D021018022

Research Inventy: International Journal Of Engineering And Science Vol.2, Issue 10 (April 2013), Pp 18-22 Issn(e): 2278-4721, Issn(p):2319-6483, Www.Researchinventy.Com

Existence Of Solutions For Nonlinear Fractional Differential Equation With Integral Boundary Conditions 1, 1,2,

Azhaar H. Sallo , 2, Afrah S. Hasan

Department of Mathematics, Faculty of Science, University of Duhok, Kurdistan Region, Iraq.

Abstract. In this paper we discuss the existence of solutions defined in C [0,T ] for boundary value problems for a nonlinear fractional differential equation with a integral condition. The results are derived by using the Ascoli-Arzela theorem and Schauder-Tychonoff fixed point theorem.

Keywords: Riemann_Liouville fractional derivative and integral, boundary value problem, nonlinear fractional differential equation, integral condition, Ascoli-Arzela theorem and Schauder-Tychonoff fixed point theorem.

I.

INTRODUCTION

Fractional boundary value problem occur in mechanics and many related fields of engineering and mathematical 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 fields such as population dynamics, blood flow models, chemical engineering and cellular systems that can be modeled to a nonlinear fractional differential equation with integral boundary conditions. Recently, many authors 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 on existence and uniqueness of solutions for nonlocal and integral boundary value problems such as Ahmad and Ntouyas [2] and Hamani, Benchohra and Graef [6].

In this paper we prove the existence of the solutions of a nonlinear fractional differential equation with an integral boundary condition at the right end point of [0,T ] in C [0,T ] , where C [0,T ] is the space of all continuous functions over Tychonoff fixed point theorem.

[0,T ] ,which

results are based on Ascoli-Arzela theorem and Schauder-

II.

PRELIMINARIES

In 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:

1 b  1 f (t ) b  t  dt  ( ) a provided that this integral exists in Lebesgue sense, where  is the gamma function.

Lemma 2.2. Assume that f C (0,1)  L (0,1) with a fractional derivative of order   0 that belongs to C (0,1)  L (0,1) , then b a

D  f 

D0 D0 f (t )  f (t )  C1t  1  C2t  2  ...  Cnt  n

 R ; i =1, 2, …, n, where n is the smallest integer greater than or equal to  .

Lemma 2.3. Let  ,   R ,   1 . If x  a , then for some C i

x a

I

 (x  a )   (x  a )    (    1) (   1)  0

;    negative int eger



Lemma 2.4. The following relation

x a

D

 x a

;    negative int eger

D f  D  (   ) f 

x a

18

holds if

Existence Of Solutions For Nonlinear Fractional Differential…

  0 ,   0 and the function f (x ) C on a closed interval [a, b ] . (n ) b.   0 or   n  0 ,   0 and the function f (x ) C on a closed interval [a, b ] . x  Lemma 2.5. If   0 and f ( x) is continuous on [a, b ] , then a D f ( x) exists and it is continuous a.

with 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 or infinite).Then F contains a uniformly convergent sequence of function f n , converging to a function

f C (I ) where C (I ) denotes the space of all continuous bounded functions on I . Thus any sequence in F 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 continuous map of Y into itself. Then T has a fixed point y Y .

III.

B

and

T

MAIN RESULT

The 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 C[0, T ] , and 1    2 , then the solution of T   D  x t   f  t , x t  ,  h (t , , x ( ))d   , 0  

x

 2 

t  (0, T )

 0  0

(3.1)

(3.2)



x 1 T   a  x( ) d

(3.3)

0

Where   (0, T ) and a is a constant, is given by



t  1 at  1 x t   f (t , x(t ),  h(t , , x( ))d ) dt  (   ) f ( , x( ),     ( ) 0 (  1)   ( )  0 0 T

T

t T   1  1 ( t   ) f  , x  , h( , s, x( s)) ds  d      ( ) 0 0  

T

 h( , s, x( s))ds) d  0

a  Where    and   ( ) . 



Proof. Operate both sides of equation (3.1) by the operator D , to obtain T   D  D  x t   D  f  t , x t  ,  h (t , , x ( ))d   0   From Lemma (2.1), we get T   x t   C 1t  1  C 2t  2  D  f  t , x t  ,  h (t , , x ( ))d   0    1 Now, operate both sides of equation (3.4) by the operator D , to have

(3.4)

T   D  1x t   D  1C 1t  1  D  1C 2t  2  D  1D  f  t , x t  ,  h (t , , x ( ))d   0  

 1

From Lemma (2.2) and Lemma (2.3), D

C1t  1  C1( ) , D  1C 2t  2  0 and

19

(3.5)

a

Existence Of Solutions For Nonlinear Fractional Differential…

T T     1 D D f  t , x  t  ,  h(t , , x( ))d   D f  t , x  t  ,  h(t , , x( ))d  0 0    

 1



Therefore equation (3.5) can be written as T   (3.6) D  1x t   C 1( )  0t D 1f  t , x t  ,  h (t , , x ( ))d   0    2 Now, operating on both sides of equation (3.4) by the operator D and using again Lemma (2.2) and Lemma (2.3), equation (3.4) takes the form

T   (3.7) D  2x t  C 1( )t  C 2(  1)  0t D 2f  t , x t  ,  h (t , , x ( ))d   0   Now from the condition (3.2) and equation (3.7), it follows that C 2  0 , and from the condition (3.3) and

equation (3.6) we get  T T   a  x( )d  C1( )   f  t , x  t  ,  h(t , , x( ))d  dt 0 0 0      T T T     a  1 aC1    1d  (   s ) f s , x s , h ( s , z , x ( z )) dz dsd   C  (  )  f t , x t , h(t , , x( ))d  dt        1      ( ) 0 0 0 0 0 0      T T T       1 a C1  (   ) f  , x   ,  h( , s, x(s))ds  d    f  t , x  t  ,  h(t , , x( ))d  dt     ( )  0  (  1) 0 0 0    

a    and   ( ) , therefore the solution of the given boundary value problem takes the form   T T t  1 at  1 x t   f (t , x(t ),  h(t , , x( ))d ) dt  (   ) f ( , x( ),    ( ) 0  (   1)    (  )  0 0

Where 

t T  1  1  h (  , s , x ( s )) ds ) d   ( t   ) f  , x  , h (  , s , x ( s )) ds     d . 0 0 ( ) 0   Theorem: Assume that f  t , x  t   and h  t , x  t   belong to C[0, T ] , then the fractional boundary value problem (3.1-3) has a unique solution on [0, T ] .

T

Proof: Let

X  {x (t ); x (t ) C [0,T ]} and the mapping T : C 0,T   C 0,T  defined by

 T t  1  T a Tx(t )  f (t , x(t ),  h(t , , x( ))d )dt  (   )      ( )  0 (  1) 0 0 T T 1 t   1 f ( , x( ),  h( , s, x( s))ds)d   ( t   ) f (  , x (  ),  0 h( , s, x( s))ds)d 0  ( ) 0

(3.8)

in order to apply the Schauder-Tychonoff fixed point theorem, we should prove the following steps Step1: 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. Thus

T

maps Y into itself Step2: T is a continuous mapping on Let

 x (t )



n

n 1

be a sequence in

X

X

.

such that

lim x n (t )  x (t ) where x (t ) C 0,T n 

 , consider

Tx n (t ) Tx (t )  T T t  1 T ( f ( t , x ( t ), h ( t ,  , x (  )) d   f ( t , x ( t ),  n n 0 0 h(t, , x( )) d  dt   ( ) 0

20

Existence Of Solutions For Nonlinear Fractional Differential…



 T T a    f (  , x (  ), h( , s, x( s))ds)d ] (    ) [ f (  , x (  ), h (  , s , x ( s )) ds ) n n    (  1) 0 0  0 t T T 1 (3.9)  (t   ) 1[ f ( , xn ( ),  h( , s, xn ( s))ds)  f ( , x( ),  h( , s, x( s))ds)]d  ( ) 0 0 0

the right hand side of the equation (3.9) tends to zero as n tends to infinity, since and the sequence

 x (t )

 n 1

n

converges to

f

is a continuous function

x (t ) , that is

xn (t )  x(t )  0 Also since

f

as

n 

is bounded hence by Lebesgue's dominated convergence theorem we have

Txn (t )  Tx(t )  0 as n   therefore 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 uniformly bounded as shown in step1, for proving the equicontinuity, let t1 , t2  (0, T ] such that t1  t2 , then T T t2 1 Tx(t2 )  Tx(t1 )  f (t2 , x( t2 ),  h(t2 , , x( )) d ) dt2    ( ) 0 0 

 T a t2 1 (   ) f ( , x( ),  h( , s, x( s)) ds) d  (  1)(  ( )) 0 0



T 1 t  1  ( t   ) f (  , x (  ), 2  0 h( , s, x( s))ds)d ( ) 0 2

T t1 1 T  f (t , x(t ), h(t , , x( ))d )dt1   ( ) 0 1 1 0 1 



 T at1 1 (    ) f (  , x (  ),  0 h( , s, x( s))ds)d (  1)(  ( )) 0

T 1 t  1 ( t   ) f (  , x (  ),  1 0 h( , s, x( s))ds)d ( ) 0 f on [0, T ] there exists a positive constant M such that 2



By continuity of

 t2 1 T at2 1 Tx(t2 )  Tx(t1 )  Mdt2  (   ) Md     ( ) 0 (  1)(  ( )) 0

=



T 1 t t1 1 (t2   ) 1 Md  Mdt1  ( ) 0   ( ) 0



 a t1 1 1 t   1 (    ) Md     (t1   ) M d (  1)(  ( )) 0 ( ) 0

2

1

 MT aM (t2 1  t1 1 )  (t2 1  t1 1 )  (   ) d   ( ) (  1)(  ( )) 0



t M t [  (t 2   ) 1d    (t 1   ) 1d  ] ( ) 0 0 2

1

MT aM  1  1  1  (t2  t1 )  (t2 1  t1 1 )   ( ) (  2)(  ( )) 21

Existence Of Solutions For Nonlinear Fractional Differential…



t M t [  [(t 2   ) 1  (t 1   ) 1 ]d    (t 2   ) 1d  ] ( ) 0 t 1

2

1



MT aM (t2 1  t1 1 )  (t2 1  t1 1 )   ( ) (  2)(  ( ))  1



M  t2 (t2  t1 ) (t2  t1 ) t1      ( )     

Tx(t2 )  Tx(t1 )  MT aM  1 M  1  1 (t2  t1 )  (t2 1  t1 1 )  (t2  t1 )   ( ) (  2)(  ( ))  (  1) when t1 tends to t 2 , with t1  t2   , we have Tx(t2 )  T x(t1 )   , which proves that the family TX is equicontinuous. Thus by Ascoli-Arzela theorem, TX has a compact closure. In view of step1, step2 and 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] [2] [3] [4] [5] [6] [7]

Ahmad B and Nieto J.J., Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions. Boundary Value Problems, 2011, 2011:36.

Ahmad B. and Ntouyas S. K., Boundary Value Problems for Fractional Differential Inclusions with Four-Point Integral Boundary Conditions, Surveys in Mathematics and its Applications, 6 (2011), 175-193.

Barrett, J.H., Differential equation of non-integer order, Canad. J. Math., 6 (4) (1954), 529-541.

Darwish M.A. and Ntouyas S.K., On initial and boundary value problems for fractional order mixed type functional differential inclusions, Comput. Math. Appl. 59 (2010), 1253-1265.

Goldberg, R.R., Methods of Real Analysis, John Wiley and Sons, Inc, USA (1976).

Hamani S., Benchohra M. and Graef J.R., Existence results for boundary value problems with nonlinear fractional differential inclusions and integral conditions, Electron. J. Differential Equations 2010, No. 20, 16 pp.

Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J., Theory and Applications of Fractional Differential Equations. Elsevier (NorthHolland), Math. Studies, 204, Amsterdam (2006).

22