Existence and Multiplicity of Periodic Solutions Generated by Impulses

Abstract

We investigate the existence and multiplicity of periodic solutions for a class of second-order differential systems with impulses. By using variational methods and critical point theory, we obtain such a system possesses at least one nonzero, two nonzero, or infinitely many periodic solutions generated by impulses under different conditions, respectively. Recent results in the literature are generalized and significantly improved.

1. Introduction

Consider the following second-order impulsive differential equations:𝑢̈(𝑡)+𝑉_𝑢(𝑡,𝑢(𝑡))=0,𝑡∈𝑠_𝑘−1,𝑠_𝑘,Δ𝑢̇𝑠_𝑘=𝜆𝑔_𝑘𝑢𝑠_𝑘,𝑢(0)−𝑢(𝑇)=𝑢̇(0)−𝑢̇(𝑇)=0, (𝑃_𝜆) where 𝑠_𝑘,𝑘=1,2,…,𝑚, are instants in which the impulses occur and 0=𝑠₀<𝑠₁<𝑠₂<⋯<𝑠_𝑚<𝑠_𝑚+1=𝑇, Δ𝑢̇(𝑠_𝑘)=𝑢̇(𝑠⁺_𝑘)−𝑢̇(𝑠⁻_𝑘) with 𝑢̇(𝑠^±_𝑘)=lim_𝑡→𝑠^±_𝑘𝑢̇(𝑡), 𝑔_𝑘(𝑢)=grad_𝑢𝐺_𝑘(𝑢), 𝐺_𝑘∈𝐶¹(ℝ^𝑁,ℝ), 𝑉∈𝐶¹([0,𝑇]×ℝ^𝑁,ℝ), 𝑉_𝑢(𝑡,𝑢(𝑡))=grad_𝑢𝑉(𝑡,𝑢), 𝜆>0 is a constant.

Impulsive differential equations can be used to describe the dynamics of processes which possess abrupt changes at certain instants. Up to now, impulsive differential systems have been widely applied in many science fields such as control theory, biology, mechanics, see [1–7] and references therein. For general theory of impulsive differential equations, we refer the readers to the monographs as [8–10].

The existence of the solutions is one of the most important topics of impulsive differential systems. Many classical methods and tools have been used to study them, such as coincidence degree theory, fixed point theory, and the method of upper and lower solutions. See [11–15] and references therein.

Recently, some authors creatively applied variational method to deal with impulsive problems, see [16–21]. The variational method is opening a new approach for dealing with discontinuity problems such as impulses. However, when the problems studied in [16–21] degenerate to the cases without impulses, plenty of the corresponding results can also be obtained. In other words, the effect of impulses was not seen evidently. As pointed out in [22], these results, in some sense, mean that the nonlinear term 𝑉_𝑢 plays a more important role than the impulsive terms 𝑔_𝑘 do in guaranteeing the existence of solutions. Due to this point, Zhang and Li [22] studied the existence of solutions for impulsive differential systems generated by impulses.

Definition 1.1 (see [22]). A solution is called a solution generated by impulses if this solution is nontrivial when impulsive terms are not zero, but it is trivial when impulsive term is zero.
For example, if problem (𝑃_𝜆) does not possess non-zero solution when 𝑔_𝑘≡0 for all 1≤𝑘≤𝑚, then nonzero solution for problem (𝑃_𝜆) is called solution generated by impulses. In detail, Zhang and Li [22] obtained the following theorem.

Theorem A (see [22]). Assume that 𝑉,𝑊 satisfy the following conditions:(𝑉₁)𝑉 is continuous differentiable and there exist positive constants 𝑏₁,𝑏₂>0 such that 𝑏₁|𝑢|²≤−𝑉(𝑡,𝑢)≤𝑏₂|𝑢|² for all (𝑡,𝑢)∈[0,𝑇]×ℝ^𝑁;(𝑉₂)−𝑉(𝑡,𝑢)≤−𝑉_𝑢(𝑡,𝑢)𝑢≤−2𝑉(𝑡,𝑢) for all (𝑡,𝑢)∈[0,𝑇]×ℝ^𝑁;(𝑔₁) there exists a 𝜃>2 such that 𝑔_𝑘(𝑢)𝑢≤𝜃𝐺_𝑘(𝑢)<0, for 𝑢∈ℝ^𝑁⧵{0} and 𝑘=1,2,…,𝑚. Then, problem (𝑃₁) possesses at least one non-zero solution generated by impulses.

Motivated by the facts mentioned above, in this paper, we will further study the existence of solution for problem (𝑃_𝜆) generated by impulses under more general conditions. In addition, we will investigate the multiple solutions and infinitely many solutions generated by impulses.

Now, we state our results.

Theorem 1.2. Assume that (𝑔₁) and the following (𝑉^′₁), (𝑉^′₂) hold. (𝑉^′₁)𝑉 is continuous differentiable and there exist positive constants 𝑏>0 and 𝛾∈(1,2] such that −𝑉(𝑡,𝑢)≥𝑏|𝑢|^𝛾∀(𝑡,𝑢)∈[0,𝑇]×ℝ^𝑁; (1.1)(𝑉^′₂) there exists a constant 𝜚∈[2,𝜃) such that −𝑉(𝑡,𝑢)≤−𝑉_𝑢(𝑡,𝑢)𝑢≤−𝜚𝑉(𝑡,𝑢)∀(𝑡,𝑢)∈[0,𝑇]×ℝ^𝑁. (1.2)
Then, problem (𝑃₁) possesses at least one non-zero solution generated by impulses.

Remark 1.3. It is easy to see that (𝑉^′₁), (𝑉^′₂) are weaker than (𝑉₁), (𝑉₂). Therefore, Theorem 1.2 improves Theorem A.
Indeed, taking 𝑉(𝑡,𝑢)=−|𝑢|^𝛾−|𝑢|^𝜍,1<𝛾<𝜍≤2,𝐺_𝑘(𝑢)=−|𝑢|^𝜃, (1.3) all conditions in Theorem 1.4 are satisfied, but conditions in Theorem A cannot be satisfied.

Theorem 1.4. Assume that (𝑉₁), (𝑉₂) hold. Moreover, the following conditions are satisfied: (𝑉₃)𝑉_𝑢(𝑡,𝑢₁−𝑢₂)=𝑉_𝑢(𝑡,𝑢₁)−𝑉_𝑢(𝑡,𝑢₂); (𝐻₁)∑^𝑚_𝑘=1𝐺_𝑘(𝑢)≤0 andmaxlimsup_𝑢→0−∑^𝑚_𝑘=1𝐺_𝑘(𝑢)|𝑢|²,limsup_{|𝑢|→+∞}−∑^𝑚_𝑘=1𝐺_𝑘(𝑢)|𝑢|²<𝐴,0<𝐴𝐶²₁<𝑏₃, (1.4) where 𝑏₃=min{1/2,𝑏₁} and 𝐶₁ is a constant which will be defined in Section 2; (𝐻₂) there exists a constant vector 𝜉=(𝜉¹,𝜉²,…,𝜉^𝑁)∈ℝ^𝑁⧵{0} such that ∑^𝑚_𝑘=1𝐺_𝑘(𝜉)<∫^𝑇₀𝑉(𝑡,𝜉)𝑑𝑡.Then, there exists 𝐵>0 such that problem (𝑃₁) possesses at least two non-zero solutions generated by impulses; moreover, their norms are less than 𝐵.

Remark 1.5. Compared with Theorem A, Theorem 1.4 deals with the multiple solutions generated by impulses. In Theorem A and Theorem 1.2, impulses are superquadratic. However, some impulses which are subquadratic can satisfy the conditions of Theorem 1.4.

Example 1.6. Let 𝑉(𝑡,𝑢)=−|𝑢|². It is easy to see that conditions (𝑉₁), (𝑉₂), and (𝑉₃) hold. Let 𝑇=1, 𝑚=1, 𝜉=(1,0,…,0), and 𝐺₁(𝑢)=−2|𝑢|³,|𝑢||𝑢||𝑢|<1,−6+4,≥1. (1.5) Then, we have limsup_𝑢→0−𝐺₁(𝑢)|𝑢|²=limsup_{|𝑢|→+∞}−𝐺₁(𝑢)|𝑢|²=0,−2=𝐺₁(𝜉)<^𝑇₀𝑉(𝑡,𝜉)𝑑𝑡=−1. (1.6) Hence, conditions (𝐻₁) and (𝐻₂) in Theorem 1.4 are satisfied.

Theorem 1.7. Assume that (𝑉₁), (𝑉₂) hold. Moreover, the following conditions are satisfied: (𝐻₃) let 𝛼=liminf_𝜌→+∞max_{|𝑢|≤𝜌}−∑^𝑚_𝑘=1𝐺_𝑘(𝑢)𝜌²,𝛽=limsup_{|𝑢|→+∞}−∑^𝑚_𝑘=1𝐺_𝑘(𝑢)|𝑢|², (1.7) and 𝑇𝑏₂𝐶²₁𝛼<𝑏₃𝛽.

Then, for every 𝜆∈Λ∶=(𝑇𝑏₂/𝛽,𝑏₃/𝐶²₁𝛼), problem (𝑃_𝜆) possesses an unbounded sequence of solutions generated by impulses.

Example 1.8. Let 𝑉(𝑡,𝑢)=(−1/2)|𝑢|²,𝑇=1,𝑚=1,𝑁=1, 𝑎_𝑛=2𝑛!(𝑛+2)!−14(𝑛+1)!,𝑏_𝑛=2𝑛!(𝑛+2)!+14(𝑛+1)!, (1.8)𝑔₁(𝑢)=𝜋(𝑛+1)!(𝑛+1)!²−𝑛!²sin2𝜋(𝑛+1)!𝑢−𝑎_𝑛,|𝑢|∈_𝑛∈ℕ𝑎_𝑛,𝑏_𝑛,0,otherwise, (1.9) and 𝑔₂=max{0,2𝑢}. Let 𝑔=𝑔₁+𝑔₂,−𝐺=^𝑢₀𝑔(𝑡)𝑑𝑡. (1.10) Then, −𝐺 is a 𝐶¹ function with −𝐺^′=𝑔. From the computation of Example  3.2 in [23], one has 𝐶₁=√2,𝑏₂=1/2,𝑏₃=1/2 in this case and lim_𝑛→+∞−𝐺𝑎_𝑛𝑎²_𝑛=1,lim_𝑛→+∞−𝐺𝑏_𝑛𝑏²_𝑛=5. (1.11) Moreover, liminf_{|𝑢|→+∞}−𝐺(𝑢)𝑢²=1,limsup_{|𝑢|→+∞}−𝐺(𝑢)𝑢²=5. (1.12) In addition, −𝐺 is nondecreasing. By the monotonicity of −𝐺, one has max_{|𝑢|≤𝜌}[−𝐺(𝑢)]=−𝐺𝜌. (1.13) Hence, we have 𝛼=1,𝛽=5. (1.14) Then, for 𝜆∈[1/10,1/4], problem 𝑢̈(𝑡)−𝑢=0,𝑡≠𝑠₁,Δ𝑢̇𝑠₁=𝜆𝑔𝑢𝑠₁,𝑢(0)−𝑢(1)=𝑢̇(0)−𝑢̇(1)=0 (1.15) possesses an unbounded sequence of solutions generated by impulses.

Remark 1.9. In Theorem 1.7, substituting 𝜌→+∞ and |𝑢|→+∞ with 𝜌→0⁺ and |𝑢|→0⁺, applying part (g) of Lemma 2.4 instead of part (f) in the proof, we can obtain a sequence of pairwise distinct solutions generated by impulses.
The paper is organized as follows. In Section 2, we present some preliminaries. In Section 3, we give the proof of our main results.

2. Preliminaries

In order to prove our main results, we give some definitions and lemmas that will be used in the proof of our main results. Let 𝐻¹_𝑇=𝑢∶[0,𝑇]→ℝ^𝑁∣𝑢isabsoltelycontinuous,𝑢(0)=𝑢(𝑇),𝑢̇∈𝐿²[0,𝑇],ℝ^𝑛. (2.1) Then, 𝐻¹_𝑇 is a Hilbert space with the inner product⟨𝑢,𝑣⟩=^𝑇₀[(𝑢(𝑡),𝑣(𝑡))+(𝑢̇(𝑡),𝑣̇(𝑡))]𝑑𝑡,∀𝑢,𝑣∈𝐻¹_𝑇, (2.2) where (⋅,⋅) denotes the inner product in ℝ^𝑁. The corresponding norm is defined by ‖𝑢‖=^𝑇₀|𝑢̇(𝑡)|²+|𝑢(𝑡)|²𝑑𝑡^1/2,∀𝑢∈𝐻¹_𝑇. (2.3) Let ‖⋅‖₂=(∫^𝑇₀|𝑢(𝑡)|²)^1/2 denote the norm of Banach space of 𝐿^𝑝([0,𝑇],ℝ^𝑁). Since (𝐻¹_𝑇,‖⋅‖) is compactly embedded in 𝐶([0,𝑇],ℝ^𝑁) (see [24]), we claim that there exists a positive constant 𝐶₁ such that ‖𝑢‖_∞≤𝐶₁‖𝑢‖, (2.4) where ‖𝑢‖_∞=max_{𝑡∈[0,𝑇]}|𝑢(𝑡)|.

To study the problem (𝑃_𝜆), we consider the functional 𝐼 defined by 𝐼(𝑢)=^𝑇₀12|𝑢̇|²−𝑉(𝑡,𝑢)𝑑𝑡+𝜆^𝑚_𝑘=1𝐺_𝑘𝑢𝑠_𝑘. (2.5) Similar to the proof of Lemma 1 of [22] (see also [20, 21]), we can easily prove the following Lemma 2.1.

Lemma 2.1. Suppose 𝑉∈𝐶¹[0,𝑇]×ℝ^𝑁→ℝ, 𝐺_𝑘∈𝐶¹(ℝ^𝑁,ℝ^𝑁), 𝑘=1,2,…,𝑚. Then, 𝐼 is Frechét differentiable with 𝐼^′(𝑢)𝑣=^𝑇₀𝑢̇𝑣̇−𝑉_𝑢(𝑡,𝑢)𝑣𝑑𝑡+𝜆^𝑚_𝑘=1𝑔_𝑘𝑢𝑠_𝑘𝑣𝑠_𝑘, (2.6) for any 𝑢 and 𝑣 in 𝐻¹_𝑇. Furthermore, 𝑢 is a solution of (𝑃_𝜆) if and only if 𝑢 is a critical point of 𝐼 in 𝐻¹_𝑇.

Lemma 2.2 (see [24, 25]). Let 𝐸 be a real Banach space and 𝐼∈𝐶¹(𝐸,ℝ) satisfying the (𝑃.𝑆.) condition. Suppose 𝐼(0)=0 and(𝐼₁) there are constants 𝜌,𝛽>0 such that 𝐼|_𝜕𝐵𝜌≥𝛽, where 𝐵_𝜌={𝑥∈𝐸∣‖𝑥‖<𝜌},(𝐼₂) there is an 𝑒∈𝐸⧵𝐵_𝜌 such that 𝐼(𝑒)≤0.Then, 𝐼 possesses a critical value 𝑐≥𝛽.

If 𝐸 is a real Banach space, denote by 𝒲_𝑋 the class of all functionals Φ∶𝐸→ℝ possessing the following property: if {𝑢_𝑛} is a sequence in 𝐸 converging weakly to 𝑢 and liminf_𝑛→∞Φ(𝑢_𝑛)≤Φ(𝑢), then {𝑢_𝑛} has a subsequence converging strongly to 𝑢. For example, if 𝐸 is uniformly convex and 𝑔∶[0,+∞)→ℝ is a continuous, strictly increasing function, then, by a classical results, the functional 𝑢→𝑔(‖𝑢‖) belongs to the class 𝒲_𝑋.

Lemma 2.3 (see [26]). Let 𝐸 be a separable and reflexive real Banach space; let Φ∶𝐸→ℝ be a coercive, sequentially weakly lower semicontinuous 𝐶¹ functional, belonging to 𝒲_𝑋, bounded on each bounded subset of 𝐸, and whose derivative admits a continuous inverse on 𝐸^∗; 𝐽∶𝐸→ℝ a 𝐶¹ functional with compact derivative. Assume that Φ has a strict local minimum 𝑢₀ with Φ(𝑢₀)=𝐽(𝑢₀)=0. Finally, setting 𝛼^′=max0,limsup_{‖𝑢‖→+∞}𝐽(𝑢)Φ(𝑢),limsup_𝑢→𝑢0𝐽(𝑢)Φ(𝑢),𝛽^′=sup_{𝑢∈Φ−1()0,+∞}𝐽(𝑢)Φ(𝑢), (2.7) assume that 𝛼^′<𝛽^′. Then, for each compact interval [𝑎,𝑏]⊂(1/𝛽^′,1/𝛼^′) (with the conventions 1/0=+∞,1/+∞=0), there exists 𝐵>0 with the following property: for every 𝜆∈[𝑎,𝑏] and every 𝐶¹ functional Ψ∶𝐸→ℝ with compact derivative, there exists 𝛿>0 such that, for each 𝜇∈[0,𝛿], the equation Φ^′(𝑥)=𝜆𝐽^′(𝑥)+𝜇Ψ^′(𝑥) (2.8) has at least three solutions in 𝐸 whose norms are less than 𝐵.

Now, we recall a result which insures infinitely critical points. For all 𝑟>inf_𝑋Φ, we put 𝜓(𝑟)=inf_{𝑥∈Φ−1((−∞,𝑟))}sup_𝑥∈Φ⁻¹_{((−∞,𝑟))}Ψ(𝑥)−Ψ(𝑥)(𝑥),𝑟−Φ𝛾∶=liminf_𝑟→+∞𝜓(𝑟),𝛿∶=liminf_{𝑟→inf𝑋Φ+}𝜓(𝑟). (2.9)

Lemma 2.4 (see [27]). Let 𝑋 be a reflexive real Banach space, let Φ∶𝑋→𝑅 be a (strongly) continuous, coercive, and sequentially weakly lower semicontinuous and G𝑎teaux differentiable function, and let Ψ∶𝑋→𝑅 be a sequentially weakly upper semicontinuous and G𝑎teaux differentiable function. One has(e)for every 𝑟>inf_𝑋Φ and every 𝜆∈(0,1/𝜓(𝑟)), the restriction of the functional Φ−𝜆Ψ to Φ⁻¹((−∞,𝑟)) admits a global minimum, which is a critical point (local minimum) of Φ−𝜆Ψ in 𝑋. (f)if 𝛾<+∞, then, for each 𝜆∈(0,1/𝛾), the following alternative holds:either(f1)Φ−𝜆Ψ possesses a global minimumor(f2)there is a sequence {𝑥_𝑛} of critical points (local minima) of Φ−𝜆Ψ such that lim_𝑛→+∞Φ𝑥_𝑛=+∞. (2.10)(g)if 𝛿<+∞, then, for each 𝜆∈(0,1/𝛿), the following alternative holds:either(g1)here is a global minimum of Φ which is a local minimum of Φ−𝜆Ψor(g2)there is a sequence of pairwise distinct critical points (local minima) of Φ−𝜆Ψ, with lim_𝑛→+∞Φ(𝑥_𝑛)=inf_𝑋Φ which weakly converges to a global minimum of Φ.

3. Proof of the Main Results

In this section, we give the proof of our main results in turn. 𝐶_𝑖,𝑖=1,2,… denote different constants.

Proof of Theorem 1.2. We firstly show that 𝐼 satisfies the 𝑃.𝑆. condition. Assume that {𝑢_𝑗}_𝑗∈ℕ⊂𝐻¹_𝑇 is a sequence such that {𝐼(𝑢_𝑗)}_𝑗∈ℕ is bounded and 𝐼^′(𝑢_𝑗)→0 as 𝑗→+∞. Then, there exists a constant 𝐶₂>0 such that 𝐼𝑢_𝑗≤𝐶₂,𝐼^′𝑢_𝑗≤𝐶₂,∀𝑗∈ℕ. (3.1) By (3.1), (2.4), (2.5), (2.6), (𝑉^′₁), (𝑉^′₂), and (g₁), we have 2𝐶₂+𝐶₃𝑢_𝑗≥2𝐼𝑢_𝑗−2𝜃𝐼^′𝑢_𝑗𝑢_𝑗=21−𝜃^𝑇₀𝑢̇_𝑗²+𝑑𝑡_𝑇02𝜃𝑉_𝑢𝑡,𝑢_𝑗𝑢_𝑗−2𝑉𝑡,𝑢_𝑗𝑑𝑡+2^𝑚_𝑘=1𝐺_𝑘𝑢_𝑗𝑠_𝑘−2𝜃^𝑚_𝑘=1𝑔_𝑘𝑢_𝑗𝑠_𝑘𝑢_𝑗𝑠_𝑘≥21−𝜃^𝑇₀𝑢̇_𝑗²𝜚𝑑𝑡+21−𝜃^𝑇₀𝑏𝑢_𝑗^𝛾≥2𝑑𝑡1−𝜃^𝑇₀𝑢̇_𝑗²𝜚𝑑𝑡+21−𝜃𝑏𝐶₁^𝛾−2𝑢_𝑗^𝛾−2𝑇₀𝑢_𝑗²2𝑑𝑡≥min1−𝜃𝜌,21−𝜃𝑏𝐶₁^𝛾−2𝑢_𝑗^𝛾−2𝑢_𝑗²2≥min1−𝜃𝑢_𝑗²𝜚,21−𝜃𝑏𝐶₁^𝛾−2𝑢_𝑗^𝛾. (3.2) It follows from (3.2) that {𝑢_𝑗}_𝑗∈ℕ is bounded in 𝐻¹_𝑇. In a similar way to Lemma  2 in [22], we can prove that {𝑢_𝑗}_𝑗∈ℕ has a convergent subsequence in 𝐻¹_𝑇. Hence, 𝐼 satisfies the 𝑃.𝑆. condition.
Second, we verify (𝐼₁) of Lemma 2.2. By (g₁) and (2.4), there exists a 𝛿₁>0 such that, for any ‖𝑢‖_∞≤𝛿₁, 𝐺_𝑘(𝑢≤1)4𝑚𝐶₁|𝑢|². (3.3) By (2.4), there exists a 0<𝛿<1 such that, for any ‖𝑢‖≤𝛿, the inequality ‖𝑢‖_∞≤𝛿₁ holds. Then for 𝑢∈𝐻¹_𝑇 with ‖𝑢‖=𝛿₀, 𝛿₀ small enough (0<𝛿₀<𝑚𝑖𝑛{(2𝑏𝐶₁^𝛾−2)^1/2−𝛾,𝛿}) such that the following inequality holds 𝐼(𝑢)≥12^𝑇₀|𝑢̇(𝑡)|²𝑑𝑡+𝑏^𝑇₀|𝑢(𝑡)|^𝛾1𝑑𝑡−4𝐶₁‖𝑢‖²_∞1≥min2‖𝑢‖²,𝑏𝐶₁^𝛾−2‖𝑢‖^𝛾−14‖𝑢‖²≥12‖𝑢‖²−14‖𝑢‖²≥14𝛿₀². (3.4) By (𝑉^′₂), there exist 𝐶₃,𝐶₄>0 such that −𝑉(𝑡,𝑢)≤𝐶₃|𝑢|^𝜚+𝐶₄,∀(𝑡,𝑢)∈[0,𝑇]×ℝ^𝑛. (3.5) By (g₁), there exist 𝐶₅,𝐶₆>0 such that 𝐺_𝑘(𝑢)≤−𝐶₅|𝑢|^𝜃+𝐶₆,∀𝑢∈ℝ^𝑛,𝑘∈{1,2,…,𝑚}. (3.6) To verify (𝐼₂) of Lemma 2.2, choose 𝑢∈𝐻¹_𝑇 such that 𝑢(𝑠_𝑘)≠0 for some 𝑘∈{1,2,…,𝑚}. Hence, we obtain 𝐼()≤𝑡𝑡𝑢²2^𝑇₀|𝑢̇(𝑡)|²𝑑𝑡+𝐶₃|𝑡|^𝜚𝑇₀|𝑢(𝑡)|^𝜚𝑑𝑡−𝐶₅|𝑡|^𝜃𝑚_𝑘=1𝑢𝑠_𝑘^𝜃+𝐶₇, (3.7) where 𝐶₇ is a positive constant. Since 𝜃>2, 𝜚∈[2,𝜃), (3.7) implies 𝐼(𝑡𝑢)→−∞ as 𝑡→∞. So, for 𝑡 large enough, 𝑒=𝑡𝑢 satisfies condition (𝐼₂). By the Mountain Pass Lemma (Lemma 2.2), 𝐼 possesses at least one non-zero critical point. Then, by Lemma 2.1, problem (𝑃₁) has at least one non-zero solution.
Finally, we verify that the solution is generated by impulses. Suppose 𝑔_𝑘≡0 and 𝑢 is a solution for problem (𝑃₁). Then, by (𝑉^′₁) and (𝑉^′₂), 0=^𝑇₀|𝑢̇|²−𝑉_𝑢()𝑢≥𝑡,𝑢𝑑𝑡_𝑇0|𝑢̇|²+𝑏|𝑢|^𝛾≥𝑑𝑡_𝑇0|𝑢̇|²𝑑𝑡+𝑏‖𝑢‖_∞^𝛾−2𝑇₀|𝑢|²≥𝑑𝑡_𝑇0|𝑢̇|²𝑑𝑡+𝑏𝐶₁^𝛾−2‖𝑢‖^𝛾−2𝑇₀|𝑢|²𝑑𝑡≥min1,𝑏𝐶₁^𝛾−2‖𝑢‖^𝛾−2‖𝑢‖²=min‖𝑢‖²,𝑏𝐶₁^𝛾−2‖𝑢‖^𝛾. (3.8) This implies that 𝑢≡0, that is, problem (𝑃₁) does not possess any non-zero solutions when impulses are zero. Hence, by Definition 1.1, the non-zero solution obtained above is generated by impulses.

Proof of Theorem 1.4. In order to apply Lemma 2.3, we let Φ(𝑢)=^𝑇₀12|𝑢̇(𝑡)|²((𝑡𝐽(𝑢)−𝑉𝑡,𝑢))𝑑𝑡,=−^𝑚_𝑘=1𝐺_𝑘𝑢𝑠_𝑘. (3.9) Obviously, 𝐸=𝐻¹_𝑇 is a separable and uniformly convex Banach space. By (𝑉₁), we have 𝑏₃‖𝑢‖²≤Φ(𝑢)≤𝑏₄‖𝑢‖², (3.10) where 𝑏₃=min{1/2,𝑏₁}and𝑏₄=min{1/2,𝑏₂}. Hence, by (3.10) and Lemma 2.1, we can obtain that Φ(𝑢) is a coercive, 𝐶¹ functional, and bounded on each bounded subset of 𝐸. For a sequence {𝑢_𝑛}⊂𝐻¹_𝑇, if 𝑢_𝑛⇀𝑢∈𝐻¹_𝑇 and liminf_𝑛→∞Φ(𝑢_𝑛)≤Φ(𝑢), then 𝑢_𝑛→𝑢∈𝐶[0,𝑇]. This means that ∫^𝑇₀[(−1/2)|𝑢(𝑡)|²−𝑉(𝑡,𝑢(𝑡))]𝑑𝑡 is weakly continuous. Hence, we have that Φ(𝑢)=(1/2)‖𝑢‖²+∫^𝑇₀[(−1/2)|𝑢(𝑡)|²−𝑉(𝑡,𝑢(𝑡))]𝑑𝑡 is sequentially weakly lower semicontinuous and liminf_𝑛→∞‖𝑢_𝑛‖²≤‖𝑢‖². Therefore, {𝑢_𝑛} has a subsequence converging strongly to 𝑢, that is, Φ(𝑢) belongs to 𝒲_𝑋. For any 𝑢∈𝐻¹_𝑇⧵{0}, we have Φ^′(𝑢),𝑢=^𝑇₀(𝑢̇(𝑡),𝑢̇(𝑡))−𝑉_𝑢((𝑡(𝑡)≥𝑡,𝑢)),𝑢𝑑𝑡_𝑇0(𝑢̇(𝑡),𝑢̇(𝑡))+𝑏₁|𝑢(𝑡)|²𝑑𝑡≥min1,𝑏₁‖𝑢‖². (3.11) So, lim_{‖𝑢‖→∞}⟨Φ^′(𝑢),𝑢⟩/‖𝑢‖=+∞, that is, Φ^′ is coercive. For any 𝑢,𝑣∈𝐻¹_𝑇, one has Φ^′(𝑢)−Φ^′(𝑣),𝑢−𝑣=^𝑇₀(𝑢̇(𝑡)−𝑣̇(𝑡),𝑢̇(𝑡)−𝑣̇(𝑡−))𝑑𝑡_𝑇0𝑉_𝑢(𝑡,𝑢(𝑡))−𝑉_𝑢((𝑡(𝑡)(𝑡≥𝑡,𝑣)),𝑢−𝑣)𝑑𝑡_𝑇0(𝑢̇(𝑡)−𝑣̇(𝑡),𝑢̇(𝑡)−𝑣̇(𝑡+))𝑑𝑡_𝑇0𝑏₁|𝑢(𝑡)−𝑣(𝑡)|²𝑑𝑡≥min1,𝑏₁‖𝑢−𝑣‖². (3.12) So, Φ^′ is uniformly monotone. By [28, Theorem  26.A(d)], we have that (Φ^′)⁻¹ exists and is continuous. For any 𝑢,𝑣∈𝐻¹_𝑇, 𝐽^′(𝑢),𝑢=−^𝑚_𝑘=1𝑔_𝑘𝑢𝑠_𝑘𝑢𝑠_𝑘. (3.13) Let 𝑢_𝑛⇀𝑢∈𝐻¹_𝑇, then 𝑢_𝑛→𝑢∈𝐶[0,𝑇]. Hence, 𝐽^′(𝑢_𝑛)→𝐽^′(𝑢), Ψ^′(𝑢_𝑛)→Ψ^′(𝑢) as 𝑛→∞. Therefore, we have that 𝐽^′ is a compact operator by [28, Proposition  26.2]. In addition, Φ has a strict local minimum 0 with Φ(0)=𝐽(0)=0. In view of (𝐻₁), there exist 𝜏₁>𝜏₂>0 such that −^𝑚_𝑘=1𝐺_𝑘(𝑢)≤𝐴|𝑢|² (3.14) for any |𝑢|∈[0,𝜏₂)∪(𝜏₁,+∞). By the continuity of 𝐺_𝑘, ∑^𝑚_𝑘=1𝐺_𝑘(𝑢) is bounded for any |𝑢|∈[𝜏₂,𝜏₁]. We can choose 𝐶₈>0 and 𝜎>2 such that −^𝑚_𝑘=1𝐺_𝑘(𝑢)≤𝐴|𝑢|²+𝐶₈|𝑢|^𝜎 (3.15) for any 𝑢∈ℝ^𝑁. Hence, by (2.4), we have 𝐽(𝑢)≤𝐴𝐶²₁‖𝑢‖²+𝐶₈𝐶^𝜎₁‖𝑢‖^𝜎 (3.16) for all 𝑢∈𝐻¹_𝑇. By (3.16) and (𝐻₁), we obtain limsup_𝑢→0𝐽(𝑢)Φ(𝑢)≤limsup_𝑢→0𝐴𝐶²₁𝑏₃+𝐶₈𝐶^𝜎₁𝑏₃‖𝑢‖^𝜎−2≤𝐴𝐶²₁𝑏₃<1. (3.17) On the other hand, by (𝐻₁), −∑^𝑚_𝑘=1𝐺_𝑘(𝑢)≥0. Then, one has −∑^𝑚_𝑘=1𝐺_𝑘𝑢𝑠_𝑘Φ(𝑢)=−∑^𝑚_𝑘=1𝐺_𝑘𝑢𝑠_𝑘|_{|𝑢(𝑠𝑘)|≤𝜏1}Φ(𝑢)+−∑^𝑚_𝑘=1𝐺_𝑘𝑢𝑠_𝑘|_{|𝑢(𝑠𝑘)|>𝜏1}Φ(𝑢)≤𝐶₉Φ(𝑢)+−∑^𝑚_𝑘=1𝐺_𝑘𝑢𝑠_𝑘|_{|𝑢(𝑠𝑘)|>𝜏1}Φ(𝑢)≤𝐶₉𝑏₃‖𝑢‖²+𝐴𝑢𝑠_𝑘²𝑏₃‖𝑢‖²≤𝐶₉𝑏₃‖𝑢‖²+𝐴𝐶²₁‖𝑢‖²𝑏₃‖𝑢‖², (3.18) where 𝐶₉ is a positive constant and 𝑓|_𝐴 denotes the restriction of the functional 𝑓 to the set 𝐴. Hence, we have limsup_{‖𝑢‖→∞}𝐽(𝑢)Φ(𝑢)≤limsup_{‖𝑢‖→∞}𝐶₉𝑏₃‖𝑢‖²+𝐴𝐶²₁𝑏₃≤𝐴𝐶²₁𝑏₃<1. (3.19) Combining (3.18) with (3.19), one has 𝛼^′<1.
By {𝐻₂}, we obtain 𝛽^′=sup_{𝑢∈Φ−1()0,+∞}𝐽(𝑢)Φ(𝑢)≥−∑^𝑚_𝑘=1𝐺_𝑘(𝜉)−∫^𝑇₀𝑉(𝑡,𝜉)𝑑𝑡>1. (3.20) Then, for each compact interval [𝑎,𝑏]⊂(1/𝛽^′,1/𝛼^′), 1∈[𝑎,𝑏], there exists 𝐵>0 with the following property: for every 𝑓(𝑡), there exists 𝛿>0 such that, for each 𝜇∈[0,𝛿], Φ^′(𝑥)=𝐽^′(𝑥)+𝜇Ψ^′(𝑥). (3.21) Let 𝜇=0, then we can obtain that problem (𝑃₁) has at least three solutions in 𝐸 whose norms are less than 𝐵.

Finally, by Theorem  4 in [22], we can prove that problem (𝑃₁) does not possess any non-zero periodic solutions when impulses are zero. By Definition 1.1, problem (𝑃₁) has at least two solutions generated by impulses.

Proof of Theorem 1.7. Let Φ(𝑢)=^𝑇₀12|𝑢̇(𝑡)|²((𝑡Ψ(𝑢)−𝑉𝑡,𝑢))𝑑𝑡,=−^𝑚_𝑘=1𝐺_𝑘𝑢𝑠_𝑘. (3.22) Similar to the proof of Theorems  3.1 and 3.3 in [27], it is easy to see that the functionals Φ,Ψ satisfy the regularity assumptions required in Lemma 2.4. Let us now verify that 𝛾<+∞. (3.23) Let {𝜌_𝑛} be a sequence of positive numbers such that 𝜌_𝑛→+∞ and lim_𝑛→+∞max_{|𝑢|≤𝜌𝑛}−∑^𝑚_𝑘=1𝐺_𝑘(𝑢)𝜌²_𝑛=liminf_𝜌→+∞max_{|𝑢|≤𝜌𝑛}−∑^𝑚_𝑘=1𝐺_𝑘(𝑢)𝜌². (3.24) If we put, for every 𝑛∈ℕ, 𝑟_𝑛=𝜌²_𝑛𝑏₃𝐶²₁, (3.25) then, by (2.4) and (3.10), Φ(𝑢)≤𝑟_𝑛 implies that ‖𝑢‖_∞≤𝜌_𝑛. Since 0∈Φ⁻¹((−∞,𝑟_𝑛)), the following inequality holds: 𝜓𝑟_𝑛=inf_{𝑥∈Φ−1−∞,𝑟𝑛}sup_𝑥∈Φ⁻¹_{((−∞,𝑟𝑛))}Ψ(𝑥)−Ψ(𝑥)𝑟_𝑛(𝑥)≤𝐶−Φ₂₁max_{|𝑢|≤𝜌𝑛}−∑^𝑚_𝑘=1𝐺_𝑘(𝑢)𝜌²_𝑛𝑏₃. (3.26) Hence, 𝐶0<𝛾≤²¹𝛼𝑏₃<+∞. (3.27) From 𝜆∈Λ∶=(𝑇𝑏₂/𝛽,𝑏₃/𝐶²₁𝛼), we have 1Λ⊂0,𝛾. (3.28) Fix 𝜆∈Λ and let us verify that the functional 𝐼=Φ−𝜆Ψ is unbounded from below. In fact, by the choice of 𝜆 and the positivity of 𝛽, one has that there exists a sequence {𝑦_𝑛} in ℝ^𝑛 such that lim_𝑛→∞|𝑦_𝑛|=+∞ and −∑^𝑚_𝑘=1𝐺_𝑘(𝑦_𝑛)/|𝑦_𝑛|²>𝑇𝑏₂/𝜆. Let 𝑣_𝑛(𝑡)=𝑦_𝑛. Then, 𝐼𝑣_𝑛=^𝑇₀−𝑉𝑡,𝑣_𝑛𝑑𝑡−𝜆−^𝑚_𝑘=1𝐺_𝑘𝑣_𝑛≤𝑇𝑏₂−∑−𝜆^𝑚𝑘=1𝐺_𝑘𝑦_𝑛𝑦_𝑛²𝑦_𝑛². (3.29) Note that 𝑇𝑏₂−𝜆(−∑^𝑚_𝑘=1𝐺_𝑘(𝑦_𝑛)/|𝑦_𝑛|²)<0, 𝐼 has no global minimum. Hence, by Lemma 2.4, there is a sequence {𝑥_𝑛} of critical points (local minima) of Φ−𝜆Ψ such that lim_𝑛→+∞Φ(𝑥_𝑛)=+∞. By Theorem  4 in [22], we can prove that problem (𝑃_𝜆) does not possess any non-zero periodic solutions when impulses are zero. Therefore, by Definition 1.1, the solutions obtained above are all generated by impulses.

Acknowledgment

The paper was supported by the Foundation for Youth Key Teacher by the Hengyang Normal University. H. Chen was supported by NFSC (10871206).