Three Solutions for Forced Duffing-Type Equations with Damping Term
© Y. Li and T. Zhang. 2011
Received: 16 December 2010
Accepted: 11 February 2011
Published: 10 March 2011
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
Lemma 2.1 (Hölder Inequality).
Assume the following condition holds.
From the proof of Lemma 2.2, we can show the following Lemma.
Assume the following condition holds.
Lemma 2.5 (see ).
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.
Assume that (f1) holds. Suppose further that
(ii) (see [16, Theorem 3.6])
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.
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.
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.
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
admits at least three distinct solutions.
This work is supported by the National Natural Sciences Foundation of People's Republic of China under Grant no. 10971183.
- 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
This article is published under license to BioMed Central Ltd. 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.