Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 193061, 21 pages
doi:10.1155/2012/193061
Research Article
Analysis of a System for Linear Fractional
Differential Equations
Fang Wang,1, 2 Zhen-hai Liu,3 and Ping Wang4
1
School of Mathematical Science and Computing Technology, Central South University,
Hunan, Changsha 410075, China
2
School of Mathematical Science and Computing Technology,
Changsha University of Science and Technology, Hunan, Changsha 410076, China
3
School of Mathematical Science and Computing Technology,
Guangxi University for Nationalities, Guangxi, Nanning 530006, China
4
Xiamen Institute of Technology, Huaqiao University, Fujian, Xiamen 361021, China
Correspondence should be addressed to Fang Wang, wangfang811209@tom.com
Received 3 March 2012; Revised 21 July 2012; Accepted 23 July 2012
Academic Editor: Chein-Shan Liu
Copyright q 2012 Fang Wang et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
The main purpose of this paper is to obtain the unique solution of the constant coefficient
q
homogeneous linear fractional differential equations Dt0 Xt P Xt, Xa B and the constant
q
coefficient nonhomogeneous linear fractional differential equations Dt0 Xt P Xt D, Xa B
if P is a diagonal matrix and Xt ∈ C1−q t0 , T × C1−q t0 , T × · · · × C1−q t0 , T and prove the
existence and uniqueness of these two kinds of equations for any P ∈ LRm and Xt ∈
C1−q t0 , T × C1−q t0 , T × · · · × C1−q t0 , T . Then we give two examples to demonstrate the main
results.
1. Introduction
System of fractional differential equations has gained a lot of interest because of the
challenges it offers compared to the study of system of ordinary differential equations.
Numerous applications of this system in different areas of physics, engineering, and
biological sciences have been presented in 1–3. The differential equations involving the
Riemman-Liouville differential operators of fractional order 0 < q < 1 appear to be
more important in modeling several physical phenomena and therefore seem to deserve an
independent study of their theory parallel to the well-known theory of ordinary differential
equations. The existence and uniqueness of solution for fractional differential equations with
2
Journal of Applied Mathematics
any Xt ∈ Ct0 , T × Ct0 , T × · · · × Ct0 , T have been studied in many papers, see 4–28. In
4 Daftradar-Gejji and Babakhani have studied the existence and uniqueness of
q
D0 Xt − X0 P Xt,
1.1
q
where D0 denotes the standard Riemman-Liouville fractional derivative, 0 < q < 1, Xt x1 t, x2 t, . . . xm tT , X0 X0 x10 , x20 , . . . , xm0 T , P ∈ LRm which is an m
dimensional linear space. They have obtained that the system 1.1 has a unique solution
defined on 0,T if P ∈ LRm and Xt ∈ Ct0 , T × Ct0 , T × · · · × Ct0 , T . In 17 Belmekki
et al. have studied the existence of periodic solution for some linear fractional differential
equation in C1−q 0, 1. In 21 Ahmad and Nieto have studied the Riemann-Liouwille
fractional differential equations with fractional boundary conditions. In comparison with the
earlier results of this type we get more general assumptions. We assume Xt ∈ C1−q t0 , T ×
C1−q t0 , T × · · · × C1−q t0 , T instead of Xt ∈ Ct0 , T × Ct0 , T × · · · × Ct0 , T and consider
the following system of fractional differential equations:
q
Dt0 Xt P Xt,
q
Dt0 Xt P Xt D,
Xa B,
Xa B,
1.2
q
where D0 denotes the standard Riemman-Liouville fractional derivative, 0 < q < 1, P ∈
LRm ,
Xt ∈ C1−q t0 , T × C1−q t0 , T × · · · × C1−q t0 , T ,
1.3
a, B ∈ t0 , T ×Rm and D is a constant vector. We completely generalize the results in 4 and
obtain the new results if Xt ∈ C1−q t0 , T × C1−q t0 , T × · · · × C1−q t0 , T . Furthermore, we also
obtain some results of the unique solution of the homogeneous and nonhomogeneous initial
value problems with the classical Mittag-Leffler special function 5 which is similar to the
ordinary differential equations. Now we introduce the first Mittag-Leffler function eq t − t0 defined by
eq t − t0 ∞
t − t0 kq−1
.
Γ kq
k1
1.4
The function eq t − t0 belongs to C1−q t0 , T . Indeed, taking the norm in C1−q t0 , T , we have
∞
T − t0 k−1q
eq t − t0 ≤
< ∞.
1−q
Γ kq
k1
1.5
Journal of Applied Mathematics
3
The formula remains valid for q → 1− . In this case, e1 t − t0 expt − t0 . Then we introduce
the second Mittag-Leffler function Eq t − t0 defined by
Eq t − t0 ∞
t − t0 kq
.
k1 Γ kq 1
1.6
The formula remains also valid for q → 1− . In this case, E1 t − t0 expt − t0 − 1.
The paper is organized as follows. In Section 2 we recall the definitions of fractional
integral and derivative and related basic properties and preliminary results used in the text.
In Section 3 we obtain the unique solution of the constant coefficient homogeneous and
nonhomogeneous linear fractional differential equations for P being the diagonal matrix. In
Section 4 we prove the existence and uniqueness of these two kinds of equations for any
P ∈ LRm . In Section 5 we give some specific examples to illustrate the results.
2. Definitions and Preliminary Results
Let us denote by Ct0 , T the space of all continuous real functions defined on t0 , T , which
turns out to be a Banach space with the norm
x max |xt|.
t∈t0 ,T 2.1
We define similarly another Banach space C1−q t0 , T , in which function xt is
continuous on t0 , T and t − t0 1−q xt is continuous on t0 , T with the norm:
x1−q max t − t0 1−q |xt|.
t∈t0 ,T 2.2
Lt0 , T is the space of real functions defined on t0 , T which are Lebesgue integrable
on t0 , T .
Obviously C1−q t0 , T ⊂ Lt0 , T .
The definitions and results of the fractional calculus reported below are not exhaustive
but rather oriented to the subject of this paper. For the proofs, which are omitted, we refer the
reader to 6 or other texts on basic fractional calculus.
Definition 2.1 see 6. The fractional primitive of order q > 0 of function xt ∈ C1−q t0 , T is
given by
1
q
It0 xt Γ q
t
t − sq−1 xsds.
2.3
t0
q
From 17 we know It0 xt exists for all q > 0, when x ∈ C1−q t0 , T ; consider also that
q
when x ∈ Ct0 , T then It0 xt ∈ Ct0 , T and moreover
q
It0 xt0 0.
2.4
4
Journal of Applied Mathematics
Definition 2.2 see 6. The fractional derivative of order 0 < q < 1 of a function xt ∈
C1−q t0 , T is given by
d
1
dx
Γ 1−q
q
Dt0 xt
t
t − s−q xsds.
2.5
t0
q q
We have Dt0 It0 xt xt for all xt ∈ C1−q t0 , T .
Lemma 2.3 see 6. Let 0 < q < 1. If one assumes xt ∈ C1−q t0 , T , then the fractional differential
equation
q
Dt0 xt 0
2.6
has xt ct − t0 q−1 , c ∈ R, as solutions.
From this lemma we can obtain the following law of composition.
Lemma 2.4 see 6. Assume that xt ∈ C1−q t0 , T with a fractional derivative of order 0 < q < 1
that belongs to C1−q t0 , T . Then
q
q
It0 Dt0 xt xt ct − t0 q−1 ,
2.7
for some c ∈ R. When the function x is in Ct0 , T , then c 0.
Lemma 2.5 see 6. Let U be a nonempty closed subset of a Banach space E, and let αn ≥ 0 for
every n and such ∞
n0 αn converges. Moreover, let the mapping A : U → U satisfy the inequality
An u − An v ≤ αn u − v,
2.8
for every n ∈ N and any u, v ∈ U. Then, A has a uniquely defined fixed point u∗ . Furthermore, for
∗
any u0 ∈ U, the sequence An u0 ∞
n1 converges to this fixed point u .
Lemma 2.6 see 12. Let P ∈ LRm and have real eigenvalues λ1 , λ2 , . . . , λr . Then there exists a
basis of Rm in which the matrix representation of P assumes Jordan form, that is, the matrix of P is
made of diagonal blocks of the form diagJ1 , J2 , . . . , Jr , where each Ji consists of diagonal blocks of the
form
Journal of Applied Mathematics
5
⎛
λi
⎜1
⎜
⎜
⎜0
⎜.
⎜.
⎝.
···
···
···
..
.
0
λi
1
..
.
⎞
0
0⎟
⎟
⎟
0 ⎟.
.. ⎟
⎟
.⎠
0
0
0
..
.
2.9
0 0 · · · 1 λi
Lemma 2.7 see 12. Let P ∈ LRm and have complex eigenvalues μj αj iβj , j 1, 2, . . . , r,
with multiplicity. Then there exists a basis of Rm , where P has matrix form diagJ1 , J2 , . . . , Jr , where
each Ji consists of diagonal blocks of the type
⎛
⎞
0 0
0 0⎟
⎟
⎟
0 0 ⎟,
.. .. ⎟
⎟
. .⎠
0 0 · · · I2 D
D
⎜I
⎜ 2
⎜
⎜0
⎜.
⎜.
⎝.
0
D
I2
..
.
···
···
···
..
.
D
αi −βi
1 0
, I2 .
βi αi
0 1
2.10
Lemma 2.8 see 12. Let P ∈ LRm . Then Rm has a basis giving P a matrix representation
composed of diagonal blocks of type Ji and/or matrices Ji , where Ji and Ji are as defined in the preceding
lemmas.
Now, we will introduce Lemma 2.9 to prove the following Theorem 4.4 in Section 4.
Lemma 2.9. Let 0 < q < 1. Assume that xt and ft belong to C1−q t0 , T . Then For the initial
value problem
q
Dt0 xt λxt ft,
xa b
2.11
has a unique solution xt ∈ C1−q t0 , T provided t0 < a < a0 , where a0 is a suitable constant
depending on t0 , q, and λ.
Proof. The initial value problem 2.11 will be solved in two steps.
1 Local existence.
Our problem is equivalent to the problem of determination of fixed points of the
following operator:
q−1
Axt ct − t0 1
Γ q
t
t − sq−1 λxs fs ds,
2.12
t0
with
c
1
b− Γ q
a
t0
a − s
q−1 λxs fs ds a − t0 1−q .
2.13
6
Journal of Applied Mathematics
It is immediate to verify that A : C1−q t0 , T → C1−q t0 , T is also well defined.
Indeed,
t − t0 1−q Axt
1 t
≤ |λ| t − sq−1 s − t0 q−1 s − t0 1−q xsds
Γ q t0
1 t
q−1
q−1
1−q
t − s s − t0 s − t0 fsds
Γ q t0
≤ |c| |λ|x1−q I q t − t0 q−1 f 1−q I q t − t0 q−1 2.14
Γ q
Γ q
q
≤ |c| |λ|x1−q t − t0 f 1−q t − t0 q ,
Γ 2q
Γ 2q
for xt and ft belong to ∈ C1−q t0 , T .
Then we can also prove A is a contraction operator. Indeed,
t − t0 Γ q
Γ q
q
Axt − Ayt ≤ |λ| a − t0 x − y1−q |λ| t − t0 q x − y1−q
Γ 2q
Γ 2q
Γ q
Γ q
q
≤ |λ| a − t0 x − y1−q |λ| T − t0 q x − y1−q ,
Γ 2q
Γ 2q
2.15
1−q for all xt, yt ∈ C1−q t0 , T . Let us assume
Γ q
1
|λ| a − t0 q < ,
2
Γ 2q
2.16
that is,
a < a0 1/q
Γ 2q
t0 .
2|λ|Γ q
2.17
Taking T − a > 0 sufficiently small, we also have
Γ q
1
|λ| T − t0 q < ,
2
Γ 2q
2.18
Axt − Ayt ≤ Lx − y ,
1−q
1−q
2.19
and then
Journal of Applied Mathematics
7
with L < 1. Therefore A is a contraction operator. This shows that initial problem
2.11 has a unique solution.
2 Continuation of solution.
Since we know the value of xt on t0 , a, then we can compute
c∗ 1
b− Γ q
a
q−1 a − s
λxs fs ds a − t0 1−q .
2.20
t − sq−1 λys fs ds,
2.21
t0
We can solve the integral problem
1
yt c∗ t − t0 q−1 Γ q
t
t0
obtaining a unique solution yt ∈ C1−q t0 , T for all T > t0 . Now xt and yt
agree on t0 , a. Thus the solution admits yt as its continuation. Hence the proof
of Lemma 2.9 is complete .
3. Initial Value Problem: Continuous Solutions on t0 , T We open this section with some basic examples, concerning the case when the solutions in
C1−q t0 , T × C1−q t0 , T × · · · × C1−q t0 , T are submitted to an initial condition.
Theorem 3.1. Let 0 < q < 1. For all a, B ∈ t0 , T × Rm the initial value problem
q
Dt0 Xt 0,
Xa B
3.1
admits
Xt Ba − t0 1−q t − t0 q−1 ,
3.2
as unique solution in C1−q t0 , T × C1−q t0 , T × · · · × C1−q t0 , T .
Proof. According to Lemma 2.4, the initial value problem 3.1 is equivalent to the following
equations:
Xt Ct − t0 q−1 ,
C Ba − t0 1−q .
3.3
Hence the proof of Theorem 3.1 is complete.
Theorem 3.2. Let 0 < q < 1. Assume
T
Ft f1 t, f2 t, . . . , fm t ∈ C1−q t0 , T × C1−q t0 , T × · · · × C1−q t0 , T .
3.4
8
Journal of Applied Mathematics
Then for all a, B ∈ t0 , T × Rm the initial value problem
q
Dt0 Xt Ft,
Xa B
3.5
has a unique solution in
C1−q t0 , T × C1−q t0 , T × · · · × C1−q t0 , T ,
3.6
Xt x1 t, x2 t, . . . , xm tT ,
3.7
given by
with
xi t 1
bi − Γ q
a
a − s
q−1
t0
1
fi sds a − t0 1−q t − t0 q−1 Γ q
t
t − sq−1 fi sds,
t0
i 1, 2, . . . , m.
3.8
Proof. According to Lemma 2.4, the initial value problem 3.5 is equivalent to the following
equations:
t
1
Xt Ct − t0 q−1 t − sq−1 Fsds,
Γ q t0
a
1
q−1
C B− a − s Fsds a − t0 1−q .
Γ q t0
3.9
Hence the proof of Theorem 3.2 is complete.
The result remains true even if q → 1− . In this case, 3.5 is reduced to the ordinary
differential equations
X t Ft,
Xa B,
3.10
which have a unique solution in
Ct0 , T × Ct0 , T × · · · × Ct0 , T ,
3.11
Xt x1 t, x2 t, . . . , xm tT ,
3.12
given by
Journal of Applied Mathematics
9
with
xi t bi −
a
fi sds t0
t
fi sds,
i 1, 2, . . . , m.
3.13
t0
Theorem 3.3. Let 0 < q < 1. For all a, B ∈ t0 , T × Rm the initial value problem
q
Dt0 Xt P Xt,
Xa B,
3.14
⎞
0 0
0 0⎟
⎟
⎟
0 0⎟
.. .. ⎟
⎟
. . ⎠
0 λm
3.15
where
⎛
λ1 0 · · ·
⎜ 0 λ ···
2
⎜
⎜
0 0 ···
P ⎜
⎜. .
⎜ . . ..
⎝. .
.
0 0 ···
has a unique solution in
C1−q t0 , T × C1−q t0 , T × · · · × C1−q t0 , T ,
3.16
Xt x1 t, x2 t, . . . , xm tT ,
3.17
given by
with
1/q
1/q
xi t bi eq−1 λi a − t0 eq λi t − t0 ,
i 1, 2, . . . , m.
3.18
Proof. We can write 3.27 in the following form:
q
Dt0 x1 t λ1 x1 t,
q
Dt0 x2 t λ2 x2 t,
..
.
3.19
q
Dt0 xm t λm xm t,
x1 a b1 , . . . , xm a bm .
According to Lemma 2.4,
q
Dt0 xi t λi xi t
3.20
10
Journal of Applied Mathematics
is equivalent to the following equations:
q
xi t ci t − t0 q−1 It0 λi xi t,
3.21
for some ci ∈ R. From 3.21 we obtain, by iteration,
xi t ci Γ q
λn−1 t − t0 nq−1
t − t0 q−1 λi t − t0 2q−1
··· i
Γ q
Γ 2q
Γ nq
nq
λni It0 xi t.
3.22
nq
Letting n → ∞, λni It0 xi t1−q → 0 if xi t ∈ C1−q t0 , T . Indeed,
n nq
λi It0 xi t
1−q
≤ λn xi t
Γ q
1−q
i
t − t0 nq .
Γ n 1q
3.23
On the other hand,
eq t − t0 ∞
t − t0 kq−1
,
Γ kq
k1
3.24
then we can obtain
1/q−1 1/q
xi t ci Γ q λi
eq λi t − t0 .
3.25
Since xi a bi ,
1/q
1/q
xi t bi eq−1 λi a − t0 eq λi t − t0 ,
i 1, 2, . . . , m.
3.26
Hence the proof of Theorem 3.3 is complete.
The result remains valid even if q → 1− . In this case,
X t P Xt,
Xa B
3.27
has a unique solution in
Ct0 , T × Ct0 , T × · · · × Ct0 , T ,
3.28
Xt x1 t, x2 t, . . . , xm tT ,
3.29
given by
Journal of Applied Mathematics
11
with
xi t bi exp−λi a − t0 expλi t − t0 ,
i 1, 2, . . . , m.
3.30
Theorem 3.4. Let 0 < q < 1. For all a, B ∈ t0 , T × Rm the initial value problem
q
Dt0 Xt P Xt D,
Xa B,
3.31
where
⎛
λ1 0 · · ·
⎜ 0 λ ···
2
⎜
⎜
0
0
···
⎜
P ⎜
⎜ .. .. . .
⎝. .
.
0 0 ···
⎞
0 0
0 0⎟
⎟
⎟
0 0 ⎟,
.. .. ⎟
⎟
. . ⎠
0 λm
3.32
and D d1 , d2 , . . . , dm has a unique solution in
C1−q t0 , T × C1−q t0 , T × · · · × C1−q t0 , T ,
3.33
Xt x1 t, x2 t, . . . , xm tT ,
3.34
given by
with
1/q
1/q
1/q
1/q
eq−1 λi a − t0 eq λi t − t0 di λ−1
xi t bi − di λ−1
i Eq λi a − t0 i Eq λi t − t0 i 1, 2, . . . , m.
3.35
Proof. We can write 3.44 in the following form:
q
Dt0 x1 t λ1 x1 t d1 ,
q
Dt0 x2 t λ2 x2 t d2 ,
..
.
q
Dt0 xm t λm xm t dm ,
x1 a b1 , . . . , xm a bm .
3.36
12
Journal of Applied Mathematics
According to Lemma 2.4, the equation
q
Dt0 xi t λi xi t di
3.37
is equivalent to the following equations:
q
q
xi t ci t − t0 q−1 It0 λi xi t It0 di ,
3.38
for some ci ∈ R. From 3.38 we obtain, by iteration,
q
q
2q
2q
It0 λi xi t It0 ci λi t − t0 q−1 It0 λ2i xi t It0 λi di ,
q
2q
q
2q
xi t ci t − t0 q−1 It0 λi ci t − t0 q−1 It0 λ2i xi t It0 di It0 λi di ,
q
q
2q
2q
It0 xi t It0 ci t − t0 q−1 It0 λi xi t It0 di ,
nq−1
− t0 Γ nq
nq
λn−1
t − t0 q λi t − t0 2q
nq
i t − t0 λni It0 xi t.
di
··· Γ q1
Γ 2q 1
Γ nq 1
xi t ci Γ q
q−1
t − t0 Γ q
2q−1
λi t − t0 Γ 2q
··· λn−1
i t
3.39
nq
Letting n → ∞, λni It0 xi t1−q → 0 if xi t ∈ C1−q t0 , T . Indeed,
n nq
λi It0 xi t
1−q
max t −
t∈t0 ,T t0 1−q λni
≤ λn xi t
i
1−q
1
Γ q
Γ q
t
t − s
nq−1
t0
xi sds
3.40
t − t0 nq .
Γ n 1q
On the other hand,
eq t − t0 ∞
t − t0 kq−1
,
Γ kq
k1
∞
t − t0 kq
Eq t − t0 .
k1 Γ kq 1
3.41
Then we can obtain
1/q
1/q−1 1/q
xi t ci Γ q λi
eq λi t − t0 di λ−1
i Eq λi t − t0 .
1/q
3.42
We know that di λ−1
Eq λi t−t0 is satisfied for the fractional nonhomogeneous linear
qi
differential equation Dt0 xi t λ1 xi t di . So we can also deduce that the general solution of
Journal of Applied Mathematics
13
the fractional nonhomogeneous linear differential equation is equal to the general solution of
the corresponding homogeneous linear differential equation plus the special solution of the
nonhomogeneous linear differential equation. If Xa B, xi a bi , then
1/q
1/q
1/q
1/q
−1
−1
e
e
d
λ
λ
λ
λ
xi t bi − di λ−1
E
λ
E
−
t
−
t
−
t
−
t
a
a
t
t
q
0
0
q
0
i
q
0
q
i
i
i
i
i
i
i 1, 2, . . . , m.
3.43
Hence the proof of Theorem 3.4 is complete.
The result remains valid even if q → 1− . In this case,
X t P Xt D,
Xa B,
3.44
where
⎛
λ1 0 · · · 0
⎜ 0 λ ··· 0
2
⎜
⎜
0
0
··· 0
⎜
P ⎜
⎜ .. .. . . ..
⎝. .
. .
0 0 ··· 0
0
0
0
..
.
⎞
⎟
⎟
⎟
⎟,
⎟
⎟
⎠
3.45
λm
and D d1 , d2 , . . . , dm has a unique solution in
Ct0 , T × Ct0 , T × · · · × Ct0 , T ,
3.46
Xt x1 t, x2 t, . . . , xm tT ,
3.47
given by
with
−1
−1
xi t bi − di λ−1
i E1 λi a − t0 e1 λi a − t0 e1 λi t − t0 di λi E1 λi t − t0 bi di λ−1
i
−1
exp−λa − t0 expλi t − t0 − dλ ,
3.48
i 1, 2, . . . , m.
4. Existence and Uniqueness of the Solution
In Section 3 we have obtained the unique solution of the constant coefficient homogeneous
and nonhomogeneous linear fractional differential equations for P being the diagonal matrix.
In the present section we will prove the existence and uniqueness of these two kinds of
equations for any P ∈ LRm .
14
Journal of Applied Mathematics
Theorem 4.1. Let 0 < q < 1 and P ∈ LRm . If the matrix P has distinct real eigenvalues, then for
all a, B ∈ t0 , T × Rm the initial value problem
q
Dt0 Xt P Xt,
Xa B
4.1
has the unique solution Xt ∈ C1−q t0 , T × C1−q t0 , T × · · · × C1−q t0 , T .
Proof. Since the matrix P has distinct real eigenvalues, there exists an invertible matrix Q such
that
⎞
⎛
λ1 0 · · · 0 0
⎜ 0 λ ··· 0 0 ⎟
2
⎟
⎜
⎟
⎜
−1
0
0
· · · 0 0 ⎟,
⎜
4.2
Q PQ ⎜
⎟
⎜ .. .. . . .. .. ⎟
⎝. .
. . . ⎠
0 0 · · · 0 λm
where λ1 , λ2 , . . . , λm are the eigenvalues of the matrix P . If we define Y t Q−1 Xt,
q
q
q
Dt0 Y t Dt0 Q−1 Xt Q−1 Dt0 Xt Q−1 P Xt RY t,
4.3
with R diagλ1 , λ2 , . . . , λm . From the above Theorem 3.3 we know the initial value problem
q
Dt0 Y t RY t,
Y a Q−1 B
4.4
has a unique solution Y t ∈ C1−q t0 , T × C1−q t0 , T × · · · × C1−q t0 , T defined on t0 , T .
Then Xt QY t uniquely solves the equations 4.1, where t ∈ t0 , T . Hence the proof of
Theorem 4.1 is complete .
Theorem 4.2. Let 0 < q < 1. For all a, B ∈ t0 , T × R2 the initial value problem
q
Dt0 Xt P Xt,
Xa B,
4.5
where
P
α −β
β α
4.6
has the unique solution Xt ∈ C1−q t0 , T × C1−q t0 , T defined on t0 , T .
Proof. Let us define
Zt x1 t ix2 t,
μ α iβ.
4.7
We can find that 4.5 is equivalent to the following equation
q
Dt0 Zt μZt,
Za x1 a x2 a b1 ib2 .
4.8
Journal of Applied Mathematics
15
Obviously, Zt ∈ C1−q t0 , T if x1 t and x2 t belong to C1−q t0 , T . From the above
Theorem 3.3 in Section 3, we know the complex Equation 4.7 has a unique solution defined
on t0 , T . Hence the proof of Theorem 4.2 is complete.
Theorem 4.3. Let 0 < q < 1 and P ∈ R2 . If P has eigenvalues α ± iβ, for all a, B ∈ t0 , T × R2 the
initial value problem
q
Dt0 Xt P Xt,
Xa B
4.9
has a unique solution Xt ∈ C1−q t0 , T × C1−q t0 , T .
Proof. Since P has eigenvalues α ± iβ, there exists an invertible matrix Q such that P QSQ−1
where
S
α −β
.
β α
4.10
Define
Y t Q−1 Xt,
4.11
then
q
q
q
Dt0 Y t Dt0 Q−1 Xt Q−1 Dt0 Xt Q−1 P Xt SY t.
4.12
From the above Theorem 4.2, we know the initial value problem
q
Y a Q−1 B
Dt0 Y t SY t,
4.13
has a unique solution defined on t0 , T . Hence the proof of result is complete.
Theorem 4.4. Let 0 < q < 1 and P ∈ Rm be an elementary Jordan matrix:
⎛
λ
⎜1
⎜
⎜
⎜0
⎜.
⎜.
⎝.
⎞
0
0⎟
⎟
⎟
0 ⎟.
.. ⎟
⎟
.⎠
0 0 ··· 1 λ
0
λ
1
..
.
···
···
···
..
.
0
0
0
..
.
4.14
The initial value problem
q
Dt0 Xt P Xt,
Xa B
4.15
has a unique solution Xt ∈ C1−q t0 , T ×C1−q t0 , T ×· · ·× C1−q t0 , T provided a, B ∈ t0 , a0 ×
Rm , where a0 is a suitable constant depending on t0 , q, and λ.
16
Journal of Applied Mathematics
Proof. From the 4.15, we can write the equations in the following form:
q
Dt0 x1 t λx1 t,
q
Dt0 x2 t x1 t λx2 t,
..
.
4.16
q
Dt0 xm t xm−1 t λxm t,
x1 a b1 , . . . , xm a bm .
Consider the first equation
q
Dt0 x1 t λx1 t,
x1 a b1 .
4.17
We can obtain the solution of this equation
x1 t b1 eq−1 λ1/q a − t0 eq λ1/q t − t0 .
4.18
Consider the second equation
q
Dt0 x2 t x1 t λx2 t,
x2 a b2 ,
4.19
where now x1 t ∈ C1−q t0 , T is a known function. Since x1 t, x2 t ∈ C1−q t0 , T , according
to Lemma 2.9, 4.19 has a unique solution in C1−q t0 , T . Now x1 t and x2 t are known
functions which will be substituted in
q
Dt0 x3 t x2 t λx3 t,
x3 a b3
4.20
and so on. Thus the system of equations given in 4.15 has unique solution in C1−q t0 , T ×
C1−q t0 , T × · · · × C1−q t0 , T .
Theorem 4.5. Let 0 < q < 1 and P ∈ LRm . The initial value problem
q
Dt0 Xt P Xt,
Xa B
4.21
has the unique solution Xt ∈ C1−q t0 , T × C1−q t0 , T × · · · × C1−q t0 , T provided a, B ∈ t0 , a0 ×
Rm , where a0 is a suitable constant depending on t0 , q, and λ.
Journal of Applied Mathematics
17
Proof. In view of Lemma 2.8, there exists an invertible matrix Q such that Q−1 P Q is composed
of diagonal blocks of the type Ji and Ji , as defined in the preceding Lemmas 2.7 and 2.8. Let
B Q−1 P Q and Y t Q−1 Xt. Consider the initial value problem:
q
q
q
Dt0 Y t Dt0 Q−1 Xt Q−1 Dt0 Xt Q−1 P Xt BY t,
Y a Q−1 Xa Q−1 B.
4.22
Then in view of Theorems 4.1–4.5, 4.22 has a unique solution: Y t ∈ C1−q t0 , T ×
C1−q t0 , T × · · · × C1−q t0 , T . Therefore 4.21 has a unique solution Q−1 Y t ∈ C1−q t0 , T ×
C1−q t0 , T × · · · × C1−q t0 , T .
Remark 4.6. All the above results are valid for q → 1− . Moreover, we can also discuss the
case if a t0 , in this case, we cannot consider the usual initial condition xt0 b, but
limt → t0 t − t0 1−q xt b. We can also obtain some similar results by the same method, So
we did not give the detailed process and conclusion in this paper.
5. Illustrative Examples
In this section, we give some specific examples to illustrate the above results.
Example 5.1. Consider the following system, where 0 < q < 1, t ∈ t0 , T , a, B ∈ t0 , T ×
R3 ,B b1 , b2 , b3 ,
q
Dt0 x1 t 3x1 t − x2 t x3 t,
q
Dt0 x2 t −x1 t 5x2 t − x3 t,
q
Dt0 x3 t x1 t − x2 t 3x3 t,
x1 a b1 ,
x2 a b2 ,
5.1
x3 a b3 .
Here
⎛
⎞
3 −1 1
P ⎝−1 5 −1⎠,
1 −1 3
5.2
having the eigenvalues 2, 3, and 6. Choose the eigenvectors g1 1, 0, −1T , g2 1, 1, 1T ,
and g3 1, −2, 1T . Then
⎛
⎞
⎞
3 −1 1
2 0 0
⎝0 3 0⎠ Q−1 ⎝−1 5 −1⎠Q,
1 −1 3
0 0 6
⎛
5.3
18
Journal of Applied Mathematics
where
⎞
1
1
0
−
⎜2
2⎟
⎟
⎜
⎟
⎜
⎜1 1 1 ⎟
⎟
⎜
⎜ 3 3 3 ⎟.
⎟
⎜
⎜1 1 1 ⎟
⎟
⎜
⎠
⎝ −
6 3 6
⎛
Q−1
5.4
Define the Y Q−1 X. Then the system of equation in Y is decoupled, namely,
q
Dt0 y1 t 2y1 t,
q
Dt0 y2 t 3y2 t,
q
Dt0 y3 t 6y3 t,
y1 a 1
1
b1 − b3 ,
2
2
y2 a 1
1
1
b1 b2 b 3 ,
3
3
3
y3 a 1
1
1
b1 − b2 b 3 .
6
3
6
5.5
In view of 3.30, we can obtain
1
1
b1 − b3 eq−1 21/q a − t0 eq 21/q t − t0 ,
2
2
1
1
1
y2 t b1 b2 b3 eq−1 31/q a − t0 eq 31/q t − t0 ,
3
3
3
1
1
1
y3 t b1 − b2 b3 eq−1 61/q a − t0 eq 61/q t − t0 .
6
3
6
y1 t 5.6
Hence
x1 t y1 t y2 t y3 t,
x2 t y2 t − 2y3 t,
x3 t −y1 t y2 t y3 t.
5.7
Journal of Applied Mathematics
19
Example 5.2. Consider the following system, where 0 < q < 1, t ∈ t0 , T , a, B ∈ t0 , T ×
R2 ,B b1 , b2 ,
q
Dt0 x1 t −2x1 t − x2 t
q
Dt0 x2 t 13x1 t 4x2 t
x1 a b1 ,
5.8
x2 a b2 .
Here
P
−2 −1
13 4
5.9
having the eigenvalues 1 ± 2i. Choose the eigenvectors g1 1, −3 − 2iT , and g2 1, −3 2iT , Then
1 2i
0
0
1 − 2i
Q−1
−2 −1
Q,
13 4
5.10
where
Q
Q
−1
1
1
,
−3 − 2i −3 2i
⎛
⎞
2 3i i
⎜ 4
4 ⎟
⎜
⎟
⎟
⎜
i ⎟.
⎜ 2 − 3i
⎝
− ⎠
4
4
5.11
Define the Y Q−1 X. Then the system of equation in Y is decoupled, namely,
q
Dt0 y1 t 1 2iy1 t,
q
Dt0 y2 t
1 − 2iy2 t,
3
1
b1 b 2 i
4
4
3
1
1
b1 b2 i.
y2 a b1 −
2
4
4
y1 a 1
b1 2
5.12
In view of 3.30, we can obtain
3
1
b1 b2 i eq−1 1 2i1/q a − t0 eq 1 2i1/q t − t0 y1 t 4
4
3
1
1
y2 t b1 −
b1 b2 i eq−1 1 − 2i1/q a − t0 eq 1 − 2i1/q t − t0 .
2
4
4
1
b1 2
5.13
20
Journal of Applied Mathematics
Hence
x1 t y1 t y2 t
x2 t −3 y1 t y2 t − 2i y1 t − y2 t .
5.14
Acknowledgments
The authors are highly grateful for the referee’s careful reading and comments on this paper.
The present paper was supported by the NNSF of China Grants no. 11271087 and no.
61263006.
References
1 B. Bonilla, M. Rivero, L. Rodrı́guez-Germá, and J. J. Trujillo, “Fractional differential equations as
alternative models to nonlinear differential equations,” Applied Mathematics and Computation, vol. 187,
no. 1, pp. 79–88, 2007.
2 T. M. Atanackovic and B. Stankovic, “On a system of differential equations with fractional derivatives
arising in rod theory,” Journal of Physics A, vol. 37, no. 4, pp. 1241–1250, 2004.
3 M. Caputo, “Linear models of dissipation whose Q is almost independent,” Geophysical Journal of the
Royal Astronomical Society, vol. 13, pp. 529–539, 1967.
4 V. Daftardar-Gejji and A. Babakhani, “Analysis of a system of fractional differential equations,”
Journal of Mathematical Analysis and Applications, vol. 293, no. 2, pp. 511–522, 2004.
5 K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations,
John Wiley & Sons, New York, NY, USA, 1993.
6 D. Delbosco and L. Rodino, “Existence and uniqueness for a nonlinear fractional differential
equation,” Journal of Mathematical Analysis and Applications, vol. 204, no. 2, pp. 609–625, 1996.
7 Y. Zhou, F. Jiao, and J. Li, “Existence and uniqueness for p-type fractional neutral differential
equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 7-8, pp. 2724–2733, 2009.
8 A. Yildirim, “Fractional variational iteration method for fractional initial-boundary value problems
arising in the application of nonlinear science,” Computers & Mathematics with Applications, vol. 62, no.
5, pp. 2273–2278, 2011.
9 S. Das, K. Vishal, P. K. Gupta, and A. Yildirim, “An approximate analytical solution of time-fractional
telegraph equation,” Applied Mathematics and Computation, vol. 217, no. 18, pp. 7405–7411, 2011.
10 A. Yıldırım, “Solution of BVPs for fourth-order integro-differential equations by using homotopy
perturbation method,” Computers & Mathematics with Applications, vol. 56, no. 12, pp. 3175–3180, 2008.
11 J. Deng, Z. Deng, and L. Yang, “An existence and uniqueness result for fractional order semilinear
functional differential equations with nondense domain,” Applied Mathematics Letters, vol. 24, no. 6,
pp. 924–928, 2011.
12 Y. Luchko and R. Gorenflo, “An operational method for solving fractional differential equations with
the Caputo derivatives,” Acta Mathematica Vietnamica, vol. 24, no. 2, pp. 207–233, 1999.
13 D. K. Diethelm and N. J. Ford, “Analysis of fractional differential equations,” Journal of Mathematical
Analysis and Applications, vol. 265, no. 2, pp. 229–248, 2002.
14 V. Lakshmikantham and A. S. Vatsala, “Basic theory of fractional differential equations,” Nonlinear
Analysis. Theory, Methods & Applications, vol. 69, no. 8, pp. 2677–2682, 2008.
15 V. Lakshmikantham, “Theory of fractional functional differential equations,” Nonlinear Analysis.
Theory, Methods & Applications, vol. 69, no. 10, pp. 3337–3343, 2008.
16 S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Drivatives, Gordon and Breach
Science Publishers, Yverdon, Switzerland, 1993.
17 M. Belmekki, J. J. Nieto, and R. Rodrı́guez-López, “Existence of periodic solution for a nonlinear
fractional differential equation,” Boundary Value Problems, vol. 2009, Article ID 324561, 18 pages, 2009.
18 S. Q. Zhang, “Positive solutions to singular boundary value problem for nonlinear fractional
differential equation,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1300–1309, 2010.
19 D. Băleanu and O. G. Mustafa, “On the global existence of solutions to a class of fractional differential
equations,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1835–1841, 2010.
Journal of Applied Mathematics
21
20 H. Jafari and V. Daftardar-Gejji, “Positive solutions of nonlinear fractional boundary value problems
using Adomian decomposition method,” Applied Mathematics and Computation, vol. 180, no. 2, pp.
700–706, 2006.
21 B. Ahmad and J. J. Nieto, “Riemann-Liouville fractional differential equations with fractional
boundary conditions ,” Fixed Point Theory. In press.
22 B. Ahmad and J. J. Nieto, “Existence results for a coupled system of nonlinear fractional differential
equations with three-point boundary conditions,” Computers & Mathematics with Applications, vol. 58,
no. 9, pp. 1838–1843, 2009.
23 L. L. Lv, J. Wang, and W. Wei, “Existence and uniqueness results for fractional differential equations
with boundary value conditions,” Opuscula Mathematica, vol. 31, no. 4, pp. 629–643, 2011.
24 G. M. N’Guérékata, “A Cauchy problem for some fractional abstract differential equation with non
local conditions,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 5, pp. 1873–1876, 2009.
25 G. Mophou and G. M. N’Guérékata, “Optimal control of a fractional diffusion equation with state
constraints,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1413–1426, 2011.
26 G. Q. Chai, “Existence results for boundary value problems of nonlinear fractional differential
equations,” Computers & Mathematics with Applications, vol. 62, no. 5, pp. 2374–2382, 2011.
27 F. Wang, “Existence and uniqueness of solutions for a nonlinear fractional differential equation,”
Journal of Applied Mathematics and Computing, vol. 39, no. 1-2, pp. 53–67, 2012.
28 H. A. H. Salem, “On the existence of continuous solutions for a singular system of non-linear
fractional differential equations,” Applied Mathematics and Computation, vol. 198, no. 1, pp. 445–452,
2008.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014