# 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

## 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 .

Lemma 2.1 (Hölder Inequality).

Lemma 2.2.

Assume the following condition holds.

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

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.

Lemma 2.4.

Assume the following condition holds.

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

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)

(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

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