Nonexistence of Homoclinic Orbits for a Class of Hamiltonian Systems

Abstract

We give several sufficient conditions under which the first-order nonlinear Hamiltonian system 𝑥^′(𝑡)=𝛼(𝑡)𝑥(𝑡)+𝑓(𝑡,𝑦(𝑡)),𝑦^′(𝑡)=−𝑔(𝑡,𝑥(𝑡))−𝛼(𝑡)𝑦(𝑡) has no solution (𝑥(𝑡),𝑦(𝑡)) satisfying condition 0<∫^+∞_−∞[|𝑥(𝑡)|^𝜈+(1+𝛽(𝑡))|𝑦(𝑡)|^𝜇]𝑑𝑡<+∞, where 𝜇,𝜈>1 and (1/𝜇)+(1/𝜈)=1, 0≤𝑥𝑓(𝑡,𝑥)≤𝛽(𝑡)|𝑥|^𝜇, 𝑥𝑔(𝑡,𝑥)≤𝛾₀(𝑡)|𝑥|^𝜈, 𝛽(𝑡),𝛾₀(𝑡)≥0, and 𝛼(𝑡) are locally Lebesgue integrable real-valued functions defined on ℝ.

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 𝑝(𝑡)𝑢^′(𝑡)^′−𝐿(𝑡)𝑢(𝑡)+∇𝑊(𝑡,𝑢(𝑡))=0, (1) and the other is the first-order Hamiltonian system 𝑥^′(𝑡)𝑦^′(𝑡)=0−𝐼𝐻𝐼0^𝑥(𝑡)(𝑡)𝐻𝑡,𝑥,𝑦_𝑦𝑡,𝑥(𝑡),𝑦(𝑡). (2) By means of variational methods, in order to seek the homoclinic solutions for system (1), one usually defines a functional 𝜑(𝑢) on the Banach space 𝐸=𝑢∈𝑊^1,2ℝ,ℝ^𝑛∶_ℝ𝑎₁(𝑡)𝑢^′(𝑡)²+𝑎₂(𝑡)|𝑢(𝑡)|²𝑑𝑡<+∞, (3) where 𝑎₁,𝑎₂∈𝐶(ℝ,(0,+∞)) associated with the coefficients 𝑝(𝑡) and 𝐿(𝑡) of system (1). And then one proves that 𝜑 possesses critical points on 𝐸 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 0<_ℝ1+𝑎₁(𝑡)𝑢^′(𝑡)²+1+𝑎₂(𝑡)|𝑢(𝑡)|²𝑑𝑡<+∞. (4) Similarly, the non-trivial homoclinic solutions of system (2) which were studied in the literature are also a class of special solutions satisfying condition 0<_ℝ1+𝑏₁(𝑡)|𝑥(𝑡)|²+1+𝑏₂(𝑡)𝑦(𝑡)²𝑑𝑡<+∞, (5) where 𝑏₁,𝑏₂∈𝐶(ℝ,(0,+∞)) associated with the potential 𝐻 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, 𝑥^′′(𝑡)+𝑞(𝑡)𝑥(𝑡)=0 (6) has no non-trivial homoclinic solutions as 𝑞(𝑡)≡ constant, but when 𝑞(𝑡)≢ 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 𝑥^′(𝑡)(𝑡)𝑥(𝑡)(𝑡),𝑦=𝛼+𝑓𝑡,𝑦_′(𝑡)=−𝑔(𝑡,𝑥(𝑡))−𝛼(𝑡)𝑦(𝑡), (7) where 𝛼(𝑡) is locally Lebesgue integrable real-valued function defined on ℝ, 𝑓,𝑔∶ℝ²→ℝ. For every 𝑡∈ℝ, 𝑓(𝑡,𝑥) and 𝑔(𝑡,𝑥) are continuous on 𝑥 in ℝ, and for every 𝑥∈ℝ, 𝑓(𝑡,𝑥) and 𝑔(𝑡,𝑥) are locally Lebesgue integrable real-valued functions on 𝑡.

For the sake of convenience, we give the following assumptions on 𝑓 and 𝑔.(F) For any 𝑐≠0, meas{𝑡∈ℝ∶𝑓(𝑡,𝑐exp(−∫^𝑡₀𝛼(𝑠)𝑑𝑠))≠0}>0, and there exist a constant 𝜇>1 and a locally Lebesgue integrable nonnegative function 𝛽(𝑡) defined on ℝ such that 0≤𝑥𝑓(𝑡,𝑥)≤𝛽(𝑡)|𝑥|^𝜇,∀(𝑡,𝑥)∈ℝ². (8)(G)𝑔(𝑡,0)=0 for 𝑡∈ℝ, and there exists a locally Lebesgue integrable nonnegative function 𝛾₀(𝑡) defined on ℝ such that 𝑥𝑔(𝑡,𝑥)≤𝛾₀(𝑡)|𝑥|^𝜈,∀(𝑡,𝑥)∈ℝ², (9) where 𝜈>1 and (1/𝜇)+(1/𝜈)=1.

Remark 1. In case 𝑔(𝑡,𝑥)=𝛾(𝑡)𝜑(𝑥), where 𝛾(𝑡) is locally Lebesgue integrable real-valued function defined on ℝ, 𝜑∈𝐶(ℝ,ℝ), and satisfies that 0≤𝑥𝜑(𝑥)≤|𝑥|^𝜈,∀(𝑡,𝑥)∈ℝ², (10) then we can choose 𝛾₀(𝑡)=𝛾⁺(𝑡)=max{𝛾(𝑡),0}.

Let 𝑢(𝑡)=(𝑥(𝑡),𝑦(𝑡))^⊤, 𝐻(𝑡,𝑥,𝑦)=∫^𝑥₀𝑔(𝑡,𝑠)𝑑𝑠+𝛼(𝑡)𝑥𝑦+∫^𝑦₀𝑓(𝑡,𝑠)𝑑𝑠, and 𝐽=01−10. (11) Then we can rewrite (7) as a standard first-order Hamiltonian system 𝑢^′(𝑡)=𝐽∇𝐻(𝑡,𝑢(𝑡)). (12)

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 𝑥^′(𝑡)(𝑡)𝑥(𝑡)(𝑡)𝑦(𝑡),𝑦=𝛼+𝛽_′(𝑡)=−𝛾(𝑡)𝑥(𝑡)−𝛼(𝑡)𝑦(𝑡) (13) and the other is the first-order quasilinear Hamiltonian system 𝑥^′(𝑡)=𝛼(𝑡)𝑥(𝑡)+𝛽(𝑡)𝑦(𝑡)^𝜇−2𝑦(𝑡),𝑦^′(𝑡)=−𝛾(𝑡)|𝑥(𝑡)|^𝜈−2𝑥(𝑡)−𝛼(𝑡)𝑦(𝑡), (14) (see [21, 22] and the references therein), where 𝜇,𝜈>1 and (1/𝜇)+(1/𝜈)=1, and 𝛽(𝑡) and 𝛾(𝑡) are locally Lebesgue integrable real-valued functions defined on ℝ. 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 𝑝(𝑡)𝑥^′(𝑡)^′+𝑞(𝑡)𝑥(𝑡)=0, (15) the second-order half-linear differential equation 𝑝(𝑡)𝑥^′(𝑡)^𝑟−2𝑥^′(𝑡)^′+𝑞(𝑡)|𝑥(𝑡)|^𝑟−2𝑥(𝑡)=0, (16) and the second-order nonlinear differential equation 𝑝(𝑡)𝜙𝑥^′(𝑡)^′+ℎ(𝑡,𝑥(𝑡))=0, (17) where 𝑟>1, 𝑝(𝑡) and 𝑞(𝑡) are locally Lebesgue integrable real-valued functions defined on ℝ and 𝑝(𝑡)>0, 𝜙∈𝐶(ℝ,ℝ), and ℎ∈𝐶(ℝ²,ℝ). Indeed, we can rewrite the above-mentioned second-order differential equations as the form of system (7). For example, let 𝑦(𝑡)=𝑝(𝑡)𝑥^′(𝑡)^𝑟−2𝑥^′(𝑡). (18) Then (16) can be written as the form of (13): 𝑥^′(𝑡)=𝑝(𝑡)^1/(1−𝑟)𝑦(𝑡)^{(2−𝑟)/(𝑟−1)}𝑦(𝑡),𝑦^′(𝑡)=−𝑞(𝑡)|𝑥(𝑡)|^𝑟−2𝑥(𝑡), (19) where 𝜇=𝑟/(𝑟−1), 𝜈=𝑟 and 𝛼(𝑡)=0, 𝛽(𝑡)=[𝑝(𝑡)]^1/(1−𝑟), and 𝛾(𝑡)=𝑞(𝑡). If 𝜙 has an inverse 𝜙⁻¹, then let 𝑦(𝑡)=𝑝(𝑡)𝜙𝑥^′(𝑡). (20) Hence, (17) can be written as the form of (7): 𝑥^′(𝑡)=𝜙⁻¹𝑦(𝑡)𝑝(𝑡),𝑦^′(𝑡)=−ℎ(𝑡,𝑥(𝑡)), (21) where 𝛼(𝑡)=0, 𝑓(𝑡,𝑥)=𝜙⁻¹(𝑥/𝑝(𝑡)), and 𝑔(𝑡,𝑥)=ℎ(𝑡,𝑥).

In Sections 2 and 3, we will give some necessary conditions for existence of homoclinic solutions of systems (7) and (13), which satisfy conditions 0<^+∞_−∞|𝑥(𝑡)|^𝜈+1+𝛽(𝑡)𝑦(𝑡)^𝜇𝑑𝑡<+∞,0<^+∞_−∞|𝑥(𝑡)|²+1+𝛽(𝑡)𝑦(𝑡)²𝑑𝑡<+∞, (22) 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 𝛼(𝑡) and 𝛽(𝑡) as follows:(A0)liminf_{|𝑡|→+∞}∫^𝑡₀𝛼(𝑠)𝑑𝑠>−∞, (A1)∫^+∞_−∞|𝛼(𝑠)|𝑑𝑠<+∞, (B0)𝛽(𝑡)≥(≢)0, ∀𝑡∈ℝ, (B1)𝛽(𝑡)>0, ∀𝑡∈ℝ, (B2)∫^+∞_−∞𝛽(𝜏)exp(−𝜇∫^𝜏₀𝛼(𝑠)𝑑𝑠)𝑑𝜏<+∞.

Denote 𝜁(𝑡)∶=^𝑡_−∞𝛽(𝜏)exp𝜇^𝑡_𝜏𝛼(𝑠)𝑑𝑠𝑑𝜏^𝜈/𝜇,𝜂(𝑡)∶=_𝑡^+∞𝛽(𝜏)exp−𝜇^𝜏_𝑡𝛼(𝑠)𝑑𝑠𝑑𝜏^𝜈/𝜇. (23)(24)

Theorem 2. Suppose that hypotheses (F), (G), (A0), (B0), and (B2) are satisfied. If system (7) has a solution (𝑥(𝑡),𝑦(𝑡)) satisfying 0<^+∞_−∞|𝑥(𝑡)|^𝜈+1+𝛽(𝑡)𝑦(𝑡)^𝜇𝑑𝑡<+∞, (25) then one has the following inequality: ^+∞_−∞𝜁(𝑡)𝜂(𝑡)𝜁(𝑡)(𝑡)𝛾+𝜂⁰(𝑡)𝑑𝑡≥1. (26)

Proof. Hypothesis (B2) implies that functions 𝜁(𝑡) and 𝜂(𝑡) are well defined on ℝ. Without loss of generality, we can assume that ^+∞_−∞𝜁(𝑡)𝜂(𝑡)𝜁(𝑡)(𝑡)𝛾+𝜂⁰(𝑡)𝑑𝑡<+∞. (27) It follows from (F), (25), and (B0) that liminf_{𝑡→−∞}|𝑥(𝑡)|=liminf_𝑡→+∞|𝑥(𝑡)|=0,liminf_{𝑡→−∞}𝑦(𝑡)=liminf_𝑡→+∞𝑦(𝑡)=0,_+∞−∞𝑓𝜏,𝑦(𝜏)𝑦(𝜏)𝑑𝜏≤^+∞_−∞𝛽(𝜏)𝑦(𝜏)^𝜇𝑑𝜏<+∞. (28)(29)(30) Set 𝐴(𝑡)={𝜏∈(−∞,𝑡]∶𝛽(𝜏)>0} for 𝑡∈ℝ, and then it follows from (F) that 𝛽(𝜏)^−𝜈/𝜇𝑓(𝜏,𝑧)^𝜈=|𝑧|^−𝜈𝛽(𝜏)^−𝜈/𝜇𝑧𝑓(𝜏,𝑧)^𝜈≤|𝑧|^−𝜈𝛽(𝜏)^−𝜈/𝜇𝛽(𝜏)|𝑧|^𝜇𝜈−1()()(𝑡)𝑧𝑓𝜏,𝑧=𝑧𝑓𝜏,𝑧,𝜏∈𝐴,𝑡∈ℝ,𝑧≠0. (31) Since 𝑓(𝜏,0)=0 for 𝜏∈ℝ, it follows that 𝛽(𝜏)^−𝜈/𝜇𝑓(𝜏,𝑧)^𝜈≤𝑧𝑓(𝜏,𝑧),𝜏∈𝐴(𝑡),𝑡∈ℝ. (32) Hence, from (F), (23), (24), (30), (32), and the Hölder inequality, one has ^𝑡_−∞𝑓𝜏,𝑦(𝜏)exp^𝑡_𝜏𝛼(𝑠)=𝑑𝑠𝑑𝜏_𝐴(𝑡)𝑓𝜏,𝑦(𝜏)exp^𝑡_𝜏𝛼(𝑠)≤𝑑𝑠𝑑𝜏_𝐴(𝑡)𝛽(𝜏)exp𝜇^𝑡_𝜏𝛼(𝑠)𝑑𝑠𝑑𝜏^1/𝜇×_𝐴(𝑡)𝛽(𝜏)^−𝜈/𝜇𝑓𝜏,𝑦(𝜏)^𝜈𝑑𝜏^1/𝜈≤_𝐴(𝑡)𝛽(𝜏)exp𝜇^𝑡_𝜏𝛼(𝑠)𝑑𝑠𝑑𝜏^1/𝜇×_𝐴(𝑡)𝑓𝜏,𝑦(𝜏)𝑦(𝜏)𝑑𝜏^1/𝜈≤^𝑡_−∞𝛽(𝜏)exp𝜇^𝑡_𝜏𝛼(𝑠)𝑑𝑠𝑑𝜏^1/𝜇×^𝑡_−∞𝑓𝜏,𝑦(𝜏)𝑦(𝜏)𝑑𝜏^1/𝜈=𝜁(𝑡)^1/𝜈𝑡_−∞𝑓𝜏,𝑦(𝜏)𝑦(𝜏)𝑑𝜏^1/𝜈<+∞,∀𝑡∈ℝ,_𝑡+∞𝑓𝜏,𝑦(𝜏)exp−^𝜏_𝑡𝛼(𝑠)≤𝑑𝑠𝑑𝜏_𝑡+∞𝛽(𝜏)exp−𝜇^𝜏_𝑡𝛼(𝑠)𝑑𝑠𝑑𝜏^1/𝜇×_𝑡^+∞𝑓𝜏,𝑦(𝜏)𝑦(𝜏)𝑑𝜏^1/𝜈=𝜂(𝑡)^1/𝜈_𝑡^+∞𝑓𝜏,𝑦(𝜏)𝑦(𝜏)𝑑𝜏^1/𝜈<+∞,∀𝑡∈ℝ. (33)(34) From (A0), (28), (33), (34), and the first equation of system (7), we have 𝑥(−∞)∶=lim_{𝑡→−∞}𝑥(𝑡)=0=lim_𝑡→+∞𝑥(𝑡)(),𝑥(𝑡)=∶=𝑥+∞_𝑡−∞𝑓𝜏,𝑦(𝜏)exp^𝑡_𝜏𝛼(𝑠)𝑥(𝑡)𝑑𝑠𝑑𝜏,∀𝑡∈ℝ,=−_𝑡^+∞𝑓𝜏,𝑦(𝜏)exp−^𝜏_𝑡𝛼(𝑠)𝑑𝑠𝑑𝜏,∀𝑡∈ℝ. (35)(36)(37) Combining (33) with (36), one has |𝑥(𝑡)|^𝜈=^𝑡_−∞𝑓𝜏,𝑦(𝜏)exp^𝑡_𝜏𝛼(𝑠)𝑑𝑠𝑑𝜏^𝜈≤𝜁(𝑡)^𝑡_−∞𝑓𝜏,𝑦(𝜏)𝑦(𝜏)𝑑𝜏,∀𝑡∈ℝ. (38) Similarly, it follows from (34) and (37) that |𝑥(𝑡)|^𝜈=_𝑡^+∞𝑓𝜏,𝑦(𝜏)exp−^𝑡_𝜏𝛼(𝑠)𝑑𝑠𝑑𝜏^𝜈≤𝜂(𝑡)_𝑡^+∞𝑓𝜏,𝑦(𝜏)𝑦(𝜏)𝑑𝜏,∀𝑡∈ℝ. (39) Hence, from (38) and (39), one has |𝑥(𝑡)|^𝜈≤𝜁(𝑡)𝜂(𝑡)𝜁(𝑡)+𝜂(𝑡)^+∞_−∞𝑓𝜏,𝑦(𝜏)𝑦(𝜏)𝑑𝜏,∀𝑡∈ℝ. (40) Now, it follows from (27), (30), and (40) that ^+∞_−∞𝛾₀(𝑡)|𝑥(𝑡)|^𝜈≤𝑑𝑡_+∞−∞𝜁(𝑡)𝜂(𝑡)𝜁(𝑡)(𝑡)𝛾+𝜂⁰(𝑡)𝑑𝑡^+∞_−∞𝑓𝑡,𝑦(𝑡)𝑦(𝑡)𝑑𝑡<+∞. (41) By (29), we can choose two sequences {𝑡_−𝑛}^∞_𝑛=1 and {𝑡_𝑛}^∞_𝑛=1 such that −∞<⋯<𝑡₋₃<𝑡₋₂<𝑡₋₁<𝑡₁<𝑡₂<𝑡₃<⋯<+∞,lim_𝑛→∞𝑡_−𝑛=−∞,lim_𝑛→∞𝑡_𝑛=+∞,lim_𝑛→∞𝑦𝑡_−𝑛=lim_𝑛→∞𝑦𝑡_𝑛=0. (42)(43) By (7), we obtain 𝑥(𝑡)𝑦(𝑡)^′=𝑓𝑡,𝑦(𝑡)𝑦(𝑡)−𝑔(𝑡,𝑥(𝑡))𝑥(𝑡). (44) Integrating the above equation from 𝑡_−𝑛 to 𝑡_𝑛, we have ^𝑡𝑛_𝑡−𝑛𝑔((𝑡𝑥(𝑡)=𝑡,𝑥))𝑑𝑡_{𝑡𝑛𝑡−𝑛}𝑓𝑡,𝑦(𝑡)𝑦(𝑡)𝑑𝑡+𝑥𝑡_−𝑛𝑦𝑡_−𝑛−𝑥𝑡_𝑛𝑦𝑡_𝑛,𝑛=1,2,3,…. (45) Let 𝑛→∞ in the above equation, and using (30), (35), and (43) we obtain lim_𝑛→∞^𝑡𝑛_𝑡−𝑛𝑔(𝑡,𝑥(𝑡))𝑥(𝑡)𝑑𝑡=^+∞_−∞𝑓𝑡,𝑦(𝑡)𝑦(𝑡)𝑑𝑡, (46) which, together with (41), implies that ^+∞_−∞𝛾₀(𝑡)|𝑥(𝑡)|^𝜈≤𝑑𝑡_+∞−∞𝜁(𝑡)𝜂(𝑡)𝜁(𝑡)(𝑡)𝛾+𝜂⁰(𝑡)𝑑𝑡^+∞_−∞𝑓(𝑡)𝑦(𝑡)=𝑡,𝑦𝑑𝑡_+∞−∞𝜁(𝑡)𝜂(𝑡)𝜁(𝑡)(𝑡)𝛾+𝜂⁰(𝑡)𝑑𝑡×lim_𝑛→∞^𝑡𝑛_𝑡−𝑛𝑔((𝑡𝑥(𝑡)≤𝑡,𝑥))𝑑𝑡_+∞−∞𝜁(𝑡)𝜂(𝑡)𝜁(𝑡)(𝑡)𝛾+𝜂⁰(𝑡)𝑑𝑡×lim_𝑛→∞^𝑡𝑛_𝑡−𝑛𝛾₀(𝑡)|𝑥(𝑡)|^𝜈=𝑑𝑡_+∞−∞𝜁(𝑡)𝜂(𝑡)𝜁(𝑡)(𝑡)𝛾+𝜂⁰(𝑡)𝑑𝑡^+∞_−∞𝛾₀(𝑡)|𝑥(𝑡)|^𝜈𝑑𝑡. (47)
We claim that ^+∞_−∞𝛾₀(𝑡)|𝑥(𝑡)|^𝜈𝑑𝑡>0. (48) If (48) is not true, then ^+∞_−∞𝛾₀(𝑡)|𝑥(𝑡)|^𝜈𝑑𝑡=0. (49) From (F), (G), (46), and (49), we have 0≤^+∞_−∞𝑓𝑡,𝑦(𝑡)𝑦(𝑡)𝑑𝑡=lim_𝑛→∞^𝑡𝑛_𝑡−𝑛𝑔((𝑡𝑥(𝑡)≤𝑡,𝑥))𝑑𝑡_+∞−∞𝛾₀(𝑡)|𝑥(𝑡)|^𝜈𝑑𝑡=0, (50) which, together with (F), implies that 𝑓𝑡,𝑦(𝑡)𝑦(𝑡)=0,a.e.𝑡∈ℝ. (51) Combining (36) with (51), we obtain that 𝑥(𝑡)≡0,∀𝑡∈ℝ, (52) which, together with (G) and the second equation of system (7), implies that 𝑦(𝑡)=𝑦(0)exp−^𝑡₀𝛼(𝑠)𝑑𝑠,∀𝑡∈ℝ. (53) From (F), (51), and the above, one has 𝑦(𝑡)≡0,∀𝑡∈ℝ. (54) 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 (𝑥(𝑡),𝑦(𝑡)) satisfying (25), then one has the following inequalities: ^+∞_−∞𝛾(𝑡)^𝑡_−∞𝛽(𝜏)𝑑𝜏_𝑡^+∞𝛽(𝜏)𝑑𝜏^𝜈/2𝜇−𝜈𝑑𝑡≥2exp2^+∞_−∞|𝛼(𝑠)|+𝜇⁻¹|𝜔(𝑠)|𝑑𝑠_+∞−∞,𝛽(𝑡)𝑑𝑡^1/𝜇+∞_−∞𝛾⁺(𝑡)𝑑𝑡^1/𝜈−1≥2exp2^+∞_−∞|𝛼(𝑠)|+𝜇⁻¹|𝜔(𝑠)|𝑑𝑠_+∞−∞,𝛾₀(𝑡)^𝑡_−∞𝛽(𝜏)𝑑𝜏_𝑡^+∞𝛽(𝜏)𝑑𝜏^𝜈/2𝜇−𝜈𝑑𝑡≥2exp2^+∞_−∞|𝛼(𝑠)|𝑑𝑠_+∞−∞,𝛽(𝑡)𝑑𝑡^1/𝜇+∞_−∞𝛾₀(𝑡)𝑑𝑡^1/𝜈−1≥2exp2^+∞_−∞|𝛼(𝑠)|𝑑𝑠, (55)(56)(57)(58) where 𝜔∈𝐿¹(ℝ) is an arbitrary function and 𝛽(𝑡)=𝛽(𝑡)exp^𝑡_𝑡0𝜔(𝑠),𝛾(𝑡)𝑑𝑠=𝛾₀(𝑡)−𝜈exp𝜇^𝑡_𝑡0𝜔(𝑠)𝑑𝑠 (59) for some 𝑡₀∈ℝ.

Proof. (A1), (B0), and (B2) imply that (A0) and ∫^+∞_−∞𝛽(𝑠)𝑑𝑠<+∞. Since 𝜁(𝑡)+𝜂(𝑡)≥2𝜁(𝑡)𝜂(𝑡)^1/2, (60) then it follows from (23), (24), (26), (56), and (57) that 1≤^+∞_−∞𝜁(𝑡)𝜂(𝑡)𝜁(𝑡)(𝑡)𝛾+𝜂⁰(𝑡)≤1𝑑𝑡2^+∞_−∞𝜁(𝑡)𝜂(𝑡)^1/2𝛾₀(𝑡)=1𝑑𝑡2^+∞_−∞𝛾₀(𝑡)^𝑡_−∞𝛽(𝜏)exp𝜇^𝑡_𝜏𝛼(𝑠)×𝑑𝑠𝑑𝜏_𝑡+∞𝛽(𝜏)×exp−𝜇^𝜏_𝑡𝛼(𝑠)𝑑𝑠𝑑𝜏^𝜈/2𝜇≤1𝑑𝑡2^+∞_−∞𝛾₀(𝑡)^𝑡_−∞𝛽(𝜏)𝑑𝜏_𝑡^+∞𝛽(𝜏)𝑑𝜏^𝜈/2𝜇𝜈𝑑𝑡×exp2^+∞_−∞|𝛼(𝑠≤1)|𝑑𝑠2^+∞_−∞𝛾(𝑡)^𝑡_−∞𝛽(𝜏)𝑑𝜏_𝑡^+∞𝛽(𝜏)𝑑𝜏^𝜈/2𝜇𝜈𝑑𝑡×exp2^+∞_−∞|𝛼(𝑠)|+𝜇⁻¹|𝜔(𝑠)|𝑑𝑠, (61) which implies that (55) holds. Note that ^𝑡_−∞𝛽(𝜏)𝑑𝜏_𝑡^+∞𝛽(𝜏)1𝑑𝜏≤4^+∞_−∞𝛽(𝜏)𝑑𝜏², (62) 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, |𝑥(𝑡)|^𝜈<𝜁(𝑡)𝜂(𝑡)𝜁(𝑡)+𝜂(𝑡)^+∞_−∞𝑓𝜏,𝑦(𝜏)𝑦(𝜏)𝑑𝜏,∀𝑡∈ℝ. (63) In fact, if (63) is not true, then there exists a 𝑡_∗∈ℝ such that 𝑥𝑡_∗^𝜈=𝜁𝑡_∗𝜂𝑡_∗𝜁𝑡_∗+𝜂𝑡_∗^+∞_−∞𝑓𝜏,𝑦(𝜏)𝑦(𝜏)𝑑𝜏. (64) Hence, from (38), (39), and (64), we obtain 𝑥𝑡_∗^𝜈=𝜁𝑡_∗^𝑡∗_−∞𝑓(𝜏)𝑦(𝜏)𝑥𝑡𝜏,𝑦𝑑𝜏,_∗𝜈=𝜂𝑡_∗𝑡^+∞_∗𝑓𝜏,𝑦(𝜏)𝑦(𝜏)𝑑𝜏. (65)(66) It follows from (23), (38), and (65) that ^𝑡∗_−∞𝑓𝜏,𝑦(𝜏)exp^𝑡∗_𝜏𝛼(𝑠)𝑑𝑠𝑑𝜏^𝜈=^𝑡∗_−∞𝛽(𝜏)exp𝜇^𝑡∗_𝜏𝛼(𝑠)𝑑𝑠𝑑𝜏^𝜈/𝜇×^𝑡∗_−∞𝑓𝜏,𝑦(𝜏)𝑦(𝜏)𝑑𝜏, (67) which, together with the Hölder inequality, implies that there exists a constant 𝑐₁ such that 𝑓𝜏,𝑦(𝜏)𝑦(𝜏)=𝑐₁𝛽(𝜏)exp𝜇^𝑡∗_𝜏𝛼(𝑠)𝑑𝑠,−∞<𝜏≤𝑡_∗. (68) Similarly, it follows from (24), (39), (66), and the Hölder inequality that there exists a constant 𝑐₂ such that 𝑓𝜏,𝑦(𝜏)𝑦(𝜏)=𝑐₂𝛽(𝜏)exp−𝜇^𝜏_𝑡∗𝛼(𝑠),𝑡𝑑𝑠_∗≤𝜏<+∞. (69) From (F), (68), and (69), one has that 𝑐₁,𝑐₂≥0. If 𝑐₁=𝑐₂=0, then 𝑓(𝜏,𝑦(𝜏))𝑦(𝜏)=0 for 𝜏∈ℝ; it follows from (36) that 𝑥(𝑡)=0 for 𝑡∈ℝ. Similar to the proof of (54), one has 𝑦(𝑡)=0 for 𝑡∈ℝ, which contradicts (25). If 𝑐₁+𝑐₂>0, then 𝑓(𝜏,𝑦(𝜏))𝑦(𝜏)>0 for 𝜏∈(−∞,𝑡_∗] or for 𝜏∈[𝑡_∗,+∞); it follows from (A0) and (36) that 𝑥(+∞)≠0, 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 (𝑥(𝑡),𝑦(𝑡)) satisfying (25), then one has the following inequality: ^+∞_−∞𝜁(𝑡)𝜂(𝑡)𝜁(𝑡)(𝑡)𝛾+𝜂⁰(𝑡)𝑑𝑡>1, (70) where 𝜁(𝑡) and 𝜂(𝑡) 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 (𝑥(𝑡),𝑦(𝑡)) satisfying (25), then ^+∞_−∞𝛾(𝑡)^𝑡_−∞𝛽(𝜏)𝑑𝜏_𝑡^+∞𝛽(𝜏)𝑑𝜏^𝜈/2𝜇−𝜈𝑑𝑡>2exp2^+∞_−∞|𝛼(𝑠)|+𝜇⁻¹|𝜔(𝑠)|𝑑𝑠_+∞−∞,𝛽(𝑡)𝑑𝑡^1/𝜇+∞_−∞𝛾(𝑡)𝑑𝑡^1/𝜈−1>2exp2^+∞_−∞|𝛼(𝑠)|+𝜇⁻¹|𝜔(𝑠)|𝑑𝑠_+∞−∞,𝛾₀(𝑡)^𝑡_−∞𝛽(𝜏)𝑑𝜏_𝑡^+∞𝛽(𝜏)𝑑𝜏^𝜈/2𝜇−𝜈𝑑𝑡>2exp2^+∞_−∞|𝛼(𝑠)|𝑑𝑠_+∞−∞,𝛽(𝑡)𝑑𝑡^1/𝜇+∞_−∞𝛾₀(𝑡)𝑑𝑡^1/𝜈−1>2exp2^+∞_−∞|𝛼(𝑠)|𝑑𝑠, (71) where 𝛽(𝑡) and 𝛾(𝑡) 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 𝑟>1, 𝑝(𝑡)>0 for 𝑡∈ℝ and ^+∞_−∞1𝑝(𝜏)^1/(𝑟−1)𝑑𝜏<+∞. (72) If (16) has a solution 𝑥(𝑡) satisfying 0<^+∞_−∞|𝑥(𝑡)|^𝑟+𝑝(𝑡)^1/(𝑟−1)1+𝑝(𝑡)𝑥^′(𝑡)^𝑟𝑑𝑡<+∞, (73) then ^+∞_−∞∫^𝑡_−∞𝑝(𝜏)^{−1/(𝑟−1)}𝑑𝜏^𝑟−1∫_𝑡^+∞𝑝(𝜏)^{−1/(𝑟−1)}𝑑𝜏^𝑟−1∫^𝑡_−∞𝑝(𝜏)^{−1/(𝑟−1)}𝑑𝜏^𝑟−1+∫_𝑡^+∞𝑝(𝜏)^{−1/(𝑟−1)}𝑑𝜏^𝑟−1×𝑞⁺(𝑡)𝑑𝑡>1,_+∞−∞𝑞⁺(𝑡)^𝑡_−∞𝑝(𝜏)^{−1/(𝑟−1)}×𝑑𝜏_𝑡+∞𝑝(𝜏)^{−1/(𝑟−1)}𝑑𝜏^(𝑟−1)/2𝑑𝑡>2. (74)

Applying Theorem 4 to the second-order nonlinear differential equation (17) (i.e., system (21)), where 𝛼(𝑡)=0, 𝑓(𝑡,𝑥)=𝜙⁻¹(𝑥/𝑝(𝑡)), and 𝑔(𝑡,𝑥)=ℎ(𝑡,𝑥), we have the following corollary.

Corollary 7. Suppose that 𝑟>1 and 𝑝(𝑡)>0 for 𝑡∈ℝ, and that (72) and the following hypothesis are satisfied:(H1) There exists a locally Lebesgue integrable nonnegative function 𝛾₀(𝑡) defined on ℝ such that 0≤𝑥𝜙⁻¹(𝑥)≤|𝑥|^{𝑟/(𝑟−1)},𝑥ℎ(𝑡,𝑥)≤𝛾₀(𝑡)|𝑥|^𝑟,∀(𝑡,𝑥)∈ℝ². (75)If (17) has a solution 𝑥(𝑡) satisfying (73), then ^+∞_−∞∫^𝑡_−∞𝑝(𝜏)^{−1/(𝑟−1)}𝑑𝜏^𝑟−1∫_𝑡^+∞𝑝(𝜏)^{−1/(𝑟−1)}𝑑𝜏^𝑟−1∫^𝑡_−∞𝑝(𝜏)^{−1/(𝑟−1)}𝑑𝜏^𝑟−1+∫_𝑡^+∞𝑝(𝜏)^{−1/(𝑟−1)}𝑑𝜏^𝑟−1×𝛾₀(𝑡)𝑑𝑡>1. (76)

3. Lyapunov-Type Inequalities for System (13)

When 𝜇=𝜈=2, assumption (B2) reduces to the following form:(B2^′)∫^+∞_−∞𝛽(𝜏)exp(−2∫^𝜏₀𝛼(𝑠)𝑑𝑠)𝑑𝜏<+∞.

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 (B2^′) are satisfied. If system (13) has a solution (𝑥(𝑡),𝑦(𝑡)) satisfying 0<^+∞_−∞|𝑥(𝑡)|²+1+𝛽(𝑡)𝑦(𝑡)²𝑑𝑡<+∞, (77) then ^+∞_−∞^𝑡_−∞𝛽(𝜏)exp2^𝑡_𝜏𝛼(𝑠)×𝑑𝑠𝑑𝜏_𝑡+∞𝛽(𝜏)exp−2^𝜏_𝑡𝛼(𝑠)×𝑑𝑠𝑑𝜏_𝑡−∞𝛽(𝜏)exp2^𝑡_𝜏𝛼(𝑠)+𝑑𝑠𝑑𝜏_𝑡+∞𝛽(𝜏)exp−2^𝜏_𝑡𝛼(𝑠)𝑑𝑠𝑑𝜏⁻¹×𝛾⁺(𝑡)𝑑𝑡≥1,_+∞−∞𝛾⁺(𝑡)^𝑡_−∞𝛽(𝜏)𝑑𝜏_𝑡^+∞𝛽(𝜏)𝑑𝜏^1/2−𝑑𝑡≥2exp^+∞_−∞|𝛼(𝑠)|𝑑𝑠. (78)

Corollary 9. Suppose that hypotheses (A0), (B1), and (B2^′) are satisfied. If system (13) has a solution (𝑥(𝑡),𝑦(𝑡)) satisfying (77), then^+∞_−∞^𝑡_−∞𝛽(𝜏)exp2^𝑡_𝜏𝛼(𝑠)×𝑑𝑠𝑑𝜏_𝑡+∞𝛽(𝜏)exp−2^𝜏_𝑡𝛼(𝑠)×𝑑𝑠𝑑𝜏_𝑡−∞𝛽(𝜏)exp2^𝑡_𝜏𝛼(𝑠)+𝑑𝑠𝑑𝜏_𝑡+∞𝛽(𝜏)exp−2^𝜏_𝑡𝛼(𝑠)𝑑𝑠𝑑𝜏⁻¹×𝛾⁺(𝑡)𝑑𝑡>1,_+∞−∞𝛾⁺(𝑡)^𝑡_−∞𝛽(𝜏)𝑑𝜏_𝑡^+∞𝛽(𝜏)𝑑𝜏^1/2−𝑑𝑡>2exp^+∞_−∞|𝛼(𝑠)|𝑑𝑠. (79)

Corollary 10. Suppose that 𝑝(𝑡)>0 for 𝑡∈ℝ and that ^+∞_−∞𝑑𝜏𝑝(𝜏)<+∞. (80) If (15) has a solution 𝑥(𝑡) satisfying 0<^+∞_−∞|𝑥(𝑡)|²+𝑝(𝑡)1+𝑝(𝑡)𝑥^′(𝑡)²𝑑𝑡<+∞, (81) then ^+∞_−∞𝑞⁺(𝑡)^𝑡_−∞𝑑𝜏𝑝(𝜏)_𝑡^+∞𝑑𝜏𝑝(𝜏)𝑑𝑡>^+∞_−∞𝑑𝜏𝑝(𝜏). (82)

Corollary 11. Suppose that 𝑝(𝑡)>0 for 𝑡∈ℝ  and that (80) and the following hypothesis are satisfied:(H2) There exists a locally Lebesgue integrable nonnegative function 𝛾₀(𝑡) defined on ℝ such that 0≤𝑥𝜙⁻¹(𝑥)≤|𝑥|²,𝑥ℎ(𝑡,𝑥)≤𝛾₀(𝑡)|𝑥|²,∀(𝑡,𝑥)∈ℝ². (83)If (17) has a solution 𝑥(𝑡) satisfying (81), then ^+∞_−∞𝛾₀(𝑡)^𝑡_−∞𝑑𝜏𝑝(𝜏)_𝑡^+∞𝑑𝜏𝑝(𝜏)𝑑𝑡>^+∞_−∞𝑑𝜏𝑝(𝜏). (84)

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 ^+∞_−∞𝜁(𝑡)𝜂(𝑡)𝜁(𝑡)(𝑡)𝛾+𝜂⁰(𝑡)𝑑𝑡<1,_+∞−∞𝛾₀(𝑡)^𝑡_−∞𝛽(𝜏)𝑑𝜏_𝑡^+∞𝛽(𝜏)𝑑𝜏^𝜈/2𝜇−𝜈𝑑𝑡<2exp2^+∞_−∞|𝛼(𝑠)|𝑑𝑠_+∞−∞,𝛽(𝑡)𝑑𝑡^1/𝜇+∞_−∞𝛾₀(𝑡)𝑑𝑡^1/𝜈−1<2exp2^+∞_−∞|𝛼(𝑠)|𝑑𝑠 (85)(86)(87) holds, then system (7) has no solution (𝑥(𝑡),𝑦(𝑡)) satisfying 0<^+∞_−∞|𝑥(𝑡)|^𝜈+1+𝛽(𝑡)𝑦(𝑡)^𝜇𝑑𝑡<+∞. (88)

Corollary 13. Suppose that hypotheses (F), (G), (A0), (B1), and (B2) are satisfied. If one of the conditions ^+∞_−∞𝜁(𝑡)𝜂(𝑡)𝜁(𝑡)(𝑡)𝛾+𝜂⁰(𝑡)𝑑𝑡≤1,_+∞−∞𝛾₀(𝑡)^𝑡_−∞𝛽(𝜏)𝑑𝜏_𝑡^+∞𝛽(𝜏)𝑑𝜏^𝜈/2𝜇−𝜈𝑑𝑡≤2exp2^+∞_−∞|𝛼(𝑠)|𝑑𝑠_+∞−∞,𝛽(𝑡)𝑑𝑡^1/𝜇+∞_−∞𝛾₀(𝑡)𝑑𝑡^1/𝜈−1≤2exp2^+∞_−∞|𝛼(𝑠)|𝑑𝑠 (89) holds, then system (7) has no solution (𝑥(𝑡),𝑦(𝑡)) satisfying (86).

Corollary 14. Suppose that hypotheses (A0), (B0), and (B2^′) are satisfied. If ^+∞_−∞^𝑡_−∞𝛽(𝜏)exp2^𝑡_𝜏𝛼(𝑠)×𝑑𝑠𝑑𝜏_𝑡+∞𝛽(𝜏)exp−2^𝜏_𝑡𝛼(𝑠)×𝑑𝑠𝑑𝜏_𝑡−∞𝛽(𝜏)exp2^𝑡_𝜏𝛼(𝑠)+𝑑𝑠𝑑𝜏_𝑡+∞𝛽(𝜏)exp−2^𝜏_𝑡𝛼(𝑠)𝑑𝑠𝑑𝜏⁻¹×𝛾₀(𝑡)𝑑𝑡<1 (90) or ^+∞_−∞𝛾⁺(𝑡)^𝑡_−∞𝛽(𝜏)𝑑𝜏_𝑡^+∞𝛽(𝜏)𝑑𝜏^1/2𝑑𝑡<2exp−^+∞_−∞|𝛼(𝑠)|𝑑𝑠 (91) holds, then system (13) has no solution (𝑥(𝑡),𝑦(𝑡)) satisfying 0<^+∞_−∞|𝑥(𝑡)|²+1+𝛽(𝑡)𝑦(𝑡)²𝑑𝑡<+∞. (92)

Corollary 15. Suppose that hypotheses (A0), (B1), and (B2^′) are satisfied. If ^+∞_−∞^𝑡_−∞𝛽(𝜏)exp2^𝑡_𝜏𝛼(𝑠)×𝑑𝑠𝑑𝜏_𝑡+∞𝛽(𝜏)exp−2^𝜏_𝑡𝛼(𝑠)×𝑑𝑠𝑑𝜏_𝑡−∞𝛽(𝜏)exp2^𝑡_𝜏𝛼(𝑠)+𝑑𝑠𝑑𝜏_𝑡+∞𝛽(𝜏)exp−2^𝜏_𝑡𝛼(𝑠)𝑑𝑠𝑑𝜏⁻¹×𝛾⁺(𝑡)𝑑𝑡≤1 (93) or ^+∞_−∞𝛾⁺(𝑡)^𝑡_−∞𝛽(𝜏)𝑑𝜏_𝑡^+∞𝛽(𝜏)𝑑𝜏^1/2𝑑𝑡≤2exp−^+∞_−∞|𝛼(𝑠)|𝑑𝑠 (94) holds, then system (13) has no solution (𝑥(𝑡),𝑦(𝑡)) satisfying (92).

Corollary 16. Suppose that 𝑝(𝑡)>0 for 𝑡∈ℝ and that (80) holds. If ^+∞_−∞𝑞⁺(𝑡)^𝑡_−∞𝑑𝜏𝑝(𝜏)_𝑡^+∞𝑑𝜏𝑝(𝜏)𝑑𝑡≤^+∞_−∞𝑑𝜏𝑝(𝜏), (95) then (15) has no solution 𝑥(𝑡) satisfying (81).

Corollary 17. Suppose that 𝑝(𝑡)>0 for 𝑡∈ℝ and that (80) and (H2) are satisfied. If ^+∞_−∞𝛾₀(𝑡)^𝑡_−∞𝑑𝜏𝑝(𝜏)_𝑡^+∞𝑑𝜏𝑝(𝜏)𝑑𝑡≤^+∞_−∞𝑑𝜏𝑝(𝜏), (96) then (17) has no solution 𝑥(𝑡) satisfying (81).

Example 18. Consider the second-order nonlinear differential equation 1+𝑡²𝑥^′(𝑡)^′+𝑞(𝑡)𝑥(𝑡)1+sin²𝑥(𝑡)=0, (97) where 𝑞(𝑡) is locally Lebesgue integrable real-valued function defined on ℝ. In view of Corollary 16, if ^+∞_−∞𝜋²4−(arctan𝑡)²𝑞⁺(𝑡)𝜋𝑑𝑡≤2, (98) then (97) has no solution 𝑥(𝑡) satisfying 0<^+∞_−∞|𝑥(𝑡)|²+1+𝑡²²𝑥^′(𝑡)²𝑑𝑡<+∞. (99)

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).