Infinitely Many Homoclinic Orbits for 2𝑛th-Order Nonlinear Functional Difference Equations Involving the 𝑝-Laplacian

Abstract

By establishing a new proper variational framework and using the critical point theory, we establish some new existence criteria to guarantee that the 2𝑛th-order nonlinear difference equation containing both advance and retardation with 𝑝-Laplacian Δ^𝑛(𝑟(𝑡−𝑛)𝜑_𝑝(Δ^𝑛𝑢(𝑡−1)))+𝑞(𝑡)𝜑_𝑝(𝑢(𝑡))=𝑓(𝑡,𝑢(𝑡+𝑛),…,𝑢(𝑡),…,𝑢(𝑡−𝑛)), 𝑛∈ℤ(3), 𝑡∈ℤ, has infinitely many homoclinic orbits, where 𝜑_𝑝(𝑠) is 𝑝-Laplacian operator;𝜑_𝑝(𝑠)=|𝑠|^𝑝−2𝑠(1<𝑝<∞)𝑟, 𝑞, 𝑓 are nonperiodic in 𝑡. Our conditions on the potential are rather relaxed, and some existing results in the literature are improved.

1. Introduction

In this paper, we shall be concerned with the existence of homoclinic orbits of the nonlinear difference equationΔ^𝑛𝑟(𝑡−𝑛)𝜑_𝑝Δ^𝑛𝑢(𝑡−1)+𝑞(𝑡)𝜑_𝑝(𝑢(𝑡(()(𝑡)((3)))=𝑓𝑡,𝑢𝑡+𝑛,…,𝑢,…,𝑢𝑡−𝑛)),𝑛∈ℤ,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)). 𝜑_𝑝(𝑠) is 𝑝-Laplacian operator 𝜑_𝑝(𝑠)=|𝑠|^𝑝−2𝑠(1<𝑝<∞). As usual, we say that a solution 𝑢(𝑡) of (1.1) is homoclinic (to 0) if 𝑢(𝑡)→0 as 𝑡→±∞. In addition, if 𝑢(𝑡)≢0, then 𝑢(𝑡) is called a nontrivial homoclinic solution. It is well known that homoclinic orbits play an important role in analyzing the chaos of dynamical systems; we refer the interested reader to [1–15].

We may think of (1.1) as being a discrete analogue of the 2𝑛th-order differential equation𝑟(𝑡)𝜑_𝑝𝑥^(𝑛)(𝑛)+𝑞(𝑡)𝜑_𝑝(𝑥)−𝑓(𝑡,𝑥(𝑡+𝑛),…,𝑥(𝑡),…,𝑥(𝑡−𝑛))=0,𝑡∈ℝ. (1.2)

If 𝑝=2, (1.1) reduces to the following equation:Δ^𝑛𝑟(𝑡−𝑛)Δ^𝑛𝑢(𝑡−𝑛)+𝑞(𝑡)𝑢(𝑡)=𝑓(𝑡,𝑢(𝑡+𝑛),…,𝑢(𝑡),…,𝑢(𝑡−𝑛)),𝑛∈ℤ(3),𝑡∈ℤ. (1.3) Recently, Chen and Tang [2] applied the critical point theory to prove the existence of homoclinic solutions for (1.3).

In some recent papers [1, 2, 8, 9, 16–18], the authors studied the existence of periodic and homoclinic solutions of second-order nonlinear difference equation by using the critical point theory. These papers show that the critical point method is an effective approach to the study of periodic solutions of second-order difference equations. Compared to one-order or second-order difference equations, the study of higher-order equations has received considerably less attention (see, e.g., [2, 3, 19–21] and references contained therein). But to the best knowledge of the authors, results on existence of homoclinic solutions of (1.1) have not been found in the literature.

Motivated by the recent papers [2, 5, 14, 22], the aim of this paper is to consider problem (1.1) in a more general sense. More exactly our results represent the extensions to equations with 𝑝-Laplacian. Throughout the paper, for a function 𝐺, we let 𝐺^′_𝑖(𝑥₁,𝑥₂,…,𝑥_𝑖,…,𝑥_𝑛) denote the partial derivative of 𝐺 on the 𝑖 variable.

2. Main Results

Theorem 2.1. Assume that 𝑞 and 𝐹 satisfy the following assumptions: (r)For every 𝑡∈ℤ, 𝑟(𝑡)>0;(q)For every 𝑡∈ℤ, 𝑞(𝑡)>0, and lim_{|𝑡|→+∞}𝑞(𝑡)=+∞; (F1)There exists a functional 𝐹(𝑡,𝑥_𝑛(𝑡),…,𝑥₀(𝑡)) which is continuously differentiable in the variable from 𝑥_𝑛 to 𝑥₀ for every 𝑡∈ℤ and satisfy ⁰_𝑖=−𝑛𝐹^′_2+𝑛+𝑖𝑡+𝑖,𝑥_𝑛+𝑖,…,𝑥_𝑖=𝑓𝑡,𝑥_𝑛,𝑥_𝑛−1,…,𝑥₀,𝑥₋₁,…,𝑥_−𝑛,𝑓𝑡,𝑥_𝑛,𝑥_𝑛−1,…,𝑥₀,𝑥₋₁,…,𝑥_−𝑛=𝑜∑^𝑛_𝑖=−𝑛𝑥_𝑖^𝑝−1,as∑^𝑛_𝑖=−𝑛𝑥_𝑖^𝑝−1𝐹⟶0,𝑡,𝑥_𝑛,…,𝑥₀=𝑜∑^𝑛_𝑖=−𝑛𝑥_𝑖^𝑝,as∑^𝑛_𝑖=−𝑛𝑥_𝑖^𝑝⟶0 (2.1) uniformly in 𝑡∈ℤ⧵𝐽, where 𝐽 is defined in (F2);(F2)𝐹(𝑡,𝑥_𝑛,…,𝑥₀)=𝑊(𝑡,𝑥₀)−𝐻(𝑡,𝑥_𝑛,…,𝑥₀), for every 𝑡∈ℤ, 𝑊,𝐻 are continuously differentiable in 𝑥₀ and 𝑥_𝑛,…,𝑥₀, respectively. Moreover, there is a bounded set 𝐽⊂ℤ such that𝐻𝑡,𝑥_𝑛,…,𝑥₀≥0; (2.2)(F3)There is a constant 𝜇>𝑝 such that0<𝜇𝑊𝑡,𝑥₀≤𝑊^′₂𝑡,𝑥₀𝑥₀,∀𝑡,𝑥₀∈ℤ×(ℝ⧵{0}); (2.3)(F4)𝐻(𝑡,0,…,0)≡0, and there is a constant 𝜚∈(𝑝,𝜇) such that⁰_𝑖=−𝑛𝐻^′_2+𝑛+𝑖𝑡,𝑥_𝑛,…,𝑥₀𝑥_−𝑖≤𝜚𝐻𝑡,𝑥_𝑛,…,𝑥₀; (2.4)(F5)There exists a constant 𝑏>0 such that𝐻𝑡,𝑥_𝑛,…,𝑥₀≤𝑏𝛾^𝜚,for𝑡∈ℤ,𝛾>1, (2.5) where 𝛾=(∑^𝑛_𝑖=0|𝑥_𝑖|^𝑝)^1/𝑝.(F6)𝐹(𝑡,−𝑥_𝑛,…,−𝑥₀)=𝐹(𝑡,𝑥_𝑛,…,𝑥₀), for all (𝑡,𝑥_𝑛,…,𝑥₀)∈ℤ×ℝ^𝑛+1.Then (1.1) possesses an unbounded sequence of homoclinic solutions.

Theorem 2.2. Assume that 𝑟, 𝑞, and 𝐹 satisfy (r), (q), (F1), (F3)–(F5) and the following assumption: (F2’)𝐹(𝑡,𝑥_𝑛,…,𝑥₀)=𝑊(𝑡,𝑥₀)−𝐻(𝑡,𝑥_𝑛,…,𝑥₀), for every 𝑡∈ℤ, 𝑊,𝐻 are continuously differentiable in 𝑥₀ and 𝑥_𝑛,…,𝑥₀, respectively, and𝐹𝑡,𝑥_𝑛,…,𝑥₀=𝑜𝛾^𝑝as𝛾⟶0, (2.6) where 𝛾=(∑^𝑛_𝑖=0|𝑥_𝑖|^𝑝)^1/𝑝 uniformly in 𝑡∈ℤ,
Then (1.1) possesses an unbounded sequence of homoclinic solutions.

Theorem 2.3. Assume that 𝑟, 𝑞, and 𝐹 satisfy (r), (q), (F1) and satisfy the following assumptions: (F7)For any 𝑡∈ℤ,𝐹𝑡,𝑥_𝑛,…,𝑥₀≥𝐹𝑡,𝑥₀≥0; (2.7)(F8)For any 𝑟>0, there exist 𝑎=𝑎(𝑟), 𝑏=𝑏(𝑟)>0, and 𝜈<𝑝 such that1𝑝+𝑎+𝑏∑^𝑛_𝑖=0𝑥_𝑖^{𝑝𝜈/𝑝}𝐹𝑡,𝑥_𝑛,…,𝑥₀≤⁰_𝑖=−𝑛𝐹^′_2+𝑛+𝑖𝑡,𝑥_𝑛,…,𝑥₀𝑥_−𝑖,∀𝑡∈ℤ,^𝑛_𝑖=0𝑥_𝑖^𝑝1/𝑝≥𝑟; (2.8)(F9)For any 𝑡∈ℤlim_𝑠→+∞𝑠^−𝑝min_|𝑥|=1𝐹(𝑡,𝑠𝑥)=+∞. (2.9) Then there exists an unbounded sequence of homoclinic solutions for (1.1).

3. Preliminaries

To apply critical point theory to study the existence of homoclinic solutions of (1.1), we shall state some basic notations and lemmas, which will be used in the proofs of our main results.

Let𝑆={𝑢(𝑡)}_𝑡∈ℤ∶𝑢(𝑡)∈ℝ,𝑡∈ℤ,𝐸=𝑢∈𝑆∶∑_𝑡∈ℤ𝑟(𝑡−1)Δ^𝑛𝑢(𝑡−1)^𝑝+𝑞(𝑡)|𝑢(𝑡)|^𝑝<+∞ (3.1) and for 𝑢∈𝐸, let‖𝑢‖=_𝑡∈ℤ𝑟(𝑡−1)Δ^𝑛𝑢(𝑡−1)^𝑝+𝑞(𝑡)|𝑢(𝑡)|^𝑝1/𝑝. (3.2) Then 𝐸 is a uniform convex Banach space with this norm.

Let 𝐼∶𝐸→ℝ be defined by𝐼(𝑢)=1𝑝‖𝑢‖^𝑝−_𝑡∈ℤ𝐹(𝑡,𝑢(𝑡+𝑛),…,𝑢(𝑡)). (3.3) If (𝑞) and (F1) hold, then 𝐼∈𝐶¹(𝐸,ℝ) and one can easily check that𝐼^′(𝑢),𝑣=_𝑡∈ℤ𝑟(𝑡−1)Δ^𝑛𝑢(𝑡−1)^𝑝−2Δ^𝑛𝑣(𝑡−1)+𝑞(𝑡)|𝑢(𝑡)|^𝑝−2𝑣(𝑡)−𝑓(𝑡),𝑢(𝑡+𝑛),…u(𝑡),…𝑢(𝑡−𝑛)𝑣(𝑡),∀𝑢,𝑣∈𝐸. (3.4) By usingΔ^𝑛𝑢(𝑡−1)=^𝑛_𝑘=0(−1)^𝑘𝑛𝑘𝑢(𝑡+𝑛−𝑘−1), (3.5) we can compute the partial derivative as𝜕𝐼(𝑢)𝜕𝑢(𝑡)=Δ^𝑛𝑟(𝑡−𝑛)𝜑_𝑝Δ^𝑛𝑢(𝑡−1)+𝑞(𝑡)𝜑_𝑝(𝑢(𝑡))−𝑓(𝑡,𝑢(𝑡+𝑛),…,𝑢(𝑡),…,𝑢(𝑡−𝑛)). (3.6) So, the critical points of 𝐼 in 𝐸 are the solutions of (1.1) with 𝑢(±∞)=0.

Lemma 3.1 (see [12]). Let 𝐸 be a real Banach space and 𝐼∈𝐶¹(𝐸,ℝ) satisfy (PS)-condition with 𝐼 even. Suppose that 𝐼 satisfies the following conditions: (i)𝐼(0)=0;(ii)There exist constants 𝜌,𝛼>0 such that 𝐼|_{𝜕𝐵𝜌(0)}≥𝛼;(iii)For each finite dimensional subspace 𝐸^′⊂𝐸, there is 𝑟=𝑟(𝐸^′)>0 such that 𝐼(𝑢)≤0 for 𝑢∈𝐸^′⧵𝐵_𝑟(0), where 𝐵_𝑟(0) is an open ball in 𝐸 of radius 𝑟 centered at 0.Then 𝐼 possesses an unbounded sequence of critical values.

Lemma 3.2. For 𝑢∈𝐸, 𝛽‖𝑢‖^𝑝_∞≤𝛽‖𝑢‖^𝑝_𝑙𝑝≤‖𝑢‖^𝑝, (3.7) where 𝛽=inf_𝑡∈ℤ𝑞(𝑡).

Lemma 3.3. Assume that (F3) holds. Then for every (𝑡,𝑥)∈ℤ×ℝ, 𝑠^−𝜇𝑊(𝑡,𝑠𝑥) is nondecreasing on (0,+∞).

The proof of Lemmas 3.2 and 3.3 is routine and so we omit it.

4. Proofs of Theorems

Proof of Theorem 2.1. It is clear that 𝐼(0)=0. Our proof is devided into three steps.
Step 1 (PS Condition). Assume that {𝑢_𝑘}_𝑘∈ℕ⊂𝐸 is a sequence such that {𝐼(𝑢_𝑘)}_𝑘∈ℕ is bounded and 𝐼^′(𝑢_𝑘)→0 as 𝑘→+∞. Then there exists a constant 𝑐>0 such that 𝐼𝑢_𝑘≤𝑐,𝐼^′𝑢_𝑘𝐸∗≤𝜚𝑐for𝑘∈ℕ. (4.1) From (2.2), (2.3), (2.4), (4.1), (F3), and (F4), we obtain 𝑝𝑐+𝑝𝑐𝑢_𝑘≥𝑝𝐼𝑢_𝑘−𝑝𝜚𝐼^′𝑢_𝑘,𝑢_𝑘=𝜚−𝑝𝜚𝑢_𝑘^𝑝−𝑝_𝑡∈ℤ𝑊𝑡,𝑢_𝑘(𝑡)−1𝜚𝑊^′₂𝑡,𝑢_𝑘(𝑡)𝑢_𝑘(𝑡)+𝑝_𝑡∈ℤ𝐻𝑡,𝑢_𝑘(𝑡+𝑛),…,𝑢_𝑘(𝑡)−𝑝𝜚⁰_{𝑡∈ℤ𝑖=−𝑛}𝐻^′_2+𝑛+𝑖𝑡+𝑖,𝑢_𝑘(𝑡+𝑛+𝑖),…,𝑢_𝑘(𝑡+𝑖)𝑢_𝑘(𝑡)=𝜚−𝑝𝜚𝑢_𝑘^𝑝−𝑝_𝑛∈ℤ𝑊𝑡,𝑢_𝑘(𝑡)−1𝜚𝑊^′₂𝑡,𝑢_𝑘(𝑡)𝑢_𝑘(𝑡)+𝑝_𝑡∈ℤ𝐻𝑡,𝑢_𝑘(𝑡+𝑛),…,𝑢_𝑘(𝑡)−𝑝𝜚⁰_{𝑡∈ℤ𝑖=−𝑛}𝐻^′_2+𝑛+𝑖𝑡,𝑢_𝑘(𝑡+𝑛),…,𝑢_𝑘(𝑡)𝑢_𝑘()≥𝑡−𝑖𝜚−𝑝𝜚𝑢_𝑘^𝑝,𝑘∈ℕ. (4.2) By (4.2), there exists a constant 𝐴>0 such that 𝑢_𝑘≤𝐴for𝑘∈ℕ. (4.3) It can be assumed that 𝑢_𝑘⇀𝑢₀ in 𝐸. For any given number 𝜀>0, by (F1), we can choose 𝜁>0 such that 𝑓(𝑡,𝑢(𝑡+𝑛),…,𝑢(𝑡),…,𝑢(𝑡−𝑛))≤𝜀𝜉^𝑝−1for𝑡∈ℤ⧵𝐽,(𝑢(𝑡+𝑛),…,𝑢(𝑡),…,𝑢(𝑡−𝑛))∈ℝ^𝑛+1, (4.4) where 𝜉=(∑^𝑛_𝑖=−𝑛|𝑢(𝑡+𝑖)|^𝑝)^1/𝑝≤𝜁.
Since 𝑞(𝑡)→∞, we can also choose an integer Π>max{|𝑘|∶𝑘∈𝐽} such that 𝑞(𝑡)≥(2𝑛+1)𝐴^𝑝𝜁^𝑝,|𝑡|≥Π. (4.5) By (4.2) and (4.4), we have 𝑢_𝑘(𝑡)^𝑝=1𝑞(𝑡)𝑞(𝑡)𝑢_𝑘(𝑡)^𝑝≤𝜁^𝑝(2𝑛+1)𝐴^𝑝𝑢_𝑘^𝑝≤𝜁^𝑝2𝑛+1for|𝑡|≥Π,𝑘∈ℕ. (4.6) Since 𝑢_𝑘⇀𝑢₀ in 𝐸, it is easy to verify that 𝑢_𝑘(𝑡) converges to 𝑢₀(𝑡) pointwise for all 𝑡∈ℤ, that is lim_𝑘→∞𝑢_𝑘(𝑡)=𝑢₀(𝑡),∀𝑡∈ℤ. (4.7) Hence, we have by (4.6) and (4.7) 𝑢₀(𝑡)^𝑝≤𝜁^𝑝2𝑛+1for|𝑡|≥Π. (4.8) It follows from (4.7) and the continuity of 𝑓(𝑡,𝑢(𝑡+1),…,𝑢(𝑡),…,𝑢(𝑡−𝑛)) on 𝑢(𝑡+1),…,𝑢(𝑡),…,𝑢(𝑡−𝑛) that there exists 𝑘₀∈ℕ such that ^Π_𝑡=−Π𝑓𝑡,𝑢_𝑘(𝑡+𝑛),…,𝑢_𝑘(𝑡),…,𝑢_𝑘(𝑡−𝑛)−𝑓𝑡,𝑢₀(𝑡+𝑛),…,𝑢₀(𝑡),…,𝑢₀(𝑡−𝑛)<𝜀for𝑘≥𝑘₀. (4.9) On the other hand, it follows from (F1), (2.4), (4.2), (4.4), (4.6), and (4.8) that _|𝑡|>Π𝑓𝑡,𝑢_𝑘(𝑡+𝑛),…,𝑢_𝑘(𝑡),…,𝑢_𝑘(𝑡−𝑛)−𝑓𝑡,𝑢₀(𝑡+𝑛),…,𝑢₀(𝑡),…,𝑢₀()×𝑢𝑡−𝑛_𝑘(𝑡)−𝑢₀(𝑡)≤_|𝑡|>Π𝑓𝑡,𝑢_𝑘(𝑡+𝑛),…,𝑢_𝑘(𝑡),…,𝑢_𝑘()+𝑓𝑡−𝑛𝑡,𝑢₀(𝑡+𝑛),…,𝑢₀(𝑡),…,𝑢₀(𝑡−𝑛)×𝑢_𝑘(𝑡)+𝑢₀(𝑡)≤𝜀_|𝑡|>Π^𝑛_𝑖=−𝑛𝑢_𝑘(𝑡+𝑖)^𝑝−1+^𝑛_𝑖=−𝑛𝑢₀(𝑡+𝑖)^𝑝−1𝑢_𝑘(𝑡)+𝑢₀(𝑡≤())2𝑛+1𝜀_𝑡∈ℤ𝑢_𝑘(𝑡)^𝑝−1+𝑢₀(𝑡)^𝑝−1𝑢_𝑘(𝑡)+𝑢₀(𝑡≤())4𝑛+2𝜀_𝑡∈ℤ𝑢_𝑘(𝑡)^𝑝+𝑢₀(𝑡)^p≤(4𝑛+2)𝜀𝛽𝐴^𝑝+𝑢₀^𝑝. (4.10) Since 𝜀 is arbitrary, combining (4.9) with (4.10), we get _𝑡∈ℤ𝑓𝑡,𝑢_𝑘(𝑡+𝑛),…,𝑢_𝑘(𝑡),…,𝑢_𝑘(𝑡−𝑛)−𝑓𝑡,𝑢₀(𝑡+𝑛),…,𝑢₀(𝑡),…,𝑢₀()𝑢𝑡−𝑛_𝑘(𝑡)−𝑢₀(𝑡)⟶0as𝑘⟶∞. (4.11) Using the Hölder’s inequality 𝑎𝑐+𝑏𝑑≤𝑎^𝑝+𝑏^𝑝1/𝑝𝑐^𝑞+𝑑^𝑞1/𝑞, (4.12) where 𝑎,𝑏,𝑐,𝑑 are nonnegative numbers and 1/𝑝+1/𝑞=1, 𝑝>1, it follows from (3.3) and (3.4) that 𝐼^′𝑢_𝑘−𝐼^′𝑢₀,𝑢_𝑘−𝑢₀=_𝑡∈ℤ𝑟(𝑡−1)Δ^𝑛𝑢_𝑘(𝑡−1)^𝑝−2Δ^𝑛𝑢_𝑘(𝑡−1),Δ^𝑛𝑢_𝑘(𝑡−1)−Δ^𝑛𝑢₀(𝑡−1)+_𝑡∈ℤ𝑞(𝑡)𝑢_𝑘(𝑡)^𝑝−2𝑢_𝑘(𝑡),𝑢_𝑘(𝑡)−𝑢₀(𝑡−_𝑡∈ℤ)𝑟(𝑡−1)Δ^𝑛𝑢₀(𝑡−1)^𝑝−2Δ^𝑛𝑢₀(𝑡−1),Δ^𝑛𝑢_𝑘(𝑡−1)−Δ^𝑛𝑢₀(𝑡−1)−_𝑡∈ℤ𝑞(𝑡)𝑢₀(𝑡)^𝑝−2𝑢₀(𝑛),𝑢_𝑘(𝑡)−𝑢₀(𝑡)−_𝑡∈ℤ𝑓𝑡,𝑢_𝑘(𝑡+𝑛),…,𝑢_𝑘(𝑡),…,𝑢_𝑘(𝑡−𝑛)−𝑓𝑡,𝑢₀(𝑡+𝑛),…,𝑢₀(𝑡),…,𝑢₀(𝑡−𝑛),𝑢_𝑘(𝑛)−𝑢₀(𝑛)=𝑢_𝑘^𝑝+𝑢₀^𝑝−_𝑛∈ℤ𝑟(𝑡−1)Δ^𝑛𝑢_𝑘(𝑡−1)^𝑝−2Δ^𝑛𝑢_𝑘(𝑡−1),Δ^𝑛𝑢₀(𝑡−1)−_𝑡∈ℤ𝑞(𝑡)𝑢_𝑘(𝑡)^𝑝−2𝑢_𝑘(𝑡),𝑢₀(𝑡−_𝑡∈ℤ)𝑟(t−1)Δ^𝑛𝑢₀(𝑡−1)^𝑝−2Δ^𝑛𝑢₀(𝑡−1),Δ^𝑛𝑢_𝑘(𝑡−1)−_𝑡∈ℤ𝑞(𝑡)𝑢₀(𝑡)^𝑝−2𝑢₀(𝑡),𝑢_𝑘(𝑡)−_𝑡∈ℤ𝑓𝑡,𝑢_𝑘(𝑡+𝑛),…,𝑢_𝑘(𝑡),…,𝑢_𝑘(𝑡−𝑛)−𝑓𝑡,𝑢₀(𝑡+𝑛),…,𝑢₀(𝑡),…,𝑢₀(𝑡−𝑛),𝑢_𝑘(𝑛)−𝑢₀(𝑛)≥𝑢_𝑘^𝑝+𝑢₀^𝑝−_𝑡∈ℤ𝑟(𝑡−1)Δ^𝑛𝑢₀(𝑡−1)^𝑝1/𝑝_𝑡∈ℤ𝑟(𝑡−1)Δ^𝑛𝑢_𝑘(𝑡−1)^𝑝1/𝑞−_𝑡∈ℤ𝑞(𝑡)𝑢₀(𝑡)^𝑝1/𝑝_𝑡∈ℤ𝑞(𝑡)𝑢_𝑘(𝑡)^𝑝1/𝑞−_𝑡∈ℤ𝑟(𝑡−1)Δ^𝑛𝑢_𝑘(𝑡−1)^𝑝1/𝑝_𝑡∈ℤ𝑟(𝑡−1)Δ^𝑛𝑢₀(𝑡−1)^𝑝1/𝑞−_𝑡∈ℤ𝑞(𝑡)𝑢_𝑘(𝑡)^𝑝1/𝑝_𝑡∈ℤ𝑞(𝑡)𝑢₀(𝑡)^𝑝1/𝑞−_𝑡∈ℤ𝑓𝑡,𝑢_𝑘(𝑡+𝑛),…,𝑢_𝑘(𝑡),…,𝑢_𝑘(𝑡−𝑛)−𝑓𝑡,𝑢₀(𝑡+𝑛),…,𝑢₀(𝑡),…,𝑢₀(𝑡−𝑛),𝑢_𝑘(𝑡)−𝑢₀(𝑡)≥𝑢_𝑘^𝑝+𝑢₀^𝑝−_𝑡∈ℤ𝑟(𝑡−1)Δ^𝑛𝑢₀(𝑡−1)^𝑝+𝑞(𝑡)𝑢₀(𝑡)^𝑝1/𝑝⋅_𝑡∈ℤ𝑟(𝑡−1)Δ^𝑛𝑢_𝑘(𝑡−1)^𝑝+𝑞(𝑡)𝑢_𝑘(𝑡)^𝑝1/𝑞−_𝑡∈ℤ𝑟(𝑡−1)Δ^𝑛𝑢_𝑘(𝑛−1)^𝑝+𝑞(𝑡)𝑢_𝑘(𝑡)^𝑝1/𝑝⋅_𝑡∈ℤ𝑟(𝑡−1)Δ^𝑛𝑢₀(𝑛−1)^𝑝+𝑞(𝑡)𝑢₀(𝑡)^𝑝1/𝑞−_𝑡∈ℤ𝑓𝑡,𝑢_𝑘(𝑡+𝑛),…,𝑢_𝑘(𝑡),…,𝑢_𝑘(𝑡−𝑛)−𝑓𝑡,𝑢₀(𝑡+𝑛),…,𝑢₀(𝑡),…,𝑢₀(𝑡−𝑛),𝑢_𝑘(𝑡)−𝑢₀(𝑡)=𝑢_𝑘^𝑝+𝑢₀^𝑝−𝑢₀𝑢_𝑘^𝑝−1−𝑢_𝑘𝑢₀^𝑝−1−_𝑡∈ℤ𝑓𝑡,𝑢_𝑘(𝑡+𝑛),…,𝑢_𝑘(𝑡),…,𝑢_𝑘(𝑡−𝑛)−𝑓𝑡,𝑢₀(𝑡+𝑛),…,𝑢₀(𝑡),…,𝑢₀(𝑡−𝑛),𝑢_𝑘(𝑡)−𝑢₀(𝑡)=𝑢_𝑘^𝑝−1−𝑢₀^p−1𝑢_𝑘−𝑢₀−_𝑡∈ℤ𝑓𝑡,𝑢_𝑘(𝑡+𝑛),…,𝑢_𝑘(𝑡),…,𝑢_𝑘(𝑡−𝑛)−𝑓𝑡,𝑢₀(𝑡+𝑛),…,𝑢₀(𝑡),…,𝑢₀(𝑡−𝑛),𝑢_𝑘(𝑡)−𝑢₀(𝑡). (4.13) Since 𝐼^′(𝑢_𝑘)→0 as 𝑘→+∞ and 𝑢_𝑘⇀𝑢₀ in 𝐸, it follows from (4.11) and (4.13) that 𝐼^′𝑢_𝑘−𝐼^′𝑢₀,𝑢_𝑘−𝑢₀⟶0as𝑘→∞, (4.14) which yields that ‖𝑢_𝑘‖→‖𝑢‖ as 𝑘→+∞. By the uniform convexity of 𝐸 and the fact that 𝑢_𝑘⇀𝑢₀ in 𝐸, it follows from the Kadec-Klee property that 𝑢_𝑘→𝑢₀ in 𝐸. Hence, 𝐼 satisfies (PS)-condition.
Step 2 (Condition (ii) of Lemma 3.1). By (F1), there exists 𝜂∈(0,1) such that |𝐹(()(𝑡≤2𝑡,𝑢𝑡+𝑛,…,𝑢))|^−𝑝𝛽4𝑝(𝑛+1)^𝑛_𝑖=0|𝑢(𝑡+𝑖)|^𝑝for𝑡∈ℤ⧵𝐽,^𝑛_𝑖=0|𝑢(𝑡+𝑖)|^𝑝1/𝑝≤𝜂. (4.15) Set 𝑀=sup{𝑊(𝑡,𝑢)∣𝑡∈𝐽,𝑢∈ℝ,|𝑢|=1}, (4.16) and 𝛿=min{(𝛽/(4𝑝𝑀+1))^{(𝜇−𝑝)},𝜂}. If ‖𝑢‖=𝛽^1/𝑝𝛿∶=𝜌, then by Lemma 3.2, |𝑢(𝑡)|≤𝛿≤𝜂<1 for 𝑡∈ℤ. By (4.16) and Lemma 3.2, we have _𝑡∈𝐽𝑊(𝑡,𝑢(𝑡))≤_{(𝑡)𝑡∈𝐽,𝑢≠0}𝑊𝑢(𝑡)𝑡,|𝑢(𝑡|𝑢(𝑡)|)|^𝜇≤𝑀_𝑡∈𝐽|𝑢(𝑡)|^𝜇≤𝑀𝛿^𝜇−𝑝_𝑡∈𝐽|𝑢(𝑡)|^𝑝≤𝑀𝛿^𝜇−𝑝𝛽_𝑡∈𝐽𝑞(𝑡)|𝑢(𝑡)|^𝑝≤14𝑝_𝑡∈𝐽𝑞(𝑡)|𝑢(𝑡)|^𝑝. (4.17) Set 𝛼=𝛽𝛿^𝑝/2𝑝. Hence, from (3.3), (4.15), (4.17), (q), (F1), and (F2), we have 𝐼(𝑢)=1𝑝‖𝑢‖^𝑝−_𝑡∈ℤ𝐹(()(𝑡=1𝑡,𝑢𝑡+𝑛,…,𝑢))𝑝‖𝑢‖^𝑝−_{𝑡∈ℤ⧵𝐽}𝐹(𝑡,𝑢(𝑡+𝑛),…,𝑢(𝑡))−_𝑡∈𝐽𝐹(()(𝑡≥1𝑡,𝑢𝑡+𝑛,…,𝑢))𝑝‖𝑢‖^𝑝−2^−𝑝𝛽4𝑝(𝑛+1)_{𝑡∈ℤ⧵𝐽}^𝑛_𝑖=0|𝑢(𝑡+𝑖)|^𝑝−_𝑡∈𝐽𝑊(𝑡,𝑢(𝑡))+_𝑡∈𝐽𝐻(𝑡,𝑢(t)(𝑡≥1+𝑛,…,𝑢))𝑝‖𝑢‖^𝑝−𝛽4𝑝_𝑡∈ℤ|𝑢(𝑡)|^𝑝−14𝑝_𝑡∈𝐽𝑞(𝑡)|𝑢(𝑡)|^𝑝≥1𝑝‖𝑢‖^𝑝−14𝑝_𝑡∈ℤ𝑞(𝑡)|𝑢(𝑡)|^𝑝−14𝑝_𝑡∈𝐽𝑞(𝑡)|𝑢(𝑡)|^𝑝≥1𝑝‖𝑢‖^𝑝−1‖𝑢‖4𝑝^𝑝−1‖𝑢‖4𝑝^𝑝=1‖𝑢‖2𝑝^𝑝=𝛼. (4.18) equation (4.18)shows that ‖𝑢‖=𝜌 implies that 𝐼(𝑢)≥𝛼, that is, 𝐼 satisfies assumption (ii) of Lemma 3.1.
Step 3 (Condition (iii) of Lemma 3.1). Let 𝐸^′ be a finite dimensional subspace of 𝐸. Assume that dim𝐸′=𝑚 and 𝑢₁,𝑢₂,…,𝑢_𝑚 is a base of 𝐸^′.
For any 𝑢∈𝐸^′, there exist 𝜆_𝑖∈ℝ, 𝑖=1,2,…,𝑚 such that 𝑢(𝑡)=^𝑚_𝑖=1𝜆_𝑖𝑢_𝑖(𝑡)for𝑡∈ℤ. (4.19) Let ‖𝑢‖_∗=^𝑚_𝑖=1𝜆_𝑖𝑢_𝑖. (4.20) It is easy to verify that ‖⋅‖_∗ defined by (4.20) is a norm of 𝐸^′. Since all the norms of a finite dimensional normed space are equivalent, so there exists a constant 𝑑>0 such that ‖𝑢‖_∗≤𝑑‖𝑢‖_∞for𝑢∈𝐸^′. (4.21) Assume that 𝑢_𝑖=𝑑,𝑖=1,2,…,𝑚. (4.22)
Since 𝑢_𝑖∈𝐸, by Lemma 3.2, we can choose an integer Π₁>0 such that 𝑢_𝑖(𝑡<𝜂)𝑚,𝑖=1,2,…,𝑚,|𝑡|>Π₁, (4.23) where 𝜂 is given in (4.15). Set Θ=^𝑚_𝑖=1𝜆_𝑖𝑢_𝑖(𝑡)∶𝜆_𝑖∈ℝ,𝑖=1,2,…,𝑚;^𝑚_𝑖=1𝜆_𝑖=1=𝑢∈𝐸^′∶‖𝑢‖_∗=𝑑. (4.24) Hence, for 𝑢∈Θ, let 𝑡₀=𝑡₀(𝑢)∈ℤ such that 𝑢𝑡₀=‖𝑢‖_∞. (4.25) Then, by (4.19)–(4.25), we have 𝑑=𝑑^𝑚_𝑖=1𝜆_𝑖=^𝑚_𝑖=1𝜆_𝑖𝑢_𝑖=‖𝑢‖_∗≤𝑑‖𝑢‖_∞=𝑑𝑢𝑡₀≤𝑑^𝑚_𝑖=1𝜆_𝑖𝑢_𝑖𝑡₀,𝑢∈Θ. (4.26) This shows that |𝑢(𝑡₀)|≥1 and there exists 𝑖₀∈{1,2,…,𝑚} such that |𝑢_𝑖0(𝑡₀)|≥1/𝑚, which, together with (4.23), implies that |𝑡₀|≤Π₁.
Set 𝜏=min𝑊(𝑡,𝑥)∶|𝑡|≤Π₁,|𝑥|=1. (4.27) Since 𝑊(𝑡,𝑥)>0 for all 𝑡∈ℤ and 𝑥∈ℝ⧵{0}, and 𝑊(𝑡,𝑥) is continuous in 𝑥, so 𝜏>0. It follows from (4.27) and Lemma 3.3 that ^Π1_𝑛=−Π1𝑊(𝑡,𝑢(𝑡))≥𝑊𝑡₀,𝑢𝑡₀≥𝑊𝑡₀,𝑢𝑡₀𝑢𝑡₀𝑢𝑡₀^𝜇≥min_|𝑥|=1𝑊𝑡₀,𝑥𝑢𝑡₀^𝜇≥𝜏for𝑢∈Θ. (4.28) For any 𝑢∈𝐸, it follows from (F5) that ^Π1_𝑡=−Π1𝐻(()(𝑡=𝑡,𝑢𝑡+𝑛,…,𝑢))_{𝑡∈ℤ−Π1,Π1,∑𝑛𝑖=0|𝑢(𝑡+𝑖)|𝑝1/𝑝>1}𝐻(()(𝑡+𝑡,𝑢𝑡+𝑛,…,𝑢))_{𝑡∈ℤ−Π1,Π1,∑𝑛𝑖=0|𝑢(𝑡+𝑖)|𝑝1/𝑝≤1}𝐻(𝑡,𝑢(𝑡+𝑛),…,𝑢(𝑡))≤𝑏_{𝑡∈ℤ−Π1,Π1,∑𝑛𝑖=0|𝑢(𝑡+𝑖)|𝑝1/𝑝>1}^𝑛_𝑖=0|𝑢(𝑡+𝑖)|^{𝑝𝜚/𝑝}+^Π1_𝑡=−Π1max_{∑𝑛𝑖=0|𝑢(𝑡+𝑖)|𝑝1/𝑝≤1}𝐻(()(𝑡≤()𝑡,𝑢𝑡+𝑛,…,𝑢))𝑛+1^𝜚/𝑝+1𝑏_𝑡∈ℤ|𝑢(𝑡)|^𝜚+^Π1_𝑡=−Π1max_{∑𝑛𝑖=0|𝑢(𝑡+𝑖)|𝑝1/𝑝≤1}𝐻(()(𝑡≤()𝑡,𝑢𝑡+𝑛,…,𝑢))𝑛+1^𝜚/𝑝+1𝛽^−𝜚/𝑝𝑏‖𝑢‖^𝜚+^Π1_𝑡=−Π1max_{∑𝑛𝑖=0|𝑢(𝑡+𝑖)|𝑝1/𝑝≤1}𝐻(𝑡,𝑢(𝑡+𝑛),…,𝑢(𝑡))=𝑀₁‖𝑢‖^𝜚+𝑀₂, (4.29) where 𝑀₁=(𝑛+1)^𝜚/𝑝+1𝛽^−𝜚/𝑝𝑏,𝑀₂=^Π1_𝑡=−Π1max_{∑𝑛𝑖=0|𝑢(𝑡+𝑖)|𝑝1/𝑝≤1}𝐻(𝑡,𝑢(𝑡+𝑛),…,𝑢(𝑡)). (4.30)
From (3.3), (3.7), (4.28), and (4.29), we have for 𝑢∈Θ and 𝜎>1𝐼()=𝜎𝜎𝑢^𝑝𝑝‖𝑢‖^𝑝−_𝑡∈ℤ𝐹(()(𝑡=𝜎𝑡,𝜎𝑢𝑡+𝑛,…,𝜎𝑢))_𝑝𝑝‖𝑢‖^𝑝−_|𝑡|>Π2𝐹(𝑡,𝜎𝑢(𝑡+𝑛),…,𝜎𝑢(𝑡))−_|𝑡|≤Π2𝐹(()(𝑡≤𝜎𝑡,𝜎𝑢𝑡+𝑛,…,𝜎𝑢))_𝑝𝑝‖𝑢‖^𝑝+_|𝑡|>Π2^𝑛_𝑖=0𝛽𝜎^𝑝()|𝑢(4𝑝𝑛+1𝑡+𝑖)|^𝑝−_|𝑡|≤Π2𝐹(()(𝑡≤𝜎𝑡,𝜎𝑢𝑡+𝑛,…,𝜎𝑢))_𝑝𝑝‖𝑢‖^𝑝+𝜎^𝑝‖𝑢‖4𝑝^𝑝−_|𝑡|≤Π2𝐹(()(𝑡=𝜎𝑡,𝜎𝑢𝑡+𝑛,…,𝜎𝑢))_𝑝𝑝‖𝑢‖^𝑝+𝜎^𝑝‖𝑢‖4𝑝^𝑝−_|𝑡|≤Π2𝑊(𝑡,𝜎𝑢(𝑡))+_|𝑡|≤Π2𝐻(()(𝑡≤𝜎𝑡,𝜎𝑢𝑡+𝑛,…,𝜎𝑢))_𝑝𝑝‖𝑢‖^𝑝+𝜎^𝑝‖𝑢‖4𝑝^𝑝+𝜎^𝜚𝑀₁‖𝑢‖^𝜚+𝑀₂−𝜏𝜎^𝜇=(𝑐𝜎)^𝑝𝑝+𝑐^𝑝𝜎^𝑝4𝑝+𝑀₁(𝑐𝜎)^𝜚+𝑀₂𝜎^𝜚−𝜏𝜎^𝜇. (4.31) Since 𝜇>𝜚>𝑝, we deduce that there is 𝜎₀=𝜎₀(𝑑,𝑀₁,𝑀₂,𝜏)=𝜎₀(𝐸′)>1 such that 𝐼(𝜎𝑢)<0for𝑢∈Θ,𝜎≥𝜎₀. (4.32) That is 𝐼(𝑢)<0for𝑢∈𝐸^′,‖𝑢‖≥𝑑𝜎₀. (4.33) This shows that (iii) of Lemma 3.1 holds. By Lemma 3.1, 𝐼 possesses an unbounded sequence {𝑑_𝑘}_𝑘∈ℕ of critical values with 𝑑_𝑘=𝐼(𝑢_𝑘), where 𝑢_𝑘 is such that 𝐼^′(𝑢_𝑘)=0 for 𝑘=1,2,…. If {‖𝑢_𝑘‖}_𝑘∈ℕ is bounded, then there exists 𝐵>0 such that 𝑢_𝑘≤𝐵for𝑘∈ℕ. (4.34) By a similar fashion for the proof of (4.15), for the given 𝜂 in (4.15), there exists Π₃>0 such that 𝑢_𝑘(𝑡)≤𝜂for|𝑡|≥Π₃,𝑘∈ℕ. (4.35) Thus, from (F2), (3.7), (4.34), and (4.35), we have 1𝑝𝑢_𝑘^𝑝=𝑑_𝑘+_𝑡∈ℤ𝐹𝑡,𝑢_𝑘(𝑡+𝑛),…,𝑢_𝑘(𝑡)=𝑑_𝑘+_|𝑡|>Π3𝐹𝑡,𝑢_𝑘(𝑡+𝑛),…,𝑢_𝑘(𝑡)+^Π3_𝑡=−Π3𝐹𝑡,𝑢_𝑘(𝑡+𝑛),…,𝑢_𝑘(𝑡)≥𝑑_𝑘−𝛽4𝑝(𝑛+1)_|𝑡|>Π3^𝑛_𝑖=0𝑢_𝑘(𝑡+𝑖)^𝑝−^Π3_𝑛=−Π3𝐻𝑡,𝑢_𝑘(𝑡+𝑛),…,𝑢_𝑘(𝑡)≥𝑑_𝑘−1𝑢4𝑝^𝑘𝑝−^Π3_𝑛=−Π3max_{𝑢𝑘≤𝐵/𝛽}𝐻𝑡,𝑢_𝑘(𝑡+𝑛),…,𝑢_𝑘(𝑡). (4.36) It follows that 𝑑_𝑘≤5𝑢4𝑝^𝑘𝑝+^Π3_𝑡=−Π3max_{|𝑢𝑘|≤𝐵/𝛽}𝐻𝑡,𝑢_𝑘(𝑡+𝑛),…,𝑢_𝑘(𝑡)<+∞. (4.37) This contradicts the fact that {𝑑_𝑘}_𝑘∈ℕ is unbounded, and so {‖𝑢_𝑘‖}_𝑘∈ℕ is unbounded.

Proof of Theorem 2.2. In the proof of Theorem 2.1, the condition that 𝐻(𝑡,𝑢(𝑡+𝑛),…,𝑢(𝑡))≥0 for (𝑡,𝑢(𝑡+𝑛),…,𝑢(𝑡))∈𝐽×ℝ^𝑛+1, 𝛾=(∑^𝑛_𝑖=0|u(𝑡+𝑖)|^𝑝)^1/𝑝≤1 in (F1) is only used in the proofs of assumption (ii) of Lemma 3.1. Therefore, we only prove that assumption (ii) of Lemma 3.1 still holds using (F2') instead of (F2). By (F2'), it follows that |𝐹(()(𝑡≤2𝑡,𝑢𝑡+𝑛,…,𝑢))|^−𝑝𝛽2𝑝(𝑛+1)^𝑛_𝑖=0|𝑢(𝑡+𝑖)|^𝑝for𝑡∈ℤ,^𝑛_𝑖=0|𝑢(𝑡+𝑖)|^𝑝1/𝑝≤𝜂. (4.38) If ‖𝑢‖=𝛽^1/𝑝𝜂∶=𝜌, then by Lemma  3.2, |𝑢(𝑡)|≤𝜂 for 𝑡∈ℤ. Set 𝛼=𝛽𝜂^𝑝/2𝑝. Hence, from (3.3), (4.38), and Lemma 3.2, we have 𝐼(𝑢)=1𝑝‖𝑢‖^𝑝−_𝑡∈ℤ𝐹(()(𝑡≥1𝑡,𝑢𝑡+𝑛,…,𝑢))𝑝‖𝑢‖^𝑝−2^−𝑝𝛽2𝑝(𝑛+1)^𝑛_𝑖=0|𝑢(𝑡+𝑖)|^𝑝≥1𝑝‖𝑢‖^𝑝−1‖𝑢‖2𝑝^𝑝=1‖𝑢‖2𝑝^𝑝=𝛼. (4.39) equation (4.39) shows that ‖𝑢‖=𝜌 implies that 𝐼(𝑢)≥𝛼, that is, assumption (ii) of Lemma 3.1 holds. The proof of Theorem 2.2 is completed.

Proof of Theorem 2.3. We first show that 𝐼 satisfies condition (C). Assume that {𝑢_𝑘}_𝑘∈ℕ⊂𝐸 is a (C) sequence of 𝐼, that is, {𝐼(𝑢_𝑘)}_𝑘∈ℕ is bounded and (1+‖𝑢_𝑘‖)‖𝐼^′(𝑢_𝑘)‖→0 as 𝑘→+∞. Then it follows from (3.3) and (3.4) that 𝐶₁≥𝑝𝐼𝑢_𝑘−𝐼^′𝑢_𝑘,𝑢_𝑘=_𝑡∈ℤ⁰_𝑖=−𝑛𝐹^′_2+𝑛+𝑖𝑡,𝑢_𝑘(𝑡+𝑛),…,𝑢_𝑘(𝑡)𝑢_𝑘(𝑡−𝑖)−𝑝𝐹𝑡,𝑢_𝑘(𝑡+𝑛),…,𝑢_𝑘(𝑡). (4.40) It follows from (F8) that there exists 𝜂∈(0,1) such that (4.15) holds. By (F7) and (F8), we have ⁰_𝑖=−𝑛𝐹^′_2+𝑛+𝑖(𝑡,𝑢(𝑡+𝑛),…,𝑢(𝑡))𝑢(𝑡−𝑖)>𝑝𝐹𝑡,𝑢_𝑘(𝑡+𝑛),…,𝑢(𝑡)≥0, (4.41) and for 𝑡∈ℤ, ∑^𝑛_𝑖=0|𝑢(𝑡+𝑖)|^𝑝≥𝜂^𝑝, we have 𝐹𝑡,𝑢_𝑘(𝑡+𝑛),…,𝑢_𝑘(𝑡)≤𝑎+𝑏^𝑛_𝑖=0|𝑢(𝑡+𝑖)|^{𝑝𝜈/𝑝}×⁰_𝑖=−𝑛𝐹^′_2+𝑛+𝑖𝑡,𝑢_𝑘(𝑡+𝑛),…,𝑢_𝑘(𝑡)𝑢_𝑘(𝑡−𝑖)−𝑝𝐹𝑡,𝑢_𝑘(𝑡+𝑛),…,𝑢_𝑘(𝑡). (4.42) It follows from (3.2), (3.3), (4.38), (4.40), (4.41), and (4.42) that 1𝑝𝑢_𝑘^𝑝=𝐼𝑢_𝑘+_𝑡∈ℤ𝐹𝑡,𝑢_𝑘()(𝑡𝑢𝑡+𝑛,…,𝑢)=𝐼_𝑘+_{∑𝑡∈ℤ𝑛𝑖=0|𝑢(𝑡+𝑖)|𝑝1/𝑝≤𝜂}𝐹𝑡,𝑢_𝑘()(𝑡)+𝑡+𝑛,…,𝑢_{∑𝑡∈ℤ𝑛𝑖=0|𝑢(𝑡+𝑖)|𝑝1/𝑝>𝜂}𝐹𝑡,𝑢_𝑘()(𝑡)𝑢𝑡+𝑛,…,𝑢≤𝐼_𝑘+2^−𝑝𝛽2𝑝(𝑛+1)_{∑𝑡∈ℤ𝑛𝑖=0|𝑢(𝑡+𝑖)|𝑝1/𝑝≤𝜂}𝑢_𝑘(𝑡+𝑖)^𝑝+_{∑𝑡∈ℤ𝑛𝑖=0|𝑢(𝑡+𝑖)|𝑝1/𝑝>𝜂}𝑎+𝑏^𝑛_𝑖=0|𝑢(𝑡+𝑖)|^{𝑝𝜈/𝑝}×⁰_𝑖=−𝑛𝐹^′_2+𝑛+𝑖𝑡,𝑢_𝑘(𝑡+𝑛),…,𝑢_𝑘(𝑡)𝑢_𝑘(𝑡−𝑖)−𝑝𝐹𝑡,𝑢_𝑘(𝑡+𝑛),…,𝑢_𝑘(𝑡)≤𝐶₂+1𝑢2𝑝^𝑘2+_𝑡∈ℤ𝑎+𝑏^𝑛_𝑖=0|𝑢(𝑡+𝑖)|^{𝑝𝜈/𝑝}×⁰_𝑖=−𝑛𝐹^′_2+𝑛+𝑖𝑡,𝑢_𝑘(𝑡+𝑛),…,𝑢_𝑘(𝑡)𝑢_𝑘(𝑡−𝑖)−𝑝𝐹𝑡,𝑢_𝑘(𝑡+𝑛),…,𝑢_𝑘(𝑡)≤𝐶₂+1𝑢2𝑝^𝑘𝑝+𝑎+𝑏𝑢_𝑘^𝜈_{∞𝑡∈ℤ}∇𝑊𝑛,𝑢_𝑘(𝑛),𝑢_𝑘(𝑛)−𝑝𝑊𝑛,𝑢_𝑘(𝑛)≤𝐶₂+1𝑢2𝑝^𝑘𝑝+𝐶₁𝑎+𝑏(𝑛+1)𝑢_𝑘^𝜈_∞≤𝐶₂+1𝑢2𝑝^𝑘𝑝+𝐶₁𝑎+𝛽^−𝜈/𝑝𝑏(𝑛+1)𝑢_𝑘^𝜈,𝑘∈ℕ. (4.43) Since 𝜈<𝑝, it follows from (4.43) that {‖𝑢_𝑘‖}_𝑘∈ℕ is bounded. Similar to the proof of Theorem 2.1, we can prove that {𝑢_𝑘} has a convergent subsequence in 𝐸. Hence, 𝐼 satisfies condition (C).
It is obvious that 𝐼 is even and 𝐼(0)=0, and so assumption (i) of Lemma 3.1 holds. The proof of assumption (ii) of Lemma 3.1 is the same as in the proof of Theorem 2.2.
Finally, it remains to show that 𝐼 satisfies assumption (iii) of Lemma 3.1. Let 𝐸^′ be a finite dimensional subspace of 𝐸. Since all norms of a finite dimensional normed space are equivalent, so there is a constant 𝑑>0 such that (4.21) holds. Assume that dim𝐸^′=𝑚 and 𝑢₁,𝑢₂,…,𝑢_𝑚 are the bases of 𝐸^′ such that (4.22) holds. Let 𝜂, Π₁, and Θ be the same as in the proof of Theorem 2.1. Then (4.23), (4.25), and (4.26) hold. For the Π₁ given in the proof of Theorem 2.1. By (F9), there exists 𝜎₀=𝜎₀(𝑑,Π₁)>1 such that 𝑠^−𝑝min_|𝑥|=1𝑊(𝑛,𝑠𝑥)≥(2𝑑)^𝑝for𝜎𝑠≥⁰2,𝑛∈ℤ−Π₁,Π₁. (4.44) It follows from (F7), (F9), (3.3), and (4.44) that 𝐼()=𝜎𝜎𝑢^𝑝𝑝‖𝑢‖^𝑝−_𝑡∈ℤ𝐹(()(𝑡≤𝜎𝑡,𝜎𝑢𝑡+𝑛,…,𝜎𝑢))_𝑝𝑝‖𝑢‖^𝑝−𝐹𝑡₀,𝜎𝑢𝑡₀≤𝜎^𝑝𝑝‖𝑢‖^𝑝−min_|𝑥|=1𝐹𝑡₀,𝜎𝑢𝑡₀𝑥≤(𝑑𝜎)^𝑝𝑝−(𝑑𝜎)^𝑝𝑢𝑡₀^𝑝≤(𝑑𝜎)^𝑝𝑝−(𝑑𝜎)^𝑝()=−𝑑𝜎^𝑝𝑞,𝑢∈Θ,𝜎≥𝜎₀. (4.45) We deduce that there is 𝜎₀=𝜎₀(𝑑,Π₁)=𝜎₀(𝐸^′)>1 such that 𝐼(𝜎𝑢)<0for𝑢∈Θ,𝜎≥𝜎₀. (4.46) That is 𝐼(𝑢)<0for𝑢∈𝐸^′,‖𝑢‖≥𝑑𝜎₀. (4.47) This shows that condition (iii) of Lemma 3.1 holds. By Lemma 3.1, 𝐼 possesses an unbounded sequence {𝑑_𝑘}_𝑘∈ℕ of critical values with 𝑑_𝑘=𝐼(𝑢_𝑘), where 𝑢_𝑘 is such that 𝐼′(𝑢_𝑘)=0 for 𝑘=1,2,…. Since 1𝑝𝑢_𝑘^𝑝=𝑑_𝑘+_𝑛∈ℤ𝐹𝑡,𝑢_𝑘(𝑡+𝑛),…,𝑢_𝑘(𝑡)≥𝑑_𝑘 (4.48) and {𝑑_𝑘}_𝑘∈ℕ is unbounded, it follows that {‖𝑢_𝑘‖}_𝑘∈ℕ is unbounded. The proof is complete.

5. Examples

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

Example 5.1. In (1.1), let 𝑞(𝑡)→+∞ as |𝑡|→+∞ and 𝐹(𝑡,𝑢(𝑡+𝑛),…,𝑢(𝑡))=|𝑢(𝑡)|^4+|𝑡|−^𝑛_𝑖=0𝑢²(𝑡+𝑖)^{(5+3|𝑡|)/2(4+|𝑡|)}. (5.1) Let 𝑝=2, 𝜇=4, 𝜚=3, 𝐽={−3,−2,−1,0,1,2,3} and 𝑊(𝑡,𝑢(𝑡))=|𝑢(𝑡)|^4+|𝑡|,𝐻(𝑡,𝑢(𝑡+𝑛),…,𝑢(𝑡))=^𝑛_𝑖=0𝑢²(𝑡+𝑖)^{(5+3|𝑡|)/2(4+|𝑡|)}. (5.2) Then it is easy to verify that all conditions of Theorem 2.1 are satisfied. By Theorem 2.1, (1.1) has a nontrivial homoclinic solution.

Example 5.2. In (1.1), let 𝑟(𝑡)>0, 𝑞(𝑡)→+∞ as |𝑡|→+∞ and 𝐹(𝑡,𝑢(𝑡+𝑛),…,𝑢(𝑡))=^𝑚1_𝑘=1𝑎_𝑘|𝑢(𝑡)|^𝜇𝑘−^𝑚2_𝑗=1𝑏_𝑗^𝑛_𝑖=0|𝑢(𝑡+𝑖)|^{𝑝𝜚𝑗/𝑝}, (5.3) where 𝜇₁>𝜇₂>⋯>𝜇_𝑚1>𝜚₁>𝜚₂>⋯>𝜚_𝑚2>𝑝, 𝑎_𝑘,𝑏_𝑗>0, 𝑘=1,2,…,𝑚₁; 𝑗=1,2,…,𝑚₂. Let 𝜇=𝜇_𝑚1, 𝜚=𝜚₁, and 𝑊(𝑡,𝑢(𝑡))=^𝑚1_𝑘=1𝑎_𝑘|𝑢(𝑡)|^𝜇𝑘,𝐻(𝑡,𝑢(𝑡+𝑛),…,𝑢(𝑡))=^𝑚2_𝑗=1𝑏_𝑗^𝑛_𝑖=0|𝑢(𝑡+𝑖)|^{𝑝𝜚𝑗/𝑝}. (5.4) Then it is easy to verify that all conditions of Theorem 2.2 are satisfied. By Theorem 2.2, (1.1) has a nontrivial homoclinic solution.

Example 5.3. In (1.1), let 𝐹(𝑡,𝑢(𝑡+𝑛),…,𝑢(𝑡))=𝑞(𝑡)^𝑛_𝑖=0|𝑢(𝑡+𝑖)|^𝑝ln1+^𝑛_𝑖=0|𝑢(𝑡+𝑖)|^𝑝1/𝑝, (5.5) where 𝑞∶ℤ→(0,∞) such that 𝑞(𝑡)→+∞ as |𝑡|→+∞. Since ⁰_𝑖=−𝑛𝐹^′_2+𝑛+𝑖(()(𝑡𝑢()(𝑡𝑝𝑡,𝑢𝑡+𝑛,…,𝑢))𝑡−𝑖=𝑞)^𝑛_𝑖=0|𝑢(𝑡+𝑖)|^𝑝ln1+^𝑛_𝑖=0|𝑢(𝑡+𝑖)|^𝑝1/𝑝+∑^𝑛_𝑖=0|𝑢(𝑡+𝑖)|^{𝑝𝑝+1/𝑝}1+∑^𝑛_𝑖=0|𝑢(𝑡+𝑖)|^𝑝1/𝑝≥1𝑝+1+∑^𝑛_𝑖=0|𝑢(𝑡+𝑖)|^𝑝1/𝑝𝐹(()(𝑡(()(𝑡𝑡,𝑢𝑡+𝑛,…,𝑢))≥0,∀𝑡,𝑢𝑡+𝑛,…,𝑢))∈ℤ×ℝ^𝑛+1. (5.6) This shows that (F8) holds with 𝑎=𝑏=𝜈=1. It is easy to verify that assumptions (q), (F1), and (F7) of Theorem 2.3 are satisfied. By Theorem 2.3, (1.1) has an unbounded sequence of homoclinic solutions.

Acknowledgment

This work is partially supported by Major Project of Science Research Fund of Education Department in Hunan (no: 11A095).