Abstract and Applied Analysis

Volume 2013 (2013), Article ID 547682, 10 pages

http://dx.doi.org/10.1155/2013/547682

Research Article

Nonexistence of Homoclinic Orbits for a Class of Hamiltonian Systems

Xiaoyan Lin,¹ Qi-Ming Zhang,² and X. H. Tang³

¹Department of Mathematics, Huaihua College, Huaihua, Hunan 418008, China

²College of Science, Hunan University of Technology, Zhuzhou, Hunan 412007, China

³School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan 410083, China

Received 31 March 2013; Accepted 22 July 2013

Academic Editor: Marco Squassina

Copyright © 2013 Xiaoyan Lin et al. 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 Hamiltonian system has no solution satisfying condition where and , , , , and are locally Lebesgue integrable real-valued functions defined on .

1. Introduction

In 1897, Poincaré [1] studied the existence of homoclinic solutions for Hamiltonian systems and realized that homoclinic solutions play a very important role in the study of the behavior of dynamical systems. Since then many methods have been developed to this study ([2–6]). Recently, the critical point theory has been successfully applied to establish the existence and multiplicity of homoclinic solutions for Hamiltonian systems; see [1, 7–20] and references therein.

Among the above-mentioned literature, there are two classes of Hamiltonian systems that have been widely investigated: one is the second-order Hamiltonian system

and the other is the first-order Hamiltonian system

By means of variational methods, in order to seek the homoclinic solutions for system (1), one usually defines a functional on the Banach space

where associated with the coefficients and of system (1). And then one proves that possesses critical points on which are homoclinic solutions of system (1). Thus, the nontrivial homoclinic solutions of system (1) which were studied in the existing work are actually a class of special solutions satisfying condition

Similarly, the non-trivial homoclinic solutions of system (2) which were studied in the literature are also a class of special solutions satisfying condition

where associated with the potential of system (2).

As mentioned earlier, the existence and multiplicity of homoclinic solutions for Hamiltonian systems have been studied extensively via critical point theory in recent years; various sufficient conditions for existence are established. However, as we know, there are no results on nonexistence of homoclinic solutions for Hamiltonian systems in the literature. For the simplest second-order Hamiltonian system,

has no non-trivial homoclinic solutions as constant, but when constant, there seem to be no results on existence or non-existence of homoclinic solutions in the literature either.

In this paper, we consider the first-order nonlinear Hamiltonian system

where is locally Lebesgue integrable real-valued function defined on , . For every , and are continuous on in , and for every , and are locally Lebesgue integrable real-valued functions on .

For the sake of convenience, we give the following assumptions on and .

(F) For any , , and there exist a constant and a locally Lebesgue integrable nonnegative function defined on such that

(G) for , and there exists a locally Lebesgue integrable nonnegative function defined on such that

 where and .

Remark 1. In case , where is locally Lebesgue integrable real-valued function defined on , , and satisfies that

then we can choose .

Let , , and

Then we can rewrite (7) as a standard first-order Hamiltonian system

There are two special forms of system (7) which have been dealt with extensively in the literature: one is the first-order linear Hamiltonian system

and the other is the first-order quasilinear Hamiltonian system

(see [21, 22] and the references therein), where and , and and are locally Lebesgue integrable real-valued functions defined on . In addition, the special forms of system (7) also contain many other well-known second-order differential equations such as the second-order linear differential equation

the second-order half-linear differential equation

and the second-order nonlinear differential equation

where , and are locally Lebesgue integrable real-valued functions defined on and , , and . Indeed, we can rewrite the above-mentioned second-order differential equations as the form of system (7). For example, let

Then (16) can be written as the form of (13):

where , and , , and . If has an inverse , then let

Hence, (17) can be written as the form of (7):

where , , and .

In Sections 2 and 3, we will give some necessary conditions for existence of homoclinic solutions of systems (7) and (13), which satisfy conditions

respectively. These necessary conditions are actually Lyapunov-type inequalities, which generalize the classical Lyapunov inequality for system (6); see [21–25]. Taking advantage of these necessary conditions, we are able to establish some criteria for non-existence of homoclinic solutions of systems (7) and (13) in Section 4.

2. Lyapunov-Type Inequalities for System (7)

In this section, we will establish some Lyapunov-type inequalities for system (7). For the sake of convenience, we list some assumptions on and as follows:

(A0), (A1), (B0), , (B1), , (B2).

Denote

Theorem 2. Suppose that hypotheses (F), (G), (A0), (B0), and (B2) are satisfied. If system (7) has a solution satisfying

then one has the following inequality:

Proof. Hypothesis (B2) implies that functions and are well defined on . Without loss of generality, we can assume that

It follows from (F), (25), and (B0) that

Set for , and then it follows from (F) that

Since for , it follows that

Hence, from (F), (23), (24), (30), (32), and the Hölder inequality, one has

From (A0), (28), (33), (34), and the first equation of system (7), we have

Combining (33) with (36), one has

Similarly, it follows from (34) and (37) that

Hence, from (38) and (39), one has

Now, it follows from (27), (30), and (40) that

By (29), we can choose two sequences and such that

By (7), we obtain

Integrating the above equation from to , we have

Let in the above equation, and using (30), (35), and (43) we obtain

which, together with (41), implies that

We claim that

If (48) is not true, then

From (F), (G), (46), and (49), we have

which, together with (F), implies that

Combining (36) with (51), we obtain that

which, together with (G) and the second equation of system (7), implies that

From (F), (51), and the above, one has

Both (52) and (54) contradict (25). Therefore, (48) holds. Hence, it follows from (47) and (48) that (26) holds.

Corollary 3. Suppose that hypotheses (F), (G), (A1), (B0), and (B2) are satisfied. If system (7) has a solution satisfying (25), then one has the following inequalities:

where is an arbitrary function and

for some .

Proof. (A1), (B0), and (B2) imply that (A0) and . Since

then it follows from (23), (24), (26), (56), and (57) that

which implies that (55) holds. Note that

which, together with (55), yields that (56) holds. It follows from (55) and (56) that (57) and (58) hold.

In case hypothesis (B0) is replaced by (B1) in the proof of Theorem 2, then (40) is strict; that is,

In fact, if (63) is not true, then there exists a such that

Hence, from (38), (39), and (64), we obtain

It follows from (23), (38), and (65) that

which, together with the Hölder inequality, implies that there exists a constant such that

Similarly, it follows from (24), (39), (66), and the Hölder inequality that there exists a constant such that

From (F), (68), and (69), one has that . If , then for ; it follows from (36) that for . Similar to the proof of (54), one has for , which contradicts (25). If , then for or for ; it follows from (A0) and (36) that , which contradicts (35). The above two cases show that (63) holds. Hence, in view of the proof of Theorem 2, we have the following theorem.

Theorem 4. Suppose that hypotheses (F), (G), (A0), (B1), and (B2) are satisfied. If system (7) has a solution satisfying (25), then one has the following inequality:

where and are defined by (23) and (24), respectively.

Similar to the proof of Corollary 3, we can drive the following corollary from Theorem 4.

Corollary 5. Suppose that hypotheses (F), (G), (A1), (B1), and (B2) are satisfied. If system (7) has a solution satisfying (25), then

where and are defined by (59).

Applying Theorem 4 and Corollary 5 to system (19) (i.e., (16)), we have immediately the following two corollaries.

Corollary 6. Suppose that , for and

If (16) has a solution satisfying

then

Applying Theorem 4 to the second-order nonlinear differential equation (17) (i.e., system (21)), where , , and , we have the following corollary.

Corollary 7. Suppose that and for , and that (72) and the following hypothesis are satisfied:

(H1) There exists a locally Lebesgue integrable nonnegative function defined on such that

If (17) has a solution satisfying (73), then

3. Lyapunov-Type Inequalities for System (13)

When , assumption (B2) reduces to the following form:

.

Applying some results obtained in the last section to the first-order linear Hamiltonian system (13), we have immediately the following corollaries.

Corollary 8. Suppose that hypotheses (A0), (B0), and are satisfied. If system (13) has a solution satisfying

then

Corollary 9. Suppose that hypotheses (A0), (B1), and are satisfied. If system (13) has a solution satisfying (77), then

Corollary 10. Suppose that for and that

If (15) has a solution satisfying

then

Corollary 11. Suppose that for   and that (80) and the following hypothesis are satisfied:

(H2) There exists a locally Lebesgue integrable nonnegative function defined on such that

If (17) has a solution satisfying (81), 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 (7) and (13) immediately.

Corollary 12. Suppose that hypotheses (F), (G), (A0), (B0), and (B2) are satisfied. If one of the conditions

holds, then system (7) has no solution satisfying

Corollary 13. Suppose that hypotheses (F), (G), (A0), (B1), and (B2) are satisfied. If one of the conditions

holds, then system (7) has no solution satisfying (86).

Corollary 14. Suppose that hypotheses (A0), (B0), and are satisfied. If

or

holds, then system (13) has no solution satisfying

Corollary 15. Suppose that hypotheses (A0), (B1), and are satisfied. If

or

holds, then system (13) has no solution satisfying (92).

Corollary 16. Suppose that for and that (80) holds. If

then (15) has no solution satisfying (81).

Corollary 17. Suppose that for and that (80) and (H2) are satisfied. If

then (17) has no solution satisfying (81).

Example 18. Consider the second-order nonlinear differential equation

where is locally Lebesgue integrable real-valued function defined on . In view of Corollary 16, if

then (97) has no solution satisfying

Acknowledgments

This work is partially supported by the NNSF (no. 11201138) of China Hunan Provincial Natural Science Foundation (no. 11JJ2005), and the Scientific Research Fund of Hunan Provincial Education Department (12B034).

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