Volume 2012, Issue 1 720139

Research Article

Open Access

Homoclinic Orbits for a Class of Noncoercive Discrete Hamiltonian Systems

Corresponding Author

Long Yuhua

[email protected]

School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China gzhu.edu.cn

Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institute, Guangzhou University, Guangzhou 510006, China gzhu.edu.cn

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Long Yuhua,

Corresponding Author

Long Yuhua

[email protected]

School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China gzhu.edu.cn

Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institute, Guangzhou University, Guangzhou 510006, China gzhu.edu.cn

Search for more papers by this author

First published: 03 October 2012

https://doi.org/10.1155/2012/720139

Citations: 3

Academic Editor: Wan-Tong Li

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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 N, Z, and R denote the set of all natural numbers, integers, and real numbers, respectively. Throughout this paper, without special statement, |·| denotes the usual norm in R^N with N ∈ N, u · v denotes the inner product of u ∈ R^N and v ∈ R^N.

Consider the noncoercive discrete Hamiltonian systems

()

where v : R^N → R^m(1 ≤ m ≤ N) is a nontrivial linear operator, v^* is its adjoint, v × v is the tensorial product of v, v : (v × v)(p, q) = (v(p), v(q))

()

with L(t) is an (N × N) symmetric matrix valued function and H : R × R^N, (t, y) ↦ H(t, y) is a continuous function, differentiable with respect to the second variable with continuous derivative H′(t, y) = (∂H/∂y)(t, y). S is the shift operator defined as

and

, where x₁, x₂ ∈ R^N. Δx_i(t) = x_i(t + 1) − x_i(t), i = 1,2, is the forward difference operator. J is the standard symplectic matrix

, where I_N is the identity matrix on R^N.

As usual, assuming that a solution x(t) = 0 an equilibrium for (1.1), we say that a solution x(t) is homoclinic to 0 if x(t) satisfies x(t) ≠ 0, and the asymptotic condition x(t) → 0 as |t | →+∞. 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 Z₂ index theory and linking theorem. In [16], when H 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 H satisfies superquadratic conditions and need not satisfy the global Ambrosetti-Rabinowitz (AR) condition:

(AR): there exist two constants μ > 2 and r > 0 such that for all t ∈ Z and x ∈ R^2N, |x | ≥ r

()

Let l(t) denotes the smallest eigenvalue of v^*L(t)v, that is,

()

For later use, we need the following assumptions:

(L₁) there exists 1 < γ < 2 such that l(t) | t|^γ−2 → +∞ as |t | →+∞;

(H₁)H(t, y)/|y|² → +∞ as |y | →+∞, t ∈ Z;

(H₂)|H′(t, y)|/|y| → 0 as |y | →+0 and t ∈ Z;

(H₃) there exist a > 0 and α > 1 such that

()

(H₄) there exist β > α, b > 0 and r > 0 such that

()

(H₅) for all t ∈ Z and all y ∈ R^2N

()

Remark 1.1. By assumption (H₁) and (H₂), we know that H(t, y) satisfies the superquadratic condition at both infinity and 0 respect to the second variable y.

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 X of S as

()

Denote u = v × v, u^* = v^* × v^*, define another subspace E of X as follows:

()

The space E is a Hilbert space with the inner product

()

and the norm introduced from the inner product as follows:

()

Define a functional F(x) on E as follows:

()

according to the definition of ∥x∥, F(x) can be written in another form as follows:

()

The functional F(x) is a well-defined C¹ on E, 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 F(x) (see (2.6) or (2.7)).

Let

()

while

()

then

()

it follows that

()

Write

, i = 1,2, for any given t ∈ Z, there holds

()

Then we can draw a conclusion that F^′(x) = 0 is true if and only if

()

which can be reformed as

()

that is

()

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

Lemma 2.1. x = {x(t)} ≠ 0 is a homoclinic orbit of (1.1) if and only if x is a critical point of functional F(x) in E.

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 E be a real Hilbert space with the norm ∥·∥. Suppose that E has an orthogonal decomposition E = E¹ ⊕ E² with both E¹ and E² being infinite dimensional. Suppose (ν_n) (resp., (ω_n)) is an orthogonal basis for E¹ (resp. E²), and set , , , and , the restriction of f on E_n. We say that f satisfies the (PS)^* condition if any sequence (x⁽ⁿ⁾) in E, (x⁽ⁿ⁾) ∈ E_n such that f_n(x⁽ⁿ⁾) ≤ C 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 F(x).

Lemma 3.1 (Generalized mountain pass lemma). Let f ∈ C¹(E, R) satisfy

(f₁) the (PS)^*condition;

(f₂) there are ρ, δ > 0 such that

()

for all x ∈ S_ρ = {x ∈ E²∣∥x∥ = ρ};

(f₃) there are r > ρ, M > 0, with ∥e∥ = 1 such that

()

where Q = {(B_r⋂ E¹)} ⊕ {se0 ≤ s ≤ r}.

Then f has a critical point x with f(x) ≥ δ.

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 σ(A), the spectrum of A, consists of eigenvalues numbered by, (counted in their multiplicities) the following:

()

with λ_k → ±∞ as k → ±∞, and denote the corresponding system of eigenfunctions of A by (e_k).

Let E⁰ = KerA, E⁺ = span {e₁, …, e_n} and

, where

stands for the orthogonal complementary subspace of S in E. Then

()

so the functional (2.7) can be rewritten as follows

()

for all x = x⁻ + x⁰ + x⁺ ∈ E⁻ + E⁰ + E⁺.

Set l² = {x = {x(t)} ∈ S∣∑_t∈Z | x(t)|² < +∞} and l^∞ = {x = {x(t)} ∈ S∣ | x(t)|<+∞, for all t ∈ Z} and their norms are defined by the following:

()

respectively. For any given 1 ≤ r < +∞, define l^r = {x = {x(t)} ∈ S∣∑_t∈Z | x(t)|^r < +∞} with the norm

()

Define a selfadjoint operator A on E by the following:

()

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

()

it is easy to get, for all x ∈ E,

()

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

Proposition 3.2. Let L satisfy (L₁). Then for all 1 ≤ p ∈ (2/(3 − γ), +∞) there exists a constant λ_p > 0 such that

()

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

Step 1. When (L₁) holds and p = 2, we prove that

()

Note that, by (L₁), l(t)→+∞ as |t | →+∞, that is, l(t) is bounded from below and so there is a such that

()

For R > 0, choose a subsequence x^(k)(t) ∈ E, one has

()

For any given ϵ > 0, by (3.18), one can take R₀ so large that

()

Without loss of generality, we can assume that x^(k)⇀0 in E. Define E_I = {x ∈ S_I∣∑_t∈Z [(JΔSx(t − 1), x(t))+(M(t)Sx(t), x(t))]<+∞}, S_I = {x = {x(t)}∣x(t) ∈ E, t ∈ I} and I = {t∣ | t | ≤ R₀}. So x^(k) is bounded in E_I, which implies that x^(k) is bounded in

. This together with the uniqueness of the weak limit in

, we have x^(k) → 0 in E_I, so there exists a k₀ such that

()

Combing (3.19) and (3.20), we have x^(k) → 0 in l². It follows that (3.16) is true.

Step 2. For all p > 2, there exists a constant λ_p > 0 such that (3.15) holds.

For any p > 2 and x ∈ E, by the Hölder inequality, we have

()

which together with (3.16) yields (3.15).

Step 3. Since (L₁) implies l(t)→+∞ as |t | → ∞, by Step 1 and 2, it remains to consider the case for 1 ≤ p ∈ (2/(3 − γ), 2).

Let

()

By (L₁),

as R → ∞.

Write α = (2 − γ)/(2 − p), then αp > 1. Set for R > 0 and x ∈ E, and |t|^α·|x(t)| > 1} and and |t|^α·|x(t)| ≤ 1}. Then

()

From (3.24), we get

()

and so

()

Since αp > 1 and

then there exists a constant s such that

()

which together with (3.26) yields

()

Give ϵ > 0, by (3.28), choose R₀ > 0 so large that

()

Denote I = {t ∈ Z∣ | t | ≤ R₀}, E_I = {x∣x(t) ∈ E, t ∈ I}. Any given subsequence (x^(k)) ∈ E, we can suppose x^(k)⇀0 on E, now x^(k) is bounded in . This together with the uniqueness of the weak limit in on I. For any x ∈ E_I, we have

()

Combining (3.29) and (3.30), it follows

()

that is,

, there has a constant λ_p > 0 such that

()

while 1 ≤ p ∈ (2/(3 − γ), 2).

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 H satisfies (L₁) and (H₁)–(H₅). Then the discrete Hamiltonian system (1.1) has a nontrivial homoclinic orbit.

Remark 4.2. Observe that if x(t) is a homoclinic solution of (1.1), then y(t) = x(−t) is a homoclinic solution of the following:

()

Moreover, −H(−t, y) satisfies (H₁)‒ (H₅) whenever H(t, y) satisfies

, (H₂), (H₃),

and

, where

H(t, y)/|y|² → −∞ as |y | →+∞, t ∈ Z;
there exist β > α, b > 0 and r > 0 such that
()
for all t ∈ Z and all y ∈ R^2N
()

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 (H₁), (H₄), and (H₅) with , , and .

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

Lemma 4.4. If H satisfies assumptions of Theorem 4.1, then there are constants ρ > 0, δ > 0 such that

()

where

Proof. For any x ∈ E, it is easy to see that there exist two constants 0 < m₀ < M₀ such that

()

By (H₁), (H₃), and (4.5), for all ϵ > 0, there exist a constant C_ϵ > 0 such that

()

Now by mean value Theorem, (4.5) and (4.6), for all x ∈ E and t ∈ Z, we have

()

on the other hand,

()

similarly,

()

it follows that

()

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

()

Choosing ϵ such that

, we obtain, for any x ∈ E²,

()

Since α > 1, then there are constants ρ > 0, δ > 0 such that

()

which completes the proof of Lemma 4.4.

Lemma 4.5. Under assumptions of Theorem 4.1, let with , there exist r₁, r₂ > 0 such that

()

where

Proof. Let with and F = E¹⊕span {e}. For x = x⁻ + x⁰ + x⁺ ∈ F − {0} and ϵ > 0, denote

()

then there exists ϵ₁ > 0 such that

()

where ♯Ω_x is the number of t in Ω_x and [·] is the greatest integer function.

By (H₁), for , there exists R₁ > 0 such that

()

where m₀ was defined by (4.5). Then it follows

()

for all x ∈ F − {0} with ∥x∥ ≥ R₁/m₀ϵ₁ and t ∈ Ω_x. Hence, from (4.16) and (4.18), one has

()

is true for all x ∈ F − {0} with

. Let r₁ > 0 and denote

()

Then by (4.19), for all r₁ > max {ρ, R₁/m₀ϵ₁}, we have

()

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

In order to verify that F(x) satisfies (f₁) of Lemma 3.1, we need the following lemma.

Lemma 4.6. Write

()

Then h(x) ∈ C¹(E, R).

Proof. Let φ(z) = h(t, x + zy), 0 ≤ z ≤ 1, for all x, y ∈ E, since H ∈ C¹(E, R)

()

where 0 < θ_t < 1, then h(x) is Gâteaux differential on E and

()

Let x^(k)⇀x weakly in E, by Proposition 3.2, one can assume that x^(k) → x strongly in l^p for p ∈ [1, +∞). By (4.24), we have

()

By (4.5) and (4.6), there exists a constant C₁ > 0 such that for any R > 0, there holds

()

so by Proposition 3.2, there exists a constant C₂ > 0 such that for any ∥y∥ = 1,

()

We deduce from (4.27) that for any ϵ > 0, there has R > 0 so large that

()

for all k ∈ N and y ∈ E with ∥y∥ = 1. On the hand, it is well known that since x^(k) → x strongly in l², then when

()

as k → ∞, where

and I_R = {t∣t ∈ Z, |t | < R}. Thus there is k₀ ∈ N such that

()

is true for all integer k ≥ k₀ and all y ∈ E with ∥y∥ = 1. Combining (4.28) and (4.30), it is easy to see

()

It follows from the arbitrariness of ϵ that h(x) ∈ C¹(E, R).

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

Lemma 4.7. With assumptions of Theorem 4.1, F(x) satisfies the (PS) ^* condition.

Proof. Let (x^(k)) be a (PS) ^* sequence, that is, x^(k) ∈ E_n, for all k ∈ N, and F(x^(k)) ≤ C, F^′(x^(k)) → 0, as k → ∞. We claim that (x^(k)) is bounded. If not, passing to a subsequence if necessary, we may assume that ∥x^(k)∥ → ∞ as k → ∞.

Denote I₁ = {t ∈ Z∣ | x^(k)(t)| ≥ r/m₀} and I₁ = {t ∈ Z∣ | x^(k)(t)| < r/m₀} for all k ∈ N. By (H₄), (H₅), (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 C₃ 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 1 < α < β, we deduce from (4.33) and (4.38) that

()

similarly,

()

Now, let

()

Then

()

that is

()

For β > 1, by (4.32), (4.41) and Proposition 3.2, there exists a constant C₄ > 0 such that

()

Since E⁰ is of finite dimension, using Hölder inequality and (4.45), for any

, we have

()

where C₅ > 0 is a constant.

Hence by (4.45) and (4.46), there exist positive constants C₆, C₇ such that

()

which implies

()

when β > 1.

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

()

which is a contradiction. Therefore, (x^(k)) must be bounded. That is, F(x) satisfies the (PS) ^* condition.

By Lemma 3.1, F(x) possesses a critical point x ∈ E such that F(x) ≥ δ > 0 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.

References

1 Poincar H., Les Mthodes Nouvelles De La Mcanique Cleste, 1899, Gauthier-Villars, Paris, France.
Google Scholar
2 Ding Y. and Girardi M., Infinitely many homoclinic orbits of a Hamiltonian system with symmetry, Nonlinear Analysis A. (1999) 38, no. 3, 391–415, https://doi.org/10.1016/S0362%2D546X(98)00204%2D1, 1705761, ZBL0938.37034.
10.1016/S0362-546X(98)00204-1
Web of Science® Google Scholar
3 Hofer H. and Wysocki K., First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems, Mathematische Annalen. (1990) 288, no. 3, 483–503, https://doi.org/10.1007/BF01444543, 1079873.
10.1007/BF01444543
Web of Science® Google Scholar
4 Ma M. and Guo Z., Homoclinic orbits for second order self-adjoint difference equations, Journal of Mathematical Analysis and Applications. (2006) 323, no. 1, 513–521, https://doi.org/10.1016/j.jmaa.2005.10.049, 2262222, ZBL1107.39022.
10.1016/j.jmaa.2005.10.049
Web of Science® Google Scholar
5 Moser J., Stable and Random Motions in Dynamical Systems, 1973, Princeton University Press, Princeton, NJ, USA, 0442980.
Google Scholar
6 Omana W. and Willem M., Homoclinic orbits for a class of Hamiltonian systems, Differential and Integral Equations. (1992) 5, no. 5, 1115–1120, 1171983, ZBL0759.58018.
10.57262/die/1370870945
Google Scholar
7 Szulkin A. and Zou W., Homoclinic orbits for asymptotically linear Hamiltonian systems, Journal of Functional Analysis. (2001) 187, no. 1, 25–41, https://doi.org/10.1006/jfan.2001.3798, 1867339, ZBL0984.37072.
10.1006/jfan.2001.3798
Web of Science® Google Scholar
8 Agarwal R. P., Difference Equations and Inequalities: Theory, Methods and Applications, 1992, Marcel Dekker, New York, NY, USA, 1155840.
Google Scholar
9 Elaydi S. N., An Introduction to Difference Equations, 1996, Springer, New York, NY, USA, 1410259.
10.1007/978-1-4757-9168-6
Web of Science® Google Scholar
10 Kelley W. G. and Peterson A. C., Difference Equations: An Introduction with Applications, 1991, Academic Press, New York, NY, USA, 1142573.
Google Scholar
11 Kocić V. L. and Ladas G., Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, 1993, Kluwer Academic, Dodrecht, The Netherlands, 1247956.
10.1007/978-94-017-1703-8
Google Scholar
12 Mickens R. E., Difference Equations: Theory and Application, 1990, Van Nostrand Reinhold, New York, NY, USA, 1158461.
Google Scholar
13 Sharkovsky A. N., Maĭstrenko Y. L., and Romanenko E. Y., Difference Equations and Their Applications, 1993, Kluwer Academic, Dodrecht, The Netherlands, https://doi.org/10.1007/978%2D94%2D011%2D1763%2D0, 1244579.
10.1007/978-94-011-1763-0
Google Scholar
14 Long Y. H., Multiplicity results for periodic solutions with prescribed minimal periods to discrete Hamiltonian systems, Journal of Difference Equations and Applications. (2010) 1–20.
Web of Science® Google Scholar
15 Guo Z. and Yu J., The existence of periodic and subharmonic solutions of subquadratic second order difference equations, Journal of the London Mathematical Society. (2003) 68, no. 2, 419–430, https://doi.org/10.1112/S0024610703004563, 1994691, ZBL1046.39005.
10.1112/S0024610703004563
Web of Science® Google Scholar
16 Zhou Z., Yu J., and Guo Z., The existence of periodic and subharmonic solutions to subquadratic discrete Hamiltonian systems, The ANZIAM Journal. (2005) 47, no. 1, 89–102, https://doi.org/10.1017/S1446181100009792, 2159854, ZBL1081.39019.
10.1017/S1446181100009792
Web of Science® Google Scholar
17 Yu J. S., Shi H. P., and Guo Z. M., Homoclinic orbits for nonlinear difference equations containing both advance and retardation, Journal of Mathematical Analysis and Applications. (2009) 352, no. 2, 799–806, https://doi.org/10.1016/j.jmaa.2008.11.043, 2501925, ZBL1160.39311.
10.1016/j.jmaa.2008.11.043
Web of Science® Google Scholar
18 Deng X. and Cheng G., Homoclinic orbits for second order discrete Hamiltonian systems with potential changing sign, Acta Applicandae Mathematicae. (2008) 103, no. 3, 301–314, https://doi.org/10.1007/s10440%2D008%2D9237%2Dz, 2430446, ZBL1153.39012.
10.1007/s10440-008-9237-z
Web of Science® Google Scholar
19 Long Y. H., Homoclinic solutions of some second-order nonperiodic discrete systems, Advances in Difference Equations. (2011) 64, 12, https://doi.org/10.1186/1687%2D1847%2D2011%2D64.
10.1186/1687-1847-2011-64
Google Scholar
20 Rabinowitz P., Minimax methods in critical point theory with applications to differential equations, 65, Conference Board of the Mathematical Sciences (CBMS ′89), 1986, Regional Conference Series in Mathematics.
Google Scholar

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Homoclinic Orbits for a Class of Noncoercive Discrete Hamiltonian Systems

Abstract

1. Introduction

2. Variational Structure

3. Preliminary Results

4. Main Results and Proofs

Acknowledgment

References

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Homoclinic Orbits for a Class of Noncoercive Discrete Hamiltonian Systems

Abstract

1. Introduction

2. Variational Structure

3. Preliminary Results

4. Main Results and Proofs

Acknowledgment

References

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