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

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,

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