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Existence of Homoclinic Solutions for a Class of Nonlinear Difference Equations
Advances in Difference Equations volumeÂ 2010, ArticleÂ number:Â 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 [1â€“3].
In general, (1.1) may be regarded as a discrete analogue of a special case of the following secondorder differential equation:
which has arose in the study of fluid dynamics, combustion theory, gas diffusion through porous media, thermal selfignition of a chemically active mixture of gases in a vessel, catalysis theory, chemically reacting systems, and adiabatic reactor (see, e.g., [4â€“6] and their references). In the case of , (1.2) has been discussed extensively in the literature; we refer the reader to the monographs [7â€“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 PoincarÃ©; 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 [1â€“3, 11â€“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 â€‰;

(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 socalled global AmbrosettiRabinowitz condition on ; see [15].
(AR) For every , is continuously differentiable in , and there is a constantâ€‰ suchâ€‰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 allâ€‰;

(q)
for allâ€‰ 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 functionssuch that
(F5) There exist two functionsandsuch that
whereas,is a positive constant.
(F6) There existand two constantsandsuch 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) â€‰â€‰, 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 Hlder 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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DOI: https://doi.org/10.1155/2010/470375