 Advances in Pure Mathematics, 2011, 1, 59-62 doi:10.4236/apm.2011.13013 Published Online May 2011 (http://www.scirp.org/journal/apm) Copyright © 2011 SciRes. APM An Arbitrary (Fractional) Orders Differential Equation with Internal Nonlocal and Integral Conditions Ahmed El-Sayed1, E. O. Bin-Taher2 1Faculty of Science, Alexandr ia University, Alexandria, Egypt 2Faculty of Science, Hadhramout University of Science and Technology , Hadhramout, Yemen E-mail: amasayed@hotma il.com, ebtsamsam@yahoo.com Received February 28, 2011; revised March 25, 2011; acce pted Ma rc h 30, 2011 Abstract In this paper we study the existence of solution for the differential equation of arbitrary ( fractional) orders d = ,, 0,1 dxftDxtt, with the general form of internal nonlocal condition =1 =1 = , pmkkj jkjax bx   ,0,1, , 0,1, kjacdbc d . The problem with nonlocal integral condition will be studied. Keywords: Internal Nonlocal Problem, Integral Condition, Fractional Calculus, Existence of Solution, Caratheodory Theorem 1. Introduction Problems with non-local conditions have been exten-sively studied by several authors in the last two decades. The reader is referred to ([1-10]), and references therein. In this work we study the existence of at least one so-lution for the nonlocal problem of the arbitrary (frac-tional) or d er different ial equation   d = ,, 0,1 and 0,1dxt ftDxttt (1) with the general nonlocal condition =1 =1 = ,pmkkj jkjax bx  (2) where  ,0,1, , 0,1, kjacdbc d  and 0 is parameter. As an app lication, we deduce the ex istence of solution for the nonlocal problem of the differential (1) with the integral cond ition   d = d. cbadxss xss (3) It must be noticed that the following nonlocal and inte-gral conditions are special cases of our nonlocal and in-tegral conditions   = , , and ,,xxac db  (4)  =1 = , , and ,,mkk kkax xacdb  (5) =1 = 0, ,,mkk kkax ac (6)   d = , ,,caxss xdb (7) and   d = 0, ,.caxss ac (8) 2. Preliminaries Let 1LI denotes the class of Lebesgue integrable functions on the interval  =,Iab , with the norm 1= dLIuutt and  CI denotes the class of con-tinuous functions on the interval I, with the norm  = suptIuut and .  denotes the gamma func-tion. Definition 2.1 The fractional-order integral of the function 1,fLab of order R is defined by (see )    1 = d.taatsIftfs s Definition 2.2 The Caputo fractional-order derivative of A. EL-SAYED ET AL. Copyright © 2011 SciRes. APM 60 order 0,1 of the absolutely continuous function ft is define d by (se e [1 1] and [ 1 2] )  1 d = .daaDft Iftt Definition 2.3 The function :0,1 fRR is called 1LCaratheodory if 1) ,tftx is measurable for each xR, 2) ,xftx is continuous for almost all 0,1t, 3) there exists 10,1,, mLDD R such that fm. Now we state Caratheodory Theorem ( ). Theorem 2.1 Let 0,1 fRR be 1LCara- theodory, then the initial-value problem  d=,, for a.e. t>0, and 0=doxtftxtx xt (9) has at least one absolutely continuous solution 0,xAC T. Here we generalize Caratheodory theorem for the nonlocal problem (1) - (2). 3. Main Results Consider firstly the fractional-order integral equation  1 = ,,ytIf tyt (10) Definition 3.1 The function y is called a solution of the fractional-order integral Equation (10), if 0,1yC and satisfies (10). Theorem 3.1 Let :0,1 fRR be 1LCara theodory. Then there exists at least one solution of the fractional-order integral Equation (10). Proof. Let   = Max: 0,1, 0 and 0,1aMImtta , then   11 , ,d d , 0.taatatsIftytfsys sts mssM a Define the sequence nyt by    10 = , d, 0 ,11 tnntsyt fsysst which can be written in the operator form 1 1 = ,.nnytI Iftyt  Then   1 101 , d1 2 2 tnntsytIIftytMstMM   For 12, 0,1tt such that 12 < tt, then        21121121211 01020211010 = , d 1 , d1 = , d (1 ) , d (1 ) , d(1 ) ,(1 )tnn ntntntnttntntsyt ytfsyssts fsy ssts fsy ssts fsy ssts fsy ssts fsy s  211210 d , d(1 ) , d.(1 )tnttnsts fsy ssts fsy ss Therefore    212211212 11221 21 d 1 d d 11 .2 tnn ttttttsyt ytmssttmMttM  Hence  2112 11 < < nnttytyt  and nyt is a sequence of equi-continuous and uni-formly bounded functions. By Arzela-Ascoli Theorem, ( and ) there exists a subsequence knyt of continuous functions which converges uniformly to a continuous function y as k. Now we show that this limit function is the required solution. Since 1, ,nkfsy sms L and ,nkfsys is continuous in the second argu-ment, A. EL-SAYED ET AL. Copyright © 2011 SciRes. APM 61.., , as ,nkiefsysfsysk therefore the sequence   ,,nktsfsys  0,1 satisfies Lebesgue dominated convergence theorem. Hence    0 0 , dlim 1 = , d = (),1tnkktts fsy ssts fsyssyt which proves the existence of at least one solution 0,1yC of the fractional-order functional integral Equation (10). For the existence of solution for the nonlocal problem (1) - (2) we have the following theorem. Theorem 3.2 Let the assumptions of Theorem 3.1 are satisfied. Then nonlocal problem (1) - (2) has at least one solution 0,1xAC. Proof. Consider the nonlocal problem (1) - (2). Let   = , yt Dxt then  1d = ,dxtyt It (11)  1 = ,ytIftyt (12) and y is the solution of the fractional-order integral Equation (10). Operating by I on both sides of Equation(11), we obtain    d = = 0 dxtIytIxt xt (13)  = 0 .xtx Iyt (14) Let =kt in Equation (13), we get    10=1 =1=1 = d0 .mm mkkkk kkkk ksaxayss xa  And let =jt in Equation (13), we get    10=1 =1=1 = d0 .pp pjjjjjjj jsbxbjy ssxb  From Equation (2), we get   10=1 =110=1 =1 d0 = d0 .mmkkkkkkppjjjjjjsayssxasbyssxb Then we get   10=110=10= d dmkkkkpjjjjsxAa ysssbyss and    10=110=110= d ddmkkkkpjjjjtsxtAaysssbyssts ys s (15) where 1=1 =1= pmjkjkAba which, by Theorem 3.1, has at least one solution 0,1 .xAC Now, from Equation (15), we have    10=1 010=10= = dlim dkjmkkktpjjjsxxtAaysssAb yss and     10 1=111100=11= = dlim1 d djmkkktkpjjjsxxtAa yssssAbyssyss from which we deduce that Equation (15) has at least one solution 0,1 .xAC To complete the proof, diffe rentiating (15), we obtain  d = = ,.dxytftDxtt Also from (15) we can prove that the solution satisfies the nonlocal condition (2). 4. Nonlocal Integral Condition Let 0,1 .xAC be the solution of the nonlocal prob-lem (1) - (2). Let 11 012=, ,, = <<,<= kkkkk kmattttattttc  A. EL-SAYED ET AL. Copyright © 2011 SciRes. APM 62 and 11 012=,(,), = <<,<= jjjjj jpbsss sdssssb  then the nonlocal condition (2) will be 11=1 =1 = .pmkkkj jjkjtt xss x From the continuity of the solution x of the nonlocal problem (1) - (2) w e can ob tain 11=1 =1 = .limlim pmkkkj jjmpkjtt xss x  and the nonlocal condition (2) transformed to the integral one   d = d .cbadxss xss (16) Now, we have the following Theorem Theorem 4.1 Let the assumptions of Theorem 3.2 are satisfied. Then there exist at least one solution 0,1 .xAC of the nonlocal problem with integral condition,  = , , 0, 1,xt ftDxt t   d = d , .cbadxssyssbd ca Letting = 0 in (16), the we can easily prove the following corollary . Theorem 4.2 Let the assumptions 1) - 2) are satisfied. 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