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Existence of Homoclinic Solutions for a Class of Nonlinear Difference Equations
Advances in Difference Equations volume 2010, Article number: 470375 (2010)
Abstract
By using the critical point theory, we establish some existence criteria to guarantee that the nonlinear difference equation has at least one homoclinic solution, where
,
and
is non periodic in
. Our conditions on the nonlinear term
are rather relaxed, and we generalize some existing results in the literature.
1. Introduction
Consider the nonlinear difference equation of the form

where is the forward difference operator defined by
,
,
is the ratio of odd positive integers,
and
are real sequences,
.
. As usual, we say that a solution
of (1.1) is homoclinic (to 0) if
as
. In addition, if
, then
is called a nontrivial homoclinic solution.
Difference equations have attracted the interest of many researchers in the past twenty years since they provided a natural description of several discrete models. Such discrete models are often investigated in various fields of science and technology such as computer science, economics, neural network, ecology, cybernetics, biological systems, optimal control, and population dynamics. These studies cover many of the branches of difference equation, such as stability, attractiveness, periodicity, oscillation, and boundary value problem. Recently, there are some new results on periodic solutions of nonlinear difference equations by using the critical point theory in the literature; see [1–3].
In general, (1.1) may be regarded as a discrete analogue of a special case of the following second-order differential equation:

which has arose in the study of fluid dynamics, combustion theory, gas diffusion through porous media, thermal self-ignition of a chemically active mixture of gases in a vessel, catalysis theory, chemically reacting systems, and adiabatic reactor (see, e.g., [4–6] and their references). In the case of , (1.2) has been discussed extensively in the literature; we refer the reader to the monographs [7–10].
It is well known that the existence of homoclinic solutions for Hamiltonian systems and their importance in the study of the behavior of dynamical systems have been already recognized from Poincaré; homoclinic orbits play an important role in analyzing the chaos of dynamical system. In the past decade, this problem has been intensively studied using critical point theory and variational methods.
In some recent papers [1–3, 11–14], the authors studied the existence of periodic solutions, subharmonic solutions, and homoclinic solutions of some special forms of (1.1) by using the critical point theory. These papers show that the critical point method is an effective approach to the study of periodic solutions for difference equations. Along this direction, Ma and Guo [13] applied the critical point theory to prove the existence of homoclinic solutions of the following special form of (1.1):

where ,
,
, and
.
Theorem A (see [13]).
Assume that and
satisfy the following conditions:
-
(p)
for all
;
-
(q)
for all
and
;
-
(f1)
there is a constant
such that
(1.4) -
(f2)
uniformly with respect to
.
Then (1.3) possesses a nontrivial homoclinic solution.
It is worth pointing out that to establish the existence of homoclinic solutions of (1.3), condition (f1) is the special form (with ) of the following so-called global Ambrosetti-Rabinowitz condition on
; see [15].
(AR) For every ,
is continuously differentiable in
, and there is a constant
such that

However, it seems that results on the existence of homoclinic solutions of (1.1) by critical point method have not been considered in the literature. The main purpose of this paper is to develop a new approach to the above problem by using critical point theory.
Motivated by the above papers [13, 14], we will obtain some new criteria for guaranteeing that (1.1) has one nontrivial homoclinic solution without any periodicity and generalize Theorem A. Especially, satisfies a kind of new superquadratic condition which is different from the corresponding condition in the known literature.
In this paper, we always assume that ,
,
. Our main results are the following theorems.
Theorem 1.1.
Assume that and
satisfy the following conditions:
-
(p)
for all
;
-
(q)
for all
and
;
(F1) , for every
,
and
are continuously differentiable in
, and there is a bounded set
such that

uniformly in ;
(F2) there is a constant such that

(F3) and there is a constant
such that

Then (1.1) possesses a nontrivial homoclinic solution.
Theorem 1.2.
Assume that ,
and
satisfy
,
,
and the following assumption:
(F1') , for every
,
and
are continuously differentiable in
and

uniformly in . Then (1.1) possesses a nontrivial homoclinic solution.
Remark 1.3.
Obviously, both conditions
and
are weaker than
. Therefore, both Theorems 1.1 and 1.2 generalize Theorem A by relaxing conditions
and
.
When is subquadratic at infinity, as far as the authors are aware, there is no research about the existence of homoclinic solutions of (1.1). Motivated by the paper [16], the intention of this paper is that, under the assumption that
is indefinite sign and subquadratic as
, we will establish some existence criteria to guarantee that (1.1) has at least one homoclinic solution by using minimization theorem in critical point theory.
Now we present the basic hypothesis on ,
and
in order to announce the results in this paper.
(F4) For every,
is continuously differentiable in
, and there exist two constants
and two functions
such that

(F5) There exist two functionsand
such that

whereas
,
is a positive constant.
(F6) There existand two constants
and
such that

Up to now, we can state our main results.
Theorem 1.4.
Assume that ,
and
satisfy
,
,
,
and
. Then (1.1) possesses at least one nontrivial homoclinic solution.
By Theorem 1.4, we have the following corollary.
Corollary 1.5.
Assume that ,
, and
satisfy
,
and the following conditions:
(F7) , where
and
,
is a constant such that
for some
.
(F8) There exist constants ,
and
such that

(F9) as
is a positive constant.
Then (1.1) possesses at least one nontrivial homoclinic solution.
2. Preliminaries
Let

and for , let

Then is a uniform convex Banach space with this norm.
As usual, for , let

and their norms are defined by

respectively.
For any with
, we let
, and for function
and
, we set

Let be defined by

If ,
and
,
or
holds, then
and one can easily check that

Furthermore, the critical points of in
are classical solutions of (1.1) with
.
We will obtain the critical points of I by using the Mountain Pass Theorem. We recall it and a minimization theorem as follows.
Let be a real Banach space and
satisfy (PS)-condition. Suppose that
satisfies the following conditions:
-
(i)
;
-
(ii)
there exist constants
such that
;
-
(iii)
there exists
such that
.
Then possesses a critical value
given by

where is an open ball in
of radius
centered at
and
Lemma 2.2.
For

where .
Proof.
Since , it follows that
. Hence, there exists
such that

So, we have

The proof is completed.
Lemma 2.3.
Assume that and
hold. Then for every
,
-
(i)
is nondecreasing on
;
-
(ii)
is nonincreasing on
.
The proof of Lemma 2.3 is routine and so we omit it.
Lemma 2.4 . (see [18]).
Let be a real Banach space and
satisfy the (PS)-condition. If
is bounded from below, then
is a critical value of
.
3. Proofs of Theorems
Proof of Theorem 1.1..
In our case, it is clear that . We show that
satisfies the (PS)-condition. Assume that
is a sequence such that
is bounded and
as
. Then there exists a constant
such that

From (2.6), (2.7), (3.1), (F2), and (F3), we obtain

It follows that there exists a constant such that

Then, is bounded in
. Going if necessary to a subsequence, we can assume that
in
. For any given number
, by (F1), we can choose
such that

Since , we can also choose an integer
such that

By (3.3) and (3.5), we have

Since in
, it is easy to verify that
converges to
pointwise for all
, that is,

Hence, we have by (3.6) and (3.7)

It follows from (3.7) and the continuity of on
that there exists
such that

On the other hand, it follows from (3.3), (3.4), (3.6), and (3.8) that

Since is arbitrary, combining (3.9) with (3.10), we get

It follows from (2.7) and the Hlder's inequality that

Since , it follows from (3.11) and (3.12) that
in
. Hence,
satisfies the (PS)-condition.
We now show that there exist constants such that
satisfies assumption (ii) of Lemma 2.1. By (F1), there exists
such that

It follows from that

Set


If , then by Lemma 2.2,
for
, we have by (q), (3.15), and Lemma 2.3(i) that

Set . Hence, from (2.6), (3.14), (3.17), (q), and (F1), we have

Equation (3.18) shows that implies that
, that is,
satisfies assumption (ii) of Lemma 2.1. Finally, it remains to show that
satisfies assumption (iii) of Lemma 2.1. For any
, it follows from (2.9) and Lemma 2.3(ii) that

where

Take such that

and for
. For
, by Lemma 2.3(i) and (3.21), we have

where . By (2.6), (3.19), (3.21), and (3.22), we have for

Since and
, (3.23) implies that there exists
such that
and
. Set
. Then
,
and
. By Lemma 2.1,
possesses a critical value
given by

where

Hence, there exists such that

Then function is a desired classical solution of (1.1). Since
,
is a nontrivial homoclinic solution. The proof is complete.
Proof of Theorem 1.2.
In the proof of Theorem 1.1, the condition that for
,
in (F1), is only used in the the proofs of assumption (ii) of Lemma 2.1. Therefore, we only proves assumption (ii) of Lemma 2.1 still hold that using (F1') instead of (F1). By (F1'), there exists
such that

Since , it follows that

If , then by Lemma 2.2,
for
. Set
. Hence, from (2.6) and (3.28), we have

Equation (3.29) shows that implies that
, that is, assumption (ii) of Lemma 2.1 holds. The proof of Theorem 1.2 is completed.
Proof of Theorem 1.4.
In view of Lemma 2.4, . We first show that
is bounded from below. By (F4), (2.6), and H
lder inequality, we have

Since , (3.30) implies that
as
. Consequently,
is bounded from below.
Next, we prove that satisfies the (PS)-condition. Assume that
is a sequence such that
is bounded and
as
. Then by (2.6), (2.9), and (3.30), there exists a constant
such that

So passing to a subsequence if necessary, it can be assumed that in
. It is easy to verify that
converges to
pointwise for all
, that is,

Hence, we have, by (3.31) and (3.32),

By (F5), there exists such that

For any given number , by (F5), we can choose an integer
such that

It follows from (3.32) and the continuity of on
that there exists
such that

On the other hand, it follows from (3.31), (3.33), (3.34), (3.35), and (F5) that

Since is arbitrary, combining (3.36) with (3.37), we get

Similar to the proof of Theorem 1.1, it follows from (3.12) that

Since , it follows from (3.38) and (3.39) that
in
. Hence,
satisfies (PS)-condition.
By Lemma 2.4, is a critical value of
, that is, there exists a critical point
such that
.
Finally, we show that . Let
and
for
. Then by (F4) and (F6), we have

Since , it follows from (3.40) that
for
small enough. Hence
, therefore
is nontrivial critical point of
, and so
is a nontrivial homoclinic solution of (1.1). The proof is complete.
Proof of Corollary 1.5..
Obviously, (F7) and (F8) imply that (F4) holds, and (F7) and (F9) imply that (F5) holds with . In addition, by (F7) and (F8), we have

This shows that (F6) holds also. Hence, by Theorem 1.4, the conclusion of Corollary 1.5 is true. The proof is complete.
4. Examples
In this section, we give some examples to illustrate our results.
Example 4.1.
In (1.1), let and

where such that
as
,
,
. Let
,
and

Then it is easy to verify that all conditions of Theorem 1.1 are satisfied. By Theorem 1.1, (1.1) has at least a nontrivial homoclinic solution.
Example 4.2.
In (1.1), let ,
for all
and
, and let

where ,
. Let
, and

Then it is easy to verify that all conditions of Theorem 1.2 are satisfied. By Theorem 1.2, (1.1) has at least a nontrivial homoclinic solution.
Example 4.3.
In (1.1), let such that
as
and

Then

We can choose such that

Let

Then

These show that all conditions of Theorem 1.4 are satisfied, where

By Theorem 1.4, (1.1) has at least a nontrivial homoclinic solution.
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Acknowledgments
The authors would like to express their thanks to the referees for their helpful suggestions. This paper is partially supported by the NNSF (no: 10771215) of China and supported by the Outstanding Doctor degree thesis Implantation Foundation of Central South University (no: 2010ybfz073).
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Chen, P., Tang, X. Existence of Homoclinic Solutions for a Class of Nonlinear Difference Equations. Adv Differ Equ 2010, 470375 (2010). https://doi.org/10.1155/2010/470375
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Keywords
- Periodic Solution
- Difference Equation
- Homoclinic Orbit
- Real Banach Space
- Critical Point Theory