Existence of Periodic Solutions for a Class of Difference Systems with p-Laplacian

Abstract

By applying the least action principle and minimax methods in critical point theory, we prove the existence of periodic solutions for a class of difference systems with p-Laplacian and obtain some existence theorems.

1. Introduction

Consider the following 𝑝-Laplacian difference system: Δ|Δ𝑢(𝑡−1)|^𝑝−2Δ𝑢(𝑡−1)=∇𝐹(𝑡,𝑢(𝑡)),𝑡∈ℤ, (1.1) where Δ is the forward difference operator defined by Δ𝑢(𝑡)=𝑢(𝑡+1)−𝑢(𝑡), Δ²𝑢(𝑡)=Δ(Δ𝑢(𝑡)), 𝑝∈(1,+∞) such that 1/𝑝+1/𝑞=1, 𝑡∈ℤ, 𝑢∈ℝ^𝑁, 𝐹∶ℤ×ℝ^𝑁→ℝ, and 𝐹(𝑡,𝑥) is continuously differentiable in 𝑥 for every 𝑡∈ℤ and 𝑇-periodic in 𝑡 for all 𝑥∈ℝ^𝑁.

When 𝑝=2, (1.1) reduces to the following second-order discrete Hamiltonian system: Δ²𝑢(𝑡−1)=∇𝐹(𝑡,𝑢(𝑡)),𝑡∈ℤ. (1.2)

Difference equations provide a natural description of many discrete models in real world. Since discrete models exist in various fields of science and technology such as statistics, computer science, electrical circuit analysis, biology, neural network, and optimal control, it is of practical importance to investigate the solutions of difference equations. For more details about difference equations, we refer the readers to the books [1–3].

In some recent papers [4–18], the authors studied the existence of periodic solutions and subharmonic solutions of difference equations by applying critical point theory. These papers show that the critical point theory is an effective method to the study of periodic solutions for difference equations. Motivated by the above papers, we consider the existence of periodic solutions for problem (1.1) by using the least action principle and minimax methods in critical point theory.

2. Preliminaries

Now, we first present our main results.

Theorem 2.1. Suppose that 𝐹 satisfies the following conditions: (F1) there exists an integer 𝑇>1 such that 𝐹(𝑡+𝑇,𝑥)=𝐹(𝑡,𝑥) for all (𝑡,𝑥)∈ℤ×ℝ^𝑁;(F2) there exist 𝑓, 𝑔∈𝑙¹([1,𝑇],ℝ⁺) and 𝛼∈[0,𝑝−1) such that |∇𝐹(𝑡,𝑥)|≤𝑓(𝑡)|𝑥|^𝛼+𝑔(𝑡),∀(𝑡,𝑥)∈ℤ[1,𝑇]×ℝ^𝑁, (2.1) where ℤ[𝑎,𝑏]∶=ℤ∩[𝑎,𝑏] for every 𝑎,𝑏∈ℤ with 𝑎≤𝑏,(F3)liminf_{|𝑥|→+∞}|𝑥|^{𝑇−𝑞𝛼}_𝑡=1𝐹()>2𝑡,𝑥^𝑞𝛼(𝑇−1)^{𝑞(2𝑝−1)/𝑝}𝑞𝑇^𝑇_𝑡=1𝑓^𝑞(𝑡),∀𝑡∈ℤ[1,𝑇]. (2.2) Then problem (1.1) has at least one periodic solution with period 𝑇.

Theorem 2.2. Suppose that 𝐹 satisfies (F1) and the following conditions: ^𝑇_𝑡=1𝑓(𝑡)<𝑇^𝑝2^𝑝−1(𝑇−1)^{𝑝(1+𝑞)/𝑞}; (2.3)(F2)' there exist 𝑓, 𝑔∈𝑙¹([1,𝑇],ℝ⁺) such that |∇𝐹(𝑡,𝑥)|≤𝑓(𝑡)|𝑥|^𝑝−1+𝑔(𝑡),∀(𝑡,𝑥)∈ℤ[1,𝑇]×ℝ^𝑁, (2.4) where ℤ[𝑎,𝑏]∶=ℤ∩[𝑎,𝑏] for every 𝑎,𝑏∈ℤ with 𝑎≤𝑏;(F4)liminf_{|𝑥|→+∞}|𝑥|^𝑇−𝑝_𝑡=1𝐹()>2𝑡,𝑥^𝑝𝑇^𝑞/𝑝(𝑇−1)^{𝑞(2𝑝−1)/𝑝}𝑇^𝑝−2^𝑝−1(𝑇−1)^{𝑝(1+𝑞)/𝑞}∑^𝑇_𝑡=1𝑓(𝑡)^{𝑇𝑞/𝑝}_𝑡=1𝑓^𝑞(𝑡),∀𝑡∈ℤ[1,𝑇]. (2.5) Then problem (1.1) has at least one periodic solution with period 𝑇.

Theorem 2.3. Suppose that 𝐹 satisfies (F1), (F2), and the following condition:
(F5) limsup_{|𝑥|→+∞}|𝑥|^{𝑇−𝑞𝛼}_𝑡=1𝐹()2𝑡,𝑥<−^𝑞𝛼(𝑇−1)^{𝑞(2𝑝−1)/𝑝}+2𝑝𝑇^𝑞𝛼(𝑇−1)^{(𝑞−1)2(2𝑝−1)/𝑝}𝑞𝑇^{(𝑞−1)2/𝑞}+2^𝑞𝛼(𝑇−1)^{2𝑝−1+(2𝑝−1)/𝑝}𝑝𝑇^{(𝑝+1)/𝑞𝑇}_𝑡=1𝑓^𝑞(𝑡)∀𝑡∈ℤ[1,𝑇]. (2.6) Then problem (1.1) has at least one periodic solution with period 𝑇.

Theorem 2.4. Suppose that 𝐹 satisfies (F1), (2.3), (F2)', and the following condition:
(F6) limsup_{|𝑥|→+∞}|𝑥|^𝑇−𝑝_𝑡=1𝐹()2𝑡,𝑥<−^𝑝𝑝𝑇^𝑞/𝑝(𝑇−1)^{𝑞(2𝑝−1)/𝑝}𝑇^𝑝+2^𝑝−1(𝑇−1)^{𝑝(1+𝑞)/𝑞}∑^𝑇_𝑡=1𝑓(𝑡)𝑝𝑇^𝑝−2^𝑝−1(𝑇−1)^{𝑝(1+𝑞)/𝑞}∑^𝑇_𝑡=1𝑓(𝑡)^𝑞+2^𝑝𝑝𝑇^1/𝑝𝑇(𝑇−1)^{2𝑝−1+(2𝑝−1)/𝑝}𝑝𝑇^𝑝−2^𝑝−1(𝑇−1)^{𝑝(1+𝑞)/𝑞}∑^𝑇_𝑡=1𝑓(𝑡)^1+1/𝑝+2^𝑝𝑝𝑇^{(𝑞−1)2/𝑝}(𝑇−1)^{(𝑞−1)2(2𝑝−1)/𝑝}𝑞𝑝𝑇^𝑝−2^𝑝−1(𝑇−1)^{𝑝(1+𝑞)/𝑞}∑^𝑇_𝑡=1𝑓(𝑡)^{(𝑞−1)2/𝑝𝑇𝑡=1}𝑓^𝑞(𝑡),∀𝑡∈ℤ[1,𝑇]. (2.7) Then problem (1.1) has at least one periodic solution with period 𝑇.

Remark 2.5. The lower bounds and the upper bounds of our theorems are more accurate than the existing results in the literature. Moreover, there are functions satisfying our results but not satisfying the existing results in the literature.
Let the Sobolev space 𝐸_𝑇 be defined by 𝐸_𝑇=𝑢∶ℤ⟶ℝ^𝑁∣𝑢(𝑡+𝑇)=𝑢(𝑡),𝑡∈ℤ. (2.8)
For 𝑢∈𝐸_𝑇, let 𝑢=(1/𝑇)∑^𝑇_𝑡=1𝑢(𝑡), 𝑢=𝑢+𝑢, and 𝐸_𝑇={𝑢∈𝐸_𝑇∣𝑢=0}, then 𝐸_𝑇=ℝ^𝑁⊕𝐸_𝑇. Let ‖𝑢‖=|𝑢|^𝑝+^𝑇_𝑡=1|Δ𝑢(𝑡)|^𝑝1/𝑝,𝑢∈𝐸_𝑇. (2.9) As usual, let ‖𝑢‖_∞=sup{|𝑢(𝑡)|∶𝑡∈ℤ[1,𝑇]},∀𝑢∈𝑙^∞ℤ[1,𝑇],ℝ^𝑁. (2.10)
For any 𝑢∈𝐸_𝑇, let 𝜑(𝑢)=1𝑝^𝑇_𝑡=1|Δ𝑢(𝑡)|^𝑝+^𝑇_𝑡=1𝐹((𝑡=1𝑡,𝑢))𝑝^𝑇_𝑡=1|Δ𝑢(𝑡)|^𝑝+^𝑇_𝑡=1𝐹(𝑡,𝑢(𝑡)). (2.11)
To prove our results, we need the following lemma.

Lemma 2.6 (see [18]). Let 𝑢∈𝐸_𝑇. If ∑^𝑇_𝑡=1𝑢(𝑡)=0, then ‖𝑢‖_∞≤(𝑇−1)^{(1+𝑞)/𝑞}𝑇‖‖,‖𝑢‖_𝑝𝑝𝑢=^𝑇_𝑡=1|𝑢(𝑡)|^𝑝≤(𝑇−1)^2𝑝−1𝑇^𝑝−1‖𝑢‖^𝑝. (2.12)(2.13)

3. Proofs

For the sake of convenience, we denote 𝑀₁=^𝑇_𝑡=1𝑓^𝑞(𝑡)^1/𝑞,𝑀₂=^𝑇_𝑡=1𝑓(𝑡),𝑀₃=^𝑇_𝑡=1𝑔(𝑡). (3.1)

Proof of Theorem 2.1. From (F3), we can choose 𝑎₁>(𝑇−1)^{(2𝑝−1)/𝑝}/𝑇^{(𝑝−1)/𝑝} such that liminf_{|𝑥|→+∞}|𝑥|^{𝑇−𝑞𝛼}_𝑡=1𝐹()>𝑎𝑡,𝑥^𝑞12^𝑞𝛼𝑞𝑀^𝑞₁. (3.2) It follows from (F2), (2.12), and (2.13) that ^𝑇_𝑡=1[𝐹(𝑡,𝑢(𝑡))−𝐹(𝑡,𝑢=)]_𝑇𝑡=110(∇𝐹(𝑡,(𝑡(𝑡≤𝑢+𝑠𝑢)),𝑢))𝑑𝑠_𝑇𝑡=110𝑓(𝑡)|𝑢+𝑠𝑢(𝑡)|^𝛼|𝑢(𝑡)|𝑑𝑠+^𝑇_𝑡=1¹₀𝑔(𝑡)|𝑢(𝑡)|𝑑𝑠≤2^𝛼𝑇_𝑡=1𝑓(𝑡)|𝑢|^𝛼+|𝑢(𝑡)|^𝛼|𝑢(𝑡)|+^𝑇_𝑡=1𝑔(𝑡)|𝑢(𝑡)|≤2^𝛼|𝑢|^𝛼𝑇_𝑡=1𝑓^𝑞(𝑡)^1/𝑞𝑇_𝑡=1|𝑢(𝑡)|^𝑝1/𝑝+2^𝛼‖𝑢‖_∞^𝑇1+𝛼_𝑡=1𝑓(𝑡)+‖𝑢‖_∞^𝑇_𝑡=1𝑔(𝑡)=2^𝛼|𝑢|^𝛼M₁‖𝑢‖_𝑝+2^𝛼𝑀₂‖𝑢‖_∞^1+𝛼+𝑀₃‖𝑢‖_∞≤1𝑝𝑎^𝑝₁‖𝑢‖^𝑝_𝑝+𝑎^𝑞₁2^𝑞𝛼𝑞|𝑢|^𝑞𝛼𝑀^𝑞₁+2^𝛼𝑀₂‖𝑢‖_∞^1+𝛼+𝑀₃‖𝑢‖_∞≤(𝑇−1)^2𝑝−1𝑝𝑎^𝑝₁𝑇^𝑝−1‖𝑢‖^𝑝+2^𝛼𝑀₂(𝑇−1)^{(1+𝑞)(1+𝛼)/𝑞}𝑇^1+𝛼‖𝑢‖^1+𝛼+𝑎^𝑞₁2^𝑞𝛼𝑞|𝑢|^𝑞𝛼𝑀^𝑞₁+𝑀₃(𝑇−1)^{(1+𝑞)/𝑞}𝑇‖𝑢‖. (3.3) Hence, we have𝜑(𝑢)=1𝑝^𝑇_𝑡=1|Δ𝑢(𝑡)|^𝑝+^𝑇_𝑡=1𝐹((𝑡=1𝑡,𝑢))𝑝^𝑇_𝑡=1|Δ𝑢(𝑡)|^𝑝+^𝑇_𝑡=1[𝐹(𝑡,𝑢(𝑡))−𝐹(𝑡,𝑢)]+^𝑇_𝑡=1𝐹(𝑡,𝑢)≥1𝑝^𝑇_𝑡=1|Δ𝑢(𝑡)|^𝑝−2^𝛼𝑀₂(𝑇−1)^{(1+𝑞)(1+𝛼)/𝑞}𝑇^1+𝛼‖𝑢‖^1+𝛼+^𝑇_t=1𝐹(𝑡,𝑢)−(𝑇−1)^2𝑝−1𝑝𝑎^𝑝₁𝑇^𝑝−1‖𝑢‖^𝑝−𝑎^𝑞₁2^𝑞𝛼𝑞|𝑢|^𝑞𝛼𝑀^𝑞₁−𝑀₃(𝑇−1)^{(1+𝑞)/𝑞}𝑇‖‖=1𝑢𝑝−(𝑇−1)^2𝑝−1𝑝𝑎^𝑝₁𝑇^𝑝−1‖𝑢‖^𝑝−2^𝛼𝑀₂(𝑇−1)^{(1+𝑞)(1+𝛼)/𝑞}𝑇^1+𝛼‖𝑢‖^1+𝛼+|𝑢|^𝑞𝛼|𝑢|^{𝑇−𝑞𝛼}_𝑡=1𝐹(𝑡,𝑢)−𝑎^𝑞₁2^𝑞𝛼𝑞𝑀^𝑞₁−𝑀₃(𝑇−1)^{(1+𝑞)/𝑞}𝑇‖𝑢‖. (3.4) The above inequality and (3.2) imply that 𝜑(𝑢)→+∞ as ‖𝑢‖→∞. Hence, by the least action principle, problem (1.1) has at least one periodic solution with period 𝑇.

Proof of Theorem 2.2. From (2.3) and (F4), we can choose a constant 𝑎₃∈ℝ such that𝑎₃>𝑇^1/𝑝(𝑇−1)^{(2𝑝−1)/𝑝}𝑇^𝑝−2^𝑝−1𝑀₂(𝑇−1)^{𝑝(1+𝑞)/𝑞1/𝑝}>0,liminf_{|𝑥|→+∞}|𝑥|^𝑇−𝑝_𝑡=1𝐹()>𝑎t,𝑥^𝑞32^𝑝𝑞𝑀^𝑞₁. (3.5) It follows from (F2)' and Lemma 2.6 that ^𝑇_t=1[𝐹(𝑡,𝑢(𝑡))−𝐹(𝑡,𝑢)]=^𝑇_𝑡=1¹₀(∇𝐹(𝑡,(𝑡(𝑡≤𝑢+𝑠𝑢)),𝑢))𝑑𝑠_𝑇𝑡=110𝑓(𝑡)|𝑢+𝑠𝑢(𝑡)|^𝑝−1|𝑢(𝑡)|𝑑𝑠+^𝑇_𝑡=1𝑔(𝑡(𝑡≤)|𝑢)|_𝑇𝑡=1102^𝑝−1𝑓(𝑡)|𝑢|^𝑝−1+𝑠^𝑝−1|𝑢(𝑡)|^𝑝−1|𝑢(𝑡)|𝑑𝑠+^𝑇_𝑡=1𝑔(𝑡(𝑡=)|𝑢)|_𝑇𝑡=12^𝑝−1𝑓(𝑡)|𝑢|^𝑝−1+1𝑝|𝑢(𝑡)|^𝑝−1|𝑢(𝑡)|+^𝑇_𝑡=1𝑔(𝑡)|𝑢(𝑡)|≤2^𝑝−1|𝑢|^𝑝−1𝑇_𝑡=1𝑓^𝑞(𝑡)^1/𝑞𝑇_𝑡=1|𝑢(𝑡)|^𝑝1/𝑝+2^𝑝−1𝑝𝑀₂‖𝑢‖^𝑝_∞+𝑀₃‖𝑢‖_∞=2^𝑝−1𝑀₁|𝑢|^𝑝−1‖𝑢‖_𝑝+2^𝑝−1𝑝𝑀₂‖𝑢‖^𝑝_∞+𝑀₃‖𝑢‖_∞≤1𝑝𝑎^𝑝₃‖𝑢‖^𝑝_𝑝+𝑎^𝑞₃𝑀^𝑞₁2^𝑝𝑞|𝑢|^𝑝+2^𝑝−1𝑝𝑀₂‖𝑢‖^𝑝_∞+𝑀₃‖𝑢‖_∞≤(𝑇−1)^2𝑝−1𝑝𝑎^𝑝₃𝑇^𝑝−1+2^𝑝−1𝑀₂(𝑇−1)^{𝑝(1+𝑞)/𝑞}𝑝𝑇^𝑝‖𝑢‖^𝑝+𝑎^𝑞₃𝑀^𝑞₁2^𝑝𝑞|𝑢|^𝑝+𝑀₃(𝑇−1)^{(1+𝑞)/𝑞}𝑇‖𝑢‖, (3.6) which implies that 𝜑(𝑢)=1𝑝^𝑇_𝑡=1|Δ𝑢(𝑡)|^𝑝+^𝑇_𝑡=1[𝐹(𝑡,𝑢(𝑡))−𝐹(𝑡,𝑢)]+^𝑇_𝑡=1𝐹(𝑡,𝑢)≥1𝑝−(𝑇−1)^2𝑝−1𝑝𝑎^𝑝₃𝑇^𝑝−1−2^𝑝−1𝑀₂(𝑇−1)^{𝑝(1+𝑞)/𝑞}𝑝𝑇^𝑝‖𝑢‖^𝑝−𝑀₃(𝑇−1)^{(1+𝑞)/𝑞}𝑇‖𝑢‖+|𝑢|^𝑝|𝑢|^𝑇−𝑝_𝑡=1𝐹(𝑡,𝑢)−𝑎^𝑞₃𝑀^𝑞₁2^𝑝𝑞. (3.7) The above inequality and (3.5) imply that 𝜑(𝑢)→+∞ as ‖𝑢‖→∞. Hence, by the least action principle, problem (1.1) has at least one periodic solution with period 𝑇.

Proof of Theorem 2.3. First we prove that 𝜑 satisfies the (PS) condition. Assume that {𝑢_𝑛} is a (PS) sequence of 𝜑; that is, 𝜑′(𝑢_𝑛)→0 as 𝑛→∞ and {𝜑(𝑢_𝑛)} is bounded. By (F5), we can choose 𝑎₂>(𝑇−1)^{(2𝑝−1)/𝑝}/𝑇^{(𝑝−1)/𝑝} such that limsup_{|𝑥|→+∞}|𝑥|^{𝑇−𝑞𝛼}_𝑡=1𝐹()2t,𝑥<−^𝑞𝛼𝑎^𝑞₂𝑝+2^𝑞𝛼𝑎^(𝑞−1)2₂𝑞+2^𝑞𝛼𝑎₂(𝑇−1)^2𝑝−1𝑝𝑇^𝑝−1𝑀^𝑞₁. (3.8) In a similar way to the proof of Theorem 2.1, we have ^𝑇_𝑡=1∇𝐹𝑡,𝑢_𝑛(𝑡),𝑢_𝑛(𝑡≤())𝑇−1^2𝑝−1𝑝𝑎^𝑝₂𝑇^𝑝−1𝑢_𝑛^𝑝+2^𝛼𝑀₂(𝑇−1)^{(1+𝑞)(1+𝛼)/𝑞}𝑇^1+𝛼𝑢_𝑛^1+𝛼+𝑎^𝑞₂2^𝑞𝛼𝑞𝑢_𝑛^𝑞𝛼𝑀^𝑞₁+𝑀₃(𝑇−1)^{(1+𝑞)/𝑞}𝑇𝑢_𝑛. (3.9) Hence, we have 𝑢_𝑛𝑝≥𝜑^′𝑢_𝑛,𝑢_𝑛=^𝑇_𝑡=1Δ𝑢_𝑛(𝑡)^𝑝+^𝑇_𝑡=1∇𝐹𝑡,𝑢_𝑛(𝑡),𝑢_𝑛(𝑡)≥()1−𝑇−1^2𝑝−1𝑝𝑎^𝑝₂𝑇^𝑝−1𝑢_𝑛^𝑝−𝑀₃(𝑇−1)^{(1+𝑞)/𝑞}𝑇𝑢_n−2^𝛼𝑀₂(𝑇−1)^{(1+𝑞)(1+𝛼)/𝑞}𝑇^1+𝛼𝑢_𝑛^1+𝛼−𝑎^𝑞₂2^𝑞𝛼𝑞𝑢_𝑛^𝑞𝛼𝑀^𝑞₁. (3.10) From (2.13), we have 𝑢_𝑛𝑝=^𝑇_𝑡=1𝑢_𝑛(𝑡)^𝑝1/𝑝≤(𝑇−1)^{(2𝑝−1)/𝑝}𝑇^{(𝑝−1)/𝑝}𝑢_𝑛. (3.11) From (3.10) and (3.11), we obtain 𝑎^𝑞₂2^𝑞𝛼𝑀^𝑞₁𝑞𝑢_𝑛^𝑞𝛼≥()1−𝑇−1^2𝑝−1𝑝𝑎^𝑝₂𝑇^𝑝−1𝑢_𝑛^𝑝−(𝑇−1)^{(2𝑝−1)/𝑝}𝑇^{(𝑝−1)/𝑝}𝑢_𝑛−2^𝛼𝑀₂(𝑇−1)^{(1+𝑞)(1+𝛼)/𝑞}𝑇^1+𝛼𝑢_𝑛^1+𝛼−𝑀₃(𝑇−1)^{(1+𝑞)/𝑞}𝑇𝑢_𝑛≥𝑝−1𝑝𝑢_𝑛^𝑝+𝐶₁=1𝑞𝑢_𝑛^𝑝+𝐶₁, (3.12) where 𝐶₁=min_{𝑠∈[0,+∞)}{(1/𝑝−(𝑇−1)^2𝑝−1/𝑝𝑎^𝑝₂𝑇^𝑝−1)s^𝑝−(2^𝛼𝑀₂(𝑇−1)^{(1+𝑞)(1+𝛼)/𝑞}/𝑇^1+𝛼)𝑠^1+𝛼 −[(𝑇−1)^{(2𝑝−1)/𝑝}/𝑇^{(𝑝−1)/𝑝}+𝑀₃(𝑇−1)^{(1+𝑞)/𝑞}/𝑇]𝑠}. Notice that 𝑎₂>(𝑇−1)^{(2𝑝−1)/𝑝}/𝑇^{(𝑝−1)/𝑝} implies −∞<𝐶₁<0. Hence, it follows from (3.12) that 𝑢_𝑛^𝑝≤2^𝑞𝛼𝑎^𝑞₂𝑀^𝑞₁𝑢_𝑛^𝑞𝛼−𝑞𝐶₁,𝑢_𝑛≤2^{𝑞𝛼/𝑝}𝑎₂^𝑞/𝑝𝑀₁^𝑞/𝑝𝑢_𝑛^{𝑞𝛼/𝑝}+𝐶₂, (3.13)(3.14) where 𝐶₂>0. By the proof of Theorem 2.1, we have ^𝑇_𝑡=1𝐹𝑡,𝑢_𝑛(𝑡)−𝐹𝑡,𝑢_𝑛≤2^𝛼𝑀₁𝑢_𝑛^𝛼𝑢_𝑛𝑝+2^𝛼𝑀₂𝑢_𝑛∞^1+𝛼+𝑀₃𝑢_𝑛∞≤(𝑇−1)^2𝑝−1𝑝𝑎₂^𝑞−1𝑇^𝑝−1𝑢_𝑛^𝑝+2^𝛼𝑀₂(𝑇−1)^{(1+𝑞)(1+𝛼)/𝑞}𝑇^1+𝛼𝑢_𝑛^1+𝛼+𝑎^(𝑞−1)2₂2^𝑞𝛼𝑞𝑢_𝑛^𝑞𝛼𝑀^𝑞₁+𝑀₃(𝑇−1)^{(1+𝑞)/𝑞}𝑇𝑢_𝑛. (3.15) It follows from the boundedness of 𝜑(𝑢_𝑛), (3.13)–(3.15) that 𝐶₃≤𝜑𝑢_𝑛=1𝑝^𝑇_𝑡=1Δ𝑢_𝑛(𝑡)^𝑝+^𝑇_𝑡=1𝐹𝑡,𝑢_𝑛(𝑡)−𝐹𝑡,𝑢_𝑛+^𝑇_𝑡=1𝐹𝑡,𝑢_𝑛≤1𝑝+(𝑇−1)^2𝑝−1𝑝𝑎₂^𝑞−1𝑇^𝑝−1𝑢_𝑛^𝑝+2^𝛼𝑀₂(𝑇−1)^{(1+𝑞)(1+𝛼)/𝑞}𝑇^1+𝛼𝑢_𝑛^1+𝛼+𝑎^(𝑞−1)2₂2^𝑞𝛼𝑞𝑢_𝑛^𝑞𝛼𝑀^𝑞₁+𝑀₃(𝑇−1)^{(1+𝑞)/𝑞}𝑇𝑢_𝑛+^𝑇_𝑡=1𝐹𝑡,𝑢_𝑛≤1𝑝+(𝑇−1)^2𝑝−1𝑝𝑎₂^𝑞−1𝑇^𝑝−12^𝑞𝛼𝑎^𝑞₂𝑀^𝑞₁𝑢_𝑛^𝑞𝛼−𝑞𝐶₁+𝑎^(𝑞−1)2₂2^𝑞𝛼𝑞𝑢_𝑛^𝑞𝛼𝑀^𝑞₁+^𝑇_𝑡=1𝐹𝑡,𝑢_𝑛+2^𝛼𝑀₂(𝑇−1)^{(1+𝑞)(1+𝛼)/𝑞}𝑇^1+𝛼2^{𝑞𝛼/𝑝}𝑎₂^𝑞/𝑝𝑀₁^𝑞/𝑝𝑢_𝑛^{𝑞𝛼/𝑝}+𝐶₂^1+𝛼×𝑀₃(𝑇−1)^{(1+𝑞)/𝑞}𝑇2^{𝑞𝛼/𝑝}𝑎₂^𝑞/𝑝𝑀₁^𝑞/𝑝𝑢_𝑛^{𝑞𝛼/𝑝}+𝐶₂≤2^𝑞𝛼𝑎^𝑞₂𝑝+𝑎^(𝑞−1)2₂2^𝑞𝛼𝑞+𝑎₂2^𝑞𝛼(𝑇−1)^2𝑝−1𝑝𝑇^𝑝−1𝑀^𝑞₁𝑢_𝑛^𝑞𝛼−1𝑝+(𝑇−1)^2𝑝−1𝑝𝑎₂^𝑞−1𝑇^𝑝−1𝑞𝐶₁+2^2𝛼𝑀₂(𝑇−1)^{(1+𝑞)(1+𝛼)/𝑞}𝑇^1+𝛼2^{𝑞𝛼(1+𝛼)/𝑝}𝑎₂^{𝑞(1+𝛼)/𝑝}𝑀₁^{𝑞(1+𝛼)/𝑝}𝑢_𝑛^{𝑞𝛼(1+𝛼)/𝑝}+𝐶₂^1+𝛼+^𝑇_𝑡=1𝐹𝑡,𝑢_𝑛+𝑀₃(𝑇−1)^{(1+𝑞)/𝑞}𝑇2^{𝑞𝛼/𝑝}𝑎₂^𝑞/𝑝𝑀₁^𝑞/𝑝𝑢_𝑛^{𝑞𝛼/𝑝}+𝐶₂=𝑢_𝑛^𝑞𝛼𝑢_𝑛^{𝑇−𝑞𝛼}_𝑡=1𝐹𝑡,𝑢_𝑛+2^𝑞𝛼𝑎^𝑞₂𝑝+𝑎^(𝑞−1)2₂2^𝑞𝛼𝑞+𝑎₂2^𝑞𝛼(𝑇−1)^2𝑝−1𝑝𝑇^𝑝−1𝑀^𝑞₁+2^{2𝛼+𝑞𝛼(1+𝛼)/𝑝}𝑎₂^{𝑞(1+𝛼)/𝑝}𝑀₁^{𝑞(1+𝛼)/𝑝}𝑀₂(𝑇−1)^{(1+𝑞)(1+𝛼)/𝑞}𝑇^1+𝛼𝑢_𝑛^{𝛼(1+𝛼−𝑝)(𝑞−1)}+2^{𝑞𝛼/𝑝}𝑎₂^𝑞/𝑝𝑀₁^𝑞/𝑝𝑀₃(𝑇−1)^{(1+𝑞)/𝑞}𝑇𝑢_𝑛^−𝛼+𝐶₄, (3.16) where 𝐶₃ is a positive constant and 𝐶₄ is a constant. The above inequality and (3.8) imply that {𝑢_𝑛} is bounded. Hence {𝑢_𝑛} is bounded by (2.13) and (3.13). Since 𝐸_𝑇 is finite dimensional, we conclude that 𝜑 satisfies (PS) condition.
In order to use the saddle point theorem ([19], Theorem 4.6), we only need to verify the following conditions:(I1)𝜑(𝑢)→−∞as|𝑢|→∞inℝ^𝑁, (I2)𝜑(𝑢)→+∞as‖𝑢‖→∞in𝐸_𝑇.
In fact, from (F5), we have ^𝑇_𝑡=1𝐹(𝑡,𝑢)⟶−∞as|𝑢|⟶∞inℝ^𝑁, (3.17) which together with (2.11) implies that 𝜑(𝑢)=^𝑇_𝑡=1𝐹(𝑡,𝑢)⟶−∞as|𝑢|⟶∞inℝ^𝑁. (3.18)
Hence, (I1) holds.
Next, for all 𝑢∈𝐸_𝑇, by (F2) and (2.12), we have ^𝑇_𝑡=1[𝐹(𝑡,𝑢(𝑡))−𝐹(𝑡,0)]=^𝑇_𝑡=1¹₀(((𝑡(𝑡≤∇𝐹𝑡,𝑠𝑢)),𝑢))𝑑𝑠_𝑇𝑡=1𝑓(𝑡)|𝑢(𝑡)|^1+𝛼+^𝑇_𝑡=1𝑔(𝑡)|𝑢(𝑡)|≤M₂‖𝑢‖_∞^1+𝛼+𝑀₃‖𝑢‖_∞≤𝑀₂(𝑇−1)^{(1+𝑞)(1+𝛼)/𝑞}𝑇^(1+𝛼)‖𝑢‖^1+𝛼+𝑀₃(𝑇−1)^{(1+𝑞)/𝑞}𝑇‖𝑢‖, (3.19) which implies that 𝜑(𝑢)=1𝑝^𝑇_𝑡=1|Δ𝑢(𝑡)|^𝑝+^𝑇_𝑡=1[𝐹(𝑡,𝑢(𝑡))−𝐹(𝑡,0)]+^𝑇_𝑡=1𝐹()≥1𝑡,0𝑝‖𝑢‖^𝑝−𝑀₂(𝑇−1)^{(1+𝑞)(1+𝛼)/𝑞}𝑇^(1+𝛼)‖𝑢‖^1+𝛼−𝑀₃(𝑇−1)^{(1+𝑞)/𝑞}𝑇‖𝑢‖+^𝑇_𝑡=1𝐹(𝑡,0), (3.20) for all 𝑢∈𝐸_𝑇. By Lemma 2.6, ‖𝑢‖→∞ in 𝐸_𝑇 if and only if ‖𝑢‖→∞, so from (3.20), we obtain 𝜑(𝑢)→+∞ as ‖𝑢‖→∞ in 𝐸_𝑇; that is, (I2) is verified. Hence, the proof of Theorem 2.3 is complete.

Proof of Theorem 2.4. First we prove that 𝜑 satisfies the (PS) condition. Assume that {𝑢_𝑛} is a (PS) sequence of 𝜑; that is, 𝜑′(𝑢_𝑛)→0 as 𝑛→∞ and {𝜑(𝑢_𝑛)} is bounded. By (2.3) and (F6), we can choose 𝑎₄∈ℝ such that 𝑎₄>𝑝^1/𝑝𝑇^1/𝑝(𝑇−1)^{(2𝑝−1)/𝑝}𝑝𝑇^𝑝−2^𝑝−1𝑀₂(𝑇−1)^{𝑝(1+𝑞)/𝑞1/𝑝},limsup_{|𝑥|→+∞}|𝑥|^𝑇−𝑝_𝑡=1𝐹()2𝑡,𝑥<−^𝑝𝑎^𝑞₄𝑇^𝑝+2^𝑝−1𝑀₂(𝑇−1)^{𝑝(1+𝑞)/𝑞}+2^𝑝𝑇𝑎₄(𝑇−1)^2𝑝−1𝑝𝑇^𝑝−2^𝑝−1𝑀₂(𝑇−1)^{𝑝(1+𝑞)/𝑞}+2^𝑝𝑎^(𝑞−1)2₄𝑞𝑀^𝑞₁. (3.21)(3.22) In a similar way to the proof of Theorem 2.2, we obtain ^𝑇_𝑡=1∇𝐹𝑡,𝑢_𝑛(𝑡),𝑢_𝑛(𝑡≤())𝑇−1^2𝑝−1𝑝𝑎^𝑝₄𝑇^𝑝−1+2^𝑝−1𝑀₂(𝑇−1)^{𝑝(1+𝑞)/𝑞}𝑝𝑇^𝑝𝑢_𝑛^𝑝+𝑎^𝑞₄𝑀^𝑞₁2^𝑝𝑞𝑢_𝑛^𝑝+𝑀₃(𝑇−1)^{(1+𝑞)/𝑞}𝑇𝑢_𝑛. (3.23) Hence, we have 𝑢_𝑛𝑝≥𝜑^′𝑢_𝑛,𝑢_𝑛=1𝑝^𝑇_𝑡=1Δ𝑢_𝑛(𝑡)^𝑝+^𝑇_𝑡=1∇𝐹𝑡,𝑢_𝑛(𝑡),𝑢_𝑛(𝑡)≥()1−𝑇−1^2𝑝−1𝑝𝑎^𝑝₄𝑇^𝑝−1−2^𝑝−1𝑀₂(𝑇−1)^{𝑝(1+𝑞)/𝑞}𝑝𝑇^𝑝𝑢_𝑛^𝑝−𝑎^𝑞₄𝑀^𝑞₁2^𝑝𝑞𝑢_𝑛^𝑝−𝑀₃(𝑇−1)^{(1+𝑞)/𝑞}𝑇𝑢_𝑛, (3.24) which together with (3.11) implies that 𝑎^𝑞₄𝑀^𝑞₁2^𝑝𝑞𝑢_𝑛^𝑝≥()1−𝑇−1^2𝑝−1𝑝𝑎^𝑝₄𝑇^𝑝−1−2^𝑝−1𝑀₂(𝑇−1)^{𝑝(1+𝑞)/𝑞}𝑝𝑇^𝑝𝑢_𝑛^𝑝−𝑀₃(𝑇−1)^{(1+𝑞)/𝑞}𝑇𝑢_𝑛−(𝑇−1)^{(2𝑝−1)/𝑝}𝑇^1/𝑞𝑢_𝑛≥1𝑞21−^𝑝−1𝑀₂(𝑇−1)^{𝑝(1+𝑞)/𝑞}𝑝𝑇^𝑝𝑢_𝑛^𝑝+𝐶₅, (3.25) where 𝐶₅=min_s∈[0,+∞){(1/𝑝−(𝑇−1)^2𝑝−1/𝑝𝑎^𝑝₄𝑇^𝑝−1−2^𝑝−1𝑀₂(𝑇−1)^{𝑝(1+𝑞)/𝑞}/𝑝²𝑇^𝑝)𝑠^𝑝−[𝑀₃(𝑇−1)^{(1+𝑞)/𝑞}/𝑇 +(𝑇−1)^{(2𝑝−1)/𝑝}/𝑇^1/𝑞]𝑠}. It follows from (3.21) that −∞<𝐶₅<0, so, we obtain 𝑢_𝑛^𝑝≤𝑝𝑇^𝑝𝑎^𝑞₄𝑀^𝑞₁2^𝑝𝑝𝑇^𝑝−2^𝑝−1𝑀₂(𝑇−1)^{𝑝(1+𝑞)/𝑞}𝑢_𝑛^𝑝−𝑝𝑇^𝑝𝑞𝐶₅𝑝𝑇^𝑝−2^𝑝−1𝑀₂(𝑇−1)^{𝑝(1+𝑞)/𝑞},𝑢_𝑛≤2𝑝^1/𝑝𝑇𝑎₄^𝑞/𝑝𝑀₁^𝑞/𝑝𝑝𝑇^𝑝−2^𝑝−1𝑀₂(𝑇−1)^{𝑝(1+𝑞)/𝑞1/𝑝}𝑢_𝑛+𝐶₆, (3.26)(3.27) where 𝐶₆ is a positive constant. By the proof of Theorem 2.2, we have ^𝑇_𝑡=1F𝑡,𝑢_𝑛(𝑡)−𝐹𝑡,𝑢_𝑛≤2^𝑝−1𝑀₁|𝑢|^𝑝−1‖𝑢‖_𝑝+2^𝑝−1𝑝𝑀₂‖𝑢‖^𝑝_∞+𝑀₃‖𝑢‖_∞≤(𝑇−1)^2𝑝−1𝑝𝑎₄^𝑞−1𝑇^𝑝−1+2^𝑝−1𝑀₂(𝑇−1)^{𝑝(1+𝑞)/𝑞}𝑝𝑇^𝑝𝑢_𝑛^𝑝+𝑎^(𝑞−1)2₄𝑀^𝑞₁2^𝑝𝑞𝑢_𝑛^𝑝+𝑀₃(𝑇−1)^{(1+𝑞)/𝑞}𝑇𝑢_𝑛. (3.28)
It follows from the boundedness of 𝜑(𝑢_𝑛), (3.26), (3.27), and the above inequality that𝐶₇≤𝜑𝑢_𝑛=1𝑝^𝑇_𝑡=1|Δ𝑢(𝑡)|^𝑝+^𝑇_𝑡=1[𝐹(𝑡,𝑢(𝑡))−𝐹(𝑡,𝑢)]+^𝑇_𝑡=1𝐹(𝑡,𝑢)≤1𝑝+(𝑇−1)^2𝑝−1𝑝𝑎₄^𝑞−1𝑇^𝑝−1+2^𝑝−1𝑀₂(𝑇−1)^{𝑝(1+𝑞)/𝑞}𝑝𝑇^𝑝𝑢_𝑛^𝑝+^𝑇_t=1𝐹𝑡,𝑢_𝑛+𝑀₃(𝑇−1)^{(1+𝑞)/𝑞}𝑇𝑢_𝑛+𝑎^(𝑞−1)2₄𝑀^𝑞₁2^𝑝𝑞𝑢_𝑛^𝑝≤1𝑝+(𝑇−1)^2𝑝−1𝑝𝑎₄^𝑞−1𝑇^𝑝−1+2^𝑝−1𝑀₂(𝑇−1)^{𝑝(1+𝑞)/𝑞}𝑝𝑇^𝑝×𝑝𝑇^𝑝𝑎^𝑞₄𝑀^𝑞₁2^𝑝𝑝𝑇^𝑝−2^𝑝−1𝑀₂(𝑇−1)^{𝑝(1+𝑞)/𝑞}𝑢_𝑛^𝑝−𝑝𝑇^𝑝𝑞𝐶₅𝑝𝑇^𝑝−2^𝑝−1M₂(𝑇−1)^{𝑝(1+𝑞)/𝑞}+^𝑇_𝑡=1𝐹(𝑡,𝑢)+𝑎^(𝑞−1)2₄𝑀^𝑞₁2^𝑝𝑞𝑢_𝑛^𝑝+𝑀₃(𝑇−1)^{(1+𝑞)/𝑞}𝑇2𝑝^1/𝑝𝑇𝑎₄^𝑞/𝑝𝑀₁^𝑞/𝑝𝑝𝑇^𝑝−2^𝑝−1𝑀₂(𝑇−1)^{𝑝(1+𝑞)/𝑞1/𝑝}𝑢_𝑛+𝐶₆=𝑢_𝑛^𝑝2^𝑝𝑎^𝑞₄𝑇^𝑝+2^𝑝−1𝑀₂(𝑇−1)^{𝑝(1+𝑞)/𝑞}+2^𝑝𝑇𝑎₄(𝑇−1)^2𝑝−1𝑝𝑇^𝑝−2^𝑝−1𝑀₂(𝑇−1)^{𝑝(1+𝑞)/𝑞}+2^𝑝𝑎^(𝑞−1)2₄𝑞𝑀^𝑞₁+𝑢_𝑛^𝑇−𝑝_𝑡=1𝐹𝑡,𝑢_𝑛+2𝑝^1/𝑝𝑇𝑎₄^𝑞/𝑝𝑀₁^𝑞/𝑝𝑀₃(𝑇−1)^{(1+𝑞)/𝑞}𝑇𝑝𝑇^𝑝−2^𝑝−1𝑀₂(𝑇−1)^{𝑝(1+𝑞)/𝑞1/𝑝}𝑢_𝑛^−𝑝+1+𝐶₈, (3.29) where 𝐶₇ is a positive constant and 𝐶₈ is a constant. The above inequality and (3.22) imply that {𝑢_𝑛} is bounded. Hence, {𝑢_𝑛} is bounded by (2.13) and (3.26).
Similar to the proof of Theorem 2.3, we only need to verify (I1) and (I2). It is easy to verify (I1) by (F6). Now, we verify that (I2) holds. For 𝑢∈𝐸_𝑇, by (F2)' and (2.12), we have ^𝑇_𝑡=1(𝐹(𝑡,𝑢(𝑡))−𝐹(𝑡,0))=^𝑇_𝑡=1¹₀(((𝑡(𝑡≤∇𝐹𝑡,𝑠𝑢)),𝑢))𝑑𝑠_𝑇𝑡=110𝑓(𝑡)𝑠^𝑝−1|𝑢(𝑡)|^𝑝𝑑𝑠+^𝑇_𝑡=1𝑔(𝑡𝑢(𝑡≤𝑀)|)|₂𝑝‖𝑢‖^𝑝_∞+𝑀₃‖𝑢‖_∞≤𝑀₂(𝑇−1)^{𝑝(1+𝑞)/𝑞}𝑝𝑇^𝑝‖𝑢‖^𝑝+𝑀₃(𝑇−1)^{(1+𝑞)/𝑞}‖‖.𝑝𝑇𝑢 (3.30) Thus, we have 𝜑(𝑢)=1𝑝^𝑇_𝑡=1|Δ𝑢(𝑡)|^𝑝+^𝑇_𝑡=1(𝐹(𝑡,𝑢(𝑡))−𝐹(𝑡,0))+^𝑇_𝑡=1𝐹()≥1𝑡,0𝑝−𝑀₂(𝑇−1)^{𝑝(1+𝑞)/𝑞}𝑝𝑇^𝑝‖𝑢‖^𝑝−𝑀₃(𝑇−1)^{(1+𝑞)/𝑞}‖‖+𝑝𝑇𝑢^𝑇_𝑡=1𝐹(𝑡,0), (3.31) for all 𝑢∈𝐸_𝑇. By Lemma 2.6, ‖𝑢‖→∞ in 𝐸_𝑇 if and only ‖𝑢‖→∞. So from the above inequality, we have 𝜑(𝑢)→+∞ as ‖𝑢‖→∞, that is (I2) is verified. Hence, the proof of Theorem 2.4 is complete.

4. Example

In this section, we give four examples to illustrate our results.

Example 4.1. Let 𝑝=5/2 and 𝐹(𝑡,𝑥)=sin2𝜋𝑡𝑇|𝑥|^5/3+sin2𝜋𝑡𝑇+1|𝑥|^4/3+(ℎ(𝑡),𝑥), (4.1) where ℎ∈𝑙¹(ℤ[1,𝑇],ℝ^𝑁) and ℎ(𝑡+𝑇)=ℎ(𝑡). It is easy to see that 𝐹(𝑡,𝑥) satisfies (F1) and |(≤5∇𝐹𝑡,𝑥)|3sin2𝜋𝑡𝑇|𝑥|^2/3+43sin2𝜋𝑡𝑇+1|𝑥|^1/3+|ℎ(𝑡≤5)|3sin2𝜋𝑡𝑇+𝜀|𝑥|^2/3+𝑎(𝜀)+|ℎ(𝑡)|,∀(𝑡,𝑥)∈ℤ[1,𝑇]×ℝ^𝑁, (4.2) where 𝜀>0, and 𝑎(𝜀) is a positive constant and is dependent on 𝜀. The above shows that (F2) holds with 𝛼=2/3 and 𝑓(𝑡)=53sin2𝜋𝑡𝑇+𝜀,𝑔(𝑡)=𝑎(𝜀)+|ℎ(𝑡)|. (4.3) Moreover, we have liminf_{|𝑥|→+∞}|𝑥|^𝑇−2𝛼_𝑡=1𝐹()2𝑡,𝑥=𝑇,_𝑞𝛼(𝑇−1)^{𝑞(2𝑝−1)/𝑝}𝑞𝑇^𝑇_𝑡=1𝑓^𝑞(𝑡)=3×2^10/9(𝑇−1)^8/3553𝜀^5/3. (4.4) We can choose 𝜀 suitable such that liminf_{|𝑥|→+∞}|𝑥|^𝑇−2𝛼_𝑡=1𝐹(𝑡,𝑥)=𝑇>3×2^10/9(𝑇−1)^8/3553𝜀^5/3=2^𝑞𝛼(𝑇−1)^{𝑞(2𝑝−1)/𝑝}𝑞𝑇^𝑇_𝑡=1𝑓^𝑞(𝑡), (4.5) which shows that (F3) holds. Then from Theorem 2.1, problem (1.1) has at least one periodic solution with period 𝑇.

Example 4.2. Let 𝑝=2, then 𝑞=2. Let 𝐹()=1𝑡,𝑥612+sin2𝜋𝑡𝑇|𝑥|²+|𝑥|^3/2+(ℎ(𝑡),𝑥), (4.6) where ℎ∈𝑙¹(ℤ[1,𝑇],ℝ^𝑁) and ℎ(𝑡+𝑇)=ℎ(𝑡). It is easy to see that 𝐹(𝑡,𝑥) satisfies (F1) and |(≤1∇𝐹𝑡,𝑥)|312+sin2𝜋𝑡𝑇𝑥|+3|2|𝑥|^1/2+|ℎ(𝑡≤1)|312+sin2𝜋𝑡𝑇+𝜀|𝑥|+𝑏(𝜀)+|ℎ(𝑡)|,∀(𝑡,𝑥)∈ℤ[1,𝑇]×ℝ^𝑁, (4.7) where 𝜀>0, and 𝑏(𝜀) is a positive constant and is dependent on 𝜀. The above shows that (F2)' holds with 𝑓(𝑡)=1312+sin2𝜋𝑡𝑇+𝜀,𝑔(𝑡)=𝑏(𝜀)+|ℎ(𝑡)|. (4.8) Observe that |𝑥|^𝑇−𝑝_𝑡=1𝐹(𝑡,𝑥)=|𝑥|^𝑇−2_𝑡=11612+sin2𝜋𝑡𝑇|𝑥|²+|𝑥|^3/2+(ℎ(𝑡)=𝑇,𝑥)|𝑥|12+𝑇^−1/2+^𝑇_𝑡=1ℎ(𝑡),|𝑥|⁻²𝑥. (4.9) On the other hand, if we let 𝑇=2, then we have ^𝑇_𝑡=1𝑓(𝑡)=2312+𝜀,^𝑇_𝑡=1𝑓²(𝑡)=19^𝑇_𝑡=112+sin2𝜋𝑡𝑇+𝜀²=2912+𝜀²,2^𝑝𝑇^𝑞/𝑝(𝑇−1)^{𝑞(2𝑝−1)/𝑝}𝑇^𝑝−2^𝑝−1(𝑇−1)^{𝑝(1+𝑞)/𝑞}∑^𝑇_𝑡=1𝑓(𝑡)^{𝑇𝑞/𝑝}_𝑡=1𝑓^𝑞(𝑡)=2+8𝜀+8𝜀².15−6𝜀 (4.10) We can choose 𝜀 sufficiently small such that ^𝑇_𝑡=1𝑓(𝑡)=2312𝑇+𝜀<2=^𝑝2^𝑝−1(𝑇−1)^𝑝−1,liminf_{|𝑥|→+∞}|𝑥|^𝑇−𝑝_𝑡=1𝐹()=1𝑡,𝑥6>2+8𝜀+8𝜀²=215−6𝜀_𝑝𝑇^𝑞/𝑝(𝑇−1)^{𝑞(2𝑝−1)/𝑝}𝑇^𝑝−2^𝑝−1(𝑇−1)^{𝑝(1+𝑞)/𝑞}∑^𝑇_𝑡=1𝑓(𝑡)^{𝑇𝑞/𝑝}_𝑡=1𝑓^𝑞(𝑡), (4.11) which shows that (2.3) and (F4) hold. Then from Theorem 2.2, problem (1.1) has at least one periodic solution with period 𝑇.

Example 4.3. Let 𝑝=2, then 𝑞=2. Let 𝐹(𝑡,𝑥)=sin2𝜋𝑡𝑇|𝑥|^7/4+sin2𝜋𝑡𝑇−1|𝑥|^3/2+(ℎ(𝑡),𝑥), (4.12) where ℎ∈𝑙¹(ℤ[1,𝑇],ℝ^𝑁) and ℎ(𝑡+𝑇)=ℎ(𝑡). It is easy to see that 𝐹(𝑡,𝑥) satisfies (F1) and |(≤7∇𝐹𝑡,𝑥)|4sin2𝜋𝑡𝑇|𝑥|^3/4+32sin2𝜋𝑡𝑇−1|𝑥|^1/2+|ℎ(𝑡≤7)|4sin2𝜋𝑡𝑇+𝜀|𝑥|^3/4+𝑐(𝜀)+|ℎ(𝑡)|,∀(𝑡,𝑥)∈ℤ[1,𝑇]×ℝ^𝑁, (4.13) where 𝜀>0 and 𝑐(𝜀) is a positive constant and is dependent on 𝜀. The above shows that (F2) holds with 𝛼=3/4 and 𝑓(𝑡)=74sin2𝜋𝑡𝑇+𝜀,𝑔(𝑡)=𝑐(𝜀)+|ℎ(𝑡)|. (4.14) Observe that |𝑥|^{𝑇−𝑞𝛼}_𝑡=1𝐹(𝑡,𝑥)=|𝑥|^𝑇−3/2_𝑡=1sin2𝜋𝑡𝑇|𝑥|^7/4+sin2𝜋𝑡𝑇−1|𝑥|^3/2+(ℎ(𝑡),𝑥)=−𝑇+^𝑇_𝑡=1ℎ(𝑡),|𝑥|^−3/2𝑥. (4.15) On the other hand, we have 2^𝑞𝛼(𝑇−1)^{q(2𝑝−1)/𝑝}+2𝑝𝑇^𝑞𝛼(𝑇−1)^{(𝑞−1)2(2𝑝−1)/𝑝}𝑞𝑇^{(𝑞−1)2/𝑞}+2^𝑞𝛼(𝑇−1)^{2𝑝−1+(2𝑝−1)/𝑝}𝑝𝑇^{(𝑝+1)/𝑞𝑇}_𝑡=1𝑓^𝑞(𝑡)=√2(𝑇−1)³𝑇+√2(𝑇−1)^3/2𝑇^1/2+√2(𝑇−1)^9/2𝑇^3/2𝑇_𝑡=14916sin2𝜋𝑡𝑇+𝜀²=49√2𝜀²(𝑇−1)^3/2𝑇^1/2(𝑇−1)^3/2+𝑇+(𝑇−1)³16𝑇^1/2. (4.16) We can choose 𝜀 suitable such that limsup_{|𝑥|→+∞}|𝑥|^{𝑇−𝑞𝛼}_𝑡=1𝐹()√𝑡,𝑥=−𝑇<−492𝜀²(𝑇−1)^3/2𝑇^1/2(𝑇−1)^3/2+𝑇+(𝑇−1)³16𝑇^1/22=−^𝑞𝛼(𝑇−1)^{𝑞(2𝑝−1)/𝑝}+2𝑝𝑇^𝑞𝛼(𝑇−1)^{(𝑞−1)2(2𝑝−1)/𝑝}𝑞𝑇^{(𝑞−1)2/𝑞}+2^𝑞𝛼(𝑇−1)^{2𝑝−1+(2𝑝−1)/𝑝}𝑝𝑇^{(𝑝+1)/𝑞𝑇}_𝑡=1𝑓^𝑞(𝑡), (4.17) which shows that (F5) holds. Then from Theorem 2.3, problem (1.1) has at least one periodic solution with period 𝑇.

Example 4.4. Let 𝑝=2, then 𝑞=2. Let 𝐹()=1𝑡,𝑥3sin2𝜋𝑡𝑇−18|𝑥|²+|𝑥|^3/2+(ℎ(𝑡),𝑥), (4.18) where ℎ∈𝑙¹(ℤ[1,𝑇],ℝ^𝑁) and ℎ(𝑡+𝑇)=ℎ(𝑡). It is easy to see that 𝐹(𝑡,𝑥) satisfies (F1) and |(≤2∇𝐹𝑡,𝑥)|3sin2𝜋𝑡𝑇−18𝑥|+3|2|𝑥|^1/2+|ℎ(𝑡≤2)|3sin2𝜋𝑡𝑇−18+𝜀|𝑥|+𝑑(𝜀)+|ℎ(𝑡)|,∀(𝑡,𝑥)∈ℤ[1,𝑇]×ℝ^𝑁, (4.19) where 𝜀>0, 𝑑(𝜀) is a positive constant and is dependent on 𝜀. The above shows that (F2)' holds with 𝑓(𝑡)=23sin2𝜋𝑡𝑇−18+𝜀,𝑔(𝑡)=𝑑(𝜀)+|ℎ(𝑡)|. (4.20) Observe that |𝑥|^𝑇−𝑝_𝑡=1𝐹(𝑡,𝑥)=|𝑥|^𝑇−2_𝑡=113sin2𝜋𝑡𝑇−18|𝑥|²+|𝑥|^3/2+(ℎ(𝑡)𝑇,𝑥)=−|𝑥|24+𝑇^−1/2+^𝑇_𝑡=1ℎ(𝑡),|𝑥|⁻²𝑥. (4.21) On the other hand, if we let 𝑇=2, then we have ^𝑇_𝑡=1𝑓(𝑡)=4318+𝜀,^𝑇_𝑡=1𝑓²(𝑡)=49^𝑇_𝑡=118+sin2𝜋𝑡𝑇+𝜀²=8918+𝜀²,−2^𝑝𝑝𝑇^𝑞/𝑝(𝑇−1)^{𝑞(2𝑝−1)/𝑝}𝑇^𝑝+2^𝑝−1(𝑇−1)^{𝑝(1+𝑞)/𝑞}∑^𝑇_𝑡=1𝑓(𝑡)𝑝𝑇^𝑝−2^𝑝−1(𝑇−1)^{𝑝(1+𝑞)/𝑞}∑^𝑇_𝑡=1𝑓(𝑡)^𝑞+2^𝑝𝑝𝑇^1/𝑝𝑇(𝑇−1)^{2𝑝−1+(2𝑝−1)/𝑝}𝑝𝑇^𝑝−2^𝑝−1(𝑇−1)^{𝑝(1+𝑞)/𝑞}∑^𝑇_𝑡=1𝑓(𝑡)^1+1/𝑝+2^𝑝𝑝𝑇^{(𝑞−1)2/𝑝}(𝑇−1)^{(𝑞−1)2(2𝑝−1)/𝑝}𝑞𝑝𝑇^𝑝−2^𝑝−1(𝑇−1)^{𝑝(1+𝑞)/𝑞}∑^𝑇_𝑡=1𝑓(𝑡)^{(𝑞−1)2/𝑝𝑇𝑡=1}𝑓^𝑞(𝑡)=192+128×(1/8+𝜀)3×(8−(8/3)(1/8+𝜀))²+16(8−(8/3)(1/8+𝜀))^3/2+8(8−(8/3)(1/8+𝜀))^1/2×8918+𝜀². (4.22)(4.23) We can choose 𝜀 sufficiently small such that ^𝑇_𝑡=1𝑓(𝑡)=4318𝑇+𝜀<2=^𝑝2^𝑝−1(𝑇−1)^𝑝−1,limsup_{|𝑥|→+∞}|𝑥|^𝑇−𝑝_𝑡=1𝐹()1𝑡,𝑥=−<()12192+128×1/8+𝜀3×(8−(8/3)(1/8+𝜀))²+16(8−(8/3)(1/8+𝜀))^3/2+8(8−(8/3)(1/8+𝜀))^1/2×8918+𝜀²2=−^𝑝𝑝𝑇^𝑞/𝑝(𝑇−1)^{𝑞(2𝑝−1)/𝑝}𝑇^𝑝+2^𝑝−1(𝑇−1)^{𝑝(1+𝑞)/𝑞}∑^𝑇_𝑡=1𝑓(𝑡)𝑝𝑇^𝑝−2^𝑝−1(𝑇−1)^{𝑝(1+𝑞)/𝑞}∑^𝑇_𝑡=1𝑓(𝑡)^𝑞+2^𝑝𝑝𝑇^1/𝑝𝑇(𝑇−1)^{2𝑝−1+(2𝑝−1)/𝑝}𝑝𝑇^𝑝−2^𝑝−1(𝑇−1)^{𝑝(1+𝑞)/𝑞}∑^𝑇_𝑡=1𝑓(𝑡)^1+1/𝑝+2^𝑝𝑝𝑇^{(𝑞−1)2/𝑝}(𝑇−1)^{(𝑞−1)2(2𝑝−1)/𝑝}𝑞𝑝𝑇^𝑝−2^𝑝−1(𝑇−1)^{𝑝(1+𝑞)/𝑞}∑^𝑇_𝑡=1𝑓(𝑡)^{(𝑞−1)2/𝑝𝑇𝑡=1}𝑓^𝑞(𝑡), (4.24)(4.25) which shows that (F6) holds. Then from Theorem 2.4, problem (1.1) has at least one periodic solution with period 𝑇.

Acknowledgment

This work is partially supported by the Scientific Research Foundation of the Guangxi Education Office of China (201203YB093 and 200911MS270).