We give several sufficient conditions under which the first-order nonlinear discrete Hamiltonian system Δx(n) = α(n)x(n + 1) + β(n) | y(n)|^μ−2y(n), Δy(n) = −γ(n) | x(n + 1)|^ν−2x(n + 1) − α(n)y(n) has no solution (x(n), y(n)) satisfying condition , where μ, ν > 1 and 1/μ + 1/ν = 1 and α(n), β(n), and γ(n) are real-valued functions defined on ℤ.

1. Introduction

In 1907, Lyapunov [1] established the first so-called Lyapunov inequality:

()

if Hill’s equation

()

has a real solution x(t) such that

()

and the constant 4 in (1) cannot be replaced by a larger number, where q(t) is a piecewise continuous and nonnegative function defined on ℝ. Since this result has found applications in the study of various properties of solutions such as oscillation theory, disconjugacy, and eigenvalue problems of (2), a large number of Lyapunov-type inequalities were established in the literature which generalized or improved (1); see [1–20].

In 1983, Cheng [3] first obtained the discrete analogy of Lyapunov inequality (1) for the second-order difference equation:

()

where, and in the sequel, Δ denotes the forward difference operator defined by Δx(n) = x(n + 1) − x(n).

When a = −∞ and b = +∞, that is, system (4) has a solution x(n) satisfying lim _|n|→∞ x(n) = 0, which is called homoclinic solution, whether one can obtain Lyapunov-type inequalities for (4)? To the best of our knowledge, there are no results.

In 2003, Sh. Guseinov and Kaymakçalan [7] partly generalized the Cheng’s result to the discrete linear Hamiltonian system:

()

where α(n), β(n), and γ(n) are real-valued functions defined on ℤ and a and b are not necessarily usual zeros, but rather, generalized zeros. Later, some better Lyapunov-type inequalities for system (5) were obtained in [19, 20].

Very recently, He and Zhang [10] further generalized the result in [19] to the following first-order nonlinear difference system:

()

where μ, ν > 1 and 1/μ + 1/ν = 1 and α(n), β(n), and γ(n) are real-valued functions defined on ℤ.

When μ = ν = 2, system (6) reduces to (5). In addition, the special forms of system (6) contain many well-known difference equations which have been studied extensively and have much applications in the literature [21–23], such as the second-order linear difference equation:

()

and the second-order half-linear difference equation:

()

where r > 1, p(n) and q(n) are real-valued functions defined on ℤ and p(n) > 0. Let

()

then (8) can be written as the form of (6):

()

where μ = r/(r − 1), ν = r and α(n) = 0, β(n) = [p(n)] ^1/(1−r) and γ(n) = q(n).

In this paper, we will establish several Lyapunov-type inequalities for systems (5) and (6) if they have a solution (x(n), y(n)) satisfying conditions

()

respectively. Taking advantage of these Lyapunov-type inequalities, we are able to establish some criteria for nonexistence of homoclinic solutions of systems (5) and (6). As we know, there are no results on non-existence of homoclinic solutions for Hamiltonian systems in previous literature.

2. Lyapunov-Type Inequalities for System (6)

In this section, we shall establish some Lyapunov-type inequalities for system (6). For the sake of convenience, we list some assumptions on α(n) and β(n) as follows:

(A0)
α(n) < 1, for all n ∈ ℤ, ;
(A1)
α(n) < 1, for all n ∈ ℤ, ;
(B0)
β(n)≥(≢) 0, for all n ∈ ℤ;
(B1)
.

Denote

()

Theorem 1. Suppose that hypotheses (A0), (B0), and (B1) are satisfied. If system (6) has a solution (x(n), y(n)) satisfying

()

then one has the following inequality:

()

where γ⁺(n) = max {γ(n), 0}.

Proof. Hypothesis (B1) implies that functions ζ(n) and η(n) are well defined on ℤ. Without loss of generality, we can assume that

()

From (14) and (B0), one has

()

It follows from (13), (18), and the Hölder inequality that

()

From (A0), (17), (19), (20), and the first equation of system (6), we have

()

Combining (19) with (21), one has

()

Similarly, it follows from (20) and (22) that

()

Combining (23) with (24), one has

()

Now, it follows from (16), (18), and (25) that

()

By (6), we obtain

()

Summing the above from −∞ to +∞ and using (17) and (18), we obtain

()

which, together with (26), implies that

()

We claim that

()

If (30) is not true, then

()

From (28) and (31), we have

()

It follows that

()

Combining (21) with (33), we obtain that

()

which, together with the second equation of system (6), implies that

()

Combining the above with (17), one has

()

Both (34) and (36) contradict with (14). Therefore, (30) holds. Hence, it follows from (29) and (30) that (15) holds.

Corollary 2. Suppose that hypotheses (A1), (B0), and (B1) are satisfied. If system (6) has a solution (x(n), y(n)) satisfying (14), then one has the following inequality:

()

where and in the sequel,

()

Proof. Obviously, (A1) implies that

()

and so (A0) holds, and which, together with (B1), implies that

. Since

()

it follows that

()

which implies that (37) holds.

Since

()

then it follows from (37) that the following corollary is true.

Corollary 3. Suppose that hypotheses (A1), (B0), and (B1) are satisfied. If system (6) has a solution (x(n), y(n)) satisfying (14), then

()

Applying Theorem 1 and Corollary 2 to system (8) (i.e., (10)), we have immediately the following two corollaries.

Corollary 4. Suppose that r > 1 and p(n) > 0 for n ∈ ℤ, and that

()

If (8) has a solution x(n) satisfying

()

then

()

Corollary 5. Suppose that r > 1 and p(n) > 0 for n ∈ ℤ, and that (44) holds. If (8) has a solution x(n) satisfying (45), then

()

3. Lyapunov-Type Inequalities for System (5)

When μ = ν = 2, assumption (B1) reduces the following form:

(B2)
.

Applying the results obtained in last section to the first-order linear Hamiltonian system (5), we have immediately the following corollaries.

Corollary 6 6. Suppose that hypotheses (A0), (B0), and (B2) are satisfied. If system (5) has a solution (x(n), y(n)) satisfying

()

then

()

Corollary 7. Suppose that hypotheses (A1), (B0), and (B2) are satisfied. If system (5) has a soldution (x(n), y(n)) satisfying (48), then

()

Corollary 8. Suppose that p(n) > 0 for n ∈ ℤ, and that

()

If (7) has a solution x(n) satisfying

()

then

()

4. Nonexistence of Homoclinic Solutions

Applying the results obtained in Sections 2 and 3, we can drive the following criteria for non-existence of homoclinic solutions of systems (5) and (6) immediately.

Corollary 9. Suppose that hypotheses (A0), (B0), and (B1) are satisfied. If

()

then system (6) has no solution (x(n), y(n)) satisfying

()

Corollary 10. Suppose that hypotheses (A1), (B0), and (B1) are satisfied. If

()

then system (6) has no solution (x(n), y(n)) satisfying (55).

Corollary 11. Suppose that hypotheses (A1), (B0), and (B1) are satisfied. If

()

then system (6) has no solution (x(n), y(n)) satisfying (55).

Corollary 12. Suppose that hypotheses (A0), (B0), and (B2) are satisfied. If

()

then system (5) has no solution (x(n), y(n)) satisfying

()

Corollary 13. Suppose that hypotheses (A1), (B0), and (B2) are satisfied. If

()

then system (5) has no solution (x(n), y(n)) satisfying (59).

Corollary 14. Suppose that p(n) > 0 for n ∈ ℤ, and that (51) holds. If

()

then (7) has no solution x(n) satisfying (52).

Example 15. Consider the second-order difference equation:

()

where q(n) is real-valued function defined on ℤ. In view of Corollary 14, if

()

then (62) has no solution x(n) satisfying

()

Acknowledgment

This work is supported by the Scientific Research Fund of Hunan Provincial Education Department (07A066).

References

1 Lyapunov A. M., Probléme général de la stabilité du mouvement, Annde La Faculté. (1907) 2, no. 9, 203–474.
Google Scholar
2 Bohner M., Clark S., and Ridenhour J., Lyapunov inequalities for time scales, Journal of Inequalities and Applications. (2002) 7, no. 1, 61–77, https://doi.org/10.1155/S102558340200005X, 1923568, ZBL1088.34503.
10.1155/S102558340200005X
Web of Science® Google Scholar
3 Cheng S. S., A discrete analogue of the inequality of Lyapunov, Hokkaido Mathematical Journal. (1983) 12, no. 1, 105–112, 689261, ZBL0535.39002.
10.14492/hokmj/1381757783
Google Scholar
4 Cheng S.-S., Lyapunov inequalities for differential and difference equations, Polytechnica Posnaniensis. (1991) no. 23, 25-–41, 1177008, ZBL0753.34017.
Google Scholar
5 Clark S. and Hinton D., A Liapunov inequality for linear Hamiltonian systems, Mathematical Inequalities & Applications. (1998) 1, no. 2, 201–209, https://doi.org/10.7153/mia-01-18, 1613448, ZBL0909.34033.
10.7153/mia-01-18
Web of Science® Google Scholar
6 Clark S. and Hinton D., Discrete Lyapunov inequalities, Dynamic Systems and Applications. (1999) 8, no. 3-4, 369–380, 1722967, ZBL0940.39013.
Google Scholar
7 Guseinov G. Sh. and Kaymakçalan B., Lyapunov inequalities for discrete linear Hamiltonian systems, Computers & Mathematics with Applications. (2003) 45, no. 6–9, 1399–1416, https://doi.org/10.1016/S0898-1221(03)00095-6, 2000605, ZBL1055.39029.
10.1016/S0898-1221(03)00095-6
Web of Science® Google Scholar
8 Guseinov G. Sh. and Zafer A., Stability criteria for linear periodic impulsive Hamiltonian systems, Journal of Mathematical Analysis and Applications. (2007) 335, no. 2, 1195–1206, https://doi.org/10.1016/j.jmaa.2007.01.095, 2346900, ZBL1128.34005.
10.1016/j.jmaa.2007.01.095
Web of Science® Google Scholar
9 Hartman P., Difference equations: disconjugacy, principal solutions, Green′s functions, complete monotonicity, Transactions of the American Mathematical Society. (1978) 246, 1–30, https://doi.org/10.2307/1997963, 515528, ZBL0409.39001.
10.2307/1997963
Web of Science® Google Scholar
10 He X. and Zhang Q.-M., A discrete analogue of Lyapunov-type inequalities for nonlinear difference systems, Computers & Mathematics with Applications. (2011) 62, no. 2, 677–684, https://doi.org/10.1016/j.camwa.2011.05.049, 2817905.
10.1016/j.camwa.2011.05.049
Web of Science® Google Scholar
11 Jiang L. and Zhou Z., Lyapunov inequality for linear Hamiltonian systems on time scales, Journal of Mathematical Analysis and Applications. (2005) 310, no. 2, 579–593, https://doi.org/10.1016/j.jmaa.2005.02.026, 2022945, ZBL1076.37053.
10.1016/j.jmaa.2005.02.026
Web of Science® Google Scholar
12 Lin S. H. and Yang G. S., On discrete analogue of Lyapunov inequality, Tamkang Journal of Mathematics. (1989) 20, no. 2, 169–186, 1038523, ZBL0697.39004.
Google Scholar
13 Wang X., Stability criteria for linear periodic Hamiltonian systems, Journal of Mathematical Analysis and Applications. (2010) 367, no. 1, 329–336, https://doi.org/10.1016/j.jmaa.2010.01.027, 2600402, ZBL1195.34079.
10.1016/j.jmaa.2010.01.027
Web of Science® Google Scholar
14 Tang X.-H. and Zhang M., Lyapunov inequalities and stability for linear Hamiltonian systems, Journal of Differential Equations. (2012) 252, no. 1, 358–381, https://doi.org/10.1016/j.jde.2011.08.002, 2852210, ZBL1242.37039.
10.1016/j.jde.2011.08.002
Web of Science® Google Scholar
15 Tang X. H., Zhang Q.-M., and Zhang M., Lyapunov-type inequalities for the first-order nonlinear Hamiltonian systems, Computers & Mathematics with Applications. (2011) 62, no. 9, 3603–3613, https://doi.org/10.1016/j.camwa.2011.09.011, 2844910, ZBL1236.34040.
10.1016/j.camwa.2011.09.011
Web of Science® Google Scholar
16 Tiryaki A., Ünal M., and Çakmak D., Lyapunov-type inequalities for nonlinear systems, Journal of Mathematical Analysis and Applications. (2007) 332, no. 1, 497–511, https://doi.org/10.1016/j.jmaa.2006.10.010, 2319679, ZBL1123.34037.
10.1016/j.jmaa.2006.10.010
Web of Science® Google Scholar
17 Ünal M., Çakmak D., and Tiryaki A., A discrete analogue of Lyapunov-type inequalities for nonlinear systems, Computers & Mathematics with Applications. (2008) 55, no. 11, 2631–2642, https://doi.org/10.1016/j.camwa.2007.10.014, 2416031, ZBL1142.39309.
10.1016/j.camwa.2007.10.014
Web of Science® Google Scholar
18 Ünal M. and Çakmak D., Lyapunov-type inequalities for certain nonlinear systems on time scales, Turkish Journal of Mathematics. (2008) 32, no. 3, 255–275, 2436257, ZBL1166.34005.
Web of Science® Google Scholar
19 Zhang Q.-M. and Tang X. H., Lyapunov inequalities and stability for discrete linear Hamiltonian systems, Applied Mathematics and Computation. (2011) 218, no. 2, 574–582, https://doi.org/10.1016/j.amc.2011.05.101, 2820519, ZBL1229.39026.
10.1016/j.amc.2011.05.101
Web of Science® Google Scholar
20 Zhang Q.-M. and Tang X. H., Lyapunov inequalities and stability for discrete linear Hamiltonian systems, Journal of Difference Equations and Applications. (2012) 18, no. 9, 1467–1484, https://doi.org/10.1080/10236198.2011.572071.
10.1080/10236198.2011.572071
Web of Science® Google Scholar
21 Agarwal R., Ahlbrandt C., Bohner M., and Peterson A., Discrete linear Hamiltonian systems: a survey, Dynamic Systems and Applications. (1999) 8, no. 3-4, 307–333, 1722960, ZBL0942.39009.
Google Scholar
22 Ahlbrandt C. D. and Peterson A. C., Discrete Hamiltonian Systems, 1996, 16, Kluwer Academic, Dordrecht, The Netherlands, Kluwer Texts in the Mathematical Sciences, 1423802.
10.1007/978-1-4757-2467-7
Google Scholar
23 Elaydi S. N., An Introduction to Difference Equations, 2004, 3rd edition, Springer, New York, NY, USA.
Google Scholar

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Nonexistence of Homoclinic Solutions for a Class of Discrete Hamiltonian Systems

Abstract

1. Introduction

2. Lyapunov-Type Inequalities for System (6)

3. Lyapunov-Type Inequalities for System (5)

4. Nonexistence of Homoclinic Solutions

Acknowledgment

References

References

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Nonexistence of Homoclinic Solutions for a Class of Discrete Hamiltonian Systems

Abstract

1. Introduction

2. Lyapunov-Type Inequalities for System (6)

3. Lyapunov-Type Inequalities for System (5)

4. Nonexistence of Homoclinic Solutions

Acknowledgment

References

References

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