Hindawi Publishing Corporation
Boundary Value Problems
Volume 2008, Article ID 937138, 16 pages
doi:10.1155/2008/937138
Research Article
Existence of Solutions of Periodic Boundary
Value Problems for Impulsive Functional Duffing
Equations at Nonresonance Case
Xingyuan Liu1 and Yuji Liu2
1
2
Department of Mathematics, Shaoyang University, Shaoyang, Hunan 422000, China
Department of Mathematics, Guangdong University of Business Studies, Guangzhou,
Guangdong 510320, China
Correspondence should be addressed to Xingyuan Liu, liuxingyuan999@sohu.com
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: x t αx t βxt ft, xt, xα1 t, . . . , xαn t, a.e. t ∈
0, T , Δxtk Ik xtk , x tk , k 1, . . . , m, Δx tk Jk xtk , x tk , k 1, . . . , m, xi 0 xi T , 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.
Copyright q 2008 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; see 1–24 and the references therein. For example, consider
the following PBVP:
x t f t, xt , t ∈ 0, 2π,
1.1
x0 x2π,
x 0 x 2π,
the well-known result is that if f satisfies the nonresonance condition
−N 12 ≤ fu t, u ≤ − − N 2 ,
1.2
where N is a nonnegative integer and is a positive constant, then PBVP1.1 has a unique
solution; see 1.
2
Boundary Value Problems
For PBVP of the Duffing equation
x t cx t g t, xt et,
x0 x2π,
t ∈ 0, 2π,
1.3
x 0 x 2π,
in 16, the authors proved the following results.
Theorem 1.1. Suppose g is a L2 -Caratheodory function, and there are a ≤ A, r < 0 < R such that
gt, x ≥ A,
x ≥ R, t ∈ 0, 2π,
gt, x ≤ a,
x ≤ r, t ∈ 0, 2π,
1.4
and further there is r ∈ L0, 2π with ||r||∞ < 1 C2 such that
lim
|x| → ∞
gt, x
< rt,
x
t ∈ 0, 2π.
1.5
Then, PBVP1.3 has at least one solution for each e ∈ L2 0, 2π with a ≤ 1/2π
2π
0
esds ≤ A.
In 17, Nieto and Rodrı́guez-López studied the following PBVP:
x t ax t bxt cx t dx t σt,
x0 xT ,
t ∈ 0, T ,
x 0 x T .
1.6
They gave Green’s function to express the unique solution for the correspondence secondorder 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.
In 2, Ding et al. studied the PBVP:
x t f t, xt, x θt 0,
x0 xT ,
t ∈ 0, T ,
x 0 x T .
1.7
Sufficient conditions for the existence of solutions of PBVP1.7 are given by using upper and
lower solution method.
The PBVPs,
x t ρ2 xt f t, xt , t ∈ 0, 2π,
1.8
x0 x2π,
x 0 x 2π,
−x t ρ2 xt f t, xt ,
x0 x2π,
t ∈ 0, 2π,
x 0 x 2π,
1.9
were studied in 8, 18–20. In 19, based upon Krasnosel skii fixed-point theorems, Jiang
proved that PBVP1.8 and PBVP1.9 with singularity have at least one positive solution
provided that ft, x is superlinear or sublinear at x 0 and x ∞. In 21, Wang, who
utilized the Schauder fixed-point theorem, proved that PBVP1.9 without singularity has at
X. Liu and Y. Liu
3
least one positive solution provided that ft, x is sublinear at x ∞. In 18, Zhang and
Wang established multiplicity results to positive solutions of PBVP1.8 and PBVP1.9 with
f being a Caratheodory function.
In paper 24, Liu and Ge studied the following equation:
x t − ptx t qtxt λf t, x t − τt rt;
1.10
they established sufficient conditions for the existence of positive periodic solutions of 1.10.
In 11, Peng studied the existence of periodic solutions of the functional Duffing
equation:
x t cx t g1 t, x t − τ1 t g2 t, x t − τ2 t pt.
1.11
The studies on the existence of solutions of the periodic boundary value problems of
the second-order impulsive differential equations,
x t f t, xt 0,
t ∈ 0, T ,
x0 xT ,
x 0 x T ,
Δxi tk Ii, k x tk , k 1, . . . , p, i 0, 1,
1.12
can be found in 3, 4, 12–14, 20, 22, 23 and 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.
H f 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:
x t αx t βxt f t, xt, x α1 t , . . . , x αn t , a.e. t ∈ 0, T ,
Δx tk Ik x tk , x tk , k 1, . . . , m,
Δx tk Jk x tk , x tk , k 1, . . . , m,
xi 0 xi T ,
1.13
i 0, 1,
where m is a positive integer, α, β ∈ R, f : 0, T × Rn1 → R is a impulsive Caratheodory
function, αi ∈ C1 0, T , 0, T , whose inverse function is denoted by βi with βi ∈ C1 0, T ,
Ik , Jk : R2 → R are continuous.
Our purpose here is to provide sufficient conditions for the existence of solutions of
1.13 at nonresonance case. This will be done by applying the well-known Schaeffer’s fixedpoint theorem, and we do not rely on the existence of upper and lower solutions and the
assumption H mentioned above.
4
Boundary Value Problems
The main results and examples in this paper are established in Section 2. The proofs of
the main results are presented in Section 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 PBVP 1.13, we introduce the following Banach spaces and definitions.
Suppose u : J 0, T → R, and 0 t0 < t1 < · · · < tm < tm1 T . For k 0, . . . , m,
define the function uk : tk , tk1 → R by uk t ut. Choose
X
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎫
lim− ut utk , ⎪
⎪
t → tk
⎪
⎪
⎪
⎪
⎪
lim ut,
⎬
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
t → 0
u : J → R uk ∈ C0 tk , tk1 , k 0, . . . , m, there exist the limits
t → tk
lim ut u0, ⎪
⎪
⎪
⎪
⎪
⎪
⎪
lim− ut uT ⎭
t→T
2.1
with the norm
||u||X ||u|| sup ut
2.2
Y X × Rm × Rm
2.3
t∈0,T for u ∈ X. Choose
with the norm
||y||Y ||y|| max
sup ut , max
t∈0,T 1≤k≤m
xk , max yk
1≤k≤m
2.4
for y {u, x1 , . . . , xm , y1 , . . . , ym } ∈ Y . Then, X and Y are real Banach spaces.
A function F : 0, 1 × Rn1 → R is called an impulsive Caratheodory function if
i F•, u0 , u1 , . . . , un ∈ X for each u u0 , . . . , un ∈ Rn1 ;
ii Ft, •, . . . , • is continuous for t /
tk k 1, . . . , m.
By a solution of PBVP 1.13, we mean a function x : 0, T → R satisfying the following
conditions:
i x ∈ X is differentiable in tk , tk1 k 0, 1, . . . , m, there exist the limits
limt → tk x t, limt → t−k x t x tk k 0, 1, . . . , m, limt → 0 x t x 0 and
limt → T − x t x T ;
ii x ∈ X is differentiable in tk , tk1 k 0, 1, . . . , m, there exist the limits
limt → tk x t, limt → t−k x t x tk k 0, 1, . . . , m, limt → 0 x t x 0, and
limt → T − x t x T ;
iii x ∈ X;
iv the equations in 1.13 are satisfied.
X. Liu and Y. Liu
5
Consider the following homogenous PBVP:
x t αx t βxt 0, t ∈ 0, T ,
Δx tk 0, k 1, . . . , m,
Δx tk 0, k 1, . . . , m,
x0 xT ,
2.5
x 0 x T .
If x ∈ X is a solution of problem 2.5, then
2
x txt αx txt β xt 0,
2
x tx t α x t βxtx t 0.
2.6
Integrating them from 0 to T , one sees that
−
T
0
2
x t dt β
T
2
xt dt 0,
0
T
2
α
x t dt 0.
2.7
0
If β > 0 and α /
0 or β < 0, we get xt ≡ 0, then problem 2.5 has unique solution xt 0 at
the cases either β > 0 and α / 0 or β < 0. We call PBVP 1.13 at nonresonance case. It suffices
to consider the following cases:
Case 1. α ≥ 0, β < 0,
Case 2. α < 0, β < 0,
Case 3. 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.
A1 xIk x, y ≥ 0 for all x, y ∈ R.
A1 Ik x, y2 xIk x, y ≤ 0 for all x, y ∈ R.
A2 xJk x, y yIk x, y λIk x, yJk x, y ≥ 0 for all x, y ∈ R, λ ∈ 0, 1.
A2 xJk x, y yIk x, y λIk x, yJk x, y ≤ 0 for all x, y ∈ R, λ ∈ 0, 1.
A3 There exist constants θk ≥ 0 such that |Ik x, y| ≤ θk |x| for all x, y ∈ R with
m
θ
< 1.
k
k1
C There exist impulsive Caratheodory functions h : 0, T × Rn → R, gi : 0, T × R → R,
and function r ∈ X and such that
i ft, x0 , . . . , xn ht, x0 , . . . , xn ni0 gi t, xi rt holds for all t, x0 , . . . , xn ∈
0, T × Rn1 .
ii There exist constants q ≥ 0 and θ > 0 such that
ht, x0 , . . . , xn x0 ≥ θ|x0 |q1
2.9
holds for all t, x0 , . . . , xn ∈ 0, T × Rn1 .
iii lim|x| → ∞ supt∈0,T |gi t, x|/|x|q ri ∈ 0, ∞ for i 0, . . . , n.
C There exist impulsive Caratheodory functions h : 0, T ×Rn → R, gi : 0, T ×R → R,
and function r ∈ X and such that Ci and Ciii hold.
6
Boundary Value Problems
ii There exist constants q ≥ 0 and θ > 0 such that
ht, x0 , . . . , xn x0 ≤ −θ|x0 |q1
2.10
holds for all t, x0 , . . . , xn ∈ 0, T × Rn1 .
Theorem 2.1. Suppose α ≥ 0 and β < 0, A1 , A2 , A3 , and C hold. Then, PBVP 1.13 has at
least one solution if
θ > r0 n
rk βk
q/q1
2.11
k1
holds.
Theorem 2.2. Suppose α < 0 and β < 0, A1 , A2 , A3 , and C hold. Then, PBVP 1.13 has at
least one solution if 2.11 holds.
Theorem 2.3. Suppose α > 0 and β > 0, A1 , A2 , A3 , and C hold. Then, PBVP 1.13 has at
least one solution if 2.11 holds.
Theorem 2.4. Suppose α < 0 and β > 0, A1 , A2 , A3 , and C hold. Then, PBVP 1.13 has at
least one solution if 2.11 holds.
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:
x t αx t βxt 2q1
k xk t rt,
t ∈ 0, T , t / tk , k 1, . . . , m,
k0
Δx tk 0, k 1, . . . , m,
3
Δx tk bk x tk , k 1, . . . , m,
x0 xT ,
2.12
x 0 x T ,
where q is a positive integer, k ≥ 0, T > 0, r is a continuous function. Corresponding to PBVP
1.13, one finds that
2q1
f t, x0 , x1 , . . . , xn k x0k rt,
k0
Ik x, y 0,
2.13
Jk x, y bk x3 ,
It is easy to see that
i A1 holds,
ii since ak ≥ 0, one gets that yIk x, y xJk x, y λIk x, yJk x, y bk x4 ≥ 0, then
A2 holds,
iii A3 holds,
X. Liu and Y. Liu
7
iv ht, x 2q1 x2q1 with α2q1 > 0 and gk t, x αk xk for k 0, 1, . . . , 2q implies that
C holds.
It follows from Theorem 2.1 that PBVP 2.12 has at least one solution.
Example 2.6. Consider the following PBVP:
x t αx t βxt 2q1
αk x t k
k0
n
βk x
2q1
k1
1
t rt,
n1
t ∈ 0, T , t /
tk , k 1, . . . , m,
Δx tk 0, k 1, . . . , m,
5
Δx tk bk x tk , k 1, . . . , m,
x0 xT ,
2.14
x 0 x T ,
where α ≥ 0, β < 0, T > 0, q is a positive integer, α2q1 > 0, m, n are positive integers,
r ∈ L1 0, T , bk ∈ R, 0 < t1 < · · · < tm < T , αk , βk ∈ R.
It is easy to see that
i A1 holds,
ii It is easy to see that
xJk x, y yIk x, y λIk x, yJk x, y bk x6 ≥ 0
2.15
if bk ≥ 0, then A2 holds,
iii A3 holds,
iv if ht, x α2q1 x2q1 with α2q1 > 0 and gk t, x βk x2q1 for k 1, . . . , n, g0 t, x 2q
α xk , then C holds.
k0 k
It follows from Theorem 2.2 that PBVP 2.14 has at least one solution if
m
bk ≥ 0, k 1, . . . , m,
|bk | < 1,
α2q1 > 0,
n
k1
βk α2q1 / 0,
n
k1
βk n 12q1/2q2 < α2q1 .
2.16
k1
3. Proofs of theorems
In this section, we prove that main theorems are presented in Section 2. We define the linear
operator L : DL ⊆ X → Y and the nonlinear operator N : X → Y by
⎛ ⎞
x t αx t βxt
⎜
⎟
⎜
⎟
Δx t1
⎜
⎟
⎜
⎟
..
⎜
⎟
⎜
⎟
.
⎜
⎟
⎜
⎟
⎟ for x ∈ DL,
3.1
Δx
t
Lxt ⎜
m
⎜
⎟
⎜
⎟
⎜
⎟
Δx t1
⎜
⎟
⎜
⎟
⎜
⎟
..
⎜
⎟
.
⎝
⎠
Δx tm
8
Boundary Value Problems
where
DL ⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
u : 0, T −→ R
⎫
u ∈ X is differentiable in tk , tk1 k 0, 1, . . . , m, ⎪
⎪
⎪
⎪
⎪
⎪
⎪
there exist the limits lim x t,
⎪
⎪
t → tk
⎪
⎪
⎪
⎪
lim− x t x tk k 0, 1, . . . , m,
⎪
⎪
⎪
t → tk
⎪
⎪
⎪
⎬
lim x t x 0, lim x t x T ,
t → 0
t → T−
x ∈ X there exist the limits lim x t,
t → tk
lim− x t x tk k 0, 1, . . . , m,
t → tk
lim x t x 0, lim− x t x T ,
t → 0
x ∈ X,
⎞
⎛ f t, xt, x α1 t , . . . , x αn t
⎟
⎜
⎟
⎜
I 1 x t 1 , x t1
⎟
⎜
⎟
⎜
..
⎟
⎜
⎟
⎜
.
⎟
⎜
⎟
⎜
⎟
x
t
,
x
t
I
Nxt ⎜
m
m
m
⎟
⎜
⎟
⎜
⎟
⎜
J 1 x t 1 , x t1
⎟
⎜
⎟
⎜
⎟
⎜
..
⎟
⎜
.
⎠
⎝
J m x t m , x tm
t→T
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎭
,
for x ∈ 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 → R are continuous. Then, the following results hold.
i Ker L {0}.
ii L is a Fredholm operator of index zero.
iii N is L-compact on Ω with Ω being 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 with Ker L {0}, N : X → Y is L-compact on any open-bounded subset of X.
If 0 ∈ Ω ⊂ X is an open-bounded subset and Lx / λNx for all x ∈ DL ∩ ∂Ω and λ ∈ 0, 1, then
there exists at least one x ∈ Ω such that Lx Nx.
Lemma 3.3. Suppose β < 0 and α ≥ 0 and A1 , A2 , A3 , and C hold. Let Ω1 {x ∈ DL :
Lx λNx, ∃λ ∈ 0, 1}. Then, Ω1 is bounded if 2.11 holds.
Proof. Suppose x ∈ Ω1 , then
x t αx t βxt λf t, xt, x α1 t , . . . , x αn t , a.e. t ∈ 0, T ,
Δx tk λIk x tk , x tk , k 1, . . . , m,
Δx tk λJk x tk , x tk , k 1, . . . , m,
x0 xT ,
x 0 x T .
3.3
X. Liu and Y. Liu
9
T
Step 1. Prove that there exists a constant M1 > 0 so that 0 |xs|q1 ds ≤ M1 for each x ∈ Ω1 .
Multiplying both sides of the first equation of 3.3 by xt, integrating it from 0 to T ,
we get from C that
T
m
2
x T xT − x 0x0 −
x tk x t k − x tk x t k −
x s ds
0
k1
T
m
2 α 2 2 2 α 2
xt dt
− x tk
−
x tk
β
xT − x0
2
2 k1
0
T
λ f s, xs, x α1 s , . . . , x αn s xsds
0 T
T
λ
h s, xs, x α1 s , . . . , x αn s xsds g0 s, xs xsds
0
0 T
n T
gi s, x αi s xsds rsxsds .
i1
0
3.4
0
It follows from A1 that
m
m
2 2 − x tk
x tk
x tk − x tk
x tk x tk
k1
k1
m
Δx tk 2x tk Δx t−k
k1
m
λ I k x t k , x tk
2x tk λIk x tk , x tk
3.5
k1
m
≥ 2λ Ik x tk , x tk x tk ≥ 0.
k1
On the other hand, A2 implies that
m
m
x tk x tk − x tk x tk x tk x tk − x tk x tk − x tk x tk
k1
k1
m
λ
x tk I k x t k , x tk x tk J k x t k , x tk
k1
λ2
m
3.6
I k x t k , x tk J k x t k , x tk
k1
≥ 0.
We get
T
T
h s, xs, x α1 s , . . . , x αn s xsds g0 s, xs xsds
0
0
T
n T
gi s, x αi s xsds rsxsds ≤ 0.
i1
0
0
3.7
10
Boundary Value Problems
It follows from C that
T
T
n T
q1
xs
θ
ds ≤ − g0 s, xs xsds −
gi s, x αi s xsds
0
0
T
≤
T
rsxsds β
T
2
xt dt
0
0
g0 s, xs
xs ds 0
0
i1
n T
gi s, x αi s
3.8
xs ds
0
i1
T
rs xs ds.
0
Let > 0 satisfy
n
θ > r0 rk βk
k1
q/q1
.
∞
3.9
For such > 0, there exists δ > 0 such that for every i 0, 1, . . . , n,
gi t, x < ri |x|q for a.e. t ∈ 0, T and all x such that |x| > δ.
3.10
Let, for i 1, . . . , n, Δ1,i {t : t ∈ 0, T , |xαi t| ≤ δ}, Δ2,i {t : t ∈ 0, T , |xαi t| > δ},
gδ,i maxt∈0,T ,|x|≤δ |gi t, x|, and Δ1 {t ∈ 0, T , |xt| ≤ δ}, Δ2 {t ∈ 0, T , |xt| > δ}, and
δ max{gδ,k : k 0, . . . , n}. Then, we get
T
q1
xs
θ
ds
0
g0 s, xs
Δ1
n Δ2,i
i1
xs ds n xs ds i1
Δ1,i
xs ds
gi s, x αi s
T
rs xs ds
0
q1
ds 0
T
Δ2
g0 s, xs
xs ds gi s, x αi s
T
≤ r0 xs
n
rk T
0
xs ds 0
T
≤ r0 xs
q1
x αk s
q
xs ds
0
k1
rs xs ds gδ,0
T
n
T
xs ds
gδ,k
0
k1
ds
0
n
rk T
x αk s
q1
T
rs
q1
1/q1
ds
0
q1/q
q/q1 T
xs
ds
0
r0 q/q1 T
xs
ds
0
k1
q1
T
1/q1
xs ds
ds
n 1δ
0
T
0
xs
q1
ds
n
rk k1
0
αk T αk 0
xu
q1
βk u du
q/q1 T
xs
0
q1
1/q1
ds
X. Liu and Y. Liu
11
T
q1/q
rs
q/q1 T
xs
ds
0
1/q1
ds
0
n 1δ T q/q1
≤ r0 q1
T
T
xs
q1
1/q1
ds
0
q1
xs
ds
0
n
rk βk
q/q1
T
xu
rs
q1/q
q/q1 T
xs
ds
T
xs
q1
1/q1
ds
q1
1/q1
ds
1/q1
ds
0
r0 n
rk q/q1
βk
k1
q1
0
n 1δ T q/q1
xs
0
0
q/q1 T
du
0
k1
T
1q
T
rs
q1/q
T
xs
q/q1 T
xs
ds
0
q1
ds
0
q1
1/q1
ds
0
n 1δ T q/q1
T
xs
q1
1/q1
ds
.
0
3.11
Then, 3.9 implies that there exists a constant M1 > 0 such that
T
0
|xs|q1 ds ≤ M1 .
Step 2. Prove that there exists a constant M2 > 0 such that ||x||∞ ≤ M2 for each x ∈ Ω1 .
It follows from Step 1 that there exists ξ ∈ 0, T such that |xξ| ≤ M1 /T 1/q1 .
Multiplying both sides of the first equation of 3.3 by xt, integrating it from 0 to T ,
we get, using A1 , A2 , and C,
T
m
m
2
2 2 1
− x tk
x s ds −
x tk
−
x tk x tk − x tk x tk
2 k1
0
k1
T
−λ
f s, xs, x α1 s , . . . , x αn s xsds λβ
0
≤ −λ
T
T
2
xt dt
0
f s, xs, x α1 s , . . . , x αn s xsds λβ
0
T
≤ −λ
T
2
xt dt
0
h s, xs, x α1 s , . . . , x αn s xsds 0
T
g0 s, xs xsds
0
T
T
n T
2
xt dt
gi s, x αi s xsds rsxsds λβ
i1
0
0
0
12
Boundary Value Problems
T
≤ −λ
T
n T
g0 s, xs xsds − λ
gi s, x αi s xsds − λ rsxsds
0
≤
T
g0 s, xs
xs ds 0
!
≤
n T
i1
r0 n
rk T
rs
q1/q
0
gi s, x αi s
xs ds T
0
T
xs
q1
ds
0
q/q1 T
xs
ds
0
rs xs ds
0
q/1q
βk ∞
k1
0
i1
q1
1/q1 "
ds
0
q/q1
n 1δ T
T
xs
q1
1/q1
ds
.
3.12
0
Then,
!
T
n
2
x s ds ≤
rk βk
r0 0
k1
q/1q
∞
M1 T
rs
q1/q
0
"
q/q1
1/q1
ds
M1
1/q1
n 1δ T q/q1 M1
: M2 .
3.13
Due to A3 , one sees that
⎧
t
⎪
⎪
⎪
xξ
λ
Ik x tk , x tk x sds
⎪
⎪
⎨
ξ
ξ≤tk <t
xt ⎪
ξ
⎪
⎪
⎪
xξ
−
λ
Ik x tk , x tk − x sds
⎪
⎩
≤
≤
M1
T
M1
T
M1
T
1/q1
m
θk x T
m
θk x T
m
1/2
T
1/2
x s ds
2
θk x T 1/2 M21/2 .
k1
It follows from A3 that
||x||∞ ≤
3.14
0
k1
1/q1
x s ds
0
k1
1/q1
if t < ξ,
t
t≤tk <ξ
≤
if t ≥ ξ,
1−
1
m
k1
θk
M1
T
1/q1
T
1/2
M21/2
.
3.15
It follows that Ω1 is bounded. This completes the proof of Lemma 3.3.
Lemma 3.4. Suppose β < 0 and α > 0 and A1 , A2 , A3 , and C hold. Let Ω1 {x ∈ DL :
Lx λNx, ∃λ ∈ 0, 1}. Then, Ω1 is bounded if 2.11 holds.
X. Liu and Y. Liu
13
Proof. The proof is similar to that of Lemma 3.3.
Lemma 3.5. Suppose β > 0 and α > 0 and A1 , A2 , A3 , and C hold. Let Ω1 {x ∈ DL :
Lx λNx, ∃λ ∈ 0, 1}. Then, Ω1 is bounded if 2.11 holds.
Proof. Suppose x ∈ Ω1 , then we get 3.3.
T
Step 1. Prove that there exists a constant M1 > 0 so that 0 |xs|q1 ds ≤ M1 for each x ∈ Ω1 .
Multiplying both sides of the first equation of 3.3 by xt, integrating it from 0 to T ,
we get from C T
m
2
x T xT − x 0x0 −
x tk x t k − x tk x t k −
x s ds
0
k1
T
m
2 α 2 2 2 α 2
xt dt
− x tk
−
x tk
β
xT − x0
2
2 k1
0
T
T
h s, xs, x α1 s , . . . , x αn s xsds g0 s, xs xsds
λ
0
0
T
n T
gi s, x αi s xsds rsxsds .
i1
0
3.16
0
It follows from A1 that
m
m
2 2 − x tk
x tk
x tk − x tk
x tk x tk
k1
k1
m
Δx tk 2x tk Δx t−k
k1
m
λ I k x t k , x tk
2x tk λIk x tk , x tk
3.17
k1
m
≤ λ I k x t k , x tk
2x tk Ik x tk , x tk
k1
≤ 0.
On the other hand, A2 implies that
m
m
x tk x t k − x tk x t k x tk x tk − x tk x tk − x tk x tk
k1
k1
m
λ
x tk I k x t k , x tk x tk J k x t k , x tk
k1
λ2
m
I k x t k , x tk J k x t k , x tk
k1
≤ 0.
3.18
14
Boundary Value Problems
We get
T
λ
h s, xs, x α1 s , . . . , xαn s xsds T
0
g0 s, xs xsds
0
n T
i1
gi s, x αi s xsds 0
3.19
T
≥ 0.
rsxsds
0
It follows from C that
T
xs
θ
q1
ds ≤
T
0
n
g0 s, xs xsds −
0
≤
i1
T
T
rsxsds β
T
0
g0 s, xs
xs ds 0
gi s, x αi s xsds
0
2
xt dt
0
T
n T
i1
gi s, x αi s
3.20
xs ds
0
T
rs xs ds.
0
The remainder of the proof is similar to that of the corresponding part of the proof of
Lemma 3.3.
Lemma 3.6. Suppose β > 0 and α < 0 and A1 , A2 , A3 , and C hold. Let Ω1 {x ∈ DL :
Lx λNx, ∃λ ∈ 0, 1}. Then, Ω1 is bounded if 2.11 holds.
Proof. The proof is similar to that of Lemma 3.5.
Proof of Theorem 2.1. It is easy to show that L : DL ⊂ X → Y is a Fredholm operator of index
zero with Ker L {0}, N : X → Y is L-compact on any open-bounded subset of X. Let Ω
satisfy 0 ∈ Ω1 ⊂ Ω ⊂ X which is a open-bounded subset, where Ω1 is given in Lemma 3.3. It
follows from Lemma 3.3 that Lx /
λNx for all x ∈ DL ∩ ∂Ω and λ ∈ 0, 1, then there exists
at least one x ∈ Ω such that Lx Nx. Hence, x is a solution of PBVP 1.13.
Proof of Theorem 2.2. Following Lemma 3.4, the proof is similar to that of Theorem 2.1.
Proof of Theorem 2.3. Following Lemma 3.5, the proof is similar to that of Theorem 2.1.
Proof of Theorem 2.4. Following Lemma 3.6, the proof is similar to that of Theorem 2.1.
Acknowledgments
The authors are very grateful to the referee for his or her suggestions to rearrange the paper
details. The first author is supported by Science Foundation of Educational Committee of
Hunan province 08C794, and both authors are supported by Natural Science Foundation
of Hunan province, China no. 06JJ5008 and Natural Science Foundation of Guangdong
province no. 7004569.
X. Liu and Y. Liu
15
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