Volume 2008, Article ID 585378,14pages 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.

Copyrightq2008 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 statessee1–4. Many literatures have been published about existence of solutions for first-order and second-order impulsive ordinary diﬀerential equations with boundary conditions5–19, 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 to20–32and 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 equations35,36. Very recently, Wang and Shen37investigated 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

*ut* *ft, ut*0*,* *t*∈*J*0*J*\ {*t*1*, . . . , tm*}*,*

Δ*utk*

*Ik*

*utk*

Δ*utk*

*I _{k}*∗

*utk*

*,* *k*1*, . . . , m,*

*u*0 *uT* 0*,* *u*0 *uT* 0*,* _{}

1.1

where*J* 0*, T*, 0*< t*1*< t*2 *<*· · ·*< tm< T*,*f* :0*, T*×*R* → *R*is continuous on*t, x*∈*J*0×*R*,

*ft _{k}, x* :lim

*t*→

*t*,

_{k}ft, x*ft*−

*:lim*

_{k}, x*t*→

*t*−

*exist,*

_{k}ft, x*ft*−

_{k}, x*ftk, x*;Δ

*utk*

*ut*−

_{k}*ut*−* _{k}*,Δ

*utk*

*ut*−

_{k}*ut*−

*;*

_{k}*Ik, I*∗∈

_{k}*CR, R*.

To the best of the authors knowledge, no one has studied the existence of solutions for impulsive antiperiodic boundary value problem1. The following Schaefer’s fixed-point theorem is fundamental in the proof of our main results.

**Lemma 1.1**see38 Schaefer. *LetE* *be a normed linear space withH* : *E* → *Ea compact*
*operator. If the set*

*S*:*x*∈*E*|*xλHx,* *for someλ*∈0*,*1 1.2

*is bounded, thenHhas at least one fixed point.*

The paper is formulated as follows. In Section 2, some definitions and lemmas are given. InSection 3, we obtain two new existence theorems by using Schaefer’s fixed point theorem. InSection 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* ∈ *J*0,*ut _{k}*,

*ut*−

*exist, and*

_{k}*ut*−

_{k}*utk, k*1*, . . . , m*},

*P C*1_{}_{J}_{{}_{u}_{:} _{J}_{→} _{R}_{:} _{u}_{is continuously di}_{ﬀ}_{erentiable for any} _{t}_{∈} _{J}

0,*ut _{k}*,

*ut*−

*exist, and*

_{k}*ut*−

_{k}*utk, k*1

*, . . . , m*}.

*P CJ*and*P C*1_{}_{J}_{}_{are Banach space with the norms}

*uP C*sup
*t*∈*J*

|*ut*|*,*

*uP C*1 max*u _{P C},u_{P C}.*

2.1

A solution to the impulsive BVP1is a function*u*∈*P C*1_{}_{J}_{}_{∩}* _{C}*2

_{}

_{J}0that satisfies1
for each*t*∈*J*.

Consider the following impulsive BVP with*λ >*0

−*ut* *λ*2*ut* *σt,* *t*∈*J*0*,*

Δ*utk*

*Ik*

*utk*

*,* *k*1*, . . . , m,*

Δ*utk*

*I _{k}*∗

*utk*

*,* *k*1*, . . . , m,*
*u*0 *uT* 0*,* *u*0 *uT* 0*,*

2.2

For convenience, we set*IkIkutk, I _{k}*∗

*I*∗

_{k}*utk*.

**Lemma 2.1.** *u*∈*P C*1_{}_{J}_{}_{∩}* _{C}*2

_{}

_{J}0*is a solution of* 2.2*if and only ifu*∈*P C*1*Jis a solution of the*

*impulsive integral equation*

*ut*

*T*

0

*Gt, sσsds*

*m*

*k*1

*Gt, tk*

−*I _{k}*∗

*Wt, tk*

*Ik* *,* 2.3

*where*

*Gt, s* 1

2*λ*

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

*e*−*λt*−*s*

1*e*−*λT* −

*eλt*−*s*

1*eλT,* 0≤*s < t*≤*T,*

*eλTt*−*s*

1*eλT* −

*e*−*λTt*−*s*

1*e*−*λT* *,* 0≤*t*≤*s*≤*T,*

*Wt, s* 1

2

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

*e*−*λt*−*s*

1*e*−*λT*

*eλt*−*s*

1*eλT,* 0≤*s < t*≤*T,*

−*eλTt*−*s*

1*eλT* −

*e*−*λTt*−*s*

1*e*−*λT* *,* 0≤*t*≤*s*≤*T.*

2.4

*Proof.* If*u*∈*P C*1_{}_{J}_{}_{∩}* _{C}*2

_{}

_{J}0is a solution of2.2, setting

*vt* *ut* *λut,* 2.5

then, by the first equation of2.2we have

*vt*−*λvt* −*σt,* *t /tk.* 2.6

Multiplying2.6by*e*−*λt*_{and integrating on}_{}_{0}_{, t}

1and*t*1*, t* *t*1*< t*≤*t*2, respectively, we get

*e*−*λt*1* _{v}_{t}*−

1

−*v*0 −

*t*1

0

*σse*−*λsds,*

*e*−*λtvt*−*e*−*λt*1_{v}_{t}

1

−

_{t}

*t*1

*σse*−*λsds,* *t*1*< t*≤*t*2*.*

2.7

So

*vt* *eλt*

*v*0−

_{t}

0

*e*−*λsσsdse*−*λt*1Δ_{v}_{t}

1

*,* *t*1*< t*≤*t*2*.* 2.8

In the same way, we can obtain that

*vt* *eλt*

*v*0−

_{t}

0

*e*−*λs _{σ}*

_{}

_{s}_{}

_{ds}_{}

0*<tk<t*

*e*−*λtk _{I}*∗

*kλIk*

where*v*0 *u*0 *λu*0. Integrating2.5, we have

*ut* *e*−*λt*

*u*0

*t*

0

*vseλsds*

0*<tk<t*

*eλtk _{I}*

*k*

*,* *t*∈*J.* 2.10

By2.9, we get

_{t}

0

*vseλsds* 1

2*λ*

*v*0*e*2*λt*−1−

_{t}

0

*e*2*λt*−*e*2*λsσse*−*λsds*

0*<tk<t*

*e*2*λt*−*e*2*λtk _{e}*−

*λtk*∗

_{I}*kλIk*

*.*

2.11

Substituting2.11into2.10, we obtain

*ut* 1

2*λ*

*λu*0−*u*0*e*−*λtu*0 *λu*0*eλt*

*t*

0

*e*−*λt*−*s*−*eλt*−*sσsds*

0*<tk<t*

*eλt*−*tk _{I}*∗

*kλIk*

−
0*<tk<t*

*e*−*λt*−*tk _{I}*∗

*k*−*λIk*

*,* *t*∈*J,*

2.12

*ut* 1

2

−*λu*0−*u*0*e*−*λtu*0 *λu*0*eλt*

−

*t*

0

*e*−*λt*−*seλt*−*sσsds*

0*<tk<t*

*eλt*−*tk _{I}*∗

*kλIk*

0*<tk<t*

*e*−*λt*−*tk _{I}*∗

*k*−*λIk*

*,* *t*∈*J.*

2.13

In view of*u*0 *uT* 0 and*u*0 *uT* 0, we have

*u*0 *λu*0 1
1*eλT*

*T*

0

*eλT*−*sσsds*−

0*<tk<T*

*eλT*−*tk _{I}*∗

*kλIk*

*,*

*λu*0−*u*0 1
1*e*−*λT*

−

_{T}

0

*e*−*λT*−*sσsds*

0*<tk<T*

*e*−*λT*−*tk _{I}*∗

*k*−*λIk*

*.*

2.14

Substituting2.14into2.12, by routine calculation, we can get2.3.

Conversely, if*u*is a solution of2.3, then direct diﬀerentiation of2.3gives−*ut*
*σt*−*λ*2*ut*,* _{t /}tk*. Moreover, we obtainΔ

*u*|

*ttk*

*Ikutk*,Δ

*u*|

*ttk*

*Ik*∗

*utk*,

*u*0

*uT*0

and*u*0 *uT* 0. Hence,*u*∈*P C*1_{}_{J}_{}_{∩}* _{C}*2

_{}

_{J}*Remark 2.2.* We call*Gt, s*above the Green function for the following homogeneous BVP:

−*ut* *λ*2*ut* 0*,* *t*∈*J,*

*u*0 *uT* 0*,* *u*0 *uT* 0*.* 2.15

Define a mapping*A*:*P C*1_{}_{J}_{} _{→} * _{P C}*1

_{}

_{J}_{}

_{by}

*Aut*

_{T}

0

*Gt, sfs, usλ*2_{u}_{}_{s}_{} _{ds}_{}*m*

*k*1

*Gt, tk*

−*I _{k}*∗

*Wt, tk*

*Ik* *,* *t*∈0*, T.*

2.16

In view ofLemma 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 problem1.
It is easy to check that

_{G}_{}_{t, s}_{}_{≤} *eλT*−1

2*λ*1*eλT,* *Wt, s*≤

1

2*.* 2.17

**Lemma 2.3.** *Ifu*∈*P C*1_{}_{J}_{}_{and}_{u}_{}_{0} _{u}_{}_{T}_{0}_{, then}

*uP C*≤

1 2

*T*

0

_{u}_{}_{s}_{}_{ds}_{}*m*

*k*1

_{Δ}_{u}

*tk*

*.* 2.18

*Proof.* Since*u*∈*P C*1_{}_{J}_{}_{, we have}

*ut* *u*0
0*<tk<t*

Δ*utk*

*t*

0

*usds.* 2.19

Set*tT*, we obtain from*u*0 *uT* 0 that

*u*0 −1
2

_{m}

*k*1

Δ*utk*

*T*

0

*usds*

*.* 2.20

Substituting2.20into2.19, we get

_{u}_{}_{t}_{}_{}_{}

1 2

*t*

0

*usds*−

_{T}

*t*

*usds*

1 2

_{}

0*<tk<t*

Δ*utk*

−

*t*≤*tk*

Δ*utk*

_{}

≤ 1

2

*t*

0

_{u}_{}_{s}_{}_{ds}_{}*T*

*t*

_{u}_{}_{s}_{}_{ds}_{}1

2

_{}

0*<tk<t*

_{Δ}_{u}

*tk*

*t*≤*tk*

_{Δ}_{u}

*tk*

1 2

*T*

0

_{u}_{}_{s}_{}_{ds}_{}*m*

*k*1

_{Δ}_{u}

*tk*

*.*

2.21

**3. Main results**

In this section, we study the existence of solutions for BVP1. For this purpose we assume
that there exist constants 0 *< η <* 1, functions *a, b, h* ∈ *CJ,*0*,*∞, and nonnegative
constants*αk, βk, γk, δkk*1*,*2*, . . . , m*such that

H1|*ft, u*| ≤*at*|*u*|*bt*|*u*|*ηht*, and

H2|*Iku*| ≤*αk*|*u*|*βk*,|*I _{k}*∗

*u*| ≤

*γk*|

*u*|

*δk*,

*k*1

*, . . . , m*

hold.

*Remark 3.1.* H1means that the nonlinearity growths at most linearly in*u*,H2implies that
the impulses areat mostlinear.

For convenience, let

*p*1
3
2

*T*

0

*atdtλ*2*T*

*m*

*i*1

*γi*

*,*

*p*2
3
2

_{T}

0

*btdt,*

*p*3 3
2

*T*

0

*htdt*

*m*

*i*1

*δi*

*.*

3.1

**Theorem 3.2.** *Suppose that conditions (H*1*and (H*2*are satisfied. Further assume that*

*T*

4*q*1

*Tq*1
2

*m*

*i*1

*αi* *m*

4

*m*

*i*1

*α*2*i* *<*1*,* 3.2

*holds, whereq*1

*T*

0*atdt*

_{m}

*i*1*p*1*αiγi* *andp*1 *as in*3.1*. Then, BVP*1*has at least one*

*solution.*

*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

*uμAu,* *μ*∈0*,*1*.* 3.3

Then,

*ut* *μAut* *μ*−*ft, ut*−*λ*2*ut* *λ*2*Aut*

−*μft, ut*−*λ*2*μ*−1*ut,*

3.4

Integrating3.4from 0 to*T*, we get that

*uT*−*u*0

_{T}

0

*utdt*

*m*

*i*1

*I _{i}*∗−

*μ*

_{T}

0

*ft, utdt*−*λ*2*μ*−1

_{T}

0

*utdt*

*m*

*i*1

*I _{i}*∗

*.*3.6

In view of*u*0 *uT* 0, we obtain by3.6that

_{u}_{}_{0}_{}_{≤} 1

2

*T*

0

_{f}

*t, utdtλ*

2

2

*T*

0

_{u}_{}_{t}_{}_{dt}_{}1
2

*m*

*i*1

* _{I}*∗

*i.* 3.7

Integrating3.4from 0 to*t*, we obtain that

*ut*−*u*0

_{t}

0

*usds*

0*<ti<t*

*I _{i}*∗−

*μ*

_{t}

0

*fs, usds*−*λ*2*μ*−1

_{t}

0

*usds*

0*<ti<t*

*I _{i}*∗

*.*3.8

From3.7and3.8, we have

_{u}_{}_{t}_{}_{≤}_{u}_{}_{0}_{}_{}*T*
0

_{f}

*s, usdsλ*2

*T*

0

_{u}_{}_{s}_{}_{ds}_{}*m*

*i*1

* _{I}*∗

*i*

≤ 3

2

*T*

0

*atutbtutηhtdt*3

2*λ*
2

*T*

0

_{u}_{}_{t}_{}_{dt}_{}3
2

*m*

*i*1

*γiuP Cδi*

≤ 3

2

*uP C*

_{T}

0

*atdtuη _{P C}*

_{T}

0

*btdt*

_{T}

0

*htdt*

3
2*λ*

2_{T}_{}_{u}_{}

*P C*

3 2

*m*

*i*1

*γiuP Cδi*

*,*

3.9

that is,

*uP C*≤

3 2

*T*

0

*atdtλ*2_{T}_{}*m*

*i*1

*γi*

*uP C*

3 2

_{T}

0

*btdtuη _{P C}*3

2

_{T}

0

*htdt* 3

2

*m*

*i*1

*δi.*

3.10

Thus,

*uP C*≤*p*1*uP Cp*2*uη _{P C}p*3

*,*3.11

where*p*1*, p*2*, p*3are as in3.1. Integrating3.5from 0 to*T*, we get that

−

_{T}

0

*ututdt*≤*μ*

_{T}

0

In view of*u*0 *uT* 0 and*u*0 *uT* 0, we have

_{T}

0

*ututdt*

_{T}

0

*utdut*

_{t}_{1}

0

*utdut*

_{t}_{2}

*t*1

*utdut*· · ·

_{T}

*tn*

*utdut*

*tut*|*t*1

0 −

_{t}_{1}

0

*ut*2*dtutut*|*t*2

*t*1−

_{t}_{2}

*t*1

*ut*2*dt*

· · ·*utut*|*Ttn*−

*T*

*tn*

*ut*2*dt*

*ut*1−0

*ut*1−0

−*u*0*u*0 *ut*2−0

*ut*2−0

−*ut*10

*ut*10

· · ·*uTuT*−*utn*0

*utn*0

−

_{T}

0

*ut*2*dt*

*ut*1−0

*ut*1−0

−*ut*10

*ut*10

· · ·*utn*−0

*utn*−0

−*utn*0

*utn*0

−

_{T}

0

*ut*2*dt*

*ut*1−0

*ut*1−0

−*ut*1−0

*ut*10

*ut*1−0

*ut*10

−*ut*10

*ut*10

· · ·*utn*−0

*utn*−0

−*utn*−0

*utn*0

*utn*−0

*utn*0

−*utn*0

*utn*0

−

*T*

0

*ut*2*dt*

−*ut*1−0*I*1∗−*ut*10*I*1− · · · −*utn*−0*In*∗−*utn*0*In*−

_{T}

0

*ut*2*dt.*

3.13

Substituting3.13into3.12, we obtain byH2*,* H3*,*and3.11that

_{T}

0

*ut*2*dt*≤*μ*

_{T}

0

*utft, utdt*−*ut*1−0

*I*_{1}∗

−*ut*10

*I*1− · · · −*u*

*tn*−0

*In*∗−*u*

*tn*0

*In*

≤*μ*

*T*

0

*utft, utdtuP C*
*m*

*i*1

* _{I}*∗

*iuP C*
*m*

*i*1

_{I}_{i}

≤

*T*

0

*atu*2*t* *btut*1*ηhtutdt*

*uP C*
*m*

*i*1

*γiuP Cδi*

*uP C*
*m*

*i*1

*αiuP Cβi*

≤ *u*2*P C*

*T*

0

*atdtu*1_{P C}η

*T*

0

*btdtuP C*

*T*

0

*htdt*

*m*

*i*1

*γiu*2*P C*
*m*

*i*1

*δiuP C*

*p*1*uP Cp*2*uη _{P C}p*3

*m*

*i*1

*αiuP Cβi*

*.*

Thus,

_{T}

0

*ut*2*dt*≤*q*1*u*2*P Cq*2*u _{P C}*1

*ηq*3

*uP Cq*4

*uη*5

_{P C}q*,*3.15

where

*q*1

_{T}

0

*atdt*

*m*

*i*1

*p*1*αiγi*

*,*

*q*2

_{T}

0

*btdtp*2

*m*

*i*1

*αi,*

*q*3

*T*

0

*htdt*

*m*

*i*1

*p*1*βip*3*αiδi*

*,*

*q*4*p*2

*m*

*i*1

*βi,*

*q*5*p*3

*m*

*i*1

*βi.*

3.16

ByLemma 2.3and3.15, we have

*u*2*P C*≤

1 4

*T*

0

_{u}_{}_{t}_{}* _{dt}*
2

1 2

*T*

0

_{u}_{}_{t}_{}_{dt}m

*i*1
|*Ii*|1

4

_{m}

*i*1
|*Ii*|

2

≤ *T*

4

_{T}

0

*ut*2*dt*

√

*T*

2

*T*

0

*ut*2*dt*

1*/*2_{}_{m}

*i*1
|*Ii*|*m*

4

*m*

*i*1
|*Ii*|2

≤ *T*

4

*q*1*u*2*P Cq*2*u*1* _{P C}ηq*3

*uP Cq*4

*uη*5

_{P C}q

√

*T*

2

*q*1*u*2* _{P C}q*2

*u*1

*3*

_{P C}ηq*uP Cq*4

*uη*5 1

_{P C}q*/*2

×*m*

*i*1

*αiuP Cβi*

*m*
4

*m*

*i*1

*α*2*iu*2*P C*2*αiβiuP Cβ*2*i*

*Tq*1

4

*Tq*1
2

*m*

*i*1

*αim*

4

*m*

*i*1

*α*2*i*

*u*2* _{P C}*· · ·

*.*

3.17

It follows from the above inequality and3.2that there exists*M*1*>*0 such that*uP C*≤*M*1.
Hence, we get by3.11that

Thus, *uP C*1 ≤ max{*M*_{1}*, M*_{2}}. It follows from Lemma 1.1 that BVP 1 has at least one
solution. The proof is complete.

**Theorem 3.3.** *Assume that (H*2*holds. Suppose that there exist a continuous and nondecreasing*

*functionψ*:0*,*∞ → 0*,*∞*and a nonnegative functionc*∈*CJwith*

_{f}_{}_{t, u}* _{λ}*2

_{u}_{≤}

_{c}_{}

_{t}_{}

_{ψ}_{}

_{|}

_{u}_{|}

_{}

_{,}

_{t}_{∈}

_{J, u}_{∈}

_{R.}_{}

_{3.19}

_{}

*Moreover suppose that*

lim sup

*u*→ ∞

*ψu*

*u* *< L* 3.20

*holds, where*

*L*: 1−

*eλT*−1*/*2*λ*1*eλTmi*1*γi*−1*/*2*mi*1*αi*

*eλT*_{−}_{1}_{/}_{2}_{λ}_{1}_{}_{e}λTT

0*csds*

*>*0*.* 3.21

*Then, BVP*1*has at least one solution.*

*Proof.* From3.20, 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*−*εvK,* *v*≥0*.* 3.23

Assume that*u*is a solution of the equation

*uμAu,* *μ*∈0*,*1*.* 3.24

Then, we have by3.19,2.17, and3.23that

_{u}_{}_{t}_{}_{}_{μ}_{}

_{T}

0

*Gt, sfs, us* *λ*2*usds*

*m*

*k*1

*Gt, tk*

−*I _{k}*∗

*Wt, tk*

*Ik*

≤ *eλT*−1

2*λ*1*eλT*

*T*

0

*csψ*|*u*|*ds* *e*

*λT*_{−}_{1}

2*λ*1*eλT*
*m*

*i*1

*γiuP Cδi*

1 2

*m*

*i*1

*αiuP Cβi*

*.*

Thus, we have

*uP C*≤ *e*
*λT*_{−}_{1}

2*λ*1*eλT*

*T*

0

*csdsL*−*εuP CK*

*eλT*_{−}_{1}

2*λ*1*eλT*
*m*

*i*1

*γi*

1 2

*m*

*i*1

*αi*

*uP C* *e*
*λT*_{−}_{1}

2*λ*1*eλT*
*m*

*i*1

*δi*

1 2

*m*

*i*1

*βi,*

3.26

that is,

*ε* *e*

*λT*_{−}_{1}

2*λ*1*eλT*

*T*

0

*csdsuP C*≤ *e*
*λT*_{−}_{1}

2*λ*1*eλTK*

*T*

0

*csds* *e*

*λT*_{−}_{1}

2*λ*1*eλT*
*m*

*i*1

*δi*1

2

*m*

*i*1

*βi,*

3.27

which implies that there exists*M*3 *>*0 such that*uP C* ≤ *M*3. By3.7,3.8, and3.23, we
get

_{u}_{}_{t}_{}_{≤} 3

2

_{T}

0

_{f}_{}_{s, u}_{}_{s}_{}_{ds}_{}3
2*λ*

2

_{T}

0

|*us*|*ds*3

2

*m*

*i*1

* _{I}*∗

*i*≤ 3 2

*0*

_{T}_{f}_{}_{s, u}_{}_{s}* _{λ}*2

_{u}_{}

_{s}_{}

_{ds}_{}

_{3}

*2*

_{λ}_{T}

0

|*us*|*ds*3

2

*m*

*i*1

* _{I}*∗

*i*≤ 3 2

*T*0

*csψ*|*us*|*ds*3*λ*2

*T*

0

|*us*|*ds*3

2

*m*

*i*1

*γiuP Cδi*

≤ 3

2

*T*

0

*csdsLuP CK*

3*λ*2*TuP C*3

2

*m*

*i*1

*γiuP Cδi*

*,*

3.28

which implies that

*uP C*≤

3*L*

2

_{T}

0

*csds*3*λ*2*T*3

2

*m*

*i*1

*γi*

*uP C*

3*K*

2

_{T}

0

*csds*3

2

*m*

*i*1

*δi*
≤ 3
2
*L*
* _{T}*
0

*csds*2*λ*2*T*

*m*

*i*1

*γi*
*M*3
3*K*
2
* _{T}*
0

*csds*3

2

*m*

*i*1

*δi*:*M*4*.*

3.29

Hence, *uP C*1 ≤ max{*M*_{3}*, M*_{4}}. It follows from Lemma 1.1that BVP 1 has at least one
solution. The proof is complete.

**4. Example**

*Example 4.1.* Consider the problem

*ut* 1

*πut*cos

2_{t}_{}1
2*e*

*t _{u}*1

*/*2

_{}

_{t}_{1}

_{}

_{tan}

_{t}_{}

_{0}

_{,}

_{t}_{∈}

0*,π*

2

\

*π*

3

*,*

Δ*ut*1

1 3sin

*ut*1

1
4*,* Δ*u*

* _{t}*
1

1
2*u*

*t*1

1
3*,* *t*1

*π*

3*,*

*u*0 *uT* 0*,* *u*0 *uT* 0*,*

4.1

Let*ft, u* 1*/πu*cos2_{t}_{1}_{/}_{2}_{}* _{e}t_{u}*1

*/*2

_{}

_{1}

_{}

_{tan}

_{t}_{,}

_{I}1*u* 1*/*3sin*u* 1*/*4,*I*_{1}∗*u* 1*/*2*u*

1*/*3,*T* *π/*2,*J* 0*, π/*2. It is easy to show that

_{f}_{}_{t, u}_{}_{≤}_{a}_{}_{t}_{}_{|}_{u}_{|}_{}_{b}_{}_{t}_{}_{|}_{u}_{|}1*/*2_{}_{h}_{}_{t}_{}_{,}_{}_{4.2}_{}

where*at* 1*/π*cos2_{t}_{,}_{b}_{}_{t}_{ }_{1}_{/}_{2}_{}_{e}t_{,}_{h}_{}_{t}_{1}_{}_{tan}_{t}_{. And}

_{I}_{1}_{}_{u}_{}_{≤} 1
3|*u*|

1

4*,* *I*

∗
1*u*≤

1
2|*u*|

1

3*.* 4.3

Thus,H1andH2hold. Obviously,*α*1 1*/*3,*β*1 1*/*4,*γ*1 1*/*2,*δ*1 1*/*3, and*m*1. Let

*λ*2_{}_{1}_{/}_{4}_{π}_{, we have}

*p*1
3
2

*T*

0

*atdt*2*λ*2*T*

*m*

*i*1

*γi*

3 2

1

4

1

4

1 2

3
2*,*

*q*1

_{T}

0

*atdt*

*m*

*i*1

*p*1*αiγi*

1

4

3 2·

1

3

1

2

5
4*.*

4.4

Therefore,

*Tq*1

4

*Tq*1
2

*m*

*i*1

*αim*

4

*m*

*i*1

*α*2* _{i}* 0

*.*7522

*<*1

*,*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 problem4.1has 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 China10771212 and the Natural Science Foundation of Jiangsu Education Oﬃce06KJB110010.

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