Journal of Applied Mathematics

Volume 2012 (2012), Article ID 720139, 21 pages

doi:10.1155/2012/720139

Research Article

Homoclinic Orbits for a Class of Noncoercive Discrete Hamiltonian Systems

Long Yuhua1,2

¹School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China

²Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institute, Guangzhou University, Guangzhou 510006, China

Received 9 June 2012; Accepted 8 August 2012

Academic Editor: Wan-Tong Li

Copyright © 2012 Long Yuhua. 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.

Abstract.

A class of first-order noncoercive discrete Hamiltonian systems are considered. Based on a generalized mountain pass theorem, some existence results of homoclinic orbits are obtained when the discrete Hamiltonian system is not periodical and need not satisfy the global Ambrosetti-Rabinowitz condition.

1. Introduction

Let , , and denote the set of all natural numbers, integers, and real numbers, respectively. Throughout this paper, without special statement, denotes the usual norm in with , denotes the inner product of and .

Consider the noncoercive discrete Hamiltonian systems

where is a nontrivial linear operator, is its adjoint, is the tensorial product of ,

with is an symmetric matrix valued function and , is a continuous function, differentiable with respect to the second variable with continuous derivative . is the shift operator defined as and , where . , , is the forward difference operator. is the standard symplectic matrix , where is the identity matrix on .

As usual, assuming that a solution an equilibrium for (1.1), we say that a solution is homoclinic to 0 if satisfies , and the asymptotic condition as . Such solutions have been found in various models of continuous dynamical systems and frequently have tremendous effects on the dynamics of such nonlinear systems. So the homoclinic orbits have been extensively studied since the time of Poincaré, see [1–7] and references therein.

In recent years, there has been much research activity concerning the theory of difference equations. To a large extent, this due to the realization that difference equations are important in applications. New applications that involve difference equations continue to arise with frequency in the modelling of computer science, economics, neural network, ecology, cybernetics, and so forth, we can refer to [8–13] for detail. Many scholars have investigated discrete Hamiltonian systems independently main for two reasons. The first one is that the behaviour of discrete Hamiltonian systems is sometimes sharply different from the behaviour of the corresponding continuous systems. The second one is that there is a fundamental relationship between solutions of continuous systems and the corresponding discrete systems by employing discrete variable methods (see [8] for detail).

The general form of (1.1) is

which was studied by many scholars in various fields. By making use of minimax theory and geometrical index theory, [14] gave results on subharmonic solutions with prescribed minimal periods. When (1.3) are superquadratic systems, Guo and Yu [15] obtained some existence and multiplicity results by index theory and linking theorem. In [16], when is subquadratic at infinity, the authors gave some existence results of periodic solutions. As to homoclinic orbits for discrete systems, [17–19] studied the second order discrete systems by critical point theory recently. While for the first order discrete systems, such as (1.1) or (1.3), to the authors' best knowledge, it seems there exists no similar results.

Moreover, we may regard (1.1) as being a discrete analogue of Hamiltonian systems

Equation (1.1) is the best approximation of (1.4) when one lets the step size not be equal to 1 but the variable's step size go to zero, so solutions of (1.1) can give some desirable numerical features for (1.4). (1.4) is one form of classical Hamiltonian systems appearing in the study of various fields and many well-known results were given.

In view of above reasons, the goal of this paper is to study the existence of homoclinic orbits for the first order discrete Hamiltonian system (1.1) when satisfies superquadratic conditions and need not satisfy the global Ambrosetti-Rabinowitz (AR) condition:

(AR): there exist two constants and such that for all and ,

Let denotes the smallest eigenvalue of , that is,

For later use, we need the following assumptions:

there exists such that as ;

as , ;

as and ;

there exist and such that

there exist , and such that

for all and all

Remark 1.1. By assumption and , we know that satisfies the superquadratic condition at both infinity and 0 respect to the second variable .

The rest of the paper is organized as follows. In Section 2, we shall establish the variational structure for (1.1) and turn the problem of looking for homoclinic orbits for (1.1) to the problem for seeking critical points of the corresponding functional. In order to apply the generalized mountain pass theorem, we give some preliminary results in Section 3. In Section 4, we shall state our main result and complete the proof of our result.

2. Variational Structure

Set is a space which is composed of the following vectors,

Define the subspace of as

Denote , , define another subspace of as follows:

The space is a Hilbert space with the inner product

and the norm introduced from the inner product as follows:

Define a functional on as follows:

according to the definition of , can be written in another form as follows:

The functional is a well-defined on , and next we prove that the problem of looking for homoclinic orbits for (1.1) can be turned to the problem for seeking critical points of the corresponding functional (see (2.6) or (2.7)).

Let

while

then

it follows that

Write , , for any given , there holds

Then we can draw a conclusion that is true if and only if

so

which can be reformed as

that is

which is just (1.1). Therefore, we obtain the following lemma.

Lemma 2.1. is a homoclinic orbit of (1.1) if and only if is a critical point of functional in .

3. Preliminary Results

In order to apply the critical point theory to look for critical points for (2.6), we give some lemmas which will be of fundamental importance in proving our main result.

Let be a real Hilbert space with the norm . Suppose that has an orthogonal decomposition with both and being infinite dimensional. Suppose (resp., ) is an orthogonal basis for (resp. ), and set , , , and , the restriction of on . We say that satisfies the condition if any sequence in , such that a constant, and possesses a convergent subsequence.

We state a basic theorem introduced in [20] by Rabinowitz which is used to obtain the critical points of the functional .

Lemma 3.1 (Generalized mountain pass lemma). Let satisfy

the condition;

there are such that

for all ;

there are , , with such that

where

Then has a critical point with .

Next we consider the eigenvalue problem.

Equation (3.3) can be reformed as follows:

Denote

then (3.3) can be expressed by the following:

Therefore a standard argument shows that , the spectrum of , consists of eigenvalues numbered by, (counted in their multiplicities) the following:

with as , and denote the corresponding system of eigenfunctions of by .

Let Ker, and , where stands for the orthogonal complementary subspace of in . Then

so the functional (2.7) can be rewritten as follows

for all .

Set and , for all and their norms are defined by the following:

respectively. For any given , define with the norm

Define a selfadjoint operator on by the following:

is the absolute value. Give another norm the domain of by the following:

it is easy to get, for all ,

Now we state a fundamental proposition, which will be used in the later.

Proposition 3.2. Let satisfy . Then for all there exists a constant such that

Proof. We complete the proof of Proposition 3.2 by 3 steps.

Step  1. When holds and , we prove that

Note that, by , as , that is, is bounded from below and so there is a such that

For , choose a subsequence , one has

For any given , by (3.18), one can take so large that

Without loss of generality, we can assume that in . Define , and . So is bounded in , which implies that is bounded in . This together with the uniqueness of the weak limit in , we have in , so there exists a such that

Combing (3.19) and (3.20), we have in . It follows that (3.16) is true.

Step  2. For all , there exists a constant such that (3.15) holds.

For any and , by the Hölder inequality, we have

which together with (3.16) yields (3.15).

Step  3. Since implies as , by Step 1 and 2, it remains to consider the case for .

Let

By , as .

Write , then . Set for and , and and and . Then

From (3.24), we get

and so

Since and then there exists a constant such that

which together with (3.26) yields

Give , by (3.28), choose so large that

Denote , . Any given subsequence , we can suppose on , now is bounded in . This together with the uniqueness of the weak limit in on . For any , we have

Combining (3.29) and (3.30), it follows

that is, , there has a constant such that

while .

4. Main Results and Proofs

In the previous section, we turned the homoclinic orbits problem of (1.1) to the corresponding critical point problem of the functional (2.6) or (2.7). Next, we state our main results and complete their proofs by Lemma 3.1.

Our main result is as follows.

Theorem 4.1. Suppose that satisfies and –. Then the discrete Hamiltonian system (1.1) has a nontrivial homoclinic orbit.

Remark 4.2. Observe that if is a homoclinic solution of (1.1), then is a homoclinic solution of the following:

Moreover, satisfies ‒ whenever satisfies , , , and , where

as , ; there exist , and such that

for all and all

So in the following, we will give another theorem and can omit its proof.

Theorem 4.3. The conclusion of Theorem 4.1 holds when replacing , , and with , , and .

With the aid of previous sections, we will verify that satisfies the assumptions of Lemma 3.1. We will proceed by successive lemmas.

Lemma 4.4. If satisfies assumptions of Theorem 4.1, then there are constants , such that

where .

Proof. For any , it is easy to see that there exist two constants such that

By , , and (4.5), for all , there exist a constant such that

Now by mean value Theorem, (4.5) and (4.6), for all and , we have

on the other hand,

similarly,

it follows that

By Proposition 3.2 and (4.10), for any , it holds

Choosing such that , we obtain, for any ,

Since , then there are constants , such that

which completes the proof of Lemma 4.4.

Lemma 4.5. Under assumptions of Theorem 4.1, let with , there exist such that

where .

Proof. Let with and . For and , denote

then there exists such that

where is the number of in and is the greatest integer function.

By , for , there exists such that

where was defined by (4.5). Then it follows

for all with and . Hence, from (4.16) and (4.18), one has

is true for all with . Let and denote

Then by (4.19), for all , we have

where is defined by Lemma 4.5, this is just (4.14). We completed the proof of Lemma 4.5.

In order to verify that satisfies of Lemma 3.1, we need the following lemma.

Lemma 4.6. Write

Then .

Proof. Let , , for all , since

where , then is Gâteaux differential on and

Let weakly in , by Proposition 3.2, one can assume that strongly in for . By (4.24), we have

By (4.5) and (4.6), there exists a constant such that for any , there holds

so by Proposition 3.2, there exists a constant such that for any ,

We deduce from (4.27) that for any , there has so large that

for all and with . On the hand, it is well known that since strongly in , then when

as , where and . Thus there is such that

is true for all integer and all with . Combining (4.28) and (4.30), it is easy to see

It follows from the arbitrariness of that .

Finally, let us complete the proof of Theorem 4.1 by verifying satisfies the Palais-Smale condition.

Lemma 4.7. With assumptions of Theorem 4.1, satisfies the condition.

Proof. Let be a sequence, that is, , for all , and , , as . We claim that is bounded. If not, passing to a subsequence if necessary, we may assume that as .

Denote and for all . By , , (4.5), and Lemma 4.6, we have

which implies that

Making use of (4.5) and (4.6), we obtain that there is a constant such that

Hence from (4.34), there holds

By Hölder inequality and Proposition 3.2, we achieve

Similarly,

Combing (4.35)–(4.37), yields

Since , we deduce from (4.33) and (4.38) that

similarly,

Now, let

Then

that is

For , by (4.32), (4.41) and Proposition 3.2, there exists a constant such that

Since is of finite dimension, using Hölder inequality and (4.45), for any , we have

where is a constant.

Hence by (4.45) and (4.46), there exist positive constants , such that

which implies

when .

By (4.39), (4.40), and (4.48), it follows

which is a contradiction. Therefore, must be bounded. That is, satisfies the condition.

By Lemma 3.1, possesses a critical point such that and (1.1) has a nontrivial homoclinic orbit.

Acknowledgment

This work was supported by the National Natural Science Foundation of China (no. 11101098) and the Xinmiao Program of Guangzhou University.

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