nonlocal conditions.

Transcription

1 ISSN prin, online Inernaional Journal of Nonlinear Science Vol No.1,pp.3-9 Boundary Value Problem for Some Fracional Inegrodifferenial Equaions wih Nonlocal Condiions Mohammed M Maar Mahemaics Deparmen, Al-Azhar Universiy-Gaza, Gaza Srip, PA. Received 17 November 29, acceped 2 January 211 Absrac: In his paper, he exisence problems of he soluion of some boundary value fracional inegrodifferenial equaions wih nonlocal condiions are invesigaed. he resuls are obained using Banach and Krasnoselkii fixed poin heorems. Keywords: Fracional calculus; exisence and uniqueness; inegrodifferenial equaions; fixed poin heorems; nonlocal condiions. 1 Inroducion Fracional differenial equaions are emerged as a new branch of applied mahemaics by which many physical and engineering approaches can be modeled. he fac ha fracional differenial equaions are considered as alernaive models o nonlinear differenial equaions which induced exensive researches in various fields including he heoreical par see [1]-[11] and references herein. he exisence and uniqueness problems of fracional nonlinear differenial and inegrodifferenial equaions as a basic heoreical par of some applicaions are invesigaed by many auhors see for examples [3], [6], and [7]. In [1], he auhors obained sufficien condiions for he exisence of soluions for a class of boundary value problem of fracional differenial equaions in he case of 1 < 2 involving he Capuo fracional derivaive and nonlocal condiions using he Banach and Schaefer s fixed poins heorems. he Cauchy problems for some fracional absrac differenial equaions in he case of < 1 wih nonlocal condiions are invesigaed by he auhors in [2] and [1] using he Banach and Krasnoselkii fixed poin heorems. he Banach fixed poin heorem is used in [4] and [8] o invesigae he exisence problems of fracional inegrodifferenial equaions in he case of < 1 in Banach spaces. Moivaed by hese works we sudy in his paper he exisence of soluion of boundary value problem for fracional inegrodifferenial equaions in he case of 1 < 2 in Banach spaces by using Banach and Krasnoselkii fixed poin heorems. 2 Preliminaries We need some basic definiions and properies of fracional calculus which will be used in his paper. Definiion 1 A real funcion f is said o be in he space C μ, μ R if here exiss a real number p > μ, such ha f p f 1, where f 1 C[, ; and i is said o be in he space C n μ if and only if f n C μ, n N. Definiion 2 A funcion f C μ, μ 1 is said o be fracional inegrable of order > if and if, hen I f f. I f I f 1 s 1 fsds <, Γ Corresponding auhor. address: mohammed maar@homail.com Copyrigh c World Academic Press, World Academic Union IJNS /434

2 4 Inernaional Journal of Nonlinear Science, Vol.11211, No.1, pp. 3-9 Nex, we inroduce he Capuo fracional derivaive. Definiion 3 he fracional derivaive in he Capuo sense is defined as D f d D f I n n f 1 d n s n 1 d n fs Γ n ds n ds for n 1 < n, n N, >, f C 1. n In paricular, if 1 < 2, hen D f 1 Γ2 s 1 f sds, where f s d2 fs ds, and f C is a funcion wih values in absrac space X. he ideniy I D f f + a + b, 1 where J, a, b are consans and oher properies of he fracional operaors used in he general heory of fracional differenial equaions can be found in [5], [9], and [11]. e Y CJ, X be a Banach space of all coninuous funcions x from a compac inerval J [, ] ino a Banach space X. e D {, s : s } be subse of R 2. Consider he fracional nonlinear inegrodifferenial equaion D x f, x, g, s, xsds, x hx, x kx, where 1 < 2, and he nonlinear funcions f, g, h, and k saisfy he following hypoheses: H1 f : J Y Y Y, g : D Y Y are joinly coninuous funcions and here exiss posiive consans B, C such ha { f, x, u f, y, v Cx y + u v g, s, x g, s, y B x y for any J,, s D, x, y, u, v Y. Moreover, le A sup J f,, H2 h : Y Y, and k : Y Y are coninuous funcions such ha for any x, y Y. { hx hy C x y kx ky C x y 2 g, s, ds, and max{a, B, C}. emma 1 he inegrodifferenial equaion D x f, g, sds, x hx, x kx, 3 is equivalen o he inegral equaion where y f, x hx + kx I y + I y 4 g, sds is a fracional inegrable of order funcion. IJNS for conribuion: edior@nonlinearscience.org.uk

3 Mohammed M Maar: Boundary Value Problem for Some Fracional Inegrodifferenial Equaions 5 Proof. Applying he fracional inegral operaor I o boh sides of equaion 3, and using he ideniy 1, we ge I D x I f, x + a + b 1 Now, if, we have a hx, and if, we have g, sds s 1 fs, gs, rdrds. which implies ha herefore kx hx + b 1 b kx + hx + 1 s 1 fs, s 1 fs, gs, rdrds gs, rdrds. x hx + kx hx s 1 fs, gs, rdrds hx + kx s 1 fs, s 1 fs, gs, rdrds s 1 fs, gs, rdrds gs, rdrds which is equaion 4. On he oher hand, applying he fracional differenial operaor D o boh sides of equaion 4, i is easily o ge equaion 3. In view of emma 1, equaion 2 is equivalen o he inegral equaion x hx + kx I F x + I F x where F x fs, xs, saisfies he following esimaes and gs, r, xrdr is a fracional inegrable of order nonlinear operaor. he operaor F I F x I F x F + I F C s 1 x + Bs x + Ads C x + CB+1 x + CA Γ x + 1 IJNS homepage: hp://

4 6 Inernaional Journal of Nonlinear Science, Vol.11211, No.1, pp. 3-9 I F x F y 1 + x y + 1 for every x, y Y, J. 3 Exisence problems We prove he exisence of he fracional nonlinear inegrodifferenial equaion 2 by using he well-known Banach fixed poin heorem. he following condiion is essenial o ge he conracion propery. H3 e < q < 1, and r be a posiive finie real number such ha 2 q 1 + Γ r 1 q 1 h + k + 22 Γ+1. Moreover, le B r {y Y : y r}. heorem 2 If he hypoheses H1-H3 are saisfied, hen he fracional inegrodifferenial equaion 2 has a unique soluion on J. Proof. Define he operaor Ψ : Y Y by Ψx hx + kx I F x + I F x. We show ha Ψ has a fixed poin on B r. his fixed poin is hen a soluion of equaion 2. Firsly, we show ha ΨB r B r. e x B r, hen Ψx hx + kx + I F x + I F x hx + kx + I F x + I F x h + C x + C x + k x x h + k q r + qr r x IJNS for conribuion: edior@nonlinearscience.org.uk

5 Mohammed M Maar: Boundary Value Problem for Some Fracional Inegrodifferenial Equaions 7 Hence, he operaor Ψ maps B r ino iself. Nex, we prove ha Ψ is a conracion mapping on B r. e x, y B r, hen Ψx Ψy hx + kx I F x + I F x hy ky + I F y I F y hx hy + kx ky + I F y F x + I F x F y x y + x y x y x y x y q x y. + 1 Hence, he operaor Ψ has a unique fixed poin which is a soluion o equaion 2. he nex resul is based on he following well-known fixed poin heorem. heorem 3 Krasnoselkii e S be a closed convex and nonempy subse of a Banach space X. e P and Q be wo operaors such ha i P x + Qy S whenever x, y S; ii P is a conracion mapping; iii Q is compac and coninuous. hen here exiss z S such ha z P z + Qz. o apply he above heorem we need he following condiion insead of he condiion H1. H4 he funcions f : J Y Y, and g : D Y Y are joinly coninuous and here exiss a posiive consan such ha for all, x, y J Y Y. f, x, y heorem 4 If he hypoheses H2 and H4 are saisfied, and if C < 1, hen he fracional inegrodifferenial equaion 2 has a soluion on J. Proof. e r 1 C 1 2 h + k + Γ+1. Define he operaors P, and Q on he compac se B r {y Y : y r} Y by { P x hx + kx Qy I F y I F y. We observe ha hence I F y Qy, 2 5 IJNS homepage: hp://

6 8 Inernaional Journal of Nonlinear Science, Vol.11211, No.1, pp. 3-9 and P x + Qy P x + Qy hx + kx + I F y I F y h + k C x + C herefore, if x, y B r, hen P x + Qy B r. On he oher hand, i is easily o show ha he operaor P is a conracion. Indeed, since P x P y hx hy + kx ky C x y + C x y C x y. By he hypohesis H4, he operaor Q is coninuous and by he inequaliy 5, i is uniformly bounded on B r. For he equiconinuiy of Qy, le 1, 2 in J, and y B r, we have x. hence I F y 1 I F y s 1 fs, ys, gs, r, yrdrds s 1 fs, ys, gs, r, yrdrds 1 2 s 1 1 s 1 ds s 1 ds Qy 1 Qy 2 I F y 1 I F y I F y 1 I F y As 2 1, he righ-hand side of he above inequaliy ends o zero which gives he equiconinuiy of Qy. So QB r is relaively compac. By he Arzela Ascoli heorem, Q is compac. Hence by he Krasnoselkii heorem here exiss a soluion o equaion 2. References [1] M. Benchohra, S. Hamania, S. K. Nouyas.Boundary value problems for differenial equaions wih fracional order and nonlocal condiions. Nonlinear Analysis, 7129: IJNS for conribuion: edior@nonlinearscience.org.uk

7 Mohammed M Maar: Boundary Value Problem for Some Fracional Inegrodifferenial Equaions 9 [2] K. Balachandran, J. Y. Park.Nonlocal Cauchy problem for absrac fracional semilinear evoluion equaions.nonlinear Analysis, 7129: [3] D. Delbosco,. Rodino.Exisence and uniqueness for a fracional differenial equaion. Journal of Mahemaical Analysis and Applicaions, : [4] O. K. Jarada, A. Al-Omari, S. Momani.Exisence of he mild soluion for fracional semilinear iniial value problem. Nonlinear Analysis, 6928: [5] A. A. Kilbas, H. M. Srivasava, J. J. rujillo. heory and applicaions of fracional differenial equaions. Elsevier, Amserdam,26. [6] V. akshmikanham.heory of fracional funcional differenial equaions.nonlinear Analysis, 6928: [7] V. akshmikanham, A. S. Vasala.Basic heory of fracional differenial equaions. Nonlinear Analysis, 6928: [8] M Maar.On exisence and uniqueness of he mild soluion for fracional semilinear inegro-differenial equaions. Journal of Inegral Equaions and Applicaions, acceped. [9] K. S. Miller, B Ross.An inroducion o he fracional calculus and fracional differenial equaions. J.Wiley & Sons, New York, [1] G. M. N guerekaa. A Cauchy problem for some fracional absrac differenial equaion wih nonlocal condiion. Nonlinear Analysis, [11] I. Podlubny. Fracional differenial equaions. Academic Press, New York, IJNS homepage: hp://