Hindawi Publishing Corporation
Boundary Value Problems
Volume 2008, Article ID 585378, 14 pages
doi:10.1155/2008/585378
Research Article
Antiperiodic Boundary Value
Problems for Second-Order Impulsive
Ordinary Differential Equations
Chuanzhi Bai
Department of Mathematics, Huaiyin Teachers College, Huaian, Jiangsu 223300, China
Correspondence should be addressed to Chuanzhi Bai, czbai8@sohu.com
Received 23 July 2008; Accepted 23 November 2008
Recommended by Raul F. Manasevich
We consider a second-order ordinary differential equation with antiperiodic boundary conditions
and impulses. By using Schaefer’s fixed-point theorem, some existence results are obtained.
Copyright q 2008 Chuanzhi Bai. 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
Impulsive differential equations, which arise in biology, physics, population dynamics,
economics, and so forth, are a basic tool to study evolution processes that are subjected
to abrupt in their states see 1–4. Many literatures have been published about existence
of solutions for first-order and second-order impulsive ordinary differential equations with
boundary conditions 5–19, which are important for complementing the theory of impulsive
equations. In recent years, the solvability of the antiperiodic boundary value problems
of first-order and second-order differential equations were studied by many authors, for
example, we refer to 20–32 and the references therein. It should be noted that antiperiodic
boundary value problems appear in physics in a variety of situations 33, 34. Recently,
the existence results were extended to antiperiodic boundary value problems for first-order
impulsive differential equations 35, 36. Very recently, Wang and Shen 37 investigated the
antiperiodic boundary value problem for a class of second-order differential equations by
using Schauder’s fixed point theorem and the lower and upper solutions method.
Inspired by 35–37, in this paper, we investigate the antiperiodic boundary value
problem for second-order impulsive nonlinear differential equations of the form
u t f t, ut 0, t ∈ J0 J \ {t1 , . . . , tm },
Δu tk Ik utk , k 1, . . . , m,
2
Boundary Value Problems
Δu tk Ik∗ utk ,
u0 uT 0,
k 1, . . . , m,
u 0 u T 0,
1.1
where J 0, T , 0 < t1 < t2 < · · · < tm < T , f : 0, T × R → R is continuous on t, x ∈ J0 × R,
ftk , x : limt → tk ft, x, ft−k , x : limt → t−k ft, x exist, ft−k , x ftk , x; Δutk utk −
ut−k , Δu tk u tk − u t−k ; Ik , Ik∗ ∈ CR, R.
To the best of the authors knowledge, no one has studied the existence of solutions
for impulsive antiperiodic boundary value problem 1. The following Schaefer’s fixed-point
theorem is fundamental in the proof of our main results.
Lemma 1.1 see 38 Schaefer. Let E be a normed linear space with H : E → E a compact
operator. If the set
S : x ∈ E | x λHx, for some λ ∈ 0, 1
1.2
is bounded, then H has at least one fixed point.
The paper is formulated as follows. In Section 2, some definitions and lemmas are
given. In Section 3, we obtain two new existence theorems by using Schaefer’s fixed point
theorem. In Section 4, an illustrative example is given to demonstrate the effectiveness of the
obtained results.
2. Preliminaries
In order to define the concept of solution for 1, we introduce the following spaces of
functions:
P CJ {u : J → R : u is continuous for any t ∈ J0 , utk , ut−k exist, and ut−k utk , k 1, . . . , m},
P C1 J {u : J → R : u is continuously differentiable for any t ∈ J0 , u tk , u t−k exist, and u t−k u tk , k 1, . . . , m}.
P CJ and P C1 J are Banach space with the norms
uP C sup|ut|,
t∈J
uP C1
max uP C , u P C .
2.1
A solution to the impulsive BVP 1 is a function u ∈ P C1 J ∩ C2 J0 that satisfies 1
for each t ∈ J.
Consider the following impulsive BVP with λ > 0
−u t λ2 ut σt, t ∈ J0 ,
Δu tk Ik utk , k 1, . . . , m,
Δu tk Ik∗ utk , k 1, . . . , m,
u0 uT 0,
where σ ∈ P CJ.
u 0 u T 0,
2.2
Chuanzhi Bai
3
For convenience, we set Ik Ik utk , Ik∗ Ik∗ utk .
Lemma 2.1. u ∈ P C1 J ∩ C2 J0 is a solution of 2.2 if and only if u ∈ P C1 J is a solution of the
impulsive integral equation
ut T
Gt, sσsds 0
m
G t, tk − Ik∗ W t, tk Ik ,
2.3
k1
where
⎧ −λt−s
eλt−s
e
⎪
⎪
⎪
−
,
0 ≤ s < t ≤ T,
1 ⎨ 1 e−λT 1 eλT
Gt, s 2λ ⎪
⎪ eλT t−s e−λT t−s
⎪
⎩
−
, 0 ≤ t ≤ s ≤ T,
1 eλT
1 e−λT
⎧ −λt−s
eλt−s
e
⎪
⎪
⎪
,
0 ≤ s < t ≤ T,
⎨
1 1 e−λT 1 eλT
Wt, s ⎪ eλT t−s e−λT t−s
2⎪
⎪
⎩−
−
, 0 ≤ t ≤ s ≤ T.
1 eλT
1 e−λT
2.4
Proof. If u ∈ P C1 J ∩ C2 J0 is a solution of 2.2, setting
vt u t λut,
2.5
then, by the first equation of 2.2 we have
v t − λvt −σt,
t
/ tk .
2.6
Multiplying 2.6 by e−λt and integrating on 0, t1 and t1 , t t1 < t ≤ t2 , respectively, we get
e−λt1 v t−1 − v0 −
e
−λt
vt − e
−λt1
v t1 −
t
t1
σse−λs ds,
0
2.7
σse
−λs
ds,
t1 < t ≤ t2 .
t1
So
t
−λs
−λt1
vt e v0 − e σsds e Δv t1 ,
λt
t1 < t ≤ t2 .
2.8
0
In the same way, we can obtain that
vt e
λt
v0 −
t
e
0
−λs
σsds e
0<tk <t
−λtk
Ik∗
λIk ,
t ∈ J,
2.9
4
Boundary Value Problems
where v0 u 0 λu0. Integrating 2.5, we have
ut e
−λt
u0 t
vse ds λs
0
e
λtk
Ik ,
t ∈ J.
2.10
0<tk <t
By 2.9, we get
t
2λt
2λt
1
v0 e − 1 −
vse ds e − e2λs σse−λs ds
2λ
0
0
−λtk ∗
2λt
2λtk
e −e
e
Ik λIk .
t
λs
2.11
0<tk <t
Substituting 2.11 into 2.10, we obtain
1 ut λu0 − u 0 e−λt u 0 λu0 eλt
2λ
t
−λt−s
− eλt−s σsds eλt−tk Ik∗ λIk
e
0
−
e
−λt−tk 0<tk <t
Ik∗
− λIk
2.12
0<tk <t
,
t ∈ J,
1
u t − λu0 − u 0 e−λt u 0 λu0 eλt
2
t
−λt−s
−
eλt−s σsds eλt−tk Ik∗ λIk
e
0
e
0<tk <t
−λt−tk Ik∗
− λIk
2.13
0<tk <t
,
t ∈ J.
In view of u0 uT 0 and u 0 u T 0, we have
T
1
λT −s
λT −tk ∗
e
σsds −
e
u 0 λu0 Ik λIk ,
1 eλT
0
0<tk <T
T
1
−λT −s
−λT −tk ∗
− e
λu0 − u 0 σsds e
Ik − λIk .
1 e−λT
0
0<tk <T
2.14
Substituting 2.14 into 2.12, by routine calculation, we can get 2.3.
Conversely, if u is a solution of 2.3, then direct differentiation of 2.3 gives −u t σt−λ2 ut, t / tk . Moreover, we obtain Δu|ttk Ik utk , Δu |ttk Ik∗ utk , u0uT 0
and u 0 u T 0. Hence, u ∈ P C1 J ∩ C2 J0 is a solution of 2.2.
Chuanzhi Bai
5
Remark 2.2. We call Gt, s above the Green function for the following homogeneous BVP:
−u t λ2 ut 0,
t ∈ J,
2.15
u 0 u T 0.
u0 uT 0,
Define a mapping A : P C1 J → P C1 J by
Aut T
0
m
Gt, s f s, us λ2 us ds G t, tk − Ik∗ W t, tk Ik ,
t ∈ 0, T .
k1
2.16
In view of Lemma 2.1, we easily see that u is a fixed point of operator A if and only if
u is a solution to the impulsive boundary value problem 1.
It is easy to check that
Gt, s ≤
eλT − 1
,
2λ 1 eλT
Wt, s ≤ 1 .
2
2.17
Lemma 2.3. If u ∈ P C1 J and u0 uT 0, then
uP C
1
≤
2
T
m u sds Δu tk .
0
2.18
k1
Proof. Since u ∈ P C1 J, we have
ut u0 Δu tk t
u sds.
2.19
0
0<tk <t
Set t T , we obtain from u0 uT 0 that
1
u0 −
2
m
k1
Δu tk T
u sds .
2.20
0
Substituting 2.20 into 2.19, we get
T
t
1
1
ut u sds − u sds Δu tk − Δu tk 2
2 0<tk <t
0
t
t≤tk
T
t
1
1
u sds u sds Δu tk Δu tk ≤
2
2 0<tk <t
0
t
t≤tk
T
m
1
u sds Δu tk .
2
0
k1
The proof is complete.
2.21
6
Boundary Value Problems
3. Main results
In this section, we study the existence of solutions for BVP 1. For this purpose we assume
that there exist constants 0 < η < 1, functions a, b, h ∈ CJ, 0, ∞, and nonnegative
constants αk , βk , γk , δk k 1, 2, . . . , m such that
H1 |ft, u| ≤ at|u| bt|u|η ht, and
H2 |Ik u| ≤ αk |u| βk , |Ik∗ u| ≤ γk |u| δk , k 1, . . . , m
hold.
Remark 3.1. H1 means that the nonlinearity growths at most linearly in u, H2 implies that
the impulses are at most linear.
For convenience, let
3
p1 2
T
atdt λ T 2
0
3
2
p2 3
p3 2
T
γi ,
i1
T
3.1
btdt,
0
htdt 0
m
m
δi .
i1
Theorem 3.2. Suppose that conditions (H1 and (H2 are satisfied. Further assume that
T
q1 4
holds, where q1 solution.
T
0
atdt m
m
m
T q1 m
αi α2 < 1,
2 i1
4 i1 i
i1 p1 αi
3.2
γi and p1 as in 3.1. Then, BVP 1 has at least one
Proof. It is easy to check by Arzela-Ascoli theorem that the operator A is completely
continuous. Assume that u is a solution of the equation
u μAu,
μ ∈ 0, 1.
3.3
Then,
u t μAu t μ − f t, ut − λ2 ut λ2 Aut
−μf t, ut − λ2 μ − 1ut,
−utu t μutf t, ut λ2 μ − 1u2 t ≤ μutf t, ut .
3.4
3.5
Chuanzhi Bai
7
Integrating 3.4 from 0 to T , we get that
T
u T − u 0 u tdt 0
m
Ii∗ −μ
T
f t, ut dt − λ2 μ − 1
T
0
i1
utdt 0
m
Ii∗ .
3.6
i1
In view of u 0 u T 0, we obtain by 3.6 that
1
u 0 ≤
2
T
2
f t, ut dt λ
2
0
T
m utdt 1 I ∗ .
2 i1 i
0
3.7
Integrating 3.4 from 0 to t, we obtain that
u t − u 0 t
Ii∗
0<ti <t
u sds 0
−μ
t
f s, us ds − λ2 μ − 1
t
0
usds 0
Ii∗ .
3.8
0<ti <t
From 3.7 and 3.8, we have
u t ≤ u 0 T
f s, us ds λ2
T
0
T
m usds I ∗ i
0
i1
T
m
η
3
3
atut btut ht dt λ2 utdt γi uP C δi
2
2
0
0
i1
T
T
T
m
3
3
3
η
uP C atdt uP C btdt htdt λ2 T uP C ≤
γi uP C δi ,
2
2
2 i1
0
0
0
3.9
≤
3
2
that is,
u P C
3
≤
2
T
atdt λ T 2
0
m
γi
i1
3
uP C 2
T
0
η
btdtuP C
3
2
T
htdt 0
m
3
δi .
2 i1
3.10
Thus,
η
u P C ≤ p1 uP C p2 uP C p3 ,
3.11
where p1 , p2 , p3 are as in 3.1. Integrating 3.5 from 0 to T , we get that
−
T
0
utu tdt ≤ μ
T
utft, utdt.
0
3.12
8
Boundary Value Problems
In view of u0 uT 0 and u 0 u T 0, we have
T
T
utu tdt
0
utd u t
0
t1
utd u t 0
tu t|t01 −
t1
0
t2
2
u t dt utu t|tt21 −
· · · utu t|Ttn −
utd u t
tn
t1
T
utd u t · · · T
t2
2
u t dt
t1
2
u t dt
tn
u t1 − 0 u t1 − 0 − u0u 0 u t2 − 0 u t2 − 0 − u t1 0 u t1 0
T
2
· · · uT u T − u tn 0 u tn 0 −
u t dt
0
u t1 − 0 u t1 − 0 − u t1 0 u t1 0
T
2
· · · u tn − 0 u tn − 0 − u tn 0 u tn 0 −
u t dt
0
u t1 − 0 u t1 − 0 − u t1 − 0 u t1 0 u t1 − 0 u t1 0
− u t1 0 u t1 0 · · · u tn − 0 u tn − 0 − u tn − 0 u tn 0
T
2
u tn − 0 u tn 0 − u tn 0 u tn 0 −
u t dt
0
−ut1 − 0I1∗ − u t1 0I1 − · · · − utn − 0In∗ − u tn 0In −
T
0
u t dt.
2
3.13
Substituting 3.13 into 3.12, we obtain by H2 , H3 , and 3.11 that
T
0
2
T
utf t, ut dt − u t1 − 0 I1∗
0
− u t1 0 I1 − · · · − u tn − 0 In∗ − u tn 0 In
T
m m ≤ μ utf t, ut dt uP C Ii∗ u P C Ii u t dt ≤ μ
≤
T
0
i1
0
uP C
m
m
γi uP C δi u P C
αi uP C βi
i1
≤ u2P C
i1
1η
atu2 t btut htut dt
m
i1
T
0
1η
atdt uP C
γi u2P C m
i1
T
0
i1
btdt uP C
T
htdt
0
m
η
δi uP C p1 uP C p2 uP C p3
αi uP C βi .
i1
3.14
Chuanzhi Bai
9
Thus,
T
2
1η
η
u t dt ≤ q1 u2P C q2 uP C q3 uP C q4 uP C q5 ,
0
3.15
where
q1 T
atdt 0
q2 T
m
p1 αi γi ,
i1
btdt p2
m
0
q3 T
htdt 0
q4 p2
αi ,
i1
m
p1 βi p3 αi δi ,
3.16
i1
m
βi ,
i1
q5 p3
m
βi .
i1
By Lemma 2.3 and 3.15, we have
u2P C
1
≤
4
T
0
u tdt
2
1
2
T
m
u tdt |Ii | 1
4
0
i1
m
|Ii |
2
i1
√ T
1/2 m
m
2
T
T 2
m
≤
|Ii | |Ii |2
u t dt u t dt
4 0
2
4
0
i1
i1
T
T
1η
η
q1 u2P C q2 uP C q3 uP C q4 uP C q5
4
√
1/2
T
1η
η
q1 u2P C q2 uP C q3 uP C q4 uP C q5
2
m
m
m
2
×
αi uP C βi αi u2P C 2αi βi uP C βi2
4 i1
i1
m
m
T q1 T q1
m
2
αi α u2P C · · · .
4
2 i1
4 i1 i
≤
3.17
It follows from the above inequality and 3.2 that there exists M1 > 0 such that uP C ≤ M1 .
Hence, we get by 3.11 that
η
u P C ≤ a1 M1 a2 M1 a3 : M2 .
3.18
10
Boundary Value Problems
Thus, uP C1 ≤ max{M1 , M2 }. It follows from Lemma 1.1 that BVP 1 has at least one
solution. The proof is complete.
Theorem 3.3. Assume that (H2 holds. Suppose that there exist a continuous and nondecreasing
function ψ : 0, ∞ → 0, ∞ and a nonnegative function c ∈ CJ with
ft, u λ2 u ≤ ctψ|u|,
t ∈ J, u ∈ R.
3.19
Moreover suppose that
ψu
<L
u→∞ u
lim sup
3.20
holds, where
L :
1−
m
m
eλT − 1 /2λ 1 eλT
i1 γi − 1/2 i1 αi
> 0.
λT
T
csds
e − 1 /2λ 1 eλT
0
3.21
Then, BVP 1 has at least one solution.
Proof. From 3.20, there exist 0 < ε < L and M > 0 such that
ψv ≤ L − εv,
v ≥ M.
3.22
Thus, there exists K > 0 such that
ψv ≤ L − εv K,
v ≥ 0.
3.23
Assume that u is a solution of the equation
u μAu,
μ ∈ 0, 1.
3.24
Then, we have by 3.19, 2.17, and 3.23 that
m
T
∗
ut μ Gt, s fs, us λ2 us ds G t, tk − Ik W t, tk Ik 0
k1
eλT − 1
≤ 2λ 1 eλT
T
0
csψ|u|ds m
m
1
eλT − 1 u
δ
γ
αi uP C βi .
i
P
C
i
λT
2
2λ 1 e
i1
i1
3.25
Chuanzhi Bai
11
Thus, we have
uP C
T
eλT − 1
≤ csds L − εuP C K
λT
2λ 1 e
0
m
m
m
m
eλT − 1 1
eλT − 1 1
u
γ
α
δ
βi ,
i
i
P
C
i
2 i1
2 i1
2λ 1 eλT i1
2λ 1 eλT i1
3.26
that is,
eλT − 1
ε 2λ 1 eλT
T
csdsuP C
0
eλT − 1
≤ K
2λ 1 eλT
T
csds 0
m
m
eλT − 1 1
δ
βi ,
i
2 i1
2λ 1 eλT i1
3.27
which implies that there exists M3 > 0 such that uP C ≤ M3 . By 3.7, 3.8, and 3.23, we
get
3
u t ≤
2
≤
3
2
3
≤
2
≤
3
2
T
fs, usds 3 λ2
2
0
T
T
|us|ds 0
fs, us λ2 usds 3λ2
0
T
0
T
T
0
csψ |us| ds 3λ2
T
m 3
I ∗ 2 i1 i
|us|ds m 3
I ∗ 2 i1 i
m
3
|us|ds γi uP C δi
2 i1
0
3.28
m
3
csds LuP C K 3λ2 T uP C γi uP C δi ,
2
0
i1
which implies that
m
m
3L T
3
3K T
3
csds 3λ2 T γi uP C csds δi
2 0
2 i1
2 0
2 i1
T
m
m
3
3K T
3
2
≤
γi M3 csds δi : M4 .
L csds 2λ T 2
2 0
2 i1
0
i1
u P C ≤
3.29
Hence, uP C1 ≤ max{M3 , M4 }. It follows from Lemma 1.1 that BVP 1 has at least one
solution. The proof is complete.
4. Example
In this section, we give an example to illustrate the effectiveness of our results.
12
Boundary Value Problems
Example 4.1. Consider the problem
1
1
u t utcos2 t et u1/2 t 1 tan t 0,
π
2
1
1
Δu t1 sin u t1 ,
3
4
u0 uT 0,
π
π
\
,
t ∈ 0,
2
3
1 1
Δu t1 u t1 ,
2
3
t1 π
,
3
4.1
u 0 u T 0,
Let ft, u 1/πu cos2 t 1/2et u1/2 1 tan t, I1 u 1/3 sin u 1/4, I1∗ u 1/2u 1/3, T π/2, J 0, π/2. It is easy to show that
ft, u ≤ at|u| bt|u|1/2 ht,
4.2
where at 1/πcos2 t, bt 1/2et , ht 1 tan t. And
I1 u ≤ 1 |u| 1 ,
3
4
∗ 1
I u ≤ |u| 1 .
1
2
3
4.3
Thus, H1 and H2 hold. Obviously, α1 1/3, β1 1/4, γ1 1/2, δ1 1/3, and m 1. Let
λ2 1/4π, we have
T
m
3 1 1 1
3
3
2
,
p1 atdt 2λ T γi 2
2
4
4
2
2
0
i1
T
m
1 3 1 1 5
q1 atdt p1 αi γi · .
4 2 3 2 4
0
i1
4.4
Therefore,
T q1
4
m
m
T q1 m
αi α2 0.7522 < 1,
2 i1
4 i1 i
4.5
which implies that 3.2 holds. So, all the conditions of Theorem 3.2 are satisfied. By
Theorem 3.2, antiperiod boundary value problem 4.1 has at least one solution.
Acknowledgments
The author would like to thank the referees for their valuable suggestions and comments.
This project is supported by the National Natural Science Foundation of China 10771212
and the Natural Science Foundation of Jiangsu Education Office 06KJB110010.
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