Volume 2010, Article ID 490741,14pages doi:10.1155/2010/490741
Research Article
Nonlinear Boundary Value Problem of First-Order
Impulsive Functional Differential Equations
Kexue Zhang
1and Xinzhi Liu
21School of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, China 2Department of Applied Mathematics, University of Waterloo, Waterloo, ON, Canada N2L 3G1
Correspondence should be addressed to Xinzhi Liu,[email protected]
Received 8 December 2009; Accepted 30 January 2010
Academic Editor: Shusen Ding
Copyrightq2010 K. Zhang and X. Liu. 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.
This paper investigates the nonlinear boundary value problem for a class of first-order impulsive functional differential equations. By establishing a comparison result and utilizing the method of upper and lower solutions, some criteria on the existence of extremal solutions as well as the unique solution are obtained. Examples are discussed to illustrate the validity of the obtained results.
1. Introduction
It is now realized that the theory of impulsive differential equations provides a general framework for mathematical modelling of many real world phenomena. In particular, it serves as an adequate mathematical tool for studying evolution processes that are subjected to abrupt changes in their states. Some typical physical systems that exhibit impulsive behaviour include the action of a pendulum clock, mechanical systems subject to impacts, the maintenance of a species through periodic stocking or harvesting, the thrust impulse maneuver of a spacecraft, and the function of the heart. For an introduction to the theory of impulsive differential equations, refer to1.
It is also known that the method of upper and lower solutions coupled with the monotone iterative technique is a powerful tool for obtaining existence results of nonlinear differential equations 2. There are numerous papers devoted to the applications of this method to nonlinear differential equations in the literature, see3–9and references therein. The existence of extremal solutions of impulsive differential equations is considered in papers
nonlinear boundary value problem of a class of first-order impulsive functional differential equations. Such equations include the retarded impulsive differential equations as special cases5,12–14.
The rest of this paper is organized as follows. In Section 2, we establish a new comparison principle and discuss the existence and uniqueness of the solution for first order impulsive functional differential equations with linear boundary condition. We then obtain existence results for extremal solutions and unique solution inSection 3by using the method of upper and lower solutions coupled with monotone iterative technique. To illustrate the obtained results, two examples are discussed inSection 4.
2. Preliminaries
LetJ 0, T,T > 0,J J− {t1, t2, . . . , tp}with 0 < t1 < t2 < · · · < tp < T. We define that
P CJ {x:J → R:xis continuous for anyt∈J;xtkandxt−kexist andxt−k xtk},
P C1J {x:J → R :xis continuously differentiable for anyt ∈J;xtk,xt−kexist and
xt−k xtk}. It is clear thatP CJandP C1Jare Banach spaces with respective norms
xP Csup
t∈J
|xt|, xP C1xP Cx
P C. 2.1
Let us consider the following nonlinear boundary value problemNBVP:
xt ft, xt,ϕxt, t∈JJ−t1, t2, . . . , tp
,
Δxt Ik
t, xt,ϕxt, ttk, k1,2, . . . , p,
gx0, xT 0,
2.2
wheref:J×R2 → Ris continuous in the second and the third variables, and for fixedx, y∈ R,f·, x, y∈P CJ,g ∈CR2,R,I
k ∈CR3,R,k 1,2, . . . , pandϕ :P CJ → P CJis
continuous.
A functionx∈P C1Jis called a solutions of NBVP2.2if it satisfies2.2.
Remark 2.1. iIfϕxt xtand the impulsesIk depend only onxtk, the equation of
NBVP2.2reduces to the simpler case of impulsive differential equations:
xt ft, xt, t∈J,
Δxtk Ikxtk, k1,2, . . . , p 2.3
which have been studied in many papers. In some situation, the impulseIkdepends also on
some other parameterse.g., the control of the amount of drug ingested by a patient at certain moments in the model for drug distribution1,3.
iiIfϕxt xθt, whereθ∈CJ, J, the equation of NBVP2.2can be regarded as retarded differential equation which has been considered in5,12–14.
Lemma 2.2see1. Asumme that
B0the sequence{tk}satisfies0≤t0< t1< t2<· · ·< tk<· · · withlimk→ ∞tk ∞, B1m∈P C1Ris left continous attkfork1,2, . . .,
B2fork1,2, . . .,t≥t0,
mt≤ptmt qt, t /tk
mtk≤dkmtk bk,
2.4
wherep, q∈CR,R,dk≥0andbkare real constants.
Then
mt≤mt0
t0<tk<t
dkexp
t
t0 psds
t
t0s<tk<t
dkexp
t
s
pσdσ
qsds
t0<tk<ttk<tj<t
djexp
t
tk
psds
bk.
2.5
In order to establish a comparison result and some lemmas, we will make the following assumptions on the functionϕ.
H1There exists a constantR >0 such that
ϕxt≥Rinf
t∈Jxt, for any x∈P CJ, ∀t∈J. 2.6
H2The functionϕsatisfies Lipschitz condition, that is, there exists aL >0 such that
ϕx−ϕyP C≤Lx−yP C, ∀x, y∈P CJ. 2.7
Inspired by the ideas in5,6, we shall establish the following comparison result.
Theorem 2.3. Letm∈P C1Jsuch that
mt≤ −Mmt−Nϕmt, t∈J,
Δmtk≤ −Lkmtk, k1,2, . . . , p,
m0≤μmT,
2.8
Suppose in addition that condition (H1) holds and
NReMTμ
μ
T
0s<tk<T
1−LkeMsds≤ p
k1
1−Lk2 2.9
thenmt≤0, fort∈J.
Proof. For simplicity, we letck1−Lk,k1,2, . . . , p. Setvt mteMt, then we have
vt≤ −NeMtϕmt, t∈J,
vtk≤ckvtk, k1,2, . . . , p
v0≤μe−MTvT.
, 2.10
Obviously,vt≤0 impliesmt≤0.
To showvt≤0, we suppose, on the contrary, thatvt>0 for somet∈J. It is enough to consider the following cases.
ithere exists at∈J, such thatvt>0, andvt≥0 for allt∈J;
iithere existt∗, t∗∈J, such thatvt∗<0,vt∗>0.
Case (i). By2.10, we havevt ≤ 0 fort /tk andΔvtk ≤ 0,k 1,2, . . . , m, hencevtis
nonincreasing inJ, that is,vT≤v0. Ifμ < eMT, thenv0< vT, which is a contradiction.
IfμeMT, thenv0≤vTwhich impliesvt≡C >0. But from2.10, we getvt<0 for
t∈J. Hence,vT< v0. It is again a contradiction.
Case (ii). Let inft∈Jvt −λ, thenλ > 0. For somei∈ {1,2, . . . , p}, there existst∗ ∈ ti, ti1
such thatvt∗ −λorvt∗ −λ. We only considervt∗ −λ, as for the casevt∗ −λ, the
proof is similar.
From2.10and conditionH1, we get
vt≤ −NeMtϕmt −NeMtϕvte−Mtt
≤ −NReMtinf
t∈J
vte−Mt≤ −NReMtinf
t∈J{vt}
≤λNReMt, t∈J.
2.11
Consider the inequalities
vt≤λNReMt, t∈J,
vtk≤ckvtk, k1,2, . . . , p.
ByLemma 2.2, we have
vt≤vt∗
t∗<tk<t
ck t t∗
s<tk<t
ck
λNReMsds, 2.13
that is
vt≤ −λ
t∗<tk<t
ck λNR t t∗
s<tk<t
ck
eMsds. 2.14
First, we assume thatt∗> t∗. Lettt∗in2.14, then
vt∗≤ −λ
t∗<tk<t∗
ck λNR t∗ t∗
s<tk<t∗
ck
eMsds. 2.15
Noting thatvt∗>0, we have
t∗<tk<t∗
ck< NR
t∗
t∗
s<tk<t∗
ck
eMsds. 2.16
Hence
p
k1 ck
2
≤ p
k1
ck< NR
T
0
s<tk<T
ck
eMsds 2.17
which is a contradiction.
Next, we assume thatt∗< t∗. ByLemma 2.2and2.10, we have
0< vt∗≤v0
0<tk<t∗
ck t∗ 0
s<tk<t∗
ck
λNReMsds
≤μe−MTvT
0<tk<t∗
ck λNR t∗ 0
s<tk<t∗
ck
eMsds,
2.18
then
0< μe−MTvT
p
k1 ck λNR T 0
s<tk<T
ck
SettingtT in2.14, we have
vT≤vt∗
t∗<tk<T
ck T t∗
s<tk<T
ck
λNReMsds
−λ
t∗<tk<T
ck λNR T t∗
s<tk<T
ck
eMsds
≤ −λ
p
k1
ckλNR
T
0
s<tk<T
ck
eMsds
2.20
with2.19, we obtain that
0< μe−MTvT
p
k1 ck λNR T 0
s<tk<T
ck
eMsds
≤
−λ
p
k1 ck λNR T 0
s<tk<T
ck
eMsds
μe−MT
p
k1 ck λNR T 0
s<tk<T
ck
eMsds
−μλe−MT
p
k1 ck
2
μλNRe−MT
p
k1 ck
T
0
s<tk<T
ck
eMsds
λNR
T
0
s<tk<T
ck
eMsds,
2.21
that is,
μe−MT
p
k1 ck
2
≤
μNRe−MT
p
k1 ck NR T 0
s<tk<T
ck
eMsds
< NRμe−MT1
T
0
s<tk<T
ck
eMsds.
2.22
Therefore,
p
k1
1−Lk2< NR
eMTμ
μ
T
0s<tk<T
1−LkeMsds 2.23
which is a contradiction. The proof ofTheorem 2.3is complete.
Corollary 2.4. Assume that there existM > 0,N ≥ 0,0 ≤ Lk < 1, fork 1,2, . . . , psuch that
m∈P C1Jsatisfies2.8with0< μe−MT≤1and
RNeMTμeMT
μ ≤
p
k11−Lk 2
T
0
s<tk<T1−Lkds
2.24
thenmt≤0, fort∈J.
Remark 2.5. Setting μ ≡ 1, Corollary 2.4 reduces to the Theorem 2.3 of Li and Shen 6. Therefore,Theorem 2.3andCorollary 2.4develops and generalizes the result in6.
Remark 2.6. We show some examples of functionϕsatisfyingH1.
i ϕxt xθt, whereθ∈CJ×J, satisfiesH1withR1,
ϕxt xθt≥inf
t∈Jxt, fort∈J. 2.25
ii ϕxt t0Txsds, satisfiesH1withRT,
ϕxt
tT
0
xsds≥tTinf
t∈Jxt≥Tinft∈Jxt, fort∈J. 2.26
Consider the linear boundary value problemLBVP
yt Myt Nϕyt σt, t∈J,
Δytk −Lkytk Ik
tk, ηtk,
ϕηtkLkηtk, k1,2, . . . , p,
gη0, ηTM1
y0−η0−M2
yT−ηT0,
2.27
whereM >0,N≥0, 0≤Lk<1,k1,2, . . . , p, andη, σ∈P CJ.
By direct computation, we have the following result.
Lemma 2.7. y∈P C1Jis a solution of LBVP2.27if and only ifyis a solution of the impulsive integral equation
yt Ce−MtBη
T
0
Gt, sσs−Nϕysds
0<tk<T
Gt, tk−Lkytk Ik
tk, ηtk,
ϕηtkLkηtk
, t∈J,
2.28
whereBη−gη0, ηT M1η0−M2ηT,C M1−M2e−MT−1,M1/M2e−MT and
Gt, s
⎧ ⎨ ⎩
CM2eMs−t−TeMs−t, 0≤s < t≤T
CM2eMs−t−T, 0≤t≤s≤T
Lemma 2.8. Let (H2) hold. Suppose further
NLT
p
k1
|Lk|
r <1, rmax{|CM1|,|CM2|}, C
M1−M2e−MT
−1
, 2.30
whereM >0,N≥0,M1/M2e−MT, then LBVP2.27has a unique solution.
ByLemma 2.7and Banach fixed point theorem, the proof ofLemma 2.8is apparent, so we omit the details.
3. Main Results
In this section, we use monotone iterative technique to obtain the existence results of extremal solutions and the unique solution of NBVP2.2. We shall need the following definition.
Definition 3.1. A functionα∈P C1Jis said to be a lower solution of NBVP2.2if it satisfies
αt≤ft, αt,ϕαt, t∈J,
Δαtk≤Ik
tk, αtk,
ϕαtk, k1,2, . . . , p,
gα0, αT≤0.
3.1
Analogously,β∈P C1Jis an upper solution of NBVP2.2if
βt≥ft, βt,ϕβt, t∈J,
Δβtk≥Ik
tk, βtk,
ϕβtk, k1,2, . . . , p,
gβ0, βT≥0.
3.2
For convenience, let us list the following conditions.
H3There exist constantsM >0,N≥0 such that
ft, x, ϕx−ft, x, ϕx≥ −Mx−x−Nϕx−ϕx 3.3
whereverα0t≤x≤x≤β0t.
H4There exist constants 0≤Lk<1 fork1,2, . . . , psuch that
Ik
tk, x, ϕx
−Ik
tk, x, ϕx
≥ −Lkx−x, k1,2, . . . , p 3.4
whereverα0tk≤x≤x≤β0tk.
H5The functionϕsatisfies
H6There exist constantsM1,M2with 0< M2e−MT ≤M1such that
gx, y−gx, y≤M1x−x−M2
y−y 3.6
whereverα00≤x≤x≤β00, andα0T≤y≤y≤β0T.
Letα0, β0 {x ∈ P C1J : α0t ≤ xt ≤ β0t, for allt ∈ J}. Now we are in the
position to establish the main results of this paper.
Theorem 3.2. Let (H1)–(H6) and inequalities2.9and2.30hold. Assume further that there exist
lower and upper solutionsα0andβ0of NBVP2.2, respectively, such thatα0 ≤β0onJ. Then there
exist monotone sequences{αn},{βn} ⊂ P C1Jwithα0 ≤ · · · ≤ αn ≤ · · · ≤ βn ≤ · · · ≤ β0, such
thatlimn→ ∞αnx∗t,limn→ ∞βn x∗tuniformly onJ. Moreover,x∗t,x∗tare minimal and
maximal solutions of NBVP2.2inα0, β0, respectively.
Proof. For anyη∈α0, β0, consider LVBP2.27with
σt ft, ηt,ϕηtMηt Nϕηt. 3.7
By Lemma 2.8, we know that LBVP 2.27 has a unique solution y ∈ P C1J. Define an
operatorA:P CJ → P CJbyyAη, then the operatorAhas the following properties:
aα0≤Aα0,Aβ0 ≤β0,
bAη1≤Aη2, ifα0≤η1≤η2≤β0.
To provea, letα1Aα0andmt α0t−α1t.
mt α0t−α1t
ft, α0t,
ϕα0
t
−−Mα1t−N
ϕα1
t ft, α0t,
ϕα0
tMα0t N
ϕα0
t
≤ −Mmt−Nϕmt,
Δmtk Δα0tk−Δα1tk
≤Ik
tk, α0tk,
ϕα0
tk−−Lkα1tk Ik
tk, α0tk,
ϕα0
tkLkα0tk
≤ −Lkmtk,
m0 α00−α10
α00−
− 1
M1gα00, α0T α00 M2
M1α1T−α0T
≤ M2
M1 mT.
3.8
To proveb, setmt x1t−x2t, wherex1 Aη1andx2 Aη2. UsingH3,H4
andH6, we get
mt x1t−x2t
Mη1t−x1t
Nϕη1
t−ϕx1
tft, η1t,
ϕη1
t
−Mη2t−x2t
−Nϕη2
t−ϕx2
t−ft, η2t,
ϕη2
t
≤ −Mmt−Nϕmt,
Δmtk Δx1tk−Δx2tk
≤Lk
η1tk−x1tk
Ik
tk, η1tk,
ϕη1
tk
−Lk
η2tk−x2tk
−Ik
tk, η2tk,
ϕη2
tk
≤ −Lkmtk,
m0 x10−x20
− 1
M1
gη10, η1T
η10 M2 M1
x1T−η1T
1
M1
gη20, η2T
−η20− M2 M1
x2T−η2T
≤ M2
M1 mT.
3.9
ByTheorem 2.3, we getmt≤0 fort∈J, that is,Aη1≤Aη2, thenbis proved.
Letαn Aαn−1and βn Aβn−1 forn 1,2,3, . . . .By the propertiesaandb, we
have
α0≤α1≤ · · · ≤αn≤ · · · ≤βn ≤ · · · ≤β1≤β0. 3.10
By the definition of operatorA, we have that{αn}and{βn}are uniformly bounded inα0, β0.
Thus{αn}and{βn}are uniformly bounded and equicontinuous inα0, β0. By Arzela-Ascoli
Theorem and3.10, we know that there existx∗,x∗inα0, β0such that
lim
n→ ∞αnt x∗t, nlim→ ∞βnt x
∗t uniformly onJ
3.11
Moreover,x∗t,x∗tare solutions of NBVP2.2inα0, β0.
To prove thatx∗,x∗ are extremal solutions of NBVP 2.2, letut ∈ α0, β0 be any
solution of NBVP2.2, that is,
ut ft, ut,ϕut, t∈J,
Δutk Ik
tk, utk,
ϕutk, k1,2, . . . , p,
gu0, uT 0.
ByTheorem 2.3and Induction, we getαnt≤ut≤βntwitht∈Jandn1,2,3, . . .which implies thatx∗t≤ut≤x∗t, that is,x∗andx∗are minimal and maximal solution of NBVP
2.2inα0, β0, respectively. The proof is complete.
Theorem 3.3. Let the assumptions ofTheorem 3.2hold and assume the following.
H7There exist constantsM > 0,N≥0such that
ft, x, ϕx−ft, x, ϕx≤ −Mx−x−Nϕx−ϕx, 3.13
whereα0t≤x≤x≤β0t.
H8There exist constants0≤Lk<1,k1,2, . . . , psuch that
Ik
tk, x, ϕx
−Ik
tk, x, ϕx
≤ −Lkx−x, k1,2, . . . , p, 3.14
whereα0tk≤x≤x≤β0tk.
H9There exist constantsM1,M2with0<M2e−MT <M1such that
gx, y−gx, y≥M1x−x−M2
y−y 3.15
wheneverα00≤x≤x≤β00, andα0T≤y≤y≤β0T.
Then NBVP2.2has a unique solution inα0, β0.
Proof. By Theorem 3.2, we know that there exist x∗, x∗ ∈ α0, β0, which are minimal and
maximal solutions of NBVP2.2withx∗t≤x∗t, t∈J.
Letmt x∗t−x∗t. UsingH7,H8, andH9, we get
mt x∗t−x∗tft, x∗t,ϕx∗t−ft, x∗t,ϕx∗t
≤ −Mm t−Nϕmt,
Δmtk Δx∗tk−Δx∗tk Ik
tk, x∗tk,
ϕx∗tk−Ik
tk, x∗tk,ϕx∗tk
≤Lkmtk,
m0 x∗0−x∗0≤
M2
M1
x∗T−x∗T
M2
M1 mT.
3.16
ByTheorem 2.3, we have thatmt< 0,t∈ J, that is,x∗t≤x∗t. Hencex∗t x∗t, this
4. Examples
To illustrate our main results, we shall discuss in this section some examples.
Example 4.1. Consider the problem
xt −1
10−|sint|x−2
5−e−2π
3
t1
t
xsds−sin t 4
2
4
3e
−2π, t∈0, T, t /t k,
Δxtk −1
2e
−π/2xtk−3
tk1
tk
xsds−sintk 4
1/7
4
17e
π/2cost
k, k1,
x0− 1 2xT−
1 6π
T
0
xsds0,
4.1
whereT 2π,k1,t1π.
Let
ϕxt
t1
t
xsds−sin t 4,
ft, x, y−1
10−|sint|x−2
5−e−2π
3 y
24
3e
−2π,
I1
t, x, y−1
2e
−π/2x−3 y1/7 4
17e
π/2cost.
4.2
Settingα0t≡2 andβ0t≡3, it is easy to verify thatα0tis a lower solution, andβ0tis an
upper solution withα0t≤β0t.
Fort∈J, and 2≤xt≤xt≤3, we have
ϕx≥1≥ 1
3inft∈Jxt,
ϕx−ϕx≥ϕx−x,
ϕx−ϕx
t1
t
xs−xsds≤ x−x.
SettingM 1/2,N e−2π/6,L1,R 1/3,L
1 1/2e−π/2andM1 1,M2 1/2, then
conditionsH1–H6are all satisfied:
2π
0 s<tk<2π
1−LkeMsds
π
0
1−L1eMsds
2π
π
eMsds≈43.4893,
p
k1
1−Lk2 1−L12
1−1 2e
−π/2 2≈0.8029,
NReMT M
2/M1
M2/M1
2eπ1
18e2π ≈0.0049,
NReMT M
2/M1
M2/M1
2π
0 s<tk<2π
1−LkeMsds≈0.2131<0.8029,
NLT
p
k1
|Lk|
r NLTL1CM1 ≈0.1082<1,
4.4
then inequalities2.9and2.30are satisfied. ByTheorem 3.2, problem4.1has extremal solutionsx∗, x∗∈α0, β0.
Example 4.2. Consider the problem
xt −1
2xt−
e−3π
eπ1
e2x−13
2, t∈0, T, t /tk,
Δxtk −2e−πxtk−2 e2xtk−11/50cost
k, k1,
x0 1 2xT,
4.5
whereT 2π,k1,t1π.
Let
ϕxe2x−1,
ft, x, y− 1
10−|sint|x−2
5− e−3π
16eπ1y
2
e4−12
16e3πeπ1,
I1
t, x, y−2e−πx−2 y1/50cost.
4.6
Settingα0t≡ 2 andβ0t ≡3, thenα0tis a lower solution, andβ0tis an upper solution
withα0t≤β0t.
Fort∈J, and 2≤xt≤xt≤3, we haveϕx e2x−1≥x,ϕx−ϕx≥ϕx−x, and
N e−3π/eπ 1,L 14e6,R 1, L
1 2e−π andM1 1,M2 1/2, then conditions
H1–H6are all satisfied:
!p
k11−Lk 2
2π
0
!
s<tk<2π1−Lkds
eπ−22
2πeπeπ−1 ≈0.1388,
NReMT M
2/M1
eMT
M2/M1
2eπ1
e2πeπ1 ≈0.0037<0.1388,
NLT
p
k1
|Lk|
r NLTL1CM1 ≈0.2095<1,
4.7
then inequalities2.24and2.30are satisfied. ByCorollary 2.4and Theorem 3.2, problem
4.5has extremal solutionsx∗, x∗∈α0, β0.
Moreover, letM1/2,Ne−3π/eπ1,L
12e−πandM11,M2 1/2. It is easy
to see that conditionsH7–H9are satisfied. By Corollary 2.4 and Theorem 3.3, problem
4.5has an unique solution inα0, β0.
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