Some classes of fractional abstract differential equations with α-integrated semi groups are studied in Banach space. The existence of a unique solution of the nonlocal Cauchy problem is studied. Some properties are given

El-Borai, Mahmoud Mohammed

El-Nadi, Khairia El-Said

English

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Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                           ISSN: 2395-0218   Volume 5, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                553  SCITECH                           Volume 5, Issue 3  RESEARCH ORGANISATION|    Published online: September 22, 2015| Journal of Progressive Research in Mathematics www.scitecresearch.com/journals     Fractional integrated semi groups and nonlocal Cauchy problem for abstract nonlinear fractional differential equations  Mahmoud M.El-Borai and Khairia El-Said El-Nadi Department of  Mathematics and Computer Science, Faculty of Science, Alexandria University, Alexandria, Egypt. m _ m _ elborai@yahoo.com khairia _ el _ said@hotmail.com  Abstract  Some classes of fractional abstract differential equations with  -integrated semi groups are studied in Banach space. The existence of a unique solution of the nonlocal Cauchy problem is studied. Some properties are given.  Key words:  -Integrated semi groups-Nonlinear fractional abstract differential equations- Nonlocal Cauchy problem.  AMS classifications: 47D60- 47D62- 35A05- 34G20.   1  Introduction Consider the following abstract fractional differential equation:   =1( )= ( ) ( , ( ) ( )) ( ) ( ),                            (1.1)ki iid u tAu t f t B t u t s t c u tdt    with the initial condition         0(0) = ,                                                                  (1.2)u u  where 10 < < kt t T  , 1,..., kc c  are real numbers, A is a linear closed operator defined on a dense set S in a Banach Space E,   1( ) = ( ( ) ,..., ( ) ), ( ), =1,...,r iB t u B t u B t u B t i r  is a family of linear closed operators defined on dense sets 1,..., rS S S , respectively in E to E , f is a given abstract function defined on rJXE  to E, 0 < 1   , 0u  is a given element in S and s is a real function , which has continuous derivative  Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                           ISSN: 2395-0218   Volume 5, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                554  ( )( ) = ,    = [0, ],  (0) = 0.d s tr t on J T sdt It is assumed that A generates  - times integrated semi groups Q(t), 0t   with the following Properties:   1: ( ) : 0C Q t t  is family of strongly continuous operator.   2C : There exist positive constants M and c such that ( ) ctQ t Me  , where    is the norm in E.   3C : The interval ( , )c   is contained in the resolvent set ( )A  of A and,   140: ( ) = ( )tC I A e Q t dt     ,  for all > c  , (I is the identity operator), 0 < 1   ,([1-9]). Let ( )SC J  be the set of all continuous functions u on J with values in S. By a strong Solution of the Cauchy problem (1.1), (1.2), we mean a function u such that:   ( ),Su C J   ( )( ),Ed u tC Jdt  u satisfies the following equation :   1001( ) = ( ) [ ( ) ( , ( ) ( )]( )tu t u t Au f B u d                        10=11( ) ( ) ( ) ,                           (1.3)( )kti iit s c u t d     Where (.)  is the gamma function. In section 2, we shall consider the linear case. In other words when f depends only on t. In this case the solution can be obtained in a closed form. Also the stability of solutions can be established. In section 3, we shall solve equation (1.3) under suitable conditions on f and the operators 1, , rB B . It is assumed that: 5 2 11: ( ) ( )i cKC B t Q t h ht   , for all 1 > 0,t h E , 6 1: ( ) , , ( )rC B t h B t h  are uniformly Holder in t J  for all iih S . It is assumed also that there exists a function g such that: 1701: ( , ( ) ( )) = ( ) ( , ( ) ( ))( )tC f t B t u t t g B u d      Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                           ISSN: 2395-0218   Volume 5, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                555 (This means that f(0,w)=0 , =d fgdt exists ) , where g  is continuous on JXE, with the following properties: 8 :C  g  satisfies a uniform Holder condition in t J  and a Lipschitz condition with respect to 1( ) , , ( )rB t u B t u . There are many important applications of the theory of integrated semi groups and the nonlocal Cauchy problem for fractional differential equation. The applications can be found in the theory of quantum mechanics and the theory of elasticity. [1-8].  2  The linear case  Let us consider the case when f depends only on t. Denote by 1( )t  and 2 ( )t  the following operator - valued functions:   10( ) = ( ) ( ) ,t d dt Q t d         120( ) = ( ) ( ) ,t Q t t d        Where   is a probability density function defined on (0, )  , see [9]. It is clear that 12( )t Kt    on (0, ]T , for some constant < 0K . Let us suppose that =11| |<kiicKMT  , where = | ( ) |supJM r t . Under this condition and the properties of the operators Q, 2  , one gets     20=1| | ( ) ( ) <1.                               (2.1)k tii iic r t d         Theorem 1. If g is continuous on J and is an element of S for every t in J and if 0u  is an element in the domain of definition of the operator 2A , then the strong solution of (1.3) is given by   0 2 001( ) = ( ) ( )(1 )tu t u t Au d                                 20=1( ) ( ) ( )kti iir t c u t d                                20( ) ( ) ,                                              (2.2)tt g d      Where   0 2 00=1 =1( ) = [ { (1 ) ( ) ( ) }]k k tii i i i ii ic u t c u c t Au d            Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                           ISSN: 2395-0218   Volume 5, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                556                       20=1( ) ( ) ,                                        (2.3)k tii iic t g d         is the inverse bonded operator:   120=1= [ ( ) ( ) ] .k tii iiI c r t d         Proof. Using our previous results [9-16], and the conditions (1C )-( 4C ) , we can write   1 0 2 20 0=1( ) = ( ) ( ) ( ) ( ) ( ) ( )kt ti iiu t t u t g d r t c u t d               Using the facts:   0( ) = ( ) ,(1 )ttQ t h h Q Ahd       > 0,  ,for all t h S  0 01 1 ( )= ( ) ( ) = ( )(1 ) (1 )t td f d dft f d t ddt dt d              for all continuous f such that f(0)=0 , 0 <1 , one gets:   0 2 001( ) = ( ) ( )(1 )tdu t u t Au ddt        2 20 0=1( ) ( ) ( ) ( ) ( ) .kt ti iir t c u t d t g d             Consequently   0 2 00=1 =1( ) = { ( ) ( ) }(1 )k k tjij j j jj icc u t c u t Au d          2 20 0=1 =1 =1( ) ( ) ( ) ( ) ( ) ,k k kt tjj j j j i ij j ic t g d c r t c u t d               From (2.2) and (2.3), we deduce that u satisfies the conditions (I), (II) and (III). Now it is easy to see that the considered strong solution is unique and more over the Cauchy problem (1.3) is correctly formulated. Is other words: If 0 0u Au g є        , for sufficiently small > 0є , then u Kє  , for some constant positive constant K.  3  The Nonlinear case Let V satisfy (formally), the following equation:  Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                           ISSN: 2395-0218   Volume 5, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                557  01( ) ( ) = ( ) ( )( )tu t t Au t t V d           01= ( ) ( , ( ) ( ))( )tt g B u d        0=11( ) ( ) ( ) .( )kti iit r c u t d     Thus we can write formally   1 0 20( ) = ( ) ( ) ( ) .                                    (3.1)tu t t u t V d       We shall solve the following equation:   =1( ) = ( , ( ) ( )) ( ) ( ).                                 (3.2)ki iiV t g t B t u t r t c u t   Theorem 2. If equation (1.3) has a strong solution, then that solution is unique.  Proof. Set   =1( ) = ( , ( ) ( )) ( ) ( ),    =1,2,kj j i j iiV t g t B t u t r t c u t j   where 1u  and 2u  are two solutions of equation (1.3) . Using conditions ( 5C  ),( 6C  ) and ( 8C  ) ,one gets:   11 2 1 20( ) ( ) ( ) ( ) ( )tV t V t M t V V d            11 20=1| | ( ) ( ) ( ) ,k tii iiM c t V V d         Where = (1 )c    and M is a positive constant Set 1 2= [ ( ) ( )]maxbtJ e V t V t   , where b is a sufficiently large positive number. It is easy to see that:   11 20( ) ( ) ( )tt V V d         11 10 1( ) (1 ) [1 1 ]t b tb bt bb e d e t d b                  Thus for some positive constant M and for sufficiently large b, one gets v  , where Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                           ISSN: 2395-0218   Volume 5, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                558 = (1 ) [1 1 ] <1v M b   . This means that 1 2( ) = ( )V t V t  on J ,so 1 2( ) = ( )u t u t  on J .  Theorem 3. Equation (1.3) has a strong unique solution.    Proof. The uniqueness is already proved. Let us prove the existence. Using the method of successive approximations, we set   =1( ) = ( , ( )) ( ) ( ).kn n i n iiV t g t Bu t r t c u t   Thus   1 1[ ( ) ( )] [ ( ) ( )] ,max maxbt btn n n nJ Je V t V t v e V t V t         So   1 1 0[ ( ) ( )] [ ( ) ( )]maxbt n btn nJe V t V t v e V t V t        Where 0 ( )V t  is the zero approximation, which can be taken the zero element in E. Thus the sequence ( )nV t  uniformly converges in the space ( )EC J  to a continuous function V (t), which satisfies equation (3.2). Since ( )EV C J ,it follows that ( )Eu C J , where u is given by equation (3.1). To prove that ( )u t S  , for all t J  , it suffices to prove that V satisfies a uniform Holder condition. Using similar arguments as in [10,15], we see that V satisfies a uniform Holder condition, [Comp 17 -29]. This completes the proof of the theorem.  Conclusion Using the theory of  -integrated semigroups, we have proved existence and uniqueness theorems for general abstract fractional nonlinear differential equations in Banach space. We have got generalizations of some of our old results. References [1] W. Arendt, Resolvent positive operators and integrated semi groups, Proc. London. Math. Soc.54 (1984),321-349. [2] M. Hieber, Integrated semi groups and differential operators in LpSpaces, Math.Ann. 201,(1991),1-16. [3] L. Hormander, Estimates for translation invariant operators in Lp space, Act.Math.104, (1960), 93-139. [4] WR.Schneider and W-wayes, Fractional diffusion and wave equation, J.Math.ics,30 (1989), 134-143. [5] W.Wayes, The fractional diffusion equation, J.Math.Phys.,27(1986),2782-2786. [6] O. Neuarander, Integrated semi group and their application to complete second order Cauchy problem, semi group Forum.,38(1989),233-251. [7] R. 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Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                           ISSN: 2395-0218   Volume 5, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                560 Math. Anal. Appl., 172(1993), 256-265. [28] A .C. Koll, Applications of fractional calculus to the theory of viscoelasticity, Trans. ASMi J.App. Each.,51(1984),307-317. [29] F. Lin, J. Veiu, Semi linear integrodifferential equations with non-locol Cauchy problem, Nonlinear Anal. TMA., 26(1996),1023-1033. 

Fractional integrated semi groups and nonlocal Cauchy problem for abstract nonlinear fractional differential equations

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Fractional integrated semi groups and nonlocal Cauchy problem for abstract nonlinear fractional differential equations

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