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Boundary Value Problems
Volume 2011, Article ID 736093, 11 pages
doi:10.1155/2011/736093
Research Article
Three Solutions for Forced Duffing-Type
Equations with Damping Term
Yongkun Li and Tianwei Zhang
Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China
Correspondence should be addressed to Yongkun Li, [email protected]
Received 16 December 2010; Revised 6 February 2011; Accepted 11 February 2011
Academic Editor: Dumitru Motreanu
Copyright q 2011 Y. Li and T. Zhang. 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.
Using the variational principle of Ricceri and a local mountain pass lemma, we study the existence
of three distinct solutions for the following resonant Duffing-type equations with damping and
perturbed term u
tσu
tft, ut λgt, ut pt,a.e.t ∈ 0,ω, u00 uω and
without perturbed term u
tσu
tft, ut pt,a.e.t ∈ 0,ω, u00 uω.
1. Introduction
In this paper, we consider the following resonant Duffing-type equations with damping and
perturbed term:
u
0 u
ω
,
1.1
where σ,λ ∈ R, f, g : 0,ω × R → R,andp : 0,ω → R are continuous. Letting λ 0in
problem 1.1 leads to
u
t
σu
t
f
t, u
t
p
t
, a.e.t∈
t
σu
t
f
t, u
t
p
t
. 1.3
Recently, many authors have studied the existence of periodic solutions of the Duffing-type
equation 1.3. By using various methods and techniques, such as polar coor dinates, the
method of upper and lower solutions and coincidence degree theory and a series of existence
results of nontrivial solutions for the Duffing-type equations such as 1.3 have been obtained;
we refer to 5–11 and references therein. There are also authors who studied the Duffing-type
equations by using the critical point theory see 12, 13.In12, by using a saddle point
theorem, Tomiczek obtained the existence of a solution of the following Duffing-type system:
u
, a.e.t∈
0,ω
,
u
0
0 u
ω
,
1.4
which is a special case of problems 1.1-1.2. However, to the best of our knowledge, there
are few results for the existence of multiple solutions of 1.3.
Our aim in this paper is to study the variational structure of problems 1.1-1.2 in
an appropriate space of functions and the existence of solutions for problems 1.1-1.2
by means of some critical point theorems. The organization of this paper is as follows. In
Section 2, we shall study the variational structure of problems 1.1-1.2 and give some
important lemmas which will be used in later section. In Section 3, by applying some critical
point theorems, we establish sufficient conditions for the existence of three distinct solutions
to problems 1.1-1.2.
2. Variational Structure
In the Sobolev space H : H
1
0
0,ω, consider the inner product
ω
0
u
s
2
ds
1/2
∀u ∈ H.
2.2
We also consider the inner product
u, v
ω
0
e
σs
s
2
ds
1/2
∀u ∈ H.
2.4
Obviously, the norm ·and the norm ·
H
are equivalent. So H is a Hilbert space with the
norm ·.
By Poincar
´
e’s inequality,
u
2
2
:
ω
0
|
u
}
ω
0
e
σs
u
s
2
ds : λ
0
u
2
∀u ∈ H,
2.5
where λ
0
: 1/λ
1
min{1,e
σω
2.6
Usually, in order to find the solution of problems 1.1-1.2, we should consider the
following functional Φ, Ψ defined on H:
Φ
u
1
2
ω
0
e
σs
|u
s
|
2
ds
ω
0
e
σs
p
s, u
s
ds,
2.7
where Fs, u
u
0
fs, μdμ, Gs, u
u
0
gs, μdμ.
4 Boundary Value Problems
Finding solutions of problem 1.1 is equivalent to finding critical points of I :ΦλΨ
in H and
I
u
,v
ω
ω
0
e
σs
f
s, u
v
s
ds −
ω
0
e
σs
λg
s, u
v
s
ds, ∀u, v ∈ H.
2.8
Lemma 2.1 H
¨
ds
1/p
·
b
a
g
s
q
ds
1/q
.
2.9
Lemma 2.2. Assume the following condition holds.
f1 There exist positive constants α, β,andγ ∈ 0, 1 such that
f
s, x
u
n
1
2
ω
0
e
σs
u
n
s
2
ds
ω
0
e
σs
p
−
ω
0
e
2σs
p
s
2
ds
1/2
u
n
2
− max
{
1,e
σω
}
2σs
p
s
2
ds
1/2
u
n
2
− α
√
ω max
{
1,e
σω
}
u
n
2
− β
e
2σs
p
s
2
ds
1/2
α
√
ω max
{
1,e
σω
}
u
n
−
λ
γ1
and β
0
such that
f
s, x
≤ α
0
β
0
|
x
|
∀
s, x
∈
0,ω
× R. 2.12
Then Φ is coercive.
Lemma 2.4. Assume the following condition holds.
f3 lim
0,ω.Putφ
K
s : sup
|x|≤K
|Fs, x| for all s ∈ 0,ω. By the continuity of f, we know that
sup
s∈0,ω
φ
K
s < ∞. Then one has
Φ
u
n
1
2
ω
0
e
σs
u
n
s
F
s, u
n
s
ds −
{
|
u
n
|
>K}
e
σs
F
s, u
n
s
ds
≥
1
2
u
ω
0
e
σs
φ
K
s
ds,
2.14
which implies that lim
n →∞
Φu
n
∞. This completes the proof.
Based on Ricceri’s variational principle in 14, 15, Fan and Deng 16 obtained the
following result which is a main tool used in our paper.
Lemma 2.5 see 16. Suppose that D is a bounded convex open subset of H, v
1
,v
2
∈ D, Φv
1
inf
D
Φc
0
and u
2
lying in D,whereu
1
∈ Φ
−1
−∞,ρ
0
∩D,
u
2
∈ Φ
−1
−∞,ρ
1
∩Bu
1
,,whereBu
1
,{u ∈ H : u −u
1
<},andu
2
∈ Bu
1
,.
3. Main Results
In this section, we will prove that problems 1.1-1.2 have three distinct solutions by using
the variational principle of Ricceri and a local mountain pass lemma.
6 Boundary Value Problems
−δ, 0
∪
0,δ
,
3.1
f5 there exists x
0
∈ H such that Φx
0
< 0.
Then there exist λ
∗
> 0 and r>0 such that, for every λ ∈ −λ
∗
,λ
∗
,problem1.1 admits at least
three distinct solutions which belong to B0,r ⊆ H.
Proof. By Lemma 2.2, condition f1 implies that the functional Φ is coercive. Since Φ is
sequentially weakly lower semicontinuous see 16, Propositions 2.5 and 2.6, Φ has a global
minimizer v
1
.Byf5,weobtainΦv
1
inf
H
Φc
s
|
≤ c
1
u
∀u ∈ H.
3.3
Choosing r
δ
<δ/c
1
, it results that
B
0,r
δ
{
u ∈ H :
u
≤ r
δ
}
⊆
s
2
ds
ω
0
e
σs
p
s
u
s
ds −
ω
0
e
σs
F
s, u
ds −
ω
0
e
σs
F
s, u
s
ds
ω
0
e
σs
|
u
s
|
2
2λ
0
e
0
.
At this point, we can apply Lemma 2.5 taking Ψ and −Ψ as perturbing terms. Then,
for ∈ 0,r
δ
small enough and any ρ
1
∈ c
0
, min {b, c
1
}, ρ
2
∈ 0, ∞, we can obtain the
following.
Boundary Value Problems 7
i There exists
λ>0suchthat,foreachλ ∈ −
λ,
λ, ΦλΨ has two distinct local
minima u
1
and u
2
satisfying
u
1
1
∪ B
0,
⊆ B
0,r
1
,
3.7
and put b sup
u≤r
1
|Φu|. Owing to the coerciveness of Φ,thereexistsr
2
>r
1
such that
inf
ur
2
Φud>b.Sinceg : 0,ω × R → R is continuous, then
sup
u
≤r
≤r
2
|
Ψ
u
|
>
b d
2
,
3.9
and when u≤r
1
Φ
u
λΨ
u
≤ b
|
λ
|
sup
u
<
θ
4
.
3.11
Therefore, by 3.6 and 3.11,
λ can be chosen small enough that
Φ
u
1
λΨ
u
1
≤ 0, Φ
u
2
λΨ
u
2
<
θ
A
ϕ ∈ C
0, 1
,H
: ϕ
0
u
1
,ϕ
1
u
2
, 3.13
and consider the real number c : inf
ϕ∈A
sup
s∈0,1
Φϕs λΨϕs.Sinceu
1
∈B0, and
each path ϕ goes through ∂B0,, one has c ≥ θ/2.
−→ c as n −→ ∞.
3.14
Applying a general mountain pass lemma without the PS condition see 17,Theorem2.8,
there exists a sequence {u
n
}⊂B0,r
2
such that Φu
n
λΨu
n
→ c and Φ
u
n
λΨ
u
n
→
0asn → ∞.Hence{u
n
} is a bounded PS
c
sequence and, taking into account the fact that
Φ
λΨ
is an S
,problem1.1 admits at least three distinct solutions which belong to B0,r ⊆ H.
Furthermore, problem 1.2 admits at least three distinct solutions.
Corollary 3.4. Assume that (f3), (f4), and (f5); hold, then there exist λ
∗
> 0 and r>0 such that, for
every λ ∈ −λ
∗
,λ
∗
,problem1.1 admits at least three distinct solutions which belong to B0,r ⊆ H.
Furthermore, problem 1.2 admits at least three distinct solutions.
4. Some Examples
Example 4.1. Consider the following r esonant Duffing-type equations with damping and
perturbed term
u
t
σu
t
f
t, u
t
s,and
f
t, x
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
∈
0, 2π
×
−1, −0.001
,
20 cos
2
s − x
1/3
for
s, x
∈
0, 2π
×
−0.001, 0.001
,
20 cos
2
s Q
1, ∞
,
4.2
in which Q
1
∈ C−1, −0.001 and Q
2
∈ C 0.001, 1 satisfy
Q
1
−1
−1 , Q
1
−0.001
0.1,Q
2
0.001
−0.1,Q
2
1
e
σs
p
s
x −
x
0
f
s, μ
dμ ≥
x
2
8e
2π
20
cos
2
s
x −
20
cos
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
0, for s 0,
10
4
sin s, for s ∈
0, 2π
,
0, for s 2π.
4.5
10 Boundary Value Problems
Clearly, x
0
∈ H.Thenweobtainthat
Φ
x
0
s
0.001
0
1
0.001
10
4
sin s
1
f
s,μ
dμ ds
e
2π
π
2
2π
0
e
sin s
1
μ
1/3
dμ ds
≤
e
2π
π
2
− 10
4
< 0.
4.6
So Φx
0
< 0, which implies that f5 is satisfied. To this end, all assumptions of Theorem 3.1
hold. By Theorem 3.1,thereexistsλ
∗
> 0, for every λ ∈ −λ
∗
,λ
∗
,problem8 admits at least
three distinct solutions.
Example 4 .2. Let λ 0. From Example 4.1, we can obtain that the following resonant Duffing-
type equations with damping:
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