By employing a fixed point theorem in a weighted Banach space, we establish the existence of a solution for a system of impulsive singular fractional differential equations. Some examples are presented to illustrate the efficiency of the results obtained.
AMS Mathematics Subject Classification : 92D25, 34A37, 34K15.
1. Introduction
Fractional differential equation is a generalization of ordinary differential equation to arbitrary noninteger orders. The origin of fractional calculus goes back to Newton and Leibniz in the seventeenth century. Recent investigations have shown that many physical systems can be represented more accurately through fractional derivative formulation
[14]
. Fractional differential equations therefore find numerous applications in different branches of physics, chemistry and biological sciences such as viscoelasticity, feed back amplifiers, electrical circuits, electro analytical chemistry, fractional multipoles and neuron modelling
[16]
. The reader may refer to the books and monographs
[15
,
6
,
8]
for fractional calculus and developments on fractional differential and fractional integrodifferential equations with applications.
On the other hand, the theory of impulsive differential equations describes processes which experience a sudden change of their state at certain moments. Processes with such characteristics arise naturally and often, for example, phenomena studied in physics, chemical technology, population dynamics, biotechnology and economics. For an introduction of the basic theory of impulsive differential equation, we refer the reader to
[11]
.
On the other hand, in recently, there have been many papers
[1
,
2
,
3
,
4
,
5
,
7
,
9
,
12
,
19
,
17
,
18]
concerned with the existence of solutions for different initial value problems involving impulsive fractional differential equations. However, there has been few papers discussed the Global existence of solutions of initial value problems for impulsive fractional differential systems on half line.
Motivated by the above reason, we discuss the following initial value problem of singular fractional differential system on the half line
where

(a)0 <α,β≤ 1,are the Caputo fractional derivatives of ordersαandβrespectively,

(b)N= {1, 2, · · · } andN0= {0, 1, 2, · · · }, 0 =t0

(c)m,n: (0, ∞) →Rsatisfym(ts,ts+1],n(ts,ts+1]∈C0(ts,ts+1] (s∈N0), bothmandnmay be singular att= 0, there exist constantsL1,L2> 0 andk,l> −1 such that m(t) ≤L1tkand n(t) ≤L2tl,t∈ (0, ∞),

(d)ϕ,ψ: (0, ∞) →Rsatisfyϕ,ψ∈L1(0, ∞), and

(e)f,g,F,Gdefined on (0, ∞) ×R×Rare Caraéodory functions,I,Jdefined on {ts:s∈N} ×R2Carathéodory sequences.
A pair of functions (
x
,
y
) with
x
: (0, ∞) →
R
and
y
: (0, ∞) →
R
is said to be a
solution
of (1.1) if
x

_{(ts,ts+1]}
,
y

_{(ts,ts+1]}
∈
C
^{0}
(
t_{s}
,
t
_{s}
_{+1}
],
s
∈
N
_{0}
,
s
∈
N
_{0}
,
exist and (
x
,
y
) satisfies all equations in (1.1).
We construct a weighted Banach space and apply the LeraySchauder nonlinear alternative to obtain the existence of at least one solution of (1.1). Our results are new and naturally complement the literature on fractional differential equations.
The paper is outlined as follows. Section 2 contains some preliminary results. The main results are presented in Section 3. Finally, in Section 4 we give two examples to illustrate the efficiency of the results obtained.
2. Preliminaries
For the convenience of the readers, we shall state the necessary definitions from fractional calculus theory.
For
h
∈
L
^{1}
(0, ∞), denote
For
r
> 0,
p
> 0,
q
> 0, let the Gamma and beta functions Γ(
r
) and
B
(
p
,
q
) be defined by
Definition 2.1
(
[6]
). Let
a
∈
R
. The
RiemannLiouville fractional integral of order α
> 0 of a function
h
: (
a
, ∞) →
R
is given by
provided that the righthand side exists.
Definition 2.2
(
[6]
). Let
a
∈
R
. The
Caputo fractional derivative of order α
> 0 of a function
h
: (
a
, ∞) →
R
is given by
where
n
− 1 ≤
α
<
n
, provided that the righthand side exists.
Remark 2.1.
Let
a
,
b
∈
R
with
a
<
b
. From Theorem 2.14 in
[6]
, we know that
almost all
t
∈ (
a
,
b
) for every
h
∈
L
^{1}
(
a
,
b
). From Theorem 2.23 in
[6]
, we have
which is the set of all absolutely continuous functions see (
[6]
, page 10), i.e. the functions
h
for which there exists (almost everywhere) a function
H
∈
L
^{1}
(
a
,
b
) such that
Definition 2.3
(
[10]
). An odd homeomorphism Φ of the real line
R
onto itself is called a
supmultiplicativelike function
if there exists a homeomorphism
ω
of [0, +∞) onto itself which
supports
Φ in the sense that for all
v
_{1}
,
v
_{2}
≥ 0,
ω
is called the
supporting function
of Φ.
Remark 2.2.
Note that any supmultiplicative function is supmultiplicativelike function. Also any function of the form
is supmultiplicativelike, provided that
c_{j}
≥ 0. Here a supporting function is defined by
ω
(
u
) := min{
u
^{k}
^{+1}
,
u
},
u
≥ 0.
Remark 2.3.
It is clear that a supmultiplicativelike function Φ and any corresponding supporting function
ω
are increasing functions vanishing at zero. Moreover, their inverses Φ
^{−1}
and
ν
respectively are increasing and such that for all
w
_{1}
,
w
_{2}
≥ 0,
ν
is called the
supporting function
of Φ
^{−1}
.
In this paper we always suppose that Φ is a supmultiplicativelike function with its supporting function
ω
. The inverse function Φ
^{−1}
has its supporting function
ν
.
Definition 2.4.
Let
σ
>
k
+
α
and
δ
>
l
+
β
. We say
K
: (0, +∞) ×
R
^{2}
→
R
is a
Carathéodory function
if it satisfies the following:

(i)t→K(t, (1+tσ)x, (1+tσ)y) is continuous on (ts,ts+1] (s= 0, 1, 2, · · · ), and for any (x,y) ∈R2there exist the limits

(ii) (x,y) →K(t, (1+tσ)x, (1+tσ)y) is continuous onR2for allt∈ (0, +∞);

(iii) for eachr> 0 there exists a constantAr> 0 such that
Definition 2.5.
{
G
: {
t_{s}
:
s
∈
N
} ×
R
^{2}
} is called a
Carathéodory sequence
if

(i)is continuous onR2for alls∈N;

(ii) for eachr> 0 there existsAr,s≥ 0 such that
To obtain the main results, we need the LeraySchauder nonlinear alternative.
Lemma 2.6
(LeraySchauder Nonlinear Alternative
[13]
).
Let X be a Banach space and T
:
X
→
X be a completely continuous operator. Suppose
Ω
is a nonempty open subset of X centered at zero. Then, either there exists x
∈
∂
Ω
and λ
∈ (0, 1)
such that x
=
λT x, or there exists
such that x
=
Tx
.
3. Main Results
In this section we shall establish the existence of at least one solution of system (1.1). Throughout, we assume that the functions and parameters in (1.1) satisfy (a)–(e) (stated in Section 1) and the following:

(A)f, g, F, Gare Carathéodory functions;

(B)I, Jare Carathéodory sequences.
Let
It is easy to show that
X
is a real Banach space. Thus, (
X
×
X
, ∥ · ∥) is a Banach space with the norm defined by (
x, y
) = max {
x

_{X}
, 
y

_{X}
}, (
x, y
) ∈
X
×
X
.
Let
x
∈
X
and
y
∈
X
. Then, there exists
r
> 0 such that (
x, y
) =
r
< +∞ From (A),
f
is a Carathéodory function, thus there exists
A_{r}
≥ 0 such that
Similarly, there exist positive constants
A
^{′}
_{r}
and
A_{r,s}
(
s
= 1, 2, · · · ) such that
Likewise,
g
,
G
and
J_{s}
are also Carathéodory functions, so there exist positive constants
B_{r}
,
B^{′}_{r}
and
B_{r,s}
(
s
= 1, 2, · · · ) such that
Lemma 3.1.
Suppose that x, y
∈
X. Then, u
∈
X is a solution of
if and only if u
∈
X satisfies the integral equation
Proof
. Let
u
∈
X
be a solution of (3.4). Then, it follows from (3.4) that for
t
∈ (
t_{i}
,
t
_{i}
_{+1}
] (
i
= 0, 1, 2, · · · ), there exist constants
c_{i}
∈
R
such that
From
we get
From ∆
u
(
t_{i}
) =
I
(
t_{i}
,
x
(
t_{i}
),
y
(
t_{i}
)), we have
c_{i}
−
c
_{i}
_{−1}
=
I
(
t_{i}
,
x
(
t_{i}
),
y
(
t_{i}
)), which leads to
On substituting
c_{i}
into (3.6), we obtain for
t
∈ (
t_{i}
,
t
_{i}
_{+1}
] (
i
= 0, 1, 2, · · · ),
which is simply the same as (3.5). Moreover, since
x
∈
X
and
y
∈
Y
, we have (3.1)–(3.3) which will lead to the expression of
u
in (3.5) is indeed in
X
.
On the other hand, if
x
∈
X, y
∈
Y
and
u
∈
X
satisfies (3.5), then we can prove that
u
∈
X
is a solution of (3.4). The proof is complete.
Lemma 3.2.
Suppose that x, y
∈
X. Then, v
∈
Y is a solution of
if and only if v
∈
Y satisfies the integral equation
Proof
. The proof is similar to that of Lemma 3.1.
Now, we define the operator
T
on
X
×
Y
by
T
(
x
,
y
)(
t
) = (
T
_{1}
(
x
,
y
)(
t
),
T
_{2}
(
x
,
y
)(
t
)) where
and
Remark 3.1.
By Lemmas 3.1 and 3.2, (
x, y
) ∈
X
×
Y
is a solution of system (1.1) if and only if (
x, y
) ∈
X
×
Y
is a fixed point of the operator
T
.
Lemma 3.3.
The operator T
:
X
×
X
→
X
×
X
is well defined and is completely continuous.
Proof
. The proof is long and will be divided into parts. First, we prove that
T
is well defined. Next, we show that
T
is continuous, and finally we prove that
T
is compact. Hence,
T
is completely continuous.
Step 1.
We shall prove that
T
:
X
×
Y
→
X
×
Y
is well defined. For (
x, y
) ∈
X
×
Y
, we have ∥(
x, y
)∥ =
r
> 0. Then, (3.1)–(3.4) hold. Hence,
It follows that
and
We see that
T
_{1}
(
x, y
)(
t
) is defined on (0, +∞) and is continuous on (
t_{s}
,
t
_{s}
_{+1}
] (
s
= 0, 1, 2, · · · ). Next, we have
exist.
Also, in view of (3.1)–(3.3) and (3.12), we find for
t
∈ (
t_{i}
,
t
_{i}
_{+1}
]
With this we have shown that
T
_{1}
(
x, y
) ∈
X
. Similarly we can show that
T
_{2}
(
x, y
) ∈
X
. Hence, (
T
_{1}
(
x, y
),
T
_{2}
(
x, y
) ∈
X
×
X
and
T
:
X
×
X
→
X
×
X
is well defined.
Step 2.
We shall prove that
T
is continuous. Let (
x_{n}
,
y_{n}
) ∈
X
×
X
with (
x_{n}
,
y_{n}
) → (
x
_{0}
,
y
_{0}
) as
n
→ ∞. We shall show that
T
(
x_{n}
,
y_{n}
) →
T
(
x
_{0}
,
y
_{0}
) as
n
→ ∞, i.e.,
T
_{1}
(
x_{n}
,
y_{n}
) →
T
_{1}
(
x
_{0}
,
y
_{0}
) and
T
_{2}
(
x_{n}
,
y_{n}
) →
T
_{2}
(
x
_{0}
,
y
_{0}
) as
n
→ ∞. In fact, there exists
r
> 0 such that ∥(
x_{n}
,
y_{n}
)∥ ≤
r
, (
n
= 0, 1, 2, · · ·). Then, (3.1)–(3.4) hold for (
x, y
) = (
x_{n}
,
y_{n}
). Also,
as
n
→ +∞. Noting
from the Lebesgue dominated convergence theorem, we get
as
n
→ +∞. Similarly,
as
n
→ +∞. Hence,
T
is continuous.
Step 3.
We shall prove that
T
is compact, i.e., for each nonempty open bounded subset Ω of
X
×
Y
, we shall prove that
is relatively compact. For this, we shall show that
is uniformly bounded, equicontinuous on each (
t_{i}
,
t
_{i}
_{+1}
] (
i
= 0, 1, 2, · · · ), both
are equiconvergent as
t
→ +∞.
Let Ω be an open bounded subset of
X
×
Y
. There exists
r
> 0 such that (3.1) holds for all (
x, y
) ∈
. Hence, (3.2)–(3.4) also hold for all (
x, y
) ∈
.
Step 3a.
We shall show that
is uniformly bounded. Let (
x, y
) ∈
For
t
∈ (
t_{i}
,
t
_{i}
_{+1}
] (
i
= 0, 1, 2, · · ·), from (3.13) we have
Similarly, we can obtain for
t
∈ (
t_{i}
,
t
_{i}
_{+1}
] (
i
= 0, 1, 2, · · ·),
Hence, it is easy to see that
is uniformly bounded.
Step 3b.
We shall prove that
s equicontinuous on each (
t_{i}
,
t
_{i}
_{+1}
] (
i
= 0, 1, 2, · · ·). We define
Then
is continuous on [
t_{i}
,
t
_{i}
_{+1}
]. So
is equicontinuous on [
t_{i}
,
t
_{i}
_{+1}
]. Thus
is equicontinuous on [
t_{i}
,
t
_{i}
_{+1}
].
Similarly, we can show that
is equicontinuous on [
t_{i}
,
t
_{i}
_{+1}
].
So
is equicontinuous on each (
t_{i}
,
t
_{i}
_{+1}
] (
i
= 0, 1, 2, · · · ).
Step 3c.
We shall show that
is equiconvergent as
t
→ +∞.
it comes from the following items:
Similarly, we can obtain for
t
∈ (
t_{i}
,
t
_{i}
_{+1}
] (
i
= 0, 1, 2, · · · ),
We have established that
is relatively compact. So
T
is completely continuous. This completes the proof.
We are now ready to present the main theorem.
Theorem 3.4.
Let (a)–(e) and (A)–(B) hold,
Φ :
R
→
R be a supmultiplicativelike function with supporting function ω, and its inverse function
Φ
^{−1}
:
R
→
R with supporting function ν. Furthermore, suppose that

(i)there exist nonnegative numbers cf, bf, af, CF, BF, AF, CI,s, BI,sand AI,ssuch thatare convergent, and the following hold for all(U, V) ∈R2andt∈ (0, ∞) :
(
ii
)
there exist nonnegative numbers c_{g}, b_{g}, a_{g}, C_{G}, B_{G}, A_{G}, C_{J,s}, B_{J,s} and A_{J,s} such that
are convergent, and the following hold for all
(
U, V
) ∈
R
^{2}
and
t
∈ (0, ∞) :
Then, the system (1.1) has at least one solution in X
×
X if
or
where
Proof
. We shall apply Lemma 2.6. From Lemma 3.3 we note that
T
is completely continuous. Let us consider the operator equation
where
λ
∈ (0, 1). We shall show that any solution (
x, y
) of (3.17) satisfies
where
M
is a constant independent of
λ
. Now, in the context of Lemma 2.1, let
In view of (3.18), it is not possible to have (
x, y
) ∈
∂
Ω satisfying (
x, y
) =
λT
(
x, y
), hence we conclude by Lemma 2.6 that there exists
such that (
x, y
) =
T
(
x, y
), i.e., the system (1.1) has a solution in
X
×
X
. This completes the proof.
We shall now proceed to prove (3.18). Let (
x, y
) be a solution of the operator equation (3.17). It follows that
x
=
λT
_{1}
(
x, y
) and
y
=
λT
_{2}
(
x, y
), i.e.,
and
It is easy to see from condition (i) that
Similarly, we get
From (3.19), using (3.21) and (3.22), we find for
t
∈ (
t_{i}
,
t
_{i}
_{+1}
] (
i
= 0, 1, 2, · · · ),
where
It follows that
or equivalently
Similarly, from (3.20) we can show that
where
Case 1.
Suppose (3.14) holds. If
by (3.24) we have
Then (3.18) holds with
If
Then, using (3.24) in (3.23) as well as (3.25) and (2.2), we get
From (3.14) we have
therefore it follows from (3.26) that
From the above discussion, we have either ∥
x
∥ ≤
W
. Substituting (3.27 into (3.24) yields
Combining (3.27) and (3.28), we have proved that (3.18) holds with
M
= max{
W
,
M
_{1}
,
M
_{1}
}.
Case 2.
Suppose (3.16) holds. If
then (3.23) implies that
Thus (3.18) holds with
If
Then, using (3.23) in (3.24) and together with (3.29) and (2.1), we find
Since
it is clear from (3.30) that
which on substituting into (3.23) gives
Coupling (3.31) and (3.32), we have shown that (3.18) holds with
M
= max{
W
′,
M
_{3}
,
M
_{4}
}. The proof is complete.
4. An example
In this section, we present an example to illustrate the main theorem.
Example 4.1.
Consider the following initial value problem of impulsive fractional differential system:
where
c
_{0}
,
b
_{0}
,
a
_{0}
,
c
_{1}
,
b
_{1}
,
a
_{1}
,
B
_{0}
and
B
_{1}
are constants.
Corresponding to system (1.1) we have

(a)

(b)

(c)are singular att= 0, m(t) = n(t) ≤L1tk=L2tlwithL1=L2= 1 and

(d)ϕ(t) =e−t=ψ(t) satisfyϕ, ψ∈L1(0, ∞), and

(e)f, g, F, G, IsandJsare defined by
Choose
Then,
σ
>
k
+ 1 and
δ
>
l
+ 1. It is easy to show that

(A)f, g, F, Gare Carathéodory functions,

(B)I(s, x, y),J(s, x, y) are Caraéodory sequences.
Furthermore, in the context of Theorem 3.1, we have Φ
^{−1}
(
x
) =
x
^{3}
with supporting function
with supporting function
ν
(
x
) =
x
^{3}
. It is easy to see that conditions (i) and (ii) in Theorem 3.1 are satisfied with
By direct computation, we get
Applying Theorem 3.1, we see that system (4.1) has at least one solution if (3.14) or (3.15) holds, i.e., if
Remark 4.1.
It is easy to see from (4.2) that system (4.1) has at least one solution for sufficiently small 
a
_{0}
, 
b
_{0}
, 
a
_{1}
, 
b
_{1}
, 
B
_{0}
 and 
B
_{1}
.
BIO
Yuji Liu
Department of Mathematics, Guangdong University of Business Studies, Guangzhou 510320, P.R.China.
email: liuyuji888@sohu.com
Patricia J.Y. Wong
School of Electrical and Electronic Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore.
email: ejywong@ntu.edu.sg
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