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

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.