Homoclinic Solutions for a Class of the Second-Order Impulsive Hamiltonian Systems

Abstract

This paper is concerned with the existence of homoclinic solutions for a class of the second order impulsive Hamiltonian systems. By employing the Mountain Pass Theorem, we demonstrate that the limit of a 2𝑘𝑇-periodic approximation solution is a homoclinic solution of our problem.

1. Introduction and Main Results

In this paper, we consider the second-order impulsive differential equation 𝑞(̈𝑡)+𝑉_𝑞𝑡,𝑞=𝑓(𝑡),𝑡≠𝑡_𝑗𝑞̇𝑡,𝑡∈ℝ,−Δ_𝑗=𝑔_𝑗𝑞𝑡_𝑗,𝑗∈ℤ, (1)(2) where 𝑞∈ℝ^𝑁, 𝑓∈(ℝ,ℝ^𝑁), 𝑉_𝑞(𝑡,𝑞)=grad_𝑞𝑉(𝑡,𝑞), 𝑔_𝑗(𝑞)=grad_𝑞𝐺_𝑗(𝑞), 𝐺_𝑗∈(ℝ^𝑁,ℝ^𝑁) for each 𝑗∈ℤ, and the operator Δ is defined as Δ𝑞(̇𝑡_𝑗)=𝑞(̇𝑡⁺_𝑗)−𝑞(̇𝑡⁻_𝑗), where 𝑞(̇𝑡⁺_𝑗)(𝑞(̇𝑡⁻_𝑗)) denotes the right-hand (left-hand) limit of 𝑞̇ at 𝑡_𝑗. There exist an 𝑚∈ℕ and a 𝑇>0 such that 0=𝑡₀<𝑡₁<⋯<𝑡_𝑚=𝑇, 𝑡_𝑗+𝑚=𝑡_𝑗+𝑇, and 𝑔_𝑗+𝑚=𝑔_𝑗, 𝑗∈ℤ. 𝑉∶ℝ×ℝ^𝑁→ℝ satisfies(V1)𝑉(𝑡,𝑞)=−𝐾(𝑡,𝑞)+𝑊(𝑡,𝑞), 𝐾,𝑊∈𝐶¹(ℝ×ℝ^𝑁,ℝ), and is 𝑇-periodic in its first variable.

We are mainly concerned with the existence of homoclinic solutions of system (1) and (2). A function 𝑞(𝑡)∈𝐶(ℝ,ℝ^𝑁) is said to be a (classical) solution of (1) and (2) if 𝑞(𝑡) satisfies (1) and (2). A (classical) solution 𝑞(𝑡) of (1) and (2) is a homoclinic solution and if 𝑞(𝑡)→0 as 𝑡→±∞ and 𝑞(̇𝑡^±)→0 as 𝑡→±∞.

When Δ𝑞(̇𝑡_𝑗)≡0, 𝑓(𝑡)=0, and 𝑉(𝑡,𝑞)=(1/2)(𝐿(𝑡)𝑞,𝑞)+𝑊(𝑡,𝑞), system (1) and (2) reduces to Hamiltonian system 𝑞(̈𝑡)+𝑉_𝑞𝑡,𝑞=0,𝑡∈ℝ. (3) Rabinowitz [1] studied the existence of nontrivial homoclinic solutions of it.

When Δ𝑞(̇𝑡_𝑗)≡0 and 𝑉(𝑡,𝑞) satisfied (V1), system (1), (2) reduces to Hamiltonian system 𝑞(̈𝑡)+𝑉_𝑞𝑡,𝑞=𝑓(𝑡),𝑡∈ℝ. (4) Izydorek and Janczewska [2] studied the existence of homoclinic solutions of it.

Some classical tools such as some fixed point theorems in cones, topological degree theory, the upper and lower solutions method combined with monotone iterative technique, and variational methods [3–20] have been widely used to get solutions of impulsive differential equations. However, the existence of homoclinic solutions for the impulsive systems is paid little attention. It is well known that the homoclinic orbit rupture phenomenon can lead to chaos, which has been interesting to the mathematicians in recent years [21–26]. In the literature, Coti-Zelati et al. [27] used dual variational methods, and Lions [28] and Hofer and Wysocki [29] employed concentration compactness method, Ekeland's variational principle, that they established the existence of homoclinic solutions of the first-order Hamiltonian systems. Rabinowitz [1] and Izydorek and Janczewska [2] obtained homoclinic solutions of a class of second order Hamiltonian systems as a limit of its periodic solutions.

In recent paper [18], Zhang and Li studied the existence of homoclinic solutions of an impulsive Hamiltonian system 𝑞^′′(𝑡)+𝑉_𝑞(𝑡,𝑞)=𝑓(𝑡), 𝑡≠𝑡_𝑗, 𝑡∈ℝ, Δ𝑞^′(𝑡_𝑗)=𝑔_𝑗(𝑞(𝑡_𝑗)), 𝑗∈ℤ as a limit of its periodic solutions. In detail, they obtained the following theorem.

Theorem A (see [18]). Assume that 𝑓∈𝐶(ℝ,ℝ^ℕ)⋂𝐿²(ℝ,ℝ^ℕ), 𝑔_𝑗 is continuous and 𝑚-periodic in 𝑗, and 𝑉, 𝑔_𝑗 satisfy the following conditions: (H1)𝑉∶ℝ×ℝ^𝑁→ℝ is continuous differentiable 𝑇-periodic, and there exist positive constants 𝑏₁,𝑏₂>0 such that 𝑏₁𝑞²≤−𝑉𝑡,𝑞≤𝑏₂𝑞²,∀𝑡,𝑞∈[0,𝑇]×ℝ^ℕ; (5)(H2)−𝑉(𝑡,𝑞)≤−𝑉_𝑞(𝑡,𝑞)𝑞≤−2𝑉(𝑡,𝑞) for all (𝑡,𝑞)∈[0,𝑇]×ℝ^ℕ; (H3)lim_|𝑞|→0(𝑔_𝑗(𝑞)/|𝑞|)=0 for 𝑗=1,2,…,𝑚; (H4) there exists a 𝜇>2 such that 𝑔_𝑗𝑞𝑞≤𝜇𝐺_𝑗𝑞<0,𝑓𝑜𝑟𝑞∈ℝ^ℕ⧵{0},𝑗=1,2,…,𝑚, (6) then the Hamiltonian 𝑞(̈𝑡)+𝑉_𝑞(𝑡,𝑞)=𝑓(𝑡), 𝑡≠𝑡_𝑗, 𝑡∈ℝ, Δ𝑞^′(𝑡_𝑗)=𝑔_𝑗(𝑞(𝑡_𝑗)), 𝑗∈ℤ, possesses at least one nonzero homoclinic solution.

Motivated by papers [1, 2, 18], in this paper, we synthesize their methods to study the existence of homoclinic solutions of systems (1), (2). In detail, firstly, we introduce the following sequence equations: 𝑞^′′(𝑡)+𝑉_𝑞𝑡,𝑞=𝑓_𝑘(𝑡),𝑡≠𝑡_𝑗,𝑡∈ℝ, (7) where for each 𝑘∈ℕ, 𝑓_𝑘∶ℝ→ℝ^𝑁 is a 2𝑘𝑇-periodic extension of the restriction of 𝑓 to the interval [−𝑘𝑇,𝑘𝑇]. Secondly, we study periodic solutions of (2) and (7) by converting the problem to the existence of critical points of some variational structure. finally, we find the homoclinic solutions of (1) and (2) as the limit of the periodic solutions of (2) and (7).

Part of the difficulty in treating (1) and (2) is subjected to the impulsive perturbation which destroys continuities of the velocity and when we apply the Mountain Theorem to prove our main result, we need the constant 𝜌, 𝛼 appearing in the theorem to be independent of 𝑘.

Our result is the following theorem.

Theorem 1. Assume that 𝑉 satisfies (V1), and 𝐾, 𝑊, and 𝑓 satisfy the following:(K1) there exist constants 𝑎₁>0 and 𝛾∈(1,2] such that for all (𝑡,𝑞)∈[0,𝑇]×ℝ^𝑁𝐾(𝑡,0)=0,𝐾𝑡,𝑞≥𝑎₁𝑞^𝛾; (8)(K2) there exists 𝜃∈(1,2] such that 𝐾𝑡,𝑞≤𝐾_𝑞𝑡,𝑞𝑞≤𝜃𝐾𝑡,𝑞,∀𝑡,𝑞∈[0,𝑇]×ℝ^𝑁; (9) (W1)𝑊(𝑡,0)=0, 𝑊_𝑞(𝑡,𝑞)=𝑜(|𝑞|), as |𝑞|→0 uniformly for 𝑡∈[0,𝑇];(W2) there exist constants 𝑟>2 and 𝑏₁>0 such that 𝑊𝑡,𝑞≤𝑏₁𝑞^𝑟,∀𝑡,𝑞∈[0,𝑇]×ℝ^𝑁; (10) (W3) there exist 𝜇>1, 𝑎₂>0, and 𝛽(𝑡)∈𝐿¹(ℝ,ℝ⁺) such that 𝜇>𝑟−𝛾 and 𝑊_𝑞𝑡,𝑞𝑞−2𝑊𝑡,𝑞≥𝑎₂𝑞^𝜇−𝛽(𝑡),∀𝑡,𝑞∈[0,𝑇]×ℝ^𝑁; (11) (W4)𝑊(𝑡,𝑞)/|𝑞|²→+∞ as |𝑞|→+∞ uniformly in 𝑡∈[0,𝑇];(G1)𝐺_𝑗(0)=0, 𝑔_𝑗(𝑞)=𝑜(|𝑞|), as |𝑞|→0, 𝑗=1,2,…,𝑚;(G2) there exists 𝑏₂>0 such that 𝐺_𝑗(𝑞)≤𝑏₂|𝑞|^𝑟, 𝑞∈ℝ^𝑁, 𝑗=1,2,…,𝑚;(G3)𝑔_𝑗(𝑞)𝑞−2𝐺_𝑗(𝑞)≥0, 𝑞∈ℝ^𝑁⧵{0}, 𝑗=1,2,…,𝑚;(F1)𝑓∈𝐶(ℝ,ℝ^𝑁)∩𝐿²(ℝ,ℝ^𝑁)∩𝐿^{𝜇/(𝜇−1)}(ℝ,ℝ^𝑁), (∫^+∞_−∞|𝑓(𝑡)|²𝑑𝑡)^1/2<min{1/2,𝑎₁−𝑏₁−𝑀}(1/𝐶), where 𝑀=sup{𝐺_𝑗(𝑞)∶𝑗=1,2,…,𝑚,|𝑞|=1}, 𝑎₁>𝑏₁+𝑀, and 𝐶 is a constant of (17). Then the system (1) and (2) possesses at least one nonzero homoclinic solution.

The rest of this paper is organized as follows. In Section 2 we present some preliminary results. Our main result's proofs are given in Section 3.

2. Preliminaries

Let 𝐻_2𝑘𝑇=𝑞∶ℝ→ℝ^𝑁∣𝑞,𝑞̇∈𝐿²([−𝑘𝑇,𝑘𝑇]),ℝ^𝑁,𝑢(𝑡)=𝑢(𝑡+2𝑘𝑇),𝑡∈ℝ. (12) Then 𝐻_2𝑘𝑇 is a Hilbert space with the norm defined by 𝑞_{𝐻2𝑘𝑇}=^𝑘𝑇_−𝑘𝑇𝑞(̇𝑡)²+𝑞²𝑑𝑡^1/2,𝑞∈𝐻_2𝑘𝑇. (13) For the norm in 𝐿^∞_2𝑘𝑇(ℝ,ℝ), which denotes a space of 2𝑘𝑇 periodic essentially bounded measurable functions from ℝ into ℝ^𝑁, ‖𝑞‖_{𝐿∞2𝑘𝑇}=esssup{|𝑞(𝑡)|∶𝑡∈[−𝑘𝑇,𝑘𝑇]}. Next we set Ω_𝑘={−𝑘𝑚+1,−𝑘𝑚+2,…,0,1,2,…,𝑘𝑚−1,𝑘𝑚} and define a functional 𝜑_𝑘 as 𝜑_𝑘𝑞=12𝜂²_𝑘𝑞−^𝑘𝑇_−𝑘𝑇𝑊𝑑𝑡−_{𝑗∈Ω𝑘}𝐺_𝑗𝑞𝑡_𝑗+^𝑘𝑇_−𝑘𝑇𝑓_𝑘𝑞𝑑𝑡,𝑞∈𝐻_2𝑘𝑇, (14) where 𝜂_𝑘𝑞=^𝑘𝑇_−𝑘𝑇𝑞(̇𝑡)²+2𝐾(𝑡,𝑞(𝑡))𝑑𝑡^1/2. (15) Note that 𝜑_𝑘 is Fréchet differentiable at any 𝑞∈𝐻_2𝑘𝑇, and for any 𝑝∈𝐻_2𝑘𝑇, we have 𝜑^′_𝑘𝑞𝑝=lim_ℎ→0𝜑_𝑘𝑞+ℎ𝑝−𝜑_𝑘𝑞ℎ=^𝑘𝑇_−𝑘𝑇𝑞(̇𝑡)𝑝(̇𝑡)−𝑉_𝑞𝑡,𝑞𝑝+𝑓_𝑘𝑝−𝑑𝑡_{𝑗∈Ω𝑘}𝑔_𝑗𝑞𝑡_𝑗𝑝𝑡_𝑗. (16) It is clear that critical points of the functional 𝜑_𝑘 are classical 2𝑘𝑇-periodic solutions of system (2) and (7).

Lemma 2 (see [2]). There is a positive constant 𝐶 such that for each 𝑘∈𝑁 and 𝑞∈𝐻_2𝑘𝑇 the following inequality holds: 𝑞_{𝐿∞2𝑘𝑇}≤𝐶𝑞_{𝐻2𝑘𝑇},∀𝑞∈𝐻_2𝑘𝑇. (17)

Lemma 3. Set 𝑚₁=inf{𝐺_𝑗(𝑞)∶|𝑞|=1,𝑗=1,2,…,𝑚}; then for every 𝜁∈ℝ⧵{0} and 𝑞∈𝐻_2𝑘𝑇⧵{0}, we have _{𝑗∈Ω𝑘}𝐺_𝑗𝜁𝑞𝑡_𝑗≥𝑚₁𝜁²_{𝑗∈Ω𝑘}𝑞(𝑡_𝑗)²−2𝑘𝑚𝑚₁. (18)

Proof. Set 𝐴_𝑘={𝑗∈Ω_𝑘∶|𝜁𝑞(𝑡_𝑗)|≤1}, 𝐵_𝑘={𝑗∈Ω_𝑘∶|𝜁𝑞(𝑡_𝑗)|≥1}, and 𝜑(𝑠)=𝑠²𝐺_𝑗(𝑞/𝑠), 𝑗∈Ω_𝑘, 𝑠>0. By (G3), we have 𝜑^′(𝑠)=𝑠2𝐺_𝑗𝑞𝑠−𝑔_𝑗𝑞𝑠𝑞𝑠≤0. (19) So we have 𝐺_𝑗(𝑞)≥|𝑞|²𝐺_𝑗(𝑞/|𝑞|), |𝑞|≥1. If 𝐵_𝑘 is empty, we have 𝑞(𝑡_𝑗)=0, 𝑗∈Ω_𝑘, which implies ∑_{𝑗∈Ω𝑘}𝐺_𝑗(𝜁𝑞(𝑡_𝑗))=0≥−2𝑘𝑚𝑚₁. Therefore, Without loss of generality, we can assume that 𝐵_𝑘 is nonempty, and we have _{𝑗∈Ω𝑘}𝐺_𝑗𝜁𝑞𝑡_𝑗≥_{𝑗∈𝐵𝑘}𝐺_𝑗𝜁𝑞𝑡_𝑗≥_{𝑗∈𝐵𝑘}𝐺_𝑗𝜁𝑞𝑡_𝑗𝜁𝑞𝑡_𝑗𝜁𝑞(𝑡_𝑗)²≥𝑚_{1𝑗∈𝐵𝑘}𝜁𝑞(𝑡_𝑗)²≥_{𝑗∈Ω𝑘}𝜁𝑞𝑡_𝑗²−_{𝑗∈𝐴𝑘}𝜁𝑞𝑡_𝑗²𝑚₁≥𝑚₁𝜁²_{𝑗∈Ω𝑘}𝑞𝑡_𝑗²−2𝑘𝑚𝑚₁. (20)

Lemma 4 (see [30]). Let 𝐸 be a real Banach space and let 𝜑∶𝐸→ℝ be a 𝐶¹-smooth functional satisfying the Palais-Smale condition and 𝜑(0)=0. If 𝜑 satisfies the following conditions:(i)there exist constants 𝜌,𝛼>0 such that 𝜑∣_𝜕𝐵𝜌≥𝛼,(ii)there exists 𝑒∈𝐸⧵𝐵_𝜌, such that 𝜑(𝑒)≤0, then 𝜑 possesses a critical value 𝑐≥𝛼 given by 𝑐=inf_𝑔∈Γmax_𝑠∈[0,1]𝜑𝑔(𝑠), (21) where Γ=𝑔∈𝐶([0,1],𝐸)∶𝑔(0)=0,𝑔(1)=𝑒. (22)

Lemma 5 (see [2]). Let 𝑞∶ℝ→ℝ^𝑁 be a continuous mapping such that 𝑞̇∈𝐿²_𝑙𝑜𝑐(ℝ,ℝ^𝑁). For every 𝑡∈ℝ the following inequality holds: 𝑞(𝑡)≤√2^𝑡+1/2_𝑡−1/2𝑞(𝑠)²+𝑞(̇𝑠)²𝑑𝑠^1/2. (23)

3. Proof of Theorem 1

We have divided the proof of Theorem 1 into a sequence of lemmas.

Lemma 6. Assume that (V1), (K1), (K2), (W1), (W2), (W3), (W4), (G1), (G2), (G3), and (F1) are satisfied; system (2), (7) possesses a 2𝑘𝑇-periodic solution.

Proof. It is clear that 𝜑_𝑘(0)=0. It is well known that Lemma 4 holds true with the (C) condition replacing the usual (PS) condition. We say the functional 𝜑_𝑘 satisfies the (C) condition; that is, for every sequence {𝑞_𝑛}⊂𝐻_2𝑘𝑇, {𝑞_𝑛} has a convergent subsequence if {𝜑_𝑘(𝑞_𝑛)} is bounded and lim_𝑛→∞(1+‖𝑞‖_{𝐻2𝑘𝑇})𝜑^′_𝑘(𝑞_𝑛)=0.
Step  1. Pick {𝑞_𝑛}⊂𝐻_2𝑘𝑇 such that {𝜑_𝑘(𝑞_𝑛)} is bounded and lim_𝑛→∞(1+‖𝑞‖_{𝐻2𝑘𝑇})𝜑^′_𝑘(𝑞_𝑛)=0 then there exists a constant 𝐶₁>0 such that 𝐶₁≥2𝜑_𝑘𝑞_𝑛−𝜑^′_𝑘𝑞_𝑛𝑞_𝑛=^𝑘𝑇_−𝑘𝑇2𝐾𝑡,𝑞_𝑛−𝐾_𝑞𝑡,𝑞𝑑𝑡−^𝑘𝑇_−𝑘𝑇2𝑊𝑡,𝑞_𝑛−𝑊_𝑞𝑡,𝑞_𝑛𝑞_𝑛𝑑𝑡−_{𝑗∈Ω𝑘}2𝐺_𝑗𝑞_𝑛𝑡_𝑗−𝑔_𝑗𝑞_𝑛𝑡_𝑗𝑞_𝑛𝑡_𝑗+^𝑘𝑇_−𝑘𝑇𝑓_𝑘𝑞_𝑛𝑑𝑡≥𝛼₁^𝑘𝑇_−𝑘𝑇𝑞_𝑛^𝜇𝑑𝑡−^+∞_−∞𝛽(𝑡)−𝑑𝑡_+∞−∞𝑓^𝜇′𝑑𝑡^{1/𝜇′𝑘𝑇}_−𝑘𝑇𝑞_𝑛^𝜇𝑑𝑡^1/𝜇, (24) where 𝜇^′>1 and 1/𝜇^′+1/𝜇=1. (24) implies (∫^𝑘𝑇_−𝑘𝑇|𝑞_𝑛|^𝜇𝑑𝑡)^1/𝜇 is bounded; that is, there exists a constant 𝐶₂>0 such that ^𝑘𝑇_−𝑘𝑇𝑞_𝑛^𝜇𝑑𝑡^1/𝜇<𝐶₂. (25) From (W2), (G2), (17), and (25), we have 12𝜂²_𝑘𝑞_𝑛=𝜑_𝑘𝑞_𝑛+^𝑘𝑇_−𝑘𝑇𝑊𝑡,𝑞_𝑛−𝑑𝑡_{𝑘𝑇−𝑘𝑇}𝑓_𝑘𝑞_𝑛𝑑𝑡+_{𝑗∈Ω𝑘}𝐺_𝑗𝑞_𝑛𝑡_𝑗≤𝐶₃+𝑏₁𝐶𝑞_{𝑛𝐻2𝑘𝑇}^{𝑟−𝜇𝑘𝑇}_−𝑘𝑇𝑞_𝑛^𝜇+𝑑𝑡_+∞−∞𝑓(𝑡)^𝜇′𝑑𝑡^1/𝜇′𝐶₂+𝑏₂𝐶𝑞_{𝑛𝐻2𝑘𝑇}^𝑟−𝜇_{𝑗∈Ω𝑘}𝑞_𝑛(𝑡_𝑗)^𝜇≤𝐶₃+𝑏₁+𝑏₂𝐶^𝜇₂𝐶^𝑟−𝜇𝑞_{𝑛𝐻2𝑘𝑇}^𝑟−𝜇+^+∞_−∞𝑓(𝑡)^𝜇′𝑑𝑡^1/𝜇′𝐶₂. (26) On the other hand, it follows from (K1) and (17) that 12𝜂²_𝑘𝑞_𝑛≥12^𝑘𝑇_−𝑘𝑇𝑞̇_𝑛²+2𝑎₁𝑞_𝑛^𝛾≥1𝑑𝑡2^𝑘𝑇_−𝑘𝑇𝑞̇_𝑛²𝑑𝑡+𝑎₁𝐶𝑞_{𝑛𝐻2𝑘𝑇}^{𝛾−2𝑘𝑇}_−𝑘𝑇𝑞_𝑛²1𝑑𝑡≥min2𝑞_𝑛²_{𝐻2𝑘𝑇},𝑎₁𝐶^𝛾−2𝑞_𝑛^𝛾_{𝐻2𝑘𝑇}. (27) Combining (26) and (27), we obtain 1min2𝑞_𝑛²_{𝐻2𝑘𝑇},𝑎₁𝐶^𝛾−2𝑞_𝑛^𝛾_{𝐻2𝑘𝑇}≤𝐶₃+𝑏₁+𝑏₂𝐶^𝜇₂𝐶^𝑟−𝜇𝑞_{𝑛𝐻2𝑘𝑇}^𝑟−𝜇+^+∞_−∞𝑓(𝑡)^{𝜇′1/𝜇′}𝐶₂. (28) Since 𝑟−𝜇<𝛾≤2, it follows that ‖𝑞_𝑛‖_{𝐻2𝑘𝑇} is bounded. In a similar way to [21, Proposition B35], we can prove that {𝑞_𝑛} has a convergent subsequence. So, the functional 𝜑_𝑘 satisfies the (C) condition.
Step  2. We show that the functional 𝜑_𝑘 satisfies the assumption (i) of Lemma 4. Set 𝜑(𝑠)=𝑠²𝐺_𝑗(𝑞/𝑠), 𝑗=1,2,…,𝑚, 𝑠>0. By (G3), we have 𝜑^′(𝑠)=𝑠2𝐺_𝑗𝑞𝑠−𝑔_𝑗𝑞𝑠𝑞𝑠≤0. (29) Hence when 0<‖𝑞‖_{𝐿∞2𝑘𝑇}≤1, we have _{𝑗∈Ω𝑘}𝐺_𝑗𝑞𝑡_𝑗≤_{𝑗∈Ω𝑘}𝐺_𝑗𝑞𝑡_𝑗𝑞𝑡_𝑗𝑞(𝑡_𝑗)²≤𝑀_{𝑗∈Ω𝑘}𝑞(𝑡_𝑗)²≤𝑀𝑞²_{𝐻2𝑘𝑇}. (30) From (K1), (W2), and (30), we have 𝜑_𝑘𝑞=12𝜂²_𝑘𝑞−^𝑘𝑇_−𝑘𝑇𝑊𝑡,𝑞𝑑𝑡−_{𝑗∈Ω𝑘}𝐺_𝑗𝑞𝑡_𝑗+^𝑘𝑇_−𝑘𝑇𝑓_𝑘1𝑞𝑑𝑡≥min2,𝑎₁−𝑏₁−𝑀𝑞²_{𝐻2𝑘𝑇}−^+∞_−∞𝑓(𝑡)²𝑑𝑡^1/2𝑞_{𝐻2𝑘𝑇}. (31) Set 1𝜌=𝐶,1𝛼=min2,𝑎₁−𝑏₁1−𝑀𝐶²−^+∞_−∞𝑓(𝑡)²𝑑𝑡^1/21𝐶>0. (32) Let ‖𝑞‖_{𝐻2𝑘𝑇}=𝜌; then 0<‖𝑞‖_{𝐿∞2𝑘𝑇}≤1 and (31) gives 𝜑_𝑘(𝑞)≥𝛼>0.
Step  3. We show that the functional 𝜑_𝑘 satisfies assumption (ii) of Lemma 4.
In order to verify (ii), we choose 𝜁∈ℝ, 𝑄∈𝐻_2𝑇⧵{0} such that 𝑄(±𝑇)=0 and 𝑀₂=max_{|𝑡|≤𝑇,|𝑞|≤1}𝐾(𝑡,𝑞). Set ℎ(𝑠)=𝑠^−𝜃𝐾(𝑡,𝑠𝑞). By (K2), we have ℎ^′(𝑠)=(𝐾_𝑞(𝑡,𝑠𝑞)𝑠𝑞−𝜃𝐾(𝑡,𝑠𝑞))/𝑠^𝜃+1≤0. So we have 𝐾≥𝑞𝑡,𝑞^𝜃𝐾𝑞𝑡,𝑞,𝑞[],≤𝑞≤1,𝑡∈0,𝑇_𝜃𝐾𝑞𝑡,𝑞,𝑞≥1,𝑡∈[0,𝑇]. (33)
Define [],𝑄=[]𝑄,𝑡∈−𝑇,𝑇0,𝑡∈−𝑘𝑇,𝑘𝑇[].−𝑇,𝑇 (34)
Take 𝐴>(1+2𝑀₂‖𝑄‖²_𝐻2𝑇)/2∫^𝑇_−𝑇𝑄(𝑡)𝑑𝑡. By (W4), there exists 𝐵>0 such that 𝑊𝑡,𝑞≥𝐴𝑞²−𝐵,𝑡∈[0,𝑇]×ℝ^𝑁. (35)
By (33), (34), (35), and Lemma 3, we have 𝜑_𝑘𝜁𝑄=12𝜂²₁(𝜁𝑄)−^𝑇_−𝑇𝑊(𝑡,𝜁𝑄)𝑑𝑡−_𝑗∈Ω1𝐺_𝑗𝜁𝑄𝑡_𝑗+𝜁^𝑇_−𝑇𝑓₁≤1𝑄𝑑𝑡21+2𝑀₂𝜁²‖𝑄‖²_𝐻2𝑇−𝐴𝜁^2𝑇_−𝑇|𝑄(𝑡)|²𝑀𝑑𝑡+2₂𝑇+𝑇𝐵+𝑚𝑚₂−𝑚₁𝜁²_𝑗∈Ω1𝑄(𝑡_𝑗)²+𝜁^𝑇_−𝑇𝑓₁²𝑑𝑡^1/2𝑇_−𝑇|𝑄|²𝑑𝑡^1/2≤𝜁²1+2𝑀₂2‖𝑄‖²_𝐻2𝑇−𝐴^𝑇_−𝑇|𝑄(𝑡)|²𝑀𝑑𝑡+2₂𝑇+𝑇𝐵+𝑚𝑚₁+𝜁^𝑇_−𝑇𝑓₁²𝑑𝑡^1/2𝑇_−𝑇|𝑄|²𝑑𝑡^1/2. (36) Clearly, 𝜑_𝑘(𝜁𝑞)→−∞ as |𝜁|→+∞, so (ii) holds. By Lemma 4, 𝜑_𝑘 possesses a critical value 𝑐_𝑘≥𝛼>0. Let 𝑞_𝑘 denote the corresponding critical point of 𝜑_𝑘 on 𝐻_2𝑘𝑇; that is, 𝜑_𝑘𝑞_𝑘=𝑐_𝑘,𝜑^′_𝑘𝑞_𝑘=0. (37) Hence the system (2), (7) possesses a nontrivial 2𝑘𝑇-periodic solution 𝑞_𝑘.

Lemma 7. Let {𝑞_𝑘} be the sequence given by (37). Then there exist a subsequence {𝑞_𝑗,𝑗} of {𝑞_𝑘} and a function 𝑞₀∈𝑊^1,2_𝑙𝑜𝑐∩𝐿^∞_𝑙𝑜𝑐(ℝ,ℝ^ℕ) such that {𝑞_𝑗,𝑗} converges to 𝑞₀ weakly in 𝑊^1,2_𝑙𝑜𝑐 and strongly in 𝐿^∞_𝑙𝑜𝑐(ℝ,ℝ^ℕ).

Proof. We assert that there is a constant 𝑀₃>0 independent of 𝑘 such that 𝑞_{𝑘𝐻2𝑘𝑇}≤𝑀₃. (38) Let 𝑒₁∈𝐻_2𝑇⧵{0} such that 𝑒₁(±𝑇)=0, 𝑒₁(𝑡_𝑘)≠0 for some 𝑡_𝑘∈(−𝑇,𝑇), and 𝜑₁(𝑒₁)≤0. Define 𝑒_𝑘=𝑒₁,|𝑡||𝑡|≤𝑇,0,𝑇≤≤𝑘𝑇. (39) We extend 𝑒_𝑘 in the way of 2𝑘𝑇-periodic to ℝ. For simplicity, we also note it again by 𝑒_𝑘. It is clear that 𝑒_𝑘∈𝐻_2𝑘𝑇 and 𝜑_𝑘(𝑒_𝑘)=𝜑₁(𝑒₁)≤0. Define 𝑔_𝑘∶[0,1]→𝐻_2𝑘𝑇 by 𝑔_𝑘(𝑠)=𝑠𝑒_𝑘 for 𝑠∈[0,1]. Then, we have 𝑐_𝑘≤max_𝑠∈[0,1]𝜑_𝑘𝑔_𝑘(𝑠)=max_𝑠∈[0,1]𝜑₁𝑔₁(𝑠)≡𝑐₀, (40) independently of 𝑘. The rest detailed argument is similar to the proof of Step 1 in Lemma 6 and we thus omit it here.
Hence, {𝑞_𝑘} is a bounded sequence in 𝑊^1,2((−𝑇,𝑇),ℝ^𝑁) and we may pick a subsequence {𝑞_1,𝑘} such that {𝑞_1,𝑘} converges weakly in 𝑊^1,2((−𝑇,𝑇),ℝ^𝑁) and strongly in 𝐿^∞((−𝑇,𝑇),ℝ^𝑁). Next {𝑞_1,𝑘} is a bounded sequence in 𝑊^1,2((−2𝑇,2𝑇),ℝ^𝑁), so we may pick a subsequence {𝑞_2,𝑘} such that {𝑞_2,𝑘} converges weakly in 𝑊^1,2((−2𝑇,2𝑇),ℝ^𝑁) and strongly in 𝐿^∞((−2𝑇,2𝑇),ℝ^𝑁). We can repeat this process and obtain, for any positive integer 𝑛, a sequence {𝑞_𝑛,𝑘} which converges weakly in 𝑊^1,2((−𝑛𝑇,𝑛𝑇),ℝ^𝑁) and strongly in 𝐿^∞((−𝑛𝑇,𝑛𝑇),ℝ^𝑁), and 𝑞_𝑘⊃𝑞_1,𝑘⊃𝑞_2,𝑘⊃⋯⊃𝑞_𝑛,𝑘⊃⋯. (41) Therefore, for any positive integer 𝑛, the sequence {𝑞_𝑘,𝑘} converges weakly in 𝑊^1,2((−𝑛𝑇,𝑛𝑇),ℝ^𝑁) and strongly in 𝐿^∞((−𝑛𝑇,𝑛𝑇),ℝ^𝑁). Hence there exists a function 𝑞∈𝑊^1,2_loc(ℝ,ℝ^𝑁)∩𝐿^∞_loc(ℝ,ℝ^𝑁) such that the sequence {𝑞_𝑘,𝑘} converges weakly in 𝑞∈𝑊^1,2_loc(ℝ,ℝ^𝑁) and strongly in 𝐿^∞_loc(ℝ,ℝ^𝑁).

Lemma 8. The function 𝑞₀ determined by Lemma 7 is a non-zero homoclinic solution of the system (1), (2).

Proof. The proof will be divided into four steps.
Firstly, we show that 𝑞₀ is a solution of the system (1), (2). Here, for simplicity, we denote {𝑞_𝑘,𝑘} by {𝑞_𝑘}. For any given interval (𝑎,𝑏)⊂(−𝑘𝑇,𝑘𝑇) and any 𝑝∈𝑊₀^1,2((𝑎,𝑏),ℝ^𝑁), define 𝑝₁=(),()⧵(),𝑝,𝑡∈𝑎,𝑏0,𝑡∈−𝑘𝑇,𝑘𝑇𝑎,𝑏 (42) so for any 𝑝∈𝑊₀^1,2((𝑎,𝑏),ℝ^𝑁), we have 0=𝜑^′_𝑘𝑞_𝑘𝑝₁=^𝑏_𝑎𝑞̇_𝑘𝑝̇−𝑉_𝑞𝑡,𝑞_𝑘𝑝+𝑓_𝑘𝑝𝑑𝑡+_{𝑡𝑗∈(𝑎,𝑏)}𝑔_𝑗𝑞_𝑘𝑡_𝑗𝑝𝑡_𝑗. (43) Therefore, one has 𝑞̇^𝑏_𝑎0𝑝̇−𝑉_𝑞𝑡,𝑞₀𝑝+𝑓𝑝𝑑𝑡+_{𝑡𝑗∈()𝑎,𝑏}𝑔_𝑗𝑞₀𝑡_𝑗𝑝𝑡_𝑗=lim_𝑘→+∞^𝑏_𝑎𝑞̇_𝑘𝑝̇−𝑉_𝑞𝑡,𝑞_𝑘𝑝+𝑓_𝑘𝑝+𝑑𝑡_{𝑡𝑗∈()𝑎,𝑏}𝑔_𝑗𝑞_𝑘𝑡_𝑗𝑝𝑡_𝑗=0. (44) The remained detailer argument is similar to the proof of Lemma 2.5 in [13] and we thus omit it here, so 𝑞₀(𝑡) is a solution of system (1) and (2).
Secondly we show that 𝑞₀(𝑡)→0, as 𝑡→±∞. {𝑞_𝑘} is weak continuity, so it is weak lower semicontinuity. One has ^+∞_−∞𝑞₀²+𝑞̇₀²𝑑𝑡=lim_𝑘→+∞^𝑘𝑇_−𝑘𝑇𝑞₀²+𝑞̇₀²𝑑𝑡≤lim_𝑘→+∞lim_𝑗→+∞inf^+𝑘𝑇_−𝑘𝑇𝑞_𝑗²+𝑞̇_𝑗²𝑑𝑡≤𝑀²₃, (45) and so _{|𝑡|≥𝑟}𝑞₀²+𝑞̇₀²𝑑𝑡→0,as𝑟⟶+∞. (46) By (23) and (46), we obtain 𝑞₀(𝑡)→0, as 𝑡→±∞.
Thirdly, we prove that 𝑞̇₀(𝑡^±)→0 as 𝑡→±∞. We have proved 𝑞₀(𝑡) is a solution of system (1) and (2), so we have 𝑞̈^𝑡𝑗_{𝑡𝑗−10}(𝑠)²=𝑑𝑠_{𝑡𝑗𝑡𝑗−1}−𝑉_𝑞(𝑠,𝑞₀(𝑠))+𝑓(𝑠)²𝑑𝑠≤2^𝑡𝑗_𝑡𝑗−1𝑉_𝑞𝑠,𝑞₀(𝑠)²+𝑓(𝑠)²𝑑𝑠. (47) By (V1), (K1), and (W1), one has 𝑉_𝑞(𝑡,0)=−𝐾_𝑞(𝑡,0)+𝑊_𝑞(𝑡,0)=0. Hence ∫^𝑡𝑗_𝑡𝑗−1|𝑞̈₀(𝑠)|²𝑑𝑠→0 as 𝑗→±∞. By (23), one has 𝑞̇₀(𝑡)²≤2^𝑡𝑗_𝑡𝑗−1𝑞̇₀(𝑠)²+𝑞̈₀(𝑠)²𝑑𝑠≤2^𝑡𝑗_𝑡𝑗−1𝑞₀(𝑠)²+𝑞̇₀(𝑠)²𝑑𝑠+2^𝑡𝑗_𝑡𝑗−1𝑞̈₀(𝑠)²𝑑𝑠,𝑡∈𝑡_𝑗−1,𝑡_𝑗. (48) Therefore one has 𝑞̇₀(𝑡^±)→0 as 𝑡→±∞.
Finally, we show 𝑞₀≠0 when 𝑓≡0. Since 0=𝑡₀<𝑡₁<⋯<𝑡_𝑚=𝑇, 𝑡_𝑗+𝑚=𝑡_𝑗+𝑇, 𝑗∈ℤ, we can let 𝛿=min_𝑗∈ℤ{𝑡_𝑗−𝑡_𝑗−1} and 𝛿=max_𝑗∈ℤ{𝑡_𝑗−𝑡_𝑗−1}. By Hölder inequality and 𝑞_𝑘(𝑡_𝑗)=𝑞_𝑘(𝜏)+∫^𝑡𝑗_𝜏𝑞̇_𝑘(𝑠)𝑑𝑠, 𝜏∈[𝑡_𝑗−1,𝑡_𝑗], 𝑗∈Ω_𝑘, we have _{𝑗∈Ω𝑘}𝑞_𝑘𝑡_𝑗²≤1𝛿_{𝑗∈Ω𝑘}^𝑡𝑗_𝑡𝑗−1𝑞_𝑘𝑡_𝑗²=1𝑑𝜏𝛿_{𝑗∈Ω𝑘}^𝑡𝑗_𝑡𝑗−1𝑞_𝑘(𝜏)+^𝑡𝑗_𝜏𝑞̇_𝑘(𝑠)𝑑𝑠²≤2𝑑𝜏𝛿_{𝑗∈Ω𝑘}^𝑡𝑗_𝑡𝑗−1𝑞_𝑘(𝜏)²+^𝑡𝑗_𝜏𝑞̇_𝑘(𝑠)𝑑𝑠²≤2𝑑𝜏𝛿^𝑘𝑇_−𝑘𝑇𝑞_𝑘(𝜏)²+2𝑑𝜏𝛿_{𝑗∈Ω𝑘}^𝑡𝑗_𝑡𝑗−1𝑡_𝑗−𝜏^𝑡𝑗_𝜏𝑞̇_𝑘(𝑠)²≤2𝑑𝑠𝑑𝜏𝛿^𝑘𝑇_−𝑘𝑇𝑞_𝑘(𝜏)²2𝑑𝜏+𝛿²𝛿^𝑘𝑇_−𝑘𝑇𝑞̇_𝑘(𝜏)²𝑑𝜏≤max1,𝛿²2𝛿𝑞_𝑘²_{𝐻2𝑘𝑇}. (49) Let 𝑀₄=max{1,𝛿²}(2/𝛿)>0, which is a constant independent of 𝑘.
It is clearly that 𝑞_𝑘(𝑡+𝑗𝑇), 𝑗∈ℤ, is a 2𝑘𝑇 periodic solution of (2), (7). So we can assume the maximum of 𝑞_𝑘 occurs in [−𝑇,𝑇]. Now we assume 𝑞₀=0, so there is ‖𝑞_𝑘‖_{𝐿∞2𝑘𝑇}=max_{𝑡∈[−𝑇,𝑇]}|𝑞_𝑘|→0 as 𝑘→+∞; therefore there exists integer 𝑁₁>0 such that ‖𝑞_𝑘‖_{𝐿∞2𝑘𝑇}<1. Combining (G1), there exists an integer 𝑁₂>0 such that when 𝑘>𝑁₂, one has 𝑔_𝑗𝑞_𝑘𝑡_𝑗𝑞_𝑘𝑡_𝑗≤12𝑀₄min1,𝑎₁,𝑗∈ℤ. (50) By (49) and (50), when 𝑘>𝑁₂, one has _{𝑗∈Ω𝑘}𝑔_𝑗𝑞_𝑘𝑡_𝑗𝑞_𝑘𝑡_𝑗≤_{𝑗∈Ω𝑘}𝑔_𝑗𝑞_𝑘𝑡_𝑗𝑞_𝑘𝑡_𝑗𝑞_𝑘(𝑡_𝑗)²≤12min1,𝑎₁𝑞_𝑘²_{𝐻2𝑘𝑇}. (51) Define a function 𝑌∶ℝ⁺→ℝ⁺ by 𝑌(𝑠)=0,𝑠=0,max_{𝑡∈[0,𝑇],|𝜉|≤𝑠}𝜉𝑊_𝑞(𝑡,𝜉)𝜉²,𝑠>0. (52) It is clear that 𝑌∈𝐶(ℝ⁺,ℝ⁺) and is monotone nondecreasing, so we have 𝑞_𝑘𝑊_𝑞𝑡,𝑞_𝑘(𝑡)𝑞_𝑘(𝑡)²≤𝑌𝑞_{𝑘𝐿∞[−𝑘𝑇,𝑘𝑇]},𝑡∈[−𝑘𝑇,𝑘𝑇]. (53) Hence we have ^𝑘𝑇_−𝑘𝑇𝑞_𝑘𝑊_𝑞𝑡,𝑞_𝑘(𝑡)𝑞𝑑𝑡≤𝑌_{𝑘𝐿∞[−𝑘𝑇,𝑘𝑇]}^𝑘𝑇_−𝑘𝑇𝑞_𝑘(𝑡)²𝑞𝑑𝑡≤𝑌_{𝑘𝐿∞[−𝑘𝑇,𝑘𝑇]}𝑞_𝑘²_{𝐻2𝑘𝑇}. (54)
Since 𝑓≡0, 𝑞_𝑘 is a solution of the system (1) and (2), so when 𝑘>max{𝑁₁,𝑁₂}, we have ^𝑘𝑇_−𝑘𝑇𝑞_𝑘𝑊_𝑞𝑡,𝑞_𝑘(𝑡)𝑑𝑡+_{𝑗∈Ω𝑘}𝑔_𝑗𝑞_𝑘𝑡_𝑗𝑞_𝑘𝑡_𝑗≥min1,𝑎₁𝑞_𝑘²_{𝐻2𝑘𝑇}. (55) Combining (51), (54), and (55) we have 𝑌𝑞_{𝑘𝐿∞[−𝑘𝑇,𝑘𝑇]}𝑞_𝑘²_{𝐻2𝑘𝑇}+12min1,𝑎₁𝑞_𝑘²_{𝐻2𝑘𝑇}≥min1,𝑎₁𝑞_𝑘²_{𝐻2𝑘𝑇}. (56) Hence, we have 𝑌𝑞_{𝑘𝐿∞[−𝑘𝑇,𝑘𝑇]}≥12min1,𝑎₁>0. (57) By the property of the function 𝑌, there exists a constant 𝑀₅>0 such that ||𝑞_𝑘||_𝐿^∞_{[−𝑘𝑇,𝑘𝑇]}≥𝑀₅>0. This is a contradiction. Hence the system (1), (2) has a nontrivial homoclinic solution even if 𝑓≡0.

Next, we give an example to illustrate our main result.

Example 9. Let 𝐾𝑡,𝑞=(4+sin𝑡)𝑞^4/3,𝑊𝑡,𝑞=(2+cos𝑡)𝑞²𝑞,𝐺ln1+_𝑗𝑞𝑡_𝑗=𝑡sin^𝑗2𝑞^𝑟𝑡_𝑗,𝑔_𝑗𝑞𝑡_𝑗𝑡=𝑟sin^𝑗2𝑞^𝑟−1𝑡_𝑗,𝑓(𝑡)=0, (58) where 𝑟>2, 𝑡_𝑗=2𝜋𝑗/𝑚, 𝑗∈ℕ. It is easy to verify that 𝐾, 𝑊, 𝐺_𝑗, 𝑔_𝑗, and 𝑓 satisfy conditions (V1), (K1), (K2), (W1), (W2), (W3), (W4), (G1), (G2), (G3), and (F1). So, system (1), (2) with 𝐾, 𝑊, 𝐺_𝑗, 𝑔_𝑗, and 𝑓 as in (58) has a nontrivial homoclinic solution.

Acknowledgments

The authors are very grateful to the referees for their very helpful comments and suggestions, which greatly improved the presentation of this paper. This work is supported by the Hunan Provincial Natural Science Foundation of China (no. 11JJ3012) and Major Project of Science Research Fund of Education Department in Hunan (no. 11A095).