- Research Article
- Open Access
Solutions of 2
th-Order Boundary Value Problem for Difference Equation via Variational Method
- Qingrong Zou1 and
- Peixuan Weng1Email author
https://doi.org/10.1155/2009/730484
© Q.Zou and P.Weng. 2009
- Received: 7 July 2009
- Accepted: 15 October 2009
- Published: 20 October 2009
Abstract
The variational method and critical point theory are employed to investigate the existence of solutions for 2
th-order difference equation
for
with boundary value condition
by constructing a functional, which transforms the existence of solutions of the boundary value problem (BVP) to the existence of critical points for the functional. Some criteria for the existence of at least one solution and two solutions are established which is the generalization for BVP of the even-order difference equations.
Keywords
- Banach Space
- Difference Equation
- Point Theorem
- Real Banach Space
- Equivalent Norm
1. Introduction
Difference equations have been applied as models in vast areas such as finance insurance, biological populations, disease control, genetic study, physical field, and computer application technology. Because of their importance, many literature deals with its existence and uniqueness problems. For example, see [1–10].
We notice that the existing results are usually obtained by various analytical techniques, for example, the conical shell fixed point theorem [1, 6], Banach contraction map method [7], Leray-Schauder fixed point theorem [2, 10], and the upper and lower solution method [3]. It seems that the variational technique combining with the critical point theory [11] developed in the recent decades is one of the effective ways to study the boundary value problems of difference equations. However because the variational method requires a "symmetrical'' functional, it is hard for the odd-order difference equations to create a functional satisfying the "symmetrical'' property. Therefore, the even-order difference equations have been investigated in most references.
Let
,
be integers, and
,
be a discrete interval in
. Inspired by [5, 8], in this paper, we try to investigate the following
th-order boundary value problem (BVP) of difference equation via variational method combining with some traditional analytical skills:
where
is the forward difference operator;
for
and
A variational functional for BVP (1.1)-(1.2) is constructed which transforms the existence of solutions of the boundary value problem (BVP) to the existence of critical points of this functional. In order to prove the existence criteria of critical points of the functional, some lemmas are given in Section 2. Two criteria for the existence of at least one solution and two solutions for BVP (1.1)-(1.2) are established in Section 3 which is the generalization for BVP of the even-order difference equations. The existence results obtained in this paper are not found in the references, to the best of our knowledge.
For convenience, we will use the following notations in the following sections:
2. Variational Structure and Preliminaries
We need two lemmas from [12] or [11].
Lemma 2.1.



Furthermore, if
has bounded linear Gâteaux derivative, then
Lemma 2.2 (mountain-pass lemma).









This means that there exists
, s.t.
,
The following lemma will be used in the proof of Lemma 2.4.
Lemma 2.3.
If
is a symmetric and positive-defined real matrix,
is a real matrix,
is the transposed matrix of
Then
is positive defined if and only if
Proof.
Let
be a Hilbert space defined by
with the norm
Hence
is an
-dimensional Hilbert space. For any
, let
then one can show that there exist constants
s.t.
; that is,
is an equivalent norm of
(see [9, page 68]).
Lemma 2.4.
Proof.







and by Lemma 2.3 with
, we know that
is positive defined. Hence all eigenvalues of
are real and positive. Let
be the minimal eigenvalue of these
eigenvalues, then
Therefore
that is,
However, how to find the
in Lemma 2.4 is a skillful and challenging task. The following lemma from [13] offers some help for the estimation of
.
Lemma 2.5 (Brualdi [13]).
In Lemma 2.5,
is the complex number set, and the denotations
,
,
,
can be found in [13]. Since
is positive defined, all eigenvalues are positive real numbers. Therefore, by Lemma 2.5, let
where
.
is a subset of
and can be calculated directly from
Define
If
, we can use this
as
in Lemma 2.4. If
, then one needs to calculate the eigenvalues directly.
Define the functional
on
by
Then
is
with
where
and
is the inner product in
. In fact, we have
The continuity of
and the right-hand side of the inequality in Lemma 2.4 lead to (2.15).
Furthermore, for any
we have
,
By using the following formula (e.g., see [14, page 28]):
we have
Repeating the above process, we obtain
Let
, that is,
Since
is arbitrary, we know that the solution of BVP (1.1)-(1.2) corresponds to the critical point of
3. Main Results
Now we present our main results of this paper.
Theorem 3.1.
then BVP (1.1)-(1.2) has at least one solution.
Proof.
Noticing that
we have
From Lemma 2.1, the conclusion of this lemma follows.
Corollary 3.2.
where
,
satisfy either
,
or
, then BVP (1.1)-(1.2) has at least one solution.
Proof.














which implies
and by Lemma 2.1, the conclusion of this lemma follows.














Let
, then we have
for
Therefore, by Theorem 3.1, the conclusion of this lemma follows.
Theorem 3.3.


- (i)
,
is defined in Lemma 2.4;
- (ii)
satisfies (3.1) in Theorem 3.1 or
satisfies the assumptions in Corollary 3.2.
Then BVP (1.1)-(1.2) has at least two solutions.
Proof.
We first show that
satisfies the P-S condition. Let
satisfy that
is bounded and
If
is unbounded, it possesses a divergent subseries, say
as
. However from (ii), we get (3.4) or (3.8), hence
as
, which is contradictory to the the fact that
is bounded.


















We denote
as its corresponding critical point.
On the other hand, by Theorem 3.1 or Corollary 3.2, we know that there exists
s.t.
If
the theorem is proved. If on the contrary,
, that is,
that implies for any
,
Taking
in
with
, by the continuity of
there exist
s.t.
,
Hence
,
are two different critical points of
that is, BVP (1.1)-(1.2) has at least two different solutions.
4. An Example
Declarations
Acknowledgments
This research is partially supported by the NSF of China and NSF of Guangdong Province.
Authors’ Affiliations
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