Hindawi Publishing Corporation
Boundary Value Problems
Volume 2011, Article ID 845413, 18 pages
doi:10.1155/2011/845413
Research Article
Lagrangian Stability of a Class of
Second-Order Periodic Systems
Shunjun Jiang, Junxiang Xu, and Fubao Zhang
Department of Mathematics, Southeast University, Nanjing 210096, China
Correspondence should be addressed to Junxiang Xu, [email protected]
Received 24 November 2010; Accepted 5 January 2011
Academic Editor: Claudianor O. Alves
Copyright q 2011 Shunjun Jiang et al. 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 following second-order differential equation: Φ
p
x
Fx, tx
ω
p
Φ
p
xα|x|
l
x
ex, t0, w here Φ
2 Boundary Value Problems
When gx satisfies
0 ≤ k ≤
g
x
x
≤ K ≤ ∞, ∀x ∈ R,
1.2
that is, the differential equation 1.1 is semilinear, similar results also hold, but the proof is
more difficult since there may be resonant case. We refer to 6–8 and the references therein.
In 8 Liu considered the following equation:
x
λ
2
x ϕ
x
e
t
,
1.3
where ϕxox as |x|→∞ and et is a 2π-periodic function. After introducing
new variables, the differential equation 1.3 can be changed into a Hamiltonian system.
Under some suitable assumptions on ϕx and et, by using a variant of Moser’s small twist
e
t
,
1.4
where fx ∈ C
5
R \ 0 ∩ C
0
R is bounded, et ∈ C
6
R \ 2πZ is periodic. The idea is
also to change the original problem to Hamiltonian system and then use a twist theorem of
area-preserving mapping to the Pioncar
´
emap.
The above differential equation essentially possess Hamiltonian structure. It is well
known that the Hamiltonian structure and reversible structure have many similar property.
Especially, they have similar KAM theorem.
Recently, Liu 6 studied the following equation:
x
F
x
x, t
x
x
, ∀x
/
0,
1.6
where Φx
x
0
ϕtdt and 0 <γ<1 <α<2. Moreover,
x
k
d
k
Φ
x
dx
k
≤ c ·
|
x
|
σ
,
x
k
∂
kl
e
x, t
∂x
k
∂t
l
ω
p
Φ
p
x
α
|
x
|
l
x e
x, t
0.
1.9
where Φ
p
s|s|
p−2
sp>1, −1 <l<p−2, and α, ω > 0 ar e constants. We want to generalize
the result in 6 to a class of p-Laplacian-type differential equations of the form 1.9.Themain
idea is similar to that in 6. We will assume that the functions F and e have some parities such
that the differential system 1.9 still has a reversible structure. After some transformations,
we change the systems 1.9 to a form of small perturbation of integrable reversible system.
Then a KAM Theorem f or reversible mapping can be applied to the Poincar
´
e mapping of this
,e
x, −t
e
x, t
.
1.10
Moreover, suppose that there exists σ<lsuch that
x
k
∂
km
F
x, t
∂x
k
∂t
m
≤ c ·
|
x
|
σ1
, 1.11
for all x
/
0, for all 0 ≤ k ≤ 6, 0 ≤ m ≤ 6. Then every solution of 1.9 is bounded.
Remark 1.2. Our main nonlinearity α|x|
l
x in 1.9 corresponds to ϕ in 1.5.Althoughitis
more special than ϕ, it makes no essential difference of proof and can simplify our proof
greatly. It is easy to see from the proof that this main nonlinearity is used to guarantee the
small twist condition.
4 Boundary Value Problems
2. The Proof of Theorem
The proof of Theorem 1.1 is based on Moser’s small twist theorem for reversible mapping. It
mainly consists of two steps. The first one is to find an equivalent system, which has a nearly
integrable form of a reversible system. The second one is to show that Pincar
´
emapofthe
equivalent system satisfies some twist theorem for reversible mapping.
2.1. Action-Angle Variables
We first recall the definitions of reversible system. Let Ω ⊂
n
be an open domain, and Z
Zz, t : Ω ×
Z
Gz, −t
−Z
z, t
, ∀z ∈ Ω, ∀t ∈ R 2.2
with DG denoting the Jacobian matrix of G.
We are interested in the special involution Gx, y → x, −y with z x, y ∈ R
2
.Let
Z Z
1
,Z
2
.Thenz
Zz, t is reversible with respect to G if and only if
Z
1
x, −y, −t
−Z
1
x, y, t
differential equation 1.9 is changed into the following planar system:
x
Φ
q
y
,
y
−ω
p
Φ
p
x
− α
|
x
|
l
x − e
x, t
− F
x, t
of action-angle variables. One of solutions for 2.5 is the function sin
p
as defined below. Let
the number π
p
defined by
π
p
2
p−1
1/p
0
ds
1 − s
p
/
p − 1
1/p
.
2.6
We define the function wt : 0,π
p
/2 → 0, p − 1
1/p
, implicitly by
p
, 0 such that sin
p
is an odd function. Finally, we extend sin
p
to R by
2π
p
-periodicity. It is not difficult to verify that sin
p
has the following properties:
i sin
p
00, sin
p
01;
iip − 1|sin
p
t|
p
|sin
p
t|
p
p − 1;
iii sin
p
t is an odd periodic function with period 2π
y r
2/q
Φ
p
ω sin
p
ωθ
.
2.9
It is easy to see that
∂
x, y
∂
r, θ
−
2
q
ω
p
r.
2.10
Since the Jacobian matrix of Θ is nonsingular for r>0, the transformation Θ is a local
homeomorphism at each point r, θ of the set R
,
θ
1 f
2
t, θ, r
1 N
2
t, θ, r
P
2
t, θ, r
,
2.11
6 Boundary Value Problems
where
N
1
t, θ, r
−α
q
−
q
2
1
ω
p−1
r
1−2/q
sin
p
θF
r
2/p
sin
p
θ, t
Φ
q
r
2/q
Φ
p
t, θ, r
α
q
p
1
ω
p
r
4/p−22/pl
sin
l
p
θ
sin
2
p
θ,
P
2
ω sin
p
θ
q
p
1
ω
p
r
−2/q
sin
p
θe
r
2/p
sin
p
θ, t
,
2.12
with
r
k
∂
ks
F
t, θ, r
∂r
k
∂t
s
≤ c · r
2/pσ
,
r
k
∂
t, θ, r
∂r
k
∂t
s
r
k
∂
ks
e
x, t
∂x
k
∂t
s
c
1
p
r
k
∂
ks
e
x, t
∂x
k
∂t
s
r
2/p−1
k
sin
k
p
θ ··· c
k
p
e
x, t
∂x
k
∂t
s
··· cx
∂
1s
e
x, t
∂x∂t
s
≤ c ·
|
x
|
σ1
≤ c · r
2/pσ1
.
D
s
t
f
t, θ, r
≤ c · Ψ
r
, ∀r ≥ r
0
, ∀
t, θ
∈ S
1
× S
1
. 2.15
Lemma 2.3 see 6. The following conclusions hold true:
i if f ∈ M
n
Ψ,thenD
j
1
Ψ
2
;
iii Suppose Ψ, Ψ
1
, Ψ
2
satisfy that, there exists c>0 such that for ∀0 ≤ ξ ≤ 2 · r,
Ψ
ξ
≤ cΨ
r
,
lim
r → ∞
r
−1
Ψ
1
lim
r →∞
Ψ
2
0.
2.16
n
1
,n
2
with n
1
n
2
min
{
n
1
,n
2
}
. 2.17
Moreover,
f
t g
1
,θ,r
2
},n
2
−1
r
−1
Ψ · Ψ
2
.
2.18
Proof. This lemma was proved in 6, but we give the proof here for reader’s convenience.
Since i and ii are easily verified by definition, so we only prove iii.Let
v
t, θ, r
t g
1
t, θ, r
,u
t, θ, r
r g
2
∂t
≤ c · Ψ
2
,
∂v
∂t
≤ c,
∂v
∂r
≤ c · r
∂
js
g
1
∂r
j
∂t
s
.
2.21
8 Boundary Value Problems
Since g
1
∈ M
n
Ψ
1
,g
2
∈ M
n
Ψ
2
, it follows that
∂
js
u
∂r
j
∈ M
n
Ψ
2
, we know that for r sufficiently
large
r
0
r g
2
≤ 2r. 2.23
By the property of Ψ,wehave
g
t, θ, r
≤ c · Ψ
u
c · Ψ
r g
2
≤ c · Ψ
1
j
1
u
∂r
j
1
∂t
j
1
···
∂
j
b
j
b
u
∂r
j
b
∂t
j
b
·
∂
i
j
1
··· j
b
i
1
··· i
m
k, j
1
··· j
b
i
1
··· i
m
s. 2.26
Without loss of generality, we assume that
j
1
j
1
1, ,j
b
1
m
1
,therearem
2
numbers which equal to 0.
Since
∂
ks
g
∂r
k
∂t
s
∂
bm
f
v, θ, u
∂r
b
∂t
m
·
∂
j
1
j
∂
j
b
2
1
j
b
2
1
u
∂r
j
b
2
1
∂t
j
b
2
1
···
∂
j
b
1
j
b
b
1
1
···
∂
j
b
j
b
u
∂r
j
b
∂t
j
b
·
∂
i
1
i
1
v
∂r
i
1
m
2
1
v
∂r
i
m
2
1
∂t
i
m
2
1
···
∂
i
m
1
i
m
1
v
∂r
i
m
1
i
m
i
m
v
∂r
i
m
∂t
i
m
,
2.28
Boundary Value Problems 9
we have
∂
ks
g
∂r
k
∂t
s
≤
c · r
−b
Ψr
−j
−m
1
2
≤ c · r
b
2
−b
1
−j
b
1
1
···j
b
m
2
−m
1
−i
m
1
1
···i
m
r
−bb
2
Ψ
. 2.30
Obviously
f
t g
1
,θ,r
− f
t, θ, r
1
0
∂f
∂t
t ηg
1
,θ,r
g
1
dη.
2.31
Since
∈ M
n
1
−1,min{n
1
,n
2
}
Ψ · Ψ
1
, 2.33
In the same way we can consider ft, θ, r g
2
− ft, θ, r and we omit the details.
2.3. Some Estimates
The following lemma gives the estimate for f
1
t, θ, r and f
2
t, θ, r.
Lemma 2.4. f
1
t, θ, r ∈ M
5,5
r
β1
, f
2
θ r
2/p
Φ
q
Φ
p
ω sin
p
θ ∈
M
5,5
r
2/p
,usingtheconclusioniii of Lemma 2.3,wehaveP
1
t, θ, r ∈ M
5,5
r
β
1
,where
β
22 − p σ/p.NotethatN
1
t, θ, r ∈ M
5,5
t, θ, r
1 f
2
t, θ, r
−1
,
dt
dθ
1 f
2
t, θ, r
−1
.
2.34
It is easy to see that f
1
−t, −θ, r−f
1
t, θ, r and f
2
t, θ, r
h
1
t, θ, r
,
dt
dθ
1 − f
2
t, θ, r
h
2
t, θ, r
1 − N
2
t, θ, r
−P
2
in 2.11. It follows h
1
−t, −θ, r−h
1
t, θ, r,h
2
−t, −θ, rh
2
t, θ, r,andso2.35 is also
reversible with respect to the involution G : r, t → r, −t. Below we prove that h
1
t, θ, r
and h
2
t, θ, r are smaller perturbations.
Lemma 2.5. h
1
t, θ, r ∈ M
5,5
r
2β1
, h
2
t, θ, r ∈ M
5,5
r
2β
.
Proof. If r
0
t, θ, r
f
1
t, θ, r
.
2.36
It is easy to verify that
∂
km
∂r
k
∂t
m
f
s1
2
f
1
|i|k,|j|m,
c
i,j
∂
i
1
i
s2
∂t
j
s2
f
2
,
2.37
where i i
1
, ,i
l2
, |i| i
1
··· i
s2
,andj and |j| are defined in the same way as i and |i|.
So, we have
∂
km
∂r
k
∂t
m
h
1
|i|k,|j|m,n≥2
i
n
j
n
∂r
i
n
∂t
j
n
f
2
,
2.38
Boundary Value Problems 11
where
∂
i
τ
j
τ
∂r
i
τ
∂t
j
τ
f
2
≤ c, τ 2, ,n for f
r
β1−i
1
r
β−i
2
···r
β−i
n
≤ c
1
r
β1
r
β
r
β
n−2
r
−i
1
···i
n
≤ cr
−k
r
2β1
dθ
1 − N
2
t, θ, r
g
2
t, θ, r
,
2.41
where g
1
t, θ, rP
1
t, θ, rh
1
t, θ, r and g
2
t, θ, r−P
2
t, θ, rh
2
t, θ, r. By the proof of
Lemma 2.4,weknowP
1
∈ M
5,5
σ − l
> 0,
2.42
with σ<l<p− 2, −1 <l.
2.4. Coordination Transformation
Lemma 2.6. There exists a transformation of the form
t t, λ r S
r, θ
, 2.43
such that the system 2.41 is changed into the form
dλ
dθ
g
1
t, θ, λ
,
dt
dθ
1 − N
2
t, θ, λ
Moreover, the system 2.44 is reversible with respect to the involution G: λ, −t → λ, t.
Proof. Let
S
r, θ
θ
0
N
1
t, θ, r
dθ
q
2
α
ω
p−1
1
l 2
sin
l2
p
r, θ
∈ M
5,5
r
β1
. 2.48
Hence the map r, θ → λ, t with λ r Sr, θ is diffeomorphism for r 1. Thus, there is
afunctionL Lλ, θ such that
r λ L
λ, θ
2.49
where
L
λ, θ 2π
p
L
λ, θ
,L
λ, −θ
N
2
t, θ, λ
− N
2
t, θ, λ L
g
2
t, θ, λ L
.
2.51
Below we estimate g
1
and g
2
.Weonlyconsiderg
2
since g
1
can be considered similarly or even
simpler.
Obviously,
lim
g
2
t, θ, λ L
∈ M
5,5
λ
β−σ
. 2.54
In the same way as the above, we have
N
2
t, θ, r
N
2
t, θ, λ L
∈ M
5,5
λ
β
2.55
M
5,5
λ
ββ
.
2.56
By 2.54 and 2.56,notingthatβ
<β, it follows that
g
2
t, θ, λ
∈ M
5,5
λ
β−σ
. 2.57
Since Lλ, −θLλ, θ, the system 2.44 is reversible in θ with respect to the involution
λ, t → λ, −t.ThusLemma 2.6 is proved.
Now we make average on the nonlinear term N
2
t, θ, λ in the second equation of
,
2.59
where N
2
α · λ
β
with α 1/2π
p
q/pα/ω
p
2/p
2π
p
/ω
0
|sin
l
p
θ|
l2
d
θ and t he new
perturbations H
1
λ, τ, θ,H
2
λ
k1
∂
ks
∂λ
k
∂t
s
H
2
λ, τ, θ
≤ C · λ
β1−σ
. 2.60
Moreover, the system 2.59 is reversible with respect to the involution G: λ, τ → λ, −τ.
14 Boundary Value Problems
Proof. We choose
S
λ, θ
S
λ, 2π
p
θ
S
λ, θ
,
S
λ, θ
∈ M
5,5
λ
β
. 2.62
Defined a transformation by
τ t
S
1
λ, τ, θ
g
1
λ, τ −
S
λ, θ
,θ
,
H
2
λ, τ, θ
g
2
λ, τ −
S
λ, θ
,H
2
λ, −τ, −θ
H
2
λ, τ, θ
, 2.66
which implies that the system 2.59 is reversible with respect to the involution G: λ, τ →
λ, −τ. In the same way as the proof of g
1
λ, t, θ and g
2
λ, t, θ,wehave
λ
k
∂
ks
∂λ
k
∂t
≤ C · λ
β1−σ
. 2.67
Thus Lemma 2.7 is proved.
Below we introduce a small parameter such that the system 2.4 is written as a form
of small perturbation of an integrable.
Let
N
2
ρ. 2.68
Boundary Value Problems 15
Since
N
2
α · λ
β
−→ 0asλ −→ ∞,
2.69
then
λ −→ ∞⇐⇒ −→ 0
1
ρ, τ, θ,
ε
−1
d
N
2
dλ
H
1
λ
, ρ
,τ,θ
,g
2
ρ, τ, θ,
H
2
λ
≤ c ·
1σ
0
,
∂
ks
∂ρ
k
∂τ
s
g
2
≤ c ·
1σ
0
,σ
0
−
σ
β
−1
λ
β1
H
1
≤ c ·
−1
λ
β−1
λ
β1−σ
≤ c ·
−1
λ
2β−σ
≤ c ·
1σ
0
.
2.75
In the same way, |g
2
Γ
A
{
L ∈ Γ : L ⊂ A
}
. 2.76
Consider a mapping f
: A ⊂ C → C, which is reversible with respect to G : ρ, τ → ρ, −τ.
Moreover, a lift of f
can be expressed in the form:
τ
1
τ ω l
1
ρ, τ
g
1
ρ, τ,
,
ρ
1
ρ l
2
l
1
∈ C
6
A
,l
1
> 0,
∂l
1
∂ρ
> 0, ∀
ρ, τ
∈ A,
l
2
·, ·
, g
1
·, ·,
, g
2
∂I
∂τ
ρ, τ
l
2
ρ, τ
·
∂I
∂ρ
ρ, τ
0, ∀
ρ, τ
∈ A.
2.79
Moreover, suppose that there are two numbers a,and
b such that a<a<
b<band
I
M
where
I
M
r
max
ρ∈S
1
I
ρ, τ
,I
m
r
min
ρ∈S
1
I
ρ, τ
.
2.81
Then there exist ς>0 and Δ > 0 such that, if <Δ and
2
,andI.
In particular, ς is independent of .
Boundary Value Problems 17
Remark 2.10. If −l
1
,l
2
, g
1
, g
2
satisfy all the conditions of Lemma 2.9,thenLemma 2.9 still holds.
Lemma 2.11 see 14,Theorem1. Assume that ω/∈ 2 πQ and l
1
·, ·, l
2
·, ·g
1
·, ·, and
g
2
·, ·, ∈ C
4
A.If
2π
0
∂l
1
·, ·,
C
4
A
<ς.
2.84
The constants ς and Δ depend on ω, l
1
,l
2
only.
We use Lemmas 2.9 and 2.11 to prove our Theorem 1.1. For the reversible mapping
2.86, l
1
−2π
p
ρ, l
2
0.
2.6. Invariant Curves
From 2.73 and 2.66,wehave
g
1
ρ, −τ, −θ,
−g
1
ρ, τ,
,
ρ
1
ρ g
2
ρ, τ,
,
2.86
where τ ∈ S
1
and ρ ∈ 1, 2.Moreover,g
1
and g
2
satisfy
∂
kl
∂ρ
k
∂τ
l
Case 1 2π
p
is rational.LetI −l
1
2π
p
ρ, it is easy to see that
l
1
ρ, τ
∈ C
6
A
,l
1
ρ, τ
−2π
p
ρ<0,
∂l
1
ρ, τ
ρ, τ
l
2
ρ, τ
∂I
∂ρ
ρ, τ
0.
2.88
Since I only depends on ρ,and∂I/∂ρρ, τ > 0, all conditions in Lemma 2.9 hold.
18 Boundary Value Problems
Case 2 2π
p
is irrational.Since
2π
p
0
∂l
1
∂ρ
τ, ρ
7 T. K
¨
upper and J. You, “Existence of quasiperiodic solutions and Littlewood’s boundedness problem of
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vol. 145, no. 1, pp. 119–144, 1998.
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