- Research Article
- Open Access

# Three Solutions for Forced Duffing-Type Equations with Damping Term

- Yongkun Li
^{1}Email author and - Tianwei Zhang
^{1}

**Received:**16 December 2010**Accepted:**11 February 2011**Published:**10 March 2011

## Abstract

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 , a.e. , and without perturbed term , a.e. , .

## Keywords

- Variational Principle
- Mountain Pass
- Critical Point Theory
- Distinct Solution
- Chaotic Synchronization

## 1. Introduction

which is a common Duffing-type equation without perturbation.

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

Obviously, the norm and the norm are equivalent. So is a Hilbert space with the norm .

where , .

Lemma 2.1 (Hölder Inequality).

Lemma 2.2.

Assume the following condition holds.

(f1) There exist positive constants , , and such that

Then is coercive.

Proof.

which implies from that . This completes the proof.

From the proof of Lemma 2.2, we can show the following Lemma.

Lemma 2.3.

Assume that and the following condition holds.

Then is coercive.

Lemma 2.4.

Assume the following condition holds.

(f3) for all .

Then is coercive.

Proof.

which implies that . 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 is a bounded convex open subset of , , , , is a strict local minimizer of , and . Then, for small enough and any , , there exists such that for each , has at least two local minima and lying in , where , , where , and .

## 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.

Theorem 3.1.

Assume that (f1) holds. Suppose further that

(f4) there exists such that

(f5) there exists such that .

Then there exist and such that, for every , problem (1.1) admits at least three distinct solutions which belong to .

Proof.

which implies that is a strict local minimum of in with .

- (i)
There exists such that, for each , has two distinct local minima and satisfying

(ii) (see [16, Theorem 3.6])

and (3.9)-(3.10) hold, for every .

and consider the real number . Since and each path goes through , one has .

Applying a general mountain pass lemma without the (PS) condition (see [17, Theorem 2.8]), there exists a sequence such that and as . Hence is a bounded sequence and, taking into account the fact that is an type mapping, admits a convergent subsequence to some . So, such turns to be a critical point of , with , hence different from and and . This completes the proof.

Taking 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 and such that, for every , problem (1.1) admits at least three distinct solutions which belong to . Furthermore, problem (1.2) admits at least three distinct solutions.

Corollary 3.4.

Assume that (f3), (f4), and (f5); hold, then there exist and such that, for every , problem (1.1) admits at least three distinct solutions which belong to . Furthermore, problem (1.2) admits at least three distinct solutions.

## 4. Some Examples

Example 4.1.

Then there exists , for every , problem (8) admits at least three distinct solutions.

Proof.

So , which implies that (f5) is satisfied. To this end, all assumptions of Theorem 3.1 hold. By Theorem 3.1, there exists , for every , problem (8) admits at least three distinct solutions.

Example 4.2.

admits at least three distinct solutions.

## Declarations

### Acknowledgment

This work is supported by the National Natural Sciences Foundation of People's Republic of China under Grant no. 10971183.

## Authors’ Affiliations

## References

- Duffing G: Erzwungene Schwingungen bei veränderlicher Eigenfrequenz und ihre Technische Beduetung. Vieweg Braunschweig, 1918Google Scholar
- Zeeman EC: Duffing's equation in brain modelling.
*Bulletin of the Institute of Mathematics and Its Applications*1976, 12(7):207-214.MathSciNetGoogle Scholar - Njah AN, Vincent UE: Chaos synchronization between single and double wells Duffing-Van der Pol oscillators using active control.
*Chaos, Solitons and Fractals*2008, 37(5):1356-1361. 10.1016/j.chaos.2006.10.038View ArticleMathSciNetGoogle Scholar - Wu X, Cai J, Wang M: Global chaos synchronization of the parametrically excited Duffing oscillators by linear state error feedback control.
*Chaos, Solitons and Fractals*2008, 36(1):121-128. 10.1016/j.chaos.2006.06.014View ArticleMathSciNetGoogle Scholar - Peng L: Existence and uniqueness of periodic solutions for a kind of Duffing equation with two deviating arguments.
*Mathematical and Computer Modelling*2007, 45(3-4):378-386. 10.1016/j.mcm.2006.05.012View ArticleMathSciNetGoogle Scholar - Chen H, Li Y: Rate of decay of stable periodic solutions of Duffing equations.
*Journal of Differential Equations*2007, 236(2):493-503. 10.1016/j.jde.2007.01.023View ArticleMathSciNetGoogle Scholar - Lazer AC, McKenna PJ: On the existence of stable periodic solutions of differential equations of Duffing type.
*Proceedings of the American Mathematical Society*1990, 110(1):125-133. 10.1090/S0002-9939-1990-1013974-9View ArticleMathSciNetGoogle Scholar - Du B, Bai C, Zhao X: Problems of periodic solutions for a type of Duffing equation with state-dependent delay.
*Journal of Computational and Applied Mathematics*2010, 233(11):2807-2813. 10.1016/j.cam.2009.11.026View ArticleMathSciNetGoogle Scholar - Wang Y, Ge W:Periodic solutions for Duffing equations with a
-Laplacian-like operator.
*Computers & Mathematics with Applications*2006, 52(6-7):1079-1088. 10.1016/j.camwa.2006.03.030View ArticleMathSciNetGoogle Scholar - Njoku FI, Omari P: Stability properties of periodic solutions of a Duffing equation in the presence of lower and upper solutions.
*Applied Mathematics and Computation*2003, 135(2-3):471-490. 10.1016/S0096-3003(02)00062-0View ArticleMathSciNetGoogle Scholar - Wang Z, Xia J, Zheng D: Periodic solutions of Duffing equations with semi-quadratic potential and singularity.
*Journal of Mathematical Analysis and Applications*2006, 321(1):273-285. 10.1016/j.jmaa.2005.08.033View ArticleMathSciNetGoogle Scholar - Tomiczek P: The Duffing equation with the potential Landesman-Lazer condition.
*Nonlinear Analysis: Theory, Methods & Applications*2009, 70(2):735-740. 10.1016/j.na.2008.01.006View ArticleMathSciNetGoogle Scholar - Li Y, Zhang T: On the existence of solutions for impulsive Duffing dynamic equations on time scales with Dirichlet boundary conditions.
*Abstract and Applied Analysis*2010, 2010:-27.Google Scholar - Ricceri B: A general variational principle and some of its applications.
*Journal of Computational and Applied Mathematics*2000, 113(1-2):401-410. 10.1016/S0377-0427(99)00269-1View ArticleMathSciNetGoogle Scholar - Ricceri B: Sublevel sets and global minima of coercive functionals and local minima of their perturbations.
*Journal of Nonlinear and Convex Analysis*2004, 5(2):157-168.MathSciNetGoogle Scholar - Fan X, Deng S-G:Remarks on Ricceri's variational principle and applications to the
-Laplacian equations.
*Nonlinear Analysis. Theory, Methods & Applications*2007, 67(11):3064-3075. 10.1016/j.na.2006.09.060View ArticleMathSciNetGoogle Scholar - Willem M:
*Minimax Theorems*. Birkhäuser, Boston, Mass, USA; 1996:x+162.View ArticleGoogle Scholar

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