Three Solutions for Forced Duffing-Type Equations with Damping Term

  • Yongkun Li1Email author and

    Affiliated with

    • Tianwei Zhang1

      Affiliated with

      Boundary Value Problems20112011:736093

      DOI: 10.1155/2011/736093

      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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq1_HTML.gif , a.e. http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq2_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq3_HTML.gif and without perturbed term http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq4_HTML.gif , a.e. http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq5_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq6_HTML.gif .

      1. Introduction

      In this paper, we consider the following resonant Duffing-type equations with damping and perturbed term:
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ1_HTML.gif
      (1.1)
      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq7_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq8_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq9_HTML.gif are continuous. Letting http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq10_HTML.gif in problem (1.1) leads to
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ2_HTML.gif
      (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 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
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ3_HTML.gif
      (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 [511] 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:
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ4_HTML.gif
      (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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq11_HTML.gif , consider the inner product
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ5_HTML.gif
      (2.1)
      inducing the norm
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ6_HTML.gif
      (2.2)
      We also consider the inner product
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ7_HTML.gif
      (2.3)
      and the norm
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ8_HTML.gif
      (2.4)

      Obviously, the norm http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq12_HTML.gif and the norm http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq13_HTML.gif are equivalent. So http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq14_HTML.gif is a Hilbert space with the norm http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq15_HTML.gif .

      By Poincaré's inequality,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ9_HTML.gif
      (2.5)
      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq16_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq17_HTML.gif is the first eigenvalue of the problem
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ10_HTML.gif
      (2.6)
      Usually, in order to find the solution of problems (1.1)-(1.2), we should consider the following functional http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq18_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq19_HTML.gif defined on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq20_HTML.gif :
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ11_HTML.gif
      (2.7)

      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq21_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq22_HTML.gif .

      Finding solutions of problem (1.1) is equivalent to finding critical points of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq23_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq24_HTML.gif and
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ12_HTML.gif
      (2.8)

      Lemma 2.1 (Hölder Inequality).

      Let f, g http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq25_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq26_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq27_HTML.gif the conjugate number of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq28_HTML.gif . Then
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ13_HTML.gif
      (2.9)

      Lemma 2.2.

      Assume the following condition holds.

      (f1) There exist positive constants http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq29_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq30_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq31_HTML.gif such that

      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ14_HTML.gif
      (2.10)

      Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq32_HTML.gif is coercive.

      Proof.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq33_HTML.gif be a sequence such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq34_HTML.gif . It follows from (f1) and Hölder inequality that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ15_HTML.gif
      (2.11)

      which implies from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq35_HTML.gif that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq36_HTML.gif . This completes the proof.

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

      Lemma 2.3.

      Assume that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq37_HTML.gif and the following condition holds.

      (f2)There exist positive constants http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq38_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq39_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ16_HTML.gif
      (2.12)

      Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq40_HTML.gif is coercive.

      Lemma 2.4.

      Assume the following condition holds.

      (f3) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq41_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq42_HTML.gif .

      Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq43_HTML.gif is coercive.

      Proof.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq44_HTML.gif be a sequence such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq45_HTML.gif . Fix http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq46_HTML.gif , from (f3), there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq47_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ17_HTML.gif
      (2.13)
      Denote by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq48_HTML.gif the set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq49_HTML.gif and by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq50_HTML.gif its complement in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq51_HTML.gif . Put http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq52_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq53_HTML.gif . By the continuity of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq54_HTML.gif , we know that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq55_HTML.gif . Then one has
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ18_HTML.gif
      (2.14)

      which implies that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq56_HTML.gif . 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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq57_HTML.gif is a bounded convex open subset of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq58_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq59_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq60_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq61_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq62_HTML.gif is a strict local minimizer of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq63_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq64_HTML.gif . Then, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq65_HTML.gif small enough and any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq66_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq67_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq68_HTML.gif such that for each http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq69_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq70_HTML.gif has at least two local minima http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq71_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq72_HTML.gif lying in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq73_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq74_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq75_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq76_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq77_HTML.gif .

      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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq78_HTML.gif such that

      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ19_HTML.gif
      (3.1)

      (f5) there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq79_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq80_HTML.gif .

      Then there exist http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq81_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq82_HTML.gif such that, for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq83_HTML.gif , problem (1.1) admits at least three distinct solutions which belong to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq84_HTML.gif .

      Proof.

      By Lemma 2.2, condition (f1) implies that the functional http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq85_HTML.gif is coercive. Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq86_HTML.gif is sequentially weakly lower semicontinuous (see [16, Propositions  2.5 and  2.6]), http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq87_HTML.gif has a global minimizer http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq88_HTML.gif . By (f5), we obtain http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq89_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq90_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq91_HTML.gif is coercive, we can choose a large enough http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq92_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ20_HTML.gif
      (3.2)
      Now we prove that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq93_HTML.gif has a strict local minimum at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq94_HTML.gif . By the compact embedding of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq95_HTML.gif into http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq96_HTML.gif , there exists a constant http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq97_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ21_HTML.gif
      (3.3)
      Choosing http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq98_HTML.gif , it results that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ22_HTML.gif
      (3.4)
      Therefore, for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq99_HTML.gif , it follows from (f4) that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ23_HTML.gif
      (3.5)

      which implies that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq100_HTML.gif is a strict local minimum of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq101_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq102_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq103_HTML.gif .

      At this point, we can apply Lemma 2.5 taking http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq104_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq105_HTML.gif as perturbing terms. Then, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq106_HTML.gif small enough and any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq107_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq108_HTML.gif , we can obtain the following.
      1. (i)

        There exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq109_HTML.gif such that, for each http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq110_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq111_HTML.gif has two distinct local minima http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq112_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq113_HTML.gif satisfying

         
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ24_HTML.gif
      (3.6)

      (ii) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq114_HTML.gif (see [16, Theorem 3.6])

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq115_HTML.gif be such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ25_HTML.gif
      (3.7)
      and put http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq116_HTML.gif . Owing to the coerciveness of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq117_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq118_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq119_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq120_HTML.gif is continuous, then
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ26_HTML.gif
      (3.8)
      Choosing http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq121_HTML.gif , hence, for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq122_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq123_HTML.gif , one has
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ27_HTML.gif
      (3.9)
      and when http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq124_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ28_HTML.gif
      (3.10)
      Further, from (3.6), we have that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq125_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq126_HTML.gif is arbitrary, letting http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq127_HTML.gif , we can obtain that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ29_HTML.gif
      (3.11)
      Therefore, by (3.6) and (3.11), http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq128_HTML.gif can be chosen small enough that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ30_HTML.gif
      (3.12)

      and (3.9)-(3.10) hold, for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq129_HTML.gif .

      For a given http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq130_HTML.gif in the interval above, define the set of paths going from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq131_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq132_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ31_HTML.gif
      (3.13)

      and consider the real number http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq133_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq134_HTML.gif and each path http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq135_HTML.gif goes through http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq136_HTML.gif , one has http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq137_HTML.gif .

      By (3.9) and (3.10), in the definition of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq138_HTML.gif , there is no need to consider the paths going through http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq139_HTML.gif . Hence, there exists a sequence of paths http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq140_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq141_HTML.gif and
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ32_HTML.gif
      (3.14)

      Applying a general mountain pass lemma without the (PS) condition (see [17, Theorem 2.8]), there exists a sequence http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq142_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq143_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq144_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq145_HTML.gif . Hence http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq146_HTML.gif is a bounded http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq147_HTML.gif sequence and, taking into account the fact that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq148_HTML.gif is an http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq149_HTML.gif type mapping, admits a convergent subsequence to some http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq150_HTML.gif . So, such http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq151_HTML.gif turns to be a critical point of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq152_HTML.gif , with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq153_HTML.gif , hence different from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq154_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq155_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq156_HTML.gif . This completes the proof.

      Taking http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq157_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq158_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq159_HTML.gif such that, for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq160_HTML.gif , problem (1.1) admits at least three distinct solutions which belong to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq161_HTML.gif . Furthermore, problem (1.2) admits at least three distinct solutions.

      Corollary 3.4.

      Assume that (f3), (f4), and (f5); hold, then there exist http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq162_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq163_HTML.gif such that, for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq164_HTML.gif , problem (1.1) admits at least three distinct solutions which belong to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq165_HTML.gif . 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
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ33_HTML.gif
      (4.1)
      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq166_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq167_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq168_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq169_HTML.gif , and
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ34_HTML.gif
      (4.2)
      in which http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq170_HTML.gif satisfy
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ35_HTML.gif
      (4.3)

      Then there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq171_HTML.gif , for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq172_HTML.gif , problem (8) admits at least three distinct solutions.

      Proof.

      Obviously, from the definitions of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq173_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq174_HTML.gif , it is easy to see that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq175_HTML.gif is continuous and (f1) holds. Taking http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq176_HTML.gif , for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq177_HTML.gif , we have that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ36_HTML.gif
      (4.4)
      which implies that (f4) is satisfied. Define
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ37_HTML.gif
      (4.5)
      Clearly, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq178_HTML.gif . Then we obtain that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ38_HTML.gif
      (4.6)

      So http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq179_HTML.gif , which implies that (f5) is satisfied. To this end, all assumptions of Theorem 3.1 hold. By Theorem 3.1, there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq180_HTML.gif , for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq181_HTML.gif , problem (8) admits at least three distinct solutions.

      Example 4.2.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq182_HTML.gif . From Example 4.1, we can obtain that the following resonant Duffing-type equations with damping:
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ39_HTML.gif
      (4.7)

      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

      (1)
      Department of Mathematics, Yunnan University

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      Copyright

      © Y. Li and T. Zhang. 2011

      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.