# The Existence and Uniqueness of Solution of Duffing Equations with Non- Perturbation Functional at Nonresonance

- Zhou Ting
^{1}and - Huang Wenhua
^{1}Email author

**2008**:859461

**DOI: **10.1155/2008/859461

© Z. Ting and H. Wenhua. 2008

**Received: **13 July 2007

**Accepted: **13 March 2008

**Published: **30 March 2008

## Abstract

This paper deals with a boundary value problem for Duffing equation. The existence of unique solution for the problem is studied by using the minimax theorem due to Huang Wenhua. The existence and uniqueness result was presented under a generalized nonresonance condition.

## 1. Introduction

In recent years, many authors are greatly attached to investigation for the existence and uniqueness of solution of Duffing equations, for example, [1–11], and so forth. Some authors ([8, 11, 12], etc.) proved the existence and uniqueness of solution of Duffing equations under perturbation functions and other conditions at nonresonance by employing minimax theorems. In 1986, Tersian investigated the equation using a minimax theorem proved by himself and reaped a result of generalized solution [13]. In 2005, Huang and Shen generalized the minimax theorem of Tersian in [13]. Using the generalized minimax theorem, Huang and Shen proved a theorem of existence and uniqueness of solution for the equation [14] under the weaker conditions than those in [13].

Stimulated by the works in [13, 14], in the present paper, we investigate the solutions of the boundary value problems of Duffing equations with non- perturbation functions at nonresonance using the minimax theorem proved by Huang in [15].

## 2. Preliminaries

Let be a real Hilbert space with inner product and norm , respectively, and be two orthogonal closed subspaces of such that . Let denote the projections from to and from to , respectively. The following theorem will be employed to prove our main theorem.

Theorem 2.1 (see [15]).

for , , , , , . Then, the following hold:

(a) has a unique critical point such that ;

It is easy to prove the following corollary of the above theorem.

Corollary 2.2.

for , , , , , , . Then, the following hold:

(a) has a unique critical point such that ;

Proof.

The conclusion of the corollary follows immediately from Theorem 2.1.

## 3. The Main Theorems

where , is a potential Carathéodory function, is a given function in .

where . Clearly, is a potential Carathéodory function, and if is a solution of (3.2), then will be a solution of (3.1).

where . Clearly, is a critical point of if and only if is a solution of the equation and hence a solution of (3.2) and thus is a solution of (3.1).

where is an identity operator.

We need to prove a lemma before presenting our main theorem.

Lemma 3.1.

Proof.

Now, we show our main theorem dealing with (3.1).

Theorem 3.2.

where is a functional defined in (3.14) and .

Proof.

where is a functional defined in (3.14) and .

We then have the following corollary of Theorem 3.2.

Corollary 3.3.

## Declarations

### Acknowledgment

The corresponding author is grateful to the referees for their helpful and valuable comments.

## Authors’ Affiliations

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