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

*Boundary Value Problems*
**volume 2008**, Article number: 859461 (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]).

Let be a real Hilbert space, let and be orthogonal closed vector subspace of such that , let be an everywhere defined functional with Gâteaux derivative, everywhere defined and hemicontinuous. Suppose that there exist two continuous functions satisfying

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

(a) has a unique critical point such that ;

(b).

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

Corollary 2.2.

Let be a real Hilbert space, let and be orthogonal closed vector subspace of such that , and let be an everywhere defined functional with second Gâteaux differential. Suppose that there exist two continuous functions satisfying

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

(a) has a unique critical point such that ;

(b).

Proof.

We note that is a second Gâteaux differentiable functional, the mean-value theorem ensures that there exists such that . Therefore, for , , , , , , we have

The conclusion of the corollary follows immediately from Theorem 2.1.

## 3. The Main Theorems

Consider the boundary value problem

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

Let , then (3.1) may be written in the form of

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

It is well known that is a Hilbert space with inner product:

and norm , respectively. The system of trigonometrical functions,

is a system of orthonormal functions in . Each can be written as the Fourier series

Define the linear operator ,

Denote

Clearly, is a self-adjoint operator, and is a Hilbert space for the inner product:

, the norm induced by this inner product is

Note that is not a space.

Since in (3.1), and hence in (3.2), is a potential Carathéodory function, there exists a function such that

and hence

and the mapping , and hence , generates a Nemytskii operator by

and hence

Define the functional by

where satisfies (3.10) and is in (3.1). We have

It is easy to see that

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).

Now, we suppose that there exists a real-bounded mapping such that

For , let

Since

equation (3.17) is equivalent to

Suppose that for ,

and for , define

which is equivalent to

Since is a self-adjoint operator, it possesses spectral resolution

with a right continuous spectral family ; and we let

for all with . Then, the operator has the spectral resolution:

where is an identity operator.

Define and by

By (3.21), we have

and hence

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

Lemma 3.1.

Suppose that in (3.2) satisfies (3.20), commutes with the linear operator and satisfies (3.21), is a continuous function defined in (3.22). Then, for

Proof.

For , let

Note that commutes with the linear operator and

By (3.22),

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

Theorem 3.2.

Let be a potential Carathéodory function satisfying (3.17). Suppose that commutes with the linear operator and satisfies

and the continuous function defined by (3.23) satisfies the conditions

Then, (3.1) has a unique solution such that

where is a functional defined in (3.14) and .

Proof.

For , by Lemma 3.1, we have

Employing Theorem 2.1, we can know that there exists a unique such that , where is a solution of (3.2) and this means that (3.1) has a unique solution such that

where is a functional defined in (3.14) and .

If the perturbation function in (3.10) is a second Gâteaux differential, (3.17), (3.34), and (3.23) become

respectively. By (3.30) in Lemma 3.1, we have

where .

We then have the following corollary of Theorem 3.2.

Corollary 3.3.

Let be a potential Carathéodory function with first Gâteaux derivative satisfying (3.39) and (3.40) and commutes with the linear operator . If the continuous function defined by (3.41) satisfies (3.35), then (3.1) has a unique solution and

where is a functional defined in (3.14) and .

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## Acknowledgment

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

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### Keywords

- Hilbert Space
- Continuous Function
- Unique Solution
- Linear Operator
- Uniqueness Result