Variational Approach to Impulsive Problems: A Survey of Recent Results

Abstract

We present a survey on the existence of nontrivial solutions to impulsive differential equations by using variational methods, including solutions to boundary value problems, periodic solutions, and homoclinic solutions.

1. Introduction

There are many processes and phenomena in the real world, which are subjected during their development to the short-term external influences. Their duration is negligible compared with the total duration of the studied phenomena and processes. Therefore, it can be assumed that these external effects are “instantaneous”; that is, they are in the form of impulses. The investigation of such “leaps and bounds” developing dynamical states is a subject of different sciences: mechanics, control theory, pharmacokinetics, epidemiology, population dynamics, economics, ecology, and so forth. We refer the reader to [1–6].

During the last two decades, the theory of the existence of solutions for impulsive differential equations has attracted much attention because of its important applications; see [3, 7–12]. Classical approaches to such problems include fixed point theory and the method of upper and lower solutions. In 2008, Tian and Ge first applied variational methods in the study of boundary value problems (BVPs) of impulsive differential equations. From then on, variational methods have been widely used to study impulsive problems, such as boundary value problems, periodic solutions, and homoclinic solutions. We refer the reader to [3, 13–22].

The main aim of this survey is to present some recent existence results for impulsive differential equations via variational methods. More precisely, we will explore the variational framework of impulsive differential equations and study the existence and multiplicity of solutions of boundary value problems, periodic solutions, and homoclinic solutions.

The rest of this paper is organized as follows. In Section 2, we will state some important results for the second-order impulsive differential equations with two kinds of boundary conditions. Periodic solutions to impulsive problems will be considered in Section 3. In Section 4, homoclinic solutions to impulsive differential equations will be studied.

Finally in this section, we must note that besides the results presented in this survey many interesting and important results on impulsive differential equations via variational methods have been obtained by other researchers; see, for example, [23–31] and the references cited therein.

2. Boundary Value Problems

In this section, we recall some results of boundary value problems for impulsive differential equations.

Firstly, we consider a class of impulsive problems with Dirichlet boundary condition −𝑢^′′(𝑡)+𝜆𝑢(𝑡)=𝑓(𝑡,𝑢(𝑡))+𝑔(𝑡),a.e.𝑡∈[0,𝑇],Δ𝑢^′𝑡_𝑗=𝑢^′𝑡⁺_𝑗−𝑢^′𝑡⁻_𝑗=𝐼_𝑗𝑢𝑡_𝑗𝑢(0)(𝑇),𝑗=1,2,…,𝑝,=𝑢=0, (1) where 𝜆 is a constant, 𝑓∶[0,𝑇]×ℝ→ℝ is continuous, 𝑔∈𝐿²[0,𝑇] and 𝐼_𝑗∶ℝ→ℝ, 𝑗=1,2,...,𝑝, are continuous.

From now on, we assume that 𝜆>−(𝜋²/𝑇²) and define 𝐿()=1𝑢,𝑢2^𝑇₀𝑢^′(𝑡)²𝜆𝑑𝑡+2^𝑇₀|𝑢(𝑡)|²𝑑𝑡,forany𝑢∈𝐻¹₀(0,𝑇). (2) Then we have the following.

Proposition 1 (see [21, Proposition 1.1]). If 𝜆>−(𝜋²/𝑇²), then there exist constants 𝑏>𝑎>0 such that 𝑎‖𝑢‖²_𝐻10≤𝐿(𝑢,𝑢)≤𝑏‖𝑢‖²_𝐻10,𝑓𝑜𝑟𝑎𝑛𝑦𝑢∈𝐻¹₀(0,𝑇). (3)

Definition 2. One says that a function 𝑢∈𝐻¹₀(0,𝑇) is a weak solution to problem (1) if ^𝑇₀𝑢^′(𝑡),𝑣^′(𝑡)+𝜆(𝑢(𝑡),𝑣(𝑡))𝑑𝑡+^𝑝_𝑗=1𝐼_𝑗𝑢𝑡_𝑗𝑣𝑡_𝑗=^𝑇₀𝑓(𝑡,𝑢)𝑣𝑑𝑡+^𝑇₀𝑔(𝑡)𝑣𝑑𝑡 (4) holds for any 𝑣∈𝐻¹₀(0,𝑇).

Theorem 3 (see [15, Theorem 4.2]). Suppose that 𝑔≡0, 𝑓 is bounded, and the impulsive functions 𝐼_𝑗 are bounded. Then problem (1) has at least one weak solution to 𝜆>−(𝜋²/𝑇²).

If 𝑓 is sublinear at infinity on 𝑢 and the impulsive functions 𝐼_𝑗 have sublinear growth, then we have the following.

Theorem 4 (see [15, Theorem 4.3]). Assume that 𝑔≡0 and the following conditions are satisfied. (𝐹1)There exist 𝑎,𝑏>0 and 𝛾∈(0,1) such that 𝑓(𝑡,𝑢)≤𝑎+𝑏|𝑢|^𝛾,𝑓𝑜𝑟𝑒𝑣𝑒𝑟𝑦(𝑡,𝑢)∈[0,𝑇]×ℝ; (5)(𝐼1)there exist 𝑎_𝑗,𝑏_𝑗>0 and 𝛾_𝑗∈[0,1)(𝑗=1,2,…,𝑝) such that 𝐼_𝑗(𝑢)≤𝑎_𝑗+𝑏_𝑗|𝑢|^𝛾𝑗,𝑓𝑜𝑟𝑒𝑣𝑒𝑟𝑦𝑢∈ℝ𝑗=1,2,…,𝑝. (6)Then problem (1) has at least one weak solution to 𝜆>−(𝜋²/𝑇²).

If 𝑓 is superlinear at infinity on 𝑢, then we have the following.

Theorem 5 (see [21, Theorem 1.2]). Assume that (𝐼1) holds and the following conditions are satisfied. (𝐹2)There is a constant 𝜇>2 such that 0<𝜇𝐹(𝑡,𝑢)≤𝑢𝑓(𝑡,𝑢) for every 𝑡∈[0,𝑇] and 𝑢∈ℝ⧵{0};(𝐺)𝑔∈𝐿²[0,𝑇] satisfies the following inequality 𝑀𝑎−𝜆₁−^𝑝_𝑗=1𝑇𝑎_𝑗+𝑏_𝑗−√𝑇𝑔_𝐿2𝜆₁>0, (7)where 𝑀∶=sup{𝐹(𝑡,𝑢)∶𝑡∈[0,𝑇],|𝑢|=1} and 𝑎 is defined in Proposition 1.
Then problem (1) has at least one weak solution to 𝜆>−(𝜋²/𝑇²).

Furthermore, we have the following result of multiple solutions.

Theorem 6 (see [21, Theorem 1.3]). Let 𝑔(𝑡)≡0. Assume that the conditions (𝐹2), (𝐼1), and (𝐺) hold. Moreover, 𝑓 and the impulsive functions 𝐼_𝑗 are odd about 𝑢; then problem (1) has infinitely many solutions.

Next, we consider another Dirichlet impulsive problem −𝑢^′′(𝑡)+𝑞(𝑡)𝑢(𝑡)=𝑓(𝑡,𝑢(𝑡)),a.e.𝑡∈[0,𝑇],Δ𝑢^′𝑡_𝑗=𝑢^′𝑡⁺_𝑗−𝑢^′𝑡⁻_𝑗=𝐼_𝑗𝑢𝑡_𝑗𝑢(0)(𝑇),𝑗=1,2,…,𝑝,=𝑢=0, (8) where 𝑞∈𝐿^∞[0,𝑇], 𝑓∶[0,𝑇]×ℝ→ℝ is continuous, and 𝐼_𝑗∶ℝ→ℝ, 𝑗=1,2,…,𝑝, are continuous.

Theorem 7 (see [20, Theorem 3.1]). Let 𝑑_𝑗 (𝑗=1,2…,𝑝) be fixed constants. If 𝑓(𝑡,𝑢)=𝜎(𝑡)∈𝐿²(0,𝑇) and 𝐼_𝑗(𝑢)=𝑑_𝑗 (𝑗=1,2,…,𝑝); then problem (8) has a unique solution 𝑢 if ess inf_{𝑡∈[0,𝑇]}𝑞(𝑡)>−(𝜋²/𝑇²).

Similar to Theorem 4, we have the following.

Theorem 8 (see [20, Theorem 3.2]). Assume that conditions (𝐹1) and (𝐼1) hold. Then problem (8) has at least one weak solution to ess inf_{𝑡∈[0,𝑇]}𝑞(𝑡)>−(𝜋²/𝑇²).

If condition (𝐼1) is replaced by the following condition:(𝐼3)𝐼_𝑗 (𝑗=1,2…,𝑝) satisfy ∫^𝑢₀𝐼_𝑗(𝑠)𝑑𝑠≥0 for all 𝑢∈ℝ,then we have the following.

Theorem 9 (see [30, Theorem 3.1]). Assume that conditions (𝐹1) and (𝐼3) hold. Then problem (8) has at least one weak solution to ess inf_{𝑡∈[0,𝑇]}𝑞(𝑡)>−(𝜋²/𝑇²).

Now we study a class of impulsive Hamiltonian systems with periodic boundary conditions −̈(𝑡)()()[],Δ𝑢̇𝑢+𝐴𝑢=𝜆∇𝐹𝑡,𝑢+𝜇∇𝐺𝑡,𝑢,a.e.𝑡∈0,𝑇_𝑖𝑡_𝑗=𝑢̇^𝑖𝑡⁺_𝑗−𝑢̇^𝑖𝑡⁻_𝑗=𝐼_𝑖𝑗𝑢^𝑖𝑡_𝑗,𝑢(0)(𝑇)=𝑢̇(0)−𝑢̇(𝑇)𝑖=1,2,…,𝑁,𝑗=1,2,…,𝑙,−𝑢=0, (9) where 𝐴∶[0,𝑇]→ℝ^𝑁×𝑁 is a continuous map from the interval [0,𝑇] to the set of 𝑁-order symmetric matrices, 𝜆, 𝜇 are two positive parameters, 𝑇 is a real positive number, 𝑢(𝑡)=(𝑢¹(𝑡),𝑢²(𝑡),…,𝑢^𝑁(𝑡)), 𝑡_𝑗,𝑗=1,2,…,𝑙, are the instants where the impulses occur and 0=𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝑙<𝑡_𝑙+1=𝑇, 𝐼_𝑖𝑗∶ℝ→ℝ (𝑖=1,2,…,𝑁,𝑗=1,2,…,𝑙) are continuous, and 𝐹,𝐺∶[0,𝑇]×ℝ^𝑁→ℝ are measurable with respect to 𝑡, for every 𝑢∈ℝ^𝑁, continuously differentiable in 𝑢, for almost every 𝑡∈[0,𝑇], and satisfy the following standard summability condition: sup_{|𝑢|≤𝑏}𝐹(,|𝐺(,|(,|(max{|⋅,𝑢)|⋅,𝑢)|∇𝐹⋅,𝑢)|∇𝐺⋅,𝑢)|}∈𝐿¹(0,𝑇),∀𝑏>0. (10)

If 𝑋 is a real Banach space, denote by 𝒲_𝑋 the class of all functionals Φ∶𝑋→ℝ possessing the following property: if {𝑢_𝑛} is a sequence in 𝑋 converging weakly to 𝑢∈𝑋 and liminf_𝑛→∞Φ(𝑢_𝑛)≤Φ(𝑢), then {𝑢_𝑛} has a subsequence converging strongly to 𝑢.

For example, if 𝑋 is uniformly convex and 𝑔∶[0,+∞)→ℝ is a continuous, strictly increasing function, then, by a classical result, the functional 𝑢→𝑔(‖𝑢‖) belongs to the class 𝒲_𝑋.

Theorem 10 (see [32]). Let 𝑋 be a separable and reflexive real Banach space; let Φ∶𝑋→ℝ be a coercive, sequentially weakly lower semicontinuous 𝐶¹ functional, belonging to 𝒲_𝑋, bounded on each bounded subset of 𝑋, whose derivative admits a continuous inverse on 𝑋^∗; 𝐽∶𝑋→ℝ  is 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), (11) assume that 𝛼<𝛽.
Then, for each compact interval [𝑎,𝑏]⊂(1/𝛽,1/𝛼) (with the conventions 1/0=+∞,1/+∞=0), there exists 𝑅>0 with the following property: for every 𝜆∈[𝑎,𝑏] and every 𝐶¹ functional Ψ∶𝑋→ℝ with compact derivative, there exists 𝛿>0 such that, for each 𝜇∈[0,𝛿], Φ^′(𝑥)=𝜆𝐽^′(𝑥)+𝜇Ψ^′(𝑥) (12) has at least three solutions in 𝑋 whose norms are less than 𝑅.

The following two results of Ricceri guarantee the existence of three solutions for a given equation.

Theorem 11 (see [33]). Let 𝑋 be a reflexive real Banach space; 𝐼⊆ℝ is an interval; let Φ∶𝑋→ℝ be a sequentially weakly lower semicontinuous 𝐶¹ functional, bounded on each bounded subset of 𝑋, whose derivative admits a continuous inverse on 𝑋^∗; −𝐽∶𝑋→ℝ is a 𝐶¹ functional with compact derivative. Assume that lim_{‖𝑥‖→+∞}(Φ(𝑥)−𝜆𝐽(𝑥))=+∞ (13) for all 𝜆∈𝐼 and that there exists 𝜌∈ℝ such that sup_𝜆∈𝐼inf_𝑥∈𝑋Φ(𝑥)+𝜆𝜌−𝐽(𝑥)<inf_𝑥∈𝑋sup_𝜆∈𝐼Φ(𝑥)+𝜆𝜌−𝐽(𝑥). (14) Then there exist a nonempty open set 𝐴⊆𝐼 and a positive number 𝑅 with the following property: for every 𝜆∈𝐴 and every 𝐶¹ functional −Ψ∶𝑋→ℝ with compact derivative, there exists 𝛿>0 such that, for each 𝜇∈[0,𝛿], Φ^′(𝑥)−𝜆𝐽^′(𝑥)−𝜇Ψ^′(𝑥)=0 (15) has at least three solutions in 𝑋 whose norms are less than 𝑅.

Proposition 12 (see [34]). Let 𝑋 be a nonempty set and Φ,𝐽 two real functions on 𝑋. Assume that there are 𝑟>0 and 𝑥₀,𝑥₁∈𝑋 such that Φ𝑥₀=𝐽𝑥₀=0,Φ𝑥₁>𝑟,sup_{𝑥∈Φ−1((−∞,𝑟])}𝐽(𝑥)𝐽𝑥<𝑟¹Φ𝑥₁. (16) Then, for each 𝜌, satisfying sup_{𝑥∈Φ−1((−∞,𝑟])}𝐽(𝑥)𝐽𝑥<𝜌<𝑟¹Φ𝑥₁, (17) one has sup_𝜆≥0inf_𝑥∈𝑋Φ(𝑥)+𝜆𝜌−𝐽(𝑥)<inf_𝑥∈𝑋sup_𝜆≥0Φ(𝑥)+𝜆𝜌−𝐽(𝑥). (18)

We always assume that the following conditions hold:(𝒜₁)𝐴(𝑡)=(𝑎_𝑖𝑗(𝑡)) is a symmetric matrix with 𝑎_𝑖𝑗∈𝐿^∞([0,𝑇]) for every 𝑡∈[0,𝑇];(𝒜₂)there exists a positive constant 𝜈 such that 𝐴(𝑡)𝑢⋅𝑢≥𝜈|𝑢|²,forevery𝑢∈ℝ^𝑁,a.e.𝑡∈[0,𝑇]; (19)(ℐ₁)for any 𝑖=1,2,…,𝑁, and 𝑗=1,2,…,𝑙, 𝐼_𝑖𝑗(𝑡) are nondecreasing in 𝑡∈ℝ and 𝐼_𝑖𝑗(𝑡)𝑡≥0,∀𝑡∈ℝ. (20)Define the Sobolev space by 𝐻¹_𝑇∶=𝑢∶[0,𝑇]⟶ℝ^𝑁𝑢(0)(𝑇)̇∶𝑢isabsolutelycontinuous,=𝑢,and𝑢∈𝐿²(0,𝑇) (21) with the inner product ⟨𝑢,𝑣⟩=^𝑇₀(𝑢(𝑡),𝑣(𝑡))𝑑𝑡+^𝑇₀(𝑢̇(𝑡),𝑣̇(𝑡))𝑑𝑡,∀𝑢,𝑣∈𝐻¹_𝑇, (22) where (⋅,⋅) denotes the inner product in ℝ^𝑁. The corresponding norm is defined by ‖𝑢‖_𝐻¹_𝑇=^𝑇₀|𝑢(𝑡)|²𝑑𝑡+^𝑇₀|𝑢̇(𝑡)|²𝑑𝑡^1/2,∀𝑢∈𝐻¹_𝑇. (23) For every 𝑢,𝑣∈𝐻¹_𝑇, we define ⟨𝑢,𝑣⟩₁=^𝑇₀(𝐴(𝑡)𝑢(𝑡)(𝑡+,𝑣))𝑑𝑡_𝑇0(𝑢̇(𝑡),𝑣̇(𝑡))𝑑𝑡,∀𝑢,𝑣∈𝐻¹_𝑇 (24) and observe that, using assumptions (𝒜₁) and (𝒜₂), (24) defines an inner product in 𝐻¹_𝑇, whose corresponding norm is ‖𝑢‖∶=⟨𝑢,𝑢⟩₁^1/2. (25) A simple computation shows that (𝐴(𝑡)𝑢,𝑢)=^𝑁_𝑖,𝑗=1𝑎_𝑖,𝑗(𝑡)𝑢^𝑖𝑢^𝑗≤^𝑁_𝑖,𝑗=1𝑎_𝑖,𝑗∞|𝑢|², (26) for every 𝑡∈[0,𝑇] and 𝑢∈ℝ^𝑁. So, putting together (𝒜₂) and (26), one has √𝑚‖𝑢‖_𝐻¹_𝑇≤‖𝑢‖≤√𝑀‖𝑢‖_𝐻¹_𝑇, (27) where 𝑚=min{1,𝜈} and 𝑀=max{1,∑^𝑁_𝑖,𝑗=1‖𝑎_𝑖,𝑗‖_∞}; that is, the norm ‖⋅‖ is equivalent to (23). At this point, since (𝐻¹_𝑇,‖⋅‖_𝐻¹_𝑇) is compactly embedded in 𝐶(0,𝑇) (see [35]), due to (27) we claim that there exists a positive number 𝑘 such that ‖𝑢‖_𝐶≤𝑘‖𝑢‖, (28) where 𝑘=(1/√𝑚)max{√𝑇,1/√𝑇} and ‖𝑢‖_𝐶=max_{𝑡∈[0,𝑇]}|𝑢(𝑡)|.

If 𝑢∈𝐻¹_𝑇, then 𝑢 is absolutely continuous and 𝑢̇∈𝐿²(0,𝑇;ℝ^𝑁). In this case, Δ𝑢̇(𝑡)=𝑢̇(𝑡⁺)−𝑢̇(𝑡⁻)=0 is not necessarily valid for every 𝑡∈(0,𝑇) and the derivative 𝑢̇ may present some discontinuities. This leads to the impulsive effects.

Following the ideas of [15], take 𝑣∈𝐻¹_𝑇 and multiply the two sides of the equality −𝑢̈+𝐴(𝑡)𝑢−𝜆∇𝐹(𝑡,𝑢)−𝜇∇𝐺(𝑡,𝑢)=0 (29) by 𝑣 and integrate from 0 to 𝑇; we have ^𝑇₀−𝑢̈+𝐴(𝑡)𝑢−𝜆∇𝐹(𝑡,𝑢)−𝜇∇𝐺(𝑡,𝑢)𝑣𝑑𝑡=0. (30) Moreover, combining 𝑢̇(0)−𝑢̇(𝑇)=0, one has ^𝑇₀(𝑢̈(𝑡),𝑣(𝑡))𝑑𝑡=^𝑙_𝑗=0^𝑡𝑗+1_𝑡𝑗(𝑢̈(𝑡)(𝑡=,𝑣))𝑑𝑡_𝑙𝑗=0𝑢̇𝑡⁻_𝑗+1,𝑣𝑡⁻_𝑗+1−𝑢̇𝑡⁺_𝑗,𝑣𝑡⁺_𝑗−^𝑙_𝑗=0^𝑡𝑗+1_𝑡𝑗(𝑢̇(𝑡),𝑣̇(𝑡=))𝑑𝑡_{𝑙𝑁𝑗=0𝑖=1}𝑢̇^𝑖𝑡⁻_𝑗+1𝑣^𝑖𝑡⁻_𝑗+1−𝑢̇^𝑖𝑡⁺_𝑗𝑣^𝑖𝑡⁺_𝑗−^𝑙_𝑗=0^𝑡𝑗+1_𝑡𝑗(𝑢̇(𝑡),𝑣̇(𝑡=𝑢̇(𝑇)𝑣(𝑇)−𝑢̇(0)𝑣(0)−))𝑑𝑡_{𝑙𝑁𝑗=1𝑖=1}Δ𝑢̇^𝑖𝑡_𝑗𝑣^𝑖𝑡_𝑗−^𝑇₀(𝑢̇(𝑡),𝑣̇(𝑡))𝑑𝑡=−^𝑙𝑁_𝑗=1𝑖=1𝐼_𝑖𝑗𝑢^𝑖𝑡_𝑗𝑣^𝑖𝑡_𝑗−^𝑇₀(𝑢̇(𝑡),𝑣̇(𝑡))𝑑𝑡. (31) Combining (30), we get ^𝑇₀(𝑢̇(𝑡),𝑣̇(𝑡+(𝐴(𝑡)𝑢(𝑡)(𝑡+)),𝑣))𝑑𝑡_{𝑙𝑁𝑗=1𝑖=1}𝐼_𝑖𝑗𝑢^𝑖𝑡_𝑗𝑣^𝑖𝑡_𝑗−𝜆^𝑇₀∇𝐹(𝑡,𝑢)𝑣𝑑𝑡−𝜇^𝑇₀∇𝐺(𝑡,𝑢)𝑣𝑑𝑡=0. (32) Considering the above equality, we introduce the following concept of a weak solution to problem (9).

Definition 13. One says that a function 𝑢∈𝐻¹_𝑇 is a weak solution to problem (9) if ^𝑇₀(𝑢̇(𝑡),𝑣̇(𝑡+(𝐴(𝑡)𝑢(𝑡)(𝑡+)),𝑣))𝑑𝑡_{𝑙𝑁𝑗=1𝑖=1}𝐼_𝑖𝑗𝑢^𝑖𝑡_𝑗𝑣^𝑖𝑡_𝑗=𝜆^𝑇₀∇𝐹(𝑡,𝑢)𝑣𝑑𝑡+𝜇^𝑇₀∇𝐺(𝑡,𝑢)𝑣𝑑𝑡 (33) holds for any 𝑣∈𝐻¹_𝑇.

In order to study problem (9), we will use the functionals Φ,𝐽,Ψ∶𝐻¹_𝑇→ℝ defined by putting Φ(𝑢)=12‖𝑢‖²+^𝑙𝑁_𝑗=1𝑖=1^{𝑢𝑖(𝑡𝑗)}₀𝐼_𝑖𝑗(𝑡)(𝑢)(𝑢),𝐽(𝑢)𝑑𝑡=𝜙+𝜓∶=^𝑇₀𝐹(𝑡,𝑢)𝑑𝑡,Ψ(𝑢)∶=^𝑇₀𝐺(𝑡,𝑢)𝑑𝑡, (34) respectively, for every 𝑢∈𝐻¹_𝑇; here, 𝜙(𝑢)=12‖𝑢‖²,𝜓(𝑢)=^𝑙𝑁_𝑗=1𝑖=1^{𝑢𝑖(𝑡𝑗)}₀𝐼_𝑖𝑗(𝑡)𝑑𝑡. (35) By the continuity of 𝐼_𝑖𝑗, 𝑖=1,2,…,𝑁, 𝑗=1,2,…,𝑙, we get that functional Φ is a continuous Gâteaux differential functional whose Gâteaux derivative is the functional Φ^′(𝑢), given by Φ^′(𝑢),𝑣=^𝑇₀(𝐴(𝑡)𝑢(𝑡)(𝑡+(𝑢̇(𝑡),𝑣̇(𝑡+,𝑣))))𝑑𝑡_{𝑙𝑁𝑗=1𝑖=1}𝐼_𝑖𝑗𝑢^𝑖𝑡_𝑗𝑣^𝑖𝑡_𝑗, (36) for any 𝑢,𝑣∈𝐻¹_𝑇. On the other hand, condition (10) implies that 𝐽 and Ψ are well defined, continuously Gâteaux differentiable in 𝐻¹_𝑇. More precisely, their Gâteaux derivatives are 𝐽^′(𝑢),𝑣=^𝑇₀()Ψ∇𝐹𝑡,𝑢𝑣𝑑𝑡,_′(𝑢),𝑣=^𝑇₀∇𝐺(𝑡,𝑢)𝑣𝑑𝑡, (37) respectively, for every 𝑢,𝑣∈𝐻¹_𝑇. Put 𝑞=^𝑁_𝑖,𝑗=1𝑎_𝑖,𝑗∞,𝜆₁‖𝑢‖=inf²+2∑^𝑙_𝑗=1∑^𝑁_𝑖=1∫^{𝑢𝑖(𝑡𝑗)}₀𝐼_𝑖𝑗(𝑡)𝑑𝑡2∫^𝑇₀𝐹(𝑡,𝑢(𝑡))𝑑𝑡∶𝑢∈𝐻¹_𝑇,_𝑇0𝐹((𝑡,𝜆𝑡,𝑢))𝑑𝑡>0₂=max0,limsup_{‖𝑢‖→+∞}2∫^𝑇₀𝐹(𝑡,𝑢(𝑡))𝑑𝑡‖𝑢‖²+2∑^𝑙_𝑗=1∑^𝑁_𝑖=1∫^{𝑢𝑖(𝑡𝑗)}₀𝐼_𝑖𝑗(𝑡),𝑑𝑡limsup_|𝑢|→02∫^𝑇₀𝐹(𝑡,𝑢(𝑡))𝑑𝑡‖𝑢‖²+2∑^𝑙_𝑗=1∑^𝑁_𝑖=1∫^{𝑢𝑖(𝑡𝑗)}₀𝐼_𝑖𝑗(𝑡)𝑑𝑡⁻¹𝐷=𝑇−𝑡_𝑙²𝑡₁𝑡²_𝑙+𝑡₁3𝑡²_𝑙𝑡²_𝑙+𝑡_𝑙𝑇+𝑇²+𝑡_𝑙−𝑡₁+𝑇−𝑡_𝑙𝑡²_𝑙+13𝑡²_𝑙𝑇³−𝑡³_𝑙>0. (38) Then we have the following.

Theorem 14 (see [16, Theorem 3.1]). Suppose that (𝒜₁)-(𝒜₂) and (ℐ₁) hold. Moreover, there exist a constant 𝑝>0 and constant vector 𝜉=(𝜉¹,…,𝜉^𝑁)∈ℝ^𝑁⧵{0} such that (ℛ₁)max{limsup_𝑢→0(max_{𝑡∈[0,𝑇]}𝐹(𝑡,𝑢)/|𝑢|²),  limsup_{|𝑢|→+∞}(max_{𝑡∈[0,𝑇]}𝐹(𝑡,𝑢)/|𝑢|²)}<𝑝;(ℛ₂)𝑝𝑇𝑘²<(∫^𝑇₀𝐹(𝑡,𝜉)𝑑𝑡/(𝑞𝑇|𝜉|²+2∑^𝑙_𝑗=1∑^𝑁_𝑖=1∫^𝜉𝑖₀𝐼_𝑖𝑗(𝑡)𝑑𝑡)).
Then, for each compact interval [𝑎,𝑏]⊂(𝜆₁,𝜆₂), there exists 𝑅>0 with the following property: for every 𝜆∈[𝑎,𝑏] and 𝐺∈𝒞 there exists 𝛿>0 such that, for each 𝜇∈[0,𝛿], problem (9) has at least three weak solutions whose norms are less than 𝑅.

Theorem 15 (see [16, Theorem 3.2]). Suppose that (𝒜₁)-(𝒜₂) and (ℐ₁) hold and there exist ℎ,𝑞∈𝐿¹(0,𝑇;ℝ⁺), three positive constants 𝑐,𝑑,𝛼 with 𝑐>𝑘𝑑𝐷𝑚/2, and 𝛼∈[0,2) such that (ℋ₁)max_{𝑡∈[0,𝑇],|𝑢|≤𝑐}𝐹(𝑡,𝑢)<(𝐷𝑚𝑑²/𝑇)(∫^𝑡𝑙_𝑡1𝐹(𝑡,𝑑ℰ)𝑑𝑡/(2𝐷𝑀𝑑²+4∑^𝑙_𝑗=1∫^𝑑₀𝐼_1𝑗(𝑡)𝑑𝑡)); here, ℰ=(1,0,…,0)∈ℝ^𝑁;(ℋ₂)𝐹(𝑡,𝑢)≥0 for each 𝑡∈[0,𝑡₁]∪[𝑡_𝑙,𝑇], |𝑢|≤𝑑𝑇/𝑡_𝑙;(ℋ₃)|𝐹(𝑡,𝑢)|≤ℎ(𝑡)|𝑢|^𝛼+𝑞(𝑡), for all 𝑢∈ℝ^𝑁 and a.e. 𝑡∈[0,𝑇].Then, there exist a nonempty open set 𝐴⊂[0,+∞) and a positive number 𝑅 with the following property: for every 𝜆∈𝐴 and every 𝐺∈𝒞 there exists 𝛿>0 such that, for each 𝜇∈[0,𝛿], problem (9) has at least three weak solutions whose norms are less than 𝑅.

As a special case of 𝐹(𝑡,𝑢), let 𝐹(𝑡,𝑢)=𝑏(𝑡)𝑓(𝑢), where 𝑏(𝑡)∈𝐿¹(0,𝑇;ℝ)⧵{0}, 𝑓(𝑢)∈𝐶¹(ℝ^𝑁;ℝ). Then we get the following two corollaries.

Corollary 16 (see [16, Corollary 3.1]). Suppose that (𝒜₁)-(𝒜₂) and (ℐ₁) hold. Moreover, there exist a constant 𝑝>0 and constant vector 𝜉=(𝜉¹,…,𝜉^𝑁)∈ℝ^𝑁⧵{0} such that (ℛ₁)^′max_{𝑡∈[0,𝑇]}𝑏(𝑡)⋅max{limsup_𝑢→0(𝑓(𝑢)/|𝑢|²),  limsup_{|𝑢|→+∞}(𝑓(𝑢)/|𝑢|²)}<𝑝;(ℛ₂)^′𝑝𝑇𝑘²<(𝑓(𝜉)∫^𝑇₀𝑏(𝑡)𝑑𝑡/(𝑞𝑇|𝜉|²+2∑^𝑙_𝑗=1∑^𝑁_𝑖=1∫^𝜉𝑖₀𝐼_𝑖𝑗(𝑡)𝑑𝑡)).Then, for each compact interval [𝑎,𝑏]⊂(𝜆₁,𝜆₂), there exists 𝑅>0 with the following property: for every 𝜆∈[𝑎,𝑏] and 𝐺∈𝒞 there exists 𝛿>0 such that, for each 𝜇∈[0,𝛿], problem (9) has at least three weak solutions whose norms are less than 𝑅.

Corollary 17 (see [16, Corollary 3.2]). Suppose that (𝒜₁)-(𝒜₂) and (ℐ₁) hold. Furthermore, there exist five positive constants 𝑐, 𝑑, 𝛼, 𝛽, and 𝛾 with 𝑐>𝑘𝑑𝐷𝑚/2 and 𝛼∈[0,2) such that (ℋ₁)^′max_{|𝑢|≤𝑐}𝑓(𝑢)<(𝐷𝑚𝑑²/‖𝑏‖_𝐿¹)  (𝑓(𝑑ℰ)∫^𝑡𝑙_𝑡1𝑏(𝑡)𝑑𝑡/(𝐷𝑀𝑑²+2∑^𝑙_𝑗=1∫^𝑑₀𝐼_1𝑗(𝑡)𝑑𝑡));(ℋ₂)^′𝑏(𝑡)𝑓(𝑢)≥0 for any 𝑡∈[0,𝑡₁]∪[𝑡_𝑙,𝑇], |𝑢|≤𝑑𝑇/𝑡_𝑙;(ℋ₃)^′|𝑓(𝑢)|≤𝛾+𝛽|𝑢|^𝛼 for every 𝑢∈ℝ^𝑁.Then, there exist a nonempty open set 𝐴⊂[0,+∞) and a positive number 𝑅 with the following property: for every 𝜆∈𝐴 and every 𝐺∈𝒞 there exists 𝛿>0 such that, for each 𝜇∈[0,𝛿], problem (9) has at least three weak solutions whose norms are less than 𝑅.

Finally, we consider the existence of infinitely many solutions for a class of impulsive systems with periodic boundary conditions −̈(𝑡)()[],Δ𝑢̇𝑢+𝐴𝑢=∇𝑊𝑡,𝑢,a.e.𝑡∈0,𝑇_𝑖𝑡_𝑗=𝑢̇^𝑖𝑡⁺_𝑗−𝑢̇^𝑖𝑡⁻_𝑗=𝐼_𝑖𝑗𝑢^𝑖𝑡_𝑗,𝑢(0)(𝑇)=𝑢̇(0)−𝑢̇(𝑇)𝑖=1,2,…,𝑁,𝑗=1,2,…,𝑙,−𝑢=0, (39) where 𝐴∶[0,𝑇]→ℝ^𝑁×𝑁 is a continuous map from the interval [0,𝑇] to the set of 𝑁-order symmetric matrices, 𝑇 is a real positive number, 𝑢(𝑡)=(𝑢¹(𝑡),𝑢²(𝑡),…,𝑢^𝑁(𝑡)), 𝑡_𝑗, 𝑗=1,2,…,𝑙, are the instants where the impulses occur and 0=𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝑙<𝑡_𝑙+1=𝑇, 𝐼_𝑖𝑗∶ℝ→ℝ (𝑖=1,2,…,𝑁,𝑗=1,2,…,𝑙) are continuous, and 𝑊∶[0,𝑇]×ℝ^𝑁→ℝ are measurable with respect to 𝑡, for every 𝑢∈ℝ^𝑁, continuously differentiable in 𝑢, for almost every 𝑡∈[0,𝑇], and satisfy the following standard summability condition: sup_{|𝑢|≤𝑏}max{|𝑊(⋅,𝑢)|,|∇𝑊(⋅,𝑢)|}∈𝐿¹(0,𝑇),∀𝑏>0. (40)

Now we state our main results.

Theorem 18 (see [36, Theorem 1.1]). Assume that 𝑊 is even in 𝑢 and the following conditions hold: (ℋ₁)𝐼_𝑖𝑗 (𝑖=1,…,𝑁,𝑗=1,2,…,𝑙) are odd and satisfy 2^𝑠₀𝐼_𝑖𝑗(𝑡)𝑑𝑡−𝐼_𝑖𝑗(𝑠)𝑠≥0,^𝑠₀𝐼_𝑖𝑗(𝑡)𝑑𝑡≥0∀𝑠∈ℝ; (41)(ℋ₂)there exist constants 𝑎_𝑖𝑗,𝑏_𝑖𝑗>0 and 𝛾_𝑖𝑗∈[0,1) such that 𝐼_𝑖𝑗(𝑠)≤𝑎_𝑖𝑗+𝑏_𝑖𝑗|𝑠|^𝛾𝑖𝑗𝑓𝑜𝑟𝑠∈ℝ; (42)(ℋ₃)lim_{|𝑢|→+∞}(𝑊(𝑡,𝑢)/𝑢²)=+∞ uniformly for 𝑡∈[0,𝑇];(ℋ₄)𝑊(𝑡,0)≡0, 0≤𝑊(𝑡,𝑢)=𝑜(|𝑢|²) as |𝑢|→0 uniformly for 𝑡∈[0,𝑇];(ℋ₅)there exist constants 𝛼>1, 1<𝛽<1+((𝛼−1)/𝛼), 𝑐₁,𝑐₂>0, and 𝐿>0 such that for every 𝑡∈[0,𝑇] and 𝑢∈ℝ^𝑁 with |𝑢|≥𝐿, (∇𝑊(𝑡,𝑢),𝑢)−2𝑊(𝑡,𝑢)≥𝑐₁|𝑢|^𝛼,|∇𝑊(𝑡,𝑢)|≤𝑐₂|𝑢|^𝛽. (43)Then problem (39) has infinitely many weak solutions.

Theorem 19 (see [36, Theorem 1.2]). Assume that the following conditions hold: (𝒮₁)𝐼_𝑖𝑗 (𝑖=1,2,…,𝑁,𝑗=1,2,…,𝑙) are odd and satisfy 𝐼_𝑖𝑗(𝑠)𝑠≥0 for any 𝑠∈ℝ;(𝒮₂)there exist constants 𝑏_𝑖𝑗>0 and 𝛾_𝑖𝑗∈[0,1) such that 𝐼_𝑖𝑗(𝑠)≤𝑏_𝑖𝑗|𝑠|^𝛾𝑖𝑗𝑓𝑜𝑟𝑠∈ℝ; (44)(𝒮₃)𝑊(𝑡,𝑢)=(𝜌/2)|𝑢|²+𝐹(𝑡,𝑢), where 𝜌 is a positive constant and not a spectrum point of −(𝑑²/𝑑𝑡²)+𝐴, 𝐹 is even in 𝑢, and 𝐹(𝑡,0)≡0;(𝒮₄)there exist 𝛿₁,𝛿₂∈[1,2) with 𝛿₁<min_{1≤𝑖≤𝑁,1≤𝑗≤𝑙}{𝛾_𝑖𝑗}+1, and 𝑑₁,𝑑₂>0 such that 𝑑₁|𝑢|^𝛿1≤𝐹(𝑡,𝑢),|∇𝐹(𝑡,𝑢)|≤𝑑₂|𝑢|^𝛿2−1,∀(𝑡,𝑢)∈[0,𝑇]×ℝ^𝑁. (45)Then problem (39) has infinitely many weak solutions.

Theorem 20 (see [36, Theorem 1.3]). Assume that 𝑊 is even in 𝑢, (𝒮₁)-(𝒮₂) and the following conditions hold: (𝒥₁)𝑊(𝑡,0)≡0 for any 𝑡∈[0,𝑇];(𝒥₂)there are constants 𝑘₁>0 and 𝜁₁∈[1,2) with 𝜁₁<min_{1≤𝑖≤𝑁,1≤𝑗≤𝑙}{𝛾_𝑖𝑗}+1 such that 𝑊(𝑡,𝑢)≥𝑘₁|𝑢|^𝜁1,𝑓𝑜𝑟𝑎𝑛𝑦(𝑡,𝑢)∈[0,𝑇]×ℝ^𝑁; (46)(𝒥₃)there exist constants 𝑘₂>0 and 𝜁₂∈[1,2) such that |∇𝑊(𝑡,𝑢)|≤𝑘₂|𝑢|^𝜁2−1,𝑓𝑜𝑟𝑎𝑛𝑦(𝑡,𝑢)∈[0,𝑇]×ℝ^𝑁. (47)Then problem (39) has infinitely many weak solutions.

3. Periodic Solutions

Firstly, we consider periodic solutions for second-order regular impulsive problems with a perturbation: 𝑢^′′(𝑡)−𝑉_𝑢(𝑡,𝑢)=𝑓(𝑡),for𝑡∈𝑠_𝑘−1,𝑠_𝑘,Δ𝑢^′𝑠_𝑘=𝑔_𝑘𝑢𝑠_𝑘, (48) where 𝑘∈ℤ, 𝑢∈ℝ^𝑛, Δ𝑢^′(𝑠_𝑘)=𝑢^′(𝑠⁺_𝑘)−𝑢^′(𝑠⁻_𝑘) with 𝑢^′(𝑠^±_𝑘)=lim𝑡→𝑠^±_𝑘, 𝑉_𝑢(𝑡,𝑢)=grad_𝑢𝑉(𝑡,𝑢), 𝑓∈𝐶(ℝ,ℝ^𝑛), 𝑔_𝑘(𝑢)=grad_𝑢𝐺_𝑘(𝑢), 𝐺_𝑘∈𝐶¹(ℝ^𝑛,ℝ^𝑛) for each 𝑘∈ℤ, and there exist 𝑚∈ℕ and 𝑇∈ℝ⁺ such that 0=𝑠₀<𝑠₁<⋯<𝑠_𝑚=𝑇, 𝑠_𝑘+𝑚=𝑠_𝑘+𝑇, and 𝑔_𝑘+𝑚≡𝑔_𝑘 for all 𝑘∈ℤ (i.e., 𝑔_𝑘 is 𝑚-periodic in 𝑘).

Theorem 21 (see [20, Theorem 1]). Assume that 𝑓 is continuous and 𝑇-periodic and 𝑔_𝑘 is continuous, 𝑚-periodic in 𝑘. Furthermore, assume that 𝑉 and 𝑔_𝑘 satisfy the following conditions: (𝑉₁)𝑉∶ℝ×ℝ^𝑛→ℝ is continuous differentiable and 𝑇-periodic, and there exist positive constants 𝑏₁,𝑏₂>0 such that 𝑏₁|𝑢|²≤−𝑉(𝑡,𝑢)≤|𝑢|²,∀(𝑡,𝑢)∈[0,𝑇]×ℝ^𝑛; (49)(𝑉₂)−𝑉(𝑡,𝑢)≤−𝑉_𝑢(𝑡,𝑢)𝑢≤−2𝑉(𝑡,𝑢) for all (𝑡,𝑢)∈[0,𝑇]×ℝ^𝑛;(𝑔₁)there exists 𝜇>2 such that 𝑔_𝑘(𝑢)𝑢≤𝜇𝐺_𝑘(𝑢)<0,𝑓𝑜𝑟𝑢∈ℝ^𝑛⧵{0},𝑘=1,2,…,𝑚. (50)Then there exists 𝑏_𝑓>0 such that, for any positive integer 𝑙, if 𝑏_𝑓,1≤𝑏_𝑓𝑙^−1/2, then problem (48) possesses at least one nonzero 𝑙𝑇-periodic solution.

Theorem 22 (see [20, Theorem 1]). Suppose that 𝑉 is even about 𝑢 and satisfies (𝑉₁)-(𝑉₂), 𝑓≡0, and 𝑔_𝑘 is continuous, odd about 𝑢, 𝑚-periodic in 𝑘, and satisfies the following: (𝑔₂)for all 𝑘=1,2,…,𝑚, liminf_|𝑢|→∞𝐺_𝑘(𝑢)|𝑢|²≥0. (51)Then there exists 𝐶_𝑔>0 such that if limsup_|𝑢|→0𝐺_𝑘(𝑢)|𝑢|²≤−𝐶_𝑔, (52) then problem (48) possesses at least 𝑙𝑚𝑛 distinct pairs of 𝑙𝑇-periodic solutions generated by impulses.

Theorem 23 (see [14, Theorem 1]). Assume that 𝑓≡0 and 𝑔_𝑘 is continuous, 𝑚-periodic in 𝑘 satisfying (𝑔₁). Furthermore, assume that 𝑉 satisfies the following condition: (𝑉₃)𝑉∶ℝ×ℝ^𝑛→ℝ is continuous differentiable and 𝑇-periodic, and 𝑉(𝑡,𝑢)≥(1/2)𝑉_𝑢(𝑡,𝑢)𝑢>0 for all 𝑡∈[0,𝑇] and 𝑢∈ℝ^𝑛⧵{0}.Then problem (48) possesses one nonzero periodic solution generated by impulses.

Corollary 24 (see [14, Corollary 1]). Assume that 𝑓≡0 and 𝑔_𝑘 is continuous, 𝑚-periodic in 𝑘 satisfying (𝑔₁). Furthermore, assume that 𝑉 satisfies the following condition: (𝑉₄)𝑉_𝑢(𝑡,𝑢)=𝛼𝑢+𝑊_𝑢(𝑡,𝑢) for all 𝑡∈[0,𝑇] and 𝑢∈ℝ^𝑛, where 𝛼>2/𝜇 for some 𝜇>2 and 𝑊(𝑡,𝑢)≥(1/2)𝑊_𝑢(𝑡,𝑢)𝑢>0 for all 𝑡∈[0,𝑇] and 𝑢∈ℝ^𝑛⧵{0}.Then problem (48) possesses one nonzero periodic solution generated by impulses.

Theorem 25 (see [14, Theorem 2]). Assume that 𝑓≡0 and 𝑉 and 𝑔_𝑘 satisfy the following conditions: (𝑉₅)𝑉(𝑡,𝑢)≥(1/2)𝑉_𝑢(𝑡,𝑢)𝑢≥0 for all (𝑡,𝑢)∈[0,𝑇]×ℝ^𝑛;(𝑔₃)𝑔_𝑘(𝑢)=2𝑢+𝑤_𝑘(𝑢), where 𝑤_𝑘=grad_𝑢𝑊_𝑘(𝑢) and satisfies the fact that there exists 𝜇>2 such that 𝑤_𝑘(𝑢)𝑢≤𝜇𝑊_𝑘(𝑢)<0,𝑓𝑜𝑟𝑢∈ℝ^𝑛⧵{0},𝑘=1,2,…,𝑚. (53)Then problem (48) possesses one nonzero periodic solution generated by impulses.

Now we discuss 𝑇-periodic solution to second-order impulsive problems with singularity as follows: 𝑢^′′(𝑡)−1𝑢^𝛼(𝑡)=𝑒(𝑡),a.e.𝑡∈(0,𝑇),Δ𝑢^′𝑡_𝑗=𝐼_𝑗𝑢𝑡_𝑗,𝑗=1,2,…,𝑝−1, (54) where 𝛼≥1, 𝑒∈𝐿¹([0,𝑇],ℝ) is 𝑇-periodic, Δ𝑢^′(𝑡_𝑗)=𝑢^′(𝑡⁺_𝑗)−𝑢^′(𝑡⁻_𝑗) with 𝑢^′(𝑡^±_𝑗)=lim_𝑡→𝑡^±_𝑗𝑢^′(𝑡); 0=𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝑝−1<𝑡_𝑝=𝑇 and 𝑡_𝑗+𝑝=𝑡_𝑗+𝑇; 𝐼_𝑗∈𝐶(ℝ,ℝ) (𝑗=1,2,…,𝑝−1) and 𝐼_𝑗+𝑝≡𝐼_𝑗.

Theorem 26 (see [18, Theorem 1.2]). Assume that(𝒮₁)𝑒∈𝐿¹([0,𝑇],ℝ) is 𝑇-periodic and ∫^𝑇₀𝑒(𝑡)𝑑𝑡≥0;(𝒮₂)there exist two constants 𝑚,𝑀 such that, for any 𝑠∈ℝ, 𝑚≤𝐼_𝑗(𝑠)≤𝑀,𝑗=1,2,…,𝑝−1, (55)where 1𝑚≤𝑀<−𝑝−1^𝑇₀𝑒(𝑡)𝑑𝑡≤0. (56) Then problem (54) has at least a positive 𝑇-periodic solution.

Set 𝐻¹_𝑇=𝑢𝑢∶ℝ⟶ℝ∣𝑢isabsolutelycontinuous,_′∈𝐿²((0,𝑇),ℝ),and𝑢(𝑡)=𝑢(𝑡+𝑇)for𝑡∈ℝ (57) with the inner product (𝑢,𝑣)=^𝑇₀𝑢(𝑡)𝑣(𝑡)𝑑𝑡+^𝑇₀𝑢^′(𝑡)𝑣^′(𝑡)𝑑𝑡,∀𝑢,𝑣∈𝐻¹_𝑇. (58) The corresponding norm is defined by ‖𝑢‖_𝐻¹_𝑇=^𝑇₀|𝑢(𝑡)|²𝑑𝑡+^𝑇₀𝑢^′(𝑡)²𝑑𝑡^1/2,∀𝑢∈𝐻¹_𝑇. (59) Then 𝐻¹_𝑇 is a Banach space (in fact it is a Hilbert space).

In order to study problem (54), for any 𝜆∈(0,1), we consider the following modified problem: 𝑢^′′(𝑡)+𝑓_𝜆(𝑢(𝑡))=𝑒(𝑡),a.e.𝑡∈(0,𝑇),Δ𝑢^′𝑡_𝑗=𝐼_𝑗𝑢𝑡_𝑗,𝑗=1,2,…,𝑝−1. (60) Following the ideas of [15], we introduce the following concept of a weak solution to problem (60).

Definition 27. One says that a function 𝑢∈𝐻¹_𝑇 is a weak solution to problem (60) if ^𝑇₀𝑢^′(𝑡)𝑣^′(𝑡)𝑑𝑡+^𝑝−1_𝑗=1𝐼_𝑗𝑢𝑡_𝑗𝑣𝑡_𝑗−^𝑇₀𝑓_𝜆(𝑢(𝑡))𝑣(𝑡)𝑑𝑡+^𝑇₀𝑒(𝑡)𝑣(𝑡)𝑑𝑡=0 (61) holds for any 𝑣∈𝐻¹_𝑇.

Let 𝐹_𝜆∈𝐶¹(ℝ,ℝ) be defined by 𝐹_𝜆(𝑠)=^𝑠₁𝑓_𝜆(𝑡)𝑑𝑡, (62) and consider the functional Φ_𝜆∶𝐻¹_𝑇⟶ℝ, (63) defined by Φ_𝜆(𝑢)1∶=2^𝑇₀𝑢^′(𝑡)²𝑑𝑡+^𝑝−1_𝑗=1^{𝑢(𝑡𝑗)}₀𝐼_𝑗(𝑠)𝑑𝑠−^𝑇₀𝐹_𝜆(𝑢(𝑡))𝑑𝑡+^𝑇₀𝑒(𝑡)𝑢(𝑡)𝑑𝑡. (64) Clearly, Φ_𝜆 is well defined on 𝐻¹_𝑇, continuously Gâteaux differentiable functional whose Gâteaux derivative is the functional Φ^′_𝜆(𝑢), given by Φ^′_𝜆(𝑢)𝑣=^𝑇₀𝑢^′(𝑡)𝑣^′(𝑡)𝑑𝑡+^𝑝−1_𝑗=1𝐼_𝑗𝑢𝑡_𝑗𝑣𝑡_𝑗−^𝑇₀𝑓_𝜆(𝑢(𝑡))𝑣(𝑡)𝑑𝑡+^𝑇₀𝑒(𝑡)𝑣(𝑡)𝑑𝑡, (65) for any 𝑣∈𝐻¹_𝑇. Moreover, it is easy to verify that Φ_𝜆 is weakly lower semicontinuous. Indeed, if {𝑢_𝑛}⊂𝐻¹_𝑇, 𝑢∈𝐻¹_𝑇, and 𝑢_𝑛⇀𝑢, then {𝑢_𝑛} converges uniformly to 𝑢 on [0,𝑇] and 𝑢_𝑛→𝑢 on 𝐿²([0,𝑇]), and combining the fact that liminf_𝑛→∞‖𝑢_𝑛‖_𝐻1𝑇≥‖𝑢_𝑛‖_𝐻1𝑇, we have liminf_𝑛→∞Φ_𝜆𝑢_𝑛=liminf_𝑛→∞12𝑢_𝑛²_𝐻1𝑇−12^𝑇₀𝑢_𝑛(𝑡)²+𝑑𝑡_{𝑝−1𝑗=1𝑢𝑛(𝑡𝑗)0}𝐼_𝑗(𝑠)𝑑𝑠−^𝑇₀𝐹_𝜆𝑢_𝑛(𝑡)𝑑𝑡+^𝑇₀𝑒(𝑡)𝑢_𝑛(𝑡)≥1𝑑𝑡2^𝑇₀𝑢^′(𝑡)²𝑑𝑡+^𝑝−1_𝑗=1^{𝑢(𝑡𝑗)}₀𝐼_𝑗(𝑠)−𝑑𝑠_𝑇0𝐹_𝜆(𝑢(𝑡))𝑑𝑡+^𝑇₀𝑒(𝑡)𝑢(𝑡)𝑑𝑡=Φ_𝜆(𝑢). (66) By the standard discussion, the critical points of Φ_𝜆 are the weak solutions of problem (1); see [15, 16].

The following version of the mountain pass theorem will be used in our argument.

Theorem 28 (see Theorem 4.10, [35]). Let 𝑋 be a Banach space and let 𝜑∈𝐶¹(𝑋,ℝ). Assume that there exist 𝑥₀,𝑥₁∈𝑋 and a bounded open neighborhood Ω of 𝑥₀ such that 𝑥₁∈𝑋⧵Ω and max𝜑𝑥₀,𝜑𝑥₁<inf_{𝑥∈𝜕Ω}𝜑(𝑥). (67) Let Γ=ℎ∈𝐶([0,1],𝑋)∶ℎ(0)=𝑥₀,ℎ(1)=𝑥₁,𝑐=inf_ℎ∈Γmax_[]𝑠∈0,1𝜑(ℎ(𝑠)). (68) If 𝜑 satisfies the (PS) condition, that is, a sequence {𝑢_𝑛} in 𝑋 satisfying 𝜑(𝑢_𝑛) is bounded and 𝜑^′(𝑢_𝑛)→0 as 𝑛→∞ has a convergent subsequence, then 𝑐 is a critical value of 𝜑 and 𝑐>max{𝜑(𝑥₀),𝜑(𝑥₁)}.

If 𝐼_𝑗(𝑢(𝑡_𝑗))≡𝑏_𝑗 (𝑗=1,2,…,𝑝−1), then we have the following result of problem (54).

Theorem 29 (see [17, Theorem 1.2]). Assume that 𝑒∈𝐿¹([0,𝑇],ℝ) is 𝑇-periodic. Then problem (54) has a positive 𝑇-periodic weak solution 𝑢∈𝐻¹_𝑇 if and only if ∑^𝑝−1_𝑗=1𝑏_𝑗+∫^𝑇₀𝑒(𝑡)𝑑𝑡<0.

Remark 30. From Theorem 29 we can see that if ∫^𝑇₀𝑒(𝑡)𝑑𝑡≥0 but ∑^𝑝−1_𝑗=1𝑏_𝑗+∫^𝑇₀𝑒(𝑡)𝑑𝑡<0, then problem (54) still admits a positive 𝑇-periodic solution. This shows that the existence of positive 𝑇-periodic solution to problem (54) depends on the forced term 𝑒 and impulsive functions 𝑏_𝑗 together, not single one.

4. Homoclinic Solutions

In this section, we are interested in the existence of homoclinic solutions for problem (48).

First we consider the case of 𝑓≡0. When 𝑉_𝑢(𝑡,𝑢) is asymptotically linear, we show that there exists a nonzero homoclinic solution as the limit of a sequence 2𝑙𝑇-periodic solutions as 𝑙 goes to ∞, and when 𝑉_𝑢(𝑡,𝑢) is sublinear, we attain a nonzero homoclinic solution in the same way by strengthening the effect of impulses. The results are as follows.

Theorem 31 (see [14, Theorem 3]). Supposing that 𝑓≡0, 𝑉 is 𝑇-periodic in 𝑡 and satisfies condition (𝑉₄), and 𝑔_𝑘 satisfies condition (𝑔₁), then problem (48) possesses at least one nonzero homoclinic solution generated by impulses.

Theorem 32 (see [14, Theorem 4]). Supposing that 𝑓≡0, 𝑉 is 𝑇-periodic in 𝑡 and satisfies condition (𝑉₅), and 𝑔_𝑘 satisfies condition (𝑔₃), then problem (48) possesses at least one nonzero homoclinic solution generated by impulses.

If 𝑛=1 and 𝑔_𝑘(𝑢)≡𝐼(𝑢) for all 𝑘=1,2,…,𝑚, then we have the following result for problem (48) when 𝑉_𝑢 satisfies the superlinear condition at infinity on 𝑢.

Theorem 33 (see [13, Theorem 1.1]). Assume that 𝑓≡0 and the following conditions hold: (𝐻₁)there exists a positive number 𝑇 such that 𝑉_𝑢(𝑡+𝑇,𝑢)=𝑉_𝑢()()()∀()𝑡,𝑢,𝑉𝑡+𝑇,𝑢=𝑉𝑡,𝑢𝑡,𝑢∈ℝ²; (69)(𝐻₂)lim_𝑢→0𝑉_𝑢(𝑡,𝑢)𝑢=0 uniformly for 𝑡∈ℝ;(𝐻₃)there exists a 𝜇>2 such that 𝑢𝑉_𝑢(𝑡,𝑢)≥𝜇𝑉(𝑡,𝑢)>0,∀(𝑡,𝑢)∈ℝ×ℝ⧵{0}; (70)(𝐻₄)there exist constants 𝑎₀>0 and 𝑎₁>0 such that 𝑉(𝑡,𝑢)≥𝑎₀|𝑢|^𝜇|𝑢|𝑉(),∀≥1,𝑡∈ℝ;𝑡,𝑢≤𝑎₀|𝑢|^𝜇,∀|𝑢|≤1,𝑡∈ℝ; (71)(𝐻₅)there exists a constant 𝑏, with 0<𝑏<((𝜇−2)/(𝜇+2)𝑇𝑚), such that |𝐼(𝑢)|≤𝑏|𝑢|,2^𝑢₀𝐼(𝑡)𝑑𝑡−𝐼(𝑢)𝑢≤0. (72)Then there exists a nontrivial weak homoclinic solution to problem (48).

The proof of Theorem 33 is based on the following version of Mountain pass theorem.

Theorem 34 (see [37]). Let 𝐸 be a Banach space and let 𝜑∈𝐶¹(𝐸,ℝ), 𝑒∈𝐸, 𝑟>0 be such that ‖𝑒‖>𝑟 and 𝑏∶=inf_{‖𝑦‖=𝑟}𝜑(𝑦)>𝜑(0)≥𝜑(𝑒). Let Γ=ℎ∈𝐶([0,1],𝑋)∶ℎ(0)=𝑥₀,ℎ(1)=𝑥₁,𝑐∶=inf_ℎ∈Γmax_[]𝑠∈0,1𝜑(ℎ(𝑠)). (73) Then, for each 𝜖>0, 𝛿>0, there exists 𝑦∈𝐸 such that (𝑀₁)𝑐−2𝜖≤𝜑(𝑦)≤𝑐+2𝜖;(𝑀₂)𝑑𝑖𝑠𝑡(𝑦,𝐸)≤2𝛿;(𝑀₃)‖𝜑^′(𝑦)‖≤8𝜖/𝛿.

Next we consider the case of 𝑓≢0 for problem (48).

Theorem 35 (see [20, Theorem 3]). Assume that 𝑉 satisfies conditions (𝑉₁)-(𝑉₂) and 𝑔_𝑘 satisfies (𝑔₁). Furthermore, 𝑔_𝑘 satisfies the following condition: (𝑔₄)𝑔_𝑘∈𝐶(ℝ) is 𝑚-periodic in 𝑘, and lim_|𝑢|→0(𝑔_𝑘(𝑢)/|𝑢|)=0 for 𝑘=1,2,…,𝑚.Then there exists 𝑏_𝑓>0 such that if ∫_ℝ|𝑓(𝑡)|²𝑑𝑡≤𝑏²_𝑓, then problem (48) possesses at least one nonzero homoclinic solution.

If 𝑉 satisfies some other conditions, then we still obtain the existence result for problem (48).

Theorem 36 (see [19, Theorem 1]). Assume that the following conditions hold: (𝑉₆)𝑉(𝑡,𝑢)=−𝐾(𝑡,𝑢)+𝑊(𝑡,𝑢), where 𝐾, 𝑊∈𝐶¹(ℝ×ℝ^𝑛,ℝ), and is 𝑇-periodic in its first variable;(𝐾₁)there exist constants 𝑎₁>0 and 𝛾∈(1,2] such that for all (𝑡,𝑢)∈[0,𝑇]×ℝ^𝑛𝐾(𝑡,0)≡0,𝐾(𝑡,𝑢)≥𝑎₁|𝑢|^𝛾; (74)(𝐾₂)there exists 𝜃∈(1,2] such that 𝐾(𝑡,𝑢)≤𝐾_𝑢(𝑡,𝑢)𝑢≤𝜃𝐾(𝑡,𝑢),∀(𝑡,𝑢)∈[0,𝑇]×ℝ^𝑛; (75)(𝑊₁)𝑊(𝑡,0)=0, 𝑊_𝑢(𝑡,𝑢)=𝑜(|𝑢|), as |𝑢|→0 uniformly for 𝑡∈[0,𝑇];(𝑊₂)there exists 𝑟>2 and 𝑏₁>0 such that 𝑊(𝑡,𝑢)≤𝑏₁|𝑢|^𝑟,∀(𝑡,𝑢)∈[0,𝑇]×ℝ^𝑛; (76)(𝑊₃)there exist 𝜇>1, 𝑎₂>0, and 𝛽(𝑡)∈𝐿¹(ℝ,ℝ⁺) such that 𝜇>𝑟−𝛾 and 𝑊_𝑢(𝑡,𝑢)𝑢−2𝑊(𝑡,𝑢)≥𝑎₂|𝑢|^𝜇(𝑡),∀()∈[]−𝛽𝑡,𝑢0,𝑇×ℝ^𝑛; (77)(𝑊₄)𝑊(𝑡,𝑢)/|𝑢|²→+∞ as |𝑢|→+∞ uniformly in 𝑡∈[0,𝑇];(𝐺₁)𝐺_𝑘(0)=0 and lim_|𝑢|→0𝑔_𝑘(𝑢)|𝑢|=0 for 𝑘=1,2,…,𝑚;(𝐺₂)there exists 𝑏₂>0 such that 𝐺_𝑘(𝑢)≤𝑏₂|𝑢|^𝑟, 𝑢∈ℝ^𝑛, 𝑘=1,2,…,𝑚;(𝐺₃)𝑔_𝑘(𝑢)𝑢−2𝐺_𝑘(𝑢)≥0, 𝑢∈ℝ^𝑛⧵{0}, 𝑘=1,2,…,𝑚;(𝐹₁)𝑓∈𝐶(ℝ,ℝ^𝑛)∩𝐿²(ℝ,ℝ^𝑛)∩𝐿^{𝜇/(𝜇−1)} satisfying _ℝ𝑓(𝑡)²𝑑𝑡^1/2<1𝐶1min2,𝑎₁−𝑏₁−𝑀, (78)where 𝑀∶=sup{𝐺_𝑘(𝑢)∶𝑘=1,2,…,𝑚,|𝑢|=1} and 𝑎₁>𝑏₁+𝑀.
Then problem (48) possesses at least one nonzero homoclinic solution.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

J. Sun was supported by the National Natural Science Foundation of China (Grant nos. 11201270 and 11271372), Shandong Natural Science Foundation (Grant no. ZR2012AQ010), and Young Teacher Support Program of Shandong University of Technology.