Antiperiodic Boundary Value Problems for Second-Order Impulsive Ordinary Differential Equations

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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, tJ0J\ {t1, . . . , tm},

Δutk

Ik

utk

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Δutk

Ikutk

, k1, . . . , m,

u0 uT 0, u0 uT 0,

1.1

whereJ 0, T, 0< t1< t2 <· · ·< tm< T,f :0, T×RRis continuous ont, xJR,

ftk, x :limttkft, x,ftk, x :limttkft, xexist,ftk, x ftk, xutk utk

utkutk utkutk;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 : EEa compact operator. If the set

S:xE|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 :JR : uis continuous for anytJ0,utk,utkexist, andutk

utk, k1, . . . , m},

P C1J {u : J R : uis continuously dierentiable for any t J

0,utk,utk exist, andutk utk, k 1, . . . , m}.

P CJandP C1Jare Banach space with the norms

uP Csup tJ

|ut|,

uP C1 maxuP C,uP C.

2.1

A solution to the impulsive BVP1is a functionuP C1JC2J

0that satisfies1 for eachtJ.

Consider the following impulsive BVP withλ >0

ut λ2ut σt, tJ0,

Δutk

Ik

utk

, k1, . . . , m,

Δutk

Ikutk

, k1, . . . , m, u0 uT 0, u0 uT 0,

2.2

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For convenience, we setIkIkutk, IkIkutk.

Lemma 2.1. uP C1JC2J

0is a solution of 2.2if and only ifuP C1Jis a solution of the

impulsive integral equation

ut

T

0

Gt, sσsds

m

k1

Gt, tk

IkWt, tk

Ik , 2.3

where

Gt, s 1

2λ

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

eλts

1eλT

eλts

1eλT, 0≤s < tT,

eλTts

1eλT

eλTts

1eλT , 0≤tsT,

Wt, s 1

2

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

eλts

1eλT

eλts

1eλT, 0≤s < tT,

eλTts

1eλT

eλTts

1eλT , 0≤tsT.

2.4

Proof. IfuP C1JC2J

0is a solution of2.2, setting

vt ut λut, 2.5

then, by the first equation of2.2we have

vtλvtσt, t /tk. 2.6

Multiplying2.6byeλtand integrating on0, t

1andt1, t t1< tt2, respectively, we get

eλt1vt

1

v0 −

t1

0

σseλsds,

eλtvteλt1vt

1

t

t1

σseλsds, t1< tt2.

2.7

So

vt eλt

v0−

t

0

eλsσsdseλtvt

1

, t1< tt2. 2.8

In the same way, we can obtain that

vt eλt

v0−

t

0

eλsσsds

0<tk<t

eλtkI

kλIk

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wherev0 u0 λu0. Integrating2.5, we have

ut eλt

u0

t

0

vseλsds

0<tk<t

eλtkI

k

, tJ. 2.10

By2.9, we get

t

0

vseλsds 1

2λ

v0e2λt−1−

t

0

e2λte2λsσseλsds

0<tk<t

e2λte2λtkeλtkI

kλIk

.

2.11

Substituting2.11into2.10, we obtain

ut 1

2λ

λu0−u0eλtu0 λu0eλt

t

0

eλtseλtsσsds

0<tk<t

eλttkI

kλIk

− 0<tk<t

eλttkI

kλIk

, tJ,

2.12

ut 1

2

λu0−u0eλtu0 λu0eλt

t

0

eλtseλtsσsds

0<tk<t

eλttkI

kλIk

0<tk<t

eλttkI

kλIk

, tJ.

2.13

In view ofu0 uT 0 andu0 uT 0, we have

u0 λu0 1 1eλT

T

0

eλTsσsds

0<tk<T

eλTtkI

kλIk

,

λu0−u0 1 1eλT

T

0

eλTsσsds

0<tk<T

eλTtkI

kλIk

.

2.14

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

Conversely, ifuis a solution of2.3, then direct differentiation of2.3gives−ut σtλ2ut,t /tk. Moreover, we obtainΔu|ttk Ikutku|ttk Ikutk,u0uT 0

andu0 uT 0. Hence,uP C1JC2J

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Remark 2.2. We callGt, sabove the Green function for the following homogeneous BVP:

ut λ2ut 0, tJ,

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

IkWt, 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. IfuP C1Jandu0 uT 0, then

uP C

1 2

T

0

usdsm

k1

Δu

tk

. 2.18

Proof. SinceuP 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

ttk

Δutk

≤ 1

2

t

0

usdsT

t

usds1

2

0<tk<t

Δu

tk

ttk

Δu

tk

1 2

T

0

usdsm

k1

Δu

tk

.

2.21

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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, hCJ,0,∞, and nonnegative constantsαk, βk, γk, δkk1,2, . . . , msuch that

H1|ft, u| ≤at|u|bt|u|ηht, and

H2|Iku| ≤αk|u|βk,|Iku| ≤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 μft, utλ2ut λ2Aut

μft, utλ2μ−1ut,

3.4

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Integrating3.4from 0 toT, we get that

uTu0

T

0

utdt

m

i1

Ii∗−μ

T

0

ft, utdtλ2μ−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

utu0

t

0

usds

0<ti<t

Ii∗−μ

t

0

fs, usdsλ2μ−1

t

0

usds

0<ti<t

Ii. 3.8

From3.7and3.8, we have

utu0T 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 Cp1uP Cp2P Cp3, 3.11

wherep1, p2, p3are as in3.1. Integrating3.5from 0 toT, we get that

T

0

ututdtμ

T

0

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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

· · ·uTuTutn0

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

ut2dtμ

T

0

utft, utdtut1−0

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 Cp2P Cp3

m

i1

αiuP Cβi

.

(9)

Thus,

T

0

ut2dtq1u2P Cq2uP C1ηq3uP Cq4P 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 Cq4P Cq5

T

2

q1u2P Cq2u1P Cηq3uP Cq4P 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 CM1. Hence, we get by3.11that

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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 functioncCJwith

ft, u λ2uctψ|u|, tJ, uR. 3.19

Moreover suppose that

lim sup

u→ ∞

ψu

u < L 3.20

holds, where

L: 1−

eλT−1/2λ1eλTmi1γi−1/2mi1αi

eλT1/2λ1eλTT

0csds

>0. 3.21

Then, BVP1has at least one solution.

Proof. From3.20, there exist 0< ε < LandM >0 such that

ψvLεv, vM. 3.22

Thus, there existsK >0 such that

ψvLεvK, 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

IkWt, tk

Ik

eλT−1

2λ1eλT

T

0

csψ|u|ds e

λT1

2λ1eλT m

i1

γiuP Cδi

1 2

m

i1

αiuP Cβi

.

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Thus, we have

uP Ce λT1

2λ1eλT

T

0

csdsLεuP CK

eλT1

2λ1eλT m

i1

γi

1 2

m

i1

αi

uP C e λT1

2λ1eλT m

i1

δi

1 2

m

i1

βi,

3.26

that is,

ε e

λT1

2λ1eλT

T

0

csdsuP Ce λT1

2λ1eλTK

T

0

csds e

λT1

2λ1eλT m

i1

δi1

2

m

i1

βi,

3.27

which implies that there existsM3 >0 such thatuP CM3. By3.7,3.8, and3.23, we get

ut 3

2

T

0

fs, usds3 2λ

2

T

0

|us|ds3

2

m

i1

Ii ≤ 3 2 T 0

fs, us λ2usds3λ2

T

0

|us|ds3

2

m

i1

Ii ≤ 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

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Example 4.1. Consider the problem

ut 1

πutcos

2t1 2e

tu1/2t 1tant0, t

0

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,I1u 1/2u

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

ft, uat|u|bt|u|1/2ht, 4.2

whereat 1cos2t,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.

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