Study for System of Nonlinear Differential Equations with Riemann-Liouville Fractional Derivative

In this work, we study existence theorem of the initial value problem for the system of fractional differential equations       1 0 , x D x t Ax t t x t b      ,  where D denotes standard Riemann-Liouville fractional derivative, 0 1,             1 2 , ,  T T 1 2 , , , , , n n x t x t x t x t b b         b b and A is a square matrix. At the same time, power-type estimate for them has been given.

where D  denotes standard Riemann-Liouville fractional derivative, 0 1 , and A is a square matrix.At the same time, power-type estimate for them has been given.

Introduction
Let n M denote the matrix over real fields or complex fields .For
The existence of solution of initial value problems for fractional order differential equations have been studied in many literatures such as [1][2][3][4].In this paper, we present the analysis of the system of fractional differential equations where D  denotes standard Riemann-Liouville fractional derivative, where To prove the main result, we begin with some definitions and lemmas.For details, see [1][2][3][4][5].
Definition 1.1 Let f be a continuous function de-

 
Then the expression That is, every square matrix A is unitarily equivalent to triangular matrix whose entries are the eigenvalues of A in a prescribed order.Further more, if and if all the eigenvalues of A are real, then U may be chosen to be real and orthogonal.
Lemma 1. 4 Assume that for some When the function then be a compact map with 0   .Then either (A 1 ) A has a fixed point in  , or (A 2 ) there is a and a Now, let's us give some hypotheses: where is a continuous function on where is a continuous function on where then , by Lemma 1.4,We are therefore reduced the initial problem to the nonlinear integral equation The existence of a solution to Problem (3) can be formulated as a fixed point equation where , in the space Clearly, it is closed, convex and nonempty.
Step I. We shall prove that we note that .

TS S 
We note that In view of the assumption the second estimate is satisfied if say   Step II.We shall prove that the operator is compact.To prove the compactness of defined by ( 5), it will be sufficient to argue on the operator defined in this way: We have where the operator Turn out to be compact from classical sufficient conditions, since   we have that Problem (3) has least a solution.
The proof is complete.
. Then Problem (6) and its associated integral equation are equivalent.Lemma 1.9 Assume that 1 2.

 
 ,  f t x satisfies H2, and such that any solution of (6) exists globally and satisfies

Theorem 2.1 Let
for all and sufficiently small 1, 2, i At the same time, the initial problem (* nto to the fol Clearly, the problem (**) is equivalent g n problems or where is the th ies of the en by 6, 0, h   s.t. the abov at least a solution e problem has Similar ly, there has at least a solution in Then by Lemma 1.7, there exists some constant Then by Lemma 1.7, there exists some constant rly, there ex Simila ist some positive constants

Definition 1 . 2
fractional derivatives of order .Let f be a continuous function defined on   , a b and 0.
This research was supported (10961020), the Science Fou of China (2012-Z-910) and the University Natural Science Research Develop Foundation of Shanxi Province of China (20111021).