Study for System of Nonlinear Differential Equations with Riemann-Liouville Fractional Derivative ()
1. Introduction
Let
denote the
matrix over real fields
or complex fields
. For ![](https://www.scirp.org/html/2-7401502\4dcb1ee3-1b0b-4d42-bea6-1b5c10172d0f.jpg)
![](https://www.scirp.org/html/2-7401502\4ce1d14c-0242-464d-ae3f-6052dc2c48c5.jpg)
here
is the usual space of continuous functions on
which is a Banach space with the norm
![](https://www.scirp.org/html/2-7401502\ce1517a1-8add-499d-a358-cb609374b268.jpg)
The space
is defined by
![](https://www.scirp.org/html/2-7401502\028b276f-08e0-4255-bdbf-6bc40c86ede7.jpg)
(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
![](https://www.scirp.org/html/2-7401502\841bd2e7-2973-4537-b60c-70c8fa9787f6.jpg)
![](https://www.scirp.org/html/2-7401502\dad1b83b-7007-47b5-bb3b-0f52d42dfcc9.jpg)
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
![](https://www.scirp.org/html/2-7401502\6aacc275-a49d-41e1-a46a-8daeaf85cf0d.jpg)
is called left-sided fractional derivatives of order ![](https://www.scirp.org/html/2-7401502\e67a7a61-82f8-438e-ae94-ec22d4369f76.jpg)
Definition 1.2 Let
be a continuous function defined on
and
Then the expression
![](https://www.scirp.org/html/2-7401502\410a9569-c0be-4297-a02f-0fb9b3dfc714.jpg)
is called left-sided fractional integral of order ![](https://www.scirp.org/html/2-7401502\955d543a-7ad5-4ccb-939d-b4353f6d6dd2.jpg)
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
![](https://www.scirp.org/html/2-7401502\9d577ba8-8417-4906-ae6a-0e0e98265b7c.jpg)
for some
When the function
then
where
and ![](https://www.scirp.org/html/2-7401502\1f6913de-d1d7-4a15-90fc-2cbc25ffe621.jpg)
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 ![](https://www.scirp.org/html/2-7401502\852cf831-5a4e-4c5e-bb65-21d16b95bbd2.jpg)
Now, let’s us give some hypotheses:
H1:
is continuous on
and is such that
(1)
where
is a continuous function on ![](https://www.scirp.org/html/2-7401502\7cc988e6-f751-4eaa-8a1d-66dbc123b822.jpg)
H2:
is continuous on
and is such that
(2)
where
is a continuous function on ![](https://www.scirp.org/html/2-7401502\a3092f6d-a196-4edc-970d-f1d7de669e96.jpg)
Lemma 1.6 Let
If we assume that
then the initial value problem
(3)
where
![](https://www.scirp.org/html/2-7401502\3af104c9-2e7e-4109-82bf-ddd4c469da45.jpg)
![](https://www.scirp.org/html/2-7401502\e6541c59-479c-4c80-9a7e-7446234aa0b2.jpg)
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
![](https://www.scirp.org/html/2-7401502\a270f0b3-5ba9-4596-8f90-0ef2b574dd80.jpg)
Clearly, it is closed, convex and nonempty.
Step I. We shall prove that we note that ![](https://www.scirp.org/html/2-7401502\7b69316a-43ab-4bee-ab1c-c254798718d3.jpg)
We note that
![](https://www.scirp.org/html/2-7401502\4a3c4e67-da22-40ea-ad26-afbc5e4cd3f6.jpg)
Since
it will be sufficient to impose
![](https://www.scirp.org/html/2-7401502\381a97b8-1ada-458f-a13d-e0af87c28737.jpg)
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
![](https://www.scirp.org/html/2-7401502\9d921bbf-e995-4947-b3fa-abb555c43175.jpg)
defined by (5), it will be sufficient to argue on the operator
![](https://www.scirp.org/html/2-7401502\8644c8f6-ff97-4e22-8f35-45f6bb5949af.jpg)
defined in this way:
![](https://www.scirp.org/html/2-7401502\6298d221-e9cc-40e1-b85e-33f69b7d8680.jpg)
We have
where the operator
![](https://www.scirp.org/html/2-7401502\cfb6e5e8-dbc2-493a-84b2-bc70ebed4a46.jpg)
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
![](https://www.scirp.org/html/2-7401502\ae73487e-1d9a-46af-9ebd-c25786e71726.jpg)
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
![](https://www.scirp.org/html/2-7401502\e5bc0b8a-6ee6-4096-8b58-8ff1deec10d5.jpg)
![](https://www.scirp.org/html/2-7401502\f9be32b7-a885-4a62-b842-9cade70e26ac.jpg)
for all
and sufficiently small ![](https://www.scirp.org/html/2-7401502\0528dc9a-5bbe-4dc0-80eb-26f5f229d135.jpg)
Proof. Given
with eigenvalues
by Lemma 1.3, there is a unitary matrix
such that
![](https://www.scirp.org/html/2-7401502\3d9eef47-5e16-4d0b-ae1a-dc03d4abd221.jpg)
is upper triangular with diagonal entries ![](https://www.scirp.org/html/2-7401502\87de5d61-e2bf-490e-8f91-6a36d4c74254.jpg)
Let
we have
![](https://www.scirp.org/html/2-7401502\bf822c0d-b295-48db-87c6-c3a728657787.jpg)
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
![](https://www.scirp.org/html/2-7401502\95b7ac98-beed-42b0-bd18-18d56c9c9f69.jpg)
for
where
is the
th entries of the vector ![](https://www.scirp.org/html/2-7401502\8c6ab663-0373-44d1-b1b9-1eee58f1f3f6.jpg)
Consider the weighed Cauchy-type problem
![](https://www.scirp.org/html/2-7401502\27863b3d-5b55-4aca-99bc-5d2ab27ed32e.jpg)
In Lemma 1.6, take
Then by lemma 1.6,
s.t. the above problem has at least a solution
![](https://www.scirp.org/html/2-7401502\64687b34-3569-4267-86c3-48650eedccf3.jpg)
Consider the following weighed Cauchy-type problem
![](https://www.scirp.org/html/2-7401502\f00cbb80-c73d-4625-928c-c06f56dccf5f.jpg)
In Lemma 1.6, take
Then by Lemma 1.6,
s.t. the above problem has at least a solution ![](https://www.scirp.org/html/2-7401502\f52aedb8-eb2d-4746-983a-7a856738e961.jpg)
Similarly, there has at least a solution in
![](https://www.scirp.org/html/2-7401502\ecb1d5eb-a9b2-47cc-9d35-b65416109734.jpg)
for the rest n-2 initial problem in (**), denote by
respectively. And therefore, there has at least a solution
![](https://www.scirp.org/html/2-7401502\44d5132e-82c0-4d5c-b9cc-6e03a50dbc90.jpg)
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, ![](https://www.scirp.org/html/2-7401502\828584fd-a49d-4c44-94e9-ff960b63352f.jpg)
and
for some
If the solution of the problems (**) denoted by
then there exists some constant
such that
for all ![](https://www.scirp.org/html/2-7401502\6e07279e-3aad-41e3-a2f0-48ba2b629b6a.jpg)
Proof. Similar to the proof of Theorem 2.1, now consider the following weighted Cauchy-type problem
![](https://www.scirp.org/html/2-7401502\d643f217-e6d6-4a4e-9d32-a51066989fcc.jpg)
Then by Lemma 1.7, there exists some constant
such that ![](https://www.scirp.org/html/2-7401502\34aea5d4-bafa-462c-a494-36b8d63cf319.jpg)
Consider the following problem
![](https://www.scirp.org/html/2-7401502\a369ec4b-7dcc-4fd5-9300-cfce0309ef33.jpg)
Then by Lemma 1.7, there exists some constant
such that ![](https://www.scirp.org/html/2-7401502\cb468065-0fcd-4a8a-ba83-d0d9b88bb3d6.jpg)
Similarly, there exist some positive constants
such that
![](https://www.scirp.org/html/2-7401502\4f6f82b0-6b9b-49cd-8c1e-0ee10c401563.jpg)
for all ![](https://www.scirp.org/html/2-7401502\2d0b0b25-486a-4f6a-afa1-3a0ac14601d4.jpg)
Let
Then we have
for all ![](https://www.scirp.org/html/2-7401502\bacad82d-0692-4c2c-9396-7ce5f1ab0326.jpg)
The proof is completed.
Theorem 2.3 Assume that
the right-hand of these equations in (9) satisfied H2, and ![](https://www.scirp.org/html/2-7401502\3dd98e0a-0aad-41f2-9a97-4aff95f5aa6e.jpg)
For some
Suppose further that
![](https://www.scirp.org/html/2-7401502\2b84c4a2-7ffd-4340-a779-5f089fff2b75.jpg)
If denote solution of the problems (**)
by
![](https://www.scirp.org/html/2-7401502\71223745-691e-44d3-8a3b-e5e4e46a3ea5.jpg)
Then there exists some constant
and
, such that
![](https://www.scirp.org/html/2-7401502\03d93066-40c8-4cda-9d16-aee95e1c1069.jpg)
for all ![](https://www.scirp.org/html/2-7401502\eb36bf69-4e2d-4163-9dff-6cbd7cc34fb8.jpg)
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).
NOTES