Volume 2011, Article ID 736093,11pages 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
Copyrightq2011 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 termut σut ft, ut λgt, ut pt, a.e.t∈0, ω,u0 0uωand without perturbed termut σut ft, ut pt, a.e.t∈0, ω,u0 0uω.
1. Introduction
In this paper, we consider the following resonant Duffing-type equations with damping and perturbed term:
uσut ft, ut λgt, ut pt, a.e. t∈0, ω,
u0 0uω, 1.1
whereσ, λ∈R,f, g : 0, ω×R → R, andp :0, ω → Rare continuous. Lettingλ 0 in problem1.1leads to
ut σut ft, ut pt, a.e. t∈0, ω,
u0 0uω, 1.2
which is a common Duffing-type equation without perturbation.
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 levels2. 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
ut σut 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 as1.3have been obtained; we refer to5–11and references therein. There are also authors who studied the Duffing-type equations by using the critical point theorysee12,13. In12, by using a saddle point theorem, Tomiczek obtained the existence of a solution of the following Duffing-type system:
ut σut
m2− σ 2
4
ut ft, ut pt, a.e. t∈0, ω,
u0 0uω,
1.4
which is a special case of problems1.1-1.2. However, to the best of our knowledge, there are few results for the existence of multiple solutions of1.3.
Our aim in this paper is to study the variational structure of problems1.1-1.2in 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. InSection 3, by applying some critical point theorems, we establish sufficient conditions for the existence of three distinct solutions to problems1.1-1.2.
2. Variational Structure
In the Sobolev spaceH:H010, ω, consider the inner product
u, vH
ω
0
inducing the norm
uH
u, vH
ω
0
us2
ds
1/2
∀u∈H. 2.2
We also consider the inner product
u, v
ω
0
eσsusvsds ∀u, v∈H, 2.3
and the norm
uu, v
ω
0
eσsus2ds
1/2
∀u∈H. 2.4
Obviously, the norm · and the norm · Hare equivalent. SoHis a Hilbert space with the norm · .
By Poincar´e’s inequality,
u2 2:
ω
0
|us|2ds≤ 1 λ1
ω
0
us2
ds
≤ 1 λ1min{1, eσω}
ω
0
eσsus2ds:λ0u2 ∀u∈H,
2.5
whereλ0:1/λ1min{1, eσω},λ1:π2/ω2is the first eigenvalue of the problem
−ut λut, t∈0, ω,
u0 0uω.
2.6
Usually, in order to find the solution of problems1.1-1.2, we should consider the following functionalΦ,Ψdefined onH:
Φu 1
2
ω
0
eσs|us|2ds
ω
0
eσspsusds−
ω
0
eσsFs, usds
Ψu −
ω
0
eσsGs, usds,
2.7
Finding solutions of problem1.1is equivalent to finding critical points ofI: ΦλΨ
inHand
Iu, v
ω
0
eσsusvsds
ω
0
eσspsvsds
−
ω
0
eσsfs, uvsds−
ω
0
eσsλgs, uvsds, ∀u, v∈H.
2.8
Lemma 2.1H ¨older Inequality. Let f, g∈Ca, b,p >1, andqthe conjugate number ofp. Then
b
a
fsgsds≤b a
fsp
ds
1/p ·
b
a
gsq
ds
1/q
. 2.9
Lemma 2.2. Assume the following condition holds.
f1There exist positive constantsα,β, andγ∈0,1such that
fs, x≤αβ|x|γ ∀s,x∈0, ω×R. 2.10
ThenΦis coercive.
Proof. Let{un}n∈N⊂Hbe a sequence such that limn→∞un ∞. It follows fromf1and
H ¨older inequality that
Φun
1 2
ω
0
eσsuns2ds
ω
0
eσspsunsds−
ω
0
eσsFs, unsds
≥ 1
2un
2−ω
0
e2σsps2ds
1/2
un2−max{1, eσω}
ω
0
α|un|β|un|γ1
ds
≥ 1
2un
2−
ω
0
e2σsps2ds
1/2 un2
−α√ωmax{1, eσω}un2−β
ω1−γmax{1, eσω}u nγ21
≥ 1
2un
2−λ0ω
0
e2σsps2ds
1/2
α√ωmax{1, eσω}
un
−λγ01βω1−γmax{1, eσω}u nγ1,
2.11
which implies fromγ∈0,1that limn→∞Φun ∞. This completes the proof.
Lemma 2.3. Assume that2βλ0max{eσω,1}<1and the following condition holds. f2There exist positive constantsα0andβ0such that
fs, x≤α0β0|x| ∀s,x∈0, ω×R. 2.12
ThenΦis coercive.
Lemma 2.4. Assume the following condition holds. f3lim|x| →∞ 0xfs, μdμ≤0for alls∈0, ω.
ThenΦis coercive.
Proof. Let{un}n∈N⊂Hbe a sequence such that limn→∞un ∞. Fix >0, fromf3, there
existsK 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. PutφKs : sup|x|≤K|Fs, x|for alls ∈0, ω. By the continuity of f, we know that
sups∈0,ωφKs<∞. Then one has
Φun 1
2
ω
0
eσsuns2ds
ω
0
eσspsunsds
−
{|un|≤K}
eσsFs, unsds−
{|un|>K}
eσsFs, unsds
≥ 1
2un
2−λ 0
ω
0
e2σsps2ds
1/2 un −
ω
0
eσsφKsds,
2.14
which implies that limn→∞Φun ∞. This completes the proof.
Based on Ricceri’s variational principle in14,15, Fan and Deng16obtained the following result which is a main tool used in our paper.
Lemma 2.5see16. Suppose thatDis a bounded convex open subset ofH,v1, v2∈D,Φv1
infDΦ c0,inf∂DΦ b > c0,v2 is a strict local minimizer ofΦ, andΦv2 c1 > c0. Then, for > 0small enough and anyρ2 > c1,ρ1 ∈ c0,min{b, c1}, there existsλ∗ > 0such that for each λ∈0, λ∗,ΦλΨhas at least two local minimau1andu2lying inD, whereu1∈Φ−1−∞, ρ0∩D, u2∈Φ−1−∞, ρ1∩Bu1, , whereBu1, {u∈H:u−u1< }, andu2∈Bu1, .
3. Main Results
Theorem 3.1. Assume that (f1) holds. Suppose further that f4there existsδ >0such that
x2
2λ0eσs psx >
x
0
fs, μdμ ∀s,x∈0, ω×−δ,0∪0, δ, 3.1
f5there existsx0∈Hsuch thatΦx0<0.
Then there existλ∗ > 0 andr > 0such that, for everyλ ∈ −λ∗, λ∗, problem1.1admits at least three distinct solutions which belong toB0, r⊆H.
Proof. By Lemma 2.2, condition f1 implies that the functional Φ is coercive. Since Φ is sequentially weakly lower semicontinuoussee16, Propositions 2.5 and 2.6,Φhas a global minimizerv1. Byf5, we obtainΦv1 infHΦ c0<0. LetD:B0, η {u∈H:u< η}.
SinceΦis coercive, we can choose a large enoughηsuch that
v1 ∈D, Φv1 inf
D Φ c0 <0, inf∂DΦ b >0> c0. 3.2
Now we prove thatΦhas a strict local minimum atv20. By the compact embedding
ofHintoC0, ω;R, there exists a constantc1>0 such that
max
s∈0,ω|us| ≤c1u ∀u∈H. 3.3
Choosingrδ< δ/c1, it results that
B0, rδ {u∈H:u ≤rδ} ⊆
u∈H: max
s∈0,ω|us|< δ
. 3.4
Therefore, for everyu∈B0, rδ\ {0}, it follows fromf4that
Φu 1
2
ω
0
eσsus2ds
ω
0
eσspsusds−
ω
0
eσsFs, usds
≥ 1
2λ0
ω
0
|us|2ds
ω
0
eσspsusds−
ω
0
eσsFs, usds
ω
0 eσs
|us|2
2λ0eσs psus−Fs, us
ds
>Φ0 0,
3.5
which implies thatv20 is a strict local minimum ofΦinHwithc1: Φv2 0> c0. At this point, we can applyLemma 2.5takingΨand−Ψas perturbing terms. Then, for ∈ 0, rδsmall enough and anyρ1 ∈ c0,min{b, c1},ρ2 ∈ 0,∞, we can obtain the
iThere exists λ > 0 such that, for each λ ∈ −λ, λ,Φ λΨhas two distinct local minimau1andu2satisfying
u1 ∈Φ−1
−∞, ρ1
, u2∈Φ−1
−∞, ρ2
∩B0, . 3.6
iiθ:infu Φu>0see16, Theorem 3.6
Letr1>0 be such that
Φ−1−∞, ρ1∪B0, ⊆B0, r1, 3.7
and putb supu≤r
1|Φu|. Owing to the coerciveness of Φ, there existsr2 > r1 such that
infur2Φu d > b. Sinceg:0, ω×R → Ris continuous, then
sup
u≤r2
|Ψu|<∞. 3.8
Choosingλ <d−b/2supu≤r2|Ψu|, hence, for everyu∈Hwithur2, one has
Φu λΨu≥d− |λ|sup
u≤r2
|Ψu|> bd
2 , 3.9
and whenu ≤r1
Φu λΨu≤b|λ|sup
u≤r2
|Ψu|< bd−b
2 :
db
2 . 3.10
Further, from3.6, we have that −∞ < Φu2 < ρ2. Sinceρ2 ∈ 0,∞is arbitrary, letting ρ2:θ/4>0, we can obtain that
Φu2< θ
4. 3.11
Therefore, by3.6and3.11,λcan be chosen small enough that
Φu1 λΨu1≤0, Φu2 λΨu2< θ
2, infu Φu λΨu≥ θ
2, 3.12
For a givenλin the interval above, define the set of paths going fromu1tou2
Aϕ∈C0,1, H:ϕ0 u1, ϕ1 u2
, 3.13
and consider the real numberc:infϕ∈Asups∈0,1Φϕs λΨϕs. Sinceu1∈B0, and
each pathϕgoes through∂B0, , one hasc≥θ/2.
By3.9and3.10, in the definition ofc, there is no need to consider the paths going through ∂B0, r2. Hence, there exists a sequence of paths{ϕn} ⊂ A such thatϕn0,1 ⊂ B0, r2and
sup
s∈0,1
Φϕns
λΨϕns
−→c asn−→∞. 3.14
Applying a general mountain pass lemma without thePSconditionsee17, Theorem 2.8, there exists a sequence{un} ⊂B0, r2such thatΦun λΨun → candΦun λΨun →
0 asn → ∞. Hence{un}is a boundedPSc sequence and, taking into account the fact that ΦλΨis anS
type mapping, admits a convergent subsequence to someu3. So, suchu3
turns to be a critical point ofΦ λΨ, withΦu3 λΨu3 c, hence different fromu1andu2
andu3/0. This completes the proof.
Takingλ0 inTheorem 3.1, we can obtain the existence of three distinct solutions for the Duffing-type equation without perturbation1.2as following.
Theorem 3.2. Assume that (f1), (f4), and (f5) hold; then problem1.2admits 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 existλ∗>0andr >0such that, for everyλ∈−λ∗, λ∗, problem1.1admits at least three distinct solutions which belong toB0, r⊆H. Furthermore, problem1.2admits at least three distinct solutions.
Corollary 3.4. Assume that (f3), (f4), and (f5); hold, then there existλ∗>0andr >0such that, for everyλ∈−λ∗, λ∗, problem1.1admits at least three distinct solutions which belong toB0, r⊆H. Furthermore, problem1.2admits at least three distinct solutions.
4. Some Examples
Example 4.1. Consider the following resonant Duffing-type equations with damping and perturbed term
ut σut ft, ut λgt, ut pt, a.e. t∈0,2π,
u0 0u2π,
whereσ1,λ∈R,gs, x sx4,ps 20 cos2s, and
ft, x
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
20 cos2sx1/3 fors, x∈0,2π×−∞,−1,
20 cos2sQ
1x fors, x∈0,2π×−1,−0.001,
20 cos2s−x1/3 fors, x∈0,2π×−0.001,0.001,
20 cos2sQ2x fors, x∈0,2π×0.001,1,
20 cos2sx1/3 fors, x∈0,2π×1,∞,
4.2
in whichQ1 ∈C−1,−0.001andQ2∈C0.001,1satisfy
Q1−1 −1, Q1−0.001 0.1, Q20.001 −0.1, Q21 1,
1
0.001
Q2sds >1.
4.3
Then there existsλ∗ > 0, for everyλ ∈ −λ∗, λ∗, problem8admits at least three distinct solutions.
Proof. Obviously, from the definitions ofQ1andQ2, it is easy to see thatf :0, ω×R → R
is continuous andf1holds. Takingδ0.001, fors, x∈0,2π×−0.001,0∪0,0.001, we have that
x2
2λ0eσs
psx−
x
0
fs, μdμ≥ x 2
8e2π 20
cos2sx−
20cos2sx−3
4x
4/3
x2
8e2π
3 4x
4/3
>0,
4.4
which implies thatf4is satisfied. Define
x0s
⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩
0, fors0,
104sins, fors∈0,2π,
0, fors2π.
Clearly,x0∈H. Then we obtain that
Φx0s
1 2
2π
0
escos2sds20 cos2t
2π
0
es104sinsds
−
2π
0 es
0.001
0
1
0.001
104sins
1
fs, μdμds
e2ππ
2
2π
0 es
0.001
0
μ1/3dμds
−
2π
0 es
1
0.001
Q2μdμds−
2π
0 es
104sins
1
μ1/3dμds
≤ e2ππ
2 −10
4
<0.
4.6
SoΦx0<0, which implies thatf5is satisfied. To this end, all assumptions ofTheorem 3.1 hold. ByTheorem 3.1, there existsλ∗ >0, for everyλ∈−λ∗, λ∗, problem8admits at least three distinct solutions.
Example 4.2. Letλ0. FromExample 4.1, we can obtain that the following resonant Duffi ng-type equations with damping:
ut ut 100e2π√x10, a.e. t∈0,2π,
u0 0u2π
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.
References
1 G. Duffing, “Erzwungene Schwingungen bei ver¨anderlicher Eigenfrequenz und ihre Technische Beduetung,” Vieweg Braunschweig, 1918.
2 E. C. Zeeman, “Duffing’s equation in brain modelling,”Bulletin of the Institute of Mathematics and Its Applications, vol. 12, no. 7, pp. 207–214, 1976.
3 A. N. Njah and U. E. Vincent, “Chaos synchronization between single and double wells Duffing-Van der Pol oscillators using active control,”Chaos, Solitons and Fractals, vol. 37, no. 5, pp. 1356–1361, 2008.
5 L. Peng, “Existence and uniqueness of periodic solutions for a kind of Duffing equation with two deviating arguments,”Mathematical and Computer Modelling, vol. 45, no. 3-4, pp. 378–386, 2007.
6 H. Chen and Y. Li, “Rate of decay of stable periodic solutions of Duffing equations,” Journal of Differential Equations, vol. 236, no. 2, pp. 493–503, 2007.
7 A. C. Lazer and P. J. McKenna, “On the existence of stable periodic solutions of differential equations of Duffing type,”Proceedings of the American Mathematical Society, vol. 110, no. 1, pp. 125–133, 1990.
8 B. Du, C. Bai, and X. Zhao, “Problems of periodic solutions for a type of Duffing equation with state-dependent delay,”Journal of Computational and Applied Mathematics, vol. 233, no. 11, pp. 2807–2813, 2010.
9 Y. Wang and W. Ge, “Periodic solutions for Duffing equations with ap-Laplacian-like operator,”
Computers & Mathematics with Applications, vol. 52, no. 6-7, pp. 1079–1088, 2006.
10 F. I. Njoku and P. Omari, “Stability properties of periodic solutions of a Duffing equation in the presence of lower and upper solutions,”Applied Mathematics and Computation, vol. 135, no. 2-3, pp. 471–490, 2003.
11 Z. Wang, J. Xia, and D. Zheng, “Periodic solutions of Duffing equations with semi-quadratic potential and singularity,”Journal of Mathematical Analysis and Applications, vol. 321, no. 1, pp. 273–285, 2006.
12 P. Tomiczek, “The Duffing equation with the potential Landesman-Lazer condition,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 2, pp. 735–740, 2009.
13 Y. Li and T. Zhang, “On the existence of solutions for impulsive Duffing dynamic equations on time scales with Dirichlet boundary conditions,”Abstract and Applied Analysis, vol. 2010, Article ID 152460, 27 pages, 2010.
14 B. Ricceri, “A general variational principle and some of its applications,”Journal of Computational and Applied Mathematics, vol. 113, no. 1-2, pp. 401–410, 2000.
15 B. Ricceri, “Sublevel sets and global minima of coercive functionals and local minima of their perturbations,”Journal of Nonlinear and Convex Analysis, vol. 5, no. 2, pp. 157–168, 2004.
16 X. Fan and S.-G. Deng, “Remarks on Ricceri’s variational principle and applications to the px -Laplacian equations,”Nonlinear Analysis. Theory, Methods & Applications, vol. 67, no. 11, pp. 3064–3075, 2007.
