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, ut0, t∈J0J\ {t1, . . . , tm},
Δutk
Ik
utk
Δutk
Ik∗utk
, k1, . . . , m,
u0 uT 0, u0 uT 0,
1.1
whereJ 0, T, 0< t1< t2 <· · ·< tm< T,f :0, T×R → Ris continuous ont, x∈J0×R,
ftk, x :limt→tkft, x,ft−k, x :limt→t−kft, xexist,ft−k, x ftk, x;Δutk utk−
ut−k,Δutk utk−ut−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 problem1. The following Schaefer’s fixed-point theorem is fundamental in the proof of our main results.
Lemma 1.1see38 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 : uis continuous for anyt ∈ J0,utk,ut−kexist, andut−k
utk, k1, . . . , m},
P C1J {u : J → R : uis continuously differentiable for any t ∈ J
0,utk,ut−k exist, andut−k utk, k 1, . . . , m}.
P CJandP C1Jare Banach space with the norms
uP Csup t∈J
|ut|,
uP C1 maxuP C,uP C.
2.1
A solution to the impulsive BVP1is a functionu∈P C1J∩C2J
0that satisfies1 for eacht∈J.
Consider the following impulsive BVP withλ >0
−ut λ2ut σt, t∈J0,
Δutk
Ik
utk
, k1, . . . , m,
Δutk
Ik∗utk
, k1, . . . , m, u0 uT 0, u0 uT 0,
2.2
For convenience, we setIkIkutk, Ik∗Ik∗utk.
Lemma 2.1. u∈P C1J∩C2J
0is a solution of 2.2if and only ifu∈P C1Jis a solution of the
impulsive integral equation
ut
T
0
Gt, sσsds
m
k1
Gt, tk
−Ik∗Wt, tk
Ik , 2.3
where
Gt, s 1
2λ
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
e−λt−s
1e−λT −
eλt−s
1eλT, 0≤s < t≤T,
eλTt−s
1eλT −
e−λTt−s
1e−λT , 0≤t≤s≤T,
Wt, s 1
2
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
e−λt−s
1e−λT
eλt−s
1eλT, 0≤s < t≤T,
−eλTt−s
1eλT −
e−λTt−s
1e−λT , 0≤t≤s≤T.
2.4
Proof. Ifu∈P C1J∩C2J
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
Multiplying2.6bye−λtand integrating on0, t
1andt1, t t1< t≤t2, respectively, we get
e−λt1vt−
1
−v0 −
t1
0
σse−λsds,
e−λtvt−e−λt1vt
1
−
t
t1
σse−λsds, t1< t≤t2.
2.7
So
vt eλt
v0−
t
0
e−λsσsdse−λt1Δvt
1
, t1< t≤t2. 2.8
In the same way, we can obtain that
vt eλt
v0−
t
0
e−λsσsds
0<tk<t
e−λtkI∗
kλIk
wherev0 u0 λu0. Integrating2.5, we have
ut e−λt
u0
t
0
vseλsds
0<tk<t
eλtkI
k
, t∈J. 2.10
By2.9, we get
t
0
vseλsds 1
2λ
v0e2λt−1−
t
0
e2λt−e2λsσse−λsds
0<tk<t
e2λt−e2λtke−λtkI∗
kλIk
.
2.11
Substituting2.11into2.10, we obtain
ut 1
2λ
λu0−u0e−λtu0 λu0eλt
t
0
e−λt−s−eλt−sσsds
0<tk<t
eλt−tkI∗
kλIk
− 0<tk<t
e−λt−tkI∗
k−λIk
, t∈J,
2.12
ut 1
2
−λu0−u0e−λtu0 λu0eλt
−
t
0
e−λt−seλt−sσsds
0<tk<t
eλt−tkI∗
kλIk
0<tk<t
e−λt−tkI∗
k−λIk
, t∈J.
2.13
In view ofu0 uT 0 andu0 uT 0, we have
u0 λu0 1 1eλT
T
0
eλT−sσsds−
0<tk<T
eλT−tkI∗
kλIk
,
λu0−u0 1 1e−λT
−
T
0
e−λT−sσsds
0<tk<T
e−λT−tkI∗
k−λIk
.
2.14
Substituting2.14into2.12, by routine calculation, we can get2.3.
Conversely, ifuis a solution of2.3, then direct differentiation of2.3gives−ut σt−λ2ut,t /tk. Moreover, we obtainΔu|ttk Ikutk,Δu|ttk Ik∗utk,u0uT 0
andu0 uT 0. Hence,u∈P C1J∩C2J
Remark 2.2. We callGt, sabove the Green function for the following homogeneous BVP:
−ut λ2ut 0, t∈J,
u0 uT 0, u0 uT 0. 2.15
Define a mappingA:P C1J → P C1Jby
Aut
T
0
Gt, sfs, usλ2us dsm
k1
Gt, tk
−Ik∗Wt, tk
Ik , t∈0, T.
2.16
In view ofLemma 2.1, we easily see thatuis a fixed point of operatorAif and only if
uis a solution to the impulsive boundary value problem1. It is easy to check that
Gt, s≤ eλT−1
2λ1eλT, Wt, s≤
1
2. 2.17
Lemma 2.3. Ifu∈P C1Jandu0 uT 0, then
uP C≤
1 2
T
0
usdsm
k1
Δu
tk
. 2.18
Proof. Sinceu∈P C1J, we have
ut u0 0<tk<t
Δutk
t
0
usds. 2.19
SettT, we obtain fromu0 uT 0 that
u0 −1 2
m
k1
Δutk
T
0
usds
. 2.20
Substituting2.20into2.19, we get
ut
1 2
t
0
usds−
T
t
usds
1 2
0<tk<t
Δutk
−
t≤tk
Δutk
≤ 1
2
t
0
usdsT
t
usds1
2
0<tk<t
Δu
tk
t≤tk
Δu
tk
1 2
T
0
usdsm
k1
Δu
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 constantsαk, βk, γk, δkk1,2, . . . , msuch that
H1|ft, u| ≤at|u|bt|u|ηht, and
H2|Iku| ≤αk|u|βk,|Ik∗u| ≤γk|u|δk,k1, . . . , m
hold.
Remark 3.1. H1means that the nonlinearity growths at most linearly inu,H2implies that the impulses areat mostlinear.
For convenience, let
p1 3 2
T
0
atdtλ2T
m
i1
γi
,
p2 3 2
T
0
btdt,
p3 3 2
T
0
htdt
m
i1
δi
.
3.1
Theorem 3.2. Suppose that conditions (H1and (H2are satisfied. Further assume that
T
4q1
Tq1 2
m
i1
αi m
4
m
i1
α2i <1, 3.2
holds, whereq1
T
0atdt
m
i1p1αiγi andp1 as in3.1. Then, BVP1has at least one
solution.
Proof. It is easy to check by Arzela-Ascoli theorem that the operator A is completely continuous. Assume thatuis a solution of the equation
uμAu, μ∈0,1. 3.3
Then,
ut μAut μ−ft, ut−λ2ut λ2Aut
−μft, ut−λ2μ−1ut,
3.4
Integrating3.4from 0 toT, we get that
uT−u0
T
0
utdt
m
i1
Ii∗−μ
T
0
ft, utdt−λ2μ−1
T
0
utdt
m
i1
Ii∗. 3.6
In view ofu0 uT 0, we obtain by3.6that
u0≤ 1
2
T
0
f
t, utdtλ
2
2
T
0
utdt1 2
m
i1
I∗
i. 3.7
Integrating3.4from 0 tot, we obtain that
ut−u0
t
0
usds
0<ti<t
Ii∗−μ
t
0
fs, usds−λ2μ−1
t
0
usds
0<ti<t
Ii∗. 3.8
From3.7and3.8, we have
ut≤u0T 0
f
s, usdsλ2
T
0
usdsm
i1
I∗
i
≤ 3
2
T
0
atutbtutηhtdt3
2λ 2
T
0
utdt3 2
m
i1
γiuP Cδi
≤ 3
2
uP C
T
0
atdtuηP C
T
0
btdt
T
0
htdt
3 2λ
2Tu
P C
3 2
m
i1
γiuP Cδi
,
3.9
that is,
uP C≤
3 2
T
0
atdtλ2Tm
i1
γi
uP C
3 2
T
0
btdtuηP C3
2
T
0
htdt 3
2
m
i1
δi.
3.10
Thus,
uP C≤p1uP Cp2uηP Cp3, 3.11
wherep1, p2, p3are as in3.1. Integrating3.5from 0 toT, we get that
−
T
0
ututdt≤μ
T
0
In view ofu0 uT 0 andu0 uT 0, we have
T
0
ututdt
T
0
utdut
t1
0
utdut
t2
t1
utdut· · ·
T
tn
utdut
tut|t1
0 −
t1
0
ut2dtutut|t2
t1−
t2
t1
ut2dt
· · ·utut|Ttn−
T
tn
ut2dt
ut1−0
ut1−0
−u0u0 ut2−0
ut2−0
−ut10
ut10
· · ·uTuT−utn0
utn0
−
T
0
ut2dt
ut1−0
ut1−0
−ut10
ut10
· · ·utn−0
utn−0
−utn0
utn0
−
T
0
ut2dt
ut1−0
ut1−0
−ut1−0
ut10
ut1−0
ut10
−ut10
ut10
· · ·utn−0
utn−0
−utn−0
utn0
utn−0
utn0
−utn0
utn0
−
T
0
ut2dt
−ut1−0I1∗−ut10I1− · · · −utn−0In∗−utn0In−
T
0
ut2dt.
3.13
Substituting3.13into3.12, we obtain byH2, H3,and3.11that
T
0
ut2dt≤μ
T
0
utft, utdt−ut1−0
I1∗
−ut10
I1− · · · −u
tn−0
In∗−u
tn0
In
≤μ
T
0
utft, utdtuP C m
i1
I∗
iuP C m
i1
Ii
≤
T
0
atu2t btut1ηhtutdt
uP C m
i1
γiuP Cδi
uP C m
i1
αiuP Cβi
≤ u2P C
T
0
atdtu1P Cη
T
0
btdtuP C
T
0
htdt
m
i1
γiu2P C m
i1
δiuP C
p1uP Cp2uηP Cp3
m
i1
αiuP Cβi
.
Thus,
T
0
ut2dt≤q1u2P Cq2uP C1ηq3uP Cq4uηP Cq5, 3.15
where
q1
T
0
atdt
m
i1
p1αiγi
,
q2
T
0
btdtp2
m
i1
αi,
q3
T
0
htdt
m
i1
p1βip3αiδi
,
q4p2
m
i1
βi,
q5p3
m
i1
βi.
3.16
ByLemma 2.3and3.15, we have
u2P C≤
1 4
T
0
utdt 2
1 2
T
0
utdtm
i1 |Ii|1
4
m
i1 |Ii|
2
≤ T
4
T
0
ut2dt
√
T
2
T
0
ut2dt
1/2m
i1 |Ii|m
4
m
i1 |Ii|2
≤ T
4
q1u2P Cq2u1P Cηq3uP Cq4uηP Cq5
√
T
2
q1u2P Cq2u1P Cηq3uP Cq4uηP Cq5 1/2
×m
i1
αiuP Cβi
m 4
m
i1
α2iu2P C2αiβiuP Cβ2i
Tq1
4
Tq1 2
m
i1
αim
4
m
i1
α2i
u2P C· · ·.
3.17
It follows from the above inequality and3.2that there existsM1>0 such thatuP C≤M1. Hence, we get by3.11that
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 (H2holds. Suppose that there exist a continuous and nondecreasing
functionψ:0,∞ → 0,∞and a nonnegative functionc∈CJwith
ft, u λ2u≤ctψ|u|, t∈J, u∈R. 3.19
Moreover suppose that
lim sup
u→ ∞
ψu
u < L 3.20
holds, where
L: 1−
eλT−1/2λ1eλTmi1γi−1/2mi1αi
eλT−1/2λ1eλTT
0csds
>0. 3.21
Then, BVP1has at least one solution.
Proof. From3.20, there exist 0< ε < LandM >0 such that
ψv≤L−εv, v≥M. 3.22
Thus, there existsK >0 such that
ψv≤L−εvK, v≥0. 3.23
Assume thatuis a solution of the equation
uμAu, μ∈0,1. 3.24
Then, we have by3.19,2.17, and3.23that
utμ
T
0
Gt, sfs, us λ2usds
m
k1
Gt, tk
−Ik∗Wt, tk
Ik
≤ eλT−1
2λ1eλT
T
0
csψ|u|ds e
λT−1
2λ1eλT m
i1
γiuP Cδi
1 2
m
i1
αiuP Cβi
.
Thus, we have
uP C≤ e λT−1
2λ1eλT
T
0
csdsL−εuP CK
eλT−1
2λ1eλT m
i1
γi
1 2
m
i1
αi
uP C e λT−1
2λ1eλT m
i1
δi
1 2
m
i1
βi,
3.26
that is,
ε e
λT−1
2λ1eλT
T
0
csdsuP C≤ e λT−1
2λ1eλTK
T
0
csds e
λT−1
2λ1eλT m
i1
δi1
2
m
i1
βi,
3.27
which implies that there existsM3 >0 such thatuP C ≤ M3. By3.7,3.8, and3.23, we get
ut≤ 3
2
T
0
fs, usds3 2λ
2
T
0
|us|ds3
2
m
i1
I∗ i ≤ 3 2 T 0
fs, us λ2usds3λ2
T
0
|us|ds3
2
m
i1
I∗ i ≤ 3 2 T 0
csψ|us|ds3λ2
T
0
|us|ds3
2
m
i1
γiuP Cδi
≤ 3
2
T
0
csdsLuP CK
3λ2TuP C3
2
m
i1
γiuP Cδi
,
3.28
which implies that
uP C≤
3L
2
T
0
csds3λ2T3
2
m
i1
γi
uP C
3K
2
T
0
csds3
2
m
i1
δi ≤ 3 2 L T 0
csds2λ2T
m
i1
γi M3 3K 2 T 0
csds3
2
m
i1
δi:M4.
3.29
Hence, uP C1 ≤ max{M3, M4}. 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
πutcos
2t1 2e
tu1/2t 1tant0, t∈
0,π
2
\
π
3
,
Δut1
1 3sin
ut1
1 4, Δu
t 1
1 2u
t1
1 3, t1
π
3,
u0 uT 0, u0 uT 0,
4.1
Letft, u 1/πucos2t 1/2etu1/21tant,I
1u 1/3sinu 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|1/2ht, 4.2
whereat 1/πcos2t,bt 1/2et,ht 1tant. And
I1u≤ 1 3|u|
1
4, I
∗ 1u≤
1 2|u|
1
3. 4.3
Thus,H1andH2hold. Obviously,α1 1/3,β1 1/4,γ1 1/2,δ1 1/3, andm1. Let
λ21/4π, we have
p1 3 2
T
0
atdt2λ2T
m
i1
γi
3 2
1
4
1
4
1 2
3 2,
q1
T
0
atdt
m
i1
p1αiγi
1
4
3 2·
1
3
1
2
5 4.
4.4
Therefore,
Tq1
4
Tq1 2
m
i1
αim
4
m
i1
α2i 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.
References
1 V. Lakshmikantham, D. D. Ba˘ınov, and P. S. Simeonov,Theory of Impulsive Differential Equations, vol. 6 ofSeries in Modern Applied Mathematics, World Scientific, Singapore, 1989.
3 S. T. Zavalishchin and A. N. Sesekin,Dynamic Impulse Systems: Theory and Application, vol. 394 of
Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997.
4 X. Liu, Ed., “Advances in impulsive differential equations,” Dynamics of Continuous, Discrete & Impulsive Systems. Series A, vol. 9, no. 3, pp. 313–462, 2002.
5 C. Bai and D. Yang, “Existence of solutions for second-order nonlinear impulsive differential equations with periodic boundary value conditions,”Boundary Value Problems, vol. 2007, Article ID 41589, 13 pages, 2007.
6 J. Li and J. Shen, “Periodic boundary value problems for second order differential equations with impulses,”Nonlinear Studies, vol. 12, no. 4, pp. 391–400, 2005.
7 J. J. Nieto, “Periodic boundary value problems for first-order impulsive ordinary differential equations,”Nonlinear Analysis: Theory, Methods & Applications, vol. 51, no. 7, pp. 1223–1232, 2002.
8 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.
9 M. Benchohra, J. Henderson, and S. Ntouyas,Impulsive Differential Equations and Inclusions, vol. 2 of
Contemporary Mathematics and Its Applications, Hindawi, New York, NY, USA, 2006.
10 T. Jankowski and J. J. Nieto, “Boundary value problems for first-order impulsive ordinary differential equations with delay arguments,”Indian Journal of Pure and Applied Mathematics, vol. 38, no. 3, pp. 203–211, 2007.
11 G. Zeng, F. Wang, and J. J. Nieto, “Complexity of a delayed predator-prey model with impulsive harvest and Holling type II functional response,”Advances in Complex Systems, vol. 11, no. 1, pp. 77– 97, 2008.
12 M. U. Akhmetov, A. Zafer, and R. D. Sejilova, “The control of boundary value problems for quasilinear impulsive integro-differential equations,”Nonlinear Analysis: Theory, Methods & Applications, vol. 48, no. 2, pp. 271–286, 2002.
13 J. J. Nieto and D. O’Regan, “Variational approach to impulsive differential equations,” Nonlinear Analysis: Real World Applications, vol. 10, no. 2, pp. 680–690, 2009.
14 H. Zhang, L. Chen, and J. J. Nieto, “A delayed epidemic model with stage-structure and pulses for pest management strategy,”Nonlinear Analysis: Real World Applications, vol. 9, no. 4, pp. 1714–1726, 2008.
15 D. Qian and X. Li, “Periodic solutions for ordinary differential equations with sublinear impulsive effects,”Journal of Mathematical Analysis and Applications, vol. 303, no. 1, pp. 288–303, 2005.
16 J. Yan, A. Zhao, and J. J. Nieto, “Existence and global attractivity of positive periodic solution of periodic single-species impulsive Lotka-Volterra systems,”Mathematical and Computer Modelling, vol. 40, no. 5-6, pp. 509–518, 2004.
17 S. Mohamad, K. Gopalsamy, and H. Akc¸a, “Exponential stability of artificial neural networks with distributed delays and large impulses,”Nonlinear Analysis: Real World Applications, vol. 9, no. 3, pp. 872–888, 2008.
18 J. J. Nieto and R. Rodr´ıguez-L ´opez, “Boundary value problems for a class of impulsive functional equations,”Computers & Mathematics with Applications, vol. 55, no. 12, pp. 2715–2731, 2008.
19 J. Li, J. J. Nieto, and J. Shen, “Impulsive periodic boundary value problems of first-order differential equations,”Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 226–236, 2007.
20 A. R. Aftabizadeh, S. Aizicovici, and N. H. Pavel, “On a class of second-order anti-periodic boundary value problems,”Journal of Mathematical Analysis and Applications, vol. 171, no. 2, pp. 301–320, 1992.
21 D. Franco, J. J. Nieto, and D. O’Regan, “Anti-periodic boundary value problem for nonlinear first order ordinary differential equations,”Mathematical Inequalities & Applications, vol. 6, no. 3, pp. 477– 485, 2003.
22 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.
23 Y. Yin, “Remarks on first order differential equations with anti-periodic boundary conditions,”
Nonlinear Times and Digest, vol. 2, no. 1, pp. 83–94, 1995.
24 Y. Yin, “Monotone iterative technique and quasilinearization for some anti-periodic problems,”
Nonlinear World, vol. 3, no. 2, pp. 253–266, 1996.
25 W. Ding, Y. Xing, and M. Han, “Anti-periodic boundary value problems for first order impulsive functional differential equations,”Applied Mathematics and Computation, vol. 186, no. 1, pp. 45–53, 2007.
27 D. Franco, J. J. Nieto, and D. O’Regan, “Existence of solutions for first order ordinary differential equations with nonlinear boundary conditions,”Applied Mathematics and Computation, vol. 153, no. 3, pp. 793–802, 2004.
28 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.
29 C. Ou, “Anti-periodic solutions for high-order Hopfield neural networks,”Computers & Mathematics with Applications, vol. 56, no. 7, pp. 1838–1844, 2008.
30 R. Wu, “An anti-periodic LaSalle oscillation theorem,”Applied Mathematics Letters, vol. 21, no. 9, pp. 928–933, 2008.
31 Y. Li and L. Huang, “Anti-periodic solutions for a class of Li´enard-type systems with continuously distributed delays,”Nonlinear Analysis: Real World Applications. In press.
32 K. Wang, “A new existence result for nonlinear first-order anti-periodic boundary value problems,”
Applied Mathematics Letters, vol. 21, no. 11, pp. 1149–1154, 2008.
33 C. Ahn and C. Rim, “Boundary flows in general coset theories,”Journal of Physics A, vol. 32, no. 13, pp. 2509–2525, 1999.
34 A. Abdurrahman, F. Anton, and J. Bordes, “Half-string oscillator approach to string field theory
ghost sector. I,”Nuclear Physics B, vol. 397, no. 1-2, pp. 260–282, 1993.
35 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.
36 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.
37 W. Wang and J. Shen, “Existence of solutions for anti-periodic boundary value problems,”Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 2, pp. 598–605, 2009.