Hindawi Publishing Corporation
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, yklie@ynu.edu.cn
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 t σu t f t, ut λgt, ut pt, a.e. t ∈ 0, ω, u0 0 uω and
without perturbed term u t σu t f t, ut pt, a.e. t ∈ 0, ω, u0 0 uω.
1. Introduction
In this paper, we consider the following resonant Duffing-type equations with damping and
perturbed term:
u σu t ft, ut λgt, ut pt,
a.e. t ∈ 0, ω,
u0 0 uω,
1.1
where σ, λ ∈ R, f, g : 0, ω × R → R, and p : 0, ω → R are continuous. Letting λ 0 in
problem 1.1 leads to
u t σu t ft, ut pt,
u0 0 uω,
a.e. t ∈ 0, ω,
1.2
which is a common Duffing-type equation without perturbation.
The Duffing equation has been used to model the nonlinear dynamics of special types
of mechanical and electrical systems. This differential equation has been named after the
2
Boundary Value Problems
studies of Duffing in 1918 1, has a cubic nonlinearity, and describes an oscillator. It is
the simplest oscillator displaying catastrophic jumps of amplitude and phase when the
frequency of the forcing term is taken as a gradually changing parameter. It has drawn
extensive attention due to the richness of its chaotic behaviour with a variety of interesting
bifurcations, torus and Arnolds tongues. The main applications have been in electronics, but
it can also have applications in mechanics and in biology. For example, the brain is full of
oscillators at micro and macro levels 2. There are applications in neurology, ecology, secure
communications, cryptography, chaotic synchronization, and so on. Due to the rich behaviour
of these equations, recently there have been also several studies on the synchronization of
two coupled Duffing equations 3, 4. The most general forced form of the Duffing-type
equation is
u t σu t ft, ut pt.
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 coordinates, 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. In 12, by using a saddle point
theorem, Tomiczek obtained the existence of a solution of the following Duffing-type system:
u t σu t σ2
m −
4
ut ft, ut pt,
2
a.e. t ∈ 0, ω,
1.4
u0 0 uω,
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 : H01 0, ω, consider the inner product
u, vH ω
0
u sv sds ∀u, v ∈ H,
2.1
Boundary Value Problems
3
inducing the norm
uH ω
1/2
2
u s ds
u, vH ∀u ∈ H.
2.2
0
We also consider the inner product
u, v ω
eσs u sv sds
∀u, v ∈ H,
2.3
0
and the norm
1/2
ω
2
σs e u s ds
u u, v ∀u ∈ H.
2.4
0
Obviously, the norm · and the norm · H are equivalent. So H is a Hilbert space with the
norm · .
By Poincaré’s inequality,
u22 :
ω
|us|2 ds ≤
0
1
≤
λ1 min{1, eσω }
1
λ1
ω
ω
2
u s ds
0
2
e u s ds : λ0 u2
σs 2.5
∀u ∈ H,
0
where λ0 : 1/λ1 min{1, eσω }, λ1 : π 2 /ω2 is the first eigenvalue of the problem
−u t λut,
t ∈ 0, ω,
2.6
u0 0 uω.
Usually, in order to find the solution of problems 1.1-1.2, we should consider the
following functional Φ, Ψ defined on H:
1
Φu 2
ω
ω
2
e |u s| ds σs
e psusds −
0
0
Ψu −
u
0
fs, μdμ, Gs, u u
0
eσs Fs, usds
0
ω
eσs Gs, usds,
0
where Fs, u ω
σs
gs, μdμ.
2.7
4
Boundary Value Problems
Finding solutions of problem 1.1 is equivalent to finding critical points of I : Φ λΨ
in H and
I u, v ω
eσs u sv sds 0
−
ω
eσs psvsds
0
ω
e fs, uvsds −
2.8
ω
σs
0
σs
e λgs, uvsds,
∀u, v ∈ H.
0
Lemma 2.1 Hölder Inequality. Let f, g ∈ Ca, b, p > 1, and q the conjugate number of p. Then
b
fsgsds ≤
b
a
fsp ds
1/p b
1/q
q
gs ds
·
.
a
2.9
a
Lemma 2.2. Assume the following condition holds.
f1 There exist positive constants α, β, and γ ∈ 0, 1 such that
fs, x ≤ α β|x|γ
∀s, x ∈ 0, ω × R.
2.10
Then Φ is coercive.
Proof. Let {un }n∈N ⊂ H be a sequence such that limn → ∞ un ∞. It follows from f1 and
Hölder inequality that
Φun ≥
1
2
ω
0
2
eσs un s ds 1
un 2 −
2
ω
0
ω
eσs psun sds −
0
2
e2σs ps ds
ω
eσs Fs, un sds
0
1/2
un 2 − max{1, eσω }
ω
α|un | β|un |γ 1 ds
0
ω
1/2
2
1
e2σs ps ds
un 2
un 2 −
2
0
√
γ 1
− α ω max{1, eσω }un 2 − β ω1−γ max{1, eσω }un 2
ω
1/2
2
√
1
2
2σs σω
ps ds
≥ un − λ0
e
α ω max{1, e } un 2
0
γ 1
− λ0 β ω1−γ max{1, eσω }un γ 1 ,
≥
which implies from γ ∈ 0, 1 that limn → ∞ Φun ∞. This completes the proof.
From the proof of Lemma 2.2, we can show the following Lemma.
2.11
Boundary Value Problems
5
Lemma 2.3. Assume that 2βλ0 max{eσω , 1} < 1 and the following condition holds.
f2 There exist positive constants α0 and β0 such that
fs, x ≤ α0 β0 |x|
∀s, x ∈ 0, ω × R.
2.12
Then Φ is coercive.
Lemma 2.4. Assume the following condition holds.
x
f3 lim|x| → ∞ 0 fs, μdμ ≤ 0 for all s ∈ 0, ω.
Then Φ is coercive.
Proof. Let {un }n∈N ⊂ H be a sequence such that limn → ∞ un ∞. Fix > 0, from f3, there
exists K K > 0 such that
Fs, x ≤ −
∀s ∈ 0, ω, |x| > K.
2.13
Denote by {|u| ≤ K} the set {s ∈ 0, ω : |us| ≤ K} and by {|u| > K} its complement in
0, ω. Put φK s : sup|x|≤K |Fs, x| for all s ∈ 0, ω. By the continuity of f, we know that
sups∈0,ω φK s < ∞. Then one has
1
Φun 2
ω
0
2
eσs un s ds
ω
eσs psun sds
0
e Fs, un sds −
−
eσs Fs, un sds
σs
{|un |≤K}
1
≥ un 2 − λ0
2
{|un |>K}
ω
e
0
2σs 2
ps ds
1/2
un −
ω
2.14
eσs φK sds,
0
which implies that limn → ∞ Φun ∞. 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, v1 , v2 ∈ D, Φv1 infD Φ c0 , inf∂D Φ b > c0 , v2 is a strict local minimizer of Φ, and Φv2 c1 > c0 . Then, for
> 0 small enough and any ρ2 > c1 , ρ1 ∈ c0 , min{b, c1 }, there exists λ∗ > 0 such that for each
λ ∈ 0, λ∗ , ΦλΨ has at least two local minima u1 and u2 lying in D, where u1 ∈ Φ−1 −∞, ρ0 ∩D,
u2 ∈ Φ−1 −∞, ρ1 ∩ Bu1 , , where Bu1 , {u ∈ H : u − u1 < }, and u2 ∈ Bu1 , .
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
Theorem 3.1. Assume that (f1) holds. Suppose further that
f4 there exists δ > 0 such that
x2
psx >
2λ0 eσs
x
f s, μ dμ
∀s, x ∈ 0, ω × −δ, 0 ∪ 0, δ,
3.1
0
f5 there exists x0 ∈ H such that Φx0 < 0.
Then there exist λ∗ > 0 and r > 0 such that, for every λ ∈ −λ∗ , λ∗ , problem 1.1 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 v1 . By f5, we obtain Φv1 infH Φ c0 < 0. Let D : B0, η {u ∈ H : u < η}.
Since Φ is coercive, we can choose a large enough η such that
v1 ∈ D,
Φv1 inf Φ c0 < 0,
D
inf Φ b > 0 > c0 .
∂D
3.2
Now we prove that Φ has a strict local minimum at v2 0. By the compact embedding
of H into C0, ω; R, there exists a constant c1 > 0 such that
max |us| ≤ c1 u
s∈0,ω
∀u ∈ H.
3.3
Choosing rδ < δ/c1 , it results that
B0, rδ {u ∈ H : u ≤ rδ } ⊆
u ∈ H : max |us| < δ .
s∈0,ω
3.4
Therefore, for every u ∈ B0, rδ \ {0}, it follows from f4 that
Φu 1
2
ω
2
eσs u s ds 0
ω
0
eσs psusds −
ω
eσs Fs, usds
0
ω
ω
1 ω
eσs psusds −
eσs Fs, usds
|us|2 ds 2λ0 0
0
0
ω
2
|us|
eσs
psus − Fs, us ds
2λ0 eσs
0
≥
3.5
> Φ0 0,
which implies that v2 0 is a strict local minimum of Φ in H with c1 : Φv2 0 > c0 .
At this point, we can apply Lemma 2.5 taking Ψ and −Ψ as perturbing terms. Then,
for ∈ 0, rδ small enough and any ρ1 ∈ c0 , min{b, c1 }, ρ2 ∈ 0, ∞, we can obtain the
following.
Boundary Value Problems
7
> 0 such that, for each λ ∈ −λ,
λ,
Φ λΨ has two distinct local
i There exists λ
minima u1 and u2 satisfying
u1 ∈ Φ−1
−∞, ρ1
,
u2 ∈ Φ−1
−∞, ρ2 ∩ B0, .
3.6
ii θ : infu Φu > 0 see 16, Theorem 3.6
Let r1 > 0 be such that
Φ−1
−∞, ρ1
3.7
∪ B0, ⊆ B0, r1,
and put b supu≤r1 |Φu|. Owing to the coerciveness of Φ, there exists r2 > r1 such that
infur2 Φu d > b. Since g : 0, ω × R → R is continuous, then
sup |Ψu| < ∞.
3.8
u≤r2
Choosing λ < d − b/2supu≤r2 |Ψu|, hence, for every u ∈ H with u r2 , one has
Φu λΨu ≥ d − |λ| sup |Ψu| >
u≤r2
bd
,
2
3.9
and when u ≤ r1
Φu λΨu ≤ b |λ| sup |Ψu| < b u≤r2
db
d−b
:
.
2
2
3.10
Further, from 3.6, we have that −∞ < Φu2 < ρ2 . Since ρ2 ∈ 0, ∞ is arbitrary, letting
ρ2 : θ/4 > 0, we can obtain that
Φu2 <
θ
.
4
3.11
Therefore, by 3.6 and 3.11, λ can be chosen small enough that
Φu1 λΨu1 ≤ 0,
Φu2 λΨu2 <
λ.
and 3.9-3.10 hold, for every λ ∈ −λ,
θ
,
2
inf Φu λΨu ≥
u
θ
,
2
3.12
8
Boundary Value Problems
For a given λ in the interval above, define the set of paths going from u1 to u2
A ϕ ∈ C0, 1, H : ϕ0 u1 , ϕ1 u2 ,
3.13
and consider the real number c : infϕ∈A sups∈0,1 Φϕs λΨϕs. Since u1 ∈ B0, and
each path ϕ goes through ∂B0, , one has c ≥ θ/2.
By 3.9 and 3.10, in the definition of c, there is no need to consider the paths going
through ∂B0, r2 . Hence, there exists a sequence of paths {ϕn } ⊂ A such that ϕn 0, 1 ⊂
B0, r2 and
sup Φ ϕn s λΨ ϕn s −→ c
as n −→ ∞.
s∈0,1
3.14
Applying a general mountain pass lemma without the PS condition see 17, Theorem 2.8,
there exists a sequence {un } ⊂ B0, r2 such that Φun λΨun → c and Φ un λΨ un →
0 as n → ∞. Hence {un } is a bounded PSc sequence and, taking into account the fact that
Φ λΨ is an S type mapping, admits a convergent subsequence to some u3 . So, such u3
turns to be a critical point of Φ λΨ, with Φu3 λΨu3 c, hence different from u1 and u2
and u3 /
0. This completes the proof.
Taking λ 0 in Theorem 3.1, we can obtain the existence of three distinct solutions for
the Duffing-type equation without perturbation 1.2 as following.
Theorem 3.2. Assume that (f1), (f4), and (f5) hold; then problem 1.2 admits at least three distinct
solutions.
Together with Lemma 2.3 and Lemma 2.4, we can easily show that the following
corollary.
Corollary 3.3. Assume that (f2), (f4), and (f5) hold; then there exist λ∗ > 0 and r > 0 such that, for
every λ ∈ −λ∗ , λ∗ , problem 1.1 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∗ , problem 1.1 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 resonant Duffing-type equations with damping and
perturbed term
u t σu t ft, ut λgt, ut pt,
u0 0 u2π,
a.e. t ∈ 0, 2π,
4.1
Boundary Value Problems
9
where σ 1, λ ∈ R, gs, x sx4 , ps 20 cos2 s, and
⎧
20 cos2 s x1/3
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪20 cos2 s Q1 x
⎪
⎪
⎪
⎪
⎨
ft, x 20 cos2 s − x1/3
⎪
⎪
⎪
⎪
⎪
⎪
⎪
20 cos2 s Q2 x
⎪
⎪
⎪
⎪
⎪
⎩
20 cos2 s x1/3
for s, x ∈ 0, 2π × −∞, −1,
for s, x ∈ 0, 2π × −1, −0.001,
for s, x ∈ 0, 2π × −0.001, 0.001,
4.2
for s, x ∈ 0, 2π × 0.001, 1,
for s, x ∈ 0, 2π × 1, ∞,
in which Q1 ∈ C−1, −0.001 and Q2 ∈ C 0.001, 1 satisfy
Q1 −1 −1,
Q1 −0.001 0.1,
Q2 0.001 −0.1,
Q2 1 1,
1
Q2 sds > 1.
0.001
4.3
Then there exists λ∗ > 0, for every λ ∈ −λ∗ , λ∗ , problem 8 admits at least three distinct
solutions.
Proof. Obviously, from the definitions of Q1 and Q2 , it is easy to see that f : 0, ω × R → R
is continuous and f1 holds. Taking δ 0.001, for s, x ∈ 0, 2π × −0.001, 0 ∪ 0, 0.001, we
have that
x2
psx −
2λ0 eσs
x
0
x2
3
f s, μ dμ ≥ 2π 20 cos2 s x − 20 cos2 s x − x4/3
4
8e
x2
3
x4/3
8e2π 4
4.4
> 0,
which implies that f4 is satisfied. Define
⎧
⎪
0,
⎪
⎪
⎪
⎨
x0 s 104 sin s,
⎪
⎪
⎪
⎪
⎩
0,
for s 0,
for s ∈ 0, 2π,
for s 2π.
4.5
10
Boundary Value Problems
Clearly, x0 ∈ H. Then we obtain that
Φx0 s 1
2
−
2π
es cos2 s ds 20 cos2 t
0
2π
0
e2π π
2
2π
0
≤
es 104 sin s ds
0
0.001 1
s
e
0
−
2π
0.001
2π
es
104 sin s f s, μ dμ ds
e
s
0
0.001
1
μ1/3 dμ ds
4.6
0
1
Q2 μ dμ ds −
2π
0.001
0
es
104 sin s
μ1/3 dμ ds
1
e2π π
− 104
2
< 0.
So Φx0 < 0, which implies that f5 is satisfied. To this end, all assumptions of Theorem 3.1
hold. By Theorem 3.1, there exists λ∗ > 0, for every λ ∈ −λ∗ , λ∗ , problem 8 admits at least
three distinct solutions.
Example 4.2. Let λ 0. From Example 4.1, we can obtain that the following resonant Duffingtype equations with damping:
√
u t u t 100e2π x 10,
u0 0 u2π
a.e. t ∈ 0, 2π,
4.7
admits at least three distinct solutions.
Acknowledgment
This work is supported by the National Natural Sciences Foundation of People’s Republic of
China under Grant no. 10971183.
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