Open Access

Three Solutions for Forced Duffing-Type Equations with Damping Term

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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq1_HTML.gif , a.e. https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq2_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq3_HTML.gif and without perturbed term https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq4_HTML.gif , a.e. https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq5_HTML.gif , https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ1_HTML.gif
(1.1)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq7_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq8_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq9_HTML.gif are continuous. Letting https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq10_HTML.gif in problem (1.1) leads to
https://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
https://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:
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq11_HTML.gif , consider the inner product
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ5_HTML.gif
(2.1)
inducing the norm
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ7_HTML.gif
(2.3)
and the norm
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ8_HTML.gif
(2.4)

Obviously, the norm https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq12_HTML.gif and the norm https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq13_HTML.gif are equivalent. So https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq15_HTML.gif .

By Poincaré's inequality,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ9_HTML.gif
(2.5)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq16_HTML.gif , https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq18_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq19_HTML.gif defined on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq20_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ11_HTML.gif
(2.7)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq21_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq23_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq24_HTML.gif and
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq25_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq26_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq27_HTML.gif the conjugate number of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq28_HTML.gif . Then
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq29_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq30_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq31_HTML.gif such that

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

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

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq33_HTML.gif be a sequence such that https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ15_HTML.gif
(2.11)

which implies from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq35_HTML.gif that https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq38_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq39_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ16_HTML.gif
(2.12)

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

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

Proof.

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

which implies that https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq58_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq59_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq60_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq61_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq62_HTML.gif is a strict local minimizer of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq63_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq64_HTML.gif . Then, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq65_HTML.gif small enough and any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq66_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq67_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq68_HTML.gif such that for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq69_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq70_HTML.gif has at least two local minima https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq71_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq72_HTML.gif lying in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq73_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq74_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq75_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq76_HTML.gif , and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq78_HTML.gif such that

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

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

Then there exist https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq81_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq82_HTML.gif such that, for every https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq85_HTML.gif is coercive. Since https://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]), https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq87_HTML.gif has a global minimizer https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq88_HTML.gif . By (f5), we obtain https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq89_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq90_HTML.gif . Since https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq92_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ20_HTML.gif
(3.2)
Now we prove that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq93_HTML.gif has a strict local minimum at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq94_HTML.gif . By the compact embedding of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq95_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq96_HTML.gif , there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq97_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ21_HTML.gif
(3.3)
Choosing https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq98_HTML.gif , it results that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ22_HTML.gif
(3.4)
Therefore, for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq99_HTML.gif , it follows from (f4) that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ23_HTML.gif
(3.5)

which implies that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq100_HTML.gif is a strict local minimum of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq101_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq102_HTML.gif with https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq104_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq105_HTML.gif as perturbing terms. Then, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq106_HTML.gif small enough and any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq107_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq109_HTML.gif such that, for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq110_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq111_HTML.gif has two distinct local minima https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq112_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq113_HTML.gif satisfying

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

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

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq115_HTML.gif be such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ25_HTML.gif
(3.7)
and put https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq116_HTML.gif . Owing to the coerciveness of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq117_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq118_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq119_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq120_HTML.gif is continuous, then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ26_HTML.gif
(3.8)
Choosing https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq121_HTML.gif , hence, for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq122_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq123_HTML.gif , one has
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ27_HTML.gif
(3.9)
and when https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq124_HTML.gif
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq125_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq126_HTML.gif is arbitrary, letting https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq127_HTML.gif , we can obtain that
https://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), https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq128_HTML.gif can be chosen small enough that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq129_HTML.gif .

For a given https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq131_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq132_HTML.gif
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq133_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq134_HTML.gif and each path https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq135_HTML.gif goes through https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq136_HTML.gif , one has https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq140_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq141_HTML.gif and
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq142_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq143_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq144_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq145_HTML.gif . Hence https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq146_HTML.gif is a bounded https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq148_HTML.gif is an https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq150_HTML.gif . So, such https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq152_HTML.gif , with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq153_HTML.gif , hence different from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq154_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq155_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq156_HTML.gif . This completes the proof.

Taking https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq158_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq159_HTML.gif such that, for every https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq162_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq163_HTML.gif such that, for every https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ33_HTML.gif
(4.1)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq166_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq167_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq168_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq169_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ34_HTML.gif
(4.2)
in which https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq170_HTML.gif satisfy
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ35_HTML.gif
(4.3)

Then there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq171_HTML.gif , for every https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq173_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq174_HTML.gif , it is easy to see that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq175_HTML.gif is continuous and (f1) holds. Taking https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq176_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq177_HTML.gif , we have that
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ37_HTML.gif
(4.5)
Clearly, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq178_HTML.gif . Then we obtain that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_Equ38_HTML.gif
(4.6)

So https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq180_HTML.gif , for every https://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 https://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:
https://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

References

  1. Duffing G: Erzwungene Schwingungen bei veränderlicher Eigenfrequenz und ihre Technische Beduetung. Vieweg Braunschweig, 1918Google Scholar
  2. Zeeman EC: Duffing's equation in brain modelling. Bulletin of the Institute of Mathematics and Its Applications 1976, 12(7):207-214.MathSciNetGoogle Scholar
  3. 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
  4. 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
  5. 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
  6. 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
  7. 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
  8. 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
  9. Wang Y, Ge W:Periodic solutions for Duffing equations with a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq183_HTML.gif -Laplacian-like operator. Computers & Mathematics with Applications 2006, 52(6-7):1079-1088. 10.1016/j.camwa.2006.03.030View ArticleMathSciNetGoogle Scholar
  10. 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
  11. 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
  12. 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
  13. 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
  14. 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
  15. 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
  16. Fan X, Deng S-G:Remarks on Ricceri's variational principle and applications to the https://static-content.springer.com/image/art%3A10.1155%2F2011%2F736093/MediaObjects/13661_2010_Article_55_IEq184_HTML.gif -Laplacian equations. Nonlinear Analysis. Theory, Methods & Applications 2007, 67(11):3064-3075. 10.1016/j.na.2006.09.060View ArticleMathSciNetGoogle Scholar
  17. Willem M: Minimax Theorems. Birkhäuser, Boston, Mass, USA; 1996:x+162.View ArticleGoogle Scholar

Copyright

© Y. Li and T. Zhang. 2011

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