Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2011, Article ID 378389, 19 pages
doi:10.1155/2011/378389
Research Article
Periodic Solutions for Autonomous q, p-Laplacian
System with Impulsive Effects
Xiaoxia Yang and Haibo Chen
School of Mathematical Sciences and Computing Technology, Central South University,
Changsha, Hunan 410083, China
Correspondence should be addressed to Haibo Chen, hbchen2003@gmail.com
Received 17 July 2011; Accepted 31 August 2011
Academic Editor: Yongkun Li
Copyright q 2011 X. Yang and H. Chen. 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.
By using the variational method, some existence theorems are obtained for periodic solutions of
autonomous q, p-Laplacian system with impulsive effects.
1. Introduction
Let B {1, 2, . . . , l}, C {1, 2, . . . , k}, l, k ∈ N.
In this paper, we consider the following system:
d
Φq u̇1 t ∇u1 Fu1 t, u2 t,
dt
a.e. t ∈ 0, T ,
d
Φp u̇2 t ∇u2 Fu1 t, u2 t,
dt
a.e. t ∈ 0, T ,
u1 0 − u1 T u̇1 0 − u̇1 T 0,
u2 0 − u2 T u̇2 0 − u̇2 T 0,
− Φq u̇1 t−j
∇Ij u1 tj ,
ΔΦq u̇1 tj Φq u̇1 t
j
ΔΦp u̇2 sm Φp u̇2 s
m − Φp u̇2 s−m ∇Km u2 sm ,
1.1
j ∈ B,
m ∈ C,
1
2
where p > 1, q > 1, T > 0, ut u1 t, u2 t u11 t, u21 t, . . . , uN
1 t, u2 t, u2 t, . . . ,
τ
N
u2 t , tj j 1, 2, . . . , l, and sm m 1, 2, . . . , k are the instants where the impulses occur
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and 0 t0 < t1 < t2 < · · · < tl < tl
1 T, 0 s0 < s1 < s2 < · · · < sk < sk
1 T ,
Ij : RN → R j ∈ B, and Km : RN → R m ∈ C are continuously differentiable
⎛ ⎞
μ−2/2 z1
N
⎜ ⎟
⎜ .. ⎟
z2i
Φμ z |z|μ−2 z ⎜ . ⎟, μ ∈ R, μ > 1,
⎝ ⎠
i1
zN
⎞
⎞
⎛
⎛
∂Ij
∂Km
⎜ ∂x1 ⎟
⎜ ∂x1 ⎟
⎟
⎟
⎜
⎜
⎜ . ⎟
⎜ . ⎟
.. ⎟,
.
⎟,
⎜
∇K
∇Ij x ⎜
x
.
m
⎟
⎟
⎜
⎜
⎟
⎟
⎜
⎜
⎝ ∂Ij ⎠
⎝ ∂Km ⎠
1.2
∂xN
∂xN
and F : RN × RN → R satisfies the following assumption.
A Fx is continuously differentiable in x1 , x2 , and there exist a1 , a2 ∈ CR
, R
such
that
|∇Fx1 , x2 | ≤ a1 |x1 | a2 |x2 |,
|Fx1 , x2 | ≤ a1 |x1 | a2 |x2 | ,
Ij x1 ≤ a1 |x1 |, ∇Ij x1 ≤ a1 |x1 |, j ∈ B,
|Km x2 | ≤ a2 |x2 |,
|∇Km x2 | ≤ a2 |x2 |,
1.3
m ∈ C,
for all x x1 , x2 ∈ RN × RN .
When p q 2, Ij ≡ 0 j ∈ B, Km ≡ 0 m ∈ C, and Fu1 , u2 F1 u1 , system 1.1
reduces to the following autonomous second-order Hamiltonian system:
ü1 t ∇u1 F1 u1 t,
a.e. t ∈ 0, T ,
u1 0 − u1 T u̇1 0 − u̇1 T 0.
1.4
There have been lots of results about the existence of periodic solutions for system 1.4 and
nonautonomous second order Hamiltonian system
ü1 t ∇u1 F1 t, u1 t,
a.e. t ∈ 0, T ,
u1 0 − u1 T u̇1 0 − u̇1 T 0,
1.5
e.g., see 1–21. Many solvability conditions have been given, for instance, coercive condition, subquadratic condition, superquadratic condition, convex condition, and so on.
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3
≡ 0 j ∈ B, Km ≡ 0 m ∈ C, and Fu1 , u2 F1 u1 , system
When p q 2, ∇Ij /
1.1 reduces to the following autonomous second-order Hamiltonian system with impulsive
effects:
ü1 t ∇u1 F1 u1 t,
a.e. t ∈ 0, T ,
u1 0 − u1 T u̇1 0 − u̇1 T 0,
u̇1 t
j − u̇1 t−j ∇Ij u1 tj .
1.6
Recently, many authors studied the existence of periodic solutions for impulsive differential
equations by using variational methods, and lots of interesting results have been obtained.
For example, see 22–28. Especially, nonautonomous second-order Hamiltonian system with
impulsive effects is considered in 25, 26 by using the least action principle and the saddle
point theorem.
When Ij ≡ 0 j ∈ B and Km ≡ 0 m ∈ C, system 1.1 reduces to the following system:
d
Φq u̇1 t ∇u1 Fu1 t, u2 t,
dt
a.e. t ∈ 0, T ,
d
Φp u̇2 t ∇u2 Fu1 t, u2 t,
dt
a.e. t ∈ 0, T ,
1.7
u1 0 − u1 T u̇1 0 − u̇1 T 0,
u2 0 − u2 T u̇2 0 − u̇2 T 0.
In 29, 30, Paşca and Tang obtained some existence results for system 1.7 by using the least
action principle and saddle point theorem. Motivated by 17, 22–30, in this paper, we are
concerned with system 1.1 and also use the least action principle and saddle point theorem
to study the existence of periodic solution. Our results still improve those in 17 even if
system 1.1 reduces to system 1.4.
A function G : RN → R is called to be λ, μ-quasiconcave if
G λ x y ≥ μ Gx G y ,
1.8
for some λ, μ > 0 and x, y ∈ RN .
Next, we state our main results.
Theorem 1.1. Let q and p be such that 1/q 1/q 1 and 1/p 1/p 1. Suppose F satisfies
assumption (A) and the following conditions:
F1 there exist
q/q
q 1
0 < r1 < q ,
T Θ q, q
p/p
p 1
0 < r2 < p ,
T Θ p, p
1.9
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such that
q
∇x1 Fx1 , x2 − ∇y1 F y1 , y2 , x1 − y1 ≥ −r1 x1 − y1 ,
p
∇x2 Fx1 , x2 − ∇y2 F y1 , y2 , x2 − y2 ≥ −r2 x2 − y2 ,
∀x1 , x2 , y1 , y2 ∈ RN × RN ,
∀x1 , x2 , y1 , y2 ∈ RN × RN ,
1.10
where
Θ q, q 1 q/q
sq 1 1 − sq 1
ds,
0
Θ p, p 1 p 1
s
p 1
1 − s
p/p
1.11
ds,
0
F2 Fx → ∞, as |x| → ∞, where x x1 , x2 ,
I1 there exists β ∈ R such that
Ij x ≥ β,
Km x ≥ β,
∀x ∈ RN , j ∈ B,
1.12
∀x ∈ RN , m ∈ C.
1,q
1,p
Then, system 1.1 has at least one solution in WT × WT , where WT1,s {u : 0, T → RN | u is
absolutely continuous, u0 uT and u̇ ∈ Ls 0, T ; RN }, s ∈ R.
Furthermore, if Ij ≡ 0 j ∈ B, Km ≡ 0 m ∈ C and the following condition holds:
F3 there exist δ > 0, a ∈ 0, q 1q/q /qT q Θq, q and b ∈ 0, p 1p/p /
pT p Θp, p such that
−a|x1 |q − b|x2 |p ≤ Fx1 , x2 ≤ 0,
∀|x1 | ≤ δ, |x2 | ≤ δ,
1,q
1.13
1,p
then system 1.7 has at least two nonzero solutions in WT × WT .
When p q 2, Fx1 , x2 F1 x1 , by Theorem 1.1, it is easy to get the following
corollary.
Corollary 1.2. Suppose F1 satisfies the following conditions:
A F1 z is continuously differentiable in z and there exists a1 ∈ CR
, R
such that
|∇F1 z| ≤ a1 |z|,
|F1 z| ≤ a1 |z|,
Ij z ≤ a1 |z|, ∇Ij z ≤ a1 |z|, j ∈ B,
for all z ∈ RN .
1.14
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5
F1 there exists 0 < r < 6/T 2 such that
∇z F1 z − ∇w F1 w, z − w ≥ −r|z − w|2 ,
∀z, w ∈ RN ,
1.15
F2 F1 z → ∞, as |z| → ∞, z ∈ RN ;
I1 there exists β ∈ R such that
Ij z ≥ β,
∀z ∈ RN , j ∈ B.
1.16
Then, system 1.6 has at least one solution in WT1,2 . Furthermore, if Ij ≡ 0 j ∈ B and the following
condition holds:
F3 there exist δ > 0 and a ∈ 0, 3/T 2 such that
−a|z|2 ≤ F1 z ≤ 0,
∀z ∈ RN , |z| ≤ δ,
1.17
then system 1.4 has at least two nonzero solutions in WT1,2 .
1,2 , one has the following sharp estimates see in 3,
For the Sobolev space W
T
Proposition 1.2:
T
0
T2 T
|u̇t|2 dt Wirtinger s inequality ,
4π 2 0
T T
|u̇t|2 dt Sobolev s inequality .
u2∞ ≤
12 0
|ut|2 dt ≤
1.18
1.19
By the above two inequalities, we can obtain the following better results than by Corollary 1.2.
Theorem 1.3. Suppose F1 satisfies assumption (A) , (F2) , (I1) and
F1 there exists 0 < r < 4π 2 /T 2 such that 1.15 holds.
Then, system 1.6 has at least one solution in WT1,2 . Furthermore, if Ij ≡ 0 j ∈ B and the following
condition holds:
F3 there exist δ > 0 and a ∈ 0, 2π 2 /T 2 such that
−a|z|2 ≤ F1 z ≤ 0,
∀z ∈ RN , |z| ≤ δ,
then system 1.4 has at least two nonzero solutions in WT1,2 .
Moreover, for system 1.6, we have the following additional result.
Theorem 1.4. Suppose F1 satisfies assumption (A) , (F1) and the following conditions:
F4 F1 z is λ, μ-quasiconcave on RN ,
1.20
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Journal of Applied Mathematics
F5 F1 z → −∞ as |z| → ∞, z ∈ RN ,
I2 there exist dj > 0 j ∈ B such that
∇Ij z ≤ dj ,
1.21
∀z ∈ RN , j ∈ B,
I3 there exist bj > 0, cj > 0, γj ∈ R, αj ∈ 0, 2j ∈ B such that
−bj |z|αj − cj ≤ Ij z ≤ γj ,
∀z ∈ RN , j ∈ B.
1.22
Then, system 1.6 has at least one solution in WT1,2 .
Remark 1.5. In 17, Yang considered the second-order Hamiltonian system with no impulsive
effects, that is, system 1.4. When Ij ≡ 0 j ∈ B, our Theorems 1.3 and 1.4 still improve those
results in 17. To be precise, the restriction of r is relaxed, and some unnecessary conditions
in 17 are deleted. In 17, the restriction of r is 0 < r < T/12, which is not right. In fact,
from his proof, it is easy to see that it should be 0 < r < 12/T 2 . Obviously, our restriction
0 < r < 4π 2 /T 2 is better. Moreover, in our Theorem 1.4, we delete such conditions of in 17,
Theorem 1: ∇F1 0 0, and there exist positive constants M, N such that
F1 z ≥ −M |z|2 − N,
z ∈ RN .
1.23
Finally, it is remarkable that Theorems 1.3 and 1.4 are also different from those results in
1–16. We can find an example satisfying our Theorem 1.3 but not satisfying the results in
1–21. For example, let
F1 z π2
π2 4
4
4
− 2 |z|2 ,
·
·
·
|z
|z
|
|
|
|z
1
2
N
2T 2
4T
1.24
where z z1 , . . . , zN τ . We can also find an example satisfying our Theorem 1.4 but not
satisfying the results in 1–21. For example, let
r
F1 z − |z|2 ,
2
1.25
where 12/T 2 < r < 4π 2 /T 2 .
2. Variational Structure and Some Preliminaries
1,p
The norm in WT is defined by
uW 1,p T
T
0
p
|ut| dt T
0
1/p
p
|u̇t| dt
.
2.1
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7
Set
up T
1/p
p
|ut| dt
2.2
u∞ max |ut|.
,
t∈0,T 0
Let
1,p
W
T
u∈
1,p
WT
|
T
utdt 0
2.3
.
0
is a subset of W and
Obviously, WT is a reflexive Banach space. It is easy to know that W
T
T
1,p
1,p
N
. In this paper, we will use the space W defined by
WT R ⊕ W
T
1,p
1,p
1,q
1,p
W WT × WT ,
1,p
ut u1 t, u2 t,
2.4
with the norm u1 , u2 W u1 W 1,q u2 W 1,p . It is clear that W is a reflexive Banach space.
T
T
1,p . Then, W W
1,q × W
1,p ⊕ RN × RN .
W
1,q × W
Let W
T
T
T
T
1,p
1,q
Lemma 2.1 see 31 or 32. Each u ∈ WT and each v ∈ WT can be written as ut u u
t
and vt v v t with
1
u
T
1
v
T
T
T
utdt,
0
u
tdt 0,
0
T
T
vtdt,
0
2.5
v tdt 0.
0
Then,
u∞ ≤
T
p 1
1/p T
1/p
p
|u̇s| ds
,
0
v ∞ ≤
T
q 1
1/q T
1/q
q
|v̇s| ds
,
0
2.6
T
T
T p Θ p, p
p
us| ds ≤ |
|u̇s|p ds,
p/p
0
0
p 1
T
T
T q Θ q, q
q
vs| ds ≤ |
|v̇s|q ds,
q/q
0
0
q 1
2.7
where
Θ p, p 1 sp 1 1 − sp 1
0
p/p
ds,
Θ q, q 1 q/q
sq 1 1 − sq 1
ds.
2.8
0
1,p
Note that if u ∈ WT , then u is absolutely continuous. However, we cannot guarantee
0,
that u̇ is also continuous. Hence, it is possible that ΔΦp u̇t Φp u̇t
− Φp u̇t− /
which results in impulsive effects.
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Journal of Applied Mathematics
1,q
Following the idea in 22, one takes v1 ∈ WT and multiplies the two sides of
d
|u̇1 t|q−2 u̇1 t − ∇x1 Fu1 t, u2 t 0,
dt
2.9
by v1 and integrate from 0 to T , one obtains
T
0
d
q−2
|u̇1 t| u̇1 t − ∇x1 Fu1 t, u2 t v1 tdt 0.
dt
2.10
Note that v1 t is continuous. So, v1 t−j v1 t
j v1 tj . Combining u̇1 0 − u̇1 T 0, one
has
T
0
l tj
1
dΦq u̇1 t
d Φq u̇1 t
, v1 t dt , v1 t dt
dt
dt
j0 tj
l Φq u̇1 t−j
1 , v1 t−j
1 − Φq u̇1 t
j , v1 t
j
dt
j0
−
l tj
1
Φq u̇1 t, v̇1 t dt
tj
j0
Φq u̇1 T , v1 T − Φq u̇1 0, v1 0
−
l
ΔΦq u̇1 tj , v1 tj −
T
0
j1
−
Φq u̇1 t, v̇1 t dt
l
∇Ij u1 tj , v1 tj −
T
Φq u̇1 t, v̇1 t dt.
0
j1
2.11
Combining with 2.10, one has
T
l
Φq u̇1 t, v̇1 t dt ∇Ij u1 tj , v1 tj
0
j1
T
0
∇x1 Fu1 t, u2 t, v1 tdt 0.
2.12
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9
Similarly, one can get
T
k
Φp u̇2 t, v̇2 t dt ∇Km u2 sm , v2 sm 0
m1
T
2.13
∇x2 Fu1 t, u2 t, v2 tdt 0,
0
1,p
for all v2 ∈ WT . Considering the above equalities, one introduces the following concept of
the weak solution for system 1.1.
1,q
1,p
Definition 2.2. We say that a function u u1 , u2 ∈ WT ×WT is a weak solution of system 1.1 if
T
l
Φq u̇1 t, v̇1 t dt ∇Ij u1 tj , v1 tj −
0
T
T
k
Φp u̇2 t, v̇2 t dt ∇Km u2 sm , v2 sm −
0
1,q
1,p
1,q
1,p
Define the functional ϕ : WT × WT
T
∇x2 Fu1 t, u2 t, v2 tdt
2.14
holds for any v v1 , v2 ∈ WT × WT .
1
q
T
0
m1
ϕu1 , u2 ∇x1 Fu1 t, u2 t, v1 tdt,
0
j1
|u̇1 t|q dt 0
1
p
→ R by
T
|u̇2 t|p dt 0
T
Fu1 t, u2 tdt
0
l
k
I j u1 t j Km u2 sm j1
2.15
m1
φu1 , u2 ψu1 , u2 ,
1,q
1,p
where u1 , u2 ∈ WT × WT ,
1
φu1 , u2 q
T
0
1
|u̇1 t| dt p
q
T
p
|u̇2 t| dt 0
T
Fu1 t, u2 tdt,
0
l
k
ψu1 , u2 I j u1 t j Km u2 sm .
j1
m1
2.16
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Journal of Applied Mathematics
1,q
1,p
By assumption A and 33, we know that φ ∈ C1 WT × WT , R. The continuity of Ij j ∈ B
and Km m ∈ C implies that ψ ∈
1,q
1,p
WT × WT , we have
ϕ u1 , u2 , v1 , v2 T
1,p
C1 WT
1,p
× WT , R.
Φq u̇1 t, v̇1 t dt T
0
T
1,p
So, ϕ ∈ C1 WT , R, and for all v1 , v2 ∈
Φp u̇2 t, v̇2 t dt
0
∇x1 Fu1 t, u2 t, v1 tdt 0
T
∇x2 Fu1 t, u2 t, v2 tdt
0
l
k
∇Ij u1 tj , v1 tj ∇Km u2 sm , v2 sm .
j1
m1
2.17
Definition 2.2 shows that the critical points of ϕ correspond to the weak solutions of system
1.1.
We will use the following lemma to seek the critical point of ϕ.
Lemma 2.3 see 3, Theorem 1.1. If ϕ is weakly lower semicontinuous on a reflexive Banach space
X and has a bounded minimizing sequence, then ϕ has a minimum on X.
Lemma 2.4 see 34. Let ϕ be a C1 function on X X1 ⊕ X2 with ϕ0 0, satisfying (PS)
condition, and assume that for some ρ > 0,
ϕu ≥ 0,
for u ∈ X1 , u ≤ ρ,
ϕu ≤ 0,
for u ∈ X2 , u ≤ ρ.
2.18
Assume also that ϕ is bounded below and infX ϕ < 0, then ϕ has at least two nonzero critical points.
Lemma 2.5 see 35, Theorem 4.6. Let X X1 ⊕ X2 , where X is a real Banach space and X1 /
{0}
and is finite dimensional. Suppose that ϕ ∈ C1 X, R satisfies (PS)-condition and
ϕ1 there is a constant α and a bounded neighborhood D of 0 in X1 such that ϕ|∂D ≤ α,
ϕ2 there is a constant β > α such that ϕ|X2 ≥ β.
Then, ϕ possesses a critical value c ≥ β. Moreover, c can be characterized as
c inf max ϕhu,
2.19
Γ h ∈ C D, X | h id on ∂D .
2.20
h∈Γ u∈D
where,
3. Proof of Theorems
1,q
1,p
Lemma 3.1. Under assumption (A), ϕ is weakly lower semicontinuous on WT × WT .
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11
Proof. Let
1 T
1 T
|u̇1 t|q dt |u̇2 t|p dt,
q 0
p 0
T
φ2 u1 , u2 Fu1 t, u2 tdt.
φ1 u1 , u2 3.1
0
Since
u v u v 1 T u̇ t v̇ t q
1 T u̇2 t v̇2 t p
1
1
2
2
1
1
,
φ1
dt p
dt
2
2
q 0 2
2
0
≤
≤
T
2q−1
q
0
2p−1
p
1
2q
T
T
0
|u̇1 t|q dt T
0
T
0
1
2p−1
|u̇2 t|p dt p
2
p
0
1
2p
1
2q−1
q
dt
u̇
t|
|
1
2q
q
1
2q
T
1
|v̇1 t|q dt
2q
T
0
1
|v̇2 t|p dt
2p
3.2
|v̇1 t|q dt
0
1
|u̇2 t| dt 2p
p
T
|v̇2 t|p dt
0
φ1 u1 , u2 φ1 v1 , v2 ,
2
then φ1 is convex. Moreover, by 33, we know that φ1 is continuous, and so, it is lower
semicontinuous. Thus, it follows from 3, Theorem 1.2 that φ1 is weakly lower continuous.
By assumption A, it is easy to verify that φ2 u1 , u2 is weakly continuous. We omit the
details. Let
ψ1 u1 l
I j u1 t j ,
ψ2 u2 j1
k
Km u2 sm .
3.3
m1
1,q
1,p
Next, we show that ψ1 and ψ2 are weakly continuous on WT and WT , respectively. In fact,
if
1,p
u1n u1 weakly in WT ,
as n −→ ∞,
3.4
then by in 3, Proposition 1.2, we know that
u1n −→ u1 strongly in C 0, T ; RN ,
as n −→ ∞.
3.5
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Journal of Applied Mathematics
So, there exists M1 > 0 such that u1 ∞ ≤ M1 and u1n ∞ ≤ M1 , for all n ∈ N. Thus, we have
l
l
ψ1 u1n − ψ1 u1 Ij u1n tj − Ij u1 tj j1
j1
l Ij u1n tj − Ij u1 tj ≤
j1
j1
∇Ij u1 tj s u1n tj − u1 tj
, u1n tj − u1 tj ds
0
l 1
≤ u1n − u1 ∞
l
j1
3.6
max a1 t −→ 0.
t∈0,3M1 1,q
Hence, ψ1 is weakly continuous on WT . Similarly, we can prove that ψ2 is also weakly
1,p
continuous on WT . Thus, we complete the proof.
Proof of Theorem 1.1. It follows from F1 and 2.7 that
T
Fu1 t, u2 t − Fu1 t, u2 0
T 1
0
0
T 1
0
≥−
0
r2
p
1
u2 t, s
u2 tds dt
∇Fx2 u1 t, u2 s
s
1
u2 t − ∇Fx2 u1 , u2 , s
u2 tds dt
∇Fx2 u1 t, u2 s
s
T
u2 t|p dt
|
0
T
r2 T p Θ p, p
≥− |u̇2 t|p dt,
p/p
0
p p 1
T
∀u1 , u2 ∈ W,
Fu1 t, u2 − Fu1 , u2 dt
0
T 1
0
0
T 1
0
0
1
u1 t, u2 , s
u1 tds dt
∇x1 Fu1 s
s
1
u1 t, u2 − ∇x1 Fu1 , u2 , s
u1 tds dt
∇x1 Fu1 s
s
3.7
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≥−
r1
q
T
13
u1 t|q dt
|
0
T
r1 T q Θ q, q
≥− |u̇1 t|q dt,
q/q
0
q q 1
∀u1 , u2 ∈ W.
3.8
Hence, by I1, 3.7, and 3.8, we have
T
1
ϕu1 , u2 q
0
T
1
|u̇1 t| dt p
q
T
p
|u̇2 t| dt T
0
Fu1 t, u2 t − Fu1 t, u2 dt
0
Fu1 t, u2 − Fu1 , u2 dt T Fu1 , u2 0
≥
l
k
I j u1 t j Km u2 sm j1
m1
T
T
1 r2 T p Θ p, p
1 r1 T q Θ q, q
p
−
−
|u̇2 t| dt |u̇1 t|q dt
p p p 1p/p
q q q 1q/q
0
0
T Fu1 , u2 − l kβ.
3.9
1,p
Note that for u ∈ WT ,
uW 1,p −→ ∞ ⇐⇒
T
p
|u| T
1/p
p
|u̇t| dt
−→ ∞,
3.10
−→ ∞.
3.11
0
1,q
and for v ∈ WT ,
vW 1,q −→ ∞ ⇐⇒
T
|v| q
T
1/q
q
|v̇t| dt
0
So, F2 and 3.9 imply that
ϕu1 , u2 −→ ∞, as u1 , u2 W −→ ∞.
3.12
Thus, by Lemma 2.3, we know that ϕ has at least one critical point which minimizes ϕ on W.
Furthermore, if Ij u1 tj ≡ 0 j ∈ B and Km u2 sm ≡ 0 m ∈ C, then system 1.1
reduces to 1.7. When F3 also holds, we will use Lemma 2.4 to obtain more critical points
W
1,q × W
1,p .
of ϕ. Let X W, X2 RN × RN and X1 W
T
T
By 3.9, we know that ϕu1 , u2 → ∞ as u1 , u2 W → ∞. So, ϕ satisfies PS
condition and is bounded below. Take ρ δ/c1 , where c1 is a positive constant such that
14
Journal of Applied Mathematics
u1 ∞ ≤ c1 u1 W 1,q ≤ c1 uW and u2 ∞ ≤ c1 u2 W 1,p ≤ c1 uW for all u1 , u2 ∈ W. It follows
T
T
from F3 and Lemma 2.1 that
1
ϕu1 , u2 q
≥
1
q
1
≥
q
T
0
T
|u̇1 t|q dt 0
T
0
T
1
|u̇1 t| dt p
q
p
|u̇2 t| dt T
0
T
1
p
0
T
1
|u̇1 t| dt p
q
0
Fu1 t, u2 tdt
0
|u̇2 t|p dt − a
T
|u1 t|q dt − b
0
T
|u2 t|p dt
0
T
T q Θ q, q
|u̇2 t| dt − a |u̇1 t|q dt
q/q
0
q 1
3.13
p
T
T p Θ p, p
− b
|u̇2 t|p dt,
p/p
0
p 1
∀u1 , u2 ∈ X1 .
Since a ≤ q 1q/q /qT q Θq, q and b ≤ p 1p/p /pT p Θp, p , 3.13 implies that
ϕu1 , u2 ≥ 0 for all u1 , u2 ∈ X1 with uW ≤ ρ. By F3, it is easy to obtain that ϕu1 , u2 ≤ 0,
for all u1 , u2 ∈ X2 with uW ≤ ρ.
If inf{ϕu1 , u2 : u1 , u2 ∈ W} 0, then from above, we have ϕu1 , u2 0 for all
u1 , u2 ∈ X2 with u1 , u2 W ≤ ρ. Hence, all u1 , u2 ∈ X2 with u1 , u2 W ≤ ρ are
minimal points of ϕ, which implies that ϕ has infinitely many critical points. If inf{ϕu1 , u2 :
u1 , u2 ∈ W} < 0, then by Lemma 2.4, ϕ has at least two nonzero critical points. Hence,
system 1.7 has at least two nontrivial solutions in W. We complete our proof.
Proof of Theorem 1.3. We only need to use 1.18 and 1.19 to replace 2.6 and 2.7 in the
proof Theorem 1.1 with p q 2, Ft, u1 , u2 F1 u1 and Km u2 ≡ 0 m ∈ C. It is easy. So,
we omit it.
Lemma 3.2. Under the assumptions of Theorem 1.4, the functional ϕ1 defined by
1
ϕ1 u1 2
T
2
|u̇1 t| dt T
0
F1 u1 tdt 0
l
Ij u1 tj dt
3.14
j1
satisfies (PS) condition.
Proof. Suppose that {u1n } is a PS sequence for ϕ1 ; that is, there exists D1 > 0 such that
ϕu1n ≤ D1 ,
∀n ∈ N,
ϕ u1n −→ 0,
as n −→ ∞.
3.15
Hence, for n large enough, we have ϕ u1n ≤ 1. It follows from F1 , I2, and 1.18 that
u1n WT1,2
≥
ϕ1 u1n , u
1n
T
2
|u̇1n t| dt 0
T
1n tdt
∇x1 F1 u1n t, u
0
l
∇Ij u1n tj , u
1n tj
j1
Journal of Applied Mathematics
T
15
|u̇1n t|2 dt 0
T
1n tdt
∇x1 F1 u1n t − ∇x1 F1 u1n t, u
0
l
∇Ij u1n tj , u
1n tj
j1
≥
T
0
T2
|u̇1n t| dt − r 2
4π
2
T2
≥ 1−r 2
4π
T
T
0
u1n ∞
|u̇1n t|2 dt − l
dj
j1
2
|u̇1n t| dt −
0
T
12
1/2 T
1/2
2
|u̇1n t| dt
0
l
dj ,
j1
3.16
for n large enough. By 1.18, we have
1/2
1/2 T
T2
2
≤
1
,
|u̇1n t| dt
4π 2
0
u1n W 1,2
T
3.17
and 3.16, 3.17, and r < 4π 2 /T 2 imply that there exists D2 , D3 > 0 such that
T
|u̇1n t|2 dt ≤ D2 ,
0
3.18
u1n W 1,2 ≤ D3 .
T
It follows from F4, 3.15, I3, 1.18, and 3.18 that
−D1 ≤ ϕ1 u1n ≤
1
2
1
2
1
2
1
2
T
|u̇1n t|2 dt 0
T
0
T
0
F1 u1n tdt 0
|u̇1n t|2 dt 0
T
T
1
μ
l
I j u1 t j
j1
T
0
F1 λu1n dt −
T
F1 −
u1n tdt 0
T
|u̇1n t| dt F1 λu1n − T F1 0 −
μ
2
|u̇1n t|2 dt l
γj
j1
T
u1n t − F1 0dt F1 −
0
l
T
F1 λu1n − T F1 0 γj
μ
j1
l
j1
γj
16
Journal of Applied Mathematics
T 1
1
u1n t − ∇F1 0, −s
u1n tds dt
∇F1 −s
0 0 s
T 1
l
1 T
T
≤
s|
u1n t|2 ds dt − T F1 0 γj
|u̇1n t|2 dt F1 λu1n r
2 0
μ
0 0
j1
−
≤
1
2
T
|u̇1n t|2 dt 0
T
r
F1 λu1n μ
2
T
u1n t|2 dt − T F1 0 |
0
l
γj
j1
≤
l
max{1, r}
T
u1n 2W 1,2 F1 λu1n − T F1 0 γj
2
μ
T
j1
≤
l
max{1, r} q T
D3 F1 λu1n − T F1 0 γj ,
2
μ
j1
3.19
for all n and 3.19 and F5 imply that {u1n } is bounded. Combining 3.18, we know that
{u1n } is a bounded sequence. Similar to the argument in 25, it is easy to obtain that ϕ satisfies
PS condition.
Proof of Theorem 1.4. From I3 and F5, it is easy to see that for x1 ∈ RN ,
ϕ1 x1 −→ −∞,
as |x1 | −→ ∞.
3.20
1,2 , by 1.18, F1 and I3, we have
For all u1 ∈ W
T
ϕ1 u1 1
2
1
2
1
2
1
2
T
|u̇1 t|2 dt 0
T
|u̇1 t|2 dt 0
T
j1
F1 u1 t − F1 0dt T F1 0 |u̇1 t|2 dt T 1
0
|u̇1 t|2 dt ∇F1x1 su1 t, u1 tds dt 0
T 1
0
l
I j u1 t j
j1
0
l
Ij u1 tj T F1 0
j1
1
∇F1x1 su1 t − ∇F1x1 0, su1 tds dt
s
l
Ij u1 tj T F1 0
j1
l
I j u1 t j
0
0
T
F1 u1 tdt 0
0
T
T
Journal of Applied Mathematics
1
≥
2
1
≥
2
T
0
T
0
≥
T
r1 T 2
|u̇1 t| dt −
4π 2
T
12
T
|u1 t|2 dt T F1 0 −
0
2
1 r1 T 2
−
2 4π 2
−
r1
|u̇1 t | dt −
2
2
17
l
l
α bj u1 tj j − cj
j1
T
|u̇1 t|2 dt T F1 0 −
0
j1
l
l
αj
bj u1 ∞ − cj
j1
j1
|u̇1 t|2 dt T F1 0
0
T
αj /2
αj /2 l
l
2
bj
− cj .
|u̇1 t| dt
j1
0
j1
3.21
1,2 , u1 1,2 is equivalent to u̇1 L2 . Then, r1 < 4π 2 /T 2 , αj < 2j ∈ B
Note that for all u1 ∈ W
WT
T
and 3.21 imply that
ϕ1 u1 −→ ∞,
as u1 W 1,2 −→ ∞,
T
1,2 .
u1 ∈ W
T
3.22
It follows from 3.20 and 3.22 that ϕ1 satisfies ϕ1 and ϕ2 in Lemma 2.5. Combining with
Lemma 3.2, Lemma 2.5 shows that ϕ1 has at least one critical point. Thus, we complete the
proof.
4. Examples
Example 4.1. Let q 4, p 2, T π, t1 1, and s1 2. Consider the following system:
d
Φ4 u̇1 t ∇u1 Fu1 t , u2 t, a.e. t ∈ 0, π,
dt
d
Φ2 u̇2 t ∇u2 Fu1 t, u2 t, a.e. t ∈ 0, π,
dt
u1 0 − u1 π u̇1 0 − u̇1 π 0,
4.1
u2 0 − u2 π u̇2 0 − u̇2 π 0,
ΔΦ4 u̇1 1 Φq u̇1 1
− Φq u̇1 1− ∇I1 u1 1,
ΔΦ2 u̇2 2 Φp u̇2 2
− Φp u̇2 2− ∇K1 u2 2,
4
4
4
4
2
2
x12
· · · x1N
1/π 2 x21
x22
· · · x2N
− 1/2π 2 |x2 |2 ,
where Fx1 , x2 x11
2
2
x1 x11 , x12 , . . . , x1N , x2 x21 , x22 , . . . , x2N , I1 x e|x| , K1 x e|x| , x ∈ RN . It
is easy to verify that all conditions of Theorem 1.1 hold so that system 4.1 has at least
4
4
4
x22
· · · x2N
− 1/2π 2 |x2 |2 ,
one weak solution. Moreover, if Fx1 , x2 1/π 2 x21
N
x2 x21 , x22 , . . . , x2N , I1 x 0 and K1 x 0, x ∈ R , then system 4.1 has at least
two nonzero solutions.
18
Journal of Applied Mathematics
Example 4.2. Let T 2, t1 1. Consider the following autonomous second-order Hamiltonian
system with impulsive effects:
üt ∇u Fut,
a.e. t ∈ 0, 2,
u0 − u2 u̇0 − u̇2 0,
u̇1
− u̇ 1− ∇I1 u1,
4.2
2
where Fz z41 z22 · · · z2N − 1/2|z|2 ,I1 z e|z| , z z1 , . . . , zN τ ∈ RN . It is easy to verify
that all conditions of Theorem 1.3 hold so that system 4.2 has at least one weak solution.
Moreover, if Fz z41 z42 · · · z4N − 1/2|z|2 and I1 z 0, z ∈ RN , then system 4.2 has at
least two nonzero solutions.
Example 4.3. Let T π, t1 2. Consider the following autonomous second-order Hamiltonian
system with impulsive effects:
üt ∇u Fut,
a.e. t ∈ 0, π,
u0 − uπ u̇0 − u̇π 0,
4.3
−
u̇2 − u̇2 ∇I1 u2,
where Fz −|z|2 , I1 z 2 sin z1 , z z1 , . . . , zN τ ∈ RN . It is easy to verify that all conditions of Theorem 1.4 hold so that system 4.3 has at least one weak solution.
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