Hindawi Publishing Corporation
Boundary Value Problems
Volume 2010, Article ID 325415, 16 pages
doi:10.1155/2010/325415
Research Article
Variational Approach to Impulsive Differential
Equations with Dirichlet Boundary Conditions
Huiwen Chen and Jianli Li
Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, China
Correspondence should be addressed to Jianli Li, ljianli@sina.com
Received 18 September 2010; Accepted 9 November 2010
Academic Editor: Zhitao Zhang
Copyright q 2010 H. Chen and J. Li. 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.
We study the existence of n distinct pairs of nontrivial solutions for impulsive differential equations
with Dirichlet boundary conditions by using variational methods and critical point theory.
1. Introduction
Impulsive effects exist widely in many evolution processes in which their states are changed
abruptly at certain moments of time. Such processes are naturally seen in control theory 1, 2,
population dynamics 3, and medicine 4, 5. Due to its significance, a great deal of work has
been done in the theory of impulsive differential equations. In recent years, many researchers
have used some fixed point theorems 6, 7, topological degree theory 8, and the method
of lower and upper solutions with monotone iterative technique 9 to study the existence of
solutions for impulsive differential equations.
On the other hand, in the last few years, some researchers have used variational
methods to study the existence of solutions for boundary value problems 10–16, especially,
in 14–16, the authors have studied the existence of infinitely many solutions by using
variational methods.
However, as far as we know, few researchers have studied the existence of n distinct
pairs of nontrivial solutions for impulsive boundary value problems by using variational
methods.
2
Boundary Value Problems
Motivated by the above facts, in this paper, our aim is to study the existence of n
distinct pairs of nontrivial solutions to the Dirichlet boundary problem for the second-order
impulsive differential equations
tj , a.e. t ∈ 0, T ,
u t λht, ut 0, t /
−Δu tj Ij u tj , j 1, 2, . . . , p,
1.1
u0 uT 0,
where 0 t0 < t1 < · · · < tp < tp1 T , λ > 0, h ∈ C0, T × R, R, Ij ∈ CR, R, j 1, 2, . . . , p,
Δu tj u tj − u t−j , u tj and u t−j denote the right and the left limits, respectively, of
u tj at t tj , j 1, 2, . . . , p.
2. Preliminaries
Definition 2.1. Suppose that E is a Banach space and ϕ ∈ C1 E, R. If any sequence {uk } ⊂ E
for which ϕuk is bounded and ϕ uk → 0 as k → ∞ possesses a convergent subsequence
in E, we say that ϕ satisfies the Palais-Smale condition.
Let E be a real Banach space. Define the set Σ {A | A ⊂ E \ {θ} as symmetric closed
set}.
Theorem 2.2 see 17, Theorem 3.5.3. Let E be a real Banach space, and let ϕ ∈ C1 E, R
be an even functional which satisfies the Palais-Smale condition, ϕ is bounded from below and
ϕ0 0; suppose that there exists a set K ⊂ Σ and an odd homeomorphism h : K → Sn−1 n −
one-dimensional unit sphere and supx∈K ϕx < 0, then ϕ has at least n distinct pairs of nontrivial
critical points.
To begin with, we introduce some notation. Denote by X the Sobolev space H01 0, T ,
and consider the inner product
u, v T
u tv tdt
2.1
0
and the norm
u T
1/2
2
u t dt
.
2.2
0
Hence, X is reflexive. We define the norm in C0, T as x∞ maxt∈0,T |xt|.
For u ∈ H 2 0, T , we have that u and u are absolutely continuous and u ∈ L2 0, T .
Hence, Δu t u t − u t− 0 for every t ∈ 0, T . If u ∈ H01 0, T , then u is absolutely
continuous and u ∈ L2 0, T . In this case, the one-sided derivatives u t− , u t may not
exist. As a consequence, we need to introduce a different concept of solution. Suppose that
u ∈ C0, T such that for every j 1, 2, . . . , p, uj u|tj ,tj1 satisfies uj ∈ H 2 tj , tj1 , and
it satisfies the equation in problem 1.1 for t /
tj , a.e. t ∈ 0, T , the limits u tj , u t−j , and
Boundary Value Problems
3
j 1, 2, . . . , p exist, and impulsive conditions and boundary conditions in problem 1.1 hold,
we say it is a classical solution of problem 1.1.
Consider the functional
ϕ : X −→ R,
2.3
defined by
1
ϕu u2 − λ
2
T
Ht, utdt −
0
p utj j1
Ij sds,
2.4
0
u
where Ht, u 0 ht, sds. Clearly, ϕ is a Fréchet differentiable functional, whose Fréchet
derivative at the point u ∈ X is the functional ϕ u ∈ X ∗ given by
ϕ uv T
u tv tdt − λ
T
0
ht, utvtdt −
0
p
I j u tj v t j ,
2.5
j1
for any v ∈ X. Obviously, ϕ is continuous.
Lemma 2.3. If u ∈ X is a critical point of the functional ϕ, then u is a classical solution of problem
1.1.
Proof. The proof is similar to the proof of 16, Lemma 2.4, and we omit it here.
√
Lemma 2.4. Let u ∈ X, then u∞ ≤ T u.
Proof. For u ∈ X, then u0 uT 0. Hence, for t ∈ 0, T , by Hölder’s inequality, we have
1/2
T
T
t
√
√
2
u s ds ≤ T
u s ds
T u,
|ut| u sds ≤
0
0
0
2.6
which completes the proof.
3. Main Results
Theorem 3.1. Suppose that the following conditions hold.
i There exist a, b > 0 and γ ∈ 0, 1 such that
|ht, u| ≤ a b|u|γ
for any t, u ∈ 0, T × R.
ii ht, u is odd about u and Ht, u > 0 for every t, u ∈ 0, T × R \ {0}.
u
iii Ij u j 1, 2, . . . , p are odd and 0 Ij sds ≤ 0 for any u ∈ R j 1, 2, . . . , p.
3.1
4
Boundary Value Problems
Then for any n ∈ N, there exists λn such that λ > λn , and problem 1.1 has at least n distinct
pairs of nontrivial classical solutions.
Proof. By 2.4, ii, and iii, ϕ ∈ C1 X, R is an even functional and ϕ0 0.
Next, we will verify that ϕ is bounded from below. In view of i, iii, and Lemma 2.4,
we have
1
ϕu u2 − λ
2
1
≥ u2 − λ
2
≥
T
Ht, utdt −
0
p utj j1
Ij sds
0
T
a|ut| b|ut|γ1 dt
0
3.2
1
u2 − λaT 3/2 u − λbT γ3/2 uγ1
2
> −∞,
for any u ∈ X. That is, ϕ is bounded from below.
In the following we will show that ϕ satisfies the Palais-Smale condition. Let {uk } ⊂ X,
such that {ϕuk } is a bounded sequence and limk → ∞ ϕ uk 0. Then, there exists M > 0
such that
ϕuk ≤ M.
3.3
In view of 3.2, we have
M≥
1
uk 2 − λaT 3/2 uk − λbT γ3/2 uk γ1 .
2
3.4
So {uk } is bounded in X. From the reflexivity of X, we may extract a weakly convergent
subsequence that, for simplicity, we call {uk }, uk u in X. Next, we will verify that {uk }
strongly converges to u in X. By 2.5, we have
2
ϕ uk − ϕ u uk − u uk − u − λ
T
ht, uk t − ht, utuk t − utdt
0
p
I j uk t j − I j u t j
uk t j − u t j .
3.5
j1
By uk u in X, we see that {uk } uniformly converges to u in C0, T . So,
T
λ
ht, uk t − ht, utuk t − utdt −→ 0,
0
p
I j uk t j − I j u t j
uk tj − u tj −→ 0 as k −→ ∞.
j1
3.6
Boundary Value Problems
5
By limk → ∞ ϕ uk 0 and uk u, we have
ϕ uk − ϕ u uk − u −→ 0 as k −→ ∞.
3.7
In view of 3.5, 3.6, and 3.7, we obtain uk − u → 0 as k → ∞. Then, ϕ satisfies the
Palais-Smale condition.
√
Let vm t 2T /mπ sinmπ/T t, m 1, 2, . . . , n, then
vm 2 1 m2 π 2
T2
T
|vm t|2 dt,
m 1, 2, . . . , n.
3.8
0
Define
Kn r n
cm vm |
m1
n
2
cm
m1
r
2
,
r > 0.
3.9
√
n−1
Then, for any√r > 0, there
√ exists an odd homeomorphism f : Kn r → S . Let 0 < r < 1/ T ,
then u∞ ≤ T u T r < 1 for any u ∈ Kn r. By ii, we have
Ht, ut ut
0
ht, sds > 0 as ut /
0,
3.10
T
Ht, utdt > 0 for any u ∈ Kn r.
T
p utj Let αn infu∈Kn r 0 Ht, utdt, βn infu∈Kn r j1 0 Ij sds, then αn > 0, βn ≤ 0.
Let λn 1/2r 2 − βn α−1
n > 0, then when λ > λn , for any u ∈ Kn r, we have
then
0
ϕu ≤
<
1 2
r − λαn − βn
2
1 2
r − λn αn − βn
2
3.11
0.
By Theorem 2.2, ϕ possesses at least n distinct pairs of nontrivial critical points. That is,
problem 1.1 has at least n distinct pairs of nontrivial classical solutions.
Corollary 3.2. Let the following conditions hold:
i ht, u is bounded,
ii ht, u is odd about u and Ht, u > 0 for every t, u ∈ 0, T × R \ {0},
u
iii Ij u j 1, 2, . . . , p are odd and 0 Ij sds ≤ 0 for any u ∈ R j 1, 2, . . . , p.
Then, for any n ∈ N, there exists λn such that λ > λn , and problem 1.1 has at least n distinct
pairs of nontrivial classical solutions.
6
Boundary Value Problems
Proof. Let γ 0 in Theorem 3.1, then Corollary 3.2 holds.
Theorem 3.3. Suppose that the following conditions hold.
i There exists a, b > 0 and γ ∈ 0, 1 such that
|ht, u| ≤ a b|u|γ
for any t, u ∈ 0, T × R.
3.12
ii There exists aj , bj > 0 and γj ∈ 0, 1 j 1, 2, . . . , p such that
Ij u ≤ aj bj |u|γj
for any u ∈ R j 1, 2, . . . , p .
3.13
iii ht, u and Ij u j 1, 2, . . . , p are odd about u and Ht, u > 0 for every t, u ∈
0, T × R \ {0}.
Then, for any n ∈ N, there exists λn such that λ > λn , and problem 1.1 has at least n distinct
pairs of nontrivial classical solutions.
Proof. By 2.4 and iii, ϕ ∈ C1 X, R is an even functional and ϕ0 0.
Next, we will verify that ϕ is bounded from below. Let M1 max{a1 , a2 , . . . , ap }, M2 max{b1 , b2 , . . . , bp }. In view of i, ii, and Lemma 2.4, we have
ϕu 1
u2 − λ
2
1
≥ u2 − λ
2
−
p T
Ht, utdt 0
p utj j1
Ij sds
0
T
a|ut| b|ut|γ1 dt
0
γ 1 aj u tj bj u tj j
j1
≥
3.14
√
1
u2 − λaT 3/2 u − λbT γ3/2 uγ1 − pM1 T u
2
− M2
p
T γj 1/2 uγj 1
j1
> −∞,
for any u ∈ X. That is, ϕ is bounded from below.
In the following, we will show that ϕ satisfies the Palais-Smale condition. As in the
proof of Theorem 3.1, by 3.3 and 3.14, we have
M≥
p
√
1
uk 2 − λaT 3/2 uk − λbT γ3/2 uk γ1 − pM1 T uk − M2 T γj 1/2 uk γj 1 .
2
j1
3.15
Boundary Value Problems
7
It follows that {uk } is bounded in X. In the following, the proof of the Palais-Smale condition
is the same as that in Theorem 3.1, and we omit it here.
exists√an odd
Take the same Kn r as in Theorem 3.1, then √for any r > 0, there
√
homeomorphism f : Kn r → Sn−1 . Let 0 < r < 1/ T , then u∞ ≤ T u T r < 1
for any u ∈ Kn r. By iii, we have
Ht, ut ut
0
ht, sds > 0 as ut / 0.
3.16
T
Ht, utdt > 0 for any u ∈ Kn r.
T
p utj Let αn infu∈Kn r 0 Ht, utdt, βn infu∈Kn r j1 0 Ij sds, then αn > 0. Let
λn max{0, 1/2r 2 − βn α−1
n }, then when λ > λn , for any u ∈ Kn r, we have
Then,
0
ϕu ≤
1 2
1
r − λαn − βn < r 2 − λn αn − βn ≤ 0.
2
2
3.17
By Theorem 2.2, ϕ possesses at least n distinct pairs of nontrivial critical points. That is,
problem 1.1 has at least n distinct pairs of nontrivial classical solutions.
Corollary 3.4. Let the following conditions hold:
i ht, u is bounded,
ii Ij u j 1, 2, . . . , p are bounded,
iii ht, u and Ij u j 1, 2, . . . , p are odd about u and Ht, u > 0 for every t, u ∈
0, T × R \ {0}.
Then, for any n ∈ N, there exists λn such that λ > λn , and problem 1.1 has at least n distinct
pairs of nontrivial classical solutions.
Proof. Let γ 0 and γj 0 j 1, 2, . . . , p in Theorem 3.3, then Corollary 3.4 holds.
Theorem 3.5. Suppose that the following conditions hold.
i There exist constants σ > 0 such that ht, σ 0, ht, u > 0 for every u ∈ 0, σ.
ii ht, u is odd about u.
iii Ij u j 1, 2, . . . , p are odd and
u
0
Ij sds ≤ 0 for any u ∈ R j 1, 2, . . . , p.
Then, for any n ∈ N, there exists λn such that λ > λn , and problem 1.1 has at least n distinct
pairs of nontrivial classical solutions.
Proof. Let
⎧
⎪
ht, σ,
⎪
⎪
⎨
h1 t, u ht, u,
⎪
⎪
⎪
⎩
ht, −σ,
u > σ,
|u| ≤ σ,
u < −σ,
3.18
8
Boundary Value Problems
then h1 t, u is continuous, bounded, and odd. Consider boundary value problem
u t λh1 t, ut 0, t /
tj , a.e. t ∈ 0, T ,
−Δu tj Ij u tj , j 1, 2, . . . , p,
3.19
u0 uT 0.
Next, we will verify that the solutions of problem 3.19 are solutions of problem 1.1. In fact,
let u0 t be the solution of problem 3.19. If max0≤t≤T u0 t > σ, then there exists an interval
a, b ⊂ 0, T such that
u0 a u0 b σ,
for any t ∈ a, b.
u0 t > σ
3.20
When t ∈ a, b, by i, we have
u0 t −λh1 t, u −λht, σ 0.
3.21
Thus, there exist constants c such that u0 t c for any t ∈ a, b. We consider the following
two possible cases.
Case 1. c ≥ 0, then u0 is nondecreasing in a, b. By u0 a ≥ 0 and u0 b ≤ 0, we have
0 ≤ u0 a ≤ u0 t ≤ u0 b ≤ 0
for every t ∈ a, b.
3.22
That is, u0 t ≡ 0 for any t ∈ a, b. So, there exists a constant d such that u0 t ≡ d, which
contradicts 3.20. Then, max0≤t≤T u0 t ≤ σ. Similarly, we can prove that min0≤t≤T u0 t ≥ −σ.
Case 2. c < 0, the arguments are analogous, then u0 t is solution of problem 1.1.
For every u ∈ X, we consider the functional
ϕ1 : X −→ R,
3.23
defined by
1
u2 − λ
2
ϕ1 u T
H1 t, utdt −
0
p utj j1
Ij sds,
3.24
0
u
where H1 t, u 0 h1 t, sds.
It is clear that ϕ1 is Fréchet differentiable at any u ∈ X and
ϕ1 uv
T
0
u tv tdt − λ
T
0
p
h1 t, utvtdt − Ij u tj v tj ,
j1
3.25
Boundary Value Problems
9
for any v ∈ X. Obviously, ϕ1 is continuous. By Lemma 2.3, we have the critical points of ϕ1 as
solutions of problem 3.19. By 3.24, ii, and iii, ϕ1 ∈ C1 X, R is an even functional and
ϕ1 0 0.
In the following, we will show that ϕ1 is bounded from below. since h1 t, u 0 for
|u| ≥ σ, thus
T
H1 t, utdt T ut
0
0
h1 t, sds dt ≤
T σ
0
0
h1 t, sds dt e > 0.
3.26
0
By iii, we have
1
ϕ1 u u2 − λ
2
T
H1 t, utdt −
0
p utj j1
Ij sds
0
3.27
1
≥ u2 − λe ≥ −λe,
2
for any u ∈ X. That is, ϕ1 is bounded from below.
In the following we will show that ϕ1 satisfies the Palais-Smale condition. Let {uk } ⊂ X
such that {ϕ1 uk } is a bounded sequence and limk → ∞ ϕ1 uk 0. Then, there exists M3 > 0
such that
ϕ1 uk ≤ M3 .
3.28
1
uk 2 ≤ M3 λe.
2
3.29
By 3.27, we have
It follows that {uk } is bounded in X. In the following, the proof of the Palais-Smale condition
is the same as that in Theorem 3.1, and we omit it here.
exists√an odd
Take the same Kn r as in Theorem 3.1, then,√for any r > 0, there
√
homeomorphism f : Kn r → Sn−1 . Let 0 < r < σ/ T , then u∞ ≤ T u T r < σ
for any u ∈ Kn r. By i and ii, we have
H1 t, ut ut
h1 t, sds 0
Then,
T
0
H1 t, utdt > 0 for any u ∈ Kn r.
ut
ht, sdt > 0
0
as ut / 0.
3.30
10
Boundary Value Problems
T
p utj Let αn infu∈Kn r 0 H1 t, utdt, βn infu∈Kn r j1 0 Ij sds, then αn > 0, βn ≤ 0.
Let λn 1/2r 2 − βn α−1
n > 0, then when λ > λn , for any u ∈ Kn r, we have
ϕ1 u ≤
<
1 2
r − λαn − βn
2
1 2
r − λn αn − βn
2
3.31
0.
By Theorem 2.2, ϕ1 possesses at least n distinct pairs of nontrivial critical points. Then,
problem 3.19 has at least n distinct pairs of nontrivial classical solutions, that is, problem
1.1 has at least n distinct pairs of nontrivial classical solutions
Theorem 3.6. Let the following conditions hold.
i There exist constants σ > 0 such that ht, σ 0, ht, u > 0 for every u ∈ 0, σ.
ii There exist aj , bj > 0, and γj ∈ 0, 1 j 1, 2, . . . , p such that
Ij u ≤ aj bj |u|γj
for any u ∈ R j 1, 2, . . . , p .
3.32
iii ht, u and Ij u j 1, 2, . . . , p are odd about u.
Then, for any n ∈ N, there exists λn such that λ > λn , and problem 1.1 has at least n distinct
pairs of nontrivial classical solutions.
Proof. The proof is similar to the proof of Theorem 3.5, and we omit it here.
Theorem 3.7. Let the following conditions hold.
i There exist constants σ1 > 0 such that ht, σ1 ≤ 0.
ii There exist aj , bj > 0, and γj ∈ 0, 1 j 1, 2, . . . , p such that
Ij u ≤ aj bj |u|γj
for any u ∈ R j 1, 2, . . . , p .
3.33
iii ht, u and Ij u j 1, 2, . . . , p are odd about u and limu → 0 ht, u/u 1 uniformly for
t ∈ 0, T .
Then, for any n ∈ N, there exists λn such that λ > λn , and problem 1.1 has at least n distinct
pairs of nontrivial classical solutions.
Proof. Let
⎧
⎪
ht, σ1 ,
⎪
⎪
⎨
h2 t, u ht, u,
⎪
⎪
⎪
⎩
ht, −σ1 ,
u > σ1 ,
|u| ≤ σ1 ,
u < −σ1 ,
3.34
Boundary Value Problems
11
then h2 t, u is continuous, bounded, and odd. Consider boundary value problem
u t λh2 t, ut 0, t /
tj , a.e. t ∈ 0, T ,
−Δu tj Ij u tj , j 1, 2, . . . , p,
3.35
u0 uT 0.
Next, we will verify that the solutions of problem 3.35 are solutions of problem 1.1. In fact,
let u0 t be the solution of problem 3.35. If max0≤t≤T u0 t > σ1 , then there exists an interval
a, b ⊂ 0, T such that
for any t ∈ a, b.
u0 a u0 b σ1 , u0 t > σ1
3.36
When t ∈ a, b, by i, we have
u0 t −λh2 t, u −λht, σ1 ≥ 0.
3.37
Thus, u0 t is nondecreasing in a, b. By u0 a ≥ 0 and u0 b ≤ 0, we have
0 ≤ u0 a ≤ u0 t ≤ u0 b ≤ 0
for every t ∈ a, b.
3.38
That is, u0 t ≡ 0 for any t ∈ a, b. So, there exists a constant d such that u0 t ≡ d, which
contradicts 3.36. Then max0≤t≤T u0 t ≤ σ1 . Similarly, we can prove that min0≤t≤T u0 t ≥ −σ1 .
Then, u0 t is solution of problem 1.1.
For every u ∈ X, we consider the functional
ϕ2 : X −→ R,
3.39
defined by
1
u2 − λ
2
ϕ2 u T
H2 t, utdt −
0
p utj j1
Ij sds,
3.40
0
u
where H2 t, u 0 h2 t, sds.
It is clear that ϕ2 is Fréchet differentiable at any u ∈ X and
ϕ2 uv
T
0
u tv tdt − λ
T
0
p
h2 t, utvtdt − Ij u tj v tj ,
3.41
j1
for any v ∈ X. Obviously, ϕ2 is continuous. By Lemma 2.3, we have the critical points of ϕ2
as solutions of problem 3.35. By 3.40 and iii, ϕ2 ∈ C1 X, R is an even functional and
ϕ2 0 0.
12
Boundary Value Problems
Next, we will show that ϕ2 is bounded from below. Let M1 max{a1 , a2 , . . . , ap },
M2 max{b1 , b2 , . . . , bp }. since uh2 t, u ≤ 0 for |u| ≥ σ1 , thus
T
H2 t, utdt 0
T ut
0
T σ1
h2 t, sds dt ≤
0
0
h2 t, sds dt e.
3.42
0
By ii and Lemma 2.4, we have
ϕ2 u ≥
1
u2 − λ
2
T
H2 t, utdt −
0
p utj j1
Ij sds
0
p
√
1
u2 − λe − pM1 T u − M2 T γj 1/2 uγj 1
2
j1
3.43
> −∞,
for any u ∈ X. That is, ϕ2 is bounded from below.
In the following we will show that ϕ2 satisfies the Palais-Smale condition. Let {uk } ⊂ X
such that {ϕ2 uk } is a bounded sequence and limk → ∞ ϕ2 uk 0. Then, there exists M4 > 0
such that
ϕ2 uk ≤ M4 .
3.44
By 3.43, we have
p
√
1
2
≤
M
λe
pM
T
T γj 1/2 uk γj 1 .
M
uk uk 4
1
2
2
j1
3.45
It follows that {uk } is bounded in X. In the following, the proof of the Palais-Smale condition
is the same as that in Theorem 3.1, and we omit it here.
Take the same Kn r as in Theorem 3.1, then for any r > 0, there exists an odd
homeomorphism f : Kn r → Sn−1 . By iii, for any 0 < ε < 1, there exists δ > 0, when
|u| ≤ δ, we have
h2 t, u ≥ u − ε|u|.
3.46
√
√
√
√
Let 0 < r < min{σ1 / T , δ/ T }, then u∞ ≤ T u T r < min{σ1 , δ} for any u ∈ Kn r.
T
T ut
T
Then, 0 H2 t, utdt 0 0 h2 t, sdt ≥ 0 1/21 − ε|ut|2 dt > 0 for any u ∈ Kn r.
Boundary Value Problems
13
T
p utj Let αn infu∈Kn r 0 H2 t, utdt, βn infu∈Kn r j1 0 Ij sds, then αn > 0. Let
λn max{1/2r 2 − βn α−1
n , 0}, then when λ > λn , for any u ∈ Kn r, we have
ϕ1 u ≤
<
1 2
r − λαn − βn
2
3.47
1 2
r − λn αn − βn
2
≤ 0.
By Theorem 2.2, ϕ2 possesses at least n distinct pairs of nontrivial critical points. Then,
problem 3.35 has at least n distinct pairs of nontrivial classical solutions, that is, problem
1.1 has at least n distinct pairs of nontrivial classical solutions.
Theorem 3.8. Let the following conditions hold.
i There exist constants σ1 > 0 such that ht, σ1 ≤ 0.
ii limu → 0 ht, u/u 1 uniformly for t ∈ 0, T .
iii ht, u and Ij u j 1, 2, . . . , p are odd about u and
1, 2, . . . , p.
u
0
Ij sds ≤ 0 for any u ∈ R j Then, for any n ∈ N, there exists λn such that λ > λn , and problem 1.1 has at least n distinct
pairs of nontrivial classical solutions.
Proof. The proof is similar to the proof of Theorem 3.7, and we omit it here.
4. Some Examples
Example 4.1. Consider boundary value problem
u t λ1 t 3 ut 0,
t
/ tj , a.e. t ∈ 0, π,
−Δu tj −u tj ,
4.1
j 1, 2,
u0 uπ 0.
It is easy to see that conditions i, ii, and iii of Theorem 3.1 hold. Let
3
αn inf
u∈Kn r 4
π
0
βn inf −
u∈Kn r
1 t|ut|
2 utj j1
0
4/3
3
dt > inf
u∈Kn r 4
sds inf −
u∈Kn r
π
|ut|2 dt >
0
2 u t 2
j
j1
2
3r 2
,
4n2
4.2
≥ −πr ,
2
2
then λn 1/2r 2 − βn α−1
n < 2 4π/3n . Applying Theorem 3.1, then for any n ∈ N,
2
when λ > 2 4π/3n , problem 4.1 has at least n distinct pairs of nontrivial classical
solutions.
14
Boundary Value Problems
Example 4.2. Consider boundary value problem
u t λ1 t 3 ut 0, t / tj , a.e. t ∈ 0, π,
−Δu tj − 3 u tj , j 1, 2,
4.3
u0 uπ 0.
√
It is easy to see that conditions i, ii, and iii of Theorem 3.3 hold. Let r 1/2 π,
3
αn inf
u∈Kn r 4
π
βn inf −
u∈Kn r
1 t|ut|
0
2 utj j1
0
1/3
s
4/3
3
dt > inf
u∈Kn r 4
π
0
|ut|2 dt >
3r 2
,
4n2
2
3 4/3
3
u tj
ds inf −
>− ,
u∈Kn r
4
2
j1
4.4
2
then λn 1/2r 2 − βn α−1
n < 2 24π/3n . Applying Theorem 3.3, then for any
2
n ∈ N, when λ > 2 24π/3n , problem 4.3 has at least n distinct pairs of nontrivial
classical solutions.
Example 4.3. Consider boundary value problem
u t λ 1 t2 ut − ut3 0,
−Δu tj −u tj ,
t
/ tj , a.e. t ∈ 0, π,
j 1, 2,
4.5
u0 uπ 0.
Let σ 1, it is easy to see that conditions i, ii, and iii of Theorem 3.5 hold. Let
1 π
1
r2
4
1t
αn inf
|ut|2 dt > 2 ,
|ut| − |ut| dt > inf
u∈Kn r 0
u∈Kn r 4 0
2
4
4n
2 utj 2 u t 2
j
≥ −πr 2 ,
βn inf −
sds inf −
u∈Kn r
u∈Kn r
2
j1 0
j1
π
2
1
2
4.6
2
then λn 1/2r 2 − βn α−1
n < 2 4πn . Applying Theorem 3.5, then for any n ∈ N,
2
when λ > 24πn , problem 4.5 has at least n distinct pairs of nontrivial classical solutions.
Example 4.4. Consider boundary value problem
u t λ ut − 1 tut3 0,
−Δu tj − 3 u tj ,
t
/ tj , a.e. t ∈ 0, π,
j 1, 2,
u0 uπ 0.
4.7
Boundary Value Problems
15
Let σ1 1, it is easy to see that conditions i, ii, and iii of Theorem 3.7 hold. Let
√
r 1/4 π,
1 π
1
r2
1
|ut|2 dt > 2 ,
|ut|2 − 1 t|ut|4 dt > inf
u∈Kn r 0 2
u∈Kn r 4 0
4
4n
2 utj 2
3 4/3
3
u tj
βn inf −
s1/3 ds inf −
>− ,
u∈Kn r
u∈Kn r
4
2
j1 0
j1
αn inf
π
4.8
2
then λn 1/2r 2 − βn α−1
n < 2 24πn . Applying Theorem 3.7, then for any n ∈
2
N, when λ > 2 24πn , problem 4.7 has at least n distinct pairs of nontrivial classical
solutions.
Acknowledgments
This work was supported by the NNSF of China no. 10871062 and a project supported by
Hunan Provincial Natural Science Foundation of China no. 10JJ6002.
References
1 R. K. George, A. K. Nandakumaran, and A. Arapostathis, “A note on controllability of impulsive
systems,” Journal of Mathematical Analysis and Applications, vol. 241, no. 2, pp. 276–283, 2000.
2 G. Jiang and Q. Lu, “Impulsive state feedback control of a predator-prey model,” Journal of
Computational and Applied Mathematics, vol. 200, no. 1, pp. 193–207, 2007.
3 S. I. Nenov, “Impulsive controllability and optimization problems in population dynamics,” Nonlinear
Analysis. Theory, Methods & Applications, vol. 36, pp. 881–890, 1999.
4 M. Choisy, J.-F. Guégan, and P. Rohani, “Dynamics of infectious diseases and pulse vaccination:
teasing apart the embedded resonance effects,” Physica D, vol. 223, no. 1, pp. 26–35, 2006.
5 S. Gao, L. Chen, J. J. Nieto, and A. Torres, “Analysis of a delayed epidemic model with pulse
vaccination and saturation incidence,” Vaccine, vol. 24, no. 35-36, pp. 6037–6045, 2006.
6 J. Li and J. J. Nieto, “Existence of positive solutions for multipoint boundary value problem on the
half-line with impulses,” Boundary Value Problems, vol. 2009, Article ID 834158, 12 pages, 2009.
7 J. Chu and J. J. Nieto, “Impulsive periodic solutions of first-order singular differential equations,”
Bulletin of the London Mathematical Society, vol. 40, no. 1, pp. 143–150, 2008.
8 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.
9 L. Chen and J. Sun, “Nonlinear boundary value problem of first order impulsive functional
differential equations,” Journal of Mathematical Analysis and Applications, vol. 318, no. 2, pp. 726–741,
2006.
10 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.
11 Y. Tian and W. Ge, “Applications of variational methods to boundary-value problem for impulsive
differential equations,” Proceedings of the Edinburgh Mathematical Society. Series II, vol. 51, no. 2, pp.
509–527, 2008.
12 Y. Tian, W. Ge, and D. Yang, “Existence results for second-order system with impulse effects via
variational methods,” Journal of Applied Mathematics and Computing, vol. 31, no. 1-2, pp. 255–265, 2009.
13 H. Zhang and Z. Li, “Variational approach to impulsive differential equations with periodic boundary
conditions,” Nonlinear Analysis. Real World Applications, vol. 11, no. 1, pp. 67–78, 2010.
14 Z. Zhang and R. Yuan, “An application of variational methods to Dirichlet boundary value problem
with impulses,” Nonlinear Analysis. Real World Applications, vol. 11, no. 1, pp. 155–162, 2010.
15 J. Zhou and Y. Li, “Existence and multiplicity of solutions for some Dirichlet problems with impulsive
effects,” Nonlinear Analysis. Theory, Methods & Applicationss, vol. 71, no. 7-8, pp. 2856–2865, 2009.
16
Boundary Value Problems
16 J. Sun and H. Chen, “Variational method to the impulsive equation with Neumann boundary
conditions,” Boundary Value Problems, vol. 2009, Article ID 316812, 17 pages, 2009.
17 D. Guo, J. Sun, and Z. Liu, Functional Methods of Nonlinear Ordinary Differential Equations, Shandong
Science and Technology Press, 2005.