (6) A B C (7) is give by G sysds V,, (8) where, s A B B s, s, G, s (9) A B B s, s, V A B C roo Assume ha is a soluio o he boudary value problem (6), (7), he usig Lemma, we have I y c, c R () From (6) ad (9), we obai Ac BI y c C, () which implies ha c A B C A B BI y () Usig () i (), we obai I y A B C A B BI y, which ca be wrie as (8) Lemma is provided Lemma 6 Assume ha C, R, R, he he ucio is soluio o racioal boudary value problem (), () i ad oly i is soluio o he racioal iegral equaio, 56

4 YA SHARIFOV e al: EXISENCE AND UNIQUENESS OF SOLUIONS FOR NONLINEAR G s s, sds A B, C (3) roo Le be a soluio o he boudary value problem (), (), he by same mehod as used i Lemma 5, we ca prove ha i is a soluio o he racioal iegral equaio (3) saisy (3) ad deoe he righ had side o equaio (3) Coversely, le byv he, by Lemmas 3 ad 4, we obai his implies ha Hece, c D v( ) G I, s s, sds A B, A B C, C D C, v D I, is a soluio o racioal diereial equaio ()Also, i is easy o veriy ha saisy he codiio () Our irs resul is based o Baach ied poi heorem heorem Assume ha: (H) here eiss a cosa L such ha, y L, ad all, y R I, y, or each L A B B (4) he he boudary value problem ()-() has uique soluio o, roo rasorm he problem ()-() io a ied poi problem Cosider he operaor : C,, R C,, R deied by G, s s, sds A B C (5) Clearly, he ied pois o he operaor are soluio o he problem ()-() We shall use he Baach coracio priciple o prove ha deied by (4) has a ied poi We shall show ha is a coracio y C,, R, we have hus Le, he, or each y G, s s, s s, ysds s s, s s, ys ds A B B s s, s s, ysds L A B B y 57

5 ADVANCED MAHEMAICAL MODELS & ALICAIONS, V3, N, 8 y L A B B y Cosequely by (4) is a coracio As a cosequece o Baach ied poi heorem, we deduce ha has a ied poi which is a soluio o he problem ()-() heorem is provided he secod resul is based o Schaeer s ied poi heorem heorem Assume ha: (H) he ucio :, R R is coiuous (H3) here eiss a cosa N such ha, N or each, ad all R he he boudary value problem ()-() has a leas oe soluio o, roowe shall use Schaeer s ied poi heorem o prove ha deied by (5) has a ied poi he proo will be give i several seps Sep : Operaor is coiuous Le be a sequece such ha i, R he or each, C, G, s s, s s s ds, s ma s, s s, s ds A B B s ma s, s s, sds Sice is coiuous ucio, we have as L A B B s, s s, s L A B B s, s s, s Sep : maps bouded ses i bouded ses ic,, show ha or ay R Ideed, i is eough o, here eiss a posiive cosa l such ha or each,, R :, we have l B C,, each Hece, hus G, s s, sds A B C N N By (H4) ad (H4) we have or A B B A B C A B B A B C l 58

6 YA SHARIFOV e al: EXISENCE AND UNIQUENESS OF SOLUIONS FOR NONLINEAR Sep 3: maps bouded ses io equicoiuous ses o C,, B be a bouded se o C,, R, R Le,,, as i Sep, ad le B he s s s s, ds, s s s ds N As, he righ-had side o he above iequaliy eds o zero As a cosequece o Seps o 3 ogeher wih he Arzela-Ascoli heorem, we ca coclude ha he operaor : C,, R C,, R is compleely coiuous Sep 4: A priori bouds Now i remais o show ha he se C,, R :, or some is bouded Le or some, we have he hus, or each G, s s, sds A B C, we have his implies by (H4) ad (H6) (as i sep) ha or each N A B B A B C hus or every,, we have N A B B A B C R his shows ha he se is bouded As a cosequece o Schaeer s ied poi heorem, we deduce ha has a ied poi which is a soluio o he problem ()-() I he ollowig heorem we shall give a eisece resul or he problem ()-() by meas o a applicaio o a Leray-Schauder ype oliear aleraive, where he codiio (H3) ad is weakeed heorem 3 Assume ha (H3) ad he ollowig codiios hold L, R ad coiuous ad o decreasig (H4) here eis, :,,, or each, such ha ad all R (H5) here eiss a umber K such ha K K I I g K A B L L he boudary value problem ()-() has a leas oe soluio o, roo Cosider he operaor deied i heorems ad I ca be easily show ha is coiuous ad compleely coiuous For, le be such ha or,, each we have he rom (H4) we have or each 59

7 ADVANCED MAHEMAICAL MODELS & ALICAIONS, V3, N, 8 hus s s s A B B s s s s s A B B s s ds ds ds I I L he, by codiio (H5), here eiss K such ha K Le U he operaor U C,, C,, R: K U such ha ds : R is coiuous ad compleely coiuous By he choice o U, here eiss o or some, As a cosequece o he oliear aleraive o Leray-Schauder ype (Graas & Dugudji, 3), we deduce ha has a ied poi i U, which is a soluio o he problem ()-() his complees o proo 4 A eample I his secio we give a eample o illusrae he useuless o our mai resuls Le us cosider he ollowig olocal boudary value problem or sysem racioal diereial equaio D si,,,,, (6) D 9 e Evidely, A B E, B, A B A ad B (7) Hece he codiios (H)-(H) hold wih L, We shall check ha codiio (4) is saisied or appropriae values o wih Ideed, (8) he by heorem 33 he boudary value problem (6)-(7) has a uique soluio o, or values o saisyig codiio (8) For eample, i, he,, 9 ad,,869 6

8 YA SHARIFOV e al: EXISENCE AND UNIQUENESS OF SOLUIONS FOR NONLINEAR Reereces Agarwal, R, Bechohra, M, & Hamai, S () A survey o eisece resuls or boudary value problems o oliear racioal diereial equaios ad iclusios Aca Applicadae Mahemaicae, 9(3), Agarwal, R, Bechohra, M, Hamai, S (9) Boudary value problems or racioal diereial equaiosgeorgia Mahemaical Joural, 6(3), 4-4 Agrawal, O () Formulaio o Euler-Lagrage equaios or racioal variaioal problems J Mah Aal Appl, 7, Ahmad, B, Nieo, JJ (9) Eisece resuls or oliear boudary value problems o racioal iegrodiereial equaios wih iegral boudary codiios Boudary value problems, 9(), Ashyralyev, A, Shariov, Y A () Eisece ad uiqueess o soluios or he sysem o oliear racioal diereial equaios wih olocal ad iegral boudary codiios I Absrac ad Applied Aalysis (Vol ) Hidawi,, Id 5948 Bagley, RI (983) A heoreical basis or he applicaio o racioal calculus o viscoelasiciyjoural o Rheology, 7(3), - Bechohra, M, Hamai, S, Nouyas, SK (8) Boudary value problems or diereial equaios wih racioal order Surveys i Mahemaics ad is Applicaios, 3, - Boucheri, A (9) Secod order boudary value problems wih iegral boudary codiios Noliear Aalysis, 7(), Caaia, G, Sorreio, S (6) Fracioal derivaive liear models or describig he viscoelasic dyamic behaviour o polymeric beams I 4h Coerece ad Eposiio o Srucural Dyamics 6, IMAC-XXIV Sociey or Eperimeal Mechaics (SEM) Dhaigude, DB, Rizqa, BH (7) Eisece ad uiqueess o soluios or racioal diereial equaios wih advaced argumes, Advaced Mahemaical Models & Applicaios, (3), 4-5 Graas, A, Dugudji, J (3) Fied poi heory Spriger Sciece & Busiess Media, New- York, 3 Ibrahim, RW, Momai, S (7) O eisece ad uiqueess o soluios o a class o racioal diereial equaios Joural o Mahemaical Aalysis ad Applicaios, 3334, - Kha, RA (5) Eisece ad approimaio o soluios o oliear problems wih iegral boudary codiios Dyamic Sysems ad Applicaios,4, 8-96 Kha, RA, Rehma, MU, Hederso, J () Eisece ad uiqueess o soluios or oliear racioal diereial equaios wih iegral boudary codiios Fracioal Diereial Equaios, (), 9-43 Kilbas, AA, Srivasava, HM, rujillo, JJ (6)heory ad Applicaios o Fracioal Diereial Equaios, Norh-Hollad Mahemaics Sudies, 4, Elsevier Sciece BV, Amserdam Lakshmikaham, V, Vasala, AS (8) Basic heory o racioal diereial equaios Noliear Aalysis, 6, Magi, R (4) Fracioal calculus i bioegieerig Cri Rev Biom Eg, 3(), -4 Maiardi, F (997) Fracioal calculus I Fracals ad racioal calculus i coiuum mechaics (pp 9-348) Spriger, Viea Mezler, R, Klaer, J () Boudary value problems or racioal diusio equaios hysica A: Saisical Mechaics ad is Applicaios, 78(-), 7-5 Oldham, KB () Fracioal diereial equaios i elecrochemisry Advaces i Egieerig Soware, 4(), 9- Origueira, M (3) Special issue o racioal sigal processig ad applicaios Sigal rocessig, 83(),

9 ADVANCED MAHEMAICAL MODELS & ALICAIONS, V3, N, 8 Rabei, EM, Alhalhol, S (4) oeials o arbirary orces wih racioal derivaives I J Mod hys A, 9(7-8), Shariov, YA () Eisece ad uiqueess o soluios orhe sys-em o oliear racioal diereial equaios wih olocal boudary codiios roceedigs o IMM o NAS o Azerbaija, XXXVI (XLIV), 5-34 Shuqi, Z (6) Eisece o soluio or boudary value problem o racioal orderaca Mahemaica Sceia, 6 B(), -8 Sorreios, G (7) Aalyic Modelig ad Eperimeal Ideiicaio o Viscoelasic Mechaical Sysems; Advaces i Fracioal Calculus u, S, Nishimoo, K, Jaw, S (993) Applicaios o racioal calculus o ordiary ad parial diereial equaios o secod order Hiroshima Mah J, 3, Viagre, BM, odluby, I, Heradez,A, Feliu, V () Some approimaios o racioal order operaors used i corol heory ad applicaios Frac Calc Appl Aal, 3(3), 3-48 Xiwei, S, Ladog, L (7) Eisece o soluio or boudary value problem o oliear racioal diereial equaio Appl Mah J Chiese Uiv Ser B, 3(3),