ABSTRACT

In this work, we study existence theorem of the initial value problem for the system of fractional differential equations where denotes standard Riemann-Liouville fractional derivative, and is a square matrix. At the same time, power-type estimate for them has been given.

1. Introduction

Let denote the matrix over real fields or complex fields. For

here is the usual space of continuous functions on which is a Banach space with the norm

The space is defined by

(see [1]).

The existence of solution of initial value problems for fractional order differential equations have been studied in many literatures such as [1-4]. In this paper, we present the analysis of the system of fractional differential equations

(*)

where denotes standard Riemann-Liouville fractional derivative, where

and is a square.

To prove the main result, we begin with some definitions and lemmas. For details, see [1-5].

Definition 1.1 Let be a continuous function defined on and. Then the expression

is called left-sided fractional derivatives of order

Definition 1.2 Let be a continuous function defined on and Then the expression

is called left-sided fractional integral of order

Lemma 1.3 Given with eigenvalues

in any prescribed order, there is a unitary matrix such that is upper triangular with diagonal entries That is, every square matrix is unitarily equivalent to triangular matrix whose entries are the eigenvalues of in a prescribed order. Further more, if and if all the eigenvalues of are real, then may be chosen to be real and orthogonal.

Lemma 1.4 Assume that with fractional derivative of order that belongs to. Then

for some When the function then

where

and

Lemma 1.5 (Schauder’s fixed theorem) Assume is a relative subset of a convex set in a normed space Let be a compact map with. Then either

(A₁) has a fixed point in, or

(A₂) there is a and a such that

Now, let’s us give some hypotheses:

H1: is continuous on and is such that

(1)

where is a continuous function on

H2: is continuous on and is such that

(2)

where is a continuous function on

Lemma 1.6 Let If we assume that then the initial value problem

(3)

where

has at least a solution for sufficiently small.

Proof. If

then, by Lemma 1.4, We are therefore reduced the initial problem to the nonlinear integral equation

(4)

The existence of a solution to Problem (3) can be formulated as a fixed point equation where

(5)

in the space.

Define

Clearly, it is closed, convex and nonempty.

Step I. We shall prove that we note that

We note that

Since it will be sufficient to impose

In view of the assumption the second estimate is satisfied if say and is chosen sufficiently small.

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. By Lemma 1.5, we have that Problem (3) has least a solution.

The proof is complete.

Lemma 1.7 Suppose that satisfies H1,

and If for some then the problem

(6)

exists a positive constant such that

Lemma 1.8 Let with Suppose further that. Then Problem (6) and its associated integral equation

(7)

are equivalent.

Lemma 1.9 Assume that satisfies H2, and for some Suppose further that then there exists and such that any solution of (6) exists globally and satisfies

(8)

2. Main Results

Theorem 2.1 Let then initial problem (*) has a solution where

for all and sufficiently small

Proof. Given with eigenvalues by Lemma 1.3, there is a unitary matrix such that

is upper triangular with diagonal entries

Let we have

At the same time, the initial problem (*) changed into

(**)

Now, let’s consider the problem (**).

Clearly, the problem (**) is equivalent to the following n problems

for where is the th entries of the vector

Consider the weighed Cauchy-type problem

In Lemma 1.6, take Then by lemma 1.6, s.t. the above problem has at least a solution

Consider the following weighed Cauchy-type problem

In Lemma 1.6, take Then by Lemma 1.6, s.t. the above problem has at least a solution

Similarly, there has at least a solution in

for the rest n-2 initial problem in (**), denote by respectively. And therefore, there has at least a solution

of the problem (**). Let it is required for us.

The proof is completed.

Since the problem (**) is equivalent to the following n problems

(9)

for where is the th entries of the vector Next, we shall discuss these equations in (9).

Theorem 2.2 Assume that the right hand of these equations in (9) satisfied H1, and for some If the solution of the problems (**) denoted by

then there exists some constant such that

for all

Proof. Similar to the proof of Theorem 2.1, now consider the following weighted Cauchy-type problem

Then by Lemma 1.7, there exists some constant such that

Consider the following problem

Then by Lemma 1.7, there exists some constant such that

Similarly, there exist some positive constants such that

for all

Let Then we have

for all

The proof is completed.

Theorem 2.3 Assume that the right-hand of these equations in (9) satisfied H2, and

For some Suppose further that

If denote solution of the problems (**)by

Then there exists some constant and, such that

for all

Using Lemmas 1.3 and 1.9, the proof is similar to Theorem 2.2. Therefore, it is omitted.

3. Acknowledgements

This research was supported by the NNSF of China (10961020), the Science Foundation of Qinghai Province of China (2012-Z-910) and the University Natural Science Research Develop Foundation of Shanxi Province of China (20111021).

REFERENCES

NOTES

^*Corresponding author.