Periodic Solutions for Second Order Hamiltonian Systems with Impulses via the Local Linking Theorem

Abstract

A class of second order impulsive Hamiltonian systems are considered. By applying a local linking theorem, we establish the new criterion to guarantee that this impulsive Hamiltonian system has at least one nontrivial T-periodic solution under local superquadratic condition. This result generalizes and improves some existing results in the known literature.

1. Introduction and Main Results

Consider the second order Hamiltonian systems with impulsive effects 𝑢̈(𝑡)(𝑡)((𝑡[],Δ𝑢̇−𝐴𝑢+∇𝐹𝑡,𝑢))=0,a.e.𝑡∈0,𝑇_𝑖𝑡_𝑗=𝐼_𝑖𝑗𝑢^𝑖𝑡_𝑗𝑢(0)(𝑇)=𝑢̇(0)−𝑢̇(𝑇),𝑖=1,2,…,𝑁,𝑗=1,2,…,𝑙,−𝑢=0, (1) where 𝑢(𝑡)=(𝑢¹(𝑡),𝑢²(𝑡),…,𝑢^𝑁(𝑡)), 𝑡₀=0<𝑡₁<𝑡₂<⋯<𝑡_𝑙<𝑡_𝑙+1=𝑇, 𝑇>0, Δ(𝑢̇^𝑖(𝑡_𝑗))=𝑢̇^𝑖(𝑡⁺_𝑗)−𝑢̇^𝑖(𝑡⁻_𝑗), where 𝑢̇^𝑖(𝑡⁺_𝑗) and 𝑢̇^𝑖(𝑡⁻_𝑗) denote the right and left limits of 𝑢̇^𝑖(𝑡) at 𝑡=𝑡_𝑗, respectively, 𝐼_𝑖𝑗∶ℝ→ℝ  (𝑖=1,2,…,𝑁,𝑗=1,2,…,𝑙) are continuous, and 𝐹∈𝐶¹([0,𝑇]×ℝ^𝑁,ℝ), ∇𝐹(𝑡,𝑢)=𝜕𝐹(𝑡,𝑢)/𝜕𝑢. 𝐴=[𝑎_𝑖𝑗] is a symmetric constant matrix.

Impulsive differential equations serve as basic models to study the dynamics of processes that are subject to sudden changes in their states. The theory of impulsive differential systems has been developed by numerous mathematicians (see [1–4]). These kinds of processes naturally occur in control theory, biology, optimization theory, medicine, and so on (see [5–9]).

In recent years, many existence results are obtained for impulsive differential systems by critical point theory, such as [10–23] and their references. In most superquadratic cases, there is so-called Ambrosetti-Rabinowitz condition (see [18–23]): 0<𝜇𝐹(𝑡,𝑢)≤(∇𝐹(𝑡,𝑢),𝑢),∀𝑡∈ℝ,𝑢∈ℝ^𝑁⧵{0}, (2) where 𝜇>2 is a constant, which implies that 𝐹(𝑡,𝑢) is of superquadratic growth as |𝑢|→∞; that is, lim_|𝑢|→∞𝐹(𝑡,𝑢)|𝑢|²=+∞,uniformlyforall𝑡. (3)

Moreover, Wu and Zhang [24] study the homoclinic solutions without any periodicity assumption under the local Ambrosetti-Rabinowitz type condition. Two key conditions of the main results of [24] are listed as follows.(A1)There exist 𝜇>2 and 𝐿₁>0 such that 𝜇𝐹(𝑡,𝑢)≤(∇𝐹(𝑡,𝑢),𝑢),∀𝑡∈ℝ,|𝑢|≥𝐿₁. (4)(A2)There exists 2<𝛼<+∞ such that liminf_{|𝑢|→+∞}(𝐹(𝑡,𝑢)/|𝑢|^𝛼)>0, uniformly in 𝑡∈ℝ.

In recent paper [25], Zhang and Tang had obtained some results of the nontrivial T-periodic solutions under much weaker assumptions instead of (A1) and (A2).(B1)There exist constants 𝜇>2, 0<𝛽₀<2, and 𝐿₂>0 and a function 𝑎₀(𝑡)∈𝐿¹(0,𝑇;ℝ⁺) such that 𝜇𝐹(𝑡,𝑢)≤∇(𝐹(𝑡,𝑢),𝑢)+𝑎₀(𝑡)|𝑢|^𝛽0,∀|𝑢|≥𝐿₂,a.e.𝑡∈[0,𝑇]. (5)(B2)There exists a subset 𝐸₀ of [0,𝑇] with meas(𝐸₀)>0 such that liminf_|𝑢|→∞𝐹(𝑡,𝑢)|𝑢|²>0,a.e.𝑡∈𝐸₀. (6)

Remark 1. Condition (B2) is weaker than (A2) because condition (A2) implies lim_|𝑢|→∞(𝐹(𝑡,𝑢)/|𝑢|²)=+∞.

Recently, applying the local linking theorem (see [26]), the works in [27–30] obtained the existence of periodic solutions or homoclinic solutions with (3) superquadratic condition under different systems. As shown in [25], condition (B2) is a local superquadratic condition; this situation has been considered only by a few authors.

Motivated by papers [24, 25, 31], in this paper, we aim to consider problem (1) under local superquadratic condition via a version of the local linking theorem (see [26]). In particular, the impulsive function 𝐼_𝑖𝑗 satisfies a kind of new superquadratic condition which is different from that in the known literature. Our main results are the following theorems.

Theorem 2. Suppose that 𝐹∈𝐶¹([0,𝑇]×ℝ^𝑁,ℝ) and 𝐼_𝑖𝑗∈𝐶(ℝ,ℝ),  𝑖=1,2,…,𝑁,𝑗=1,2,…,𝑙, satisfies (B2) and consider the following.(H1)There exists a positive constant 𝜃 such that (𝐴𝑢,𝑢)≥𝜃|𝑢|²,∀𝑢∈ℝ^𝑁. (7)(H2)limsup_|𝑢|→0(|∇𝐹(𝑡,𝑢)|/|𝑢|)=0 uniformly for 𝑡∈[0,𝑇].(H3)There exist constants 𝑑>1,𝑐₁>0, and 𝐿₃>0 such that, for every 𝑡∈[0,𝑇] and 𝑢∈ℝ^𝑁 with |𝑢|≥𝐿₃, |∇𝐹(𝑡,𝑢)|≤𝑐₁|𝑢|^𝑑+1. (8)(H4)There exist constants 𝜇>2,𝐿₄>0, and 𝑏₁∈(0,(𝜇/2−1)𝜃) such that, for every 𝑡∈[0,𝑇] and 𝑢∈ℝ^𝑁 with |𝑢|≥𝐿₄, 𝜇𝐹(𝑡,𝑢)≤(∇𝐹(𝑡,𝑢),𝑢)+𝑏₁|𝑢|². (9)(I1)There exist constants 𝑏_𝑖𝑗>0 and 𝑟_𝑖𝑗∈(1,+∞) such that 𝐼_𝑖𝑗(𝑢)≤𝑏_𝑖𝑗|𝑢|^𝑟𝑖𝑗,∀𝑢∈ℝ. (10)(I2)There are two constants 𝑏₂>0 and 𝛾∈[0,2) such that 𝐼_𝑖𝑗(𝑢)𝑢≤𝜇^𝑢₀𝐼_𝑖𝑗(𝑠)𝑑𝑠+𝑏₂|𝑢|^𝛾,∀𝑢∈ℝ. (11)(I3)𝐼_𝑖𝑗 satisfies ∫^𝑢₀𝐼_𝑖𝑗(𝑠)𝑑𝑠≤0, for all 𝑢∈ℝ.Then problem (1) has at least one nontrivial T-periodic solution.

Remark 3. Noting (3), obviously, conditions (B2) and (H4) are weaker than those of (2). From (B2), we only need lim_|𝑢|→∞(𝐹(𝑡,𝑢)/|𝑢|²)>0 to hold in a subset 𝐸₀ of [0,𝑇]. What is more, 𝐹 in (2) is asked to be positive globally. Here 𝐹 need not be nonnegative globally; we also generalized Theorems 1.3 and 1.4 in [25]. For example, let 𝐹()=1𝑡,𝑢8𝑔(𝑡)|𝑢|⁴−1,∀(𝑡,𝑢)∈[0,𝑇]×ℝ^𝑁, (12) where 𝑔(𝑡)=sin2𝜋𝑡𝑇𝑇,𝑡∈0,2,𝑇0,𝑡∈2,𝑇. (13) Let 𝐸₀=[𝑇/8,𝑇/4]; then 𝐹 satisfies our Theorem 2 but does not satisfy (2) and (3) and does not satisfy the corresponding conditions in [25].

Theorem 4. Suppose that 𝐹∈𝐶¹([0,𝑇]×ℝ^𝑁,ℝ) and 𝐼_𝑖𝑗∈𝐶(ℝ,ℝ),   𝑖=1,2,…,𝑁,𝑗=1,2,…,𝑙, satisfies (H1), (H2), (H3), (I1), (I2), and (I3) and the following condition holds.(H5)There exist constants 𝜇>2 and 𝑏₃∈(0,min{𝑚₁(𝜇−2),(𝜇/2−1)𝜃}) such that 𝜇𝐹(𝑡,𝑢)≤(∇𝐹(𝑡,𝑢),𝑢)+𝑏₃|𝑢|²,∀(𝑡,𝑢)∈[0,𝑇]×ℝ^𝑁, (14) where 𝑚₁=min𝐹(𝑡,𝑢)∣𝑡∈[0,𝑇],𝑢∈ℝ^𝑁,|𝑢|=1. (15)Then problem (1) has at least one nontrivial T-periodic solution.

Theorem 5. Suppose that 𝐹∈𝐶¹([0,𝑇]×ℝ^𝑁,ℝ) and 𝐼_𝑖𝑗∈𝐶(ℝ,ℝ),   𝑖=1,2,…,𝑁,𝑗=1,2,…,𝑙, satisfies (B2), (H1), (H2), (H3), (I1), (I2), and (I3) and the following condition holds.(H6)There exist constants 𝜇>2, 𝛽∈[0,2), and 𝐿₅>0 and a function 𝑎(𝑡)∈𝐿¹(0,𝑇) and 𝑎(𝑡)≥0 such that, for every 𝑡∈[0,𝑇] and 𝑢∈ℝ^𝑁 with |𝑢|≥𝐿₅, 𝜇𝐹(𝑡,𝑢)≤∇(𝐹(𝑡,𝑢),𝑢)+𝑎(𝑡)|𝑢|^𝛽. (16)Then problem (1) has at least one nontrivial T-periodic solution.

The remaining of this paper is organized as follows. In Section 2, some fundamental facts are given. In Section 3, the main results of this paper are presented.

2. Preliminaries

Let 𝑋 be a real Banach space with direct sum decomposition 𝑋=𝑋¹⊕𝑋². Consider two sequences of subspaces 𝑋¹₀⊂𝑋¹₁⊂⋯⊂𝑋¹, 𝑋²₀⊂𝑋²₁⊂⋯⊂𝑋² such that 𝑋^𝑗=⋃_𝑛∈𝑁𝑋^𝑗_𝑛,𝑗=1,2. For every multi-index 𝛼=(𝛼₁,𝛼₂)∈𝑁², let  𝑋_𝛼=𝑋_𝛼1⊕𝑋_𝛼2; we define 𝛼≤𝛽⇔𝛼₁≤𝛽₁,𝛼₂≤𝛽₂. A sequence {𝛼_𝑛}⊂𝑁² is admissible if for every 𝛼∈𝑁² there is 𝑚∈𝑁 such that 𝑛≥𝑚⇒𝛼_𝑛≥𝛼. For every 𝜑∶𝑋→ℝ, we define by 𝜑_𝛼 the function 𝜑 restricted to 𝑋_𝛼.

Definition 6 (see [26]). Let 𝜑∈𝐶¹(𝑋,ℝ). The function 𝜑 satisfies the (𝑃𝑆)^∗ condition if every sequence {𝑢_𝛼𝑛}, such that {𝛼_𝑛} is admissible and 𝑢_𝛼𝑛∈𝑋_𝛼𝑛,sup_𝑛𝜑𝑢_𝛼𝑛<∞,𝜑^′_𝛼𝑛𝑢_𝛼𝑛⟶0,as𝑛⟶∞, (17) possesses a subsequence which converges to a critical point of 𝜑.

Definition 7 (see [26]). Let 𝑋 be a Banach space with direct sum decomposition 𝑋=𝑋¹⊕𝑋². The function 𝜑∈𝐶¹(𝑋,ℝ) has local linking at 0 with respect to (𝑋¹,𝑋²), if there exists 𝑟>0 such that 𝜑(𝑢)≥0,∀𝑢∈𝑋¹‖𝑢‖𝜑(𝑢)with≤𝑟,≤0,∀𝑢∈𝑋²with‖𝑢‖≥𝑟. (18)

Theorem 8 (see [26]). Suppose that 𝜑∈𝐶¹(𝑋,ℝ) satisfies the following assumptions:(A1)𝜑 has local linking at 0 and 𝑋¹≠{0},(A2)𝜑 satisfies (𝑃𝑆)^∗ condition,(A3)𝜑 maps bounded sets into bounded sets,(A4)for every 𝑚∈𝑁 and 𝑢∈𝑋¹_𝑚⊕𝑋², 𝜑(𝑢)→−∞ as ‖𝑢‖→∞.Then 𝜑 has at least three critical points.

Let us recall some basic notation. In the Sobolev space 𝑋∶=𝐻¹₀(0,𝑇), consider the inner product (𝑢,𝑣)=^𝑇₀(𝑢(𝑡),𝑣(𝑡))𝑑𝑡+^𝑇₀(𝑢̇(𝑡),𝑣̇(𝑡))𝑑𝑡, (19) for any 𝑢,𝑣∈𝑋. The corresponding norm is defined by ‖𝑢‖=^𝑇₀|𝑢(𝑡)|²𝑑𝑡+^𝑇₀|𝑢̇(𝑡)|²𝑑𝑡^1/2, (20) for any 𝑢∈𝑋. Moreover, it is well known that 𝑋 is compactly embedded in 𝐶[0,𝑇], which implies that ‖𝑢‖_∞≤𝐶‖𝑢‖, (21) for some constant 𝐶>0, where ‖𝑢‖_∞=max_{𝑡∈[0,𝑇]}|𝑢(𝑡)|.

Define the corresponding functional 𝜑 on 𝑋 by 𝜑(𝑢)=12^𝑇₀|𝑢̇(𝑡)|²1𝑑𝑡+2^𝑇₀((𝑡)(𝑡+𝐴𝑢,𝑢))𝑑𝑡_{𝑙𝑁𝑗=1𝑖=1𝑢𝑖(𝑡𝑗)0}𝐼_𝑖𝑗(𝑠)𝑑𝑠−^𝑇₀𝐹(𝑡,𝑢(𝑡))𝑑𝑡. (22)

By the conditions of 𝐹 and 𝐼_𝑖𝑗, 𝑖=1,2,…,𝑁,𝑗=1,2,…,𝑙, we get that functional 𝜑 is a continuously Gáteaux differential functional whose Gáteaux derivative is the functional 𝜑^′(𝑢), given by 𝜑^′(𝑢),𝑣=^𝑇₀(𝑢̇(𝑡),𝑣̇(𝑡))𝑑𝑡+^𝑇₀((𝑡)(𝑡+𝐴𝑢,𝑣))𝑑𝑡_{𝑙𝑁𝑗=1𝑖=1}𝐼_𝑖𝑗𝑢^𝑖𝑡_𝑗𝑣^𝑖𝑡_𝑗−^𝑇₀(∇𝐹(𝑡,𝑢(𝑡)),𝑣(𝑡))𝑑𝑡. (23)

If 𝑢∈𝐻¹₀(0,𝑇), then 𝑢 is absolutely continuous and 𝑢̇∈𝐿²(0,𝑇). In this case, Δ𝑢̇(𝑡)=𝑢̇(𝑡⁺)−𝑢̇(𝑡⁻)=0 may not hold for some 𝑡∈(0,𝑇); this leads to impulsive effects.

Following the ideas of [11, 12], we can prove that the critical points of 𝜑 are the weak solutions of problem (1).

To prove our main results, we have the following facts (see [32]).

Letting 𝜙(𝑢)=^𝑇₀12|𝑢̇(𝑡)|²+(𝐴𝑢(𝑡),𝑢(𝑡))𝑑𝑡, (24) we see that 𝜙(𝑢)=12‖𝑢‖²−12^𝑇₀)𝑢(𝑡)(𝑡=1((𝐼−𝐴,𝑢))𝑑𝑡2((𝐼−𝐾)𝑢,𝑢), (25) where 𝐾∶𝐻¹_𝑇→𝐻¹_𝑇 is the liner self-adjoint operator defined and 𝐼 is the 𝑁×𝑁 identity matrix. By the Riesz representation theorem, we have ((𝐼−𝐴)^𝑇₀𝑢(𝑡),𝑣(𝑡))𝑑𝑡=(𝐾𝑢,𝑣). (26)

The compact imbedding of 𝐻¹_𝑇 into 𝐶([0,𝑇],𝑅^𝑁) implies that 𝐾 is compact. Summing up the above discussion, 𝜑(𝑢) can be rewritten as 𝜑(𝑢)=12((𝐼−𝐾)𝑢,𝑢)+^𝑙𝑁_𝑗=1𝑖=1^{𝑢𝑖(𝑡𝑗)}₀𝐼_𝑖𝑗(𝑠)𝑑𝑠+^𝑇₀𝐹(𝑡,𝑢(𝑡))𝑑𝑡. (27)

By classical spectral theory, we can decompose 𝑋 into the orthogonal sum of invariant subspace for 𝐼−𝐾𝑋=𝐻⁻⊕𝐻⁰⊕𝐻⁺, (28) where 𝐻⁰=ker(𝐼−𝐾) and 𝐻⁻ and 𝐻⁺ are such that, for some 𝛿>0, ((𝐼−𝐾)𝑢,𝑢)≤−𝛿‖𝑢‖²,if𝑢∈𝐻⁻,((𝐼−𝐾)𝑢,𝑢)≥𝛿‖𝑢‖²,if𝑢∈𝐻⁺. (29) Notice that 𝐻⁻ is finite dimensional.

In this paper, we set 𝑎∶=max_{1≤𝑖,𝑗≤𝑁}|𝑎_𝑖𝑗|.

3. Proof of Main Results

3.1. The Proof of Theorem 2

Let 𝑋¹=𝐻⁺ and 𝑋²=𝐻⁰⊕𝐻⁻; then 𝑋=𝑋¹⊕𝑋². Suppose {𝑒_𝑛}_𝑛≥1 is an orthogonal basis of 𝐻⁺. Correspondingly, let 𝑋¹_𝑛=span𝑒₁,𝑒₂,…,𝑒_𝑛,𝑋²_𝑛=𝑋²,𝑛∈𝑁; (30) then 𝑋^𝑗=⋃_𝑛∈𝑁𝑋^𝑗_𝑛, 𝑗=1,2. We divide our proof into four steps.

Step 1. 𝜑 has local linking at 0.
In view of (I1), we obtain ^𝑢₀𝐼_𝑖𝑗(𝑠)≤𝑏𝑑𝑠^𝑖𝑗𝑟_𝑖𝑗|𝑢|+1^{𝑟𝑖𝑗+1}. (31) Combining this inequality, we have ∫^𝑢₀𝐼_𝑖𝑗(𝑠)𝑑𝑠|𝑢|²≤𝑏_𝑖𝑗𝑟_𝑖𝑗|𝑢|+1^{𝑟𝑖𝑗−1}⟶0,as𝑢⟶0. (32) Since 𝑟_𝑖𝑗>1, this implies ∫^𝑢₀𝐼_𝑖𝑗(𝑠)𝑑𝑠|𝑢|²⟶0,as𝑢⟶0. (33) Applying (H2) and (33), for any 𝜀>0, there exists 𝑟₁>0 such that |∇𝐹(𝑡,𝑢)|≤2𝜀|𝑢|,^𝑢₀𝐼_𝑖𝑗(𝑠)𝑑𝑠≤𝜀|𝑢|²,∀|𝑢|≤𝑟₁,𝑡∈[0,𝑇], (34) which implies that |𝐹(𝑡,𝑢)|=¹₀(∇𝐹(𝑡,𝑠𝑢),𝑢)𝑑𝑠≤¹₀|(𝑢|≤∇𝐹𝑡,𝑠𝑢)||𝑑𝑠₁₀2𝑠𝜀|𝑢|²𝑑𝑠=𝜀|𝑢|². (35)
On one hand, by (34) and (35), for all 𝑢∈𝑋¹=𝐻⁺ with ‖𝑢‖≤𝑟₂∶=𝑟₁/𝐶. Choose 𝜀=𝛿/4(𝑙𝑁+𝑇)𝐶²; then one has 𝜑(𝑢)=12((𝐼−𝐾)𝑢,𝑢)+^𝑙𝑁_𝑗=1𝑖=1^{𝑢𝑖(𝑡𝑗)}₀𝐼_𝑖𝑗(𝑠)𝑑𝑠−^𝑇₀𝐹((𝑡≥𝛿𝑡,𝑢))𝑑𝑡2‖𝑢‖²−^𝑙𝑁_𝑗=1𝑖=1𝜀𝑢^𝑖𝑡_𝑗²−𝜀^𝑇₀|𝑢|²≥𝛿𝑑𝑡2‖𝑢‖²−𝑙𝑁𝐶²𝜀‖𝑢‖²−𝑇𝐶²𝜀‖𝑢‖²=𝛿4‖𝑢‖²≥0. (36)
On the other hand, since dim𝑋² is finite, there exists a constant 𝐾₁>0 such that ‖𝑢‖≤𝐾₁‖𝑢⁻‖,∀𝑢∈𝑋². (37) For all 𝑢∈𝑋²=𝐻⁻⊕𝐻⁰ with ‖𝑢‖≤𝑟₃∶=𝑟₁/𝐶. Choose 𝜀=𝛿/4(𝑙𝑁+𝑇)𝐶²𝐾²₁; by (34), (35), and (37), we obtain 𝜑(𝑢)=12((𝐼−𝐾)𝑢⁻,𝑢⁻)+^𝑙𝑁_𝑗=1𝑖=1^{𝑢𝑖(𝑡𝑗)}₀𝐼_𝑖𝑗(𝑠)𝑑𝑠−^𝑇₀𝐹((𝑡𝛿𝑡,𝑢))𝑑𝑡≤−2‖𝑢⁻‖²+^𝑙𝑁_𝑗=1𝑖=1𝜀𝑢^𝑖𝑡_𝑗²+𝜀^𝑇₀|𝑢|²𝛿𝑑𝑡≤−2‖𝑢⁻‖²+(𝑙𝑁+𝑇)𝐶²𝜀‖𝑢‖²𝛿≤−2‖𝑢⁻‖²+(𝑙𝑁+𝑇)𝐶²𝐾²₁𝜀‖𝑢⁻‖²𝛿=−4‖𝑢⁻‖²≤0. (38) Let 𝑟=min{𝑟₂,𝑟₃}; one has 𝜑(𝑢)≥0,𝑢∈𝑋¹,‖𝑢‖𝜑(𝑢)≤𝑟,≤0,𝑢∈𝑋²,‖𝑢‖≥𝑟. (39)

Step 2. 𝜑 maps bounded sets into bounded sets.
By (H3) and 𝐹∈𝐶¹([0,𝑇]×ℝ^𝑁,ℝ), there exists 𝑐₂>0 such that |∇𝐹(𝑡,𝑢)|≤𝑐₂+𝑐₁|𝑢|^𝑑,∀(𝑡,𝑢)∈[0,𝑇]×ℝ^𝑁, (40) which implies that |𝐹(𝑡,𝑢)|≤¹₀|∇𝐹(𝑡,𝑠𝑢)||𝑢|𝑑𝑠≤¹₀𝑐₂|𝑢|+𝑐₁|𝑢|^𝑑+1𝑑𝑠=𝑐₂|𝑢|+𝑐₁|𝑢|^𝑑+1,∀(𝑡,𝑢)∈[0,𝑇]×ℝ^𝑁. (41) Note that 𝜑(𝑢=1)2^𝑇₀|𝑢̇(𝑡)|²1𝑑𝑡+2^𝑇₀((𝑡)(𝑡+𝐴𝑢,𝑢))𝑑𝑡_{𝑙𝑁𝑗=1𝑖=1𝑢𝑖(𝑡𝑗)0}𝐼_𝑖𝑗(𝑠)𝑑𝑠−^𝑇₀𝐹((𝑡≤1𝑡,𝑢))𝑑𝑡2^𝑇₀|𝑢̇(𝑡)|²1𝑑𝑡+2𝑁𝑎^𝑇₀|𝑢(𝑡)|²+𝑑𝑡_{𝑙𝑁𝑗=1𝑖=1}𝑏_𝑖𝑗𝑟_𝑖𝑗𝑢+1^𝑖𝑡_𝑗^{𝑟𝑖𝑗+1}+𝑐₂^𝑇₀|𝑢|𝑑𝑡+𝑐₁^𝑇₀|𝑢|^𝑑+1≤1𝑑𝑡2+12𝑁𝑎‖𝑢‖²+^𝑙𝑁_𝑗=1𝑖=1𝑏_𝑖𝑗𝑟_𝑖𝑗𝐶+1^{𝑟𝑖𝑗+1}‖𝑢‖^{𝑟𝑖𝑗+1}+𝑐₂𝐶𝑇‖𝑢‖+𝑐₁𝐶^𝑑1+1𝑇‖𝑢‖^𝑑+1. (42) It implies that 𝜑 maps bounded sets into bounded sets.

Step 3. 𝜑 satisfies the (𝑃𝑆)^∗ condition.
Consider a (𝑃𝑆)^∗ sequence {𝑢_𝛼𝑛} such that {𝛼_𝑛} is admissible. Then there exists a constant 𝑀₁>0 such that 𝜑𝑢_𝛼𝑛≤𝑀₁,𝜑^′_𝛼𝑛𝑢_𝛼𝑛≤𝑀₁. (43) By (41), for |𝑢|≤𝐿₄, one has |𝐹(𝑡,𝑢)|≤𝑐₂|𝑢|+𝑐₁|𝑢|^𝑑+1≤𝑐₂𝐿₄+𝑐₁𝐿₄^𝑑+1; (44) together with (H4), one has 𝜇𝐹(𝑡,𝑢)≤(∇𝐹(𝑡,𝑢),𝑢)+𝑏₁|𝑢|²+𝜇𝑐₂𝐿₄+𝑐₁𝐿₄^𝑑+1, (45) for all (𝑡,𝑢)∈[0,𝑇]×ℝ^𝑁.
It follows from (21), (43), (45), (H1), and (I2) that 𝑀₁+𝑀₁𝜇𝑢_𝛼𝑛≥𝜑𝑢_𝛼𝑛−1𝜇𝜑^′𝑢_𝛼𝑛,𝑢_𝛼𝑛=12−1𝜇^𝑇₀𝑢̇_𝛼𝑛²1𝑑𝑡+2−1𝜇^𝑇₀𝐴𝑢_𝛼𝑛,𝑢_𝛼𝑛+1𝑑𝑡𝜇^𝑇₀∇𝐹𝑡,𝑢_𝛼𝑛,𝑢_𝛼𝑛−𝜇𝐹𝑡,𝑢_𝛼𝑛+1𝑑𝑡𝜇^𝑙𝑁_𝑗=1𝑖=1𝜇^{𝑢𝑖𝛼𝑛(𝑡𝑗)}₀𝐼_𝑖𝑗(𝑠)𝑑𝑠−𝐼_𝑖𝑗𝑢^𝑖_𝛼𝑛𝑡_𝑗𝑢^𝑖_𝛼𝑛𝑡_𝑗≥12−1𝜇^𝑇₀𝑢̇_𝛼𝑛²𝜃𝑑𝑡+2−𝜃𝜇−𝑏₁𝜇^𝑇₀𝑢_𝛼𝑛²𝑐𝑑𝑡−𝜇𝑇₂𝐿₄+𝑐₁𝐿₄^𝑑+1−𝑏₂𝜇^𝑙𝑁_𝑗=1𝑖=1𝑢_𝛼𝑛^𝛾_∞1≥min2−1𝜇,12−1𝜇𝑏𝜃−¹𝜇𝑢_𝛼𝑛²−𝑏₂𝑙𝑁𝜇𝐶^𝛾𝑢_𝛼𝑛^𝛾−𝜇𝑇𝑐₂𝐿₄+𝑐₁𝐿₄^𝑑+1. (46)
Since 𝜇>2, 𝛾∈[0,2), and 𝑏₁∈(0,(𝜇/2−1)𝜃), (46) shows that {𝑢_𝛼𝑛} is bounded in 𝐻¹_𝑇. By a method similar to that of [10], we can prove that {𝑢_𝛼𝑛} has a convergent subsequence. Thus 𝜑 satisfies (𝑃𝑆)^∗ condition.

Step 4. For every 𝑚∈𝑁 and 𝑢∈𝑋¹_𝑚⊕𝑋², 𝜑(𝑢)→−∞ as ‖𝑢‖→∞.
Since dim(𝑋¹_𝑚⊕𝑋²) is finite, there exists 𝑀₂>0 such that ‖𝑢‖≤𝑀_2𝐸0|𝑢|²𝑑𝑡^1/2,∀𝑢∈𝑋¹_𝑚⊕𝑋². (47) In fact, for 𝑀₃=𝑀²₂(max{1/2,(1/2)𝑁𝑎}+𝛿/2)>0, by (B2), there exists 𝑀₄>0 such that 𝐹(𝑡,𝑢)≥𝑀₃|𝑢|²−𝑀₄,∀𝑢∈ℝ^𝑁,a.e.𝑡∈𝐸₀. (48) Hence, it follows from (47), (48), and (I3) that 𝜑(𝑢)=12((𝐼−𝐾)𝑢,𝑢)+^𝑙𝑁_𝑗=1𝑖=1^{𝑢𝑖(𝑡𝑗)}₀𝐼_𝑖𝑗(𝑠)𝑑𝑠−^𝑇₀𝐹()=1𝑡,𝑢𝑑𝑡2((𝐼−𝐾)𝑢⁻,𝑢⁻)+12(𝐼−𝐾)𝑢⁺,𝑢⁺+^𝑙𝑁_𝑗=1𝑖=1^{𝑢𝑖(𝑡𝑗)}₀𝐼_𝑖𝑗(𝑠)𝑑𝑠−^𝑇₀𝐹()𝛿𝑡,𝑢𝑑𝑡≤−2‖𝑢⁻‖²+12^𝑇₀𝑢̇⁺²1𝑑𝑡+2^𝑇₀𝐴𝑢⁺,𝑢⁺−𝑑𝑡_𝐸0𝐹()𝛿𝑡,𝑢𝑑𝑡≤−2‖𝑢⁻‖²1+max2,12𝑁𝑎𝑢⁺²−𝑀_3𝐸0|𝑢|²𝑑𝑡+𝑀₄𝑇𝛿≤−2‖𝑢⁻‖²1+max2,12𝑁𝑎𝑢⁺²−1max2,12𝑁𝑎+𝛿2‖𝑢‖²+𝑀₄𝑇𝛿≤−2‖𝑢⁻‖²1+max2,12𝑁𝑎𝑢⁺²−1max2,12𝑁𝑎+𝛿2𝑢⁺²+𝑢⁰²+𝑀₄𝑇𝛿≤−2‖𝑢‖²+𝑀₄𝑇, (49) for 𝑢∈𝑋¹_𝑚⊕𝑋² and a.e. 𝑡∈𝐸₀, which implies that 𝜑(𝑢)⟶−∞,as‖𝑢‖⟶∞on𝑢∈𝑋¹_𝑚⊕𝑋². (50) Therefore, all the assumptions of Theorem 8 are verified. Then, the proof of Theorem 2 is completed.

3.2. The Proof of Theorem 4

Following the same procedures in the proof of Theorem 2, we can prove that 𝜑 satisfies (A1), (A2), and (A3) in Theorem 8.

To prove (A4), set 𝜙(𝜉)=𝐹(𝑡,𝜉𝑢)𝜉^−𝜇+(𝑏₃/(𝜇−2))(1−𝜉^2−𝜇)|𝑢|², 𝜉∈(0,+∞); then by (H5), we have 𝜙^′(𝜉)=(∇𝐹(𝑡,𝜉𝑢),𝑢)𝜉^−𝜇−𝜇𝜉^−𝜇−1𝐹(𝑡,𝜉𝑢)+𝑏₃𝜉^1−𝜇|𝑢|²=(∇𝐹(𝑡,𝜉𝑢),𝜉𝑢)−𝜇𝐹(𝑡,𝜉𝑢)𝜉^−𝜇−1+𝑏₃𝜉^1−𝜇|𝑢|²≥0. (51) When 0<𝜉≤1, it follows from (51) that 𝜙(1)=𝐹(𝑡,𝑢)≥𝜙(𝜉)=𝐹(𝑡,𝜉𝑢)𝜉^−𝜇+𝑏₃𝜇−21−𝜉^2−𝜇|𝑢|²; (52) this implies 𝐹()≥𝐹𝑢𝑡,𝑢𝑡,|𝑢|−𝑏₃|𝑢|𝜇−2^𝜇+𝑏₃|𝑢|𝜇−2²,if|𝑢|≥1. (53) Set 𝑚₂=max{𝐹(𝑡,𝑢)∣𝑡∈[0,𝑇],𝑢∈ℝ^𝑁,|𝑢|≤1}; then by (53), we have 𝐹(𝑡,𝑢)≥𝑚₁−𝑏₃|𝑢|𝜇−2^𝜇+𝑏₃|𝑢|𝜇−2²−𝑚₂, (54)∀(𝑡,𝑢)∈[0,𝑇]×ℝ^𝑁; since  dim(𝑋¹_𝑚⊕𝑋²) is finite, there exists 𝑀₅>0 such that ‖𝑢‖^𝜇≤𝑀₅^𝑇₀|𝑢|^𝜇𝑑𝑡,∀𝑢∈𝑋¹_𝑚⊕𝑋². (55) By (54), (55), and (I3), we have 𝜑(𝑢)=12^𝑇₀|𝑢̇|²1𝑑𝑡+2^𝑇₀()+𝐴𝑢,𝑢𝑑𝑡_{𝑙𝑁𝑗=1𝑖=1𝑢𝑖(𝑡𝑗)0}𝐼_𝑖𝑗(𝑠)𝑑𝑠−^𝑇₀𝐹()≤1𝑡,𝑢𝑑𝑡2^𝑇₀|𝑢̇|²1𝑑𝑡+2𝑁𝑎^𝑇₀|𝑢|²−𝑚𝑑𝑡₁−𝑏₃𝜇−2^𝑇₀|𝑢|^𝜇𝑏𝑑𝑡+³𝜇−2^𝑇₀|𝑢|²𝑑𝑡−𝑚₂𝑇1≤max2,12𝑁𝑎‖𝑢‖²−1𝑀₅𝑚₁−𝑏₃𝑢‖𝜇−2‖^𝜇+𝑏₃𝑀₅‖𝑢‖𝜇−2²−𝑚₂𝑇. (56) Since 𝜇>2 and 𝑚₁−(𝑏₃/(𝜇−2))>0, (56) implies 𝜑(𝑢)⟶−∞,as‖𝑢‖⟶∞on𝑢∈𝑋¹_𝑚⊕𝑋². (57) Consequently, the conclusion follows from Theorem 8. This completes the proof.

3.3. The Proof of Theorem 5

Similar to the proof of Theorem 2, 𝜑 satisfies all conditions of Theorem 8. Thus, problem (1) has at least one nontrivial T-periodic solution.

Conflict of Interests

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

Acknowledgment

This work is supported by the National Natural Science Foundation of China (no. 11271371).