Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 616427, 8 pages
doi:10.1155/2012/616427
Research Article
Periodic Solutions of Some Impulsive Hamiltonian
Systems with Convexity Potentials
Dezhu Chen and Binxiang Dai
School of Mathematical Sciences and Computing Technology, Central South University, Hunan,
Changsha 410083, China
Correspondence should be addressed to Binxiang Dai, binxiangdai@126.com
Received 31 August 2012; Accepted 18 November 2012
Academic Editor: Juntao Sun
Copyright q 2012 D. Chen and B. Dai. 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 periodic solutions of some second-order Hamiltonian systems with
impulses. We obtain some new existence theorems by variational methods.
1. Introduction
Consider the following systems:
üt ft, ut, a.e. t ∈ 0, T ,
Δu̇tk gk u t−k , k 1, 2, . . . , m,
1.1
uT − u0 u̇T − u̇0 0,
where k ∈ Z, u ∈ Rn , Δu̇tk u̇tk − u̇t−k with u̇t±k limt → t±k u̇t, gk u gradu
Gk u, Gk ∈ C1 Rn , Rn for each k ∈ Z, there exists an m ∈ Z such that 0 t0 < t1 < · · · < tm <
tm1 T , and we suppose that ft, u gradu Ft, u satisfies the following assumption.
A Ft, x is measurable in t for x ∈ Rn and continuously differentiable in x for a.e.
t ∈ 0, T , and there exist a ∈ CR , R , b ∈ L1 0, T ; R such that
|Ft, x| ft, x ≤ a|x|bt,
for all x ∈ Rn and t ∈ 0, T .
1.2
2
Abstract and Applied Analysis
Many solvability conditions for problem 1.1 without impulsive effect are obtained,
such as, the coercivity condition, the convexity conditions see 1–4 and their references,
the sublinear nonlinearity conditions, and the superlinear potential conditions. Recently,
by using variational methods, many authors studied the existence of solutions of some
second-order differential equations with impulses. More precisely, Nieto in 5, 6 considers
linear conditions, 7–10 the sublinear conditions, and 11–16 the sublinear conditions
and the other conditions. But to the best of our knowledge, except 7 there is no result
about convexity conditions with impulsive effects. By using different techniques, we obtain
different results from 7.
We recall some basic facts which will be used in the proofs of our main results. Let
HT1 u : 0, T −→ Rn absolutely continuous; u0 uT , u̇t ∈ L2 0, T ; Rn ,
1.3
with the inner product
u, v T
ut, vtdt T
0
1.4
∀u, v ∈ HT1 ,
u̇t, v̇tdt,
0
where ·, · denotes the inner product in Rn . The corresponding norm is defined by
u T
ut, utdt 0
T
1/2
u̇t, u̇tdt
,
0
The space HT1 has some important properties. For u ∈ HT1 , let u 1/2T u
ut − u. Then one has Sobolev’s inequality see Proposition 1.3 in 1:
u2∞
T
≤
12
T
1.5
∀u ∈ HT1 .
T
0
utdt, and
1.6
|u̇t|2 dt.
0
Consider the corresponding functional ϕ on HT1 given by
ϕu 1
2
T
|u̇t|2 dt 0
T
0
Ft, utdt m
Gk utk .
1.7
k1
It follows from assumption A and the continuity of gk one has that ϕ is continuously
differentiable and weakly lower semicontinuous on HT1 . Moreover, we have
ϕ
u, v T
0
u̇t, v̇tdt T
0
m
ft, ut, vt dt gk utk , vtk ,
1.8
k1
for u, v ∈ HT1 and ϕ
is weakly continuous and the weak solutions of problem 1.1 correspond
to the critical points of ϕ see 8.
Abstract and Applied Analysis
3
Theorem 1.1 2, Theorem 1.1. Suppose that V and W are reflexive Banach spaces, ϕ ∈ C1 V ×
W, R, ϕv, · is weakly upper semi-continuous for all v ∈ V , and ϕ·, w : V → R is convex for all
w ∈ W and ϕ
is weakly continuous. Assume that
ϕ0, w −→ −∞
1.9
ϕv, w −→ ∞,
1.10
as w → ∞ and for every M > 0,
as v → ∞ uniformly for w ≤ M. Then ϕ has at least one critical point.
2. Main Results
Theorem 2.1. Assume that assumption A holds. If further
H1 Ft, · is convex for a.e. t ∈ 0, T , and
H2 there exist η, θ > 0 such that Gk x ≥ η|x| θ, for all x ∈ Rn , then 1.1 possesses at least
one solution in HT1 .
Remark 2.2. H1 implies there exists a point x for which
T
2.1
∇Ft, xdt 0.
0
Proof of Theorem 2.1. It follows Remark 2.2, 1.6, and H2 that
ϕu ≥
1
2
1
2
1
2
1
≥
2
1
≥
2
T
|u̇t|2 dt 0
T
T
Ft, ut − Ft, xdt 0
|u̇t|2 dt T
Ft, xdt T
0
0
0
T
T
0
T
|u̇t| dt −
0
0
2
|u̇t| dt − C0
m
Gk utk k1
m
Gk utk ft, x, ut − x dt k1
m
dt η|
u u| mθ
ft, x, u
0
T
0
T
Ft, xdt 0
2
Ft, xdt 0
T
|u̇t|2 dt T
2.2
k1
ft, xdt u∞ mη|u| − mη
u∞ mθ
T
1/2
2
|u̇t| dt
mη|u| mθ,
0
for all u ∈ HT1 and some positive constant C0 . As u → ∞ if and only if |u|2 u̇22 1/2 →
∞, we have ϕu → ∞ as u → ∞. By Theorem 1.1 and Corollary 1.1 in 1, ϕ has a
minimum point in HT1 , which is a critical point of ϕ. Hence, problem 1.1 has at least one
weak solution.
4
Abstract and Applied Analysis
Theorem 2.3. Assume that assumption A and H1 hold. If further
H3 there exist η, θ > 0 and α ∈ 0, 2 such that Gk x ≤ η|x|α θ for all x ∈ Rn and
H4 there exist some β > α and γ > 0 such that
|x|−β
T
2.3
Ft, xdt ≤ −γ,
0
for |x| ≥ M and t ∈ 0, T , where M is a constant, then 1.1 possesses at least one solution in HT1 .
Remark 2.4. We can find that our condition H4 is very different from condition vii in 7
since we prove this by the saddle point theorem substituted for the least action principle.
Proof of Theorem 2.3. We prove ϕ satisfies the PS condition at first. Suppose {un } is such an
sequence that {ϕun } is bounded and limn → ∞ ϕ
un 0. We will prove it has a convergent
subsequence. By H3 and 1.6, we have
m
Gk utk ≤
k1
m
η|
utk u|α mθ
k1
2.4
≤ 4mη |
utk |α |u|α mθ
≤ C1 u̇α2 C2 |u|α C3 ,
for some positive constants C1 , C2 , C3 . By Remark 2.2, 1.6, and 2.4, we have
1
ϕun 2
1
2
1
2
1
≥
2
1
≥
2
T
2
|u̇n t| dt 0
T
T
Ft, un t − Ft, xdt 0
|u̇n t|2 dt T
Ft, xdt T
0
0
0
T
T
0
T
|u̇n t| dt −
0
2
|u̇n t| dt − C5
m
Gk un tk k1
m
Gk un tk ft, x, un t − x dt k1
m
n dt Gk un tk ft, x, u
0
T
0
T
Ft, xdt 0
2
Ft, xdt 0
T
|u̇n t|2 dt T
2.5
k1
ft, xdt un ∞ − C1 u̇n α2 − C2 |un |α − C4
T
0
1/2
2
|u̇n t| dt
0
− C1 u̇n α2 − C2 |un |α − C4 ,
for some positive constants C4 , C5 , which implies that
C|un |
α/2
≥
T
0
1/2
2
|u̇n t| dt
− C6 ,
2.6
Abstract and Applied Analysis
5
for some positive constants C, C6 . By 1.6, the above inequality implies that
un ∞ ≤ C7 |un |α/2 1 ,
2.7
for the positive constant C7 . The one has
un | ≥ |un | − un ∞ ≥ |un | − C7 |un |α/2 1 ,
|un t| ≥ |un | − |
∀t ∈ 0, T .
2.8
If {|un |} is unbounded, we may assume that, going to a subsequence if necessary,
|un | −→ ∞
as n −→ ∞.
2.9
By 2.8 and 2.9, we have
|un t| ≥
1
|un |,
2
2.10
for all large n and every t ∈ 0, T . By 2.10 and H4 , one has |un t| ≥ M for all large n. It
follows from H4 , 2.4, 2.6, 2.7, and above inequality that
2 T
γ|un t|β dt C2 ϕun ≤ C|un |α/2 C6 −
uα∞ C2 |u|α C3
0
α/2
≤ C|un |
C6
2
−β
β
− 2 |un | T γ C8 |un |
α/2
α
1 C2 |u|α C3 ,
2.11
for large n and the positive constant C8 , which contradicts the boundedness of ϕun since
β > α. Hence |un | is bounded. Furthermore, un is bounded by 2.6. A similar calculation
to Lemma 3.1 in 9 shows that ϕ satisfies the PS condition. We now prove that ϕ satisfies
1 {u ∈ H 1 : u 0}, then
the other conditions of the saddle point theorem. Assume that H
T
T
1 ⊕ Rn . From above calculation, one has
HT1 H
T
1
ϕu ≥
2
T
2
|u̇t| dt − C5
0
T
1/2
2
|u̇t| dt
0
− C1 u̇α2 − C4 ,
2.12
1 , which implies that
for all u ∈ H
T
ϕu −→ ∞,
2.13
6
Abstract and Applied Analysis
1 . Moreover, by H3 and H4 we have
as u → ∞ in H
T
T
ϕx m
Ft, xdt 0
Gk x
2.14
k1
β
α
≤ − T γ|x| mη|x| mθ,
for |x| > M, which implies that
ϕx −→ −∞,
2.15
as |x| → ∞ in Rn since β > α. Now Theorem 2.3 is proved by 2.13, 2.15, and the saddle
point theorem.
Theorem 2.5. Assume that assumption A holds. Suppose that Ft, ·, Gk x are concave and satisfy
H5 Gk x ≤ −η|x| θ for some positive constant η, θ > 0, then 1.1 possesses at least one
solution in H1T .
1 given by
Proof of Theorem 2.5. Consider the corresponding functional ϕ on Rn × H
T
1
ϕu −
2
T
2
|u̇t| dt −
0
T
Ft, utdt −
m
0
Gk utk ,
2.16
k1
which is continuously differentiable, bounded, and weakly upper semi-continuous on HT1 .
Similar to the proof of Lemma 3.1 in 2, one has that ϕx w is convex in x ∈ Rn for every
1 . By the condition, we have −Gk x w ≥ −2Gk 1/2xGk −w. Similar to the proof
w∈H
T
of Theorem 3.1, we have
1
ϕx w −
2
1
≥ −
2
1
≥ −
2
T
2
|ẇ| dt −
0
T
T
0
m
0
2
|ẇ| dt −
T
0
T
Ft, x wdt −
0
2
|ẇ| dt − C0
Gk x w
k1
m
ft, xdt w −
Gk x w C9
∞
k1
T
0
1/2
2
|ẇ| dt
− 2Gk
1
x Gk −w C9 ,
2
2.17
Abstract and Applied Analysis
7
1 with w ≤ M by H5 and
which means ϕx w → ∞ as |x| → ∞, uniformly for w ∈ H
T
1.6. On the other hand,
ϕw −
≤ −
1
2
1
2
T
0
T
0
|ẇ|2 dt −
T
Ft, wdt −
0
|ẇ|2 dt C0
m
Gk w
k1
T
0
2.18
1/2
|ẇ|2 dt
mηw∞ C9 ,
1 by H5 and 1.6. We complete our proof
which implies that ϕw → −∞ as w → ∞ ∈ H
T
by Theorem 1.1.
Acknowledgment
The first author was supported by the Postgraduate Research and Innovation Project of
Hunan Province CX2011B078.
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