- Research Article
- Open Access

# 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

https://doi.org/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.

## Keywords

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

## 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 ;

(b) .

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 ;

(b) .

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).

Note that is not a space.

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 .

where .

We then have the following corollary of Theorem 3.2.

Corollary 3.3.

where is a functional defined in (3.14) and .

## Declarations

### Acknowledgment

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

## Authors’ Affiliations

## References

- Loud WS:
**Periodic solutions of nonlinear differential equations of Duffing type.**In*Proceedings of the U.S.-Japan Seminar on Differential and Functional Equations (Minneapolis, Minn., 1967)*. Benjamin, New York, NY, USA; 1967:199-224.Google Scholar - Ge W:
-periodic solutions of
-dimensional Duffing type equation
*Chinese Annals of Mathematics A*1988,**9**(4):498-505. (Chinese)MATHGoogle Scholar - Shen Z:
**On the periodic solution to the Newtonian equation of motion.***Nonlinear Analysis: Theory, Methods & Applications*1989,**13**(2):145-149. 10.1016/0362-546X(89)90040-0MathSciNetView ArticleMATHGoogle Scholar - Li W:
**A necessary and sufficient condition on existence and uniqueness of****-periodic solution of Duffing equation.***Chinese Annals of Mathematics B*1990,**11**(3):342-345.MathSciNetMATHGoogle Scholar - Ma R:
**Solvability of periodic boundary value problems for semilinear Duffing equations.***Chinese Annals of Mathematics A*1993,**14**(4):393-399. (Chinese)MathSciNetGoogle Scholar - Mawhin J, Ward JR Jr.:
**Nonuniform nonresonance conditions at the two first eigenvalues for periodic solutions of forced Liénard and Duffing equations.***The Rocky Mountain Journal of Mathematics*1982,**12**(4):643-654. 10.1216/RMJ-1982-12-4-643MathSciNetView ArticleMATHGoogle Scholar - Ding T, Zanolin F:
**Time-maps for the solvability of periodically perturbed nonlinear Duffing equations.***Nonlinear Analysis: Theory, Methods & Applications*1991,**17**(7):635-653. 10.1016/0362-546X(91)90111-DMathSciNetView ArticleMATHGoogle Scholar - Manásevich RF:
**A nonvariational version of a max-min principle.***Nonlinear Analysis: Theory, Methods & Applications*1983,**7**(6):565-570. 10.1016/0362-546X(83)90045-7MathSciNetView ArticleMATHGoogle Scholar - Li W, Shen Z:
**A globally convergent method for finding periodic solutions to the Duffing equation.***Journal of Nanjing University*1997,**33**(3):328-336. (Chinese)MathSciNetGoogle Scholar - Huang W, Cao J, Shen Z:On the solution of nonlinear two-point boundary value problem
,
*Applied Mathematics and Mechanics*1998,**19**(9):889-894. 10.1007/BF02458244MathSciNetView ArticleMATHGoogle Scholar - Huang W, Shen Z:
**On the existence of solutions of boundary value problem of Duffing type systems.***Applied Mathematics and Mechanics*2000,**21**(8):971-976. 10.1007/BF02428368MathSciNetView ArticleMATHGoogle Scholar - Manásevich RF:
**A min max theorem.***Journal of Mathematical Analysis and Applications*1982,**90**(1):64-71. 10.1016/0022-247X(82)90044-0MathSciNetView ArticleMATHGoogle Scholar - Tersian SA:
**A minimax theorem and applications to nonresonace problems for semilinear equations.***Nonlinear Analysis: Theory, Methods & Applications*1986,**10**(7):651-668. 10.1016/0362-546X(86)90125-2MathSciNetView ArticleMATHGoogle Scholar - Huang W, Shen Z:
**Two minimax theorems and the solutions of semilinear equations under the asymptotic non-uniformity conditions.***Nonlinear Analysis: Theory, Methods & Applications*2005,**63**(8):1199-1214. 10.1016/j.na.2005.02.125MathSciNetView ArticleMATHGoogle Scholar - Huang W:
**Minimax theorems and applications to the existence and uniqueness of solutions of some differential equations.***Journal of Mathematical Analysis and Applications*2006,**322**(2):629-644. 10.1016/j.jmaa.2005.09.046MathSciNetView ArticleMATHGoogle Scholar

## Copyright

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.