Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2009, Article ID 567329, 14 pages
doi:10.1155/2009/567329
Research Article
Antiperiodic Boundary Value Problem for
Second-Order Impulsive Differential Equations
on Time Scales
Yepeng Xing, Qiong Wang, and De Chen
Department of Applied Mathematics, Shanghai Normal University, Shanghai 200234, China
Correspondence should be addressed to Yepeng Xing, ypxing-jason@hotmail.com
Received 3 April 2009; Accepted 20 July 2009
Recommended by Alberto Cabada
We prove the existence results for second-order impulsive differential equations on time scales
with antiperiodic boundary value conditions in the presence of classical fixed point theorems.
Copyright q 2009 Yepeng Xing 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.
1. Introduction and Preliminaries
Many evolution processes are characterized by the fact that at certain moments of time
they experience a change of state abruptly. Consequently, it is natural to assume that
these perturbations act instantaneously, that is, in the form of impulses. It is known that
many biological phenomena involving threshold, bursting rhythm models in medicine and
biology, optimal control models in economics, pharmacokinetics, and frequency modulated
systems do exhibit impulse effects. The branch of modern applied analysis known as
“impulsive” differential equations provides a natural framework to mathematically describe
the aforementioned jumping processes. The reader is referred to monographs 1–4 and
references therein for some nice examples and applications to the above areas.
The study of dynamic equations on time scales goes back to Stefan Hilger 5. Now it
is still a new area of fairly theoretical exploration in mathematics. In the recent years, there
has been much progress on the qualitative properties of dynamic systems on time scales,
see 6, 7. However, there is not so much work on antiperiodic boundary value problems
for differential equations on time scales except that in 8 by Cabada In 9–12, the authors
studied antiperiodic boundary value problem for ordinary differential equations without
impulses, while in 13–18 the authors considered antiperiodic boundary value problem for
impulsive differential equations. In this paper, we study two types of second-order impulsive
2
Advances in Difference Equations
differential equations with antiperiodic boundary value conditions on time scales. Firstly, we
use Schauder’s fixed point theorem to study the existence results of the following boundary
value problem:
−xΔΔ t f t, xt, xΔ t ,
x t
k xtk Ik xtk ,
t ∈ 0, T \ Ω : {t1 , t2 , . . . , tm },
xΔ t
k xΔ tk Jk xtk , xΔ tk ,
x0 −xσT ,
k 1, 2, . . . , m,
1.1
xΔ 0 −xΔ σT .
Secondly, we employ Schaefer’s fixed point to investigate the following antiperiodic
boundary value problem on time scales:
xΔΔ t f t, xσ t, xΔ t ,
x t
k xtk Ik xtk ,
t ∈ 0, T \ {t1 , t2 , . . . , tm },
xΔ t
k xΔ tk Jk xtk , xΔ tk ,
x0 −xσT ,
k 1, 2, . . . , m,
1.2
xΔ 0 −xΔ σT .
We assume throughout this paper that f : 0, T × Rn × Rn → Rn is continuous and Ik : Rn →
Rn , Jk : Rn × Rn → Rn are also continuous k 1, 2, . . . , m.
We should mention that 1.1, 1.2 are more general than equations considered in
13, 18 since f, J both contain x and its derivative. So our result is still new even when
1.1, 1.2 reduce to equations studied in 13, 18. The paper is organized as follows. In
Section 2 we present the expression of Green’s functions of related linear operator in the space
of piecewise continuous functions. Section 3 contains the main results of the paper and is
devoted to the existence of solutions to 1.1 and 1.2.
To understand the concept of time scales and the above notation, some definitions are
useful.
A time scale is an arbitrary nonempty closed subset of the real numbers. Throughout
this paper, we will denote a time scale by the symbol T. And the forward and backward jump
operators σ, ρ : T → T are defined by
σt inf{s ∈ T : s > t},
ρt sup{s ∈ T : s < t},
1.3
respectively. The point t ∈ T is called left dense, left scattered, right dense, or right-scattered
if ρt t, ρt < t, σt t or σt > t, respectively. Points that are right dense and left dense
at the same time are called dense. We denote σσt by σ 2 t.
If T has a left scattered maximum m, define Tk : T − {m}; otherwise, set Tk T. The
symbols a, b, a, b, and so on, denote time scales intervals, for example,
a, b {t ∈ T : a ≤ t ≤ b},
where a, b ∈ T with a < ρb.
1.4
Advances in Difference Equations
3
Definition 1.1. A vector function f : T → Rn is rd-continuous provided that it is continuous
at each right dense point in T and has a left-sided limit at each left dense point in T. The set
of rd-continuous functions f : T → Rn will be denoted in this paper by Crd T Crd T, Rn .
Definition 1.2. Assume f : T → R is a function and let t ∈ Tk . Then we define f Δ t to
be the number provided it exists with the property that given any ε > 0 there exists a
neighborhood U of t i.e., U t − δ, t δ ∩ T for some δ > 0 such that
fσt − fs − f Δ tσt − s ≤ ε|σt − s|,
∀s ∈ U.
1.5
We call f Δ t the delta or Higher derivative of f at t.
Definition 1.3. A function F : T → R is called an antiderivative of f : T → R provided
F Δ t ft holds ∀t ∈ T.
1.6
Theorem 1.4 existence of antiderivatives. Every rd-continuous function has an antiderivative.
In particular if t0 ∈ T, then F defined by
Ft :
t
t0
ftΔt
for t ∈ T
1.7
is an antiderivative of f.
We assume that for ft
k , x, y : limt → t
k ft, x, y and ft−k , x, y : limt → t−k ft, x, y
both exist with ft−k , x, y ftk , x, y, k 1, 2, . . . , m. In order to define a solution of 1.1
and 1.2, we introduce and denote the Banach space P C0, T , Rn by
P C0, T ; Rn : {u : 0, T −→ Rn , u ∈ C0, T \ Ω, Rn ,
u is left continuous at t tk , the right-hand limit u t
k exists
1.8
with the norm uP C : supt∈0,T ut where · is the usual Euclidean norm and ·, · will
be the Euclidean inner product.
In a similar fashion to the above, define and denote the Banach space P C1 0, T , Rn by
P C1 0, T ; Rn : u ∈ P C0, T ; Rn : uΔ t ∈ C0, T \ Ω, Rn ,
the limits uΔ t−k , uΔ t
k exist with uΔ t−k uΔ tk 1.9
with the norm uP C1 : supt∈0,T {utP C , uΔ tP C }.
The following fixed point theorem is our main tool to prove the existence of at least
one solution to 1.2.
4
Advances in Difference Equations
Theorem 1.5 Schaefer’s fixed point theorem 19. Let X be a Banach space and let A : X → X
be a completely continuous operator. Then either:
i the operator equation x λAx has a solution for λ 1. or
ii the set S : {x ∈ X, x λAx, λ ∈0, 1} is unbounded.
2. Expression of Green’s Function
In this part, we present the expression of Green’s functions for second-order impulsive
equations with antiperiodic conditions.
Lemma 2.1. For any ht ∈ P C0, T , Rn , xt solves
−xΔΔ ht,
t ∈ 0, T \ Ω,
xΔ t
k xΔ tk Jk xtk , xΔ tk ,
x t
k xtk Ik xtk ,
k 1, 2, . . . , m,
2.1
xΔ 0 −xΔ σT ,
x0 −xσT ,
if and only if xt is the solution of integral equation
xt σT Gt, σshsΔs 0
−
m
Ht, tk Ik xtk k1
m
Δ
Gt, tk Jk xtk , x tk ,
∀t ∈ 0, σ T ,
2.2
2
k1
where
Gt, s ⎧11
⎪
⎪
σT − t s
0 ≤ s ≤ t ≤ σ 2 T ,
⎪
⎨2 2
⎪
⎪
⎪
⎩ 1 1 σT t − s
0 ≤ t < s ≤ σT ,
2 2
Ht, s ⎧
1
⎪
⎪
⎨2,
0 ≤ s ≤ t ≤ σ 2 T ,
⎪
⎪
⎩− 1 ,
2
0 ≤ t < s ≤ σT .
G
H
Proof. Assume xt is a solution of 2.1. Then by integrating −xΔΔ ht, t / tk , k 1, 2, . . . , m step by step from 0 to t we have
Δ
Δ
x t x 0 −
t
0
hsΔs xΔ t
k − xΔ tk ,
0<tk <t
∀t ∈ 0, σT .
2.3
Advances in Difference Equations
5
Similarly, we Integrate 2.3 from 0 to t step by step to get
xt x0 t
xΔ sΔs x t
k − xtk ,
∀t ∈ 0, σ 2 T .
2.4
0<tk <t
0
Substituting 2.3 into 2.4, we obtain
Δ
xt x0 x 0t −
t
t − σshsΔs
0
Δ x t
k − xtk x tk − xΔ tk t − tk ,
0<tk <t
∀t ∈ 0, σ 2 T .
2.5
0<tk <t
By xΔ 0 −xΔ σT and xΔ t
k xΔ tk Jk xtk , xΔ tk , we have from 2.3 that
xΔ 0 1
2
σT 0
hsΔs −
m
1
Jk xtk , xΔ tk .
2 k1
2.6
Note that x0 −xσT , xt
k xtk Ik xtk and 2.6 . It then follows from 2.4 that
σT x0 −
4
1
2
σT σT 0
m
hsΔs − Jk xtk , xΔ tk 0
k1
m
m
1
1
Jk xtk , xΔ tk σT − tk −
Ik xtk .
σT − shsΔs−
2 k1
2 k1
2.7
Then we substitute 2.6, 2.7 into 2.5 to obtain
xt t
1
1
1
σT − σs t hsΔs − t − σshsΔs
4
2
2
0
0
m
1
1
1
Δ
Jk xtk , x tk − σT tk − t
4
2
2
k1
σT 2.8
m
1
Ik xtk Ik xtk .
t − tk Jk xtk , xΔ tk −
2 k1
0<tk <t
0<tk <t
If we choose Gt, s, Ht, s as GH, then 2.2 is true.
Now assume xt is a solution of 2.2. Then for t /
tk , k 1, 2, . . . , m. we have by 6,
Theorem 1.117, page 46 that
Δ
x t T
0
GΔ t, σshsΔs m
k1
H Δ t, tk Ik xtk −
m
k1
GΔ t, tk Ji xtk , xΔ tk ,
2.9
6
Advances in Difference Equations
where GΔ , f Δ denotes the derivatives of G, H with respect to the first variable. That is
1
x t −
2
Δ
t
1
hsΔs 2
0
σT t
hsΔs m
Jk xtk , xΔ tk .
2.10
k1
Compute straightforwardly to get
xΔΔ t −ht.
2.11
Since
m
m
1
Ik xtk Ik xtk ,
2 k1
0<tk <t
Ht, tk Ik xtk −
k1
2.12
we have
m
m
H t
k , tk Ik xtk − Htk , tk Ik xtk k1
k1
k
m
k−1
m
1 1 Ii xti −
Ii xti −
Ii xti − Ii xti Ik xtk .
2 i0
2 i0
ik
1
ik
2.13
Similarly,
m
m
G t
k , tk Jk xtk , xΔ tk − Gtk , tk Jk xtk , xΔ tk 0,
k1
m
k1
k1
m
GΔ t
k , tk Jk xtk , xΔ tk − GΔ tk , tk Jk xtk , xΔ tk −Jk xtk , xΔ tk .
2.14
k1
Thus, the proof is completed.
By Lemma 2.1, we have for every t ∈ 0, σT max
|Gt, σs| ≤
t,s∈0,σT ×0,T 3
σT .
4
2.15
If t σ 2 T > σT , then
1 G σ 2 T , σs ≤ σT 2σ 2 T .
4
2.16
Advances in Difference Equations
7
Therefore we have the upper bounds
max
|Gt, σs| ≤
t,s∈0,σ 2 T ×0,σ T 1
σT 2σ 2 T : G0 ,
4
|Ht, s| max
t,s∈0,σ 2 T ×0,σ T 1
,
2
2.17
1
Δ
G t, s .
2
t,s∈0,σ 2 T ×0,σ T max
Recall that a mapping between Banach spaces is compact if it is continuous and carries
bounded sets into relatively compact sets.
Lemma 2.2. Suppose that f : 0, T × Rn × Rn → Rn , Ik : Rn → Rn , and Jk : Rn × Rn → Rn are
continuous. Define an operator A : P C1 0, σ 2 T , Rn → P C1 0, σ 2 T , Rn as
Axt :
σT m
Gt, σsf s, xs, xΔ s Δs Ht, tk Ik xtk 0
−
m
k1
Δ
2.18
Gt, tk Jk xtk , x tk ,
k1
where Gt, s and Ht, s are as given in Lemma 2.1. Then A is a compact map.
Proof. From 2.10 we know
1
Ax t −
2
Δ
t m
1 σT Δ
f s, xs, x s Δs f s, xs, xΔ s Δs Jk xtk , xΔ tk .
2 t
0
k1
2.19
Then the continuality of f, Ik , Jk implies A is a continuous map from P CJ, Rn to P CJ, Rn .
On the other hand, for any bounded subset S ⊂ P CJ, Rn . 2.19 implies {AxΔ t | xt ∈ S}
is also a bounded subset of P CJ, Rn . Deducing in a similar way as proving 20, Lemma 2.4,
we have A is a compact map.
Lemma 2.3. x ∈ P C1 J, Rn ∩ P C2 J, Rn is a solution of 1.1 if and only if xt ∈ P C1 J, Rn is
a fixed point of A.
Proof. It can be easily obtain from Lemma 2.1, and we omit the proof here.
3. Main Results
In this section, we prove the existence results for 1.1, 1.2 in presence of Schauder’s fixed
point theorem and Schaefer’s fixed point theorem, respectively.
8
Advances in Difference Equations
We first set
f t, x, y ,
max
t∈0,T x y α
lim
x
y → ∞
Ik x
βk lim
k 1, 2, . . . , m,
x → ∞
x
Jk x, y γk lim
k 1, 2, . . . , m.
x y
x
y → ∞
3.1
Theorem 3.1. Suppose that f : 0, T × Rn × Rn → Rn , Ik : Rn → Rn and Jk : Rn × Rn → Rn are
continuous k 1, 2, . . . , m. If
η max η1 , η2 < 1,
3.2
where
m
m
1
1
ασT γk σT 2σ 2 T βk ,
η1 2
2 k1
k1
m
η2 ασT γk .
3.3
k1
then 1.1 has at least one solution in P C1 J, Rn ∩ P C2 J, Rn .
Proof. By Lemma 2.2 it is sufficient to show that A has at least one solution in P C1 J, Rn .
First, we can choose by 3.2 α > α, βk > βk , γk > γk , k 1, 2, . . . , m such that
η1
m
m
1
1 α σT γk σT 2σ 2 T βk < 1,
2
2
k1
k1
η2 m
α σT γk < 1.
3.4
k1
By 3.1 we can choose a positive number N such that
f t, x, y < α x y ,
∀t ∈ 0, σT , x y ≥ N.
3.5
Then
f t, x, y ≤ α x y M,
∀t ∈ 0, σT , x, y ∈ RN ,
3.6
Advances in Difference Equations
9
where
M
max
t∈0,σT ,x
y≤N
f t, x, y < ∞.
3.7
Similarly, there exist positive constants Mk , Mk , k 1, 2, . . . , m such that
Ik xβk x Mk , ∀x ∈ Rn ,
Jk x, y ≤ γ x y Mk ,
k
3.8
∀x, y ∈ Rn .
3.9
It then follows by 2.17,2.18,3.6–3.9 that
1
σT σT 2σ 2 T α xt xΔ t M
4
m
m
1 β x Mk
2 k1 k
k1
m
m
1
Δ 2
γk xt x t Mk
σT σT 2σ T 4
k1
k1
≤ η1 xP C1 M1 , ∀t ∈ 0, σ 2 T ,
3.10
m
m
1 1
2
σT σT 2σ T M Mk Mk
4
2 k1
k1
3.11
Axt ≤
where
M
1
is a constant.
Similarly, we can prove that
AxΔ t ≤ η2 xP C1 M2 ,
∀t ∈ 0, σ 2 T ,
3.12
where M2 1/2M m
k1 Mk is a constant.
Consequently, by 3.10 and 3.12 we have
AxP C1 ≤ η xP C1 M ,
∀x ∈ P C1 0, σ 2 T , Rn ,
3.13
M max M1 , M2 .
3.14
where
η max η1 , η2 < 1,
10
Advances in Difference Equations
Thus we can choose r > 0 such that
ABr ⊂ Br ,
3.15
where Br {x ∈ P C1 0, σ 2 T , Rn | xP C1 ≤ r}. On the other hand, from Lemma 2.3 A is a
complete continuous operator. Hence by Schauder’s fixed point theorem A has fixed point in
Br . This completes the proof.
Remark 3.2. Assumption 3.2 is true if there hold
f t, x, y ⇒ 0,
x y
Jk x, y −→ 0,
x y
Ik x
−→ 0,
x
as p q −→ ∞,
3.16
as x −→ ∞ k 1, 2, . . . , m.
We now go on to study the existence results for 1.2. We should mention that the idea
used in the following theorem is initiated by Tisdell 21, 22.
Theorem 3.3. Suppose that f : 0, T × Rn × Rn → Rn , Ik : Rn → Rn and Jk : Rn × Rn → Rn are
continuous k 1, 2, . . . , m. If there exist nonnegative constants γ, δk , ζk , Lk , Nk and M such that
f t, x, y ≤ γ x, f t, x, y y2 M,
∀ t, x, y ∈ 0, σT \ Ω × Rn × Rn ,
Ik x ≤ δk x Lk , Jk x, y ≤ ζk x y Nk ,
"
!
m
m
m
1
max
δi 2G0 ζi , ζi < 1,
2 k1
i1
i1
3.17
∀x, y ∈ Rn ,
3.18
where · is the Euclidean inner product. Then 1.2 has at least one solution.
Proof. Consider the mapping
A : P C1
Axt σT 0, σ 2 T , Rn −→ P C1 0, σ 2 T , Rn
3.19
m
Gt, σsf s, xσ s, xΔ s Δs Ht, tk Ik xtk 0
m
Gt, tk Jk xtk , xΔ tk ,
k1
3.20
k1
where
⎧ 1 1
⎪
⎪
−
σT
−
t
s
⎪
⎨ 2 2
Gt, s ⎪
1 1
⎪
⎪
⎩−
σT t − s
2 2
0 ≤ s ≤ t ≤ σ 2 T ,
G∗ 0 ≤ t < s ≤ σT ,
Advances in Difference Equations
11
Δ
Ht, s G t, s
⎧
1
⎪
⎪
⎨2,
0 ≤ s ≤ t ≤ σ 2 T ,
⎪
⎪
⎩− 1 ,
2
0 ≤ t < s ≤ σT .
H ∗ In a similar way as we prove Lemmas 2.1, 2.2 and 2.3 we obtain
i x ∈ P C1 J, Rn ∩ P C2 J, Rn is a solution of 1.2 if and only if xt ∈ P C1 J, Rn is a
fixed point of A;
ii A is a compact operator.
Consider the equation
x Ax.
3.21
To show A has at least one fixed point, we apply Schaefer’s theorem by showing that all
potential solutions to
x λAx,
λ ∈ 0, 1
3.22
are bounded a priori, with the bound being independent of λ. With this in mind, let xt be a
solution of 3.22. Note that xt is also a solution to
xΔΔ t λf t, xσ t, xΔ t ,
t/
tk , k 1, 2, . . . , m,
x t
k xtk λIk xtk , xΔ t
k xΔ tk λJk xtk , xΔ tk ,
x0 −xσT ,
k 1, 2, . . . , m,
3.23
xΔ 0 −xΔ σT .
On one hand, we see that for λ ∈ 0, 1
# $
&
2 % σ
Δ
λf t, x t, x t ≤ λ γ xσ t, f t, xσ t, xΔ t xΔ t M
γ
γ
≤γ
$
$
σ
σ
Δ
x t, λf t, x t, x t
%
Δ 2
λx t λM
%
xσ t, xΔΔ t λxΔ t, xΔ t λM
$
3.24
%
xσ t, xΔΔ t xΔ t, xΔ t M
$
%Δ
γ xt, xΔ t M.
On the other hand, by the antiperiodic boundary condition we have
σT $
0
% $
%
$
%Δ xt, x t
Δt xσT , xΔ σT − x0, xΔ 0 0.
Δ
3.25
12
Advances in Difference Equations
It therefore follows that
σT λf t, xσ t, xΔ t Δt ≤ MσT .
3.26
0
Consequently,
xt λAxt
σT Gt, σsλf s, xσ s, xΔ s Δs
0
m
m
Δ
Ht, tk λIk xtk Gt, tk λJk xtk , x tk k1
k1
≤ G0 MσT m
m
m
m 1
1
Lk G0 Nk δi xti G0 ζi xti xΔ ti 2 k1
2
i1
k1
k1
m
m
m
m
1
1
≤ G0 MσT Lk G0 Nk δi 2G0 ζi xP C1 .
2 k1
2 k1
i1
k1
3.27
Differentiating both sides of 3.11, we can easily have
m
m
1
Δ 1
Nk ζi xP C1 .
x t ≤ MσT 2
2 k1
i1
3.28
Choose
"
m
m
1
1
max G0 MσT Lk G0 Nk , MσT ,
2 k1
2
k1
!
N
1
"
m
m
m
1
max
δi 2G0 ζi , ζi .
2 k1
i1
i1
!
N 2
3.29
Thus we have by 3.18 that
xP C1 ≤
Then the proof is completed.
N 1
.
1 − N 2
3.30
Advances in Difference Equations
13
Example 3.4. Consider antiperiodic value problem on T uΔΔ t uσt uΔ t2 cos t,
1
u1
u1 u1 4,
8
u0 −u3,
'∞
k0 3k
'
1, 3k 2 {3k 2}
t ∈ 0, 1, t /
1,
1
uΔ 1
uΔ 1 − u1 10,
8
3.31
uΔ 0 −uΔ 3.
We claim 4.2 has at least one solution.
Proof. Let T 3 and ft, x, y x xy2 cos t in Theorem 3.3. Choosing M 4, we have for
'
t, x, y ∈ 0, 1 {2, 3} × R2 that
f t, x, y |x| |x|y2 1,
x, f t, x, y y2 x2 x2 y2 y2 x cos t ≥ x2 x2 y2 y2 − |x|.
3.32
Since minv≥0 {v2 − 2v} ≥ −1, we have x2 y2 y2 − |x|y2 y2 x2 − |x| 1 > 0. Thus, for
γ 1 and M 2
γ x, f t, x, y y2 4 ≥ f t, x, y ,
∀ t, x, y ∈ 0, 1 × R2 .
3.33
Moreover, G0 9/4, 1/2δ1 2G0 ξ1 1/2 × 1/8 2 × 9/4 × 1/8 5/8 < 1, Then the
conclusion follows from Theorem 3.3.
Acknowlegments
The research was Supported by Leading Academic Discipline Project of Shanghai Normal
University DZL805, the National Natural Science Foundation of China 10671127, the
National Ministry of Education of China 20060270001, and the National Natural Science
Foundation of Shanghai 08ZR1416000.
References
1 D. Baı̆nov and P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, vol. 66
of Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Scientific & Technical,
Harlow, UK, 1993.
2 M. Benchohra, J. Henderson, and S. Ntouyas, Impulsive Differential Equations and Inclusions, vol. 2 of
Contemporary Mathematics and Its Applications, Hindawi Publishing Corporation, New York, NY, USA,
2006.
3 V. Lakshmikantham, D. D. Baı̆nov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6
of Series in Modern Applied Mathematics, World Scientific Publishing, Teaneck, NJ, USA, 1989.
4 A. M. Samoı̆lenko and N. A. Perestyuk, Impulsive Differential Equations, vol. 14 of World Scientific Series
on Nonlinear Science. Series A: Monographs and Treatises, World Scientific, River Edge, NJ, USA, 1995.
5 S. Hilger, “Analysis on measure chains—a unified approach to continuous and discrete calculus,”
Results in Mathematics, vol. 18, no. 1-2, pp. 18–56, 1990.
6 M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applcations,
Birkhäuser, Boston, Mass, USA, 2001.
14
Advances in Difference Equations
7 M. Bohner and A. Peterson, Eds., Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston,
Mass, USA, 2003.
8 A. Cabada and D. R. Vivero, “Existence and uniqueness of solutions of higher-order antiperiodic
dynamic equations,” Advances in Difference Equations, no. 4, pp. 291–310, 2004.
9 S. Aizicovici, M. McKibben, and S. Reich, “Anti-periodic solutions to nonmonotone evolution
equations with discontinuous nonlinearities,” Nonlinear Analysis: Theory, Methods & Applications, vol.
43, no. 2, pp. 233–251, 2001.
10 Y. Chen, J. J. Nieto, and D. O’Regan, “Anti-periodic solutions for fully nonlinear first-order differential
equations,” Mathematical and Computer Modelling, vol. 46, no. 9-10, pp. 1183–1190, 2007.
11 T. Jankowski, “Ordinary differential equations with nonlinear boundary conditions of antiperiodic
type,” Computers & Mathematics with Applications, vol. 47, no. 8-9, pp. 1419–1428, 2004.
12 K. Wang and Y. Li, “A note on existence of anti-periodic and heteroclinic solutions for a class of
second-order odes,” Nonlinear Analysis: Theory, Methods and Applications, vol. 70, no. 4, pp. 1711–1724,
2009.
13 C. Bai, “Antiperiodic boundary value problems for second-order impulsive ordinary differential
equations,” Boundary Value Problems, vol. 2008, Article ID 585378, 14 pages, 2008.
14 A. Cabada, E. Liz, and S. Lois, “Green’s function and maximum principle for higher order ordinary
differential equations with impulses,” The Rocky Mountain Journal of Mathematics, vol. 30, no. 2, pp.
435–446, 2000.
15 D. Franco and J. J. Nieto, “First-order impulsive ordinary differential equations with anti-periodic
and nonlinear boundary conditions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 42, no. 2,
pp. 163–173, 2000.
16 Z. Luo, J. Shen, and J. J. Nieto, “Antiperiodic boundary value problem for first-order impulsive
ordinary differential equations,” Computers & Mathematics with Applications, vol. 49, no. 2-3, pp. 253–
261, 2005.
17 J. J. Nieto and C. C. Tisdell, “Existence and uniqueness of solutions to first-order systems of nonlinear
impulsive boundary-value problems with sub-, super-linear or linear growth,” Electronic Journal of
Differential Equations, no. 105, pp. 1–14, 2007.
18 Y. Xing and V. Romanovski, “On the solvability of second-order impulsive differential equations with
antiperiodic boundary value conditions,” Boundary Value Problems, vol. 2008, Article ID 864297, 18
pages, 2008.
19 D. R. Smart, Fixed Point Theorems, Cambridge University Press, New York, NY, USA, 1980.
20 Y. Xing, M. Han, and G. Zheng, “Initial value problem for first-order integro-differential equation of
Volterra type on time scales,” Nonlinear Analysis: Theory, Methods & Applications, vol. 60, no. 3, pp.
429–442, 2005.
21 J. Chen, C. C. Tisdell, and R. Yuan, “On the solvability of periodic boundary value problems with
impulse,” Journal of Mathematical Analysis and Applications, vol. 331, no. 2, pp. 902–912, 2007.
22 M. Rudd and C. C. Tisdell, “On the solvability of two-point, second-order boundary value problems,”
Applied Mathematics Letters, vol. 20, no. 7, pp. 824–828, 2007.