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Existence Theorems of Periodic Solutions for Second-Order Nonlinear Difference Equations
Advances in Difference Equations volume 2008, Article number: 247071 (2007)
Abstract
The authors consider the second-order nonlinear difference equation of the type using critical point theory, and they obtain some new results on the existence of periodic solutions.
1. Introduction
We denote by the set of all natural numbers, integers, and real numbers, respectively. For
, define
when
.
Consider the nonlinear second-order difference equation

where the forward difference operator is defined by the equation
and

In (1.1), the given real sequences satisfy
for any
,
is continuous in the second variable, and
for a given positive integer
and for all
.
and
is the ratio of odd positive integers. By a solution of (1.1), we mean a real sequence
, satisfying (1.1).
In [1, 2], the qualitative behavior of linear difference equations of type

has been investigated. In [3], the nonlinear difference equation

has been considered. However, results on periodic solutions of nonlinear difference equations are very scarce in the literature, see [4, 5]. In particular, in [6], by critical point method, the existence of periodic and subharmonic solutions of equation

has been studied. Other interesting contributions can be found in some recent papers [7–11] and in references contained therein. It is interesting to study second-order nonlinear difference equations (1.1) because they are discrete analogues of differential equation

In addition, they do have physical applications in the study of nuclear physics, gas aerodynamics, infiltrating medium theory, and plasma physics as evidenced in [12, 13].
The main purpose here is to develop a new approach to the above problem by using critical point method and to obtain some sufficient conditions for the existence of periodic solutions of (1.1).
Let be a real Hilbert space,
,
, which implies that
is continuously Fréchet differentiable functional defined on
.
is said to be satisfying Palais-Smale condition (P-S condition) if any sequence
is bounded, and
as
possesses a convergent subsequence in
. Let
be the open ball in
with radius
and centered at
, and let
denote its boundary.
Lemma 1.1 (mountain pass lemma, see [14]).
Let be a real Hilbert space, and assume that
satisfies the P-S condition and the following conditions:
(I1) there exist constants and
such that
for all
, where
(I2) and there exists
such that
Then is a positive critical value of
, where

Lemma 1.2 (saddle point theorem, see [14, 15]).
Let be a real Banach space,
where
and is finite dimensional. Suppose
satisfies the P-S condition and
(I3) there exist constants ,
such that
(I4) there is and a constant
such that
.
Then possesses a critical value
and

where
2. Preliminaries
In this section, we are going to establish the corresponding variational framework for (1.1).
Let be the set of sequences

that is,

For any ,
is defined by

Then is a vector space. For given positive integer
is defined as a subspace of
by

Clearly, is isomorphic to
, and can be equipped with inner product

by which the norm can be induced by

It is obvious that with the inner product defined by (2.5) is a finite-dimensional Hilbert space and linearly homeomorphic to
. Define the functional
on
as follows:

where . Clearly,
, and for any
, by using
, we can compute the partial derivative as

Thus is a critical point of
on
if and only if

By the periodicity of and
in the first variable
, we have reduced the existence of periodic solutions of (1.1) to that of critical points of
on
. In other words, the functional
is just the variational framework of (1.1). For convenience, we identify
with
. Denote
and
such that
. Denote other norm
on
as follows (see, e.g., [16]):
, for all
and
. Clearly,
. Due to
and
being equivalent when
there exist constants
,
,
, and
such that
,
, and


for all ,
and
.
3. Main Results
In this section, we will prove our main results by using critical point theorem. First, we prove two lemmas which are useful in the proof of theorems.
Lemma 3.1.
Assume that the following conditions are satisfied:
(F1) there exist constants ,
, and
such that

(F2)

Then the functional

satisfies P-S condition.
Proof.
For any sequence with
being bounded and
as
there exists a positive constant
such that
Thus, by
,

Set

Then . Also, by the above inequality, we have

In view of

we have

Then we get

Therefore, for any ,

Since the above inequality implies that
is a bounded sequence in
Thus
possesses convergent subsequences, and the proof is complete.
Theorem 3.2.
Suppose that and following conditions hold:
for each
,



Then there exist at least two nontrivial -periodic solutions for (1.1).
Proof.
We will use Lemma 1.1 to prove Theorem 3.2. First, by Lemma 3.1, satisfies P-S condition. Next, we will prove that conditions
and
hold. In fact, by
, there exists
such that for any
and

where Thus for any
for all
we have

Taking we have

and the assumption is verified. Clearly,
For any given
with
and a constant

Thus we can easily choose a sufficiently large such that
and for
Therefore, by Lemma 1.1, there exists at least one critical value
We suppose that
is a critical point corresponding to
, that is,
and
By a similar argument to the proof of Lemma 3.1, for any
there exists
such that
. Clearly,
If
and the proof is complete; otherwise,
and
By Lemma 1.1,

where Then for any
By the continuity of
in
,
and
show that there exists some
such that
If we choose
such that the intersection
is empty, then there exist
such that
Thus we obtain two different critical points
,
of
in
. In this case, in fact, we may obtain at least two nontrivial critical points which correspond to the critical value
The proof of Theorem 3.2 is complete. When
, we have the following results.
Theorem 3.3.
Assume that the following conditions hold:
(G1)

(G2)

where is a constant in (2.10), and
is the minimal positive eigenvalue of the matrix

Then equation

possesses at least one -periodic solution.
First, we proved the following lemma.
Lemma 3.4.
Assume that holds, then the functional

satisfies P-S condition on .
Proof.
For any sequence with
being bounded and
as
there exists a positive constant
such that
In view of
and

we have

By , the above inequality implies that
is a bounded sequence in
. Thus
possesses a convergent subsequence, and the proof of Lemma 3.4 is complete. Now we prove Theorem 3.3 by the saddle point theorem.
Proof of Theorem 3.3.
For any we have

Take then

Set

then we have

On the other hand, for any we have

where
Clearly, is an eigenvalue of the matrix
and
is an eigenvector of
corresponding to
, where
. Let
be the other eigenvalues of
. By matrix theory, we have
for all
. Without loss of generality, we may assume that
then for any

as one finds by minimizing with respect to That is

Set

then by , we have

This implies that the assumption of saddle point theorem is satisfied. Thus there exists at least one critical point of on
, and the proof is complete. When
we have the following result.
Theorem 3.5.
Assume that the following conditions are satisfied:
(G3)
(G4)
where
Then (3.21) possesses at least one -periodic solution.
Before proving Theorem 3.5, first, we prove the following result.
Lemma 3.6.
Assume that holds, then
defined by (3.22) satisfies P-S condition.
Proof.
For any sequence with
being bounded and
as
there exists a positive constant
such that
Thus

That is,

By , the above inequality implies that
is a bounded sequence in
Thus
possesses convergent subsequences, and the proof is complete.
Proof of Theorem 3.5.
For any we have

Take then

Set

then for all
On the other hand, for any
we have

Set then
Thus
satisfies the assumption of saddle point theorem, that is, there exists at least one critical point of
on
This completes the proof of Theorem 3.5.
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Acknowledgment
This project is supported by specialized research fund for the doctoral program of higher education, Grant no. 20020532014.
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Cai, X., Yu, J. Existence Theorems of Periodic Solutions for Second-Order Nonlinear Difference Equations. Adv Differ Equ 2008, 247071 (2007). https://doi.org/10.1155/2008/247071
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DOI: https://doi.org/10.1155/2008/247071