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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 
.
for all 
;
for all 
and 
;
such that
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 
;
for all 
;
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 functions
and
such that
where
as
,
is a positive constant.
(F6) There exist
and 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 
.
;
such that 
;
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 
.
is nondecreasing on 
;
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 H
lder'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
