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The Solution of Two-Point Boundary Value Problem of a Class of Duffing-Type Systems with Non-
Perturbation Term
Boundary Value Problems volume 2009, Article number: 287834 (2009)
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

and

for all , and

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,

we have

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

Noticed that , we have
, and hence

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

where are the eigenvalues of
.
Let , then
is an invertible matrix such that
and

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

and norm , respectively.
Now, we consider the boundary value problem

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

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

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

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

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

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

Define the linear operator

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

and the norm induced by the inner product is

Define


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

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

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

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


If

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

where is a functional defined in (2.24) and
.
Proof.
First, by virtue of (2.21) and (2.22), we have



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

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

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

for Define




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

If and
satisfy (3.3), the problem

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

Corollary 3.4.
Let ,
,
,
, and
be as in Corollary 3.3. The eigenvalues of
satisfy
Denote

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

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

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.
References
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Ting Z, Wenhua H: The existence and uniqueness of solution of Duffing equations with non-
perturbation functional at nonresonance. Boundary Value Problems 2008, 2008:-9.
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Zhengxian, J., Wenhua, H. The Solution of Two-Point Boundary Value Problem of a Class of Duffing-Type Systems with Non- Perturbation Term.
Bound Value Probl 2009, 287834 (2009). https://doi.org/10.1155/2009/287834
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DOI: https://doi.org/10.1155/2009/287834