Boundary Value Problems
Volume 2008, Article ID 937138,16pages doi:10.1155/2008/937138
Research Article
Existence of Solutions of Periodic Boundary
Value Problems for Impulsive Functional Duffing
Equations at Nonresonance Case
Xingyuan Liu1and Yuji Liu2
1Department of Mathematics, Shaoyang University, Shaoyang, Hunan 422000, China 2Department of Mathematics, Guangdong University of Business Studies, Guangzhou,
Guangdong 510320, China
Correspondence should be addressed to Xingyuan Liu,[email protected]
Received 19 June 2008; Accepted 27 August 2008
Recommended by Zhitao Zhang
This paper deals with the existence of solutions of the periodic boundary value problem of the impulsive Duffing equations: xt αxt βxt ft, xt, xα1t, . . . , xαnt,a.e.t ∈
0, T,Δxtk Ikxtk, xtk, k 1, . . . , m,Δxtk Jkxtk, xtk, k 1, . . . , m, xi0
xiT, i 0,1. Sufficient conditions are established for the existence of at least one solution of
above-mentioned boundary value problem. Our method is based upon Schaeffer’s fixed-point theorem. Examples are presented to illustrate the efficiency of the obtained results.
Copyrightq2008 X. Liu and Y. 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.
1. Introduction
In recent years, many authors studied the solvability of the periodic boundary value problems PBVPs for short for second-order ordinary or functional differential equations with or without impulse effects; see1–24and the references therein. For example, consider the following PBVP:
xt ft, xt, t∈0,2π,
x0 x2π, x0 x2π, 1.1
the well-known result is that iffsatisfies the nonresonance condition
−N12≤fut, u≤ −−N2, 1.2
For PBVP of the Duffing equation
xt cxt gt, xtet, t∈0,2π,
x0 x2π, x0 x2π, 1.3
in16, the authors proved the following results.
Theorem 1.1. Supposegis aL2-Caratheodory function, and there area≤A, r <0< Rsuch that
gt, x≥A, x≥R, t∈0,2π, gt, x≤a, x≤r, t∈0,2π, 1.4
and further there isr ∈L0,2πwith||r||∞<1C2such that
lim
|x| → ∞
gt, x
x < rt, t∈0,2π. 1.5
Then, PBVP1.3has at least one solution for eache∈L20,2πwitha≤1/2π2π
0 esds≤A.
In17, Nieto and Rodr´ıguez-L ´opez studied the following PBVP:
xt axt bxt cxtdxtσt, t∈0, T,
x0 xT, x0 xT. 1.6
They gave Green’s function to express the unique solution for the correspondence second-order functional differential equation with periodic boundary conditions and the functional dependence given by the piecewise constant function. Using upper and lower solution methods, they presented sufficient conditions to assure the existence of solutions of PBVP1.6. The authors in 3,21 also studied the solvability of PBVP1.6 by the similar method.
In2, Ding et al. studied the PBVP:
xt ft, xt, xθt0, t∈0, T,
x0 xT, x0 xT. 1.7
Sufficient conditions for the existence of solutions of PBVP1.7are given by using upper and lower solution method.
The PBVPs,
xt ρ2xt ft, xt, t∈0,2π,
x0 x2π, x0 x2π, 1.8
−xt ρ2xt ft, xt, t∈0,2π,
x0 x2π, x0 x2π, 1.9
were studied in 8,18–20. In 19, based upon Krasnoselskii fixed-point theorems, Jiang proved that PBVP1.8 and PBVP1.9 with singularity have at least one positive solution provided thatft, xis superlinear or sublinear atx 0 andx ∞. In 21, Wang, who
least one positive solution provided thatft, xis sublinear atx ∞. In18, Zhang and
Wang established multiplicity results to positive solutions of PBVP1.8and PBVP1.9with
fbeing a Caratheodory function.
In paper24, Liu and Ge studied the following equation:
xt−ptxt qtxt λft, xt−τtrt; 1.10
they established sufficient conditions for the existence of positive periodic solutions of1.10.
In 11, Peng studied the existence of periodic solutions of the functional Duffing
equation:
xt cxt g1
t, xt−τ1t
g2
t, xt−τ2t
pt. 1.11
The studies on the existence of solutions of the periodic boundary value problems of the second-order impulsive differential equations,
xt ft, xt0, t∈0, T, x0 xT, x0 xT,
Δxitk
Ii, k
xtk
, k1, . . . , p, i0,1,
1.12
can be found in3,4,12–14,20,22,23and the references therein. The methods used there are of lower and upper solutions methods, the monotone iterative technique.
In all above-mentioned papers, the results are based upon the following assumptions. Hf satisfies either the Lipschitzian condition, left Lipschitzian condition, right Lipschitzian condition, or Nagumo conditions.
To the best of our knowledge, the existence of solutions of periodic boundary value problems for impulsive Duffing functional differential equations has not been well studied till now.
In this paper, we investigate the following periodic boundary value problem for the impulsive functional Duffing equation:
xt αxt βxt ft, xt, xα1t
, . . . , xαnt
, a.e.t∈0, T,
Δxtk Ik
xtk
, xtk
, k1, . . . , m,
Δxtk Jk
xtk
, xtk
, k1, . . . , m,
xi0 xiT, i0,1,
1.13
where mis a positive integer,α, β ∈ R,f : 0, T×Rn1→R is a impulsive Caratheodory
function,αi ∈ C10, T,0, T, whose inverse function is denoted byβi withβi ∈ C10, T,
Ik, Jk:R2→Rare continuous.
Our purpose here is to provide sufficient conditions for the existence of solutions of 1.13at nonresonance case. This will be done by applying the well-known Schaeffer’s
The main results and examples in this paper are established inSection 2. The proofs of the main results are presented inSection 3.
2. Main results and examples
In this section, we establish the main results and examples to illustrate the main theorems. To define solutions of PBVP1.13, we introduce the following Banach spaces and definitions.
Supposeu : J 0, T→R, and 0 t0 < t1 < · · · < tm < tm1 T. Fork 0, . . . , m,
define the functionuk:tk, tk1→Rbyukt ut. Choose
X
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
u:J→R
uk∈C0
tk, tk1
, k0, . . . , m, there exist the limits lim
t→t−kut utk,
lim
t→tkut,
lim
t→0ut u0,
lim
t→T−ut uT ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
2.1
with the norm
||u||X||u|| sup
t∈0,T
ut 2.2
foru∈X. Choose
Y X×Rm×Rm 2.3
with the norm
||y||Y ||y||max
sup
t∈0,T
ut , max
1≤k≤m xk ,1max≤k≤m yk
2.4
fory{u, x1, . . . , xm, y1, . . . , ym} ∈Y. Then,XandY are real Banach spaces.
A functionF:0,1×Rn1→Ris called an impulsive Caratheodory function if
iF•, u0, u1, . . . , un∈Xfor eachu u0, . . . , un∈Rn1;
iiFt,•, . . . ,•is continuous fort /tk k1, . . . , m.
By a solution of PBVP1.13, we mean a functionx:0, T→Rsatisfying the following conditions:
ix ∈ X is differentiable in tk, tk1 k 0,1, . . . , m, there exist the limits
limt→tkxt, limt→t−kxt xtk k 0,1, . . . , m, limt→0xt x0 and limt→T−xt xT;
iix ∈ X is differentiable in tk, tk1 k 0,1, . . . , m, there exist the limits
limt→tkxt, limt→t−
kx
t xt
k k 0,1, . . . , m, limt→0xt x0, and limt→T−xt xT;
iiix∈X;
Consider the following homogenous PBVP:
xt αxt βxt 0, t∈0, T,
Δxtk
0, k1, . . . , m,
Δxtk
0, k1, . . . , m, x0 xT, x0 xT.
2.5
Ifx∈Xis a solution of problem2.5, then
xtxt αxtxt βxt2 0,
xtxt αxt2βxtxt 0. 2.6
Integrating them from 0 toT, one sees that
−
T
0
xt2dtβ
T
0
xt2dt0, α
T
0
xt2dt0. 2.7
Ifβ >0 andα /0 orβ <0, we getxt≡0, then problem2.5has unique solutionxt 0 at the cases eitherβ >0 andα /0 orβ <0. We call PBVP1.13at nonresonance case. It suffices
to consider the following cases:
Case 1. α≥0, β <0,
Case 3. α >0, β >0,
Case 2. α <0, β <0,
Case 4. α <0, β >0. 2.8
We set the following assumptions which should be used in the main results. A1xIkx, y≥0 for allx, y∈R.
A1 Ikx, y2xIkx, y≤0 for allx, y∈R.
A2xJkx, y yIkx, y λIkx, yJkx, y≥0 for allx, y∈R, λ∈0,1.
A2xJkx, y yIkx, y λIkx, yJkx, y≤0 for allx, y∈R, λ∈0,1.
A3 There exist constantsθk ≥ 0 such that|Ikx, y| ≤ θk|x|for all x, y ∈ R with
m
k1θk<1.
CThere exist impulsive Caratheodory functionsh:0, T×Rn→R,g
i:0, T×R→R,
and functionr ∈Xand such that ift, x0, . . . , xn ht, x0, . . . , xn
n
i0git, xirtholds for allt, x0, . . . , xn∈
0, T×Rn1.
iiThere exist constantsq≥0 andθ >0 such that
ht, x0, . . . , xnx0≥θ|x0|q1 2.9
holds for allt, x0, . . . , xn∈0, T×Rn1.
iiilim|x| →∞supt∈0,T|git, x|/|x|
q r
i∈0,∞fori0, . . . , n.
CThere exist impulsive Caratheodory functionsh:0, T×Rn→R,g
i:0, T×R→R,
iiThere exist constantsq≥0 andθ >0 such that
ht, x0, . . . , xnx0 ≤ −θ|x0|q1 2.10
holds for allt, x0, . . . , xn∈0, T×Rn1.
Theorem 2.1. Supposeα≥0andβ <0,A1,A2,A3, andChold. Then, PBVP1.13has at
least one solution if
θ > r0 n
k1
rk βk
q/q1 2.11
holds.
Theorem 2.2. Supposeα <0andβ <0,A1,A2,A3, andChold. Then, PBVP1.13has at
least one solution if2.11holds.
Theorem 2.3. Supposeα >0andβ >0,A1,A2,A3, andChold. Then, PBVP1.13has at
least one solution if2.11holds.
Theorem 2.4. Supposeα <0andβ >0,A1,A2,A3, andChold. Then, PBVP1.13has at
least one solution if2.11holds.
To illustrate our main results, we present two boundary value problems that our results can readily apply, whereas the known results in the current literature do not cover.
Example 2.5. Consider the following PBVP:
xt αxt βxt
2q1
k0
kxkt rt, t∈0, T, t /tk, k1, . . . , m,
Δxtk
0, k1, . . . , m,
Δxtk bk
xtk 3
, k1, . . . , m, x0 xT, x0 xT,
2.12
whereqis a positive integer,k≥0,T >0,ris a continuous function. Corresponding to PBVP
1.13, one finds that
ft, x0, x1, . . . , xn
2q1
k0
kxk0rt,
Ikx, y 0, Jkx, y bkx3,
2.13
It is easy to see that
i A1holds,
iisinceak ≥0, one gets thatyIkx, y xJkx, y λIkx, yJkx, y bkx4 ≥0, then
A2holds,
ivht, x 2q1x2q1withα2q1>0 andgkt, x αkxkfork0,1, . . . ,2qimplies that
Cholds.
It follows fromTheorem 2.1that PBVP2.12has at least one solution.
Example 2.6. Consider the following PBVP:
xt αxt βxt
2q1
k0
αkxkt n
k1
βkx2q1
1
n1t
rt,
t∈0, T, t /tk, k1, . . . , m,
Δxtk
0, k1, . . . , m,
Δxtk
bk
xtk 5
, k1, . . . , m,
x0 xT, x0 xT,
2.14
where α ≥ 0, β < 0, T > 0, q is a positive integer,α2q1 > 0, m, n are positive integers,
r∈L10, T,b
k∈R, 0< t1<· · ·< tm< T,αk, βk∈R.
It is easy to see that i A1holds,
iiIt is easy to see that
xJkx, y yIkx, y λIkx, yJkx, y bkx6≥0 2.15
ifbk≥0, thenA2holds,
iii A3holds,
ivif2ht, x α2q1x2q1withα2q1 >0 andgkt, x βkx2q1fork1, . . . , n,g0t, x q
k0αkxk, thenCholds.
It follows fromTheorem 2.2that PBVP2.14has at least one solution if
bk≥0, k1, . . . , m, m
k1
|bk|<1,
α2q1>0, n
k1
βkα2q1/0, n
k1
βkn12q1/2q2< α2q1.
2.16
3. Proofs of theorems
In this section, we prove that main theorems are presented inSection 2. We define the linear operatorL:DL⊆X→Y and the nonlinear operatorN:X→Yby
Lxt
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
xt αxt βxt
Δxt1
.. . Δxtm
Δxt1
.. . Δxtm
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
where
DL
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
u:0, T−→R
u∈X is differentiable intk, tk1
k0,1, . . . , m,
there exist the limits lim
t→tkx t,
lim
t→t−kx
t xt k
k0,1, . . . , m,
lim
t→0x
t x0, lim t→T−x
t xT,
x∈X there exist the limits lim
t→tkx t,
lim
t→t−kx
t xtk k0,1, . . . , m,
lim
t→0x
t x0, lim
t→T−x
t xT,
x∈X,
⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ,
Nxt
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
ft, xt, xα1t
, . . . , xαnt
I1
xt1
, xt1
.. .
Im
xtm
, xtm
J1
xt1
, xt1
.. .
Jm
xtm
, xtm ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
forx∈X.
3.2 Lemma 3.1. Suppose α /0, β /0, or α 0 and β < 0, f : 0, T×Rn1→R is an impulsive
Caratheodory function,Ik, Jk:R2→Rare continuous. Then, the following results hold.
iKerL{0}.
iiLis a Fredholm operator of index zero.
iiiNisL-compact onΩwithΩbeing open and bounded.
Lemma 3.2 15. Let X and Y be Banach spaces. Suppose L : DL ⊂ X→Y is a Fredholm operator of index zero withKerL{0},N:X→Y isL-compact on any open-bounded subset ofX. If0 ∈Ω⊂Xis an open-bounded subset andLx /λNxfor allx∈DL∩∂Ωandλ∈0,1, then there exists at least onex∈Ωsuch thatLxNx.
Lemma 3.3. Supposeβ <0andα≥0andA1,A2,A3, andChold. LetΩ1 {x∈DL:
LxλNx,∃λ∈0,1}.Then,Ω1is bounded if2.11holds.
Proof. Supposex∈Ω1, then
xt αxt βxt λft, xt, xα1t
, . . . , xαnt
, a.e.t∈0, T,
Δxtk
λIk
xtk
, xtk
, k1, . . . , m,
Δxtk
λJk
xtk
, xtk
, k1, . . . , m, x0 xT, x0 xT.
Step 1. Prove that there exists a constantM1>0 so that T
0|xs|
q1ds≤M
1for eachx∈Ω1.
Multiplying both sides of the first equation of3.3byxt, integrating it from 0 toT, we get fromCthat
xTxT−x0x0−
m
k1
xtkxtk−xtk
xtk
−
T
0
xs2ds
α 2
xT2−x02−α
2
m
k1
xtk2−xtk
2
β
T
0 xt 2
dt
λ
T
0
fs, xs, xα1s
, . . . , xαns
xsds
λ
T
0
hs, xs, xα1s
, . . . , xαns
xsds
T
0
g0
s, xsxsds
n
i1 T
0
gi
s, xαis
xsds
T
0
rsxsds
.
3.4
It follows fromA1that
m
k1
xtk2−xtk 2
m
k1
xtk−xtk
xtk xtk
m
k1
Δxtk
2xtk
Δxt−k
λ
m
k1
Ik
xtk
, xtk
2xtk
λIk
xtk
, xtk
≥2λ
m
k1
Ik
xtk
, xtk
xtk
≥0.
3.5
On the other hand,A2implies that
m
k1
xtkxtk−xtk
xtk
m
k1
xtkxtk−xtk
xtk−xtk
xtk
λ
m
k1
xtk
Ik
xtk
, xtk
xtk
Jk
xtk
, xtk
λ2m k1
Ik
xtk
, xtk
Jk
xtk
, xtk
≥0.
3.6
We get
T
0
hs, xs, xα1s
, . . . , xαns
xsds
T
0
g0
s, xsxsds
n
i1 T
0
gi
s, xαis
xsds
T
0
rsxsds≤0.
It follows fromCthat
θ
T
0
xs q1ds≤ −
T
0
g0
s, xsxsds−
n
i1 T
0
gi
s, xαis
xsds
T
0
rsxsdsβ
T
0
xt 2dt
≤
T
0 g0
s, xs xs ds
n
i1 T
0 gi
s, xαis xs ds
T
0
rs xs ds.
3.8
Let >0 satisfy
θ >r0
n
k1
rk βk ∞q/q1. 3.9
For such >0, there existsδ >0 such that for everyi0,1, . . . , n,
git, x <
ri
|x|q for a.e.t∈0, Tand all xsuch that|x|> δ. 3.10
Let, fori1, . . . , n,Δ1,i {t: t∈0, T,|xαit| ≤δ}, Δ2,i {t: t∈0, T,|xαit|> δ},
gδ,i maxt∈0,T,|x|≤δ|git, x|, andΔ1{t∈0, T,|xt| ≤δ},Δ2 {t∈0, T,|xt|> δ},and
δmax{gδ,k:k0, . . . , n}. Then, we get
θ
T
0
xs q1
ds Δ1 g0
s, xs xs ds
Δ2 g0
s, xs xs ds
n
i1
Δ1,i
gi
s, xαis xs ds
n
i1
Δ2,i
gi
s, xαis xs ds T
0
rs xs ds
≤r0
T
0
xs q1
ds
n
k1
rk
T
0 x
αks q xs ds
T
0
rs xs dsgδ,0 T
0
xs ds
n
k1
gδ,k T
0
xs ds
≤r0
T
0
xs q1ds
n
k1
rk
T
0
xαks q1ds
q/q1T
0
xs q1ds
1/q1
T
0
rs q1/qds
q/q1T
0
xs q1ds
1/q1
n1δ
T
0
xs ds
r0
T
0
xs q1ds
n
k1
rk αkT
αk0
xu q1 βku du
q/q1T
0
xs q1ds
T
0
rs q1/q
ds
q/q1T
0
xs q1
ds
1/q1
n1δTq/q1
T
0
xs q1
ds
1/q1
≤r0
T
0
xs q1ds
n
k1
rk βk q/q1
T
0
xu 1q
du
q/q1T
0
xs q1
ds
1/q1
T
0
rs q1/q
ds
q/q1T
0
xs q1
ds
1/q1
n1δTq/q1
T
0
xs q1ds
1/q1
r0
n
k1
rk βk q/q1
T
0
xs q1ds
T
0
rs q1/q
ds
q/q1T
0
xs q1
ds
1/q1
n1δTq/q1
T
0
xs q1
ds
1/q1
.
3.11
Then,3.9implies that there exists a constantM1>0 such that T
0|xs|
q1ds≤M 1.
Step 2. Prove that there exists a constantM2>0 such that||x||∞≤M2for eachx∈Ω1.
It follows from Step1that there existsξ∈0, Tsuch that|xξ| ≤M1/T1/q1.
Multiplying both sides of the first equation of3.3byxt, integrating it from 0 toT, we get, usingA1,A2, andC,
T
0
xs2ds−1
2
m
k1
xtk2−xtk 2
−m k1
xtkxtk−xtkx
tk
−λ
T
0
fs, xs, xα1s
, . . . , xαns
xsdsλβ
T
0 xt 2
dt
≤ −λ
T
0
fs, xs, xα1s
, . . . , xαns
xsdsλβ
T
0
xt 2dt
≤ −λ
T
0
hs, xs, xα1s
, . . . , xαns
xsds
T
0
g0
s, xsxsds
n
i1 T
0
gi
s, xαis
xsds
T
0
rsxsds
λβ
T
0 xt 2
≤ −λ
T
0
g0
s, xsxsds−λ
n
i1 T
0
gi
s, xαis
xsds−λ
T
0
rsxsds
≤
T
0 g0
s, xs xs ds
n
i1 T
0 gi
s, xαis xs ds T
0
rs xs ds
≤ ! r0 n
k1
rk βk q/1q
∞
T
0
xs q1ds
T
0
rs q1/qds
q/q1T
0
xs q1ds
1/q1"
n1δTq/q1
T
0
xs q1ds
1/q1
. 3.12
Then,
T
0
xs2ds≤
!
r0
n
k1
rk βk q/1q
∞
M1
T
0
rs q1/qds
q/q1
M11/q1
"
n1δTq/q1M11/q1
:M2.
3.13 Due toA3, one sees that
xt ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
xξ λ
ξ≤tk<t
Ik
xtk
, xtk
t
ξ
xsds ift≥ξ,
xξ−λ
t≤tk<ξ
Ik
xtk
, xtk
−
ξ
t
xsds
ift < ξ,
≤
M1
T
1/q1 m
k1
θk x T
0
xs ds
≤
M1
T
1/q1 m
k1
θk x T1/2
T
0 xs 2
ds
1/2
≤
M1
T
1/q1
m
k1
θk x T1/2M12/2.
3.14
It follows fromA3that
||x||∞≤ 1
1−mk1θk
M
1
T
1/q1
T1/2M12/2
. 3.15
It follows thatΩ1is bounded. This completes the proof ofLemma 3.3.
Lemma 3.4. Supposeβ <0andα >0andA1,A2,A3, andChold. LetΩ1 {x∈DL:
Proof. The proof is similar to that ofLemma 3.3.
Lemma 3.5. Supposeβ >0andα >0andA1,A2,A3, andChold. LetΩ1 {x∈DL:
LxλNx,∃λ∈0,1}.Then,Ω1is bounded if2.11holds.
Proof. Supposex∈Ω1, then we get3.3.
Step 1. Prove that there exists a constantM1>0 so that T
0|xs|
q1ds≤M
1for eachx∈Ω1.
Multiplying both sides of the first equation of3.3byxt, integrating it from 0 toT, we get fromC
xTxT−x0x0−
m
k1
xtkxtk−xtk
xtk
−
T
0
xs2ds
α 2
xT2−x02−α
2
m
k1
xtk2−xtk
2
β
T
0
xt 2dt
λ
T
0
hs, xs, xα1s
, . . . , xαns
xsds
T
0
g0
s, xsxsds
n
i1 T
0
gi
s, xαis
xsds
T
0
rsxsds
.
3.16
It follows fromA1that
m
k1
xtk2−xtk
2m
k1
xtk−xtk
xtkxtk
m
k1
Δxtk
2xtk
Δxt−k
λ
m
k1
Ik
xtk
, xtk
2xtk
λIk
xtk
, xtk
≤λ
m
k1
Ik
xtk
, xtk
2xtk
Ik
xtk
, xtk
≤0.
3.17
On the other hand,A2implies that
m
k1
xtkxtk−xtk
xtk
m
k1
xtkxtk−xtk
xtk−xtk
xtk
λ
m
k1
xtk
Ik
xtk
, xtk
xtk
Jk
xtk
, xtk
λ2
m
k1
Ik
xtk
, xtk
Jk
xtk
, xtk
≤0.
We get
λ
T
0
hs, xs, xα1s
, . . . , xαns
xsds
T
0
g0
s, xsxsds
n
i1 T
0
gi
s, xαis
xsds
T
0
rsxsds
≥0.
3.19
It follows fromCthat
θ
T
0
xs q1
ds≤
T
0
g0
s, xsxsds−
n
i1 T
0
gi
s, xαis
xsds
T
0
rsxsdsβ
T
0 xt 2
dt
≤
T
0 g0
s, xs xs ds
n
i1 T
0 gi
s, xαis xs ds
T
0
rs xs ds.
3.20
The remainder of the proof is similar to that of the corresponding part of the proof of
Lemma 3.3.
Lemma 3.6. Supposeβ >0andα <0andA1,A2,A3, andChold. LetΩ1 {x∈DL:
LxλNx,∃λ∈0,1}.Then,Ω1is bounded if2.11holds.
Proof. The proof is similar to that ofLemma 3.5.
Proof ofTheorem 2.1. It is easy to show thatL:DL⊂X→Y is a Fredholm operator of index zero with KerL {0},N : X→Y isL-compact on any open-bounded subset ofX. Let Ω satisfy 0∈Ω1⊂Ω⊂Xwhich is a open-bounded subset, whereΩ1is given inLemma 3.3. It
follows fromLemma 3.3thatLx /λNxfor allx∈DL∩∂Ωandλ∈0,1, then there exists
at least onex∈Ωsuch thatLxNx. Hence,xis a solution of PBVP1.13.
Proof ofTheorem 2.2. FollowingLemma 3.4, the proof is similar to that ofTheorem 2.1.
Proof ofTheorem 2.3. FollowingLemma 3.5, the proof is similar to that ofTheorem 2.1.
Proof ofTheorem 2.4. FollowingLemma 3.6, the proof is similar to that ofTheorem 2.1.
Acknowledgments
References
1 T. R. Ding, “Nonlinear oscillations at a point of resonance,”Scientia Sinica Series A, vol. 25, no. 9, pp. 918–931, 1982.
2 W. Ding, M. Han, and J. Yan, “Periodic boundary value problems for the second order functional differential equations,”Journal of Mathematical Analysis and Applications, vol. 298, no. 1, pp. 341–351, 2004.
3 D. Jiang, J. J. Nieto, and W. Zuo, “On monotone method for first and second order periodic boundary value problems and periodic solutions of functional differential equations,”Journal of Mathematical Analysis and Applications, vol. 289, no. 2, pp. 691–699, 2004.
4 J. J. Nieto and R. Rodr´ıguez-L ´opez, “Existence and approximation of solutions for nonlinear functional differential equations with periodic boundary value conditions,”Computers & Mathematics with Applications, vol. 40, no. 4-5, pp. 433–442, 2000.
5 J. J. Nieto and R. Rodr´ıguez-L ´opez, “Remarks on periodic boundary value problems for functional differential equations,”Journal of Computational and Applied Mathematics, vol. 158, no. 2, pp. 339–353, 2003.
6 D. Jiang and J. Wei, “Monotone method for first- and second-order periodic boundary value problems and periodic solutions of functional differential equations,”Nonlinear Analysis: Theory, Methods & Applications, vol. 50, no. 7, pp. 885–898, 2002.
7 J. J. Nieto and N. Alvarez-Noriega, “Periodic boundary value problems for nonlinear first order ordinary differential equations,”Acta Mathematica Hungarica, vol. 71, no. 1-2, pp. 49–58, 1996. 8 S. Ma and J. Wu, “A small twist theorem and boundedness of solutions for semilinear Duffing
equations at resonance,”Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 1, pp. 200– 237, 2007.
9 R. Liang and J. Shen, “Periodic boundary value problem for second-order impulsive functional differential equations,”Applied Mathematics and Computation, vol. 193, no. 2, pp. 560–571, 2007. 10 S. Ma, Z. Wang, and J. Yu, “Coincidence degree and periodic solutions of Duffing equations,”
Nonlinear Analysis: Theory, Methods & Applications, vol. 34, no. 3, pp. 443–460, 1998.
11 L. Peng, “Existence and uniqueness of periodic solutions for a kind of Duffing equation with two deviating arguments,”Mathematical and Computer Modelling, vol. 45, no. 3-4, pp. 378–386, 2007. 12 A. Cabada, “The method of lower and upper solutions for second, third, fourth, and higher order
boundary value problems,”Journal of Mathematical Analysis and Applications, vol. 185, no. 2, pp. 302– 320, 1994.
13 P. Omari and M. Trombetta, “Remarks on the lower and upper solutions method for second- and third-order periodic boundary value problems,”Applied Mathematics and Computation, vol. 50, no. 1, pp. 1–21, 1992.
14 I. Kiguradze and S. Stanˇek, “On periodic boundary value problem for the equationu ft, u, u with one-sided growth restrictions onf,”Nonlinear Analysis: Theory, Methods & Applications, vol. 48, no. 7, pp. 1065–1075, 2002.
15 R. E. Gaines and J. L. Mawhin,Coincidence Degree, and Nonlinear Differential Equations, Lecture Notes in Mathematics, Vol. 568, Springer, Berlin, Germany, 1977.
16 C. P. Gupta, J. J. Nieto, and L. Sanchez, “Periodic solutions of some Lienard and Duffing equations,”
Journal of Mathematical Analysis and Applications, vol. 140, no. 1, pp. 67–82, 1989.
17 J. J. Nieto and R. Rodr´ıguez-L ´opez, “Green’s function for second-order periodic boundary value problems with piecewise constant arguments,”Journal of Mathematical Analysis and Applications, vol. 304, no. 1, pp. 33–57, 2005.
18 Z. Zhang and J. Wang, “On existence and multiplicity of positive solutions to periodic boundary value problems for singular nonlinear second order differential equations,”Journal of Mathematical Analysis and Applications, vol. 281, no. 1, pp. 99–107, 2003.
19 D. Jiang, “On the existence of positive solutions to second order periodic BVPs,”Acta Mathematica Sinica, vol. 18, no. 1, pp. 31–35, 1989.
20 J.-M. Belley and M. Virgilio, “Periodic Duffing delay equations with state dependent impulses,”
Journal of Mathematical Analysis and Applications, vol. 306, no. 2, pp. 646–662, 2005.
22 H. Wang and H. Chen, “Boundary value problem for second-order impulsive functional differential equations,”Applied Mathematics and Computation, vol. 191, no. 2, pp. 582–591, 2007.
23 X. Yang and J. Shen, “Periodic boundary value problems for second-order impulsive integro-differential equations,”Journal of Computational and Applied Mathematics, vol. 209, no. 2, pp. 176–186, 2007.