D021018022

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D021018022

  1. 1. Research Inventy: International Journal Of Engineering And ScienceVol.2, Issue 10 (April 2013), Pp 18-22Issn(e): 2278-4721, Issn(p):2319-6483, Www.Researchinventy.Com18Existence Of Solutions For Nonlinear Fractional DifferentialEquation With Integral Boundary Conditions1,Azhaar H. Sallo , 2,Afrah S. Hasan1,2,Department of Mathematics, Faculty of Science, University of Duhok, Kurdistan Region, Iraq.Abstract. In this paper we discuss the existence of solutions defined in [0, ]C T for boundary value problemsfor a nonlinear fractional differential equation with a integral condition. The results are derived by using theAscoli-Arzela theorem and Schauder-Tychonoff fixed point theorem.Keywords: Riemann_Liouville fractional derivative and integral, boundary value problem, nonlinearfractional differential equation, integral condition, Ascoli-Arzela theorem and Schauder-Tychonoff fixed pointtheorem.I. INTRODUCTIONFractional boundary value problem occur in mechanics and many related fields of engineering andmathematical physics, see Ahmad and Ntouyas [2], Darwish and Ntouyas [4], Hamani, Benchohra and Graef[6], Kilbas, Srivastava and Trujillo [7] and references therein. Various problems has faced in different fieldssuch as population dynamics, blood flow models, chemical engineering and cellular systems that can bemodeled to a nonlinear fractional differential equation with integral boundary conditions. Recently, manyauthors focused on boundary value problems for fractional differential equations, see Ahmad and Nieto [1],Darwish and Ntouyas [4] and the references therein. Some works has been published by many authors onexistence and uniqueness of solutions for nonlocal and integral boundary value problems such as Ahmad andNtouyas [2] and Hamani, Benchohra and Graef [6].In this paper we prove the existence of the solutions of a nonlinear fractional differential equation withan integral boundary condition at the right end point of [0, ]T in [0, ]C T , where [0, ]C T is the space ofall continuous functions over [0, ]T ,which results are based on Ascoli-Arzela theorem and Schauder-Tychonoff fixed point theorem.II. PRELIMINARIESIn this section we introduce definitions, lemmas and theorems which are used throughout this paper.For references see Barrett [3], Kilbas, Srivastava and Trujillo [7] and references therein.Definition 2.1. Let f be a function which is defined almost everywhere on[ , ]a b . For 0 , we define:ba D f 11( )( )baf t b t dt  provided that this integral exists in Lebesgue sense, where  is the gamma function.Lemma 2.2. Assume that f (0,1) (0,1)C L  with a fractional derivative of order 0 that belongsto (0,1) (0,1)C L , then1 21 20 0( ) ( ) ... nnD D f t f t C t C t C t            for some iC R ; i =1, 2, …, n, where n is the smallest integer greater than or equal to  .Lemma 2.3. Let R,  , 1 . If ax  , then( ); int( )( 1)( 1)0 ; intxax anegative egerx aInegative eger               Lemma 2.4. The following relation( )x x xa a aD D f D f       holds if
  2. 2. Existence Of Solutions For Nonlinear Fractional Differential…19a. 0 , 0  and the function ( )f x C on a closed interval [ , ]a b .b. 0  or 0n   , 0  and the function( )( ) nf x C on a closed interval [ , ]a b .Lemma 2.5. If 0 and ( )f x is continuous on [ , ]a b , then ( )xa D f xexists and it is continuouswith respect to x on [ , ]a b .Theorem 2.6. (The Arzela Ascoli Theorem)Let F be an equicontinuous, uniformly bounded family of real valued functions f on an interval I (finite orinfinite).Then F contains a uniformly convergent sequence of function nf , converging to a function( )f C I where ( )C I denotes the space of all continuous bounded functions on I . Thus any sequence inF contains a uniformly bounded convergent subsequence on I and consequently F has a compact closure in( )C I .Theorem 2.7. (Shauder-Tychnoff Fixed Point Theorem)Let B be a locally convex, topological vector space. Let Y be a compact, convex subset of B and T acontinuous map ofY into itself. Then T has a fixed point y Y .III. MAIN RESULTThe statements and proofs for the main results are carried out in this section.Lemma 3.1. Let   ,f t x t and   ,h t x t belong to [0, ]C T , and 1 2  , then the solution of   0, , ( , , ( ))TD x t f t x t h t x d       , (0, )t T (3.1)  20 0x (3.2)  10( )x T a x d   (3.3)Where (0, )T  and a is a constant, is given by  1 10 0 0( , ( ), ( , , ( )) ) ( ) ( , ( ),( ) ( 1) ( )T Tt atx t f t x t h t x d dt f x                    10 0 01( , , ( )) ) ( ) , , ( , , ( ))( )T t Th s x s ds d t f x h s x s ds d              Wherea   and ( )   .Proof. Operate both sides of equation (3.1) by the operator D , to obtain   0, , ( , , ( ))TD D x t D f t x t h t x d         From Lemma (2.1), we get   1 21 20, , ( , , ( ))Tx t C t C t D f t x t h t x d             (3.4)Now, operate both sides of equation (3.4) by the operator1D, to have   1 1 1 1 2 11 20, , ( , , ( ))TD x t D C t D C t D D f t x t h t x d                     (3.5)From Lemma (2.2) and Lemma (2.3),1 11 1 ( )D C t C    ,1 22 0D C t   and
  3. 3. Existence Of Solutions For Nonlinear Fractional Differential…20   1 10 0, , ( , , ( )) , , ( , , ( ))T TD D f t x t h t x d D f t x t h t x d                  Therefore equation (3.5) can be written as   1 11 00( ) , , ( , , ( ))TtD x t C D f t x t h t x d           (3.6)Now, operating on both sides of equation (3.4) by the operator2Dand using again Lemma (2.2) and Lemma(2.3), equation (3.4) takes the form   2 21 2 00( ) ( 1) , , ( , , ( ))TtD x t C t C D f t x t h t x d               (3.7)Now from the condition (3.2) and equation (3.7), it follows that 2 0C  , and from the condition (3.3) andequation (3.6) we get 10 0 0( ) ( ) , , ( , , ( ))T Ta x d C f t x t h t x d dt                1 11 10 0 0 0 0 0( ) , , ( , , ( )) ( ) , , ( , , ( ))( )T T TaaC d s f s x s h s z x z dz dsd C f t x t h t x d dt                                10 0 0 01, , ( , , ( )) ( ) , , ( , , ( ))( ) ( 1)T T TaC f t x t h t x d dt f x h s x s ds d                              Wherea   and ( )   , therefore the solution of the given boundary value problem takes the form  1 10 0 0( , ( ), ( , , ( )) ) ( ) ( , ( ),( ) ( 1) ( )T Tt atx t f t x t h t x d dt f x                    10 0 01( , , ( )) ) ( ) , , ( , , ( )) .( )T t Th s x s ds d t f x h s x s ds d              Theorem: Assume that   ,f t x t and   ,h t x t belong to [0, ]C T , then the fractional boundary valueproblem (3.1-3) has a unique solution on [0, ]T .Proof: Let { ( ); ( ) [0, ]}X x t x t C T  and the mapping    : 0, 0,T C T C T defined by10 0 0( ) ( , ( ), ( , , ( )) ) ( )( ) ( 1)T Tt aTx t f t x t h t x d dt              10 0 01( , ( ), ( , , ( )) ) ( ) ( , ( ), ( , , ( )) )( )T t Tf x h s x s ds d t f x h s x s ds d             (3.8)in order to apply the Schauder-Tychonoff fixed point theorem, we should prove the following stepsStep1: T maps X into itself.Let x X , since f is continuous on [0, ]T , it guarantees that all the terms on (3.8) are continuous. ThusT maps Y into itselfStep2: T is a continuous mapping on X .Let   1( )n nx tbe a sequence in X such that lim ( ) ( )nnx t x t where  ( ) 0,x t C T , consider( ) ( )nTx t Tx t 10 0 0( ( , ( ), ( , , ( )) ( , ( ), ( , , ( ))( )T T Tn ntf t x t h t x d f t x t h t x d dt          
  4. 4. Existence Of Solutions For Nonlinear Fractional Differential…210 0( ) [ ( , ( ), ( , , ( )) )( 1)Tn naf x h s x s ds        0( , ( ), ( , , ( )) ) ]Tf x h s x s ds d    10 0 01( ) [ ( , ( ), ( , , ( )) ) ( , ( ), ( , , ( )) )]( )t T Tn nt f x h s x s ds f x h s x s ds d            (3.9)the right hand side of the equation (3.9) tends to zero as n tends to infinity, since f is a continuous functionand the sequence   1( )n nx tconverges to ( )x t , that is( ) ( ) 0nx t x t  as nAlso since f is bounded hence by Lebesgues dominated convergence theorem we have( ) ( ) 0nTx t Tx t  as ntherefore T is a continuous mapping on X .Step3: The closure of { ( ) ; ( ) }TX Tx t x t X  is compact.To prove step3 we will prove that the family TX is uniformly bounded and equicontinuous. TX is uniformlybounded as shown in step1, for proving the equicontinuity, let 1 2, (0, ]t t T such that 1 2t t , then122 1 2 2 2 20 0( ) ( ) ( , ( ), ( , , ( )) )( )T TtTx t Tx t f t x t h t x d dt      120 0( ) ( , ( ), ( , , ( )) )( 1)( ( ))Tatf x h s x s ds d             2120 01( ) ( , ( ), ( , , ( )) )( )t Tt f x h s x s ds d      111 1 1 10 0( , ( ), ( , , ( )) )( )T Ttf t x t h t x d dt     110 0( ) ( , ( ), ( , , ( )) )( 1)( ( ))Tatf x h s x s ds d             2110 01( ) ( , ( ), ( , , ( )) )( )t Tt f x h s x s ds d      By continuity of f on[0, ]T there exists a positive constant M such that1 12 22 1 20 0( ) ( ) ( )( ) ( 1)( ( ))Tt atTx t Tx t Mdt Md                  2 11 12 10 01( )( ) ( )t Ttt Md Mdt        111110 01( ) ( )( 1)( ( )) ( )tatMd t Md                 = 1 1 1 12 1 2 10( ) ( ) ( )( ) ( 1)( ( ))M T a Mt t t t d                     2 11 12 10 0[ ( ) ( ) ]( )t tMt d t d          11 1 1 12 1 2 1( ) ( )( ) ( 2)( ( ))MT aMt t t t                  
  5. 5. Existence Of Solutions For Nonlinear Fractional Differential…221 211 1 12 1 20[ [( ) ( ) ] ( ) ]( )t ttMt t d t d               11 1 1 12 1 2 1( ) ( )( ) ( 2)( ( ))MT aMt t t t                  2 2 1 2 1 1( ) ( )( )M t t t t t t               2 1( ) ( )Tx t Tx t 11 1 1 12 1 2 1 2 1( ) ( ) ( )( ) ( 2)( ( )) ( 1)MT aM Mt t t t t t                        when 1t tends to 2t , with 1 2t t   , we have 2 1( ) ( )Tx t T x t   , which proves that the familyTX is equicontinuous. Thus by Ascoli-Arzela theorem, TX has a compact closure. In view of step1, step2and step3, the Schauder-Tychonoff fixed point theorem guarantees that T has at least one fixed point x X ,that is ( ) ( )Tx t x t .REFERENCES[1] Ahmad B and Nieto J.J., Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundaryconditions. Boundary Value Problems, 2011, 2011:36.[2] Ahmad B. and Ntouyas S. K., Boundary Value Problems for Fractional Differential Inclusions with Four-Point IntegralBoundary Conditions, Surveys in Mathematics and its Applications, 6 (2011), 175-193.[3] Barrett, J.H., Differential equation of non-integer order, Canad. J. Math., 6 (4) (1954), 529-541.[4] Darwish M.A. and Ntouyas S.K., On initial and boundary value problems for fractional order mixed type functional differentialinclusions, Comput. Math. Appl. 59 (2010), 1253-1265.[5] Goldberg, R.R., Methods of Real Analysis, John Wiley and Sons, Inc, USA (1976).[6] Hamani S., Benchohra M. and Graef J.R., Existence results for boundary value problems with nonlinear fractional differentialinclusions and integral conditions, Electron. J. Differential Equations 2010, No. 20, 16 pp.[7] Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J., Theory and Applications of Fractional Differential Equations. Elsevier (North-Holland), Math. Studies, 204, Amsterdam (2006).

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