Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2011, Article ID 915689, 17 pages
doi:10.1155/2011/915689
Research Article
Existence of Solutions to Anti-Periodic Boundary
Value Problem for Nonlinear Fractional Differential
Equations with Impulses
Anping Chen
1, 2
and Yi Chen
2
1
Department of Mathematics, Xiangnan University, Chenzhou, Hunan 423000, China
2
School of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan 411005, China
Correspondence should be addressed to Anping Chen, [email protected]
Received 20 October 2010; Revised 25 December 2010; Accepted 20 January 2011
Academic Editor: Dumitru Baleanu
Copyright q 2011 A. Chen and Y. Chen. 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.
This paper discusses the existence of solutions to antiperiodic boundary value problem f or
nonlinear impulsive fractional differential equations. By using Banach fixed point theorem,
Schaefer fixed point theorem, and nonlinear alternative of Leray-Schauder type theorem, some
existence results of solutions are obtained. An example is given to illustrate the main result.
1. Introduction
In this paper, we consider an antiperiodic boundary value problem for nonlinear fractional
differential equations with impulses
C
D
, Δu
|
tt
k
J
k
u
t
k
,k 1, 2, ,p,
u
0
u
T
0,u
0
u
ut
k
− ut
−
k
, Δu
|
tt
k
u
t
k
− u
t
−
k
, ut
k
and ut
−
k
represent the right
and left limits of ut at t t
k
To the best of our knowledge, few papers exist in the literature devoted to the
antiperiodic boundary value problem for fractional differential equations with impulses. This
paper studies the existence of solutions of antiperiodic boundary value problem for fractional
differential equations with impulses.
The organization of this paper is as follows. In Section 2, we recall some definitions of
fractional integral and derivative and preliminary results which will be used in this paper. In
Section 3, we will consider the existence results for problem 1.1.Wegivethreeresults,the
first one is based on Banach fixed theorem, the second one is based on Schaefer fixed point
theorem, and the third one is based on the nonlinear alternative of Leray-Schauder type. In
Section 4, we will give an example to illustrate the main result.
2. Preliminaries
In this section, we present some basic notations, definitions, and preliminary results which
will be used throughout this paper.
Definition 2.1 see 4 . The Caputo fractional derivative of order α of a function f : 0, ∞ →
R is defined as
C
D
α
f
t
1
Γ
n − α
t
t
0
t − s
α−1
f
s
ds,
2.2
provided that the right side is pointwise defined on 0, ∞.
Definition 2.3 see 4. T he Riemann-Liouville fractional derivative of order α>0ofa
continuous function f : 0, ∞ → R is given by
D
α
f
t
1
Γ
n − α
,i 1, 2, ,p−1,J
p
t
p
,T. J
J\{t
1
,t
2
, ,t
p
}.
We define PCJ{u : 0,T → R | u ∈ CJ
,ut
k
and ut
−
k
exists, and ut
−
k
ut
k
, 1 ≤
k ≤ p}. Obviously, PCJ is a Banach space with the norm u sup
t∈J
|ut|.
α
C
D
α
u
t
u
t
c
0
c
1
t c
2
t
2
··· c
n−1
t
n−1
,
2.4
for some c
i
∈ R, i 0, 1, 2, ,n− 1, n α1.
Lemma 2.6 nonlinear alternative of Leray-Schauder type 21. Let E be a Banach space with
t
,t∈
0,T
,t
/
t
k
,k 1, 2, ,p,
Δu|
tt
k
I
k
u
t
k
, Δu
|
tt
k
J
k
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
1
Γ
α
t
0
t − s
α−1
y
s
ds −
1
2Γ
α
p1
i1
t
t
i−1
t
i
− s
α−2
y
s
ds
T − 2t
4Γ
α − 1
p1
i1
t
i
t
i−1
t
i
i1
J
i
u
t
i
−
1
2
p
i1
I
i
u
t
i
,t∈
0,t
1
,
1
t
i
− s
α−1
y
s
ds
−
1
2Γ
α
p1
i1
t
i
t
i−1
t
i
− s
y
s
ds
−
1
2Γ
α − 1
p
i1
T − t
i
t
i
t
i−1
t
i
− s
α−2
y
i1
t − t
i
J
i
u
t
i
−
1
2
p
i1
T − t
i
J
i
u
t
2
p
i1
I
i
u
t
i
,t∈
t
k
,t
k1
, 1 ≤ k ≤ p.
2.7
Proof. Assume that y satisfies 2.6.UsingLemma 2.5, for some constants c
0
,c
1
∈ R, we have
u
t
1
t, t ∈
0,t
1
.
2.8
Advances in Difference Equations 5
Then, we obtain
u
t
1
Γ
α − 1
t
0
t − s
α−2
y
t − s
α−1
y
s
ds − d
0
− d
1
t − t
1
,
u
t
1
Γ
α − 1
t
t
1
0
t
1
− s
α−1
y
s
ds − c
0
− c
1
t
1
,
u
t
1
−d
0
,
u
t
1
−d
1
.
2.11
In view of Δu|
tt
1
ut
1
− ut
−
1
I
1
ut
1
and Δu
|
tt
1
− s
α−1
y
s
ds − c
0
− c
1
t
1
I
1
u
t
1
,
−d
1
1
Γ
α − 1
1
Γ
α
t
t
1
t − s
α−1
y
s
ds
1
Γ
α
t
1
0
ds
t − t
1
J
1
u
t
1
I
1
u
t
1
− c
0
− c
1
t, t ∈
t
1
s
ds
1
Γ
α
k
i1
t
i
t
i−1
t
i
− s
α−1
y
s
ds
1
t − t
i
J
i
u
t
i
k
i1
I
i
u
t
i
− c
0
− c
1
t, t ∈
ds
1
Γ
α
p
i1
t
i
t
i−1
t
i
− s
α−1
y
s
ds
1
Γ
α − 1
J
i
u
t
i
p
i1
I
i
u
t
i
− c
0
− c
1
T,
u
T
i
t
i−1
t
i
− s
α−2
y
s
ds
p
i1
J
i
u
t
i
− c
1
.
2.15
ds
1
2Γ
α − 1
p
i1
T − t
i
t
i
t
i−1
t
i
− s
α−2
y
s
ds
J
i
u
t
i
1
2
p
i1
T − t
i
J
i
u
t
i
1
2
p
− s
α−2
y
s
ds
1
2
p
i1
J
i
u
t
i
.
2.16
Substituting the values of c
0
and c
1
into 2.8, 2.14, respectively, we obtain 2.7.
Conversely, we assume that u is a solution of the integral equation 2.7.Bya
direct computation, it follows that the solution given by 2.7 satisfies 2.6. The proof is
If
L
1
3p 5
T
α
2Γ
α 1
7
p 1
T
α
4Γ
α
p
3
2
L
f
s, u
s
ds
1
Γ
α
0<t
k
<t
t
k
t
k−1
t
k
− s
α−1
f
s, u
ds
1
Γ
α − 1
0<t
k
<t
t − t
k
t
k
t
k−1
t
k
− s
α−2
f
s, u
s
ds
T − 2t
4Γ
α − 1
p1
i1
t
i
t
i−1
t
i
− s
α−2
f
s, u
s
u
t
i
T − 2t
4
p
i1
J
i
u
t
i
0<t
k
<t
I
k
u
−
Tv
t
|
≤
1
Γ
α
t
t
k
t − s
α−1
f
s, u
s
− f
f
s, u
s
− f
s, v
s
ds
1
2Γ
α
p1
i1
t
i
t
i−1
0<t
k
<t
t − t
k
t
k
t
k−1
t
k
− s
α−2
f
s, u
s
− f
s, v
f
s, u
s
− f
s, v
s
ds
|
T − 2t
|
4Γ
α − 1
p1
i1
t
i
t − t
k
|
J
k
u
t
k
− J
k
v
t
k
|
1
2
p
i1
T − t
i
u
t
i
− J
i
v
t
i
|
0<t
k
<t
|
I
k
u
t
k
i
|
≤
L
1
u − v
Γ
α
t
t
k
t − s
α−1
ds
3L
1
u − v
2Γ
α
p1
i1
i
− s
α−2
ds
3p
2
L
2
u − v
7Tp
4
L
3
u − v
≤
T
α
L
1
Γ
α 1
u − v
3
p 1
T
3
u − v
L
1
3p 5
T
α
2Γ
α 1
7
p 1
T
α
4Γ
α
p
7
p 1
T
α
4Γ
α
p
3
2
L
2
7T
4
L
3
u − v. 3.4
Since
L
1
3
< 1, 3.5
consequently T is a contraction; as a consequence of Banach fixed point theorem, we deduce
that T has a fixed point which is a solution of the problem 1.1.
Theorem 3.2. Assume that
H3 the function f : J × R → R is continuous and there exists a constant N
1
> 0 such that
|ft, u|≤N
1
for each t ∈ J and all u ∈ R;
H4 the functions I
k
,J
k
: R → R are continuous and there exist constants N
2
,N
3
> 0 such
that |I
k
u|≤N
2
, |J
k
u|≤N
3
, for all u ∈ R, k 1, 2, ,p.
t
t
k
t − s
α−1
f
s, u
n
s
− f
s, u
s
ds
1
Γ
α
ds
1
2Γ
α
p1
i1
t
i
t
i−1
t
i
− s
α−1
f
s, u
n
k−1
t
k
− s
α−2
f
s, u
n
s
− f
s, u
s
ds
1
2Γ
α − 1
s, u
s
ds
10 Advances in Difference Equations
|
T − 2t
|
4Γ
α − 1
p1
i1
t
i
t
i−1
t
i
− s
α−2
u
n
t
k
− J
k
u
t
k
|
1
2
p
i1
T − t
i
|
J
i
t
i
− J
i
u
t
i
|
0<t
k
<t
|
I
k
u
n
t
k
− I
k
|
≤
1
Γ
α
t
t
k
t − s
α−1
f
s, u
n
s
− f
s, u
s
s
− f
s, u
s
ds
7T
4Γ
α − 1
p1
i1
t
i
t
i−1
t
i
− s
n
t
i
− I
i
u
t
i
|
7T
4
p
i1
|
J
i
u
n
t
i
Γ
α
t
t
k
t − s
α−1
f
s, u
s
ds
1
Γ
α
p1
i1
t
i
t
i−1
t
i
− s
α−1
f
s, u
s
ds
1
Γ
ds
1
2Γ
α − 1
p
i1
T − t
i
t
i
t
i−1
t
i
− s
α−2
f
f
s, u
s
ds
0<t
k
<t
t − t
k
|
J
k
u
t
k
|
i
u
t
i
|
0<t
k
<t
|
I
k
u
t
k
|
1
2
p
i1
|
p1
i1
t
i
t
i−1
t
i
− s
α−1
ds
7TN
1
4Γ
α − 1
p1
i1
t
i
t
7
p 1
T
α
4Γ
α
p
3
2
N
2
7T
4
N
3
.
3.8
Therefore,
Tu≤
2
7T
4
N
3
: L. 3.9
Step 3. T maps bounded sets into equicontinuous sets in PCJ.
Let Ω
r
be a bounded set of PCJ as in Step 2,andletu ∈ Ω
r
. For each t ∈ J, we can
estimate the derivative Tu
t:
Tu
t
≤
0<t
k
<t
t
k
t
k−1
t
k
− s
α−2
f
s, u
s
ds
1
2Γ
k
<t
|
J
k
u
t
k
|
1
2
p
i1
|
J
i
u
t
i
|
12 Advances in Difference Equations
≤
− s
α−2
ds
3p
2
N
3
≤
N
1
T
α−1
Γ
α
3
p 1
N
1
T
α−1
2Γ
α
∈ J, t
<t
; we have
Tu
t
−
Tu
t
t
t
{
u ∈ PC
J
: u λTu for some 0 <λ<1
}
3.12
is bounded. Let u λTu for some 0 <λ<1. Thus, f or each t ∈ J, we have
u
t
λ
Γ
α
t
t
k
t − s
α−1
f
ds
−
λ
2Γ
α
p1
i1
t
i
t
i−1
t
i
− s
α−1
f
s, u
s
ds
ds
−
λ
2Γ
α − 1
p
i1
T − t
i
t
i
t
i−1
t
i
− s
α−2
f
s, u
s
ds
λ
0<t
k
<t
t − t
k
J
k
u
t
k
−
λ
2
p
i1
T − t
i
J
k
u
t
k
−
λ
2
p
i1
I
i
u
t
i
.
3.13
Advances in Difference Equations 13
For each t ∈ J,byH3 and H4, we have
u≤N
1
3p 5
. 3.14
This shows that the set ζT is bounded. As a consequence of Schaefer fixed-point theorem,
we deduce that T has a fixed point which is a solution of the problem 1.1.
In the following theorem we give an existence result for the problem 1.1 by applying
the nonlinear alternative of Leray-Schauder type and by which the conditions H3 and H4
are weakened.
Theorem 3.3. Assume that H2 and the following conditions hold.
H5 There exists φ ∈ CJ and ψ : 0, ∞ → 0, ∞ continuous and nondecreasing such that
f
t, u
≤ φ
t
ψ
|
u
|
,t∈ J, u ∈ R. 3.15
H6 There exist ψ
∗
u
|
,u∈ R.
3.16
H7 There exists a number M
∗
> 0 such that
M
∗
φ
∗
ψ
M
∗
3p 5
T
α
/2Γ
α 1
7
p1
3.17
where φ
∗
sup{φt : t ∈ J}.
Then 1.1 has at least one solution on J.
Proof. Consider the operator T defined in Theorem 3.1. It can be easily shown that T is
continuous and completely continuous. For λ ∈ 0, 1 and each t ∈ J,letu λTu. Then
from H5 and H6, and we have
|
u
t
|
≤
1
Γ
α
t
t
k
t − s
α−1
f
s, u
s
ds
1
2Γ
α
p1
i1
t
i
t
i−1
t
i
− s
t
k−1
t
k
− s
α−2
f
s, u
s
ds
1
2Γ
α − 1
p
i1
T − t
α − 1
p1
i1
t
i
t
i−1
t
i
− s
α−2
f
s, u
s
ds
0<t
u
t
i
|
|
T − 2t
|
4
p
i1
|
J
i
u
t
i
|
0<t
k
<t
|
t
t
k
t − s
α−1
φ
s
ψ
|
u
s
|
ds
1
Γ
α
2Γ
α
p1
i1
t
i
t
i−1
t
i
− s
α−1
φ
s
ψ
|
u
s
|
ψ
|
u
s
|
ds
1
2Γ
α − 1
p
i1
T − t
i
t
i
t
i−1
t
i
t
i−1
t
i
− s
α−2
φ
s
ψ
|
u
s
|
ds
0<t
k
<t
t
i
|
|
T − 2t
|
4
p
i1
ψ
∗
|
u
t
i
|
0<t
k
<t
ψ
u
T
α
Γ
α 1
φ
∗
ψ
u
3
p 1
T
α
2Γ
α 1
φ
∗
ψ
ψ
u
3p 5
T
α
2Γ
α 1
7
p 1
T
α
4Γ
α
p
3
2
/2Γ
α 1
7
p1
T
α
/4Γ
α
p
3/2
ψ
∗
u
7T/4
ψ
4. Example
Let α 3/2, T 2π, p 1. We consider the following boundary value problem:
C
D
3/2
u
t
f
t, u
t
, 0 ≤ t ≤ 2π, t
/
1
2
,
Δu|
t1/2
I
u
1
2
0,
4.1
where
f
t, u
cos tu
t 20
2
1 u
,
t, u
∈ J ×
0, ∞
,
I
u
7
p 1
T
α
4Γ
α
p
3
2
L
2
7T
4
L
3
1
400
32
√
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˘
ınov,andP.S.Simeonov,Theory of Impulsive Differential Equations, vol. 6
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ılenko and N. A. Perestyuk, Impulsive Differential Equations, vol. 14 of World Scientific Series
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ıguez-L
´
opez, “Boundary value problems for a class of impulsive functional
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