The Solution of Two-Point Boundary Value Problem of a Class of Duffing-Type Systems with Non- Perturbation Term

Abstract

This paper deals with a two-point boundary value problem of a class of Duffing-type systems with non- perturbation term. Several existence and uniqueness theorems were presented.

1. Introduction

Minimax theorems are one of powerful tools for investigation on the solution of differential equations and differential systems. The investigation on the solution of differential equations and differential systems with non- perturbation term using minimax theorems came into being in the paper of Stepan A.Tersian in 1986 [1]. Tersian proved that the equation exists exactly one generalized solution under the operators related to the perturbation term being selfadjoint and commuting with the operator and some other conditions in [1]. Huang Wenhua extended Tersian's theorems in [1] in 2005 and 2006, respectively, and studied the existence and uniqueness of solutions of some differential equations and differential systems with non- perturbation term [2–4], the conditions attached to the non- perturbation term are that the operator related to the term is self-adjoint and commutes with the operator (where is a selfadjoint operator in the equation ). Recently, by further research, we observe that the conditions imposed upon can be weakened, the self-adjointness of can be removed and is not necessarily commuting with the operator .

In this note, we consider a two-point boundary value problem of a class of Duffing-type systems with non- perturbation term and present a result as the operator related to the perturbation term is not necessarily a selfadjoint and commuting with the operator . We obtain several valuable results in the present paper under the weaker conditions than those in [2–4].

2. Preliminaries

Let be a real Hilbert space with inner product and norm , respectively, let 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 ([2]).

Let    be a real Hilbert space,   an everywhere defined functional with Gâteaux derivative    everywhere defined and hemicontinuous. Suppose that there exist two closed subspaces    and    such that  and two nonincreasing functions    satisfying

(2.1)

and

(2.2)

for all , and

(2.3)

for all . Then

(a) has a unique critical point such that ;

(b).

We also need the following lemma in the present work. To the best of our knowledge, the lemma seems to be new.

Lemma 2.2.

Let    and    be two diagonalization    matrices, let   and be the eigenvalues of and , respectively, where each eigenvalue is repeated according to its multiplicity. If commutes with , that is, , then is a diagonalization matrix and are the eigenvalues of .

Proof.

Since is a diagonalization matrix, there exists an inverse matrix such that , where are the distinct eigenvalues of , are the identity matrices. And since , that is,

(2.4)

we have

(2.5)

Denote , where are the submatrices such that and are defined, then, by (2.5),

(2.6)

Noticed that , we have , and hence

(2.7)

where and are the same order square matrices. Since is a diagonalization matrix, there exists an invertible matrix such that

(2.8)

where are the eigenvalues of .

Let , then is an invertible matrix such that and

(2.9)

is a diagonalization matrix and are the eigenvalues of .

The proof of Lemma 2.2 is fulfilled.

Let denote the usual inner product on and denote the corresponding norm by , where . Let denote the inner product on . It is known very well that is a Hilbert space with inner product

(2.10)

and norm , respectively.

Now, we consider the boundary value problem

(2.11)

where , is a real constant diagonalization matrix with real eigenvalues (each eigenvalue is repeated according to its multiplicity), is a potential Carathéodory vector-valued function , is continuous, , , .

Let , , then (2.11) may be written in the form

(2.12)

where , . Clearly, is a potential Carathéodory vector-valued function, . Clearly, if is a solution of (2.12), will be a solution of (2.11).

Assume that there exists a real bounded diagonalization matrix such that for a.e. and

(2.13)

where , commutes with and is possessed of real eigenvalues . In the light of Lemma 2.2, is a diagonalization matrix with real eigenvalues (each eigenvalue is repeated according to its multiplicity). Assume that there exist positive integers such that for

(2.14)

Let be linearly independent eigenvectors associated with the eigenvalues and let be the orthonormal vectors obtained by orthonormalizing to the eigenvectors of . Then for every

(2.15)

And let the set be a basis for the space , then for every ,

(2.16)

It is well known that each can be represented by the absolutely convergent Fourier series

(2.17)

Define the linear operator

(2.18)

Clearly, is a selfadjoint operator and is a Hilbert space for the inner product

(2.19)

and the norm induced by the inner product is

(2.20)

Define

(2.21)

(2.22)

Clearly, and are orthogonal closed subspaces of and .

Define two projective mappings and by and , , then is a selfadjoint operator.

Using the Riesz representation theorem , we can define a mapping by

(2.23)

We observe that in (2.23) is defined implicity. Let in (2.23), we have

(2.24)

Clearly, and hence is defined implicity by (2.24). It can be proved that is a solution of (2.11) if and only if satisfies the operator equation

(2.25)

3. The Main Theorems

Now, we state and prove the following theorem concerning the solution of problem (2.11).

Theorem 3.1.

Assume that there exists a real diagonalization matrix with real eigenvalues satisfying (2.14) and commuting with . Denote

(3.1)

(3.2)

If

(3.3)

problem (2.11) has a unique solution , and satisfies , and

(3.4)

where is a functional defined in (2.24) and .

Proof.

First, by virtue of (2.21) and (2.22), we have

(3.5)

(3.6)

(3.7)

Denote

By (2.24), (2.13), (3.5), (3.6), (3.7), (3.1), and (3.2), for all , let , , , , , , we have

(3.8)

for all , let , , , , , , we have

(3.9)

By (3.3), Clearly, and are nonincreasing. Now, all the conditions in the Theorem 2.1 are satisfied. By virtue of Theorem 2.1, there exists a unique such that and where is a functional defined implicity in (2.24) and . is just a unique solution of (2.12) and is exactly a unique solution of (2.11). The proof of Theorem 3.1 is completed.

Now, we assume that there exists a positive integer such that

(3.10)

for Define

(3.11)

(3.12)

(3.13)

(3.14)

Replace the condition (2.14) by (3.10) and replace (2.21), (2.22), (3.1), and (3.2) by (3.11), (3.12), (3.13), and (3.14), respectively. Using the similar proving techniques in the Theorem 3.1, we can prove the following theorem.

Theorem 3.2.

Assume that there exists a real diagonalization matrix with real eigenvalues satisfying (2.13) and (3.10) and commuting with . If the functions and defined in (3.13) and (3.14) satisfy (3.3), problem (2.11) has a unique solution , and satisfies and (3.4).

It is also of interest to the case of .

Corollary 3.3.

Let , , and be as in (2.11). Assume that there exists a real diagonalization matrix with real eigenvalues satisfying (2.13) and Denote

(3.15)

If and satisfy (3.3), the problem

(3.16)

has a unique solution , and satisfies and (3.4), where is a functional defined in

(3.17)

Corollary 3.4.

Let , , , , and be as in Corollary 3.3. The eigenvalues of satisfy Denote

(3.18)

If and satisfy (3.3), problem (3.16) has a unique solution , and satisfies and (3.4), where is a functional defined in (3.17).

If there exists a functional such that , then (2.13) should be

(3.19)

where is just a Hessian of . In this case, the following corollary follows from Theorem 3.1.

Corollary 3.5.

Let the eigenvalues of satisfy (2.14). If and defined in (3.1) and (3.2) satisfy (3.3), problem (2.11)(where ) has a unique solution , and satisfies and (3.4).

Using the similar techniques of the present paper, we can also investigate the two-point boundary value problem

(3.20)

where , , , , and are as in problem (2.11). The corresponding results are similar to the results in the present paper.

The special case of and in problem (3.20) has been studied by Zhou Ting and Huang Wenhua [5]. Zhou and Huang adopted the techniques different from this paper to achieve their research.