Existence and Multiplicity of Fast Homoclinic Solutions for a Class of Damped Vibration Problems with Impulsive Effects

Abstract

This paper is concerned with the existence and multiplicity of fast homoclinic solutions for a class of damped vibration problems with impulsive effects. Some new results are obtained under more relaxed conditions by using Mountain Pass Theorem and Symmetric Mountain Pass Theorem in critical point theory. The results obtained in this paper generalize and improve some existing works in the literature.

1. Introduction

Consider fast homoclinic solutions of the following problem: 𝑢^′′(𝑡)+𝑞(𝑡)𝑢^′(𝑡)(𝑡)𝑢(𝑡)((𝑡𝑡−𝑎+∇𝑉𝑡,𝑢))=0,a.e.𝑡∈_𝑗,𝑡_𝑗+1,𝑗∈ℤ,Δ𝑢^′𝑡_𝑗=𝑢^′𝑡⁺_𝑗−𝑢^′𝑡⁻_𝑗=𝐼𝑢𝑡_𝑗,𝑗∈ℤ, (1) where 𝑉∶ℝ×ℝ→ℝ is of class 𝐶¹, ∇𝑉(𝑡,𝑢(𝑡))=𝜕𝑉(𝑡,𝑢(𝑡))/𝜕𝑢, 𝐼∶ℝ→ℝ, 𝑎∈𝐶(ℝ,(0,+∞)), 𝑎(𝑡)→+∞ as |𝑡|→+∞, ℤ denotes the sets of integers, and 𝑡_𝑗(𝑗∈ℤ) are impulsive points. Moreover, there exist a positive integer 𝑝 and a positive constant 𝑇 such that 0<𝑡₀<𝑡₁<⋯<𝑡_𝑝−1<𝑇, 𝑡_{𝑙+𝑘𝑝}=𝑡_𝑙+𝑘𝑇, ∀𝑘∈ℤ, 𝑙=0,1,…,𝑝−1, 𝑢^′(𝑡⁺_𝑗)=lim_ℎ→0⁺𝑢^′(𝑡_𝑗+ℎ) and 𝑢^′(𝑡⁻_𝑗)=lim_ℎ→0⁻𝑢^′(𝑡_𝑗−ℎ) represent the right and left limits of 𝑢^′(𝑡) at 𝑡=𝑡_𝑗, respectively, 𝑞∶ℝ→ℝ is a continuous function, and 𝑄(𝑡)=∫^𝑠₀𝑞(𝑠)𝑑𝑠 with lim_{|𝑡|→+∞}𝑄(𝑡)=+∞. (2) When 𝐼≡0, problem (1) becomes the following damped vibration problem: 𝑢^′′(𝑡)+𝑞(𝑡)𝑢^′(𝑡)(𝑡)𝑢(𝑡)((𝑡−𝑎+∇𝑉𝑡,𝑢))=0,a.e.𝑡∈ℝ. (3)Chen et al. [1] investigated problem (3) and obtained some results of fast homoclinic solutions by critical point theory.

When 𝑞(𝑡)≡0 and 𝑎(𝑡)≡0, problem (1) becomes the following problem: 𝑢^′′(𝑡)+∇𝑉(𝑡,𝑢(𝑡))=0,a.e.𝑡∈𝑡_𝑗,𝑡_𝑗+1,𝑗∈ℤ,Δ𝑢^′𝑡_𝑗=𝑢^′𝑡⁺_𝑗−𝑢^′𝑡⁻_𝑗=𝐼𝑢𝑡_𝑗,𝑗∈ℤ. (4) Fang and Duan [2] obtained the following result of homoclinic solutions for (4) by employing Mountain Pass Theorem, a weak convergence argument, and a weak version of Lieb’s methods.

Theorem A (see [2]). Assume that the following conditions hold:(V1)there exists a positive number 𝑇 such that ()(),𝑉()(),∀()∇𝑉𝑡+𝑇,𝑥=∇𝑉𝑡,𝑥𝑡+𝑇,𝑥=𝑉𝑡,𝑥𝑡,𝑥∈ℝ²; (5)(V2)lim_𝑥→0(∇𝑉(𝑡,𝑥)/𝑥)=0 uniformly for 𝑡∈ℝ;(V3)there exists a constant 𝜇>0 such that 𝜇𝑉(𝑡,𝑥)≤(∇𝑉(𝑡,𝑥),𝑥),∀(𝑡,𝑥)∈ℝ×ℝ⧵{0}; (6)(V4)there exist constants 𝑏₀>0 and 𝑏>0 such that 𝑉(𝑡,𝑥)≥𝑏₀|𝑥|^𝜇|𝑥|𝑉()|𝑥|,∀𝑡∈ℝ,≥1;𝑡,𝑥≤𝑏^𝜇,∀𝑡∈ℝ,|𝑥|≤1; (7) (I)there exists a constant 𝑏₁ with 0<𝑏₁<(𝜇−2)/(𝜇+2)𝑝𝑇 such that |𝐼(𝑥)|≤𝑏₁|𝑥|,2^𝑥₀𝐼(𝑡)𝑑𝑡−𝐼(𝑥)𝑥≤0. (8)Then, problem (4) possesses a nontrivial weak homoclinic orbit.

For 𝐼≠0, problem (1) involves impulsive effects. Impulsive differential equations are suitable for the mathematical simulation of evolutionary processes in which the parameters undergo relatively long periods of smooth variation followed by a short-term rapid change (that is jumps) in their values. Impulsive differential equations are often investigated in various fields of science and technology, for example, many biological phenomena involving thresholds, bursting rhythm models in medicine and biology, optimal control models in economics, pharmacokinetics, and frequency modulated systems, and so on. For more details of impulsive differential equations, we refer the readers to the books [3, 4].

In recent years, some researchers have paid attention to the existence and multiplicity of solutions for impulsive differential equations via variational methods. See, for example, [1, 5–17] and references therein. However, there are few papers [2, 18, 19] concerning homoclinic solutions of impulsive differential equations by variational methods. So, it is a novel method to employ variational methods to investigate the existence of homoclinic solutions for impulsive differential equations.

Motivated by the above papers, we will establish some new results for (1). In order to introduce the concept of fast homoclinic solutions for (1), we first state some properties of the weighted Sobolev space 𝐸 on which certain variational functional associated with (1) is defined and the fast homoclinic solutions are the critical points of the certain functional.

Let 𝐸=𝑢∈𝐻^1,2(ℝ,ℝ)∣_ℝ𝑒^𝑄(𝑡)𝑢^′(𝑡)²+𝑎(𝑡)|𝑢(𝑡)|²𝑢𝑡𝑑𝑡<+∞,_𝑗^+∞_𝑗=−∞∈𝑙²,𝑢(±∞)=0,𝑢(𝑘𝑇)=0,𝑘∈ℤ, (9) where 𝑄(𝑡) is defined in (2) and 𝑙² denotes the space of sequences whose second powers are summable on ℤ; that is, _𝑗∈ℤ𝑎_𝑗²<+∞,∀𝑎=𝑎_𝑗^+∞_𝑗=−∞∈𝑙². (10) The space 𝑙² is equipped with the following norm: ‖𝑎‖_𝑙²=_𝑗∈ℤ𝑎_𝑗^21/2. (11) For 𝑢,𝑣∈𝐸, let ⟨𝑢,𝑣⟩=_ℝ𝑒^𝑄(𝑡)𝑢^′(𝑡),𝑣^′(𝑡)+(𝑎(𝑡)𝑢(𝑡),𝑣(𝑡))𝑑𝑡. (12) Similar to [2], it is easy to check that 𝐸 is a Hilbert space with the norm given by ‖𝑢‖=_ℝ𝑒^𝑄(𝑡)𝑢^′(𝑡)²+𝑎(𝑡)|𝑢(𝑡)|²𝑑𝑡^1/2. (13) It is obvious that 𝐸⊂𝐿²𝑒^𝑄(𝑡) (14) with the embedding being continuous. Here, 𝐿^𝑝(𝑒^𝑄(𝑡))(2≤𝑝<+∞) denotes the Banach spaces of functions on ℝ with values in ℝ under the norm ‖𝑢‖_𝑝=_ℝ𝑒^𝑄(𝑡)|𝑢(𝑡)|^𝑝𝑑𝑡^1/𝑝. (15) Similar to [1], we have the following definition of fast homoclinic solutions.

Definition 1. If (2) holds, a solution of (1) is called a fast homoclinic solution if 𝑢∈𝐸.

Here and in subsequence, (,) and |⋅| denote the inner product and norm in ℝ, respectively. 𝐶_𝑖(𝑖=0,1,…) denote different positive constants. Now, we state our main results.

Theorem 2. Suppose that 𝑞, 𝑎, 𝐼, and 𝑉 satisfy (2), (V1), and the following conditions: (A)𝑎∈𝐶(ℝ,(0,+∞)) and 𝑎(𝑡)→+∞ as |𝑡|→+∞,(V2)′𝑉(𝑡,𝑥)=𝑉₁(𝑡,𝑥)−𝑉₂(𝑡,𝑥), 𝑉₁,𝑉₂∈𝐶¹(ℝ×ℝ,ℝ), and there exists a constant 𝑅>0 such that |(𝑥(]∪[),∇𝑉𝑡,𝑥)|=𝑜(||)as𝑥⟶0uniformlyin𝑡∈−∞,−𝑅𝑅,+∞ (16)(V3)′there is a constant 𝜇>2 such that 0<𝜇𝑉₁(𝑡,𝑥)≤∇𝑉₁(𝑡,𝑥),𝑥,∀(𝑡,𝑥)∈ℝ×ℝ⧵{0}, (17)(V5)𝑉₂(𝑡,0)=0 and there exists a constant 𝜚∈(2,𝜇) such that 𝑉₂(𝑡,𝑥)≥0,∇𝑉₂(𝑡,𝑥),𝑥≤𝜚𝑉₂(),∀()𝑡,𝑥𝑡,𝑥∈ℝ×ℝ, (18)(I)′𝐼∈𝐶(ℝ,ℝ) and there exists a constant 𝑐 with 0<𝑐<(𝜚−2)𝑒₀/(𝜚+2)𝑝𝑇 such that |𝐼(𝑥)|≤𝑐|𝑥|,𝐼(𝑥)−𝐼𝑦,𝑥−𝑦≥0,∀𝑥,𝑦∈ℝ, (19) where 𝑒₀=𝑒^{min{𝑄(𝑡)∶𝑡∈ℝ}}.
Then, problem (1) has at least one nontrivial fast homoclinic solution.

Theorem 3. Suppose that 𝑞, 𝑎, 𝑉, and 𝐼 satisfy (2), (V1), (A), (V3)′, (I)′, and the following conditions:(V2)′′𝑉(𝑡,𝑥)=𝑉₁(𝑡,𝑥)−𝑉₂(𝑡,𝑥), 𝑉₁,𝑉₂∈𝐶¹(ℝ×ℝ,ℝ), and |∇𝑉(𝑡,𝑥)|=𝑜(|𝑥|)as𝑥⟶0uniformlyin𝑡∈ℝ, (20)(V5)′𝑉₂(𝑡,0)=0 and there exists a constant 𝜚∈(2,𝜇) such that ∇𝑉₂(𝑡,𝑥),𝑥≤𝜚𝑉₂(𝑡,𝑥),∀(𝑡,𝑥)∈ℝ×ℝ. (21) Then, problem (1) has at least one nontrivial fast homoclinic solution.

Theorem 4. Suppose that 𝑞, 𝑎, 𝑉, and 𝐼 satisfy (2), (V1), (A), (V2)′, (V3)′, (V5), (I)′, and the following condition: (V6)𝐼(−𝑡)=−𝐼(𝑡),𝑉(𝑡,−𝑥)=𝑉(𝑡,𝑥),∀(𝑡,𝑥)∈ℝ×ℝ. Then, problem (1) has an unbounded sequence of fast homoclinic solutions.

Theorem 5. Suppose that 𝑞, 𝑎, 𝑉, and 𝐼 satisfy (2), (V1), (A), (V2)′′, (V3)′, (V5)′, (I)′, and (V6). Then, problem (1) has an unbounded sequence of fast homoclinic solutions.

The rest of this paper is organized as follows. In Section 2, some preliminaries are presented. In Section 3, we give the proofs of our results. In Section 4, some examples are given to illustrate our results.

2. Preliminaries

Let 𝐸 and ‖⋅‖ be given in Section 1. By a similar argument in [2, 20], we have the following important lemma.

Lemma 6. For any 𝑢∈𝐸, ‖𝑢‖_∞≤12𝑒₀√𝑎₀‖𝑢‖=12𝑒₀√𝑎₀^ℝ𝑒^𝑄(𝑠)𝑢^′(𝑠)²+𝑎(𝑠)|𝑢(𝑠)|²𝑑𝑠^1/2,|𝑢(𝑡)|≤_𝑡^+∞𝑒^{−𝑄(𝑠)}[𝑎(𝑠)]^−1/2×𝑒^𝑄(𝑠)𝑢^′(𝑠)²+𝑎(𝑠)|𝑢(𝑠)|²𝑑𝑠^1/2,|𝑢(𝑡)|≤^𝑡_−∞𝑒^{−𝑄(𝑠)}[𝑎(𝑠)]^−1/2×𝑒^𝑄(𝑠)𝑢^′(𝑠)²+𝑎(𝑠)|𝑢(𝑠)|²𝑑𝑠^1/2,_{∞𝑗=−∞}𝑢𝑡_𝑗²≤𝑝𝑇𝑒₀𝑢^′2₂, (22)(23)(24)(25) where ‖𝑢‖_∞=esssup_𝑡∈ℝ|𝑢(𝑡)|, 𝑎₀=min_𝑡∈ℝ{𝑎(𝑡)}, and 𝑒₀ is the same as that in assumption (I)′.

The following two lemmas are Mountain Pass Theorem and Symmetric Mountain Pass Theorem, which are useful in the proofs of our theorems.

Lemma 7 (see [21]). Let 𝐸 be a real Banach space and 𝜑∈𝐶¹(𝐸,ℝ) satisfying (PS)-condition. Suppose that 𝜑(0)=0 and(i)there exist constants 𝜌, 𝛼>0 such that 𝜑_{𝜕𝐵𝜌(0)}≥𝛼;(ii)there exists an 𝑒∈𝐸⧵𝐵_𝜌(0) such that 𝜑(𝑒)≤0. Then, 𝜑 possesses a critical value 𝑐≥𝛼 which can be characterized as 𝑐=inf_ℎ∈Φmax_𝑠∈[0,1]𝜑(ℎ(𝑠)), where Φ={ℎ∈𝐶([0,1],𝐸)∣ℎ(0)=0,ℎ(1)=𝑒} and 𝐵_𝜌(0) is an open ball in 𝐸 of radius 𝜌 centered at 0.

Lemma 8 (see [21]). Let 𝐸 be a real Banach space and 𝜑∈𝐶¹(𝐸,ℝ) with 𝐼 even. Assume that 𝜑(0)=0 and 𝜑 satisfies (PS)-condition, assumption (i) of Lemma 7, and the following condition:(iii)for each finite dimensional subspace 𝐸^′⊂𝐸, there is 𝑟=𝑟(𝐸^′)>0 such that 𝜑(𝑢)≤0, for 𝑢∈𝐸^′⧵𝐵_𝑟(0), and 𝐵_𝑟(0) is an open ball in 𝐸 of radius 𝑟 centered at 0. Then, 𝜑 possesses an unbounded sequence of critical values.

Remark 9. Since it is very difficult to check condition (iii) of Lemma 8, few results about infinitely many homoclinic solutions can be seen in the literature by using Lemma 8, let alone infinitely many fast homoclinic solutions obtained by this lemma. Motivated by the idea of [22], we will use Lemma 8 to prove that problem (1) has infinitely many homoclinic fast solutions.

Lemma 10. Assume that (V3)′ and (V5) or (V5)′ hold. Then, for every (𝑡,𝑥)∈ℝ×ℝ, (i)𝑠^−𝜇𝑉₁(𝑡,𝑠𝑥) is nondecreasing on (0,+∞);(ii)𝑠^−𝜚𝑉₂(𝑡,𝑠𝑥) is nonincreasing on (0,+∞).

The proof of Lemma 10 is routine and we omit it.

The functional 𝜑 corresponding to (1) on 𝐸 is given by 𝜑(𝑢)=_ℝ12𝑒^𝑄(𝑡)𝑢^′(𝑡)²+𝑎(𝑡)|𝑢(𝑡)|²−𝑑𝑡_ℝ𝑉(𝑡,𝑢(𝑡))𝑑𝑡+^∞_𝑗=−∞^{𝑢(𝑡𝑗)}₀𝐼(𝑠)𝑑𝑠,𝑢∈𝐸. (26) We now show that 𝜑∈𝐶¹(𝐸,ℝ) and, for 𝑢,𝑣∈𝐸, 𝜑^′(𝑢)=,𝑣_ℝ𝑒^𝑄(𝑡)𝑢^′(𝑡),𝑣^′(𝑡)+(𝑎(𝑡)𝑢(𝑡)(𝑡−(((𝑡(𝑡+,𝑣))∇𝑉𝑡,𝑢)),𝑣))]𝑑𝑡_{∞𝑗=−∞}𝐼𝑢𝑡_𝑗,𝑣𝑡_𝑗. (27) Firstly, we show that 𝜑∶𝐸→ℝ. By (V2)′, for any given 𝜀₀>0, there exists 𝛾₀>0 such that |∇𝑉(𝑡,𝑥)|≤2𝜀𝑒₀𝑎₀|𝑥|,𝑡∈(−∞,−𝑅]∪[𝑅,+∞),|𝑥|≤𝛾₀, (28) where 𝑎₀=min_𝑡∈ℝ{𝑎(𝑡)}, 𝑒₀ is the same as that in assumption (I)′. Then, by 𝑉(𝑡,0)=0 and (28), we have |𝑉(𝑡,𝑥)|=¹₀(∇𝑉(𝑡,𝑠𝑥),𝑥)𝑑𝑠≤𝜀𝑒₀𝑎₀|𝑥|²,∀𝑡∈(−∞,−𝑅]∪[𝑅,+∞),𝑥∈ℝ. (29) Therefore, from (29), we have _{ℝ⧵(−𝑅,𝑅)}𝑉((𝑡≤𝑡,𝑢))𝑑𝑡_{ℝ⧵(−𝑅,𝑅)}|𝑉((𝑡≤𝑡,𝑢))|𝑑𝑡_{ℝ⧵(−𝑅,𝑅)}𝜀𝑒₀𝑎₀|𝑢(𝑡)|²≤𝑑𝑡_{ℝ⧵(−𝑅,𝑅)}𝜀𝑒^𝑄(𝑡)𝑎(𝑡)|𝑢(𝑡)|²𝑑𝑡≤_ℝ𝜀𝑒^𝑄(𝑡)𝑎(𝑡)|𝑢(𝑡)|²𝑑𝑡≤𝜀‖𝑢‖²,𝑢∈𝐸. (30) From (I)′ and Lemma 6, we have ^∞_𝑗=−∞^{𝑢(𝑡𝑗)}₀𝐼(𝑠)≤𝑑𝑠_{∞𝑗=−∞max{0,𝑢(𝑡𝑗𝑡)}min0,𝑢𝑗}|𝐼(𝑠≤1)|𝑑𝑠2^∞_𝑗=−∞𝑐𝑢𝑡_𝑗²≤𝑐𝑝𝑇2𝑒₀𝑢^′2₂≤𝑐𝑝𝑇2𝑒₀‖𝑢‖². (31) It follows from (26), (30), and (31) that 𝜑∶𝐸→ℝ. Next, we prove that 𝜑∈𝐶¹(𝐸,ℝ). Rewrite 𝜑 as the following: 𝜑(𝑢)=𝜑₁(𝑢)−𝜑₂(𝑢)+𝜑₃(𝑢), (32) where 𝜑₁(𝑢)∶=_ℝ12𝑒^𝑄(𝑡)𝑢^′(𝑡)²+𝑎(𝑡)|𝑢(𝑡)|²𝜑𝑑𝑡,₂(𝑢)∶=_ℝ𝑉(𝑡,𝑢(𝑡))𝑑𝑡,𝜑₃(𝑢)∶=^∞_𝑗=−∞^{𝑢(𝑡𝑗)}₀𝐼(𝑠)𝑑𝑠. (33) It is easy to check that 𝜑₁,𝜑₃∈𝐶¹(𝐸,ℝ) and 𝜑^′₁(𝑢),𝑣=_ℝ𝑒^𝑄(𝑡)𝑢^′(𝑡),𝑣^′(𝑡)+(𝑎(𝑡)𝑢(𝑡)(𝑡𝜑,𝑣))𝑑𝑡,_′3(𝑢),𝑣=^∞_𝑗=−∞𝐼𝑢𝑡_𝑗,𝑣𝑡_𝑗,∀𝑢,𝑣∈𝐸. (34) Next, we prove that 𝜑₂∈𝐶¹(𝐸,ℝ) and 𝜑^′₂(𝑢),𝑣=_ℝ(∇𝑉(𝑡,𝑢(𝑡)),𝑣(𝑡))𝑑𝑡,∀𝑢,𝑣∈𝐸. (35) Let 𝑢_𝑛→𝑢 in 𝐸, without loss of generality, and we can assume that ‖𝑢_𝑛‖≤𝛾₀. Since 𝑉∈𝐶¹(ℝ×ℝ,ℝ), we have ∇𝑉𝑡,𝑢_𝑛(𝑡),𝑣(𝑡)≤max_{|𝑥|≤𝛾0/2𝑒0√𝑎0}|∇𝑉(𝑡,𝑥)||𝑣(𝑡)|∶=𝐶₀|𝑣(𝑡)|,∀𝑡∈[−𝑅,𝑅]. (36) By (29) and (36), we have ∇𝑉𝑡,𝑢_𝑛(𝑡),𝑣(𝑡)≤𝜀𝑒₀𝑎₀𝑢_𝑛(𝑡)|𝑣(𝑡)|+𝐶₀|𝑣(𝑡)|≤𝜀𝑒^𝑄(𝑡)𝑎(𝑡)𝑢_𝑛(𝑡)−𝑢(𝑡)+|𝑢(𝑡)||𝑣(𝑡)|+𝐶₀|𝑣(𝑡)|∶=𝑔_𝑛(𝑡),𝑡∈ℝ. (37) Since 𝑢_𝑛(𝑡)→𝑢(𝑡), for almost every 𝑡∈ℝ, we have lim_𝑛→+∞𝑔_𝑛(𝑡)=𝜀𝑒^𝑄(𝑡)𝑎(𝑡)|𝑢(𝑡)||𝑣(𝑡)|+𝐶₀|𝑣(𝑡)|∶=𝑔(𝑡),lim_{𝑛→+∞ℝ}𝑔_𝑛(𝑡)𝑑𝑡=lim_{𝑛→+∞ℝ}𝜀𝑒^𝑄(𝑡)𝑎(𝑡)𝑢_𝑛(𝑡)−𝑢(𝑡)+|𝑢(𝑡)||𝑣(𝑡)|+𝐶₀|𝑣(𝑡)|𝑑𝑡=lim_{𝑛→+∞ℝ}𝜀𝑒^𝑄(𝑡)𝑎(𝑡)𝑢_𝑛(𝑡)−𝑢(𝑡)𝑑𝑡+_ℝ𝜀𝑒^𝑄(𝑡)𝑎(𝑡)|𝑢(𝑡)||𝑣(𝑡)|+𝐶₀|𝑣(𝑡)|𝑑𝑡=_ℝ𝜀𝑒^𝑄(𝑡)𝑎(𝑡)|𝑢(𝑡)||𝑣(𝑡)|+𝐶₀|𝑣(𝑡)|𝑑𝑡∶=_ℝ𝑔(𝑡)𝑑𝑡<+∞. (38) Then, by (37), (38), and Lebesgue’s dominated convergence theorem, we have lim_{𝑛→+∞ℝ}∇𝑉𝑡,𝑢_𝑛(𝑡)(𝑡)=,𝑣𝑑𝑡_ℝ(∇𝑉(𝑡,𝑢(𝑡)),𝑣(𝑡))𝑑𝑡. (39) Therefore, for any 𝑢,𝑣∈𝐸 and for any function 𝜃∶ℝ→(0,1), from (39), we have 𝜑^′₂(𝑢),𝑣=lim_ℎ→0+𝜑₂(𝑢+ℎ𝑣)−𝜑₂(𝑢)ℎ=lim_ℎ→0+1ℎ_ℝ[𝑉(𝑡,𝑢(𝑡)+ℎ𝑣(𝑡))−𝑉(𝑡,𝑢(𝑡))]𝑑𝑡=lim_ℎ→0+ℝ(((𝑡)(𝑡)(𝑡(𝑡=∇𝑉𝑡,𝑢+𝜃ℎ𝑣)),𝑣))𝑑𝑡_ℝ(∇𝑉(𝑡,𝑢(𝑡)),𝑣(𝑡))𝑑𝑡,∀𝑢,𝑣∈𝐸. (40) Finally, we prove that 𝜑₂∈𝐶¹(𝐸,ℝ). From (40), 𝑢_𝑛→𝑢 in 𝐸, and 𝑉∈𝐶¹(ℝ×ℝ,ℝ), we have lim_𝑛→+∞𝜑^′₂𝑢_𝑛−𝜑^′₂(𝑢),𝑣=lim_{𝑛→+∞ℝ}∇𝑉𝑡,𝑢_𝑛(𝑡)−∇𝑉(𝑡,𝑢(𝑡)),𝑣(𝑡)𝑑𝑡≤lim_{𝑛→+∞ℝ}∇𝑉𝑡,𝑢_𝑛(𝑡)((𝑡𝑣(𝑡=−∇𝑉𝑡,𝑢))|)|𝑑𝑡_ℝlim_𝑛→+∞∇𝑉𝑡,𝑢_𝑛(𝑡)((𝑡×|𝑣(𝑡−∇𝑉𝑡,𝑢)))|𝑑𝑡=0,∀𝑣∈𝐸. (41) This shows that 𝜑₂∈𝐶¹(𝐸,ℝ). Therefore, 𝜑∈𝐶¹(𝐸,ℝ) and (27) holds. Similarly, we can prove 𝜑∈𝐶¹(𝐸,ℝ) and (27) holds by (V1), (V2)′′, and (I)′. Furthermore, the critical points of 𝜑 in 𝐸 are classical solutions of (1) with 𝑢(±∞)=0.

3. Proofs of Theorems

Proof of Theorem 2. It is clear that 𝜑(0)=0. We first show that the functional 𝜑 satisfies the (PS)-condition. Let {𝑢_𝑛}⊂𝐸 satisfying 𝜑(𝑢_𝑛) which is bounded and let 𝜑^′(𝑢_𝑛)→0 as 𝑛→∞. Then, there exists a constant 𝐶₁>0 such that 𝜑𝑢_𝑛≤𝐶₁,𝜑^′𝑢_𝑛𝐸∗≤𝜇𝐶₁. (42) From (I)′ and (25), we have ^∞_𝑗=−∞𝐼𝑢𝑡_𝑗,𝑢𝑡_𝑗≤^∞_𝑗=−∞𝐼𝑢𝑡_𝑗𝑢𝑡_𝑗≤^∞_𝑗=−∞𝑐𝑢𝑡_𝑗²≤𝑐𝑝𝑇𝑒₀𝑢^′2₂. (43) From (26), (27), (31), (42), (43), (V3)′, and (V5), we have 2𝐶₁+2𝐶₁𝑢_𝑛≥2𝜑𝑢_𝑛−2𝜇𝜑^′𝑢_𝑛,𝑢_𝑛=𝜇−2𝜇𝑢_𝑛²+2^∞_𝑗=−∞^{𝑢(𝑡𝑗)}₀𝐼(𝑠)2𝑑𝑠−𝜇^∞_𝑗=−∞𝐼𝑢_𝑛𝑡_𝑗,𝑢_𝑛𝑡_𝑗−2_ℝ𝑒^𝑄(𝑡)𝑉₁𝑡,𝑢_𝑛(𝑡)−1𝜇∇𝑉₁𝑡,𝑢_𝑛(𝑡),𝑢_𝑛(𝑡)𝑑𝑡+2_ℝ𝑒^𝑄(𝑡)𝑉₂𝑡,𝑢_𝑛(𝑡)−1𝜇∇𝑉₂𝑡,𝑢_𝑛(𝑡),𝑢_𝑛(𝑡)≥𝑑𝑡𝜇−2𝜇𝑢_𝑛²−𝑐𝑝𝑇𝑒₀𝑢^′_𝑛²₂−2𝑐𝑝𝑇𝜇𝑒₀𝑢^′_𝑛²₂≥𝜇−2𝜇−𝜇+2𝑐𝑝𝑇𝜇𝑒₀𝑢_𝑛². (44) Since 𝜇>𝜚>2 and 0<𝑐<(𝜚−2)𝑒₀/(𝜚+2)𝑝𝑇, the above inequalities imply that there exists a constant 𝐶₂>0 such that 𝑢_𝑛≤𝐶₂,𝑛∈ℕ. (45) Now, we prove that 𝑢_𝑛→𝑢₀ in 𝐸. Passing to a subsequence if necessary, it can be assumed that 𝑢_𝑛⇀𝑢₀ in 𝐸. For any given number 𝜀>0, by (V2)′, we can choose 𝜉>0 such that |∇𝑉(𝑡,𝑥)|≤𝜀𝑎₀|𝑥|for|𝑡|≥𝑅,|𝑥|≤𝜉. (46) Since 𝑄(𝑡)→∞ as |𝑡|→∞, we can choose 𝑅₀>𝑅 such that 𝑄(𝑡)𝐶≥ln²𝜉for|𝑡|≥𝑅₀. (47) From (A), we can choose 𝑅^′₀>𝑅 such that 𝑎(𝑡)≥𝐶₂𝜉for|𝑡|≥𝑅^′₀. (48) It follows from (23), (45), (47), and (48) that 𝑢_𝑛(𝑡)²≤_𝑡^+∞𝑒^{−𝑄(𝑠)}[𝑎(𝑠)]^−1/2×𝑒^𝑄(𝑠)𝑢^′(𝑠)²+𝑎(𝑠)|𝑢(𝑠)|²𝑑𝑠^1/2≤𝜉²𝐶²₂𝑢_𝑛²≤𝜉²for𝑡≥𝑅₁,𝑛∈ℕ, (49) where 𝑅₁=max{𝑅₀,𝑅^′₀}. Similarly, by (24), (45), (47), and (48), we have 𝑢_𝑛(𝑡)²≤𝜉²for𝑡≤−𝑅₁,𝑛∈ℕ. (50) Since 𝑢_𝑛⇀𝑢₀ in 𝐸, it is easy to verify that 𝑢_𝑛(𝑡) converges to 𝑢₀(𝑡) pointwise for all 𝑡∈ℝ. Hence, it follows from (49) and (50) that 𝑢₀(𝑡)≤𝜉for𝑡∈−∞,−𝑅₁∪𝑅₁,+∞. (51) Since 𝑒^𝑄(𝑡)≥𝑒₀>0 on [−𝑅₁,𝑅₁]=𝐽, the operator defined by 𝑆∶𝐸→𝐻¹(𝐽)∶𝑢→𝑢|_𝐽 is a linear continuous map. So, 𝑢_𝑛→𝑢₀ in 𝐻¹(𝐽). Sobolev theorem implies that 𝑢_𝑛→𝑢₀ uniformly on 𝐽, so there is 𝑛₀∈ℕ such that ^𝑅1_−𝑅1𝑒^𝑄(𝑡)∇𝑉𝑡,𝑢_𝑛(𝑡)−∇𝑉𝑡,𝑢₀(𝑡)×𝑢_𝑛(𝑡)−𝑢₀(𝑡)𝑑𝑡<𝜀for𝑛≥𝑛₀. (52) From (45), (46), (49), (50), and (51), we have _{ℝ⧵[−𝑅1,𝑅1]}𝑒^𝑄(𝑡)∇𝑉𝑡,𝑢_𝑛(𝑡)−∇𝑉𝑡,𝑢₀(𝑡)×𝑢_𝑛(𝑡)−𝑢₀(𝑡≤)𝑑𝑡_{ℝ⧵[−𝑅1,𝑅1]}𝑒^𝑄(𝑡)∇𝑉𝑡,𝑢_𝑛(𝑡)+∇𝑉𝑡,𝑢₀(𝑡)×𝑢_𝑛(𝑡)+𝑢₀(𝑡)𝑑𝑡≤𝜀_{ℝ⧵[−𝑅1,𝑅1]}𝑒^𝑄(𝑡)𝑎₀𝑢_𝑛(𝑡)+𝑢₀(𝑡×𝑢_𝑛)(𝑡)+𝑢₀(𝑡)𝑑𝑡≤2𝜀_{ℝ⧵[−𝑅1,𝑅1]}𝑒^𝑄(𝑡)𝑎₀𝑢_𝑛(𝑡)²+𝑢₀(𝑡)²𝑑𝑡≤2𝜀_{ℝ⧵[−𝑅1,𝑅1]}𝑒^𝑄(𝑡)𝑎(𝑡)𝑢_𝑛(𝑡)²+𝑎(𝑡)𝑢₀(𝑡)²𝑢𝑑𝑡≤2𝜀_𝑛²+𝑢₀²≤2𝜀𝐶²₂+𝑢₀²,𝑛∈ℕ. (53) It follows from (52) and (53) that _ℝ𝑒^𝑄(𝑡)∇𝑉𝑡,𝑢_𝑛(𝑡)−∇𝑉𝑡,𝑢₀(𝑡)𝑢_𝑛(𝑡)−𝑢₀(𝑡)𝑑𝑡⟶0as𝑛⟶∞. (54) From (27) and (I)′, as 𝑛→∞, we have 0⟵𝜑^′𝑢_𝑛−𝜑^′𝑢₀,𝑢_𝑛−𝑢₀=𝑢_𝑛−𝑢₀²+^∞_𝑗=−∞𝐼𝑢_𝑛𝑡_𝑗−𝐼𝑢₀𝑡_𝑗,𝑢_𝑛𝑡_𝑗−𝑢₀𝑡_𝑗−_ℝ𝑒^𝑄(𝑡)∇𝑉𝑡,𝑢_𝑛(𝑡)−∇𝑉𝑡,𝑢₀(𝑡),𝑢_𝑛(𝑡)−𝑢₀(𝑡≥𝑢)𝑑𝑡_𝑛−𝑢₀²−_ℝ𝑒^𝑄(𝑡)∇𝑉𝑡,𝑢_𝑛(𝑡)−∇𝑉𝑡,𝑢₀(𝑡),𝑢_𝑛(𝑡)−𝑢₀(𝑡)𝑑𝑡. (55) It follows from (54) and (55) that 𝑢_𝑛−𝑢₀²⟶0as𝑛⟶∞. (56)
Hence, 𝑢_𝑛→𝑢₀ in 𝐸 by (56). This shows that 𝜑 satisfies (PS)-condition.
We now show that there exist constants 𝜌,𝛼>0 such that assumption (i) of Lemma 7 holds. From (V2)′, there exists 𝛿∈(0,1) such that |(≤𝑎∇𝑉𝑡,𝑥)|⁰2|𝑥|for|𝑡|≥𝑅,|𝑥|≤𝛿. (57) By 𝑉(𝑡,0)=0 and (57), we have |𝑉(≤𝑎𝑡,𝑥)|⁰4|𝑥|²for|𝑡|≥𝑅,|𝑥|≤𝛿. (58) Let 𝐶₃𝑉=sup¹(𝑡,𝑥)𝑎₀∣𝑡∈[−𝑅,𝑅],𝑥∈ℝ,|𝑥|=1. (59) Setting 𝜎=min{1/(4𝐶₃+1)^1/(𝜇−2),𝛿} and ‖𝑢‖=2𝑒₀√𝑎₀𝜎∶=𝜌, it follows from Lemma 6 that |𝑢(𝑡)|≤𝜎≤𝛿<1 for 𝑡∈ℝ. From Lemma 10 (i) and (59), we have ^𝑅_−𝑅𝑒^𝑄(𝑡)𝑉₁((𝑡≤𝑡,𝑢))𝑑𝑡_{{𝑡∈[−𝑅,𝑅]∶𝑢(𝑡)≠0}}𝑒^𝑄(𝑡)𝑉₁𝑢(𝑡)𝑡,|𝑢(𝑡|𝑢(𝑡)|)|^𝜇𝑑𝑡≤𝐶₃^𝑅_−𝑅𝑒^𝑄(𝑡)𝑎₀|𝑢(𝑡)|^𝜇𝑑𝑡≤𝐶₃𝜎^𝜇−2𝑅_−𝑅𝑒^𝑄(𝑡)𝑎₀|𝑢(𝑡)|²𝑑𝑡≤𝐶₃𝜎^𝜇−2𝑅_−𝑅𝑒^𝑄(𝑡)𝑎(𝑡)|𝑢(𝑡)|²≤1𝑑𝑡4^𝑅_−𝑅𝑒^𝑄(𝑡)𝑎(𝑡)|𝑢(𝑡)|²𝑑𝑡. (60) By (V5), (31), (58), and (60), we have 𝜑(𝑢)=12‖𝑢‖²+^∞_𝑗=−∞^{𝑢(𝑡𝑗)}₀𝐼(𝑠)𝑑𝑠−_ℝ𝑒^𝑄(𝑡)𝑉((𝑡≥1𝑡,𝑢))𝑑𝑡2‖𝑢‖²+^∞_𝑗=−∞^{𝑢(𝑡𝑗)}₀𝐼(𝑠)−𝑑𝑠_{ℝ⧵[−𝑅,𝑅]}𝑒^𝑄(𝑡)𝑉((𝑡−𝑡,𝑢))𝑑𝑡_𝑅−𝑅𝑒^𝑄(𝑡)𝑉₁((𝑡≥1𝑡,𝑢))𝑑𝑡2‖𝑢‖²−𝑐𝑝𝑇2𝑒₀𝑢^′2₂−14_{ℝ⧵[−𝑅,𝑅]}𝑎₀𝑒^𝑄(𝑡)|𝑢(𝑡)|²−1𝑑𝑡4^𝑅_−𝑅𝑒^𝑄(𝑡)𝑎(𝑡)|𝑢(𝑡)|²≥1𝑑𝑡2‖𝑢‖²−𝑐𝑝𝑇2𝑒₀𝑢^′2₂−14_{ℝ⧵[−𝑅,𝑅]}𝑒^𝑄(𝑡)𝑎(𝑡)|𝑢(𝑡)|²−1𝑑𝑡4^𝑅_−𝑅𝑒^𝑄(𝑡)𝑎(𝑡)|𝑢(𝑡)|²=1𝑑𝑡2‖𝑢‖²−𝑐𝑝𝑇2𝑒₀𝑢^′2₂−14_ℝ𝑒^𝑄(𝑡)𝑎(𝑡)|𝑢(𝑡)|²1𝑑𝑡≥min2−𝑐𝑝𝑇2𝑒₀,14‖𝑢‖². (61) Therefore, we can choose a constant 𝛼>0 depending on 𝜌 such that 𝜑(𝑢)≥𝛼 for any 𝑢∈𝐸 with ‖𝑢‖=𝜌, which shows that 𝜑 satisfies assumption (i) of Lemma 7.
Finally, it remains to show that 𝜑 satisfies assumption (ii) of Lemma 7. From Lemma 10 (ii) and (22), we have for any 𝑢∈𝐸³₋₃𝑒^𝑄(𝑡)𝑉₂((𝑡=𝑡,𝑢))𝑑𝑡_{{𝑡∈[−3,3]∶|𝑢(𝑡)|>1}}𝑒^𝑄(𝑡)𝑉₂(𝑡,𝑢(𝑡))𝑑𝑡+_{{𝑡∈[−3,3]∶|𝑢(𝑡)|≤1}}𝑒^𝑄(𝑡)𝑉₂((𝑡≤𝑡,𝑢))𝑑𝑡_{{𝑡∈[−3,3]∶|𝑢(𝑡)|>1}}𝑒^𝑄(𝑡)𝑉₂𝑢(𝑡)𝑡,|𝑢(𝑡𝑢(𝑡)||)|^𝜚𝑑𝑡+³₋₃𝑒^𝑄(𝑡)max_|𝑥|≤1𝑉₂()≤‖𝑢‖𝑡,𝑥𝑑𝑡_𝜚∞3−3𝑒^𝑄(𝑡)max_|𝑥|=1𝑉₂()+𝑡,𝑥𝑑𝑡₃₋₃𝑒^𝑄(𝑡)max_|𝑥|≤1𝑉₂()≤1𝑡,𝑥𝑑𝑡2𝑒₀√𝑎₀^𝜚‖𝑢‖^𝜚3₋₃𝑒^𝑄(𝑡)max_|𝑥|=1𝑉₂()+𝑡,𝑥𝑑𝑡₃₋₃𝑒^𝑄(𝑡)max_|𝑥|≤1𝑉₂(𝑡,𝑥)𝑑𝑡=𝐶₄‖𝑢‖^𝜚+𝐶₅, (62) where 𝐶₄=(1/2𝑒₀√𝑎₀)^𝜚∫³₋₃𝑒^𝑄(𝑡)max_|𝑥|=1𝑉₂(𝑡,𝑥)𝑑𝑡, 𝐶₅=∫³₋₃𝑒^𝑄(𝑡)max_|𝑥|≤1𝑉₂(𝑡,𝑥)𝑑𝑡. Take 𝜔∈𝐸 such that |𝜔(𝑡=|𝑡||𝑡|)|1,for≤1,0,for≥3, (63) and |𝜔(𝑡)|≤1 for |𝑡|∈(1,3]. For 𝑠>1, from Lemma 10 (i) and (63), we get ¹₋₁𝑒^𝑄(𝑡)𝑉₁(𝑡,𝑠𝜔(𝑡))𝑑𝑡≥𝑠^𝜇1₋₁𝑒^𝑄(𝑡)𝑉₁(𝑡,𝜔(𝑡))𝑑𝑡=𝐶₆𝑠^𝜇, (64) where 𝐶₆=∫¹₋₁𝑒^𝑄(𝑡)𝑉₁(𝑡,𝜔(𝑡))𝑑𝑡>0. From (26), (31), (62), (63), and (64), we get for 𝑠>1𝜑()=𝑠𝑠𝜔²2‖𝜔‖²+^∞_𝑗=−∞0^{𝑠𝜔(𝑡)}𝐼(𝑡)𝑑𝑡+_ℝ𝑒^𝑄(𝑡)𝑉₂(𝑡,𝑠𝜔(𝑡))−𝑉₁((𝑡≤𝑠𝑡,𝑠𝜔))𝑑𝑡₂2‖𝜔‖²+𝑐𝑝𝑇𝑠²2𝑒₀𝜔^′2₂+³₋₃𝑒^𝑄(𝑡)𝑉₂((𝑡−𝑡,𝑠𝜔))𝑑𝑡₁₋₁𝑒^𝑄(𝑡)𝑉₁((𝑡≤𝑒𝑡,𝑠𝜔))𝑑𝑡₀+𝑐𝑝𝑇2𝑒₀𝑠²‖𝜔‖²+𝐶₄𝑠^𝜚‖𝜔‖^𝜚+𝐶₅−𝐶₆𝑠^𝜇. (65) Since 𝜇>𝜚>2 and 𝐶₆>0, it follows from (65) that there exists 𝑠₁>1 such that ‖𝑠₁𝜔‖>𝜌 and 𝜑(𝑠₁𝜔)<0. Set 𝑒=𝑠₁𝜔(𝑡), and then 𝑒∈𝐸, ‖𝑒‖=‖𝑠₁𝜔‖>𝜌, and 𝜑(𝑒)=𝜑(𝑠₁𝜔)<0. By Lemma 7, 𝜑 has a critical value 𝑐>𝛼 given by 𝑐=inf_𝑔∈Φmax_[]𝑠∈0,1𝜑𝑔(𝑠), (66) where Φ=𝑔∈𝐶([0,1],𝐸)∶𝑔(0)=0,𝑔(1)=𝑒. (67) Hence, there exists 𝑢^∗∈𝐸 such that 𝜑𝑢^∗=𝑐,𝜑^′𝑢^∗=0. (68) The function 𝑢^∗ is a desired solution of problem (1). Since 𝑐>0, 𝑢^∗ is a nontrivial fast homoclinic solution. The proof is complete.

Proof of Theorem 3. In the proof of Theorem 2, the condition 𝑉₂(𝑡,𝑥)≥0 in (V5) is only used in the proofs of (45) and assumption (i) of Lemma 7. Therefore, we only need to prove that (45) and assumption (i) of Lemma 7 still hold if we use (V2)′′ and (V5)′ instead of (V2)′ and (V5), respectively. We first prove that (45) holds. From (V3)′, (V5)′, (26), (27), (31), (42), and (43), we have 2𝐶₁+2𝐶₁𝜇𝜚𝑢_𝑛≥2𝜑𝑢_𝑛−2𝜚𝜑^′𝑢_𝑛,𝑢_𝑛=𝜚−2𝜚𝑢_𝑛²+2_ℝ𝑒^𝑄(𝑡)𝑉₂𝑡,𝑢_𝑛(𝑡−1)𝜚∇𝑉₂𝑡,𝑢_𝑛(𝑡),𝑢_𝑛(𝑡)𝑑𝑡−2_ℝ𝑒^𝑄(𝑡)𝑉₁𝑡,𝑢_𝑛(𝑡−1)𝜚∇𝑉₁𝑡,𝑢_𝑛(𝑡),𝑢_𝑛(𝑡)𝑑𝑡+2^∞_𝑗=−∞^{𝑢(𝑡𝑗)}₀𝐼(𝑡)2𝑑𝑡−𝜚^∞_𝑗=−∞𝐼𝑢𝑡_𝑗,𝑢𝑡_𝑗≥𝜚−2𝜚𝑢_𝑛²−𝑐𝑝𝑇𝑒₀𝑢^′_𝑛²₂−2𝑐𝑝𝑇𝜚𝑒₀𝑢^′_𝑛²₂≥𝜚−2𝜚𝑢_𝑛²−𝑐𝑝𝑇𝑒₀𝑢_𝑛²−2𝑐𝑝𝑇𝜚𝑒₀𝑢_𝑛²=𝜚−2𝜚−𝜚+2𝑐𝑝𝑇𝜚𝑒₀𝑢_𝑛², (69) which implies that there exists a constant 𝐶₂>0 such that (45) holds. Next, we prove that assumption (i) of Lemma 7 still holds. From (V2)′′, there exists 𝛿∈(0,1) such that |(≤𝑎∇𝑉𝑡,𝑥)|⁰2|𝑥|for𝑡∈ℝ,|𝑥|≤𝛿. (70) By 𝑉(𝑡,0)=0 and (70), we have |𝑉(≤𝑎𝑡,𝑥)|⁰4|𝑥|²for𝑡∈ℝ,|𝑥|≤𝛿. (71) Let ‖𝑢‖=2𝑒₀√𝑎₀𝛿∶=𝜌, and it follows from Lemma 6 that |𝑢(𝑡)|≤𝛿. It follows from (31) and (71) that 𝜑(𝑢)=12‖𝑢‖²+^∞_𝑗=−∞^{𝑢(𝑡𝑗)}₀𝐼(𝑡)𝑑𝑡−_ℝ𝑒^𝑄(𝑡)𝑉((𝑡≥1𝑡,𝑢))𝑑𝑡2‖𝑢‖²−𝑐𝑝𝑇2𝑒₀𝑢^′2₂−14_ℝ𝑎₀𝑒^𝑄(𝑡)|𝑢(𝑡)|²≥1𝑑𝑡2‖𝑢‖²−𝑐𝑝𝑇2𝑒₀𝑢^′2₂−14_ℝ𝑎(𝑡)𝑒^𝑄(𝑡)|𝑢(𝑡)|²1𝑑𝑡≥min2−𝑐𝑝𝑇2𝑒₀,14‖𝑢‖². (72) Therefore, we can choose a constant 𝛼>0 depending on 𝜌 such that 𝜑(𝑢)≥𝛼 for any 𝑢∈𝐸 with ‖𝑢‖=𝜌. The proof of Theorem 3 is complete.

Proof of Theorem 4. Condition (V6) shows that 𝜑 is even. In view of the proof of Theorem 2, we know that 𝜑∈𝐶¹(𝐸,ℝ) and satisfies (PS)-condition and assumption (i) of Lemma 7. Now, we prove that (iii) of Lemma 8 holds. Let 𝐸^′ be a finite dimensional subspace of 𝐸. Since all norms of a finite dimensional space are equivalent, there exists 𝑑>0 such that ‖𝑢‖≤𝑑‖𝑢‖_∞,𝑢∈𝐸^′. (73) Assume that dim𝐸^′=𝑚 and {𝑢₁,𝑢₂,…,𝑢_𝑚} is a base of 𝐸^′ such that 𝑢_𝑖=𝑑,𝑖=1,2,…,𝑚. (74) For any 𝑢∈𝐸^′, there exist 𝜆_𝑖∈ℝ, 𝑖=1,2,…,𝑚 such that 𝑢(𝑡)=^𝑚_𝑖=1𝜆_𝑖𝑢_𝑖(𝑡)for𝑡∈ℝ. (75) Let ‖𝑢‖_∗=^𝑚_𝑖=1𝜆_𝑖𝑢_𝑖. (76) It is easy to see that ‖⋅‖_∗ is a norm of 𝐸^′. Hence, there exists a constant 𝑑^′>0 such that 𝑑^′‖𝑢‖_∗≤‖𝑢‖. Since 𝑢_𝑖∈𝐸, by Lemma 6, we can choose 𝑅₂>𝑅 such that 𝑢_𝑖(𝑡<𝑑)^′𝛿𝑚+𝑑^′,|𝑡|>𝑅₂,𝑖=1,2,…,𝑚, (77) where 𝛿 is given in (71). Let Θ=^𝑚_𝑖=1𝜆_𝑖𝑢_𝑖(𝑡)∶𝜆_𝑖∈ℝ,𝑖=1,2,…,𝑚;^𝑚_𝑖=1𝜆_𝑖==1𝑢∈𝐸^′∶‖𝑢‖_∗=𝑑. (78) Hence, for 𝑢∈Θ, let 𝑡₀=𝑡₀(𝑢)∈ℝ such that 𝑢𝑡₀=‖𝑢‖_∞. (79) Then, by (73)–(76), (78), and (79), we have 𝑑𝑑^′=𝑑𝑑^′𝑚_𝑖=1𝜆_𝑖=𝑑^′𝑚_𝑖=1𝜆_𝑖𝑢_𝑖=𝑑^′‖𝑢‖_∗≤‖𝑢‖≤𝑑‖𝑢‖_∞=𝑑𝑢𝑡₀=𝑑^𝑚_𝑖=1𝜆_𝑖𝑢_𝑖𝑡₀≤𝑑^𝑚_𝑖=1𝜆_𝑖𝑢_𝑖𝑡₀,𝑢∈Θ. (80) This shows that |𝑢(𝑡₀)|≥𝑑^′ and there exists 𝑖₀∈{1,2,…,𝑚} such that |𝑢_𝑖0(𝑡₀)|≥𝑑^′/𝑚, which together with (77) implies that |𝑡₀|≤𝑅₂. Let 𝑅₃=𝑅₂+1 and 𝛾=min𝑒^𝑄(𝑡)𝑉₁(𝑡,𝑥)∶−𝑅₃≤𝑡≤𝑅₃,𝑑^′√2≤|𝑥|≤𝑑2𝑒₀√𝑎₀. (81) Since 𝑉₁(𝑡,𝑥)>0 for all 𝑡∈ℝ and 𝑥∈ℝ⧵{0} and 𝑉₁∈𝐶¹(ℝ×ℝ,ℝ), it follows that 𝛾>0. For any 𝑢∈𝐸, from Lemmas 6 and 10 (i), we have ^𝑅3_−𝑅3𝑒^𝑄(𝑡)𝑉₂((𝑡=𝑡,𝑢))𝑑𝑡_{{𝑡∈[−𝑅3,𝑅3]∶|𝑢(𝑡)|>1}}𝑒^𝑄(𝑡)𝑉₂((𝑡+𝑡,𝑢))𝑑𝑡_{{𝑡∈[−𝑅3,𝑅3]∶|𝑢(𝑡)|≤1}}𝑒^𝑄(𝑡)𝑉₂((𝑡≤𝑡,𝑢))𝑑𝑡_{{𝑡∈[−𝑅3,𝑅3]∶|𝑢(𝑡)|>1}}𝑒^𝑄(𝑡)𝑉₂𝑢(𝑡)𝑡,|𝑢(𝑡𝑢(𝑡)||)|^𝜚+𝑑𝑡_{𝑅3−𝑅3}𝑒^𝑄(𝑡)max_|𝑥|≤1𝑉₂()≤‖𝑢‖𝑡,𝑥𝑑𝑡_{𝜚∞𝑅3−𝑅3}𝑒^𝑄(𝑡)max_|𝑥|=1𝑉₂()+𝑡,𝑥𝑑𝑡_{𝑅3−𝑅3}𝑒^𝑄(𝑡)max_|𝑥|≤1𝑉₂()≤1𝑡,𝑥𝑑𝑡2𝑒₀√𝑎₀^𝜚‖𝑢‖^𝜚𝑅3_−𝑅3𝑒^𝑄(𝑡)max_|𝑥|=1𝑉₂()+𝑡,𝑥𝑑𝑡_{𝑅3−𝑅3}𝑒^𝑄(𝑡)max_|𝑥|≤1𝑉₂(𝑡,𝑥)𝑑𝑡=𝐶₇‖𝑢‖^𝜚+𝐶₈, (82) where 𝐶₇=(1/2𝑒₀√𝑎₀)^𝜚∫^𝑅3_−𝑅3𝑒^𝑄(𝑡)max_|𝑥|=1𝑉₂(𝑡,𝑥)𝑑𝑡 and 𝐶₈=∫^𝑅3_−𝑅3𝑒^𝑄(𝑡)max_|𝑥|≤1𝑉₂(𝑡,𝑥)𝑑𝑡. Since 𝑢^′_𝑖∈𝐿²(𝑒^𝑄(𝑡)), 𝑖=1,2,…,𝑚, it follows that there exists 𝜀₁∈(0,(𝑑^′)²𝑒₀/32𝑚²𝑑²) such that ^𝑡+𝜀1_𝑡−𝜀1𝑢^′_𝑖(𝑠=)𝑑𝑠_{𝑡+𝜀1𝑡−𝜀1}𝑒^{−𝑄(𝑠)/2}𝑒^𝑄(𝑠)/2𝑢^′_𝑖(𝑠≤1)𝑑𝑠√𝑒₀^𝑡+𝜀1_𝑡−𝜀1𝑒^𝑄(𝑠)/2𝑢^′_𝑖(𝑠≤1)𝑑𝑠√𝑒₀2𝜀₁^{1/2𝑡+𝜀1}_𝑡−𝜀𝑒^𝑄(𝑠)𝑢^′_𝑖(𝑠)²𝑑𝑠^1/2≤2𝜀₁𝑒₀^1/2𝑢^′_𝑖2≤𝑑^′4𝑚for𝑡∈ℝ,𝑖=1,2,…,𝑚. (83) Then, for 𝑢∈Θ with |𝑢(𝑡₀)|=‖𝑢‖_∞ and 𝑡∈[𝑡₀−𝜀₁,𝑡₀+𝜀₁], it follows from (75), (78), (79), (80), and (83) that |𝑢(𝑡)|²=𝑢𝑡₀²+2^𝑡_𝑡0𝑢^′(𝑠)(𝑠)≥𝑢𝑡,𝑢𝑑𝑠₀₂−2^𝑡0+𝜀1_{𝑡0−𝜀1}|𝑢(𝑠)|𝑢^′(𝑠≥𝑢𝑡)𝑑𝑠₀₂−2𝑢𝑡₀^𝑡0+𝜀1_{𝑡0−𝜀1}𝑢^′(𝑠≥𝑢𝑡)𝑑𝑠₀₂−2𝑢𝑡₀^𝑚_𝑖=1𝜆_𝑖^𝑡0+𝜀1_{𝑡0−𝜀1}𝑢^′_𝑖(𝑠≥𝑑)𝑑𝑠_′22. (84) On the other hand, since ‖𝑢‖≤𝑑 for 𝑢∈Θ, then |𝑢(𝑡)|≤‖𝑢‖_∞≤𝑑2𝑒₀√𝑎₀,𝑡∈ℝ,𝑢∈Θ. (85) Therefore, from (81), (84), and (85), we have ^𝑅3_−𝑅3𝑒^𝑄(𝑡)𝑉₁((𝑡≥𝑡,𝑢))𝑑𝑡_{𝑡0+𝜀1𝑡0−𝜀1}𝑒^𝑄(𝑡)𝑉₁(𝑡,𝑢(𝑡))𝑑𝑡≥2𝜀₁𝛾for𝑢∈Θ. (86) By (77) and (78), we have |𝑢(𝑡)|≤^𝑚_𝑖=1𝜆_𝑖𝑢_𝑖(𝑡)≤𝛿for|𝑡|≥𝑅₂,𝑢∈Θ. (87) By (26), (31), (58), (82), (86), (87), and Lemma 10, we have for 𝑢∈Θ and 𝑟>1𝜑()=𝑟𝑟𝑢₂2‖𝑢‖²+^∞_𝑗=−∞^{𝑟𝑢(𝑡𝑗)}₀𝐼(𝑡)+𝑑𝑡_ℝ𝑒^𝑄(𝑡)𝑉₂(𝑡,𝑟𝑢(𝑡))−𝑉₁((𝑡≤𝑟𝑡,𝑟𝑢))𝑑𝑡₂2‖𝑢‖²+𝑐𝑝𝑇𝑟²2𝑒₀𝑢^′2₂+𝑟^𝜚_ℝ𝑒^𝑄(𝑡)𝑉₂(𝑡,𝑢(𝑡))𝑑𝑡−𝑟^𝜇_ℝ𝑒^𝑄(𝑡)𝑉₁((𝑡=𝑟𝑡,𝑢))𝑑𝑡₂2‖𝑢‖²+𝑐𝑝𝑇𝑟²2𝑒₀𝑢^′2₂+𝑟^𝜚_{ℝ⧵(−𝑅3,𝑅3)}𝑒^𝑄(𝑡)𝑉₂(𝑡,𝑢(𝑡))𝑑𝑡−𝑟^𝜇𝑅3_−𝑅3𝑒^𝑄(𝑡)𝑉₁(𝑡,𝑢(𝑡))𝑑𝑡−𝑟^𝜇_{ℝ⧵(−𝑅3,𝑅3)}𝑒^𝑄(𝑡)𝑉₁(𝑡,𝑢(𝑡))𝑑𝑡+𝑟^𝜚𝑅3_−𝑅3𝑒^𝑄(𝑡)𝑉₂((𝑡≤𝑟𝑡,𝑢))𝑑𝑡₂2‖𝑢‖²+𝑐𝑝𝑇𝑟²2𝑒₀𝑢^′2₂−𝑟^𝜚_{ℝ⧵(−𝑅3,𝑅3)}𝑒^𝑄(𝑡)𝑉(𝑡,𝑢(𝑡))𝑑𝑡−𝑟^𝜇𝑅3_−𝑅3𝑒^𝑄(𝑡)𝑉₁(𝑡,𝑢(𝑡))𝑑𝑡+𝑟^𝜚𝑅3_−𝑅3𝑒^𝑄(𝑡)𝑉₂((𝑡≤𝑟𝑡,𝑢))𝑑𝑡₂2‖𝑢‖²+𝑐𝑝𝑇𝑟²2𝑒₀𝑢^′2₂+𝑟^𝜚4_{ℝ⧵(−𝑅3,𝑅3)}𝑎₀𝑒^𝑄(𝑡)|𝑢(𝑡)|²𝑑𝑡+𝑟^𝜚𝐶₇‖𝑢‖^𝜚+𝐶₈−2𝜀₁𝛾𝑟^𝜇≤𝑟²2‖𝑢‖²+𝑐𝑝𝑇𝑟²2𝑒₀𝑢^′2₂+𝑟^𝜚4_{ℝ⧵(−𝑅3,𝑅3)}𝑎(𝑡)𝑒^𝑄(𝑡)|𝑢(𝑡)|²𝑑𝑡+𝑟^𝜚𝐶₇‖𝑢‖^𝜚+𝐶₈−2𝜀₁𝛾𝑟^𝜇≤12+𝑐𝑝𝑇2𝑒₀𝑟²‖𝑢‖²+𝑟^𝜚4‖𝑢‖²+𝑟^𝜚𝐶₇‖𝑢‖^𝜚+𝐶₈−2𝜀₁𝛾𝑟^𝜇≤12+𝑐𝑝𝑇2𝑒₀𝑟²𝑑²+𝑟^𝜚4𝑑²+𝐶₇(𝑟𝑑)^𝜚+𝐶₈𝑟^𝜚−2𝜀₁𝛾𝑟^𝜇. (88) Since 𝜇>𝜚>2, there exists 𝑟₀=𝑟₀(𝑐,𝑝,𝑇,𝑒₀,𝑑,𝑑^′,𝐶₇,𝐶₈,𝑅₂,𝑅₃,𝜀₁,𝛾)=𝑟₀(𝐸^′)>1 such that 𝜑(𝑟𝑢)<0for𝑢∈Θ,𝑟≥𝑟₀. (89) It follows that 𝜑(𝑢)<0for𝑢∈𝐸^′,‖𝑢‖≥𝑑𝑟₀, (90) which shows that (iii) of Lemma 8 holds. By Lemma 8, 𝜑 possesses an unbounded sequence {𝑐_𝑛}^∞_𝑛=1 of critical values with 𝑐_𝑛=𝜑(𝑢_𝑛), where 𝑢_𝑛 is such that 𝜑^′(𝑢_𝑛)=0 for 𝑛=1,2,… . If {‖𝑢_𝑛‖} is bounded, then there exists 𝐶₉>0 such that 𝑢_𝑛≤𝐶₉for𝑛∈ℕ. (91) By a similar fashion for the proof of (49) and (50), for the given 𝛿 in (58), there exists 𝑅₄>𝑅 such that 𝑢_𝑛(𝑡)≤𝛿for|𝑡|≥𝑅₄,𝑛∈ℕ. (92) Hence, by (22), (26), (31), (58), (91), and (92), we have 12𝑢_𝑛²=𝑐_𝑛+_ℝ𝑒^𝑄(𝑡)𝑉𝑡,𝑢_𝑛(𝑡)𝑑𝑡−^∞_𝑗=−∞^{𝑢𝑛(𝑡𝑗)}₀𝐼(𝑡)𝑑𝑡=𝑐_𝑛+_{ℝ⧵(−𝑅4,𝑅4)}𝑒^𝑄(𝑡)𝑉𝑡,𝑢_𝑛(𝑡)+𝑑𝑡_{𝑅4−𝑅4}𝑒^𝑄(𝑡)𝑉𝑡,𝑢_𝑛(𝑡)𝑑𝑡−^∞_𝑗=−∞^{𝑢𝑛(𝑡𝑗)}₀𝐼(𝑡)𝑑𝑡≥𝑐_𝑛−14_{ℝ⧵(−𝑅4,𝑅4)}𝑎₀𝑒^𝑄(𝑡)𝑢_𝑛(𝑡)²𝑑𝑡−^𝑅4_−𝑅4𝑒^𝑄(𝑡)𝑉𝑡,𝑢_𝑛(𝑡)𝑑𝑡−𝑐𝑝𝑇2𝑒₀𝑢^′_𝑛²₂≥𝑐_𝑛−14_{ℝ⧵(−𝑅4,𝑅4)}𝑒^𝑄(𝑡)𝑎(𝑡)𝑢_𝑛(𝑡)²𝑑𝑡−^𝑅4_−𝑅4𝑒^𝑄(𝑡)𝑉𝑡,𝑢_𝑛(𝑡)𝑑𝑡−𝑐𝑝𝑇2𝑒₀𝑢^′_𝑛²₂≥𝑐_𝑛−14𝑢_𝑛²−^𝑅4_−𝑅4𝑒^𝑄(𝑡)max_{|𝑥|≤2𝑒0√𝑎0𝐶9}|𝑉(−𝑡,𝑥)|𝑑𝑡𝑐𝑝𝑇2𝑒₀𝑢_𝑛². (93) It follows from (93) that 𝑐_𝑛≤34+𝑐𝑝𝑇2𝑒₀𝑢_𝑛²+^𝑅3_−𝑅3𝑒^𝑄(𝑡)max_{|𝑥|≤2𝑒0√𝑎0𝐶9}|𝑉(𝑡,𝑥)|𝑑𝑡<+∞. (94) This contradicts the fact that {𝑐_𝑛}^∞_𝑛=1 is unbounded, and so {‖𝑢_𝑛‖} is unbounded. The proof is complete.

Proof of Theorem 5. In view of the proofs of Theorems 3 and 4, the conclusion of Theorem 5 holds. The proof is complete.

4. Examples

Example 1. Consider the following system: 𝑢^′′(𝑡)+𝑡³𝑢^′(𝑡)(𝑡)𝑢(𝑡)((𝑡𝑡−𝑎+∇𝑉𝑡,𝑢))=0,a.e.𝑡∈_𝑗,𝑡_𝑗+1,𝑗∈ℤ,Δ𝑢^′𝑡_𝑗=𝑢^′𝑡⁺_𝑗−𝑢^′𝑡⁻_𝑗=𝐼𝑢𝑡_𝑗,𝑗∈ℤ, (95) where 𝑞(𝑡)=𝑡³, 𝑡∈ℝ, 𝑢∈ℝ, 𝑎∈𝐶(ℝ,(0,+∞)), and 𝑎(𝑡)→+∞ as |𝑡|→+∞. Let 𝑉(𝑡,𝑥)=^𝑚_𝑖=1𝑎_𝑖+1+sin𝑡|𝑥|^𝜇𝑖−^𝑛_𝑗=1𝑏_𝑗+1+cos𝑡|𝑥|^𝜚𝑗,𝐼(𝑥)=𝜚₁−2𝑥4𝜚₁,+2𝑝𝜋 (96) where 𝜇₁>𝜇₂>⋯>𝜇_𝑚>𝜚₁>𝜚₂>⋯>𝜚_𝑛>2, 𝑎_𝑖, 𝑏_𝑗>0, 𝑖=1,…,𝑚, and 𝑗=1,…,𝑛. Let 𝑉₁(𝑡,𝑥)=^𝑚_𝑖=1𝑎_𝑖+1+sin𝑡|𝑥|^𝜇𝑖,𝑉₂(𝑡,𝑥)=^𝑛_𝑗=1𝑏_𝑗+1+cos𝑡|𝑥|^𝜚𝑗. (97) Then, it is easy to check that all the conditions of Theorem 4 are satisfied with 𝜇=𝜇_𝑚 and 𝜚=𝜚₁. Hence, problem (95) has an unbounded sequence of fast homoclinic solutions.

Example 2. Consider the following system: 𝑢^′′(𝑡)+𝑡+𝑡³𝑢^′(𝑡)(𝑡)𝑢(𝑡)((𝑡𝑡−𝑎+∇𝑉𝑡,𝑢))=0,a.e.𝑡∈_𝑗,𝑡_𝑗+1,𝑗∈ℤ,Δ𝑢^′𝑡_𝑗=𝑢^′𝑡⁺_𝑗−𝑢^′𝑡⁻_𝑗=𝐼𝑢𝑡_𝑗,𝑗∈ℤ, (98) where 𝑞(𝑡)=𝑡+𝑡³, 𝑡∈ℝ, 𝑢∈ℝ, 𝑎∈𝐶(ℝ,(0,+∞)), and 𝑎(𝑡)→+∞ as |𝑡|→+∞. Let 𝑉(𝑡,𝑥)=𝑎₁+1+sin𝑡|𝑥|^𝜇1+𝑎₂+2+sin𝑡|𝑥|^𝜇2−𝑏₁(cos𝑡)|𝑥|^𝜚1−𝑏₂+1+cos𝑡|𝑥|^𝜚2, (99) where 𝜇₁>𝜇₂>𝜚₁>𝜚₂>2, 𝑎₁, 𝑎₂>0, 𝑏₁, and 𝑏₂>0. Let 𝑉₁(𝑡,𝑥)=𝑎₁+1+sin𝑡|𝑥|^𝜇1+𝑎₂+2+sin𝑡|𝑥|^𝜇2,𝑉₂(𝑡,𝑥)=𝑏₁(cos𝑡)|𝑥|^𝜚1+𝑏₂+1+cos𝑡|𝑥|^𝜚2,𝐼(𝑥)=𝜚₁−2𝑥6𝜚₁.+2𝑝𝜋 (100) Then, it is easy to check that all the conditions of Theorem 5 are satisfied with 𝜇=𝜇₂ and 𝜚=𝜚₁. Hence, by Theorem 5, problem (98) has an unbounded sequence of fast homoclinic solutions.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the NNSF of China (no. 11301108), Guangxi Natural Science Foundation (no. 2013GXNSFBA019004), and the Scientific Research Foundation of Guangxi Education Office (no. 201203YB093).