Local Existence Theorem of Fractional Differential Equations in L p Space

We proved the existence of P-integrable solution in ) , ( b a L -space, where 1 p    for the fractional differential equation which has the form: )) ( , ( ) ( t y t f t y D c =  0 1    with boundary condition ( ) ( ) y a y b c   + = where  D is the Caputo fractional derivative, ,   and c are positive constants with 0   +  . The contraction mapping principle has been used to establish our main result.


Introduction
Fractional differential equations have gained importance and popularity during the past three decades or so, due to mainly its demonstrated applications in numerous seemingly diverse fields of science and engineering. Fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes. This is the main advantage of fractional derivatives in comparison with classical integer-order models, in which such effects are in fact neglected. The advantages of fractional derivatives become apparent in modeling mechanical and electrical properties of real materials, and in many other fields see ( [6], [7]). Arora [4] studied the existence of solutions for boundary value problems , for fractional differential and satisfying the boundary condition (0) ( ) y y T c  += By using Schaefer's fixed point theorem.
In this paper we impose some conditions on the existence of the solution of fractional differential equation ( , Our method in this paper is by using the contraction mapping principle.

Preliminaries
In this section ,we shall give a collection of definitions and lemmas which are needed in various places in this work.

Definition 2.1 [2]
Let f be a function which is defined almost everywhere (a.e) on [a,b]. for 0   , we define Provided that this integral (lebsegue) exists, where  is gamma function.

Definition 2.2 [6]
For a function f defined on the interval [a,b], the  th Caputo fractional derivative of f is defined by

Lemma 2.4 [9] (Hölder's inequality)
Let X be a measurable space, let p and q satisfy 1 < p < , 1<q<  , and 11 1 pq += . If p f L (X)  and q g L (X)  , then ( f g ) belongs to L(X) and satisfies 11 pq pq

Local Existence of P-integrable solution in
In this section we prove some existence and uniqueness theorems in ( , ) p L a b space for the boundary value problems(1.3), with condition (1.5).

Lemma 3.1 [4]
Let 01   and f be continuous function . A function y is defined as For proof see [4]. The mean result is given in the following theorem and it is P-intgerable on (a,b) . There exists a P-intgerable solution for the fractional differential equation (1.3) with the boundary condition (1.5).
Proof. Let the mapping T on b) (a, L p be defined as:

Tg t t s f s g s ds b s f s g s ds
we have to prove that T maps every function b) (a, L g p  into a function which belongs to b) (a, L p . Let: by the lemma 2.1, the first and the second terms in the right hand side equation (3.3) is continuous for all , and for all 0 and it is measurable .
We have show that () is Lebsegue integrable.   Since ( , ) f t y satisfies lipschitz condition on D with respect to y. Therefore from inequality (3.9), we get: ( )

 
Is the only solution for the given fractional differential equation which satisfies the given boundary condition.