Periodic Solutions for a System of Difference Equations

Abstract

This paper deals with the second-order nonlinear systems of difference equations, we obtain the existence theorems of periodic solutions. The theorems are proved by using critical point theory.

1. Introduction

Let ℕ,ℤ,ℝ be the set of all natural numbers, integers, and real numbers, respectively. For 𝑎,𝑏∈ℤ, note that ℤ[𝑎,𝑏]={𝑎,𝑎+1,…,𝑏}, where 𝑎≤𝑏.

In this paper, we consider the existence of periodic solutions for the system of difference equations of the formΔ(𝑝_𝑛1(Δ𝑥_(𝑛−1)1)^𝛿)+𝑞_𝑛1(𝑥_𝑛1)^𝛿=𝑓₁(𝑛,𝑋_𝑛),Δ(𝑝_𝑛2(Δ𝑥_(𝑛−1)2)^𝛿)+𝑞_𝑛2(𝑥_𝑛2)^𝛿=𝑓₂(𝑛,𝑋_𝑛⋮),Δ(𝑝_𝑛𝑘(Δ𝑥_{(𝑛−1)𝑘})^𝛿)+𝑞_𝑛𝑘(𝑥_𝑛𝑘)^𝛿=𝑓_𝑘(𝑛,𝑋_𝑛), (1.1)which can be recorded asΔ(𝑃_𝑛(Δ𝑋^𝑇_𝑛−1)^𝛿)+𝑄_𝑛(𝑋^𝑇_𝑛)^𝛿=𝑓(𝑛,𝑋_𝑛),𝑛∈ℤ, (1.2)where 𝑘 is a positive integer,𝑃_𝑛=𝑝_𝑛10⋯00𝑝_𝑛2⋯⋯000⋯𝑝_𝑛𝑘,𝑄_𝑛=𝑞_𝑛10⋯00𝑞_𝑛2⋯⋯000⋯𝑞_𝑛𝑘, (1.3)and 𝑃_𝑛+𝜔=𝑃_𝑛>0 (i.e., 𝑝_𝑛1>0,𝑝_𝑛2>0,…,𝑝_𝑛𝑘>0), 𝑄_𝑛+𝜔=𝑄_𝑛, 𝑓=(𝑓₁,𝑓₂,…,𝑓_𝑘)^𝑇, 𝑓_𝑖=𝑓_𝑖(𝑛,𝑋_𝑛)=𝑓_𝑖(𝑛,𝑥_𝑛1,𝑥_𝑛2,…,𝑥_𝑛𝑘), 𝑓(𝑛+𝜔,𝑈)=𝑓(𝑛,𝑈) for any (𝑛,𝑈)∈ℤ×ℝ^𝑘, 𝜔>0 is a positive integer, (−1)^𝛿=−1, 𝛿 is the ratio of odd positive integers, Δ𝑋^𝑇_𝑛=𝑋^𝑇_𝑛+1−𝑋^𝑇_𝑛=(𝑥_(𝑛+1)1−𝑥_𝑛1,𝑥_(𝑛+1)2−𝑥_𝑛2,…,𝑥_(𝑛+1)𝑘−𝑥_𝑛𝑘)^𝑇,Δ²𝑋^𝑇_𝑛−1=Δ(Δ𝑋^𝑇_𝑛−1)=Δ𝑋^𝑇_𝑛−Δ𝑋^𝑇_𝑛−1. For 𝑈=(𝑢₁,𝑢₂,…,𝑢_𝑘)∈ℝ^𝑘, define 𝑈^𝛿=(𝑢^𝛿₁,𝑢^𝛿₂,…,𝑢^𝛿_𝑘). |𝑈|=(|𝑢₁|,|𝑢₂|,…,|𝑢_𝑘|), |𝑈|^𝛿=(|𝑢₁|^𝛿,|𝑢₂|^𝛿,…,|𝑢_𝑘|^𝛿). A sequence 𝑋={𝑋_𝑛}_𝑛∈ℤ is a 𝜔-periodic solution of (1.2) if substitution of it into (1.2) yields an identity for all 𝑛∈ℤ.

In [1, 2], the qualitative behavior of linear difference equationsΔ(𝑝_𝑛Δ𝑥_𝑛)+𝑞_𝑛𝑥_𝑛=0 (1.4)has been investigated. In [3], the nonlinear difference equationΔ(𝑝_𝑛Δ𝑥_𝑛−1)+𝑞_𝑛𝑥_𝑛=𝑓(𝑛,𝑥_𝑛) (1.5)has been considered. In [4], by critical point method, the existence of periodic and subharmonic solutions of equationΔ²𝑥_𝑛−1+𝑓(𝑛,𝑥_𝑛)=0,𝑛∈ℤ (1.6)has been studied. Other interesting results can been found in [5–8]. In [9], the authors consider the existence of periodic solutions for second-order nonlinear difference equationΔ(𝑝_𝑛(Δ𝑥_𝑛−1)^𝛿)+𝑞_𝑛𝑥^𝛿_𝑛=𝑓(𝑛,𝑥_𝑛),𝑛∈ℤ, (1.7)using critical point theory, obtaining some new results. It is a discrete analogues of differential equation(𝑝(𝑡)𝜙(𝑢^′))^′+𝑓(𝑡,𝑢)=0. (1.8)They do have physical applications in the study of nuclear physics, gas aerodynamics, and so on (see [10, 11]). In this paper, we obtain some new results of existence of periodic solution for the second-order nonlinear system of difference equations by using critical point theory. We remark, however, the result in [9] is only good for (1.7) which is much less general than our results in what follows.

2. Some Basic Lemmas

Let 𝐸 be a real Hilbert space, 𝐼∈𝐶¹(𝐸,ℝ) mean that 𝐼 is continuously Fréchet differentiable functional defined on 𝐸. 𝐼 is said to be satisfying Palais-Smale condition (P-S condition) if any bounded sequence {𝐼(𝑢_𝑛)} and 𝐼^′(𝑢_𝑛)→0(𝑛→∞) possess a convergent subsequence in 𝐸. Let 𝐵_𝜌 be the open ball in 𝐸 with radius 𝜌 and centered at 𝜃, and let 𝜕𝐵_𝜌 denote its boundary, 𝜃 is null element of 𝐸.

Lemma 2.1 (see [12]). Let 𝐸 be a real Hilbert space, and assume that 𝐼∈𝐶¹(𝐸,ℝ) satisfies the P-S condition and the following conditions:
(I₁)there exist constants 𝜌>0 and 𝑎>0 such that 𝐼(𝑥)≥𝑎 for all 𝑥∈𝜕𝐵_𝜌, where 𝐵_𝜌={𝑥∈𝐸∶‖𝑥‖<𝜌};(I₂)𝐼(0)≤0 and there exists 𝑥₀∉𝐵_𝜌 such that 𝐼(𝑥₀)≤0.

Then 𝑐=inf_ℎ∈Γsup_𝑠∈[0,1]𝐼(ℎ(𝑠)) is a positive critical value of 𝐼, whereΓ={ℎ∈𝐶([0,1],𝑋)∶ℎ(0)=𝜃,ℎ(1)=𝑥₀}. (2.1)

Let Ω_∗ be the set of sequences𝑋={𝑋_𝑛}_𝑛∈ℤ={…,𝑋_−𝑛,…,𝑋₋₁,𝑋₀,𝑋₁,…,𝑋_𝑛,…}, (2.2)where 𝑋_𝑛=(𝑥_𝑛1,𝑥_𝑛2,…,𝑥_𝑛𝑘)∈ℝ^𝑘, that is,Ω_∗={𝑋={𝑋_𝑛}_𝑛∈ℤ∶𝑋_𝑛∈ℝ^𝑘,𝑛∈ℤ}. (2.3)For any 𝑋,𝑌∈Ω_∗, 𝑎,𝑏∈ℝ, 𝑎𝑋+𝑏𝑌 is defined by𝑎𝑋+𝑏𝑌={𝑎𝑋_𝑛+𝑏𝑌_𝑛}^+∞_𝑛=−∞, (2.4)then Ω_∗ is a vector space. For given positive integer 𝜔, 𝐸_𝜔 is defined as a subspace of Ω_∗ by𝐸_𝜔={𝑋={𝑋_𝑛}∈Ω_∗∶𝑋_𝑛+𝜔=𝑋_𝑛,𝑛∈ℤ}. (2.5)Obviously, 𝐸_𝜔 is isomorphic to ℝ^𝑘𝜔, for any 𝑋,𝑌∈𝐸_𝜔, defined inner product⟨𝑋,𝑌⟩=^𝜔_𝑖=1⟨𝑋_𝑖,𝑌_𝑖⟩, (2.6)by which the norm ‖⋅‖ can be induced by‖𝑋‖=(^𝜔_𝑖=1‖𝑋_𝑖‖²)^1/2,𝑋∈𝐸_𝜔. (2.7)where ‖𝑋_𝑖‖=(∑^𝑘_𝑗=1|𝑥_𝑖𝑗|²)^1/2. It is obvious that 𝐸_𝜔 with the inner product defined by (2.6) is a finite-dimensional Hilbert space and linearly homeomorphic to ℝ^𝑘𝜔. Define the functional 𝐽 on 𝐸_𝜔 as follows:1𝐽(𝑋)=𝛿+1^𝜔_𝑛=1⟨𝑃_𝑛,(Δ𝑋_𝑛−1)^𝛿+11⟩−𝛿+1^𝜔_𝑛=1⟨𝑄_𝑛,𝑋_𝑛^𝛿+1⟩+^𝜔_𝑛=1𝐹(𝑛,𝑋_𝑛),𝑋∈𝐸_𝜔, (2.8)where 𝐹(𝑛,𝑋_𝑛) such that ∇_𝑈𝐹(𝑛,𝑈)=𝑓(𝑛,𝑈), that is,𝑓_𝑖(𝑛,𝑈)=𝑓_𝑖(𝑛,𝑢₁,𝑢₂,…,𝑢_𝑘𝜕)=𝜕𝑢_𝑖𝐹(𝑛,𝑢₁,𝑢₂,…,𝑢_𝑘) (2.9)for any (𝑛,𝑈)∈ℤ[1,𝜔]×ℝ^𝑘, 𝑃_𝑛=(𝑝_𝑛1,𝑝_𝑛2,…,𝑝_𝑛𝑘), 𝑄_𝑛=(𝑞_𝑛1,𝑞_𝑛2,…,𝑞_𝑛𝑘). Clearly 𝐽∈𝐶¹(𝐸_𝜔,ℝ), and for any 𝑋={𝑋_𝑛}_𝑛∈ℤ∈𝐸_𝜔, by 𝑋₀=𝑋_𝜔 and 𝑋₁=𝑋_𝜔+1, we have𝜕𝐽(𝑋)𝜕𝑥_𝑛𝑙=−Δ(𝑝_𝑛𝑙(Δ𝑥_{(𝑛−1)𝑙})^𝛿)−𝑞_𝑛𝑙(𝑥_𝑛𝑙)^𝛿+𝑓_𝑙(𝑛,𝑋_𝑛),𝑙∈ℤ[1,𝑘],𝑛∈ℤ[1,𝜔]. (2.10)Thus 𝑋={𝑋_𝑛}_𝑛∈ℤ is a critical point of 𝐽 on 𝐸_𝜔 (𝐽^′(𝑋)=0) if and only ifΔ(𝑝_𝑛𝑙(Δ𝑥_{(𝑛−1)𝑙})^𝛿)+𝑞_𝑛𝑙(𝑥_𝑛𝑙)^𝛿=𝑓_𝑙(𝑛,𝑋_𝑛),𝑙∈ℤ[1,𝑘],𝑛∈ℤ[1,𝜔]. (2.11)That is,Δ(𝑃_𝑛(Δ𝑋^𝑇_𝑛−1)^𝛿)+𝑄_𝑛(𝑋^𝑇_𝑛)^𝛿=𝑓(𝑛,𝑋_𝑛),𝑛∈ℤ. (2.12)By the periodicity of 𝑋_𝑛 and 𝑓(𝑛,𝑋_𝑛) in the first variable 𝑛, we know that if 𝑋={𝑋_𝑛}_𝑛∈ℤ∈𝐸_𝜔 is a critical point of the real functional 𝐽 defined by (2.8), then it is a periodic solution of (1.2).

For 𝑋={𝑋_𝑛}_𝑛∈ℤ∈𝐸_𝜔, 𝑋_𝑛=(𝑥_𝑛1,𝑥_𝑛2,…,𝑥_𝑛𝑘)∈ℝ^𝑘, 𝑟>1, denote‖𝑋‖_𝑟=(^𝜔_𝑖=1‖𝑋_𝑖‖^𝑟)^1/𝑟,‖𝑋_𝑛‖_𝑟=(^𝑘_𝑖=1‖𝑥_𝑛𝑖‖^𝑟)^1/𝑟. (2.13)Clearly, ‖𝑋‖₂=‖𝑋‖, ‖𝑋_𝑛‖₂=‖𝑋_𝑛‖. Because of ‖⋅‖_𝑟1 and ‖⋅‖_𝑟2 being equivalent when 𝑟₁,𝑟₂>1, so there exist constants 𝑐₁,𝑐₂,𝑐₃,𝑐₄,ℏ₁,ℏ₂,ℏ₃, and ℏ₄ such that 𝑐₂≥𝑐₁>0, 𝑐₄≥𝑐₃>0, ℏ₂≥ℏ₁>0, and ℏ₄≥ℏ₃>0,𝑐₁‖𝑋‖≤‖𝑋‖_𝛿+1≤𝑐₂𝑐‖𝑋‖,₃‖𝑋‖≤‖𝑋‖_𝛽≤𝑐₄ℏ‖𝑋‖,₁‖𝑋_𝑛‖≤‖𝑋_𝑛‖_𝛿+1≤ℏ₂‖𝑋_𝑛ℏ‖,₃‖𝑋_𝑛‖≤‖𝑋_𝑛‖_𝛽≤ℏ₄‖𝑋_𝑛‖, (2.14)for all 𝑋∈𝐸_𝜔, 𝛿>0, and 𝛽>1.

Lemma 2.2. Suppose that (F₁)there exist constants 𝑎₁>0, 𝑎₂>0, 𝛽>𝛿+1 such that𝐹(𝑛,𝑈)≤−𝑎₁‖𝑈‖^𝛽+𝑎₂ (2.15)for any (𝑛,𝑈)∈ℤ[1,𝜔]×𝑅^𝑘;(F₂)𝑞_𝑛𝑖≤0,𝑛∈ℤ,𝑖∈ℤ[1,𝑘]. (2.16)
Then1𝐽(𝑋)=𝛿+1^𝜔_𝑛=1⟨𝑃_𝑛,(Δ𝑋_𝑛−1)^𝛿+11⟩−𝛿+1^𝜔_𝑛=1⟨𝑄_𝑛,𝑋_𝑛^𝛿+1⟩+^𝜔_𝑛=1𝐹(𝑛,𝑋_𝑛) (2.17)satisfies P-S condition.

Proof. For any sequence {𝑋^(𝑙)}={…,𝑋^(𝑙)_−𝑛,…,𝑋^(𝑙)₋₁,𝑋₀^(𝑙),𝑋₁^(𝑙),…,𝑋_𝑛^(𝑙),…}∈𝐸_𝜔, 𝐽(𝑋^(𝑙)) is bounded and 𝐽^′(𝑋^(𝑙))→0(𝑙→∞). Then there exists a positive constant 𝑀>0, such that |𝐽(𝑋^(𝑙))|≤𝑀. From (F₁), we have−𝑀≤𝐽(𝑋^(𝑙))=1𝛿+1^𝜔_𝑛=1[⟨𝑃_𝑛,(𝑋_𝑛^(𝑙)−𝑋^(𝑙)_𝑛−1)^𝛿+1⟩−⟨𝑄_𝑛,(𝑋_𝑛^(𝑙))^𝛿+1+⟩]_𝜔𝑛=1𝐹(𝑛,𝑋_𝑛^(𝑙))≤1𝛿+1^𝜔_𝑛=12^𝛿+1⟨𝑃_𝑛,(|𝑋_𝑛^(𝑙)|^𝛿+1+|𝑋^(𝑙)_𝑛−1|^𝛿+1−1)⟩𝛿+1^𝜔_𝑛=1⟨𝑄_𝑛,|𝑋_𝑛^(𝑙)|^𝛿+1⟩+^𝜔_𝑛=1𝐹(𝑛,𝑋_𝑛^(𝑙))≤2^𝛿+1𝛿+1^𝜔_𝑛=1⟨𝑃_𝑛+𝑃_𝑛+1,|𝑋_𝑛^(𝑙)|^𝛿+11⟩−𝛿+1^𝜔_𝑛=1⟨𝑄_𝑛,|𝑋_𝑛^(𝑙)|^𝛿+1⟩+^𝜔_𝑛=1𝐹(𝑛,𝑋_𝑛^(𝑙))≤2^𝛿+1𝛿+1^𝜔_𝑛=1⟨𝑃_𝑛+𝑃_𝑛+1,|𝑋_𝑛^(𝑙)|^𝛿+11⟩−𝛿+1^𝜔_𝑛=1⟨𝑄_𝑛,|𝑋_𝑛^(𝑙)|^𝛿+1⟩−𝑎₁^𝜔_𝑛=1‖𝑋_𝑛^(𝑙)‖^𝛽+𝑎₂𝜔=1𝛿+1^𝜔_𝑛=1⟨2^𝛿+1(𝑃_𝑛+𝑃_𝑛+1)−𝑄_𝑛,|𝑋_𝑛^(𝑙)|^𝛿+1⟩−𝑎₁‖𝑋^(𝑙)‖^𝛽_𝛽+𝑎₂𝜔. (2.18) Set𝐴₀=max_{𝑛∈ℤ[1,𝜔],𝑖∈ℤ[1,𝑘]}{2^𝛿+1(𝑝_𝑛𝑖+𝑝_(𝑛+1)𝑖)−𝑞_𝑛𝑖}. (2.19)Then 𝐴₀>0, and−𝑀≤𝐽(𝑋^(𝑙))≤𝐴₀𝛿+1^𝜔_𝑛=1‖𝑋_𝑛^(𝑙)‖^𝛿+1_𝛿+1−𝑎₁‖𝑋^(𝑙)‖^𝛽_𝛽+𝑎₂𝜔≤𝐴₀ℏ₂^𝛿+1𝛿+1^𝜔_𝑛=1‖𝑋_𝑛^(𝑙)‖^𝛿+1−𝑎₁‖𝑋^(𝑙)‖^𝛽_𝛽+𝑎₂𝜔=𝐴₀ℏ₂^𝛿+1𝛿+1‖𝑋^(𝑙)‖^𝛿+1_𝛿+1−𝑎₁‖𝑋^(𝑙)‖^𝛽_𝛽+𝑎₂𝜔. (2.20) Because of 𝛽>𝛿+1, and (𝛽−𝛿−1)/𝛽+(𝛿+1)/𝛽=1, in view of Hölder inequality, we have^𝜔_𝑛=1‖𝑋_𝑛^(𝑙)‖^𝛿+1≤𝜔^{(𝛽−𝛿−1)/𝛽}(^𝜔_𝑛=1‖𝑋_𝑛^(𝑙)‖^𝛽)^{(𝛿+1)/𝛽}. (2.21)Thus‖𝑋^(𝑙)‖^𝛽_𝛽≥𝜔^{(𝛿+1−𝛽)/(𝛿+1)}‖𝑋^(𝑙)‖^𝛽_𝛿+1. (2.22)Then we have−𝑀≤𝐽(𝑋^(𝑙))≤𝐴₀ℏ₂^𝛿+1𝛿+1‖𝑋^(𝑙)‖^𝛿+1_𝛿+1−𝑎₁‖𝑋^(𝑙)‖^𝛽_𝛽+𝑎₂𝜔≤𝐴₀ℏ₂^𝛿+1𝛿+1‖𝑋^(𝑙)‖^𝛿+1_𝛿+1−𝑎₁𝜔^{(𝛿+1−𝛽)/(𝛿+1)}‖𝑋^(𝑙)‖^𝛽_𝛿+1+𝑎₂𝜔. (2.23) Thus, for any 𝑙∈ℕ,𝑎₁𝜔^{(𝛿+1−𝛽)/(𝛿+1)}‖𝑋^(𝑙)‖^𝛽_𝛿+1−𝐴₀ℏ₂^𝛿+1𝛿+1‖𝑋^(𝑙)‖^𝛿+1_𝛿+1≤𝑀+𝑎₂𝜔. (2.24)Because of 𝛽>𝛿+1, it is easily seen that the inequality (2.24) implies that {𝑋^(𝑙)} is a bounded sequence in 𝐸_𝜔. Thus {𝑋^(𝑙)} possesses convergent subsequences. The proof is complete.

3. Main Result

Theorem 3.1. Suppose that condition (F₁) holds, and
(F₃)for each 𝑛∈ℤ,lim_{‖𝑈‖→0}𝐹(𝑛,𝑈)‖𝑈‖^𝛿+1=0; (3.1)(F₄)for any 𝑖∈ℤ[1,𝑘],𝑛∈ℤ[1,𝜔],𝑞_𝑛𝑖<0; (3.2)(F₅)𝐹(𝑛,𝜃)=0.
Then (1.2) has at least two nontrivial 𝜔-periodic solutions.

Proof. By Lemma 2.2, 𝐽 satisfies P-S condition. Next, we will verify the conditions (I₁) and (I₂) of Lemma 2.1. By (F₃), there exists 𝜌>0, such that𝑞|𝐹(𝑛,𝑈)|≤−^maxℏ₁^𝛿+12(𝛿+1)‖𝑈‖^𝛿+1 (3.3)for any ‖𝑈‖<𝜌 and 𝑛∈ℤ[1,𝜔], where 𝑞_max=max_{𝑛∈ℤ[1,𝜔],𝑖∈ℤ[1,𝑘]}𝑞_𝑛𝑖<0. Thus1𝐽(𝑋)≥−𝛿+1^𝜔_𝑛=1⟨𝑄_𝑛,𝑋_𝑛^𝛿+1⟩+^𝜔_𝑛=1𝐹(𝑛,𝑋_𝑛)𝑞≥−^max𝛿+1^𝜔_𝑛=1‖𝑋_𝑛‖^𝛿+1_𝛿+1+𝑞_maxℏ₁^𝛿+12(𝛿+1)^𝜔_𝑛=1‖𝑋_𝑛‖^𝛿+1𝑞≥−^maxℏ₁^𝛿+1𝛿+1^𝜔_𝑛=1‖𝑋_𝑛‖^𝛿+1+𝑞_maxℏ₁^𝛿+12(𝛿+1)^𝜔_𝑛=1‖𝑋_𝑛‖^𝛿+1𝑞=−^maxℏ₁^𝛿+12(𝛿+1)‖𝑋‖^𝛿+1_𝛿+1𝑞≥−^maxℏ₁^𝛿+1𝑐₁^𝛿+12(𝛿+1)‖𝑋‖^𝛿+1 (3.4) for any 𝑋∈𝐸_𝜔 with ‖𝑋‖≤𝜌. We choose 𝑎=−ℏ₁^𝛿+1𝑐₁^𝛿+1(𝑞_max/2(𝛿+1))𝜌^𝛿+1, then we have𝐽(𝑋)|_𝜕𝐵𝜌≥𝑎>0, (3.5)that is, the condition (I₁) of Lemma 2.1 holds.
Obviously, 𝐽(0)=0. For any given 𝑉∈𝐸_𝜔 with ‖𝑉‖=1 and constant 𝛼>0,1𝐽(𝛼𝑉)=𝛿+1^𝜔_𝑛=1⟨𝑃_𝑛,(𝛼𝑉_𝑛−𝛼𝑉_𝑛−1)^𝛿+11⟩−𝛿+1^𝜔_𝑛=1⟨𝑄_𝑛,(𝛼𝑉_𝑛)^𝛿+1⟩+^𝜔_𝑛=1𝐹(𝑛,𝛼𝑉_𝑛)=1𝛿+1^𝜔_𝑛=1{𝑝_𝑛1(𝛼𝑣_𝑛1−𝛼𝑣_(𝑛−1)1)^𝛿+1+𝑝_𝑛2(𝛼𝑣_𝑛2−𝛼𝑣_(𝑛−1)2)^𝛿+1+⋯+𝑝_𝑛𝑘(𝛼𝑣_𝑛𝑘−𝛼𝑣_{(𝑛−1)𝑘})^𝛿+1}−1𝛿+1^𝜔_𝑛=1{𝑞_𝑛1(𝛼𝑣_𝑛1)^𝛿+1+𝑞_𝑛2(𝛼𝑣_𝑛2)^𝛿+1+⋯+𝑞_𝑛𝑘(𝛼𝑣_𝑛𝑘)^𝛿+1}+^𝜔_𝑛=1𝐹(𝑛,𝛼𝑉_𝑛)≤1𝛿+1^𝜔_𝑛=1{𝑝_𝑛1(2𝛼)^𝛿+1+𝑝_𝑛2(2𝛼)^𝛿+1+⋯+𝑝_𝑛𝑘(2𝛼)^𝛿+1}−1𝛿+1^𝜔_𝑛=1{𝑞_𝑛1𝛼^𝛿+1+𝑞_𝑛2𝛼^𝛿+1+⋯+𝑞_𝑛𝑘𝛼^𝛿+1}−𝑎₁^𝜔_𝑛=1‖𝛼𝑉_𝑛‖^𝛽+𝑎₂𝜔≤1𝛿+1^𝜔_𝑛=1{2^𝛿+1‖𝑃_𝑛‖₁+‖𝑄_𝑛‖₁}𝛼^𝛿+1−𝑎₁𝛼^𝛽𝜔_𝑛=1‖𝑉_𝑛‖^𝛽+𝑎₂𝜔≤1𝛿+1^𝜔_𝑛=1{2^𝛿+1‖𝑃_𝑛‖₁+‖𝑄_𝑛‖₁}𝛼^𝛿+1−𝑎₁𝛼^𝛽‖𝑉‖^𝛽_𝛽+𝑎₂𝜔≤1𝛿+1^𝜔_𝑛=1{2^𝛿+1‖𝑃_𝑛‖₁+‖𝑄_𝑛‖₁}𝛼^𝛿+1−𝑎₁𝑐^𝛽₃𝛼^𝛽+𝑎₂𝜔→−∞,(𝛼→+∞). (3.6) Thus we can choose a sufficiently large 𝛼 such that 𝛼>𝜌, and 𝑋=𝛼𝑉∈𝐸_𝜔, 𝐽(𝑋)<0. According to Lemma 2.1, there exists at least one critical value 𝑐≥𝑎>0. We suppose that 𝑋^∗ is a critical point corresponding to 𝑐, then 𝐽(𝑋^∗)=𝑐 and 𝐽^′(𝑋^∗)=0.
By similar argument of Lemma 2.2, we know that 𝐽(𝑋) is bounded from above, so there exists 𝑋^∗∗∈𝐸_𝜔 such that 𝐽(𝑋)≤𝐽(𝑋^∗∗)=𝑐_max for any 𝑋∈𝐸_𝜔. Obviously, 𝑋^∗∗≠0. If 𝑋^∗∗≠𝑋^∗, then the proof is complete. Otherwise, 𝑋^∗∗=𝑋^∗, 𝑐=𝑐_max. In view of Lemma 2.1,𝑐=inf_ℎ∈Γsup_𝑠∈[0,1]𝐽(ℎ(𝑠)), (3.7)where Γ={ℎ∈𝐶([0,1],𝐸_𝜔)∶ℎ(0)=𝜃,ℎ(1)=𝑋}. Then 𝑐_max=max_𝑠∈[0,1]𝐽(ℎ(𝑠)) for any ℎ∈Γ holds. In view of the continuity of 𝐽(ℎ(𝑠)) in 𝑠, 𝐽(𝜃)≤0, and 𝐽(𝑋)<0, we know that there exists some 𝑠₀∈(0,1) such that 𝐽(ℎ(𝑠₀))=𝑐_max. If we choose ℎ₁,ℎ₂∈Γ such that{ℎ₁(𝑠)∶𝑠∈(0,1)}∩{ℎ₂(𝑠)∶𝑠∈(0,1)}=𝜙, (3.8)then there exist 𝑠₁,𝑠₂∈(0,1) such that 𝐽(ℎ₁(𝑠₁))=𝐽(ℎ₂(𝑠₂))=𝑐_max. Then 𝐽 possesses two different critical points 𝑌=ℎ₁(𝑠₁) and Ž=ℎ₂(𝑠₂) in 𝐸_𝜔, hence, we obtain at least two nontrivial critical points which correspond to the critical value 𝑐_max. Thus (1.2) possesses at least two nontrivial 𝜔-periodic solutions. The proof is complete.

Acknowledgments

This work is supported by Natural Science Foundation of Shanxi Province (2008011002-1) and Shanxi Datong University and by the Development Foundation of Higher Education Department of Shanxi Province.