Abstract and Applied Analysis

Volume 2013 (2013), Article ID 398681, 6 pages

http://dx.doi.org/10.1155/2013/398681

Research Article

Nonexistence of Homoclinic Solutions for a Class of Discrete Hamiltonian Systems

Xiaoping Wang

Department of Mathematics, Xiangnan College, Chenzhou, Hunan 423000, China

Received 3 September 2012; Accepted 27 November 2012

Academic Editor: Jinde Cao

Copyright © 2013 Xiaoping Wang. 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.

We give several sufficient conditions under which the first-order nonlinear discrete Hamiltonian system has no solution satisfying condition , where and and and are real-valued functions defined on .

1. Introduction

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

if Hill’s equation

has a real solution such that

and the constant 4 in (1) cannot be replaced by a larger number, where is a piecewise continuous and nonnegative function defined on . 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, denotes the forward difference operator defined by .

When and , that is, system (4) has a solution satisfying , 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 , , and are real-valued functions defined on and and 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 and and , , and are real-valued functions defined on .

When , 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 , and are real-valued functions defined on and . Let

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

where , and , and .

In this paper, we will establish several Lyapunov-type inequalities for systems (5) and (6) if they have a solution 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 and as follows:

(A0), for all , ; (A1), for all , ; (B0), for all ; (B1).

Denote

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

then one has the following inequality:

where .

Proof. Hypothesis (B1) implies that functions and are well defined on . 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 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 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 and for , and that

If (8) has a solution satisfying

then

Corollary 5. Suppose that and for , and that (44) holds. If (8) has a solution satisfying (45), then

3. Lyapunov-Type Inequalities for System (5)

When , 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 satisfying

then

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

Corollary 8. Suppose that for , and that

If (7) has a solution 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 satisfying

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

then system (6) has no solution satisfying (55).

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

then system (6) has no solution satisfying (55).

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

then system (5) has no solution satisfying

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

then system (5) has no solution satisfying (59).

Corollary 14. Suppose that for , and that (51) holds. If

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

Example 15. Consider the second-order difference equation:

where is real-valued function defined on . In view of Corollary 14, if

then (62) has no solution satisfying

Acknowledgment

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

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