Research Article | Open Access

# Existence of Homoclinic Orbits for a Class of Asymptotically -Linear Difference Systems with -Laplacian

**Academic Editor:**Yong Zhou

#### Abstract

By applying a variant version of Mountain Pass Theorem in critical point theory, we prove the existence of homoclinic solutions for the following asymptotically -linear difference system with -Laplacian , where , , , are not periodic in , and W is asymptotically -linear at infinity.

#### 1. Introduction

Consider the following -Laplacian difference system:
where is the forward difference operator defined by , , , , , *, *: are not periodic in , is asymptotically -linear at infinity, and and are continuously differentiable in . 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.

When , (1.1) can be regarded as a discrete analogue of the following second-order Hamiltonian system:

The existence of homoclinic orbits for Hamiltonian systems is a classical problem and its importance in the study of the behavior of dynamical systems has been recognized by Poincaré [1]. If a system has the transversely intersected homoclinic orbits, then it must be chaotic. If it has the smoothly connected homoclinic orbits, then it cannot stand the perturbation and its perturbed system probably produces chaotic phenomenon. For the existence of homoclinic solutions of problem (1.2), one can refer to the papers [2–5].

Difference equations usually describe evolution of certain phenomena over the course of time. For example, if a certain population has discrete generations, the size of the th generation is a function of the th generation . In fact, difference equations provide a natural description of many discrete models in real world. Since discrete models exist in various fields of science and technology such as statistics, computer science, electrical circuit analysis, biology, neural network, and optimal control, it is of practical importance to investigate the solutions of difference equations. For more details about difference equations, we refer the readers to the books [6–8].

In some recent papers [9–20], the authors studied the existence of periodic solutions and subharmonic solutions of difference equations by applying critical point theory. These papers show that the critical point theory is an effective method to the study of periodic solutions for difference equations. Along this direction, several authors [21–28] used critical point theory to study the existence of homoclinic orbits for difference equations. Motivated by the above papers, we consider the existence of homoclinic orbits for problem (1.1) by using the variant version of Mountain Pass Theorem. Our result is new, which seems not to have been considered in the literature. Here is our main result.

Theorem 1.1. *Suppose that and satisfy the following conditions. *(K1)*There are two positive constants and such that
*(K2)*There is a positive constant such that
*(W1)*, as uniformly for .*(W2)*There exists a constant such that
*(W3)*There exists a function such that
*(W4)*,
and for any fixed ,
**Then problem (1.1) has at least one nontrivial homoclinic solution.*

*Remark 1.2. *The function in this paper is asymptotically -linear at infinity. The behavior of the gradient of at infinity is like that of a function , where is a real function but not a matrix function. To the best of our knowledge, similar results of this kind of -Laplacian difference systems with asymptotically -linear at infinity cannot be found in the literature. From this point, our result is new.

#### 2. Preliminaries

Let and for , let Then is a uniform convex Banach space with this norm. As usual, for , let and their norms are given by respectively.

For any , let

To prove our results, we need the following generalization of Lebesgue's dominated convergence theorem.

Lemma 2.1 (see [29]). *Let and be two sequences of measurable functions on a measurable set , and let
**
If
**
then
*

Lemma 2.2. *For ,
*

*Proof. *Since , it follows that . Hence, there exists such that
Hence, we have

Lemma 2.3. * Suppose that (K1), (K2), and (W2) hold. If in , then and in , where satisfies .*

*Proof. *From (K1) and (K2), we have
Hence, from (2.12), we have
Moreover, since in and for almost every , hence,
It follows from Lemma 2.1, (2.13), and the previous equations that
This shows that in . By a similar proof, we can prove that in . The proof is complete.

Lemma 2.4. *Under the conditions of Theorem 1.1, one has
**
for , which yields that
**
Moreover, is continuously Fréchet-differential defined on ; that is, and any critical point of on is classical solution of (1.1) with .*

*Proof. *Firstly, we show that . Let , by (2.9) and (K1), we have
By (W2), we get
Hence, from (2.9) and (2.19), we have
It follows from (2.5), (2.18), and (2.20) that . Next we prove that . Rewrite as follows:
where
It is easy to check that and
Next, we prove that , and
For any and for any function , by (K2), we have
Then by the previous equations and Lebesgue's dominated convergence theorem, we have
Similarly, we can prove that (2.25) holds by using (W2) instead of (K2). Finally, we prove that . Let in ; then by Lemma 2.3, we have
This shows that . Similarly, we can prove that . Furthermore, by a standard argument, it is easy to show that the critical points of in are classical solutions of (1.1) with . The proof is complete.

Lemma 2.5 (see [30]). *Let be a real Banach space with its dual space and suppose that satisfies
**
for some , , and with . Let be characterized by
**
where is the set of continuous paths joining 0 to ; then there exists such that
*

#### 3. Proof of Theorem 1.1

*Proof of Theorem 1.1. *We divide the proof of Theorem 1.1 into three steps.*Step 1. *From (W1), there exists such that
where . From (3.1), we have
Let and ; then from (2.9), we obtain
which together with (2.9), (3.2), and (K1) implies that
*Step 2. *From (K1), we have
By (W2) and (W3), we get
Let ; it follows from (W2), (W3), (2.19), and (3.6) that
Define with
By an easy calculation, we have
In what follows, we prove that for some with , as . Otherwise, there exist a sequence with as and a positive constant such that for all . From (3.5), we obtain
It follows from (3.7) that
Hence, from Lebesgue's dominated theorem and (3.11), we have
It follows from (3.8), (3.9), (3.10), and (3.12) that
which is a contradiction. Hence, there exists with such that .*Step 3. *From Step 1, Step 2, and Lemma 2.5, we know that there is a sequence such that
where is the dual space of . In the following, we will prove that is bounded in . Otherwise, assume that as . Let ; we have . It follows from (2.5), (2.16), (3.14), and (K2) that
Set for . Then from (3.15), we have
From (K1), (K2), and (3.14), we get
which implies that
Let . From (W1), there exists such that
Since , it follows from (2.9) and (3.19) that
For , let
Thus, from (W4), we have as , which together with (3.16) implies that
Hence, we can take sufficiently large such that
The previous inequality and (W2) imply that
Next, for the previous , let
From (W4), we have and
From (3.15) and (3.26), we get
which implies that
Therefore, there exists such that
It follows from (3.20), (3.24), and (3.29) that
which implies that
but this contradicts to (3.18). Hence, is bounded in .

Going to a subsequence if necessary, we may assume that there exists such that as . In order to prove our theorem, it is sufficient to show that . For any with , let for and let for . Then from (2.16), we have
Since as and in , it follows from (3.14) that
It follows from (3.32) and (3.33) that as .

For any , and assume that for some with , . Since
then, we have
as . Noting that
Hence, we have
which implies that ; that is, is a critical point of . From (K1) and (W1), we know that . In fact, if , we have from (2.5), (K1), and (W1) that . On the other hand, from Step 1, Step 2, and Lemma 2.5, we know that . This is a contradiction. The proof of Theorem 1.1 is complete.

#### 4. An Example

*Example 4.1. *In problem (1.1), let , and
where with . One can easily check that satisfies conditions (K1) and (K2) with , , and . An easy computation shows that
Then it is easy to check that satisfies (W1)–(W4). Hence, and satisfy all the conditions of Theorem 1.1 and then problem (1.1) has at least one nontrivial homoclinic solution.

#### Acknowledgment

This work is partially supported by the NNSF (no. 10771215) of China.

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#### Copyright

Copyright © 2011 Qiongfen Zhang and X. H. Tang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.