Research Article | Open Access
Long Yuhua, "Homoclinic Orbits for a Class of Noncoercive Discrete Hamiltonian Systems", Journal of Applied Mathematics, vol. 2012, Article ID 720139, 21 pages, 2012. https://doi.org/10.1155/2012/720139
Homoclinic Orbits for a Class of Noncoercive Discrete Hamiltonian Systems
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.
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  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,  gave results on subharmonic solutions with prescribed minimal periods. When (1.3) are superquadratic systems, Guo and Yu  obtained some existence and multiplicity results by index theory and linking theorem. In , 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  by Rabinowitz which is used to obtain the critical points of the functional .
Lemma 3.1 (Generalized mountain pass lemma). Let satisfy
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,